Fabius function

Caution

Never start Lean or Lake builds in parallel, and never start a second build while one is already running. Run exactly one lake build +FabiusFunction.Module at a time, with one target. Do not launch background build loops, pass a batch of targets, or use parallel runners such as xargs -P.

One invocation is not by itself one process: Lake sizes its worker pool to hardware concurrency and starts one lean.exe per core whenever the target has a stale dependency set. On this machine several agent sessions share 13 GB, so that fan-out starves all of them. Build a single leaf whose dependencies are already compiled:

Analysis/FabiusFunction/scripts/build_one.sh Basic

Correction (2026-09-02): LAKE_JOBS=1 does not bound Lake's workers in this repository. Lake 5 accepts neither -j nor --jobs, and the environment variable is not read. A stale umbrella target can therefore launch a burst of workers even when LAKE_JOBS=1 is set. The supported control is structural: one module per invocation, in dependency order, so there is no independent stale breadth to schedule. Use scripts/affected_modules.py and scripts/validate_affected.sh for a change-set closure, or scripts/build_one.sh for a focused leaf. Never run lake build +FabiusFunction when the facade has stale dependencies.

Do not set LEAN_NUM_THREADS=0: it serializes elaboration inside each worker and was measured to make focused builds about thirty times slower.

Starvation does not look like starvation. It surfaces as errors that read like corruption:

failed to read file '...\Mathlib\...\Basic.olean'
libc++abi: terminating due to uncaught exception of type std::bad_alloc

These are out-of-memory symptoms, not broken proofs -- the same module built by itself succeeds. Never "fix" them by editing Lean sources.

Before starting, check that nothing else -- including another agent session in a sibling worktree -- is already building; after interrupting a build, check for survivors, because stopping a task does not reliably kill the processes it spawned:

Get-Process lean,lake -ErrorAction SilentlyContinue

If a build is running, wait for it rather than racing it.

Multi-agent coordination: OFF. A single switch file, AGENTS/STATUS.md, states whether the coordination framework is in effect; flipping it — plus creating or deleting the off-main board branch it names — is the entire enable/disable procedure. The lightweight protocol it switches is AGENTS/PROTOCOL.md. The engineering policies in AGENTS.md (documentation, Lean builds, invariants) apply at all times.

This project formalizes the Fabius function and the results in both papers by Juan Arias de Reyna:

• An infinitely differentiable function with compact support: Definition and Properties; repository copies of the TeX source and published PDF are available locally;
• Arithmetic of the Fabius function, version 3; repository copies of the TeX source and published PDF are available locally.

A self-contained human-readable synthesis of the formal development is also available as LaTeX source and as a rendered PDF. That primary exposition is deliberately proof-backed: every mathematical claim in it must have a proved counterpart in the Lean development.

The asymptotic PDF refresh audit records the September 2026 source repairs, three-pass PDF builds, effective input hashes, and validation limits for articles collected by the Asymptotic project.

Artifact status (2026-09-04). The authoritative live lexical census and zero-gap result are recorded in the documentation audit and pinned in docs/doc_audit_baseline.json; this overview does not duplicate their live numeric receipt. The frozen upstream tip's 957 modules and 11,920 declarations are a historical waypoint. The retained unconditional public complexQPochhammerInf_eq_qPochhammerInfIn bridge in RvachevPochhammerFactorization gives the historical 957/11,921 merge-union checkpoint, and exists_eq_in_residual_interval in MeanValueBracket gives the historical 957/11,922 residual-certificate checkpoint. Four Bell declarations, nine repeated-differential declarations, ten Cauchy declarations, and the reverse-row Stirling theorem are included in that authoritative receipt. The documentation audit reports no missing module headers or declaration comments; the exact-dyadic inverse, Jacobi two-square, Lagrange--Rvachev Matrix, geometric Richardson, Gaussian-binomial second-moment, fixed-column rate, and half-base root simplicity, Rvachev--Appell Hasse, arbitrary-space geometric-uniform realization, regular central q-binomial sum, and Lambert branch-gap Bernoulli additions, together with the geometric-uniform moment-polynomial, its real-MGF normalization bridge, Lagrange-nodes-only, greater-than-one Gaussian asymptotic, Thue--Morse Gamma-tower differential, local Thue--Morse corner-integral, central Rvachev--Legendre cancellation, and finite Legendre--Rvachev biorthogonality, finite-prefix Thue--Morse Appell collapse, the base-two Prouhet bridge, dyadic Fourier-boundary with its quotient form, strengthened Newman self-similarity, infinite central q-Vandermonde, q-Pochhammer Lambert-form, triangular-power-product, and mean-value-bracket leaves, together with the strengthened affine finite-polynomial functional and the complex geometric-uniform moment-product bridge with its public entire-function theorem, the exterior reciprocal-germ and combined reciprocity bridges, the sharp geometric-uniform moment-polynomial degree leaf, the global geometric-uniform moment RatFunc bridge, the centered-MGF Laurent-leading theorem, finite-prefix Appell recovery, and the strengthened closed up-tail and direct positive-moment closure, are included in that clean census. Its q-series union retains QPochhammerEntire 0+5, GeometricPochhammerNormalConvergence 0+3, QPochhammerDissection 0+2, QPochhammerInfinite 1+29, QPochhammerLambertForm 0+5, GaussianBinomialAtNegOneDerivative 0+5, HalfQBinomialRootSimplicity 0+1, GaussianBinomialContinuity 0+3, GaussianBinomialCumulants 2+24, GaussianBinomialPalindromic 0+14, GaussianBinomialPolynomialStructure 0+5, CentralQBinomialReduction 0+6, RegularCentralQBinomialSum 2+1, CentralQVandermondeInfinite 0+4, CyclotomicFactorization 0+7, JacobiTripleProduct 2+25, QBinomialTheoremInfinite 1+29, GaussianBinomialFixedColumnRate 0+9, GaussianBinomialGreaterOneAsymptotics 0+2, QPascalSummation 0+4, QuantumBinomial 0+2, RogersSzegoPolynomial 1+9, QPochhammerInfiniteBounds 0+5, HeineTransformation 2+5, QGaussSummation 0+2, QPochhammerComplexOrder 1+4, BasicHypergeometricSeries 2+5, QMultinomial 1+9, JacksonIntegral 1+7, QExponential 3+8, ThetaQuasiPeriodicity 1+6, JacobiCubic 0+2, QPochhammerLogDerivative 0+10, QPochhammerOrderDerivative 0+3, PrimitiveRootBlock 0+3, QLucas 0+7, CyclotomicDivisibility 0+3, QCatalan 1+11, NewtonInterpolation 3+19, QBetaIntegral 1+8, GaussianBinomialInteger 1+10, GaussianBinomialComplexOrder 1+5, QPfaffSaalschutz 0+3, TwoPhiOneReversal 2+12, QChuVandermonde 0+10, JacobiTwoSquareCount 0+4, QuantumMultinomial 0+5, and GaussianBinomialBounds 0+6. The geometric-interpolation union also includes GeometricRichardsonGenerating 3+7, RvachevAppellHasse 1+14, RvachevLagrangeNodesOnly 1+14, and FinitePrefixAppellRecovery 11+17. The Fourier/inverse union also includes RvachevLaurentLeading 1+6. The Lambert branch-gap union adds LambertWBranchGapBernoulli 0+5: summable_norm_bernoulli_mul_pow_div_factorial proves absolute convergence for real |z| < 2π; summable_bernoulli_mul_pow_div_factorial_iff proves, for every complex z, summability exactly when ‖z‖ < 2π, hence divergence on the boundary and throughout the exterior; hasSum_bernoulli_mul_pow_div_factorial evaluates the series to z/(exp z-1) when z ≠ 0; hasSum_bernoulli_mul_pow_div_factorial_complex_iff gives the canonical removable complex value (complexExpm1Div z)⁻¹ exactly when ‖z‖ < 2π, including value 1 at z = 0; and principalLambertW_lowerLambertW_eq_bernoulliSeries gives both real-branch identities for x ∈ (-exp(-1),0) when branchGap x < 2π. The quotient theorem excludes the removable origin, while the complex theorem does not identify the sum with Lean's literal totalized quotient at zero or assert holomorphy. The branch theorem excludes both endpoints, and no remainder or higher-Puiseux claim is made. Together with the three finite branch-coordinate modules, the four-module Lambert union is 4+37, forty-one declarations. Exact historical publication receipts remain centralized in the draft manifest. Where this merge changes a TeX closure, its earlier PDF receipt is historical and a fresh synchronized render remains pending; no earlier receipt is promoted to the merged source bytes. The geometric-uniform moment-polynomial union consists of GeometricUniformMomentPolynomial 1+8 and its real-MGF normalization bridge GeometricUniformMomentPolynomialBridge 0+1, together with the inner complex-product bridge GeometricUniformComplexMomentProduct 1+3, the exterior reciprocal-germ bridge GeometricUniformExteriorComplexMomentGerm 1+2, and the sharp coefficient-and-degree leaf GeometricUniformMomentPolynomialDegree 0+3. In the target chronology the historical complex-product checkpoint 918/11,568 was followed by HalfQBinomialRootSimplicity 0+1 at 919/11,569, the exterior leaf at 920/11,572, the sharp leaf at 921/11,575, RvachevLaurentLeading at 922/11,582, and FinitePrefixAppellRecovery at the target 923/11,610 checkpoint. Retaining the unconditional public Pochhammer bridge shifts the declaration count by one and yielded the historical b899 923/11,611 semantic union. The separate incoming-branch inner/exterior checkpoints 906/11,461 and 907/11,464 remain historical. The final exact-closure tranche also retains ThueMorseCornerIntegral 1+4, RvachevLegendreCentralSum 0+3, and the sixteenth theorem in the existing zero-definition FinitePolynomialFunctional module. The separate five-module polynomial q-calculus increment is retained in full: PolynomialQDerivative 2+17, PolynomialQLeibniz 0+4, QPochhammerDerivative 0+3, LambertSeriesLog 0+4, and QGamma 2+10, totaling four definitions and thirty-eight theorems, forty-two public declarations; their exact algebraic, analytic, nonvanishing, and boundary hypotheses are recorded in the audit and walkthrough. The bounds module reuses finiteQPochhammerIn_self_pos from GeneralQConditionNumber; it does not redeclare that theorem. The rigorous forward q-monograph ledger is 181 Exact, 79 Partial, 14 None, and 8 interface rows; its source concordance is 103 Lean, 375 human, 60 N/A, and 9 conjecture rows, and the concordance extractor passes. In particular, prop:gaussian-bound is Exact, while thm:q-lucas remains Partial because Lean proves the evaluated primitive-root identity rather than the manuscript's polynomial congruence modulo Φ_d; cor:babbage-derivative remains Partial because only its value is formalized.

Deferred publication status. The fixed-26 publication check—14 fresh build cycles and 12 retained verified pairs—is recorded once in the draft manifest. A row still marked pending makes no synchronization claim. The exact older receipts below remain historical evidence; under the user-directed deferral, the fixed-26 table is an inventory rather than a merged-current parity receipt.

Four exact direct receipts record the last synchronized pre-9135 source/PDF pairs and are now historical because the live sources include the new q-Chu/reversal, geometric-generating, Gaussian second-moment, Lambert branch-gap Bernoulli, and geometric-uniform moment-polynomial APIs. The 14,037-line, 702,119-byte primary TeX has SHA-256 6a20e02cf300c0b29ba8d175831b4f86e4b336601cc5bd5f5752d5c5889be69a; its 197-page, 1,602,500-byte PDF has SHA-256 f083cd78308aba99d23d42372786c4b0a946ea8f5d47445c44d664fccfdde5e3. The 6,598-line, 465,231-byte Lean-walkthrough TeX has SHA-256 796dd849fa423ba07413eaf0a1f30dc608355c5a3cd877aa7409ad089c54794e; its 149-page, 1,231,442-byte PDF has SHA-256 bc6e3e716a1a10daf24a065f6c97e2d00cbc95071ada777050a95f91598db4a0. The 17,954-line, 813,297-byte canonical-frontier TeX has SHA-256 bcd9eefce2ead08e2cbb283e091a859aa31f36c67416543e994e10e8f9db3075; its 262-page, 1,885,642-byte PDF has SHA-256 7f7e1279e38c766a465e640638ea7e0079a942de0bc84a5c22be497af27c7bab. The 16,834-line, 837,715-byte q-series master has SHA-256 4785625c1399558f3ca59481888fc76514e0a327a1faa16945c61851f874f3d5; its 395-page, 2,494,961-byte PDF has SHA-256 89159b2635f489a42d4c972fac95332808b1d637dee7921085db1ed7d6e055af. Their exact successful three-pass page sequences were respectively 194→197→197, 144→149→149, 254→262→262, and 386→395→395. Primary and walkthrough logs and publication gates are clean; the frontier retains only expected underfull diagnostics, and the q-series master retains one harmless, readable 32.5659 pt overfull line. All page, metadata, font, render, text, and representative-visual gates passed. These receipts certify their named pre-9135 pairs only and are superseded by the synchronized 2026-09-04 receipts below.

The formerly independent Sequence publication receipt is historical: its material is now absorbed into the consolidated Transseries_And_Inversion root listed in the fixed-26 checkpoint. The Lambert guide's preceding synchronized receipt is a 4,829-line, 174,423-byte TeX with SHA-256 724dfe5b1effcda29325a5bdfb066ff970eb74ab460f650185339fefce40ebc1; its 69-page, 952,929-byte PDF has SHA-256 0b5f28dbfe590658e74150e8ccff6f023ecd0b8fb4e3e978ec275d9ddd244de6. Its successful page sequence was 67→69→69; machine and visual gates passed, with expected underfull diagnostics and one harmless readable 0.825 pt internal overfull line. The Bernoulli-series source overlay made that Lambert PDF historical; the synchronized 2026-09-04 receipt below supersedes it. The Sequence inversion/transseries volume's 16,705-line, 778,477-byte TeX has SHA-256 4aa038c10ddd931b7c1248095ddfdf0ce8769c69cc0df4f344f6365d0e45e8e1; its 205-page, 2,198,655-byte PDF has SHA-256 ec1f4d2ac608786f33be97d040fdfd03b6f74494dee74f044fd2e6631217d4fb. Its successful page sequence was 198→205→205; corrected title/author metadata, machine gates, and extensive visual checks passed. The final log retains one duplicate-page-destination notice, nine PDF-string notices, 47 overfull and 12 underfull diagnostics; sampled largest cases are clean and unclipped.

Historical synchronized publication receipts (2026-09-04). Each pass tuple below is pages/bytes. At their named source checkpoints, all six TeX/PDF pairs completed exactly three successful serial halt-on-error passes from absent sidecars. Final-log reference/rerun/error checks, metadata, A4 rotation zero, every-page render and nonblank-text checks, embedded/subset fonts with Libertinus and no Type 3, and representative visual checks all passed; generated sidecars were cleaned and forbidden checksum-ledger basenames were absent. These six pairs remain historical receipts for their named source checkpoints; the later historical b899 receipts below superseded them where a rebuilt root was listed.

• Primary exposition: TeX 14,328 lines / 715,760 bytes / SHA-256 60c0a6ff4e75ec37e6928067859671d87622ad8f430a1006dd4c71c7e7b25674; passes 197/1,579,558 → 200/1,621,473 → 200/1,621,467; final PDF 200 pages / 1,621,467 bytes / SHA-256 50febffeb7dda743330bd346b8f5fd45f85668db97c19fb52d4cd741d1692826; fonts 29 total / 6 Libertinus / 0 Type 3.
• Lean walkthrough: TeX 6,855 lines / 482,759 bytes / SHA-256 1c48c54b194eb9e99dae64ddca70e2aa5d2edd995160ee2d5bc6455b545683f7; passes 149/1,225,017 → 154/1,262,552 → 154/1,262,574; final PDF 154 pages / 1,262,574 bytes / SHA-256 f9cba79348ffb81c41fc08b6523548effcc45ded9a2eb2b18e618ac9d59d0648; fonts 30 / 7 / 0.
• Lambert Guide: TeX 4,876 lines / 177,511 bytes / SHA-256 d852a345685dd61335a89fc4fd1092680bdc597a5d1e6ac612883946ad0d99ea; passes 68/963,230 → 70/986,865 → 70/986,865; final PDF 70 pages / 986,865 bytes / SHA-256 0b8801649a6dd43d9f02dcfc2f60cac50b5c8f88bd782645bf97d30cc3dfbd41; fonts 42 / 5 / 0; one harmless readable 0.82504 pt overfull and 133 underfull diagnostics.
• Canonical frontier: TeX 18,173 lines / 826,738 bytes / SHA-256 844842bf699a24651f660bd7d81d814f6396b4fe6fc6de66a04908904221860b; passes 257/1,822,725 → 265/1,904,567 → 265/1,904,551; final PDF 265 pages / 1,904,551 bytes / SHA-256 dcaa7ac1e5397912c97a474b4023521e49d0785eb6ef67d83d0ce002d9cbb6e6; fonts 40 / 8 / 0; one readable unclipped 9.43108 pt overfull at source lines 1032–1043 and 299 underfull diagnostics.
• Geometric q-frontier: TeX 27,598 lines / 1,270,870 bytes / SHA-256 6db4e211b0588ed75a0e89e13d97306f1d5d38b42a2bf941914ea16b9ca93dae; passes 386/8,157,293 → 403/8,339,780 → 403/8,339,736; final PDF 403 pages / 8,339,736 bytes / SHA-256 4d909b5e228e2053d473dc75da502382c7a4fe2b096f798e124e6530d3a15027; fonts 43 / 11 / 0; zero overfull and 37 underfull diagnostics.
• Canonical q-series synthesis: TeX 16,910 lines / 842,514 bytes / SHA-256 196f219d5e1efba463ebabb69659697b1afb28989ef1a8da6219226d3262ad32; passes 390/2,386,364 → 398/2,501,624 → 398/2,501,638; final PDF 398 pages / 2,501,638 bytes / SHA-256 e8094b054f52b1fb71c7540f0834155fae0eac17887cb7cac1567848bd65d3b3; fonts 43 / 5 / 0. Every pass's index run accepted 164 entries, rejected none, produced 254 lines, and emitted no warning. The sole retained 32.5659 pt overfull paragraph at source lines 590–598 is readable and unclipped; the final log has zero underfull diagnostics.

Historical b899 synchronized publication receipts (2026-09-04). Each pass tuple below is pages/bytes. All ten roots were frozen and built in exactly three serial halt-on-error passes from absent sidecars. Subsequent merged source changes make these receipts historical; current rebuild state is recorded only in the fixed-26 checkpoint linked above.

• Primary exposition: TeX 15,148 lines / 759,509 bytes / SHA-256 721cb901de2254ef48991452c4831762f54a36e0a405b4bbeb7f812653e71754; passes 208/1,640,077 → 210/1,683,143 → 210/1,683,141; final PDF 210 pages / 1,683,141 bytes / SHA-256 afe85efec5716fe85cc7d8a5d6af459fd72775526f4bda11df04e6ff275b36c9; fonts 29 total / 6 Libertinus / 0 Type 3.
• Lean walkthrough: TeX 7,260 lines / 526,929 bytes / SHA-256 2005d4a70a66a1d8f3eac9be6d83585ea70c9312b09b098f2c65025db9fca814; passes 160/1,281,609 → 165/1,319,585 → 165/1,319,594; final PDF 165 pages / 1,319,594 bytes / SHA-256 b5e886d7c76db56fd9e9e1552bdd78fbd65ea9075ea1e04d62c00a7c04e948fb; fonts 30 / 7 / 0.
• Geometric q-frontier: driver 27,671 lines / 1,275,367 bytes / SHA-256 d47c0ad93eb359d13e7e9772668f16dbc98bcb4d880f3679366e1d461451bbcd; recursive TeX closure 8 files / 27,777 lines / 1,281,413 bytes / digest 39f7cd41e706314f2cafb903c2da2e6e83d2b17f5bb0612492204d15c1a28d91; passes 388/8,163,847 → 405/8,346,265 → 405/8,346,247; final PDF 405 pages / 8,346,247 bytes / SHA-256 fef7d8260543ad1d20d69e9e41fa0cfc31603de7961f6aeb97a50740aecd596c; fonts 43 / 11 / 0.
• Canonical q-series synthesis: driver 17,265 lines / 864,659 bytes / SHA-256 4dd3f7fb22387d8e3d039e8d49cd870a63ebe0881f7f215c7074854825a27bb9; recursive TeX closure 14 files / 26,762 lines / 1,210,902 bytes / digest b567430fdd64f6d50bd24fcb070216c27f7e3e81e8b0c76c3228767ebdf980c6; passes 397/2,417,476 → 405/2,533,717 → 405/2,533,715; final PDF 405 pages / 2,533,715 bytes / SHA-256 055eb1fc26467857394a5b3bd8cd327f6985ea5d2f966ab5f099ac20bb2b8fb2; fonts 43 / 5 / 0; every pass's index run accepted 164 entries, rejected none, produced 254 lines, and emitted no warning.
• Inverse-theory synthesis: driver 293 lines / 11,514 bytes / SHA-256 92fab1fae38bbcf86a45b51bfe7ff34e2801361df9d2f3d6aa3de4dc966eaa3c; recursive TeX closure 17 files / 10,682 lines / 431,748 bytes / digest 6e4e6fde424fd5046467b1f1cec0c19b6c10eb681fae4ba7cc53e14b6a5bf61e; passes 132/1,983,313 → 137/2,045,485 → 137/2,045,486; final PDF 137 pages / 2,045,486 bytes / SHA-256 cee0de894656562fbdb75d6304055fc03fae06203985119419e465a5cd213995; fonts 31 / 6 / 0.
• Comb-interpolation synthesis: driver 187 lines / 6,724 bytes / SHA-256 a4c1e33165ff7291682cd890f23fe4af98e9f11f7ad1d9a7f8b68c78d53f9a56; recursive TeX closure 15 files / 12,597 lines / 477,163 bytes / digest 9e22455b3f65eb48306ad21c57445b6052a56498cb363666ffb9b160f5cc8090; passes 153/2,383,950 → 160/2,467,995 → 160/2,468,000; final PDF 160 pages / 2,468,000 bytes / SHA-256 ad8587049580e6fde371f534b6f8b4e56fa4c929173f87d3021ed369e5225d4c; fonts 33 / 7 / 0.
• Lambert Guide: TeX 4,940 lines / 181,577 bytes / SHA-256 2e6a4782fc4e4b945869f5fb45b39cf94e8dc34296866edf26b4cdfe19b1898b; passes 68/968,083 → 70/991,847 → 70/991,848; final PDF 70 pages / 991,848 bytes / SHA-256 f802d78299f8f6aca7d31b935a4884f9343389a7307decb04c18b5159c8a4f04; fonts 42 / 5 / 0.
• Up Polynomial Synthesis: driver 2,368 lines / 98,609 bytes / SHA-256 95d293e34559e910cca2df4547e6e181a8d26bc8e8cf61c4445cf12c57ed8e0e; four-file TeX closure 5,434 lines / 211,270 bytes / digest 62aa76428089cd164705b1d31e038d4e48545681eedc01cb491e6a94f07b0e41; passes 61/1,045,488 → 62/1,071,179 → 62/1,071,181; final PDF 62 pages / 1,071,181 bytes / SHA-256 99c5d8256b983652755fe8e46ef015277e61b94941a4ca6c875bddaf0493b101; fonts 27 / 4 / 0.
• Thue--Morse Atlas: TeX 10,553 lines / 481,614 bytes / SHA-256 cced4128c359ec467baaf1a55c21c68424397f783a39ea7fe2af5a94975b9dd5; passes 139/1,681,559 → 144/1,739,891 → 144/1,739,884; final PDF 144 pages / 1,739,884 bytes / SHA-256 1c81863b0976017fab1b7f5972c50cd541b3ffb05306bf85994548a56a782fc0; fonts 38 / 8 / 0.
• Canonical semi-formalized frontier: TeX 18,651 lines / 858,502 bytes / SHA-256 140256058b7a01bcdb4f1592cfab9e6c2ac170f5f0863572627a9b2f93ab7793; passes 265/1,868,249 → 273/1,950,120 → 273/1,950,112; final PDF 273 pages / 1,950,112 bytes / SHA-256 17525c7623bf774f515ecf1a949d533bbe125fde036356c4bb9f787eedad0322; fonts 40 / 8 / 0.

Across all ten roots, required final-log error/reference/rerun gates, metadata, A4/rotation-zero checks, every-page render and nonblank-text checks, embedded/subset fonts with Libertinus and no Type 3, representative visual checks, sidecar cleanup, and the forbidden-checksum-basename search all passed. Diagnostics are clean except for five minor q-series horizontal boxes (maximum 10.14 pt), two inverse horizontal boxes (2.42 and 2.45 pt), one Lambert horizontal box (0.83 pt), and one canonical-frontier horizontal box (9.43108 pt), all nonblocking. The final aggregate TeX closure contains 76 files / 140,223 lines / 6,439,569 bytes, with direct aggregate digest ae8690ad8d160055cbae36eff96d858f87572d171e7aacf7540d67543998af21. All ten rows were synchronized at the named b899 checkpoint; they and the earlier receipts below remain historical provenance for their named sources.

For provenance, the superseded pre-d8b pairs remain historical receipts: primary TeX/PDF 938517a92565685ac9f7194b879cfe752ce783f258bde8b7b685aee41aed13dc / bf26d78dd2cc49feb87a85413ef9c04c7a8a3dac4f793cf86e3436f7502cb2a7 (694,350 / 1,593,577 bytes; 195 PDF pages); walkthrough e598aa02d4d10eda8bcfdafe3731f4a663bdcba58407f454485fae6796b41050 / 5ff79c24fbced37dfaa5eb9c34447d0e7661b2b2bc5a0597687e43f93d7e189a (456,855 / 1,219,336 bytes; 145 pages); frontier 7dd140370a0ac68522364a83a3c6423df93570741eafaef2ee8c1fac17670e2f / 9d38ab9d43befd6e26fd06ab9680b4a761365fb9d8a9f0de18489c243bd62d3e (808,185 / 1,877,159 bytes; 260 pages); and q-series d8f730b8eb6602d4d16112aea77a3e67dfbeadf46bcd28c1cdf3b12450b7d4fb / 5d25df07e6df1cd32118ee87e64c1cc54ad32da7c578a182231f98dd9fee9d5c (837,715 / 2,494,949 bytes; 395 pages). Representation Frontiers and the Integration-and-Transform master retain exact historical 301- and 377-page receipts in the frontier manifest and their local group records; those receipts certify only their named source checkpoints. Replacement parity is recorded only in the fixed-26 checkpoint linked above. Filed New Frontiers and the notation catalogue retain historical 41- and 88-page receipts; the comb-interpolation 158-page receipt is historical and was superseded by the later historical 160-page receipt above. Checksum ledgers remain abolished and hardened repository-wide; no SHA256SUMS* files exist or participate in validation.

Post-b899 upstream API chronology. The geometric-uniform union adds the complex-product bridge GeometricUniformComplexMomentProduct 1+3 and the exterior reciprocal-germ bridge GeometricUniformExteriorComplexMomentGerm 1+2, the combined inner/exterior germ leaf GeometricUniformMomentReciprocity 1+5, followed by the sharp coefficient-and-degree leaf GeometricUniformMomentPolynomialDegree 0+3 and the global rational-coefficient leaf GeometricUniformMomentRatFunc 1+4. In the merged chronology, the historical complex-product checkpoint 918/11,568 was followed by HalfQBinomialRootSimplicity 0+1 at the merged-main pre-local checkpoint 919/11,569, the exterior leaf at 920/11,572, and the sharp leaf at the historical 921/11,575 checkpoint. RvachevLaurentLeading 1+6 then gives 922/11,582, and FinitePrefixAppellRecovery 11+17 gives the historical pre-RatFunc 923/11,610 checkpoint. GeometricUniformMomentRatFunc 1+4 gives the historical RatFunc checkpoint 924/11,615. The two new ProbabilityLaplaceMoments theorems then give 924/11,617, and RvachevLegendreBiorthogonality 1+1 gives the historical 925/11,619 checkpoint. Subsequent merged source work adds the zero-definition QPochhammerLambertForm 0+5, CentralQVandermondeInfinite 0+4, TriangularPowerProduct 0+2, and MeanValueBracket 0+6 leaves, the 1+12 ThueMorseNewmanSelfSimilarity leaf, and a net twenty-nine further public declarations across existing modules, reaching the immediate pre-reciprocity checkpoint 930/11,678. Promoting complex-product differentiability/entireness and adding GeometricUniformMomentReciprocity 1+5 contributes one module and seven declarations, giving the historical reciprocity checkpoint 931/11,685. The subsequently merged DyadicBoundaryIdentity 0+2 and FinitePrefixThueMorseCollapse 0+8 leaves add two modules and ten declarations, giving the historical 933/11,695 checkpoint. Finally, ProuhetBaseTwoBridge 0+6 adds one module and six theorems, while one new theorem in DyadicBoundaryIdentity and seven new theorems in ThueMorseNewmanSelfSimilarity bring those existing modules to 0+3 and 1+19. This incoming tranche adds one module and fourteen public declarations in total, giving the historical 934/11,709 census. The next series/transseries tranche added nine facade modules carrying 72 explicit public declarations; two further additive Neumann names are generated by to_additive and therefore are not part of the lexical declaration count. Six other declarations arrived in concurrent merged work. The resulting historical checkpoint was 943/11,787, with no documentation gaps. The next transseries tranche adds nine more facade modules and 90 explicit public declarations: DerangementNearestInteger 1+7, LinLogCoreInversion 4+18, OrdinaryPartialBell 2+4, PowerLogCoreInversion 3+6, RemainderTransport 0+3, StaircaseInversion 0+7, TransseriesFlat 4+16, TransseriesHarmonicIncrement 0+2, and WrightOmega 1+12. Four integer-exponent theorems added to the existing TransseriesDifferentialBlock module bring it from 0+5 to 0+9. Thus this tranche contributes nine modules and 94 lexical declarations, giving the historical incoming checkpoint 952/11,881. A subsequent exactness overlay adds the two explicit OrderDual Neumann wrappers and Wright-omega analyticity, bringing TransseriesWellBased to 0+7 written declarations (plus two to_additive names), WrightOmega to 1+13, and the historical exactness checkpoint to 952/11,884, again with no documentation gaps. The merged successor tranche adds fifteen facade modules and 112 declarations, while five net declarations enter existing modules (QBinomialTheoremInfinite +5, RemainderTransport +1, and GaussianBinomialFixedColumnRate -1 after ownership transfer). This gives the historical 967/12,001 checkpoint. The next merge adds StirlingSeriesCoefficients 3+12 and WrightOmegaTwoOrders 0+8, plus 24 declarations in existing modules, giving the historical intermediate 969/12,048 checkpoint. UnitSeriesPowerRecurrence 0+3 then gives the historical 970/12,051 checkpoint. The overlapping branches reconcile at the historical 970/12,056 checkpoint. The next seven-module overlay is AbelPolynomialSeries 2+9, AssociahedronFaceNumbers 4+23, BernoulliFormalLog 0+5, ExponentialRescaling 0+4, PochhammerFalling 1+13, RaneyNumbers 4+12, and UnitSeriesPowerRecurrence 0+3. Its 80 declarations, offset by three declarations moved from NorlundDiagonal to ExponentialRescaling, give the historical 977/12,133 checkpoint. GridEvaluationCertificate 0+4 and IntegerCRTCertificate 0+5 then give the historical 979/12,142 checkpoint. The incoming successor consists of NorlundGeneralized 3+18, StirlingSymmetricFunctions 0+4, LagrangeInversionUniqueness 0+6, NewtonReciprocal 1+5, StirlingSecondReverseRowIdentity 0+2, and TransseriesWrightOmegaTerms 0+10, plus eight declarations in existing modules. Its +6-module/+57-declaration delta gives the historical 985/12,199 checkpoint. These overlapping receipts are not additive; the authoritative current union is computed by scripts/doc_audit.py and pinned in docs/doc_audit_baseline.json. The frozen upstream tree's 957/11,920 inventory, the retained Pochhammer bridge's historical 957/11,921 merge-union checkpoint, and the residual-existence certificate are all included in the current semantic union computed by scripts/doc_audit.py and pinned in docs/doc_audit_baseline.json. The separate incoming-branch inner/exterior checkpoints 906/11,461 and 907/11,464 remain historical. The final exact-closure tranches include ThueMorseCornerIntegral 1+4, RvachevLegendreCentralSum 0+3, the sixteenth theorem in the existing zero-definition FinitePolynomialFunctional module, the two new theorems in ProbabilityLaplaceMoments, and RvachevLegendreBiorthogonality 1+1. At a later named publication checkpoint, the retained primary exposition, Lean walkthrough, canonical frontier, Representation Frontiers, filed New Frontiers, notation catalogue, Integration-and-Transform master, comb-interpolation, and q-series synthesis PDFs contain respectively 210, 165, 273, 301, 41, 88, 377, 160, and 389 A4 pages. Their current TeX sources contain post-render unions, including the centered Appell/deconvolution, arbitrary-phase synthesis, Lagrange--Rvachev, prime-power companion-row, outer Pochhammer normal convergence, total rational integer-index zero-row Wigner-square, and finite/infinite q-Pochhammer material, as well as the fixed-depth effective inverse realizer, total inverse computability theorem, sharp exact endpoint-mass denominator, and actual Lambert branch-gap Bernoulli series. Those page counts are historical receipts, not source/PDF-parity claims for the merged bytes. The fixed-26 checkpoint linked above records the deferred publication batch and makes no current-parity claim for pending rows. The older Representation Frontiers, filed New Frontiers, notation catalogue, and Integration-and-Transform artifacts are separately historical or unsynchronized; they are not members of this exact ten-root merge set and retain the pending status recorded by their package notices.

The formally proved small-argument hierarchy—including the corrected sharp asymptotic, the general coefficient algebra for the recursive all-orders expansion, and the first two explicit periodic saddle corrections—is integrated into the primary exposition. Exploratory derivations, the small-argument notebook, and the primary-exposition gap register are preserved in the canonical research-frontier LaTeX volume (PDF). That volume labels claims still awaiting literal Lean counterparts and records their exact outstanding proof obligations.

Inverse analyticity, non-elementarity, endpoint asymptotics, dyadic self-sampling, and effective inverse computation are treated together in Inverse Fabius Theory: Analyticity, Asymptotics, Computability, and Dyadic Sampling (PDF). Its theorem concordance distinguishes exact Lean counterparts from human-proved frontier results, conjectures, open problems, and non-theorem source material; its provenance ledger records the five retired source packages and their immutable recovery points.

On the formal side, the class of elementary functions of one real variable is proved real analytic on a dense open subset of the line, and this is combined with nowhere analyticity to show that no elementary function agrees with the Fabius function on any subset of [0,1] with nonempty interior. The same conclusion holds for the stated class of continuous algebraic branches, which also reaches algebraic functions not expressible by radicals. The inverse Fabius function is nowhere real analytic on [0,1], hence is not elementary; neither it nor the Fabius function becomes representable after closing the class under continuous inverse branches at any depth, a closure that includes the Lambert W function.

The development contains executable exact arithmetic. The evaluator and its analytic correctness at every dyadic, the canonical function's existence and uniqueness, the moment and denominator arithmetic, the global differential identities, normalized integer-order and positive-real fractional Volterra calculus, exact integer primitive ladders and first fractional Fabius--Rvachev shifts, unconditionally summable analytic finite-series filters, Taylor reduction, the Fourier and entire-series identities, probability and weak-convergence constructions, polynomial step approximants, Poisson summation, and every theorem, lemma, corollary, and prose proposition in both papers are checked without sorry. The asymptotic layer additionally proves the corrected sharp small-argument expansion with its nonconstant Gamma--zeta periodic term, together with its complete all-orders saddle expansion.

Several agents sometimes develop this directory concurrently in separate worktrees. If you are one of them, read AGENTS.md first; whether the multi-agent coordination framework is currently in effect is stated by the single switch file AGENTS/STATUS.md.

Design

The formalization separates two functions that the sources both call F:

• BoundedFabius = ℝ → Set.Icc 0 1 is the CDF-style function requested for this project. IsFabius F says that it is zero on (-∞,0], one on [1,∞), smooth, symmetric on [0,1], and satisfies the differential equation on [0,1/2]. The existence/uniqueness theorem selects the canonical fabius, constructed as the fixed point of an integral contraction on continuous symmetric unit-interval-valued functions.
• extendedFabius F : ℝ → ℝ is the signed global extension used in the paper. It is defined by the locally finite Thue--Morse translate sum in equation (1). It agrees with the bounded function on [0,1] but can be negative outside it.

Rvachev's compactly supported up function is represented by rvachevUp F, which folds the bounded candidate about zero. Its evenness is structural: rvachevUp_even holds for every BoundedFabius, without the Fabius equations. Support does use IsFabius: Basic.lean gives the lightweight inclusion support_rvachev_subset_Ioo, Monotonicity.lean strengthens it to the exact pointwise identity support_rvachevUp, and tsupport_rvachev identifies the topological support with the closed interval [-1,1].

The arithmetic layer is independent of real analysis:

• moment, halfMoment : ℕ → ℚ are the rational sequences c_n, d_n.
• momentNumerator, halfMomentNumerator : ℕ → ℕ are F_n, G_n, defined by division-free recurrences.
• fabiusDyadicValue n a : ℚ computes the bounded Fabius function exactly at the signed dyadic argument a / 2^n; extendedFabiusDyadicValue computes the paper's signed global extension.
• evalFabiusDyadic : ℚ → Option ℚ is the convenient rational-input wrapper. It returns none exactly when the reduced denominator is not a power of two.
• fabiusDyadic remains the independent closed formula from equation (32), while rvachevDyadic evaluates exact dyadic values of up.
• reshetnikov : ℕ → ℚ remains rational until its integrality is proved.
• dyadicDenominator : ℕ → ℕ is the finite LCM D_n.

This makes denominator, divisibility, parity, and valuation proofs live in ℚ and ℕ; named bridge theorems connect them to the analytic functions. The Fourier transform, sinc product, inversion integral, moment series, and complex exponential generating function are also represented explicitly. At formal source checkpoint 71ab6f6728fceb753c88d8b0573077a59acf2682, summable-scale products additionally give the entire geometric reciprocal-Gamma family, its Mahler/zero/reflection laws, the exact dyadic Rvachev bridge, and the dyadic zero and meromorphic pole orders. The post-checkpoint jet/tower tranche at 0ba35abd4 adds all five public declarations of ReciprocalGammaJets.lean and the first eight public declarations of ThueMorseGammaTower.lean; the current integrated tree adds that module's ninth declaration, the master-product/tower-ratio bridge. Thus the entire reciprocal Gamma function now has exact first jets, simple analytic orders, and punctured local coefficients at every nonpositive integer, and the Thue--Morse continuation has exact first jets, Mellin/integral GammaLog levels, and dyadic log/tower laws at every natural level. The GammaLog is a chosen coordinate, not a proved Complex.log identity. Its and the tower's definitions are total in the real parameter a, while their analytic identification laws assume 0 < a; no derivative value is assigned to raw Gamma at a pole. The parameter-a differential and iterated ladder are now closed by the ThueMorseGammaTowerDifferential row below, for the chosen GammaLog coordinate and the same positive-parameter domain.

Using the Lean library

From the repository root, the complete public surface is checked with

LAKE_JOBS=1 lake build +FabiusFunction

Use import FabiusFunction when downstream code needs the entire development. For a smaller dependency footprint, the following imports are useful entry points:

The current New Frontiers finite Gram--Legendre crosswalk has eleven modules, twenty public definitions, and 109 public theorems, hence 129 declarations. Its former nine-module 18+81=99 subtotal is extended by LegendreGauntClosedForm (2+25) and FabiusLegendreGauntClosedForm (0+3). Thus the integer-index zero-row square datum and finite Wigner-square Gram route are closed; only signed/general Wigner and the later infinite spectral layers remain outside this tranche.

Precisely, the directly defined square datum is not a bridge to a separately implemented general Wigner symbol. There is no signed value or phase convention, half-integer or nonzero-magnetic-index API, general 3j/6j/9j, orthogonality, recoupling, or named Wigner-symmetry theorem. The Gaunt factorial form and product-coefficient nonnegativity/zero criteria are available by composing the listed results but have no separate named wrappers. Infinite Legendre interchange, Christoffel reconstruction, roots/quadrature, Padé/J-fractions, infinite Jacobi theory, and asymptotics also remain open.

Purpose	Focused import	Good starting declarations
Generic asymptotic scales, vector-valued Poincaré expansions, and corrected flatness (q0:def:scale, q0:def:poincare, q0:prop:uniqueness, q0:def:flat, q0:prop:invisible)	FabiusFunction.TransseriesScale, FabiusFunction.TransseriesFlat	Exact. TransseriesScale is 3+6: IsAsymptoticScale, poincarePartialSum, IsPoincareExpansion; poincarePartialSum_zero, poincarePartialSum_succ, IsPoincareExpansion.isLittleO_succ_remainder, .tendsto_coeff, .tendsto_coeff_div, .coeff_unique. TransseriesFlat is 4+22 in the merged union. Its vector API is IsFlat; isFlat_zero, IsFlat.add, .neg, .sub, .const_smul, isFlat_exp_neg_rpow_atTop, IsPoincareExpansion.add_flat, .sub_same_coeff_isFlat, .iff_sub_isFlat, IsFlat.smul_of_scale_absorption, .smul_of_isBigO_inv_pow. The retained scalar compatibility API is flatSubmodule, AbsorbsScale, powScale; mem_flatSubmodule_iff, IsFlat.mul_absorbsScale, absorbsScale_const, IsPoincareExpansion.add_isFlat, isFlat_sub_of_isPoincareExpansion, isPoincareExpansion_iff_isFlat_sub, isPoincareExpansion_zero_iff, powScale_eq_rpow, absorbsScale_of_isBigO_pow, isFlat_exp_neg, isPoincareExpansion_add_exp_neg. Variable multiplication requires explicit scale absorption; the power specialization assumes eventual nonvanishing.
Well-based supports, power--logarithmic scales, and height comparison (q0:lem:dickson, q0:lem:neumann, plt:lem:mot-dominance, q0:prop:height)	FabiusFunction.TransseriesWellBased, FabiusFunction.TransseriesScaleDominance, FabiusFunction.TransseriesPolyLogScale, FabiusFunction.TransseriesHeight	TransseriesWellBased is 0+7 written theorems: dickson_isPWO, dickson_antichain_finite, dickson_isPWO_pi, neumann_isPWO, neumann_finite_factorizations, neumann_isPWO_orderDual, neumann_finite_factorizations_orderDual; to_additive also generates two names outside the lexical census. Dickson and Neumann are Exact, with the partial-order/OrderDual boundary stated in the dedicated row below. TransseriesScaleDominance is 1+7, TransseriesPolyLogScale 0+4, and TransseriesHeight 0+3; their exact sequence-scale and displayed comparison results do not construct an unordered maximal scale or recursive global height taxonomy.
Polynomial block operators and Laurent-block differentiation (plt:eq:mot-block-derivative, plt:lem:mot-block-antiderivative)	FabiusFunction.TransseriesBlockAntiderivative, FabiusFunction.TransseriesDifferentialBlock	TransseriesBlockAntiderivative remains 3+12. TransseriesDifferentialBlock is now 0+12: derivation_pow_t, derivation_block, derivation_zpow_block, exists_zpow_block_primitive, existsUnique_zpow_block_primitive, exists_block_primitive, derivation_block_zero, exists_block_primitive_resonant, derivation_val_inv, derivation_pow_inv, derivation_zpow_t, derivation_block_zpow. The integer Laurent block formula is Exact. The compound antiderivative lemma remains Partial because generic uniqueness assumes injective evaluation at L and no concrete Laurent-polynomial model or resonant uniqueness-up-to-constants wrapper is constructed.
Bell-polynomial coefficients of powers, logarithms, and exponentials of unit series (p0:lem:bell-conversion, p0:lem:power-log, p0:cor:exp-log-jets)	FabiusFunction.UnitSeriesBellCoefficients	Exact as formal power-series algebra; no analytic convergence or logarithm branch is asserted. Exhaustive 0+16 surface: ordPartialBell_eq_factorialRatio_partialBell, factorial_mul_ordPartialBell_eq_factorial_mul_partialBell, coeff_fallingSeries_subst_eq_sum_ordPartialBell, coeff_fallingSeries_subst_eq_sum_ordPartialBell_of_pos, coeff_fallingSeries_subst_eq_sum_partialBell, coeff_negBinomSeries_subst_eq_sum_ordPartialBell, coeff_negBinomSeries_subst_eq_sum_ordPartialBell_of_pos, coeff_logOf_eq_sum_ordPartialBell, egfA_factorialDenormalize_coeff_eq, bellWeightSeries_factorialDenormalize_coeff_eq, coeff_logOf_eq_sum_partialBell, coeff_exp_subst_eq_completeBell, coeff_exp_subst_eq_partitionExpSum, coeff_exp_subst_eq_sum_weightedPartitions, coeff_exp_subst_eq_sum_div_weightedPartitions, coeff_exp_subst_recurrence.
Formal arbitrary-power coefficient recurrence (alg:merged-exp-log-power)	FabiusFunction.UnitSeriesPowerRecurrence	Exhaustive 0+3 surface: coeff_recurrence_of_mul_derivative_eq, mul_derivative_fallingSeries_subst_sub_one, coeff_fallingSeries_subst_sub_one_recurrence. The first theorem is denominator-free over every commutative ring and assumes neither F(0)=1 nor invertibility; the falling-series specialization is over a commutative rational algebra with F(0)=1. Together with the existing logarithm and exponential recurrences this makes all of alg:merged-exp-log-power Exact. These are formal-series identities, not analytic powers or branch choices.
Deterministic grid and integer residue certificates	FabiusFunction.GridEvaluationCertificate, FabiusFunction.IntegerCRTCertificate	Exhaustive 0+4 and 0+5 surfaces. Grid API: mvPolynomial_eq_of_eval_eq_on_grid, mvPolynomial_eq_of_eval_eq_on_grid_of_degreeOf_sub_le, mvPolynomial_eq_of_eval_eq_on_grid_of_degreeOf_le, mvPolynomial_grid_eval_injective. It works over any commutative integral domain. Integer API: int_prod_dvd_of_pairwise_coprime, int_eq_zero_of_modEq_zero_of_natAbs_lt_prod, int_eq_of_modEq_of_natAbs_sub_lt_prod, int_eq_of_modEq_of_natAbs_add_lt_prod, int_eq_of_modEq_of_two_mul_natAbs_lt_prod. Signed or composite pairwise-coprime moduli and the empty divisibility family are included; equality retains strict magnitude bounds. No field, characteristic-zero, primality, reconstruction, or probabilistic claim is made.
Weighted Appell translation and inversion	FabiusFunction.AppellSequence	The +11 extension is Bell.binomialConv_unitSeq, Bell.binomialConv_four_swap, Appell.translate_zero, Appell.translate_translate, Appell.binomialConv_translate, Appell.translate_injective, Appell.translate_neg_translate, Appell.translate_translate_neg, Appell.translate_eq_iff, Appell.weighted_binomial_inversion_iff, and Appell.binomialConv_translate_neg_translate. thm:merged-weighted-binomial-translation is Exact over a commutative semiring, with additive left cancellation only for injectivity; cor:merged-weighted-binomial-inversion is Exact over a commutative ring. No analytic EGF claim is made.
Stirling-series coefficients (q2:eq:stirling-cj)	FabiusFunction.StirlingSeriesCoefficients	Exhaustive 3+12 surface. Definitions: stirlingKernelCoeff, stirlingKernel, stirlingCoeff. Theorems: coeff_stirlingKernel, constantCoeff_stirlingKernel, stirlingCoeff_zero, stirlingCoeff_recurrence, bernoulli_three, bernoulli_four, stirlingKernelCoeff_one, stirlingKernelCoeff_two, stirlingKernelCoeff_three, stirlingCoeff_one, stirlingCoeff_two, stirlingCoeff_three. The formal definition, recurrence, and displayed values c₀=1, c₁=1/12, c₂=1/288, c₃=-139/51840 are Exact; the surrounding analytic Fubini-sector expansion is not claimed.
Two Wright-omega orders (plt:prop:mot-two-orders)	FabiusFunction.WrightOmegaTwoOrders	Exhaustive 0+8 surface: wrightOmega_lt_self, tendsto_log_wrightOmega_div_atTop_zero, tendsto_wrightOmega_div_atTop_one, tendsto_log_wrightOmega_sub_log_atTop_zero, tendsto_log_wrightOmega_div_log_atTop_one, self_sub_wrightOmega_isEquivalent_log, wrightOmega_sub_self_isEquivalent_neg_log, wrightOmega_residual_isEquivalent. The two concluding real equivalences ω(X)-X∼-log X and ω(X)-X+log X∼(log X)/X are Exact at atTop. The proposition remains Partial at its four-term quantitative expansion and explicit envelope.
Catalan algebra of the quadratic core (p6:prop:quadratic-core-catalan, p6:lem:quadratic-core, p6:thm:deepest-pole)	FabiusFunction.QuadraticCoreCatalan	Exhaustive 3+8 surface. Definitions: quadHalf, halfBinom, quadCoef. Theorems: catalan_two_step, quadHalf_zero, quadHalf_antidiagonal, halfBinom_step, quadHalf_rat, quadCoef_rat, quadCoef_zero, quadCoef_rec. The Catalan coefficient proposition is Exact. The quadratic-core lemma is Partial: the coefficients satisfy the displayed recursion order by order, but the module does not package the full power-series quadratic identity, positive-valuation uniqueness, or formal square root. The deepest-pole identification inside the Gamma and Barnes inversions remains unformalized.
Simple dyadic roots of the half-base Gaussian polynomial (cor:halfbase-root-locus)	FabiusFunction.HalfQBinomialRootSimplicity	Exhaustive zero-definition/one-theorem surface: halfQBinomial_sum_rootMultiplicity_two_pow. Over ℚ, every root 2^j with j<n of the coefficientwise half-base q-binomial polynomial has root multiplicity exactly one. Composed with halfQBinomial_sum_eq_zero_iff and gaussianBinomial_half_eq_halfQBinomial, this proves that the manuscript polynomial has exactly the simple roots 1,2,…,2^(n-1) and no others, so cor:halfbase-root-locus is Exact by composition. This leaf does not generalize simplicity to arbitrary characteristic or classify roots for an arbitrary base.
Leading Laurent term of the reciprocal centered Rvachev MGF (is:p2:thm:Laurent-leading)	FabiusFunction.RvachevLaurentLeading	Exhaustive public surface: one definition, rvachevCenteredMGF, and six theorems, rvachevCenteredMGF_eq_rvachevFourierProduct, rvachevCenteredMGF_pi_mul_I_int, rvachevCenteredMGF_pi_mul_I_int_ne_zero_of_odd, tendsto_sub_pow_mul_inv_rvachevFourierProduct_int, tendsto_rvachevCenteredMGF_laurent_int, and tendsto_rvachevCenteredMGF_laurent_two_pow_mul_odd. The definition uses the repository's half-scale generating coordinate as centeredComplexGeneratingFunction F (2*t), and the rotation theorem gives M(t)=Φ(i*t/(2π)). At every nonzero integer zero the generic theorem cancels the exact order `v₂(
Finite dyadic Appell-prefix expansions and exact recovery	FabiusFunction.FinitePrefixAppellRecovery	Exhaustive 11-definition/17-theorem surface. Definitions: unitUniformRawMomentRat, centeredUnitUniformRawMomentRat, dyadicPrefixScaleRat, dyadicPrefixMomentRat, uncenteredDyadicPrefixMomentRat, centeredDyadicPrefixMomentRat, kabayaIriAppellPolynomialRat, uncenteredDyadicPrefixAppellPolynomialRat, centeredDyadicPrefixAppellPolynomialRat, uncenteredDyadicPrefixAppellScalePolynomialRat, and centeredDyadicPrefixAppellScalePolynomialRat. Theorems: Appell.poly_binomialConv, Appell.binomialConv_dilate, Appell.dilate_dilate, dyadicPrefixMomentRat_zero, uncenteredDyadicPrefixMomentRat_zero, centeredDyadicPrefixMomentRat_zero, dyadicPrefixMomentRat_binomialConv_tail, binomialConv_uncenteredDyadicPrefixMomentRat_tail, binomialConv_centeredDyadicPrefixMomentRat_tail, uncenteredDyadicPrefixAppellPolynomialRat_eq_sum, centeredDyadicPrefixAppellPolynomialRat_eq_sum_even, uncenteredDyadicPrefixAppellPolynomialRat_eq_eval_scale, centeredDyadicPrefixAppellPolynomialRat_eq_eval_scale, natDegree_uncenteredDyadicPrefixAppellScalePolynomialRat, natDegree_centeredDyadicPrefixAppellScalePolynomialRat, kabayaIriAppellPolynomialRat_eq_sum_prefix, and rvachevAppellPolynomialRat_eq_sum_prefix. Finite binomial convolution constructs the prefix moments and proves the exact uncentered 2^-N and centered even 4^-N expansions. Their outer polynomial-valued degrees are respectively n and ⌊n/2⌋; this does not assert the same degree after fixing the inner evaluation point, where the centered leading coefficient can vanish (notably for odd degree at x=0). The last two theorems recover the full rational Kabaya--Iri polynomial from any n+1 consecutive prefixes at base 1/2, and the full centered Rvachev--Appell polynomial from any ⌊n/2⌋+1 consecutive prefixes at base 1/4, with no limit. Hence is:p2:thm:finite-prefix-expansion and is:p2:thm:exact-recovery are both Exact; the API is exact rational coefficient algebra and makes no analytic-MGF convergence or universal fixed-x degree claim.
Complete finite-prefix Thue--Morse Appell collapses (is:p2:thm:TM-uncentered, is:p2:cor:Prouhet-canonical, is:p2:thm:TM-centered)	FabiusFunction.FinitePrefixThueMorseCollapse	Exhaustive zero-definition/eight-theorem surface. Appell.sum_thueMorseSign_mul_eval_poly diagonalizes an arbitrary rational Appell polynomial against the signed power moments. Fabius.sum_thueMorseSign_mul_uncenteredDyadicPrefixAppellPolynomialRat uses the signed grid x+k/2^N and gives (-1)^N 2^{-((N+1).choose 2)} n.descFactorial N x^(n-N); its _of_lt and _self corollaries give zero for n<N and the first response (-1)^N N! 2^{-((N+1).choose 2)}. Fabius.sum_thueMorseSign_mul_centeredDyadicPrefixAppellPolynomialRat uses the total grid x+(1-2^-N)-2k/2^N and gives the sign-free response 2^{-(N.choose 2)} n.descFactorial N x^(n-N); its _succ form uses the manuscript's literal positive-depth grid x+(1-2^{-(m+1)})-k/2^m, while its _of_lt and _self corollaries give zero for n<N and N! 2^{-(N.choose 2)}. The two primary formulas and both first-response formulas include N=0; the successor form deliberately parametrizes positive depth. Thus the uncentered main theorem closes is:p2:thm:TM-uncentered, its cancellation/first-response corollaries close is:p2:cor:Prouhet-canonical by composition, and the centered successor theorem closes is:p2:thm:TM-centered, with the total theorem strengthening it at depth zero. The exact eight names are Appell.sum_thueMorseSign_mul_eval_poly, Fabius.sum_thueMorseSign_mul_uncenteredDyadicPrefixAppellPolynomialRat, Fabius.sum_thueMorseSign_mul_uncenteredDyadicPrefixAppellPolynomialRat_of_lt, Fabius.sum_thueMorseSign_mul_uncenteredDyadicPrefixAppellPolynomialRat_self, Fabius.sum_thueMorseSign_mul_centeredDyadicPrefixAppellPolynomialRat, Fabius.sum_thueMorseSign_mul_centeredDyadicPrefixAppellPolynomialRat_succ, Fabius.sum_thueMorseSign_mul_centeredDyadicPrefixAppellPolynomialRat_of_lt, and Fabius.sum_thueMorseSign_mul_centeredDyadicPrefixAppellPolynomialRat_self. This is rational coefficient-model algebra only: it introduces no random variable or HasLaw statement and proves no analytic MGF or Barnes-function identification.
Dyadic-boundary composition for the Rvachev sinc product	FabiusFunction.DyadicBoundaryIdentity	Exhaustive zero-definition/three-theorem surface: prod_complexSinc_prefix_mul_rvachevFourierProduct, rvachevFourierProduct_dyadic_boundary, and norm_rvachevFourierProduct_dyadic_boundary. The first clears the finite sinc prefix against the rescaled product; the second gives the entire denominator-cleared identity for every natural shell and complex displacement. The third is the printed norm-quotient formula on exactly 0 < z < 2^k, where central-lobe positivity supplies the nonzero denominator.
Base-two bridge from digit-weighted Prouhet sums to Thue--Morse power sums	FabiusFunction.ProuhetBaseTwoBridge	Exhaustive zero-definition/six-theorem surface over every commutative ring: thueMorseSign_cast_eq_neg_one_pow_digits_sum, digitPowerSum_neg_one_two, sum_range_two_neg_one_pow, digitPowerSum_neg_one_two_eq_zero_of_lt, digitPowerSum_neg_one_two_self, and sub_one_pow_mul_thueMorsePowerSumRing_self. The bridge identifies the base-two digit sum at root -1 with the Thue--Morse sign, transfers strict-low-degree Prouhet cancellation, and gives both the closed sharp moment (-1)^m 2^(m.choose 2) m! and its division-free general-machine form. It changes no existing source-result status.
Explicit recurrent values and oscillation of the Newman ratio	FabiusFunction.ThueMorseNewmanSelfSimilarity	The strengthened module is now 1+19. Its seven added theorems are eight_rpow_logb_four_three, sqrt_three_div_three_lt_two_div_three, newman_ratio_eight, newman_ratio_eight_lt_two_div_three, frequently_newmanRatio_eq_two_div_three, frequently_newmanRatio_eq_sqrt_three_div_three, and newmanRatio_oscillates. They identify the second recurrent value as sqrt 3 / 3, prove it is below 2/3, show both values occur frequently at atTop, and package the explicit oscillation. Existing source-result statuses are unchanged.
Definitions, the bounded characterization, folded up, and the global first-jet reflection law	FabiusFunction.Basic, FabiusFunction.Differential	BoundedFabius, IsFabius, rvachevUp, rvachevUp_even, rvachevUp_eq_zero_of_not_mem_Ioo, support_rvachev_subset_Ioo, rvachev_hasDerivAt, fabius_hasDerivAt, deriv_fabiusReal, deriv_fabiusReal_one_sub
Sharp bounded derivatives and the exact zero-interleaved Thue--Morse pattern on every matched dyadic grid	FabiusFunction.BoundedDerivatives	iteratedDeriv_fabiusReal_of_lt_one, iteratedDeriv_fabiusReal_dyadicGrid_eq_ite, iteratedDeriv_fabiusReal_dyadicGrid_eq_zero_iff, abs_iteratedDeriv_fabiusReal_dyadicGrid_of_odd, abs_iteratedDeriv_fabiusReal_le, isGreatest_abs_iteratedDeriv_fabiusReal
Exact dyadic derivative filtration for the up-function	FabiusFunction.DyadicDerivativeFiltration	Exhaustive public surface (zero definitions and six theorems): rvachevUp_eq_zero_of_one_le_abs, iteratedDeriv_rvachevUp_dyadic_eq_zero, iteratedDeriv_rvachevUp_dyadic_critical, dyadic_depth_eq_max_nonzero_iteratedDeriv, iteratedDeriv_rvachevUp_eq_extendedFabius, and iteratedDeriv_rvachevUp_dyadic_below. The first four give support vanishing, vanishing above a reduced dyadic point's depth, the signed Thue--Morse critical derivative, and exact depth detection. The final two are the new below-depth closure: for every order and every x<1, the up derivative is 2^((m+1).choose 2) times the signed global Fabius value at 2^m(x+1); for m<n and a<2^n, this specializes at a/2^n to the volume's denominator-2^(n-m) formula.
Exact derivative cells, signed natural moments, and normalized signed/absolute-value distributions	FabiusFunction.RvachevDerivativeDistribution	Exhaustive public surface (one definition and 18 theorems): rvachevDerivativeCell; rvachevDerivativeCell_eq_div_add, rvachevDerivativeCell_one_eq_succ_neg_one, rvachevDerivativeCell_zero_neg_one, rvachevDerivativeCell_two_pow_neg_one, rvachevDerivativeCell_mem_Icc, iteratedDeriv_rvachev_cell, iteratedDeriv_rvachev_cell_zero, abs_iteratedDeriv_rvachev_cell, intervalIntegral_comp_iteratedDeriv_rvachev, intervalIntegral_iteratedDeriv_rvachev_pow, intervalIntegral_iteratedDeriv_rvachev_pow_of_even, intervalIntegral_iteratedDeriv_rvachev_pow_eq_zero_of_odd, intervalIntegral_comp_normalized_iteratedDeriv_rvachev, intervalIntegral_comp_abs_iteratedDeriv_rvachev, intervalIntegral_comp_normalized_abs_iteratedDeriv_rvachev, map_normalized_abs_iteratedDeriv_rvachev_restrict_Icc, map_normalized_iteratedDeriv_rvachev_restrict_Icc, and intervalIntegral_abs_iteratedDeriv_rvachev_rpow. The signed-moment theorems take F : BoundedFabius, hF : IsFabius F, and n m : ℕ. With T_n=(n+1).choose 2, the hypothesis-free general formula multiplies the base moment by ((2 : ℝ)^n)⁻¹ * (1 + (-1 : ℝ)^m)^n * ((2 : ℝ)^T_n)^m; at n=0 and odd m, its Boolean-cube factor is deliberately 0^0=1, so it recovers the generally nonzero original odd moment. The even corollary assumes exactly Even m and holds for every n, including n=0 and m=0; the odd vanishing corollary assumes exactly 0<n and Odd m. At every positive derivative order, intervalIntegral_comp_normalized_iteratedDeriv_rvachev gives the sharply normalized symmetric half-mixture of up and -up for a continuous H : ℝ → E valued in any real Banach space: the normalization is ((2 : ℝ)^T_n)⁻¹ and the mixture scalar is (2 : ℝ)⁻¹. Under the same exact hypothesis 0<n, map_normalized_iteratedDeriv_rvachev_restrict_Icc gives the corresponding equality of Borel pushforwards of volume.restrict (Set.Icc (-1) 1), with measure scalar (2 : NNReal)⁻¹. This positivity hypothesis is essential: at n=0 the normalized signed derivative has the unsymmetrized original rvachevUp law. Separately, the absolute-moment theorem uses Real.rpow for every real p>=0, and the normalized absolute derivative has the same restricted-Lebesgue pushforward as up at every order, including zero. No separate product-space Rademacher realization, eLpNorm or general rearrangement-invariant norm ladder, inverse-Fabius level-set formula, packaged beta theorem, or spectral layer is asserted.
Existence, uniqueness, and the canonical functions	FabiusFunction.PaperStatements	existsUnique_fabius, fabius, fabius_spec, globalFabius
Original compact-support characterization and bounded/original bridge	FabiusFunction.OriginalUniqueness	IsOriginalFabius, IsOriginalFabius.mk_of_derivative_law, IsFabius.isOriginalFabius_rvachevUp, rvachevUp_eq_iff_eqOn_Iic_one, isFabius_iff_isOriginalFabius_rvachevUp_and_rightTail, isOriginalFabius_iff_existsUnique_isFabius
Generic affine-difference iterates and derivative orbits	FabiusFunction.AffineDifferenceOrbit	affineDifference_iterate_apply, iteratedDeriv_eq_affineDifference_iterate_on, affineDifference_iterate_two_one_apply; the module assumes a one-step derivative identity and does not prove the up-law resolvent equation
Finite polynomial functionals, moment extraction, and affine transport	FabiusFunction.FinitePolynomialFunctional	Exhaustive zero-definition/sixteen-theorem surface: sum_weight_mul_eval₂_eq_sum_coeff_mul_moment, sum_weight_mul_eval₂_eq_eval₂_of_moments, sum_weight_mul_eval₂_eq_coeff_mul_moment, sum_weight_mul_eval₂_eq_topCoeff_mul_moment, sum_weight_mul_eval₂_eq_constantCoeff_mul_sum, sum_weight_mul_eval₂_eq_constantCoeff, sum_weight_mul_eval₂_eq_map_coeff_mul_of_moments, sum_weight_mul_eval₂_eq_map_coeff_mul_top_moment, sum_weight_mul_eval₂_eq_zero_of_degree_lt, sum_weight_mul_eval₂_congr_of_map_coeff_eq, sum_weight_mul_eval_eq_eval_of_moments, sum_weight_mul_eval_eq_coeff_mul_of_moments, sum_weight_mul_eval_eq_coeff_mul_top_moment, sum_weight_mul_eval_eq_zero_of_degree_lt, sum_weight_mul_eval_congr_of_coeff_eq, and sum_weight_mul_eval_affine_of_topCoeff_extractor. The scalar-extension forms work from an arbitrary semiring into a commutative semiring; the same-ring and affine forms require only a commutative semiring. Moment expansion yields polynomial reproduction, selected/top/constant-coefficient extraction, strict-degree cancellation, and congruence. Nodes may repeat and the selected moment may vanish. The affine theorem transports any supplied top-coefficient extractor from node i to a + b * node i, multiplying by b^n; it includes b=0 and n=0 and requires neither distinct transformed nodes nor subtraction. Composed with the half-base Gaussian Prouhet extractor, it makes cor:geometric-prouhet-affine Exact under the established ℚ[X] convention.
Central-binomial valuation and the Thue--Morse sign	FabiusFunction.CentralBinomialValuation	Exhaustive public surface: padicValNat_two_centralBinom, thueMorseSign_eq_neg_one_pow_centralBinom, padicValNat_two_centralBinom_eq_zero_iff; for every natural n, the valuation is binaryWeight n, the sign is its (-1)-power, and valuation zero is equivalent to binary weight zero (hence occurs only at n = 0, not at positive powers of two)
Prime-power Pascal-row valuations and the dyadic-comb weights	FabiusFunction.PrimePowerBinomialValuation	Exhaustive zero-definition/six-theorem surface: primePowerChoose_padicValNat_add, primePowerChoose_padicValNat, primePowerSubOneChoose_padicValNat, primePowerSubTwoChoose_padicValNat, twoPowChoose_padicValNat, and twoPowSubTwoChoose_padicValNat. For every prime p, j ≤ p^m, and j ≠ 0, the first two give the truncation-free additive valuation identity and its natural-subtraction form, including j = p^m and m = 0. Every column j < p^m of row p^m-1 is a p-adic unit, including the m = 0 boundary. Under exactly 0 < j < p^m, the endpoint-flat companion is v_p(C(p^m-2,j-1)) = v_p(j); the two dyadic wrappers give the ordinary row formula and this companion for 0 < j < 2^m. The strict upper bound is essential because the companion binomial coefficient is zero at j=p^m; only valuation-histogram counts remain open.
Rademacher sine signs and the Thue--Morse product	FabiusFunction.RademacherSine	Exhaustive nine-theorem surface: sin_pi_mul_eq_neg_one_zpow_floor, sin_pi_mul_fract_pos, sign_sin_pi_mul, floor_rademacherPoint, fract_rademacherPoint_ne_zero, sign_sin_rademacherPoint, sign_sin_rademacherPoint_eq_one_of_lt, thueMorseSign_eq_prod_sign_sin, and thueMorseSign_eq_tprod_sign_sin. The factored sine identity is unconditional; the two general sign statements assume exactly a nonzero fractional part; the half-shifted point is never integral; the finite product assumes n < 2^m; and the tprod identity is total because all sufficiently late factors are proved to equal one. This is finite-support sign bookkeeping, not convergence of a genuinely nontrivial infinite product or an unshifted sine formula.
Binary digits as differences of dyadic floors	FabiusFunction.BinaryDigitFloor	Exhaustive public surface: div_two_pow_succ_eq_div_div, sub_two_mul_div_two, div_two_pow_sub_two_mul_div_two_pow_succ, testBit_toNat_eq_div_sub_two_mul_div; the identities are total in their natural-number inputs and give the atlas's exact floor-difference digit formula, without an analytic or real-floor generalization
General-base cumulative scale multiplicities and digit recovery	FabiusFunction.BaseDigitMultiplicity	Exhaustive public inventory: zero definitions and five theorems, sum_range_weightedScaleMultiplicity_of_log_lt, sum_range_weightedScaleMultiplicity_log, sum_range_div_pow_log_eq_self_add_tail, sub_one_mul_sum_padicValNat_succ_add_digitSum, and sum_range_padicValNat_succ_eq_sub_digitSum_div. The first two hold over every additive commutative monoid; the explicit-height form assumes 1 < b and Nat.log b N < H, while the sharp-height form assumes only 1 < b. The remaining natural-number identities also assume exactly 1 < b, include N = 0, and require no primality. This is the finite count (b-1) * sum_(n=1)^N (1+nu_b(n)) + s_b(N) = bN and its quotient form, not an analytic zero-multiplicity theorem.
Total complex finite Thue--Morse sinc and negative-Laplace bridges	FabiusFunction.ThueMorseComplexProductBridge	shiftedComplexSincPrefix, complexLaplacePrefix, sum_thueMorseSign_cexp_eq_sin_prod, thueMorseBlock_cexp_eq_sincPrefix, thueMorseBlock_cexp_eq_sincPrefix_of_pos, thueMorseBlock_exp_neg_eq_laplacePrefix, shiftedComplexSincPrefix_eq_thueMorseBlock_cexp_of_pos, complexLaplacePrefix_eq_thueMorseBlock_exp_neg, complexExpm1Div_neg_eq_exp_mul_complexSinc, complexLaplacePrefix_eq_exp_mul_shiftedComplexSincPrefix, shiftedComplexSincPrefix_apply_zero, complexLaplacePrefix_apply_zero; the primary equalities and the finite Fourier--Laplace rotation hold at every level and at the removable origin, while the quotient forms assume a nonzero free variable
Centered mixed differences and symmetric Thue--Morse blocks	FabiusFunction.ThueMorseSymmetricDifference	Exhaustive public surface (two definitions and 11 theorems): symmetricMixedDifference, symmetricMixedDifference_empty, symmetricMixedDifference_singleton, symmetricMixedDifference_insert, symmetricMixedDifference_eq_sum_powerset_smul, symmetricMixedDifference_polynomial_eq_coeff_card, symmetricMixedDifference_polynomial_of_degree_lt, symmetricMixedDifference_pow_card, symmetricDyadicMixedDifference, symmetricDyadicMixedDifference_zero, symmetricDyadicMixedDifference_eq_sum_thueMorseSign_smul, symmetricDyadicMixedDifference_inv_two_pow_eq_sum_thueMorseSign_smul, and symmetricDyadicMixedDifference_inv_two_pow_succ_eq_sum_thueMorseSign_smul. The general centered-cube layer needs only additive commutative groups and an additive action, permits repeated half-step values, and gives the exact powerset expansion. Over a commutative ring, a polynomial of degree at most s.card is reduced to its degree-s.card coefficient times the sign-free factor s.card! * ∏_(i∈s) (2*a_i); degree below s.card is annihilated, including the zero-polynomial/empty-support boundary, and the first surviving power has the displayed top factor. The dyadic block identity is group-valued; the two increasing-grid wrappers require a characteristic-zero field in the argument and give x-(1-2^-m)+2*n*2^-m for every m, including m=0. At positive order m+1, the full point is x-(1-2^-(m+1))+n/2^m; its varying term is n/2^m and its mesh spacing is 1/2^m. This closes the Boolean-cube, polynomial, and affine-grid clauses of the continuous-chaos report's Thue--Morse corner theorem; the companion row supplies its analytic repeated-integral clause.
Local centered-box integral formula for Thue--Morse corners (thm:TM-corner)	FabiusFunction.ThueMorseCornerIntegral	Exhaustive public surface: one definition, centeredBoxIntegral, and four theorems, centeredBoxIntegral_zero, centeredBoxIntegral_succ, symmetricMixedDifference_range_eq_centeredBoxIntegral, and symmetricMixedDifference_univ_eq_centeredBoxIntegral. For nonnegative half-steps, an open order-connected set I, ContDiffOn ℝ N g I, and containment in I of the full symmetric segment from x-∑i<N,a i to x+∑i<N,a i, the range and Fin N forms identify the corner sum with the centered nested integral of iteratedDeriv N g. This is genuinely local, not a global-ContDiff shortcut; it permits zero half-steps and N=0, strengthening the printed positive-step case. The proof inducts on N, applies the local interval FTC at every Boolean-cube corner, interchanges a finite sum and the outer integral, and recurses on deriv g. Together with the preceding algebraic API, this makes thm:TM-corner Exact. Signed half-steps are not claimed, and the following Walsh conditional-expectation corollary still lacks its sign--magnitude probability construction and 2^-N normalization.
Finite uniform-digit characteristic functions as Thue--Morse blocks	FabiusFunction.UniformDigitThueMorseBridge	charFun_uniformDigitPrefix_eq_shiftedComplexSincPrefix, thueMorseBlock_cexp_eq_charFun_uniformDigitPrefix, charFun_uniformDigitPrefix_eq_thueMorseBlock_cexp; the first two identities hold at every natural level and real frequency, including the empty prefix and frequency zero, while solving for the characteristic function assumes the real frequency is nonzero; these are finite-prefix identities and assert no infinite-product or random-tail limit
Exact first jets and simple zeros of reciprocal Gamma	FabiusFunction.ReciprocalGammaJets	deriv_Gamma_inv_neg_nat, hasDerivAt_Gamma_inv_neg_nat, hasDerivAt_Gamma_inv_zero, analyticOrderAt_Gamma_inv_neg_nat, tendsto_Gamma_inv_div_add_nat; all five statements hold for every natural zero index, concern the entire reciprocal function, and assign no derivative to raw Gamma at a pole
Thue--Morse continuation jets and Gamma tower	FabiusFunction.ThueMorseGammaTower	hasDerivAt_dirichletMellinContinuation_neg_nat, deriv_dirichletMellinContinuation_neg_nat, thueMorseGammaLog, thueMorseGammaTower, thueMorseGammaLog_eq_mellin, thueMorseGammaLog_eq_integral, thueMorseGammaLog_dyadic, thueMorseGammaTower_dyadic, ofReal_exp_mpLimit_eq_gammaTower_div; the two definitions are total in a, all analytic laws assume 0 < a (and the ratio bridge also 0 < b), and GammaLog is a chosen coordinate rather than a proved Complex.log identity; the parameter-differential closure is recorded in the next row
Positive-parameter Thue--Morse GammaLog differential ladder (p2:thm:gamma-tower)	FabiusFunction.ThueMorseGammaTowerDifferential	Exhaustive zero-definition/three-theorem surface: hasDerivAt_mellin_mellinKernel_parameter, hasDerivAt_thueMorseGammaLog_succ, and iteratedDeriv_thueMorseGammaLog. For every complex Mellin exponent and positive real damping parameter, differentiation under the integral shifts s to s+1; the specializations prove L_(r+1)'(a)=(r+1)L_r(a) and the falling-factorial iteration through exactly k ≤ r. Together with the preceding Gamma-tower API this makes p2:thm:gamma-tower Exact on its stated 0 < a domain. Every result concerns the chosen thueMorseGammaLog coordinate: no equality with a branch or the principal value of Complex.log (thueMorseGammaTower r a), and no nonpositive-parameter differential law, is asserted.
Generic normalized Volterra calculus over real normed spaces (Banach only for the FTC/Taylor layer)	FabiusFunction.NormalizedVolterra	volterraPrimitive, iteratedPrimitive, normalizedVolterra, normalizedVolterra_affine, normalizedVolterra_comp_affine, normalizedVolterra_basepoint_shift, normalizedVolterra_succ_eq_taylor_of_eq_zero, iteratedPrimitive_add, iteratedPrimitive_succ_hasStrictDerivAt, iteratedPrimitive_eq_normalizedVolterra, normalizedVolterra_succ_hasStrictDerivAt, iteratedDeriv_normalizedVolterra_add, contDiff_normalizedVolterra, normalizedVolterra_add, normalizedVolterra_succ_iteratedDeriv_eq_sub_taylor, intervalIntegrable_normalizedVolterraKernel_add, normalizedVolterra_succ_polynomial_of_taylor_support_kernel_intervalIntegrable, normalizedVolterra_succ_polynomial_of_kernel_intervalIntegrable, normalizedVolterra_polynomial, normalizedVolterra_monomial
Positive-real fractional Volterra calculus over real normed spaces	FabiusFunction.FractionalVolterra, FabiusFunction.FractionalVolterraSemigroup	fractionalVolterra, fractionalVolterra_self, fractionalVolterra_congr, fractionalVolterra_smul, intervalIntegrable_fractionalVolterra_kernel, intervalIntegral_eq_integral_min_of_eq_zero, fractionalVolterra_eq_intervalIntegral_min_of_eq_zero, fractionalVolterra_add_input, fractionalVolterra_one, fractionalVolterra_nat_succ, intervalIntegral_fractionalVolterra_betaKernel, intervalIntegrable_fractionalVolterra_betaKernel, intervalIntegral_fractionalVolterra_normalizedBetaKernel, fractionalVolterra_normalized_rpow_smul, fractionalVolterra_rpow_smul, fractionalVolterra_const, fractionalVolterra_add; the definition and algebraic endpoint/integer rules use oriented interval integrals, while kernel integrability and input additivity assume 0 < α, a ≤ x, and continuity on [a,x]; if a ≤ b, a ≤ x, and an interval-integrable kernel vanishes on Ioo b x, its integral cuts off at min x b, and for 0 < α the same support truncation holds for a continuous fractional input vanishing on that open tail; for 0 < α, 0 < β, the raw shifted beta kernel is interval-integrable when s ≤ x, and its raw and Gamma-normalized values are evaluated when s < x; on a complete target, normalized shifted powers (α, β > 0, a < x), general shifted powers (α > 0, ρ > -1, a < x), and constants (α > 0, a ≤ x) have their exact Gamma-quotient values, and fractionalVolterra_add proves additive composition of two positive orders when a ≤ x and the input is continuous on [a,x]; no order-zero law, reversed-endpoint fractional interpretation, semigroup theorem for merely interval-integrable inputs or noncomplete targets, fractional derivative or Caputo theorem, or complex order is claimed
Increasing-affine covariance, ordinary-derivative order raising, causal Rvachev fractional primitives, and one-step Fabius--Rvachev shifts	FabiusFunction.FractionalVolterraCalculus, FabiusFunction.FabiusFractionalVolterra	fractionalVolterra_affine, fractionalVolterra_comp_affine, fractionalVolterra_add_one_deriv, fractionalVolterra_add_one_deriv_of_eq_zero, rvachevFractionalPrimitive, rvachevFractionalPrimitive_eq_intervalIntegral_min, rvachevFractionalPrimitive_nat_succ, rvachevFractionalPrimitive_add, fractionalVolterra_add_one_extendedFabius_of_nonneg, fractionalVolterra_add_one_fabiusReal, fractionalVolterra_add_one_rvachevUp; affine covariance holds for every real order, positive scale, and ordered endpoints without regularity, integrability, or completeness assumptions; on a complete target, order raising assumes 0 < α, a ≤ x, continuity of the primitive on [a,x], an interval-integrable displayed derivative, and its right derivative on the open interval, records the exact Gamma-normalized left-boundary term, and includes the degenerate interval; rvachevFractionalPrimitive is total, its classical support-truncated formula assumes 0 < β and -1 ≤ x, its positive-natural-order bridge is total in the endpoint, and its additive semigroup assumes α, β > 0 and -1 ≤ x; the three shift specializations give I₀^(α+1) 𝓕(x) = 2^α I₀^α 𝓕(x/2) for α > 0, x ≥ 0, its bounded form for x ∈ [0,1], and I₋₁^(α+1) up(x) = 2^α I₀^α F((x+1)/2) for x ≥ -1; no nonpositive-scale or reversed-endpoint covariance, negative- or complex-order fractional-calculus extension, fractional derivative or Caputo theorem, shifted dyadic-lattice or endpoint-moment wrapper, transform/tail-series theorem, or inverse-function specialization is claimed
Exact signed-global and bounded Fabius primitive ladders with finite polynomial weights	FabiusFunction.FabiusAntiderivatives	normalizedVolterra_extendedFabius, normalizedVolterra_fabiusReal_of_le_one, normalizedVolterra_polynomial_mul_extendedFabius, normalizedVolterra_pow_mul_extendedFabius, normalizedVolterra_polynomial_mul_fabiusReal_of_le_one, normalizedVolterra_pow_mul_fabiusReal_of_le_one, integral_cube_mul_fabiusReal_eq; signed formulas are global, while bounded formulas assume x ≤ 1
Absolutely summable uniform-coordinate series and their canonical pushforward laws	FabiusFunction.WeightedUniformSeries	weightedUniformSeries, weightedUniformSeries_smul_weights, weightedUniformSeries_split, weightedUniformDistribution, isProbabilityMeasure_weightedUniformDistribution, weightedUniformDistribution_split, weightedUniformDistribution_reflection, ae_weightedUniformDistribution_mem_Icc, weightedUniformDistribution_restrict_Icc, weightedUniformDistribution_Icc
Compatibility names, exact barycenters, scalar/unit-mass refinements, and absolute continuity for weighted real laws	FabiusFunction.WeightedUniformDistribution	Exhaustive zero-definition/eleven-theorem surface: uniformProduct_map_head_tail_function, uniformScaledAdd_absolutelyContinuous, weightedUniformDistribution_isProbabilityMeasure, integral_id_weightedUniformDistribution, weightedUniformDistribution_smul_weights, weightedUniformDistribution_absolutelyContinuous_of_head_ne_zero, weightedUniformDistribution_absolutelyContinuous, weightedUniformDistribution_nullSingletonClass, uniformProduct_map_head_tail_weightedUniformSeries, weightedUniformDistribution_unitInterval, and weightedUniformDistribution_compl_unitInterval. In every complete Borel real normed space, norm-summable weights have barycenter (1 / 2) • ∑' n, w n; this uses no sign, order, independence expansion, or termwise integration hypothesis.
Banach-valued L1 survival-kernel calculus for finite measures, clipped endpoints, and scalar probability integration by parts	FabiusFunction.ProbabilityLaplaceMoments	integral_Icc_intervalIntegral_eq_intervalIntegral_smul_survival, integral_Icc_intervalIntegral_min_eq_intervalIntegral_smul_survival, integral_Icc_intervalIntegral_eq_intervalIntegral_smul_rvachevUp, integral_Icc_intervalIntegral_min_eq_intervalIntegral_smul_rvachevUp, intervalIntegral_mul_rvachevUp_eq_integral_Icc_intervalIntegral, intervalIntegral_mul_rvachevUp_eq_integral_Icc_intervalIntegral_min, integral_Icc_eq_left_add_intervalIntegral_deriv_mul_survival, integral_unit_eq_zero_add_integral_deriv_mul_rvachevUp
Closed up tail and positive raw moments (prop:up-tail, cor:up-moments)	FabiusFunction.ProbabilityLaplaceMoments	The module's exhaustive new +2 surface is weightedSumDistribution_real_Ici_eq_rvachevUp_of_nonneg and integral_pow_weightedSumDistribution_eq_mul_intervalIntegral_rvachevUp. The first uses the already-proved atomlessness of weightedSumDistribution to turn the strict-Ioi survival identity into P(X >= t) = up(t) for every t >= 0; with rvachevUp_eq_fabiusReal_one_sub_abs and rvachevUp_eq_one_sub_fabiusReal_of_nonneg, this makes prop:up-tail Exact, on domains at least as strong as printed. The second gives the manuscript's full-law formula integral x^n dμ = n * integral_0^1 t^(n-1) up(t) dt for every natural n >= 1, making cor:up-moments Exact. These are canonical-law measure identities; they do not introduce a new arbitrary-random-variable wrapper, and the positive-degree theorem does not itself assert the n=0 case.
Exact midpoint--endpoint value and first-jet transfer, complete higher midpoint jet, centered integral, and weighted primitive kernels	FabiusFunction.MidpointEndpointTransfer	fabiusReal_midpoint_add_eq, fabiusReal_midpoint_sub_eq, deriv_fabiusReal_midpoint_add_eq, deriv_fabiusReal_midpoint_sub_eq, iteratedDeriv_fabiusReal_half_eq_zero_of_two_le, intervalIntegral_fabiusReal_centered, intervalIntegral_mul_fabiusReal_midpoint_add_defect_eq_neg, intervalIntegral_mul_fabiusReal_midpoint_sub_defect_eq, intervalIntegral_fabiusReal_midpoint_add_defect_eq_neg, intervalIntegral_repeatedPrimitiveKernel_fabiusReal_midpoint_add_defect_eq_neg
Exact inverse-midpoint offset and defect fixed points, endpoint normalizations, positive-cell enclosures, and global oddness	FabiusFunction.InverseMidpointDefect	fabiusInvMidpointOffset, fabiusInvMidpointDefect, fabiusInvMidpointOffset_zero, fabiusInvMidpointDefect_zero, fabiusInvMidpointOffset_half, fabiusInvMidpointDefect_half, fabiusInvMidpointOffset_mem_Icc, fabiusInvMidpointOffset_equation, fabiusInvMidpointOffset_fixedPoint, fabiusInvMidpointDefect_eq_half_fabiusReal, fabiusInvMidpointDefect_fixedPoint, fabiusInvMidpointDefect_mem_Icc, fabiusInvMidpointOffset_neg, fabiusInvMidpointDefect_neg
Exact finite-spline cell around 1/4, with two-sided reflection, curvature, and conditional inverse identities	FabiusFunction.QuarterSplineLocalPolynomial, FabiusFunction.QuarterSplineTwoSided	reportFiniteFabiusApproximant_quarter_twoSided, reportFiniteFabiusApproximant_quarter_reflection, reportFiniteFabiusApproximant_quarter_centralSecondDifference, strictMonoOn_reportFiniteFabiusApproximant_quarter_twoSided, reportFiniteFabiusApproximant_quarterPrefix_value, reportFiniteFabiusApproximant_quarterPrefix_quantile
Full-order centered Rvachev moment, logarithmic-coefficient, and cumulant parity, with Bernoulli--Mersenne formulas at every positive even order	FabiusFunction.CenteredMomentParity, FabiusFunction.SinhDivBernoulliLog	centeredRvachevFullMoment_even, centeredRvachevFullMoment_odd, centeredRvachevFullLogCoefficient_even, centeredRvachevFullLogCoefficient_odd, centeredRvachevFullCumulant_even, centeredRvachevFullCumulant_odd, centeredRvachevEvenCumulant_eq_bernoulliMersenne
Totalized real hyperbolic sinc, activation odds, and activation probability	FabiusFunction.HyperbolicActivation	Exhaustive public surface (four definitions and 58 theorems): realSinhc, realSinhc_zero, realSinhc_of_ne_zero, realSinhc_eq_dslope, continuous_realSinhc, realSinhc_neg, realSinhc_even, realSinhc_pos, realSinhc_ne_zero, realSinhc_two_mul, sinh_lt_mul_cosh, realSinhc_lt_cosh_of_pos, realSinhc_le_cosh, realSinhc_lt_cosh_iff, tanhDiv, tanhDiv_zero, tanhDiv_of_ne_zero, tanhDiv_neg, tanhDiv_even, continuous_tanhDiv, tanhDiv_pos, tanhDiv_ne_zero, tanhDiv_le_one, tanhDiv_lt_one_iff, tanhDiv_le_inv_of_pos, hasDerivAt_tanh, tanhDiv_eq_dslope, tanh_nonneg_of_nonneg, tanh_pos_of_pos, tanh_lt_self_of_pos, tanh_le_self_of_nonneg, tanh_cubic_lower, activationOdds, activationProbability, activationOdds_zero, activationProbability_zero, activationProbability_of_ne_zero, activationOdds_neg, activationOdds_even, activationProbability_neg, activationProbability_even, continuous_activationOdds, continuous_activationProbability, one_add_activationOdds, one_add_activationOdds_pos, activationOdds_of_ne_zero, activationOdds_nonneg, activationOdds_pos_iff, tanhDiv_mul_one_add_activationOdds, activationProbability_mul_one_add_activationOdds, activationProbability_eq_odds_div, one_sub_activationProbability, one_sub_activationProbability_pos, one_sub_activationProbability_le_one, one_sub_activationProbability_le_inv_of_pos, one_sub_activationProbability_le_min_one_inv_of_pos, activationProbability_nonneg, activationProbability_pos_iff, activationProbability_lt_one, activationProbability_le_sq_div_three_of_nonneg, activationProbability_le_sq_div_three, and activationProbability_le_min_one_sq_div_three. The module totalizes sinh(x)/x and tanh(x)/x at the origin, proves continuity, evenness, strict positivity, the double-angle law, the exact comparison with cosh, elementary real-tanh estimates, the identification of tanhDiv with the divided slope of tanh, and denominator-free and quotient forms of the odds/probability bridge. Globally, 0 ≤ activationProbability x < 1, positivity is equivalent to x ≠ 0, and activationProbability x ≤ min 1 (x^2/3); for x>0, its complement is positive and bounded by min 1 x⁻¹. This is a real totalized-kernel API, not an identification with the complex sinc functions, an analyticity theorem, or an all-order Taylor-series theorem.
Square-summable activation-series bounds	FabiusFunction.ActivationSeries	Exhaustive public surface (three theorems): activationProbability_mul_le_quadratic, summable_activationProbability_mul_of_summable_sq, and tsum_activationProbability_mul_le. The first packages the sharp one-coordinate rescaling bound. For an arbitrary index type and every real family w with summable squares, the series ∑' i, activationProbability (w i * t) is genuinely summable for every real t and is at most (t^2/3) * ∑' i, w i^2. This is a deterministic analytic statement and requires no countability, sign, or probability-law hypothesis beyond what Summable itself entails.
Sharp local activation asymptotics	FabiusFunction.ActivationAsymptotics	Exhaustive public surface (two theorems): tendsto_activationProbability_div_sq and tendsto_activationProbability_mul_div_sq. The first proves the punctured-neighborhood limit activationProbability x / x^2 → 1/3 as x → 0, so the coefficient in the global bound activationProbability x ≤ x^2/3 is optimal. The second transports it through every real dilation: activationProbability (a*x) / x^2 → a^2/3, including a=0. The base proof uses one l'Hopital step and the continuous totalization tanhDiv 0 = 1; neither theorem asserts a higher Taylor coefficient or remainder.
Finite activation Taylor jet	FabiusFunction.ActivationTaylor	Exhaustive public surface (three theorems): tanh_sub_taylor_nine_isBigO, activationProbability_sub_taylor_eight_isBigO, and activationProbability_sub_taylor_eight_isBigO_pow. The first proves tanh x = x - x^3/3 + 2*x^5/15 - 17*x^7/315 + 62*x^9/2835 + O(‖x‖^11) at the origin. Dividing its removable remainder by x gives the exact report expansion activationProbability x = x^2/3 - 2*x^4/15 + 17*x^6/315 - 62*x^8/2835 + O(‖x‖^10), together with a literal O(x^10) wrapper. The proof obtains the derivative table structurally from tanh * cosh = sinh and iteratedDeriv_mul, then invokes the analytic power-series remainder theorem; it does not claim the all-order Bernoulli series or its radius of convergence.
Sharp square-summable activation-series asymptotics	FabiusFunction.ActivationSeriesAsymptotics	Exhaustive public surface (one theorem): tendsto_tsum_activationProbability_mul_div_sq. For every square-summable real family on an arbitrary index type, (∑' i, activationProbability (w i*t))/t^2 tends through nonzero t to (∑' i, w i^2)/3. Tannery's theorem justifies the interchange of limit and tsum, with the global quadratic estimate supplying exactly the limiting summable majorant; consequently the preceding global series budget is coefficient-sharp.
Geometric and dyadic effective activation dimensions	FabiusFunction.GeometricActivationDimension	Exhaustive public surface (two definitions and 30 theorems): activationProbability_pow_mul_le, summable_activationProbability_pow_mul, tsum_activationProbability_pow_mul_le, hasSum_normalizedGeometricWeight_sq, geometricActivationDimension, summable_geometricActivationDimension_terms, geometricActivationDimension_zero, geometricActivationDimension_nonneg, geometricActivationDimension_even, geometricActivationDimension_refinement, geometricActivationDimension_sub_refinement, geometricActivationDimension_zero_ratio, activationProbability_scale_le_geometricActivationDimension, geometricActivationDimension_pos_iff, geometricActivationDimension_le_quadratic, geometricActivationDimension_le_normalized_quadratic, geometricActivationDimension_eq_sum_range_add, geometricActivationDimension_tail_le, sum_range_activationProbability_le_geometricActivationDimension, geometricActivationDimension_le_sum_range_add_tail, dyadicEffectiveDimension, dyadicEffectiveDimension_eq_tsum, dyadicEffectiveDimension_zero, dyadicEffectiveDimension_nonneg, dyadicEffectiveDimension_even, dyadicEffectiveDimension_half_refinement, dyadicEffectiveDimension_two_mul, dyadicEffectiveDimension_refinement, dyadicEffectiveDimension_pos_iff, dyadicEffectiveDimension_le_sq_div_nine, dyadicEffectiveDimension_eq_sum_range_add, and dyadicEffectiveDimension_tail_le. The HasSum theorem isolates the exact squared mass (1-q)/(1+q) of the normalized geometric weights and supplies both convergence and the evaluated tsum. For every real q with `
Sharp geometric and dyadic activation asymptotics	FabiusFunction.GeometricActivationAsymptotics	Exhaustive public surface (two theorems): tendsto_geometricActivationDimension_div_sq and tendsto_dyadicEffectiveDimension_div_sq. For every real q with `
Exact topological support of weighted-uniform laws	FabiusFunction.WeightedUniformSupport	isOpenPosMeasure_infinitePi, uniformProduct_isOpenPosMeasure, support_map_eq_closure_range_of_continuous, weightedUniformSeries_constCoordinates, weightedUniformDistribution_support_eq_range, range_weightedUniformSeries_eq_Icc_min_max, weightedUniformDistribution_support_eq_Icc_min_max, range_weightedUniformSeries_eq_Icc, weightedUniformDistribution_support_eq_Icc, weightedUniformDistribution_support_eq_unitInterval
Continuous CDF calculus for atomless real probability laws	FabiusFunction.ContinuousCDF	continuous_cdf_of_nullSingleton, cdf_reflection_sub, measure_eq_withDensity_of_cdf_hasDerivAt; the last theorem turns an everywhere pointwise CDF derivative into the exact Lebesgue withDensity representation without assuming derivative continuity or prior absolute continuity, because CDF monotonicity supplies nonnegativity and local integrability
Contractive affine independent-copy probability laws	FabiusFunction.AffineIndependentCopy	At compiled checkpoint d312c0603: affineIndependentCopyLaw, affineIndependentCopyLaw_isProbabilityMeasure, affineIndependentCopyLaw_eq_map_prod, charFun_affineIndependentCopyLaw, charFun_eq_mul_charFun_of_affineIndependentCopy_fixedPoint, charFun_iterate_of_affineIndependentCopy_fixedPoint, eq_of_charFun_affine_recurrence, affineIndependentCopyLaw_fixedPoint_unique, affineIndependentCopy_map_fixedPoint_unique; the digit space is an arbitrary measurable space and the target is a second-countable Borel real inner-product space; a measurable digit map and probability digit/candidate laws suffice for the operator-level fixed-point API, while completeness is assumed only by the characteristic-recurrence and two fixed-point uniqueness theorems; uniqueness requires `
Geometrically weighted uniform laws, their exact mean, and their characterization	FabiusFunction.GeometricUniformLaw, FabiusFunction.GeometricUniformUniqueness	The exhaustive GeometricUniformLaw surface now has 24 declarations. Its integral_id_geometricUniformDistribution_eq_one_half specialization proves ∫ x, x ∂geometricUniformDistribution q = 1 / 2 under exactly `
Fixed half--quarter geometric multisection	FabiusFunction.GeometricUniformMultisection	evenCoordinates, oddCoordinates, geometricUniformSeries_one_half_multisection, geometricUniformDistribution_one_half_multisection, geometricUniformDistribution_one_half_conv_one_quarter; the pointwise normalized series splits exactly as Y_(1/2)(ω) = (2/3) Y_(1/4)(ω_even) + (1/3) Y_(1/4)(ω_odd), and under the product-uniform law the parity processes are independent copies, yielding both the exact product-map law and the convolution of the 2/3- and 1/3-scaled quarter laws; this fixed theorem needs no user hypotheses and does not claim general q/multisection, MGF or cumulant identities, centered-density formulas, or spectral dissection
Geometric-uniform moment polynomials, sharp degree, their global rational coefficient, and their real, inner-complex, and exterior-complex analytic normalizations	FabiusFunction.GeometricUniformMomentPolynomial, FabiusFunction.GeometricUniformMomentPolynomialDegree, FabiusFunction.GeometricUniformMomentPolynomialBridge, FabiusFunction.GeometricUniformComplexMomentProduct, FabiusFunction.GeometricUniformExteriorComplexMomentGerm, FabiusFunction.GeometricUniformMomentRatFunc	The algebra leaf has the exhaustive public surface of one definition and eight theorems: geometricUniformMomentPolynomial, geometricUniformMomentPolynomial_zero, geometricUniformMomentPolynomial_succ, geometricUniformMomentPolynomial_natDegree_le, geometricUniformMomentPolynomial_eval_zero, geometricUniformMomentPolynomial_one, geometricUniformMomentPolynomial_two, geometricUniformMomentPolynomial_three, and geometricUniformMomentPolynomial_four. Its total family over ℚ[X] satisfies the division-free residual finite q-Pochhammer recurrence, natDegree P_n ≤ n.choose 2, P_n(0)=1/(n+1)!, and the displayed values P_0 through P_4. The sharp-degree leaf has no public definitions and exactly three public theorems: coeff_geometricUniformMomentPolynomial_choose_two, coeff_geometricUniformMomentPolynomial_choose_two_sub_one, and geometricUniformMomentPolynomial_natDegree_eq. For every n, the coefficient at n.choose 2 is bernoulli' n / n!, equivalently (-1)^n B_n/n!; for n ≥ 2, the coefficient one below it is -bernoulli' n/n! + bernoulli' (n-1)/(2*(n-1)!). Consequently the exact natural degree is n.choose 2 for n=1 and even n (including n=0), and n.choose 2-1 for odd n>1. This makes prop:qF-P-degree-sharp Exact and supplies the formerly missing leading-coefficient and strict odd-degree clauses, so p7:thm:Pn is Exact. The real bridge has no public definitions and exactly one public theorem, geometricUniformMomentPolynomial_eval₂_eq_mgf_taylorCoefficient: for every real q with `
Combined geometric-uniform complex moment-germ reciprocity	FabiusFunction.GeometricUniformMomentReciprocity	Exhaustive public surface (one definition and five theorems): geometricUniformComplexMomentGerm, geometricUniformComplexMomentGerm_of_norm_lt_one, geometricUniformComplexMomentGerm_of_one_lt_norm, analyticAt_geometricUniformComplexMomentGerm, geometricUniformComplexMomentGerm_reciprocity, and geometricUniformComplexMomentGerm_moment_convolution. The combined function selects the inner product for ‖q‖<1 and the exterior reciprocal germ for 1<‖q‖, and is analytic at zero whenever ‖q‖≠1. Under exactly q≠0 and ‖q‖≠1, the reciprocity theorem gives M_q(z)·M_{q⁻¹}(-z)=1 locally as an EventuallyEq in 𝓝 0, and the convolution theorem gives the exact all-order binomial convolution of the two germs' iterated derivatives. This local boundary is deliberate: the inner product may have remote zeros and Lean's inverse is total, so no global reciprocal identity or unit-circle continuation is asserted. Canonical monograph label thm:qF-reciprocity is Exact.
Geometric tail dictionary and sinc-prefix factorization	FabiusFunction.GeometricUniformDictionary, FabiusFunction.GeometricSincFactorization	charFun_geometricUniformDigit, charFun_geometricUniformDistribution_prefix, charFun_geometricUniformDistribution_prefix_sinc, tendsto_prefix_sinc_charFun, charFun_weightedSumDistribution_prefix_sinc, tendsto_prefix_sinc_charFun_weightedSumDistribution; the digit formula is unconditional, while for every real q with `
Geometric sinc/Gamma characteristic-function bridge and phase-prefix convergence	FabiusFunction.GeometricSincCharacteristicFunction	Exhaustive public surface (zero definitions and four theorems): charFun_geometricUniformDistribution_eq_phase_mul_geometricSincProduct, charFun_geometricUniformDistribution_eq_phase_mul_geometricReciprocalGamma, tendstoLocallyUniformly_prefix_sinc_charFun, and tendstoUniformlyOn_prefix_sinc_charFun. For every real q with the sharp hypothesis `
CDF and explicit density of the geometric uniform law	FabiusFunction.GeometricUniformCDF	geometricUniformCDF, monotone_geometricUniformCDF, geometricUniformCDF_nonneg, geometricUniformCDF_le_one, measurable_geometricUniformCDF, continuous_geometricUniformCDF, geometricUniformCDF_reflection, geometricUniformCDF_one_half, geometricUniformCDF_zero_of_nonpos, geometricUniformCDF_one_of_one_le, geometricUniformCDF_eq_integral, geometricUniformCDF_eq_intervalIntegral, geometricUniformDensity, geometricUniformDensity_zero, geometricUniformDensity_nonpos_of_neg, volume_withDensity_geometricUniformDensity_eq_zero_of_nonpos, geometricUniformDistribution_ne_withDensity_geometricUniformDensity_of_nonpos, geometricUniformCDF_hasDerivAt, deriv_geometricUniformCDF, continuous_geometricUniformDensity, geometricUniformDensity_nonneg, geometricUniformDensity_zero_of_nonpos, geometricUniformDensity_zero_of_one_le, support_geometricUniformDensity_subset_Ioo, support_geometricUniformDensity_subset_Icc, tsupport_geometricUniformDensity_subset_Icc, geometricUniformDensity_hasCompactSupport, geometricUniformDensity_reflection, geometricUniformDistribution_eq_withDensity, contDiff_geometricUniformCDF, contDiff_geometricUniformDensity; continuity and CDF reflection assume `
Arbitrary-space realizations of geometric uniform laws	FabiusFunction.GeometricUniformRealization	Exhaustive public surface: one definition, geometricUniformRealization, and seventeen theorems, geometricUniformRealization_eq_tsum, geometricUniformRealization_split, uniformProcess_hasLaw_uniformProduct, weightedUniformSeries_hasLaw_of_iIndep_uniform, geometricUniformRealization_hasLaw, summable_norm_geometricUniformRealization_terms, geometricUniformRealization_mem_Icc, map_geometricUniformRealization_support_eq_Icc, integral_geometricUniformRealization_eq_one_half, one_sub_geometricUniformRealization_hasLaw, geometricUniformRealization_identDistrib_one_sub, affine_uniform_geometric_hasLaw, geometricUniformRealization_identDistrib_affine, measureReal_geometricUniformRealization_le_eq_cdf, measureReal_geometricUniformRealization_le_eq_integral, measureReal_geometricUniformRealization_le_eq_zero_of_nonpos, and measureReal_geometricUniformRealization_le_eq_one_of_one_le. For any supplied measurable space and measure carrying independent Icc 0 1-valued coordinates with uniform marginal laws, the realized series has the canonical law, exact mean 1/2, reflection symmetry, support/CDF conclusions, and the stated independent-copy affine identity; absolute convergence and the pointwise split are also exposed. The law, mean, reflection, and general CDF transfer need `
Product-probability and CDF representations	FabiusFunction.ProbabilityRepresentation	weightedCoordinateSum_eq_weightedUniformSeries, weightedCoordinateSum_eq_geometricUniformSeries_one_half, weightedSumDistribution_eq_geometricUniformDistribution_one_half, ae_weightedSumDistribution_mem_Icc, weightedSumDistribution_restrict_Icc, weightedSumCDF_eq_geometricUniformCDF_one_half, weightedSumCDF_eq_fabiusReal, geometricUniformCDF_one_half_eq_fabiusReal, geometricUniformDensity_one_half_eq_rvachevUp, fabiusReal_eq_weightedSum_probability, rvachevUp_eq_weightedSumCDF, rvachevUp_eq_weightedSum_probability_global; the dyadic smoothing, continuity, exterior-value, and reflection proofs route through the half-base geometric CDF API
Generic finite moment functionals and Hankel Gram forms	FabiusFunction.FiniteMomentGram	momentFunctional, momentFunctional_of_linearMap, momentFunctional_map, momentPairing, momentHankelMatrix, momentHankelMatrix_succ_submatrix, momentHankelDet, map_momentHankelDet, finiteMomentPairing_toMatrix, finiteMomentPairing_nondegenerate_iff; this measure-free layer works over the stated semiring, commutative-ring, and integral-domain hypotheses and by itself asserts no positivity
Generic fraction-free and normalized Gram--Stieltjes polynomials	FabiusFunction.GramStieltjes	gramStieltjesNumerator, momentPairing_gramStieltjesNumerator_eq_coeff_mul_det, momentPairing_gramStieltjesNumerator_self, gramStieltjesPolynomial, gramStieltjesPolynomial_isMonicOfDegree, momentPairing_gramStieltjesPolynomial_eq_zero, eq_gramStieltjesPolynomial_of_isMonicOfDegree_of_orthogonal, momentPairing_gramStieltjesPolynomial_self; the fraction-free construction is over a commutative ring, while field normalization and uniqueness assume the displayed Hankel minor is nonzero
Generic finite Jacobi coefficients and three-term recurrence	FabiusFunction.FiniteMomentJacobi	momentPairing_X_mul_left, gramStieltjesNorm, gramStieltjesJacobiDiagonal, gramStieltjesJacobiSubdiagonal, gramStieltjesJacobiSubdiagonal_eq_det_ratio, gramStieltjesPolynomial_three_term_zero, gramStieltjesPolynomial_three_term; over a field, a nonzero first Hankel minor gives the degree-zero base equation and three consecutive nonzero Hankel minors give every higher finite recurrence, with no measure, positivity, root, quadrature, continued-fraction, or convergence assumption
Polynomial-basis moment Gram determinants	FabiusFunction.PolynomialMomentGramDeterminant	Exhaustive public inventory: two definitions, polynomialCoefficientMatrix and polynomialMomentGramMatrix; and seven theorems, polynomialCoefficientMatrix_apply, polynomialMomentGramMatrix_apply, polynomialMomentGramMatrix_eq_transpose_mul_hankel_mul, polynomialMomentGramMatrix_det_eq_coefficient_det_sq_mul, polynomialCoefficientMatrix_det_eq_prod_coeff, polynomialMomentGramMatrix_det_eq_prod_coeff_sq_mul, and gramStieltjesJacobiSubdiagonal_eq_polynomialMomentGramMatrix_det_ratio. For a family with natDegree (p k) ≤ k, its coefficient matrix C is upper triangular and direct bilinear expansion gives G = Cᵀ H C; determinant multiplicativity and the diagonal product give det G = (∏ k, coeff (p k) k)^2 det H, and a nonzero diagonal transports the zero-based Jacobi subdiagonal to the corresponding Gram-determinant cross-ratio. No Hankel-nonvanishing hypothesis is imposed on that last Lean equality: division in a field is total, so if the middle Hankel determinant is zero then both cross-ratios are zero; that singular case is not a genuine nonsingular Jacobi recurrence. The coefficient matrix and its entry formula are semiring-level; the Gram matrix, its entry formula, and the congruence are commutative-semiring-level; determinant identities are commutative-ring-level; and the quotient identity is field-level. This generic module alone asserts no measure, positivity, orthogonality, entry rationality, or Wigner datum. The downstream Gaunt and closed-form leaves provide the finite rational entry formulas and the total squared zero-row integer-index datum. Signed 3j phase, half-integer or nonzero magnetic indices, orthogonality, recoupling, Christoffel reconstruction, and infinite Jacobi theory remain outside the current surface.
Scalar naturality of finite Gram--Stieltjes and Jacobi data	FabiusFunction.GramStieltjesNaturality	Exhaustive public inventory: zero definitions and six theorems, momentPairing_map, map_gramStieltjesNumerator, map_gramStieltjesPolynomial, map_gramStieltjesNorm, map_gramStieltjesJacobiDiagonal, and map_gramStieltjesJacobiSubdiagonal. The pairing theorem is over commutative semirings, the fraction-free numerator theorem over commutative rings, and the normalized polynomial, norm, and Jacobi theorems over fields. This is finite scalar base change only, with no measure, positivity, computation, or convergence claim.
Exact all-degree rational Jacobi system of the Rvachev up law	FabiusFunction.RvachevRationalJacobi	Exhaustive public inventory: four definitions, rvachevHankelDetRat, rvachevOrthoPolynomialRat, rvachevOrthoNormRat, and rvachevJacobiSubdiagonalRat; and thirteen theorems, upMoment_eq_rvachevRawMomentRat_cast, rvachevHankelDetRat_cast, rvachevHankelDetRat_pos, rvachevOrthoPolynomialRat_cast, rvachevOrthoPolynomialRat_isMonicOfDegree, momentPairing_rvachevOrthoPolynomialRat_eq_zero, rvachevOrthoNormRat_cast, rvachevOrthoNormRat_pos, gramStieltjesJacobiDiagonal_rvachevRawMomentRat_eq_zero, rvachevJacobiSubdiagonalRat_cast, rvachevJacobiSubdiagonalRat_pos, rvachevJacobiSubdiagonalRat_eq_det_ratio, and rvachevOrthoPolynomialRat_three_term. These noncomputable finite definitions give positive rational Hankel determinants, monic orthogonal polynomials, positive norms, zero diagonal, positive subdiagonals, cast comparison with every analytic Fabius representative, the determinant cross-ratio, and the exact rational recurrence in every degree. The zero-based rvachevJacobiSubdiagonalRat n is the conventional coefficient beta_(n+1). This module itself remains noncomputable and is not a native evaluator; the downstream executable rational Legendre Gram/value modules now supply the explicit report values H_4 and beta_4. Root/quadrature, Christoffel, continued-fraction, Padé, and asymptotic results remain outside this layer.
Executable rational ordinary Legendre polynomials	FabiusFunction.LegendrePolynomialRational	Exhaustive public inventory: two definitions, legendrePolynomialCoeffRat and legendrePolynomialRat; and six theorems, legendrePolynomialRat_cast, coeff_legendrePolynomialRat, natDegree_legendrePolynomialRat, coeff_legendrePolynomialRat_self, coeff_legendrePolynomialRat_self_ne_zero, and coeff_legendrePolynomialRat_self_div_succ. The scalar coefficient function is an executable finite sum, while the polynomial wrapper is noncomputable; all public statements are unconditional in their natural indices and identify its real cast, every coefficient, exact degree, top coefficient, nonvanishing, and consecutive top-coefficient quotient. Two private construction helpers are excluded.
Legendre Gram/Hankel determinant bridge for the up law	FabiusFunction.FabiusLegendreHankelDeterminant	Exhaustive public inventory: two definitions, upLegendreGramMatrix and upLegendreGramDet; and seven theorems, upLegendreGramMatrix_apply_eq_integral, upLegendreGramDet_eq_prod_leadingCoeff_sq_mul_hankelDet, upLegendreGramDet_zero, upLegendreGramDet_pos, coeff_legendrePolynomial_self_div_succ, gramStieltjesJacobiSubdiagonal_upMoment_eq_upLegendreGramDet_ratio, and rvachevJacobiSubdiagonalRat_cast_eq_upLegendreGramDet_ratio. Writing D_n for the determinant of the first n ordinary Legendre polynomials in the up-moment pairing and L_j = 2^(-j) * choose (2*j) j, the determinant identity D_n = (∏ j<n, L_j^2) * hankelDet F n, the empty 0×0 convention D_0 = 1, the leading-coefficient quotient, and the real Gram cross-ratio hold for every F : BoundedFabius. Identifying an entry with an integral, strict positivity, and the rational-cast bridge additionally require IsFabius F. Its zero-based index n is the conventional beta_(n+1) and satisfies beta_(n+1) = ((n+1)/(2*n+1))^2 * D_(n+2) * D_n / D_(n+1)^2; in the arbitrary-BoundedFabius real theorem this is the same totalized-division equality, while IsFabius supplies nonvanishing through positivity and makes it a genuine Jacobi formula. This module itself does not give a Gaunt or Wigner-square entry expansion or entrywise rationality by that route; the downstream Gaunt and closed-form modules below do. Signed 3j phase, half-integer or nonzero magnetic indices, recoupling, Christoffel reconstruction, and an infinite product remain outside the development. The unrelated rvachevTranslateGram is the Gram kernel of shifted-up atoms under unweighted interval integration, not this polynomial-basis moment Gram matrix.
Executable rational Legendre Gram data for the up law	FabiusFunction.FabiusLegendreRationalGram	Exhaustive public inventory: three executable definitions, rvachevLegendreGramEntryRat, rvachevLegendreGramMatrixRat, and rvachevLegendreGramDetRat; and eleven theorems, rvachevLegendreGramEntryRat_eq_momentPairing, rvachevLegendreGramMatrixRat_apply, rvachevLegendreGramMatrixRat_eq_polynomialMomentGramMatrix, rvachevLegendreGramEntryRat_cast, rvachevLegendreGramMatrixRat_cast, rvachevLegendreGramDetRat_cast, rvachevLegendreGramDetRat_eq_prod_leadingCoeff_sq_mul_rvachevHankelDetRat, rvachevLegendreGramDetRat_zero, rvachevLegendreGramDetRat_pos, rvachevOrthoNormRat_eq_rvachevLegendreGramDetRat_ratio, and rvachevJacobiSubdiagonalRat_eq_rvachevLegendreGramDetRat_ratio. The entry is a bounded double sum of executable rational Legendre coefficients and rvachevRawMomentRat; its finite matrix and determinant agree with the abstract polynomial moment Gram objects, and their real casts agree with the up-law entry, matrix, and determinant for every F : BoundedFabius satisfying IsFabius F. Over ℚ, the determinant is the rational Hankel determinant times the product of squared Legendre leading coefficients, is one in order zero and strictly positive in every order; exact rational Gram-determinant ratios recover rvachevOrthoNormRat n and the zero-based rvachevJacobiSubdiagonalRat n = beta_(n+1), with prefactor ((n+1)/(2*n+1))^2 in the latter. This module closes executable rational coefficient/entry/matrix/determinant data and cast bridges. The downstream Gaunt modules supply its finite Gaunt entry expansions, and FabiusLegendreGauntClosedForm supplies the rational entry/matrix and real matrix sums against the total squared zero-row integer datum. This layer itself makes no such identification. Signed 3j phase, half-integer or nonzero magnetic indices, recoupling, Christoffel reconstruction, root or quadrature theory, an infinite Jacobi product/continued fraction, and asymptotics remain outside the development.
Executable rational Legendre Gaunt coefficients and finite product linearization	FabiusFunction.LegendreGaunt	Exhaustive public inventory: four definitions, legendreLebesgueMomentRat, legendreGauntRat, legendreGaunt, and legendreProductLinearizationCoeffRat; and twelve theorems, legendreLebesgueMomentRat_even, legendreLebesgueMomentRat_odd, legendreLebesgueMomentRat_cast, legendreGauntRat_eq_momentFunctional, legendreGauntRat_cast, legendreGauntRat_swap_left, legendreGauntRat_swap_right, legendrePolynomial_mul_eq_sum_gaunt, legendrePolynomialRat_mul_eq_sum_gaunt, legendreGauntRat_eq_zero_of_odd_sum, legendreGauntRat_eq_zero_of_add_lt, and legendreGauntRat_eq_zero_of_triangle_violation. For every n : ℕ, legendreLebesgueMomentRat n is 2/(n+1) at even n and zero at odd n, and its real cast is ∫ x in (-1)..1, x^n. For every i j k : ℕ, legendreGauntRat i j k is the bounded triple coefficient–moment sum for P_i P_j P_k, legendreGaunt i j k = ∫ x in (-1)..1, P_i(x) P_j(x) P_k(x), and legendreProductLinearizationCoeffRat i j k = ((2*k+1)/2) * legendreGauntRat i j k. The rational sum equals the moment functional and casts to the real integral, is invariant under the displayed first-two and last-two swaps, and gives the exact real and rational identities P_i P_j = ∑ k ∈ range(i+j+1), legendreProductLinearizationCoeffRat i j k • P_k. The only hypotheses are Odd (i+j+k) for the parity zero, i+j<k for the one-sided degree zero, and i+j<k ∨ i+k<j ∨ j+k<i for the triangle-violation zero; all definitions and other theorems are total at arbitrary natural indices. This module itself proves only the necessary support zeros and makes no converse support, Wigner-square, factorial, positivity, or asymptotic claim; the next closed-form leaf proves the total squared zero-row integer-index case and sharpens those support statements.
Total zero-row Wigner-square Gaunt closed form, sharp support, positivity, and product coefficients	FabiusFunction.LegendreGauntClosedForm	Exhaustive public inventory: two definitions, legendreGauntAdmissible and legendreWignerThreeJZeroSqRat; and twenty-five theorems, legendreGauntAdmissible_iff_exists_pairwise_add, legendreGauntAdmissible_pairwise_add, legendreWignerThreeJZeroSqRat_pairwise_add, legendreWignerThreeJZeroSqRat_pairwise_add_factorial, legendreWignerThreeJZeroSqRat_eq_factorial_of_halfSum, legendreWignerThreeJZeroSqRat_eq_zero_of_not_admissible, legendreGauntRat_add_boundary, legendreGauntRat_add_boundary_eq_two_mul_wignerThreeJZeroSqRat, legendreGauntRat_zero_left, legendreGauntRat_zero_left_eq_two_mul_wignerThreeJZeroSqRat, legendreGauntRat_pairwise_add_eq_two_mul_wignerThreeJZeroSqRat, legendreGauntRat_eq_zero_of_not_admissible, legendreGauntRat_eq_two_mul_wignerThreeJZeroSqRat, legendreGaunt_eq_two_mul_wignerThreeJZeroSqRat, legendreWignerThreeJZeroSqRat_pos_iff_admissible, legendreWignerThreeJZeroSqRat_nonneg, legendreWignerThreeJZeroSqRat_eq_zero_iff_not_admissible, legendreGauntRat_pos_iff_admissible, legendreGauntRat_eq_zero_iff_not_admissible, legendreGaunt_pos_iff_admissible, legendreGaunt_eq_zero_iff_not_admissible, legendreGauntRat_nonneg, legendreGaunt_nonneg, legendreProductLinearizationCoeffRat_eq_mul_wignerThreeJZeroSqRat, and legendreProductLinearizationCoeffRat_pos_iff_admissible. Admissibility is even total degree plus the three weak triangle inequalities, equivalently a triple of pairwise-sum coordinates. The total rational square datum is zero off this support and has both central-binomial and factorial forms on it; the half-sum factorial theorem assumes the displayed half-sum identity and domination of all three indices. At arbitrary natural indices the rational and real Gaunt coefficients are twice this datum, are positive exactly on the admissible support and otherwise zero, and are nonnegative; the coefficient of P_k in P_i P_j is (2*k+1) times the datum and has the same sharp positivity support. This is only an integer-index, zero-magnetic-row square datum defined by its rational formula: no signed symbol or phase convention, half-integer indices, nonzero magnetic indices, general 3j/6j/9j symbols, Wigner orthogonality, or recoupling identity is supplied.
Finite Gaunt sums for rational and real up-law Legendre Gram entries	FabiusFunction.FabiusLegendreGaunt	Exhaustive public inventory: one definition, canonicalRvachevFullLegendreCoefficientRat; and eight theorems, canonicalRvachevFullLegendreCoefficientRat_even, canonicalRvachevFullLegendreCoefficientRat_odd, canonicalRvachevFullLegendreCoefficientRat_cast, canonicalRvachevFullLegendreCoefficientRat_eq_normalized_moment, rvachevLegendreGramEntryRat_eq_sum_full_gaunt, rvachevLegendreGramEntryRat_eq_sum_gaunt, rvachevLegendreGramMatrixRat_apply_eq_sum_gaunt, and upLegendreGramMatrix_apply_eq_sum_gaunt. The full coefficient is canonicalRvachevLegendreCoefficientRat (k/2) when 2 ∣ k and zero otherwise, hence it restricts to the canonical coefficient at 2*n and vanishes at 2*n+1. Unconditionally in k, it equals ((2*k+1)/2) * momentFunctional rvachevRawMomentRat (legendrePolynomialRat k). For all natural i,j, the rational Gram entry is both ∑ k ∈ range(i+j+1), canonicalRvachevFullLegendreCoefficientRat k * legendreGauntRat i j k and the reindexed even sum ∑ r ∈ range((i+j)/2+1), canonicalRvachevLegendreCoefficientRat r * legendreGauntRat i j (2*r); the latter is also the exact entry formula for every i j : Fin n. Only the cast theorem and the real matrix formula require F : BoundedFabius and hF : IsFabius F; then the full rational coefficient casts to rvachevFullLegendreCoefficient F k, and the real entry is the same even finite sum with rvachevLegendreCoefficient F r and legendreGaunt. This module itself proves finite polynomial and finite Gaunt-sum identities; the next leaf substitutes the total squared zero-row integer datum. It does not choose a signed 3j phase or treat half-integer/nonzero magnetic indices, orthogonality, recoupling, infinite Legendre-series interchange, Christoffel reconstruction, named values for the displayed G_3 entries, or a new asymptotic result.
Finite zero-row Wigner-square sums for rational Rvachev and real up-law Legendre Gram entries	FabiusFunction.FabiusLegendreGauntClosedForm	Exhaustive public inventory: zero definitions and exactly three theorems, rvachevLegendreGramEntryRat_eq_two_mul_sum_wignerThreeJZeroSqRat, rvachevLegendreGramMatrixRat_apply_eq_two_mul_sum_wignerThreeJZeroSqRat, and upLegendreGramMatrix_apply_eq_two_mul_sum_wignerThreeJZeroSqRat. The first gives every executable rational entry as twice the finite sum of canonicalRvachevLegendreCoefficientRat r times legendreWignerThreeJZeroSqRat i j (2*r); the second is its entrywise finite rational matrix form. For F : BoundedFabius with IsFabius F, the third gives the real up-law matrix entry as twice the corresponding finite sum with rvachevLegendreCoefficient F r and the real cast of that datum. These are finite rational entry/matrix and real matrix sums against the total squared zero-row integer datum. No signed 3j phase, half-integer or nonzero magnetic index, general Wigner-symbol API, orthogonality, or recoupling theorem is claimed.
Central-binomial cancellation in even Rvachev--Legendre synthesis (cor:leg-central-sum)	FabiusFunction.RvachevLegendreCentralSum	Exhaustive zero-definition/three-theorem surface: eval_legendrePolynomial_even_zero, eval_rvachevLegendreDeconvolutionPolynomial_even, and rvachevLegendreCentralSum. The first evaluates the Rodrigues-normalized P_(2n)(0); the second proves pointwise evenness of the deconvolved even mode; and the third proves, for every F : BoundedFabius with IsFabius F and M=4^n, the printed identity Q_(2n)(0)+2∑_(0<k<M) Q_(2n)(k/M) up(k/M)=(-1)^n choose(2n,n). It includes n=0. The proof evaluates the existing finite synthesis at zero, uses compact support to truncate `
Finite Legendre--up dyadic biorthogonality (thm:leg-biorthogonality)	FabiusFunction.RvachevLegendreBiorthogonality	Exhaustive one-definition/one-theorem surface: rvachevLegendreAnalysisKernel is exactly ((2*m+1)/2) * integral_(-1)^1 up(x-c) P_m(x) dx, and rvachevLegendreBiorthogonality proves `M⁻¹ * ∑_(
Exact low-order rational Legendre Gram and Jacobi values	FabiusFunction.FabiusLegendreRationalGramValues	Exhaustive public inventory: zero definitions and eleven theorems, moment_four, rvachevLegendreGramDetRat_one, rvachevLegendreGramDetRat_two, rvachevLegendreGramDetRat_three, rvachevLegendreGramDetRat_four, rvachevLegendreGramDetRat_five, rvachevOrthoNormRat_four, rvachevJacobiSubdiagonalRat_three, hankelRatio_four, integral_sq_upOrthoPolynomial_four, and hankelRatio_four_div_three. The first theorem gives the raw eighth moment 132809/32531625; the five determinant theorems give orders one through five as 1, 1/9, 8/2025, 39616/602791875, and 16544275456/27453718922765625. The rational norm is H_4 = rvachevOrthoNormRat 4 = 26727424/55791736875, while the zero-based index-three subdiagonal is the conventional beta_4 = rvachevJacobiSubdiagonalRat 3 = 835232/4640643. The rational computations have no analytic input; each of hankelRatio_four, integral_sq_upOrthoPolynomial_four, and hankelRatio_four_div_three transports these values to a real squared norm, squared orthogonal-polynomial integral, or consecutive-ratio quotient only for F : BoundedFabius with IsFabius F. The separate Gaunt and closed-form modules close the finite entry and total zero-row square expansions, but this values leaf supplies no named evaluation of the displayed G_3 entries. Signed 3j phase, half-integer or nonzero magnetic indices, recoupling, Christoffel reconstruction, roots or quadrature, infinite Jacobi products/continued fractions, and asymptotics remain open.
Comparison of generic Gram--Stieltjes algebra with the up measure	FabiusFunction.OrthogonalPolynomialGramBridge	momentFunctional_upMoment_eq_integral, momentPairing_upMoment_eq_integral, momentHankel_eq_momentHankelMatrix, hankelDet_eq_momentHankelDet, hankelOrthoPolynomial_eq_gramStieltjesNumerator, upOrthoPolynomial_eq_gramStieltjesPolynomial, hankelRatio_eq_gramStieltjesNorm, gramStieltjesJacobiDiagonal_upMoment_eq_zero, gramStieltjesJacobiSubdiagonal_upMoment_eq; these theorems identify both determinant constructions, their monic normalizations, and their finite Jacobi data exactly
Fabius-measure Hankel positivity and finite orthogonal-polynomial recurrence	FabiusFunction.MomentHankelMatrix, FabiusFunction.MomentHankelValues, FabiusFunction.OrthogonalPolynomialConstruction, FabiusFunction.OrthogonalPolynomialParity, FabiusFunction.OrthogonalPolynomialRecurrence, FabiusFunction.OrthogonalPolynomialJacobi	momentHankel_posDef, hankelDet_pos, hankelRatio_pos, upOrthoPolynomial_monic, integral_upOrthoPolynomial_sq, eq_upOrthoPolynomial_of_monic_of_orthogonal, upOrthoPolynomial_comp_neg_X, upOrthoPolynomial_three_term, upOrthoPolynomial_two, upOrthoPolynomial_three, upOrthoPolynomial_four; roots and Gaussian/Lobatto quadrature, finite or infinite continued-fraction identification, and convergence remain separate
Generic oriented Cauchy calculus for finite real measures	FabiusFunction.MeasureCauchyTransform	Exhaustive public surface: measureCauchyDomain, measureCauchyTransform, measureCauchyPower, measureCauchyTransform_apply, measureCauchyPower_one, measureCauchyPower_map_affine, measureCauchyTransform_map_affine, isOpen_measureCauchyDomain, hasDerivAt_measureCauchyTransform, analyticOn_measureCauchyTransform, measureCauchyPower_succ_of_uniformAffineFixedPoint, hasDerivAt_measureCauchyTransform_of_uniformAffineFixedPoint. The affine-map identities are total and allow negative nonzero scale. The two fixed-point theorems require only a finite measure, support in an affine-invariant carrier, nonzero uniform and tail scales, and the displayed uniform affine equality in law; they require neither probability normalization nor topological or measurable hypotheses on the carrier.
Generic centered moment/Laurent calculus for bounded finite real measures	FabiusFunction.MeasureCauchyMomentLaurent	Exhaustive public surface (one definition and fifteen theorems): measureCauchyMoment, inv_sub_eq_sum_range_add, measureCauchyMoment_zero, measurable_inv_sub_rclike, integrable_centered_pow_div_sub_of_ae_norm_sub_le, integrable_inv_sub_of_ae_norm_sub_le, norm_measureCauchyMoment_le, integral_inv_sub_eq_sum_range_measureCauchyMoment_add, norm_integral_inv_sub_sub_sum_range_measureCauchyMoment_le, summable_measureCauchyMoment_laurent, integral_inv_sub_eq_tsum_measureCauchyMoment, hasSum_measureCauchyMoment_laurent, measureCauchyTransform_eq_sum_range_measureCauchyMoment_add, norm_measureCauchyTransform_sub_sum_range_measureCauchyMoment_le, measureCauchyTransform_eq_tsum_measureCauchyMoment, hasSum_measureCauchyTransform_measureCauchyMoment_laurent. Apart from the field-generic kernel identity, the API works over any RCLike 𝕜. For a finite real measure, arbitrary c z : 𝕜, 0 ≤ R, almost-everywhere support ‖(x : 𝕜) - c‖ ≤ R, and R < ‖z-c‖, it gives the positive-sign expansion in measureCauchyMoment μ c k / (z-c)^(k+1), exact remainder, mass-weighted error μ.real Set.univ * (‖z-c‖-R)⁻¹ * (R/‖z-c‖)^N, and Summable/tsum/HasSum forms. The last four theorems are the complex wrappers for the named oriented transform; no probability normalization or topological-support theorem is assumed.
Geometric-uniform Cauchy--Stieltjes hierarchy	FabiusFunction.GeometricUniformCauchy	Exhaustive public surface: geometricUniformStieltjesDomain, geometricUniformStieltjesTransform, geometricUniformStieltjesPower, geometricUniformStieltjesTransform_apply, geometricUniformStieltjesPower_one, isOpen_geometricUniformStieltjesDomain, analyticOn_geometricUniformStieltjesTransform, hasDerivAt_geometricUniformStieltjesTransform_refinement, geometricUniformStieltjesPower_succ. Definitions are total in real q; holomorphy assumes `
Ordinary Cauchy--Stieltjes transforms and powers of the canonical up and unit-interval laws	FabiusFunction.CauchyTransform	Exhaustive public surface: rvachevCauchyDomain, fabiusStieltjesDomain, rvachevCauchyTransform, fabiusStieltjesTransform, rvachevCauchyPower, fabiusStieltjesPower, rvachevCauchyTransform_apply, fabiusStieltjesTransform_apply, rvachevCauchyPower_one, fabiusStieltjesPower_one, rvachevCauchyTransform_eq_integral_rvachevUp, isOpen_rvachevCauchyDomain, isOpen_fabiusStieltjesDomain, hasDerivAt_rvachevCauchyTransform, hasDerivAt_fabiusStieltjesTransform, analyticOn_rvachevCauchyTransform, analyticOn_fabiusStieltjesTransform, fabiusStieltjesTransform_eq_two_mul_rvachevCauchyTransform, rvachevCauchyTransform_eq_inv_two_mul_fabiusStieltjesTransform, fabiusStieltjesPower_eq_two_pow_mul_rvachevCauchyPower, hasDerivAt_fabiusStieltjesTransform_refinement, fabiusStieltjesPower_succ, rvachevCauchyPower_succ. The integral definitions and affine transform/power bridges are total under Lean's Bochner-integral convention; holomorphy, derivative, DDE, and adjacent-order statements use the named slit domains. The unit DDE and both complex-domain adjacent-power recurrences are direct named theorems; the centered DDE and its Thue--Morse derivative orbit remain the direct named results of CauchyRenormalization.
Atom-exact compact-support Cauchy--CDF integration by parts	FabiusFunction.CauchyCDF	integral_inv_sub_eq_mass_smul_sub_intervalIntegral_measureReal_Iic, integral_inv_sub_eq_sub_intervalIntegral_cdf, fabiusStieltjesTransform_eq_inv_sub_one_sub_intervalIntegral_fabiusReal; the generic results assume an ordered compact interval, almost-everywhere support there, and a spectral parameter off its complexification, with finite-measure and probability normalizations respectively; the Fabius wrapper is on fabiusStieltjesDomain
Survival and all-power Cauchy integration by parts	FabiusFunction.CauchySurvival, FabiusFunction.CauchyHigherPowers	intervalIntegral_inv_sub_sq, integral_inv_sub_eq_mass_smul_add_intervalIntegral_measureReal_Ioi, fabiusStieltjesTransform_eq_inv_add_intervalIntegral_rvachevUp, hasDerivAt_pow_inv_sub, intervalIntegral_pow_inv_sub, integral_pow_inv_sub_eq_mass_smul_sub_intervalIntegral_measureReal_Iic, integral_pow_inv_sub_eq_mass_smul_add_intervalIntegral_measureReal_Ioi; these are direct named CDF/survival formulas, including atoms, and do not by themselves identify the all-order Thue--Morse derivative orbit with a single named higher-kernel formula.
Rvachev Cauchy-transform renormalization and all-order affine orbit	FabiusFunction.CauchyRenormalization	mapsTo_rvachevCauchyDomain_two_mul_add_one, mapsTo_rvachevCauchyDomain_two_mul_sub_one, hasDerivAt_rvachevCauchyTransform_affineDifference, deriv_rvachevCauchyTransform, iteratedDeriv_rvachevCauchyTransform_eq_thueMorse_sum; for every bounded Fabius solution and every point off [-1,1], the exact DDE and all complex derivative orbits hold.
Real logarithmic and resolvent-order fixed-point calculus	FabiusFunction.StieltjesLogFixedPoint, FabiusFunction.StieltjesResolventHierarchy, FabiusFunction.StieltjesGeneralizedOrder	The direct named theorems prove the logarithmic fixed-point representation and every positive integer-order hierarchy for real z > 1, and the order-lowering hierarchy for real order α > 1 and real z > 1. These results do not provide a complex logarithm/branch continuation or complex order.
Up-measure Laurent, Herglotz, and integrated Stieltjes--Perron layers	FabiusFunction.StieltjesMomentLaurent, FabiusFunction.StieltjesCauchyTransform, FabiusFunction.StieltjesHerglotz, FabiusFunction.StieltjesInversion, FabiusFunction.StieltjesPerron	The radius-one, center-zero up-measure Laurent layer is connected to the generic API by ae_norm_sub_zero_le_one_rvachevMeasure and measureCauchyMoment_rvachevMeasure_zero and retains the report-facing upMoment results in real exterior and complex ‖z‖ > 1 form. The remaining direct named theorems give conjugation and upper/lower-half-plane sign and quantitative bounds, exact finite-height Poisson/conjugate-Poisson representations, approximate-identity estimates, and integrated interval Stieltjes--Perron inversion. Pointwise or nontangential Sokhotski--Plemelj boundary values and principal-value Hilbert-transform identities remain open.
Initial exact Jacobi data	FabiusFunction.OrthogonalPolynomialJacobi	Direct named theorems compute additional low moments, Hankel determinants and ratios, and the monic degree-three and degree-four orthogonal polynomials with norm/evaluation corollaries. A full J-fraction/Padé convergence theory and asymptotic Jacobi analysis remain frontier work.
Support-free quantile transport, compact inverse-CDF transport, and exact Fabius substitution	FabiusFunction.QuantileTransport	map_quantile_eq, map_inverseCDF_volume_restrict_Icc, map_fabiusInv_restrict_Icc_eq_weightedSumDistribution, integral_comp_fabiusInv_restrict_Icc_eq_weightedSumDistribution
Weighted subgraph/supergraph Fubini, generic survival layer cake, and exact Rvachev stopped primitives	FabiusFunction.SubgraphFubini, FabiusFunction.SurvivalLayerCake, FabiusFunction.ProbabilityLaplaceMoments	integral_smul_setIntegral_subgraph, integral_smul_setIntegral_supergraph, intervalIntegral_survival_smul_eq_integral_clamp, intervalIntegral_survival_smul_eq_integral_min_of_ae_mem_Icc, intervalIntegral_survival_smul_eq_integral_of_ae_mem_Icc, intervalIntegral_rvachevUp_smul_eq_integral_min, intervalIntegral_rvachevUp_smul_eq_integral
Atom-preserving lower-CDF layer cake and exact positive-real Fabius fractional integrals	FabiusFunction.CDFLayerCake, FabiusFunction.FractionalCDFLayerCake, FabiusFunction.FabiusFractionalIntegral	intervalIntegral_cdf_smul_eq_integral_clamp, intervalIntegral_cdf_smul_eq_integral_max_of_ae_le, intervalIntegral_cdf_smul_eq_integral_max_of_ae_mem_Icc, intervalIntegral_cdf_smul_eq_integral_of_ae_mem_Icc, fractionalVolterra_measureReal_Iic_eq_integral_clamp, fractionalVolterra_measureReal_Iic_eq_integral_posPart, fractionalVolterra_cdf_eq_integral_posPart, fractionalVolterra_measureReal_Iic_eq_integral_rpow_of_ae_mem_Icc, fractionalVolterra_fabiusReal_eq_integral_posPart, fractionalVolterra_fabiusReal_eq_uniformProduct_integral_posPart, fractionalVolterra_fabiusReal_one_eq_integral_rpow, fractionalVolterra_fabiusReal_one_eq_uniformProduct_integral_rpow; the partial terminal identity needs only c ≤ b and almost-everywhere upper support z ≤ b, the probability wrapper is stated directly with Mathlib's CDF, and the Fabius stopped-power formulas hold for every real endpoint and every positive real order; no fractional derivative or complex-order continuation is asserted
Generic inverse-clock and exact Fabius weighted layer-cake identities	FabiusFunction.InverseLayerCake	galoisConnection_Icc_restrict_of_lt_iff_lt, intervalIntegral_smul_intervalIntegral_of_lt_iff_lt, intervalIntegral_smul_comp_of_lt_iff_lt, intervalIntegral_mul_comp_sub_of_lt_iff_lt_of_absolutelyContinuousOnInterval, intervalIntegral_mul_comp_of_lt_iff_lt_of_absolutelyContinuousOnInterval, intervalIntegral_smul_intervalIntegral_fabiusInv, intervalIntegral_smul_comp_fabiusInv, intervalIntegral_mul_comp_fabiusInv_of_absolutelyContinuousOnInterval, intervalIntegral_mul_fabiusInv_eq, intervalIntegral_fabiusInv_eq_intervalIntegral_rvachevUp, intervalIntegral_fabiusInv_eq_one_half; the strict order adjunction makes the interval restrictions a Galois connection and supplies measurable clamped representatives, so none of the public generic layer-cake theorems assumes clock measurability; the explicit-primitive and pointwise-C¹ forms allow real or complex weights and Banach-valued primitives, while the absolutely-continuous forms allow real or complex L¹ weights and a real-valued primitive, include degenerate ordered rectangles, and need no separate right-hand-side integrability hypothesis
Absolutely-continuous two-function and variable-upper calculus for inverse-order pairs	FabiusFunction.InversePairIntegral	intervalIntegral_deriv_mul_comp_add_comp_mul_deriv_of_lt_iff_lt, intervalIntegral_deriv_mul_comp_add_comp_mul_deriv_to_of_lt_iff_lt, intervalIntegral_deriv_mul_fabiusReal_add_fabiusInv_mul_deriv, intervalIntegral_deriv_mul_fabiusReal_add_fabiusInv_mul_deriv_to, intervalIntegral_fabiusReal_add_fabiusInv_eq_one, intervalIntegral_fabiusInv_to_eq_mul_sub_intervalIntegral_fabiusReal; the generic theorems derive monotonicity and measurable clamped extensions of both clocks from the strict order equivalence, include degenerate ordered intervals, and cover every proper compact cut y ∈ [a,b]; reversed endpoints still require swapping the corresponding interval, and improper singular endpoints, higher or fractional inverse primitives, complex-valued arbitrary-AC factors, and complex Mellin continuation are not claimed
Weighted-partition exponential coefficients over commutative ℚ-algebras	FabiusFunction.ExponentialPartition, FabiusFunction.ExponentialBell	partitionExpSum_recurrence, partitionExpSum_succ, partitionExpSum_eq_sum_div, partitionExpSum_eq_expCoeff
Pairwise partition defects, sharp fixed-block bound, equality profiles, and first positive shell	FabiusFunction.PartitionDefect	Exhaustive public inventory: three definitions, pairSum, blockPairDefect, and partitionDefect, plus 33 theorems. Unconditional group: pairSum_nil, pairSum_cons, pairSum_one, pairSum_add, pairSum_congr, pairSum_map, choose_add_two, choose_list_sum_two, partitionDefect_nonneg, pairSum_add_eq, pairSum_map_add_eq, partitionDefect_eq_linear_add_pairwise_excess, pairSum_eq_zero_iff_pairwise. Positive-pair group: blockPairDefect_eq_mul_sub_one, blockPairDefect_eq_zero_iff, sub_one_mul_sub_one_eq_zero_iff, add_sub_one_le_mul_of_pos, mul_eq_add_sub_one_iff. Positive-list group, assuming exactly ∀ x ∈ r, 0 < x: partitionDefect_eq_pairSum_mul_sub_one, choose_sum_two_eq_choose_length_add_sum_add_partitionDefect, partitionDefect_eq_choose_sum_sub_choose_length_sub_sum, length_le_sum_of_pos, sum_map_sub_one, partitionDefect_lower_bound, partitionDefect_eq_zero_iff, partitionDefect_eq_lower_bound_iff. partitionDefect_fixed_block_bound and partitionDefect_fixed_block_eq_iff additionally assume r.sum = m and r.length = k; firstShell_le_fixedBlockProduct and fixedBlockProduct_eq_firstShell_iff assume exactly 2 ≤ k and k < m; firstShell_le_partitionDefect and partitionDefect_eq_firstShell_iff assume positivity, r.sum = m, and 2 ≤ r.length < m; partitionDefect_twoBlock_firstShell assumes 3 ≤ m. The list statements concern arbitrary positive lists, not labelled set partitions; one private zero-sum helper is excluded.
Complete Bell and moment--cumulant transforms over commutative ℚ-algebras	FabiusFunction.MomentCumulantAlgebra	factorialNormalize, completeBellPolynomial, momentCumulant, completeBellPolynomial_succ, completeBellPolynomial_momentCumulant, momentCumulant_completeBellPolynomial
Unit-series Bell conversion, powers/logarithms, and exponential jets	FabiusFunction.UnitSeriesBellCoefficients	Exhaustive zero-definition/sixteen-theorem surface: ordPartialBell_eq_factorialRatio_partialBell, factorial_mul_ordPartialBell_eq_factorial_mul_partialBell, coeff_fallingSeries_subst_eq_sum_ordPartialBell, coeff_fallingSeries_subst_eq_sum_ordPartialBell_of_pos, coeff_fallingSeries_subst_eq_sum_partialBell, coeff_negBinomSeries_subst_eq_sum_ordPartialBell, coeff_negBinomSeries_subst_eq_sum_ordPartialBell_of_pos, coeff_logOf_eq_sum_ordPartialBell, egfA_factorialDenormalize_coeff_eq, bellWeightSeries_factorialDenormalize_coeff_eq, coeff_logOf_eq_sum_partialBell, coeff_exp_subst_eq_completeBell, coeff_exp_subst_eq_partitionExpSum, coeff_exp_subst_eq_sum_weightedPartitions, coeff_exp_subst_eq_sum_div_weightedPartitions, and coeff_exp_subst_recurrence. These make p0:lem:bell-conversion, p0:lem:power-log, and p0:cor:exp-log-jets Exact, with all-index strengthenings where the API permits. This is formal power-series coefficient algebra over the indicated commutative rational algebras/fields; it asserts neither analytic convergence nor a choice of analytic logarithm branch.
Dickson and Neumann well-basedness	FabiusFunction.TransseriesWellBased	Exhaustive written zero-definition/seven-theorem surface: dickson_isPWO, dickson_antichain_finite, dickson_isPWO_pi, neumann_isPWO, neumann_finite_factorizations, neumann_isPWO_orderDual, and neumann_finite_factorizations_orderDual; to_additive additionally generates neumann_add_isPWO and neumann_finite_decompositions, which the lexical census deliberately does not count. The manuscript labels q0:lem:dickson and q0:lem:neumann are Exact: the two explicit OrderDual wrappers state Neumann's lemma in the manuscript's orientation. In a total order this is its reverse-well-order/no-strict-growth convention; over a merely partial order the precise claim is Set.IsPWO in the dual order, not a greatest-element characterization. The results use an ordered cancel commutative monoid and do not construct a Hahn-series type.
Elementary transseries height comparisons	FabiusFunction.TransseriesHeight	Exhaustive zero-definition/three-theorem surface: isLittleO_log_pow_rpow, isLittleO_log_pow_id, and isLittleO_pow_mul_log_pow_exp. They make the two printed real atTop comparisons in q0:prop:height Exact for natural powers and positive real comparison exponents. The module does not define a global recursive height/depth order on arbitrary nested transmonomials.
Sequence-indexed asymptotic scales and Poincaré coefficients	FabiusFunction.TransseriesScale	Exhaustive 3-definition-or-structure/6-theorem surface: IsAsymptoticScale, poincarePartialSum, IsPoincareExpansion, poincarePartialSum_zero, poincarePartialSum_succ, IsPoincareExpansion.isLittleO_succ_remainder, IsPoincareExpansion.tendsto_coeff, IsPoincareExpansion.tendsto_coeff_div, IsPoincareExpansion.coeff_unique. The scalar scale and vector-valued coefficient/remainder API makes q0:def:scale, q0:def:poincare, q0:eq:coefficients, and q0:prop:uniqueness Exact; uniqueness assumes [l.NeBot], and no convergence or maximal-set scale is claimed.
Real power--logarithmic dominance	FabiusFunction.TransseriesScaleDominance	Exhaustive one-definition/seven-theorem surface: plMonomial, tendsto_plMonomial_atTop_zero, plMonomial_div_eventuallyEq, tendsto_plMonomial_div_atTop_zero, tendsto_plMonomial_div_atTop_one, plMonomial_pos, tendsto_plMonomial_div_atTop, and plMonomial_generators_dominance. The analytic zero/one/atTop trichotomy and the integer-generator rule of plt:lem:mot-dominance are Exact. This does not by itself package the manuscript's full unordered set of monomials or its maximal-expansion notion.
Power--logarithmic scale sequences	FabiusFunction.TransseriesPolyLogScale	Exhaustive zero-definition/four-theorem surface: isLittleO_plMonomial, isAsymptoticScale_plMonomial, isAsymptoticScale_plMonomial_pow, and isAsymptoticScale_plMonomial_log. It gives the exact sequence-indexed scale consequence for strictly lexicographically decreasing exponent pairs and the pure inverse-power and fixed-power/logarithmic ladders. It does not prove the whole set-indexed/maximal package in plt:def:mot-scale.
Polynomial logarithmic-block antidifferentiation	FabiusFunction.TransseriesBlockAntiderivative	Exhaustive three-definition/twelve-theorem surface. Definitions: blockOperator, blockAntiderivative, resonantAntiderivative. Theorems: sum_sub_sum_shift, blockOperator_zero, blockOperator_sub, blockOperator_blockAntiderivative, blockOperator_surjective, natDegree_C_mul_of_ne_zero, natDegree_blockOperator, blockOperator_injective, blockOperator_bijective, derivative_resonantAntiderivative, derivative_surjective, and natDegree_resonantAntiderivative. The polynomial-operator dichotomy in plt:lem:mot-block-antiderivative is Exact: for nonzero c, ∂_L-c has the displayed explicit finite inverse and preserves nonzero degree; at resonance, differentiation is surjective with an explicit primitive raising degree by one. This is a theorem about K[L], not a construction of the full Laurent ambient ring K[t,t⁻¹][L].
Abstract differential-block bridge	FabiusFunction.TransseriesDifferentialBlock	Exhaustive zero-definition/twelve-theorem union: derivation_pow_t, derivation_block, derivation_zpow_block, exists_zpow_block_primitive, existsUnique_zpow_block_primitive, exists_block_primitive, derivation_block_zero, exists_block_primitive_resonant, derivation_val_inv, derivation_pow_inv, derivation_zpow_t, derivation_block_zpow. The integer Laurent formula and conditional generic primitive/uniqueness API are formal; the concrete Laurent model and unconditional faithful-evaluation uniqueness remain absent.
Flat functions and the invisible-function problem	FabiusFunction.TransseriesFlat	Exhaustive merged 4+22 surface, partitioned into the vector-valued 1+11 core and retained scalar 3+11 compatibility layer inventoried above. This makes q0:def:flat and all three clauses of q0:prop:invisible Exact, including the same-coefficient iff and concrete exponential remainder. Variable multiplication uses explicit scale absorption; no unrestricted multiplier theorem is claimed.
Wright omega through the principal real Lambert branch	FabiusFunction.WrightOmega	Exhaustive one-definition/thirteen-theorem surface: wrightOmega, analyticAt_wrightOmega, wrightOmega_pos, wrightOmega_add_log, principalLambertW_eq_wrightOmega_log, wrightOmega_leftInverse, wrightOmega_strictMono, wrightOmega_one, one_le_wrightOmega, wrightOmega_le_self, sub_log_le_wrightOmega, wrightOmega_envelope, add_one_div_two_le_wrightOmega, and tendsto_wrightOmega_atTop. These make all clauses of plt:prop:mot-omega-basic Exact over the reals, including AnalyticAt ℝ wrightOmega X for every real X. This formal real-analytic statement does not assert a complex Wright-omega branch or complex holomorphy.
Integer-leading harmonic increment asymptotics	FabiusFunction.TransseriesHarmonicIncrement	Exhaustive zero-definition/two-theorem surface: tendsto_div_atTop_of_tendsto_sub and tendsto_div_atTop_of_harmonic_increment. The generic Stolz--Cesàro step and the conclusion w_n/n → c₀ are exact. plt:lem:mot-harmonic remains partial: the logarithmic correction (c₁/c₀) log n, the limiting constant, and the final o(1) remainder are not formalized.
Ordinary partial Bell normalization	FabiusFunction.OrdinaryPartialBell	Exhaustive two-definition/four-theorem surface. Definitions: ordinarySeries, ordinaryPartialBell. Theorems: ordinaryPartialBell_pow, bellWeightSeries_eq_ordinarySeries, factorial_mul_ordinaryPartialBell, and ordinaryPartialBell_eq_zero_of_lt. The ordinary generating-series characterization and the factorial normalization bridge to the exponential partial Bell family are exact over commutative rational algebras. The whole printed plt:lem:bell-normalizations remains partial because the explicit multinomial-sum presentation and the final identification with the manuscript's convolution polynomial are not separate Lean theorems.
Real linear--logarithmic core inversion	FabiusFunction.LinLogCoreInversion	Exhaustive four-definition/eighteen-theorem surface. Definitions: linLogCoreArg, linLogCoreRoot, linLogCoreThreshold, linLogCoreRootLower. Theorems: linLogCore_eq_iff, principalLambertW_linLogCoreArg_pos, linLogCoreRoot_pos, linLogCore_linLogCoreRoot, strictMonoOn_linLogCore, linLogCoreRoot_unique, hasDerivAt_linLogCore, linLogCore_slope_eq, linLogCore_critical, linLogCoreArg_mem_Ioo_iff, principalLambertW_linLogCoreArg_neg, linLogCoreRoot_pos_of_neg, linLogCoreRootLower_pos, linLogCore_linLogCoreRoot_of_neg, linLogCore_linLogCoreRootLower, linLogCoreRoot_lt_critical, critical_lt_linLogCoreRootLower, and linLogCoreRoot_ne_linLogCoreRootLower. The branch-free Lambert equivalence, the unique positive principal root for b>0, both separated real roots above the sharp threshold for b<0, and the slope identities are exact. p0:thm:lambert-core remains partial only at its large-L asymptotic clause and its complex general-branch reading.
Real r=1 power--logarithmic core inversion	FabiusFunction.PowerLogCoreInversion	Exhaustive three-definition/six-theorem surface. Definitions: powerLogCore, powerLogCoreArg, powerLogCoreRoot. Theorems: powerLogCore_exp, log_powerLogCoreRoot_sub, powerLogCore_of_lambert, powerLogCore_powerLogCoreRoot, hasDerivAt_powerLogCore, and hasDerivAt_powerLogCore_root. The substitution, arbitrary-real-Lambert-solution bridge, principal-root solve law, and slope identity make p6:eq:core-r1 exact. The full p6:lem:core remains partial because general r, its root determination, and the complex W_k reading are not formalized.
Linear--logarithmic remainder transport	FabiusFunction.RemainderTransport	Exhaustive zero-definition/four-theorem surface: lipschitzOn_of_abs_deriv_le, transport_bound_mul, transport_bound, and transport_first_order. The derivative-to-Lipschitz bridge and both forms of p0:eq:transport-bound are exact; transport_first_order also proves part (2) with an explicit error under the weaker Lipschitz hypothesis on the core derivative. Thus p0:thm:remainder-transport is Exact. The perturbation's two-sided derivative bound remains distinct from the core's one-sided derivative floor.
Residual-certificate backward-error existence	FabiusFunction.BackwardErrorExistence	Exhaustive one-definition/six-theorem surface: HasSlopeLowerBound, hasSlopeLowerBound_of_le_deriv, exists_eq_of_residual, injOn_of_hasSlopeLowerBound, existsUnique_eq_of_residual, exists_eq_and_abs_sub_le, and exists_eq_of_le_deriv. This makes q3:prop:transfer Exact by composition with MeanValueBracket; the slope formulation is weaker than differentiability, and the certified ball must fit inside [a,b].
Real Cayley function and kernel	FabiusFunction.CayleyTreeFunction, FabiusFunction.CayleyKernel	CayleyTreeFunction is 1+8: cayleyTree, principalLambertW_eq_neg_cayleyTree, cayleyTree_eq_mul_exp, cayleyTree_zero, cayleyTree_nonneg, cayleyTree_exp_neg_one, cayleyTree_strictMonoOn, cayleyTree_le_one, cayleyTree_unique. This proves the real functional, boundary, monotonicity, and branch-conditioned uniqueness clauses (w≤e⁻¹, unique c≤1), so p1:def:cayley remains Partial only at complex-disc analyticity. CayleyKernel is 1+14: cayleyKernelCoeff, factorial_pos_real, cayleyKernelCoeff_eq_factorial, cayleyKernelCoeff_zero, cayleyKernelCoeff_one, cayleyKernelCoeff_two, cayleyKernelCoeff_three, exp_eq_tsum_pow_div_factorial, tsum_exp_shift_one, tsum_exp_shift_two, cayleyKernelCoeff_mul_pow_eq_sub, summable_cayleyKernelTerm, tsum_cayleyKernelTerm_mul_sq, tsum_cayleyKernelTerm, tsum_cayleyKernelTerm_zero. These are exact real coefficient, convergence, closed-form, and removable-origin identities for p1:def:cayley-kernel, not a formal complex Analytic theorem.
Cayley local coordinate	FabiusFunction.CayleyLocalCoordinate	Exhaustive one-definition/seven-theorem surface: one_sub_mul_exp_le_one, cayleyUpsilon, cayleyUpsilon_zero, cayleyUpsilon_nonneg, cayleyUpsilon_eq_half_mul_kernel, cayleyUpsilon_eq_half_mul_sq_kernel, one_sub_upsilon_div_exp_one_le, and cayleyTree_eq_one_sub. The identity υ=½v²Φ(v), including v=0, and its nonnegative branch substitution are formal; p1:thm:omega remains Partial because square-root reversion, the Bell master formula, and the Puiseux expansion are absent.
Divisor transform and Lambert correction coefficients	FabiusFunction.DivisorTransform, FabiusFunction.LambertCorrectionEquation	DivisorTransform is 2+9: divisorTransform, divisorTransform_zero, divisorTransform_eq_add_properDivisors, divisorTransform_sub, two_mul_le_of_mem_properDivisors, disturbanceCoeff, disturbanceCoeff_eq_divisorTransform, disturbanceCoeff_one, disturbanceCoeff_congr, disturbanceCoeff_prime, disturbanceCoeff_nonneg. Thus p1:def:divisor is Exact; p1:lem:R-coefficients remains Partial at its composite numerical values, Otter recurrence, and tree sequence. LambertCorrectionEquation is 2+9: corrB, corrCoeff, corrCoeff_zero, corrCoeff_one, corrB_zero, hasDerivAt_corrB, hasDerivAt_corrB_zero, deriv_corrB_zero_pos, summable_corrExpTerm, tsum_corrExpTerm, corrB_eq_tsum. The explicit real analytic content of B is exact for `
Exponential-series recurrence and Touchard Euler operator	FabiusFunction.ExpSeriesRecurrence, FabiusFunction.TouchardEulerOperator	ExpSeriesRecurrence is 1+3: expCoeff, expCoeff_zero, succ_mul_expCoeff_succ, natCast_mul_expCoeff; its general formal recurrence is exact, but p1:lem:gamma-ratio remains Partial at Bernoulli data, explicit coefficients, degree, and all-orders Stirling asymptotics. TouchardEulerOperator is 2+8: touchardPolynomial_zero, coeff_touchardPolynomial, touchardPolynomial_succ_eq_X_mul_add_derivative, eulerOp, iterate_eulerOp_exp, exp_neg_mul_iterate_eulerOp_exp, eval_touchardPolynomial_succ_eq_sum_Icc, touchardQ, touchardQ_mul, touchardQ_one. This makes q2:def:touchard Exact, strengthened to j=0 and arbitrary real r; the normalized form assumes r≠0 and 1+r>0.
Endpoint two-term estimate and sharp Laplace-moment bounds	FabiusFunction.FabiusEndpointTwoTerm, FabiusFunction.LaplaceMomentBoundsSharp	FabiusEndpointTwoTerm is 0+2: fabiusReal_le_exp_endpoint, log_fabiusReal_le_endpoint; it makes plt:prop:mot-fabius-endpoint Exact for every 0<x≤1/4 in exponential and logarithmic form. LaplaceMomentBoundsSharp is 0+7: normalizedLaplaceMoment_three_le_sharp, normalizedLaplaceMoment_four_le_sharp, dyadicHigherLaplaceMoments_le_sharp, abs_negativeLaplaceTailError_nat_le_four_div, abs_dyadicEndpointLaplaceLogError_add_secondOrder_le_sharp, abs_dyadicEndpointLaplaceLogError_add_secondOrder_le_sharp', abs_log_fabius_dyadic_sub_cumulantMain_le_sharp. It retains the square-root denominator for the third/fourth moments, gives an O(n⁻³⁄²) higher term, and provides the explicit threshold 5329 and constants 3374/n and 40537/(12n); no optimality claim is made.
Lambert-shift concavity and certificates	FabiusFunction.LambertShiftConcavity	Exhaustive zero-definition/five-theorem surface: strictConcaveOn_lambertShift, strictConcaveOn_lambertShift_Ici_zero, abs_sub_lambertShiftInv_sandwich, residual_sign_agrees, and residual_eq_zero_iff. q1:prop:certificate is Exact for z,x≥0, including the sign and sandwich. q1:prop:brackets remains Partial only at the one-sided endpoint derivative values f'(0)=2 and g'(0)=1/2; strict concavity on the closed domain and all interior bounds are formal.
Least-term Gevrey index	FabiusFunction.LeastTermIndex	Exhaustive one-definition/six-theorem surface: gevreyTerm, gevreyTerm_succ, gevreyTerm_pos, gevreyTerm_succ_lt, gevreyTerm_lt_succ, gevreyTerm_antitoneOn, and gevreyTerm_monotoneOn. The exact ratio and unimodality mechanism is formal, while p0:thm:optimal-truncation remains Partial at its Stirling envelope, transported inverse error, and exponential floor.
Power--log block classes and Wright-omega monomial rigidity	FabiusFunction.TransseriesBlockClasses, FabiusFunction.TransseriesMonomialUniqueness, FabiusFunction.TransseriesWrightOmegaTerms	TransseriesBlockClasses is 0+3 and makes plt:prop:mot-blocks Exact without constructing a quotient type or full normal form. TransseriesMonomialUniqueness is now 0+4: tendsto_const_mul_plMonomial_div_one_iff, isEquivalent_const_mul_plMonomial_iff, tendsto_plMonomial_div_const_mul_one_iff, isEquivalent_plMonomial_const_mul_iff; the first two compare arbitrary nonzero coefficients. The exhaustive 0+10 Wright-omega leaf is plMonomial_one_zero_eventuallyEq, plMonomial_zero_one_eventuallyEq, plMonomial_neg_one_one_eventuallyEq, exponents_of_wrightOmega, exponents_of_wrightOmega_sub, exponents_of_wrightOmega_residual, not_pure_of_wrightOmega_three_terms, not_isEquivalent_pure_power_wrightOmega_sub, tendsto_wrightOmega_div_plMonomial_zero_atTop, isLittleO_wrightOmega_residual_plMonomial_zero. It proves the unique exponent/coefficient triples for the first three terms and both pure-scale failures; plt:cor:mot-both-generators-needed and plt:prop:mot-one-generator-fails are Exact for the stated real atTop formulations.
Smallest differential algebra	FabiusFunction.TransseriesDifferentialClosure	Exhaustive two-definition/nine-theorem surface: IsDerivationStable, isDerivationStable_top, isDerivationStable_iInf, isDerivationStable_adjoin, derivationOrbit, subset_derivationOrbit, derivation_mem_derivationOrbit, isLeast_adjoin_derivationOrbit, isLeast_adjoin_singleton_of_derivation_eq_one, isLeast_adjoin_pair_of_derivation_log, and isLeast_adjoin_triple_of_derivation_log. The algebraic minimality clauses (i)--(iii) of plt:thm:mot-smallest-differential-algebra are exact in an arbitrary commutative-ring/algebra derivation, and the generic orbit construction strengthens them. The full theorem remains Partial at the growth clause of (i) and the algebraic-independence/unique-representation clause (iv); part (v) is already exact through TransseriesDifferentialBlock.
Order-theoretic staircase recovery	FabiusFunction.StaircaseInversion	Exhaustive zero-definition/seven-theorem surface: isLeast_ceil, staircase_ceil, staircase_separation, staircase_separation_fails, staircase_round, isLeast_residue_class, and exists_half_error_of_jump. These formalize the five recovery clauses of p0:thm:staircase under the exact StrictMono/inverse hypotheses actually used and prove that the separation hypothesis cannot be removed. The analytic construction of admissible interpolations and the Fourier expansion of the periodic layer remain outside the module.
Subfactorial defect and nearest-integer derangements	FabiusFunction.DerangementNearestInteger	Exhaustive one-definition/seven-theorem surface: subfactorialDefect, subfactorialDefect_zero, subfactorialDefect_succ, subfactorialDefect_pos, subfactorialDefect_lt, numDerangements_sub_eq, abs_numDerangements_sub_lt_half, and round_factorial_mul_exp_neg_one. The integral recursion, signed defect identity, integer-argument bounds, and nearest-integer conclusion for every n≥1 are exact. The surrounding p8:cor:nearest-integer remains partial at the real-argument bound x>-1, and p8:thm:branch-splitting and its interpolation family remain unformalized.
Catalan quadratic-core coefficients	FabiusFunction.QuadraticCoreCatalan	Exhaustive three-definition/eight-theorem surface. Definitions: quadHalf, halfBinom, quadCoef. Theorems: catalan_two_step, quadHalf_zero, quadHalf_antidiagonal, halfBinom_step, quadHalf_rat, quadCoef_rat, quadCoef_zero, and quadCoef_rec. The Catalan closed form and convolution make p6:prop:quadratic-core-catalan Exact over characteristic-zero fields. p6:lem:quadratic-core is only Partial: Lean proves the coefficient family and coefficientwise recursion, not a packaged power-series existence/uniqueness theorem, square-root identity, or exact denominator-exponent clause. p6:thm:deepest-pole is Absent: no Gamma/Barnes identification is formalized.
Euler product transform with natural multiplicities	FabiusFunction.WeightedEulerTransform	Exhaustive public surface: summable_sigma_fin_iff, tprod_sigma_fin_eq_tprod_pow, tsum_sigma_fin_eq_tsum_nsmul, tprod_one_sub_pow_eq_cexp_powerSum. The base index is countable. The summability equivalence assumes a nonnegative real family; the product and sum transfers assume the corresponding sigma-indexed family is multipliable or summable. The Euler transform assumes ∀ i, ‖f i‖ < 1 and summability of i ↦ (c i : ℝ) * ‖f i‖; it is branch-free and asserts no logarithm-of-a-power or principal-log identity.
Natural-weight linearity and iterated shift--refinement of generalized Rvachev products	FabiusFunction.WeightLinearityProducts	Exhaustive public inventory: zero definitions and nine theorems, summable_weight_natMul, summable_weight_add, summable_weight_linearCombination, generalizedRvachevProduct_natMul, generalizedRvachevProduct_linearCombination, shiftExponent_iterate, summable_shiftExponent_iterate, generalizedRvachevProduct_two_pow_mul, and generalizedRvachevProduct_shift_factorization. For every natural weight a : ℕ → ℕ satisfying exactly Summable fun h : ℕ => (a h : ℝ) / 2 ^ h, every m : ℕ, and every z : ℂ, the all-depth law is Φ_a(2^m z) = (∏ h ∈ range m, complexSinc(π * (2^(m-h) * z))^(a h)) * Φ_(S^m a)(z), where S^m a(h) = a(h+m) and the shifted weight remains admissible. The formula is global in z; at m = 0 its prefix is the empty product and it reduces to reflexivity, so it also includes the zero-depth and zero-frequency boundaries without side conditions. The linear-combination and finite-difference interfaces use natural coefficients and natural-valued admissible component weights satisfying the displayed Newton reconstruction: they turn every such nonnegative weight identity into a product identity, but do not claim a signed-exponent, analytic-germ, or pole-cancellation theorem.
Dyadic order-divisor identifiability for generalized Rvachev products	FabiusFunction.GeneralizedRvachevIdentifiability	Exhaustive public inventory: zero definitions and six theorems, weightSequence_eq_of_weightedScaleMultiplicity_base_pow_eq, analyticOrderAt_generalizedRvachevProduct_two_pow, exponent_zero_eq_toNat_analyticOrderAt_generalizedRvachevProduct, exponent_succ_eq_toNat_analyticOrderAt_generalizedRvachevProduct, exponentSequence_eq_of_analyticOrderAt_two_pow_eq, and generalizedRvachevProduct_eq_iff. The arithmetic theorem works over every additive cancellative commutative weight monoid and every base b > 1: equality of weighted multiplicities at all powers b^n forces equality of the weight sequences. For natural exponent sequences satisfying exactly Summable fun h : ℕ => (a h : ℝ) / 2 ^ h, the analytic order of Φ_a at 2^n is the inclusive prefix through n; a 0 is read from the order at 1, later exponents are consecutive differences after ENat.toNat, equality of every dyadic order determines the sequence, and two admissible entire products are equal exactly when their exponent sequences are equal. The identifying datum is the multiplicity/order divisor. A bare zero set, or merely the product values at those zero points, is insufficient—for example, a and 2 • a have the same zero set—and identifiability from spectral-zeta data, cumulant samples, or a generalized probability law remains open.
General-weight Euler--zeta expansion of the generalized sinc product	FabiusFunction.GeneralizedSincZeta	Exhaustive public surface: weightedScaleSeries, summable_weightedScaleSeries_real, summable_weightedScaleSeries, tsum_weighted_div_two_pow_even_pow, weighted_sinc_pair_powerSum, generalizedRvachevProduct_eq_cexp. The series definition is total in the natural weight a and natural index k. Every theorem assumes admissibility Summable fun h : ℕ => (a h : ℝ) / 2 ^ h; the two scale-series summability results and the weighted scale-collapse theorem additionally require k ≠ 0. The pair power-sum theorem holds for every complex z and natural r, while the product expansion assumes exactly ‖z‖ < 1. This is an analytic exponential expansion, not a principal-log, characteristic-function, probabilistic-cumulant, or support theorem.
Alternating Newton Euler--zeta kernel	FabiusFunction.AlternatingNewtonCumulantKernel	Exhaustive public surface: tsum_alternatingNewtonWeight_inv_four_pow, weightedScaleSeries_alternatingNewton, alternatingNewton_eq_cexp. The two kernel evaluations hold for every natural d and require k ≠ 0; the exponential theorem holds for every natural d under exactly ‖z‖ < 1. The natural weight exists for every d, but agreement with the source volume's signed generalized-binomial convention requires even d. These are analytic identities only, with no characteristic-function, probabilistic-cumulant, or variance interpretation.
Nonmonic Hensel lifting and formal implicit roots	FabiusFunction.ImplicitPowerSeries	FormalImplicitRoot.exists_isRoot_sub_mem, FormalImplicitRoot.eq_of_isRoot_of_sub_mem, FormalImplicitRoot.existsUnique_isRoot_sub_mem, PowerSeries.Implicit.existsUnique_isRoot_constantCoeff, PowerSeries.Implicit.existsUnique_zeroConstant_root, PowerSeries.Implicit.root, PowerSeries.Implicit.constantCoeff_root, PowerSeries.Implicit.eval_root, PowerSeries.Implicit.eq_root; this is a generic formal-series root engine over complete adic commutative rings and arbitrary commutative coefficient rings, with no concrete inverse-Fabius germ, analytic convergence, plateau localization, flat-remainder, or quantile theorem
Quarter Catalan formal germ and dyadic-rescaling bridge	FabiusFunction.QuarterCatalanGerm	Exhaustive public surface (two definitions and thirteen theorems): quarterCatalanCoefficient, quarterCatalanCoefficient_zero, quarterCatalanCoefficient_succ_eq_report, quarterCatalanGermSeries, quarterCatalanGermSeries_coeff, quarterCatalanGermSeries_coeff_succ, quarterCatalanGermSeries_constantCoeff, quarterCatalanGermSeries_equation, powerSeries_quadratic_injectiveOn_zeroConstant, eq_quarterCatalanGermSeries_of_equation, existsUnique_quarterCatalanGermSeries, dyadicGermTwo_functionalEquation, rescale_dyadicGermTwo_eq_quadraticInverse, dyadicGermTwo_eq_rescale_quadraticInverse, coeff_dyadicGermTwo_succ. The explicit Catalan coefficient sequence and its rational power series give the unique zero-constant solution of D + 4D² = (4/9)X. Rescaling the dyadic parameter by 9/4 identifies the distinguished dyadic germ exactly with the Catalan inverse of X + 4X², and every positive coefficient is (4/9)^(m+1) (-4)^m C_m. This module is formal power-series algebra only; the downstream actual-jet bridge is supplied separately.
Actual quarter inverse Catalan jet	FabiusFunction.FabiusInverseQuarterJet	Exhaustive public surface: iteratedDeriv_centeredFabiusInv_quarter_eq_quadraticInverse, iteratedDeriv_fabiusInv_five_seventy_two_succ. For every bounded Fabius solution, the full centered derivative jet at 5/72 = F(1/4) equals the factorial-scaled coefficient sequence of QuadraticInverse.inverse 4; in particular G^(m+1)(5/72) = (m+1)! (-4)^m C_m. This is equality of all jets, not local analytic equality: it neither erases the known nonanalytic flat defect nor proves that defect is nonzero by a named remainder theorem.
Finite polynomial integrals from raw moments and formal cumulants	FabiusFunction.PolynomialExpectationCumulant	integral_eval₂_eq_sum_moment, integral_eval₂_eq_sum_completeBell_momentCumulant_with_mass_correction, integral_eval₂_eq_sum_completeBell_momentCumulant_of_moment_zero_eq_one, integral_eval₂_eq_sum_completeBell_momentCumulant
Rvachev raw moments, centered Appell convolution, and triangular injective polynomial deconvolution	FabiusFunction.RvachevMomentAppell	Exhaustive public surface: six definitions and exactly 33 theorems, enumerated below in source order. It packages rational raw and reciprocal moments, rational and real monic Appell families of exact degree, and coefficientwise deconvolution as an injective real linear map preserving the top coefficient, natural degree, and leading coefficient. Smoothing recovers every polynomial in both the additive and centered x-y forms; positive-degree Appell polynomials have Rvachev mean zero. It proves no analytic reciprocal-MGF or Appell generating-series identity, literal differential-operator expansion, parity theorem for the reciprocal/deconvolution families, or displayed low-coefficient table.
Exact shifted-up polynomial synthesis, including arbitrary-phase self-sampling	FabiusFunction.RvachevPolynomialSynthesis	Exhaustive public surface: zero definitions and exactly five theorems, tsum_rvachevDeconvolvedPolynomial_mul_shifted_rvachevUp, normalized_tsum_rvachevDeconvolvedPolynomial_mul_shifted_rvachevUp, sum_Ioo_rvachevDeconvolvedPolynomial_mul_shifted_rvachevUp, normalized_sum_Ioo_rvachevDeconvolvedPolynomial_mul_shifted_rvachevUp, and normalized_tsum_shifted_rvachevDeconvolvedPolynomial_mul_rvachevUp. For every nonzero natural mesh M and P.natDegree ≤ v₂(M), the first four give global and exact finite k ∈ (-2M,2M) synthesis on [-1,1]; the fifth reconstructs P.eval x for arbitrary real phase and real x. At M=2^N this arbitrary-phase layer reaches every degree at most N; the adjacent parity-selected layer adds one degree at its selected phases.
Parity-selected one-extra-degree Rvachev quadrature and Appell synthesis	FabiusFunction.RvachevSuperconvergentSynthesis	Exhaustive public surface: one definition, IsRvachevSuperconvergentPhase, and exactly eight theorems, isRvachevSuperconvergentPhase_two_pow_iff, tsum_quarter_monomial_eq_integral_of_even_deg, tsum_three_quarters_monomial_eq_integral_of_even_deg, tsum_shifted_monomial_eq_integral_superconvergent, tsum_shifted_polynomial_eq_integral_superconvergent, integral_polynomial_mul_rvachevUp_eq_normalized_tsum_superconvergent, normalized_tsum_shifted_rvachevDeconvolvedPolynomial_mul_rvachevUp_superconvergent, and normalized_tsum_shifted_rvachevAppellPolynomial_mul_rvachevUp_superconvergent. For every nonzero natural mesh M, the selected endpoint or quarter phases give exactness through degree v₂(M)+1, physical-coordinate quadrature, deconvolved-polynomial reconstruction, and the Appell monomial specialization. On M=2^N, even N selects 0,1/2 and odd N selects 1/4,3/4. This generic-mesh theorem is stronger than the dyadic manuscript form. The predicate records exact real representatives, not integer translates or a complete classification; no maximality, positivity, or rationality theorem is claimed.
Shifted dyadic polynomial comb exactness and normalized self-sampling quadrature	FabiusFunction.PolynomialCombExactness	Exhaustive public surface: zero definitions and exactly three theorems, finite_support_comb, tsum_shifted_polynomial_eq_integral, and integral_polynomial_mul_rvachevUp_eq_dyadic_tsum. For every bounded Fabius solution, natural level m, real phase theta, and arbitrary real weight function g, the sampled product g(theta+k) * up(2^-m * (theta+k)) has finite integer support. Every real polynomial P with P.natDegree <= m therefore satisfies the corresponding whole-line shifted comb identity. In physical coordinates, for every natural N, arbitrary real phase, and P.natDegree <= N, its integral against up equals 2^-N times the integer sum over nodes 2^-N * (theta+k) weighted by up at those same nodes. The statements include level zero; the sums are finite by compact support, and no phase rationality, positivity, infinite-support convergence, or optimal-mesh claim is imposed.
Generic finite-node Lagrange--Rvachev decoder, cardinal biorthogonality, and exact interpolation loop	FabiusFunction.LagrangeRvachevSynthesis	Exhaustive public surface: two definitions, lagrangeRvachevDecoder and lagrangeRvachevAtomCoefficient; and seven theorems, natDegree_lagrangeBasis_le_card_sub_one, natDegree_lagrangeInterpolate_le_card_sub_one, normalized_sum_Ioo_lagrangeRvachevDecoder_mul_shifted_rvachevUp, normalized_sum_Ioo_lagrangeRvachevDecoder_eval_node, lagrangeRvachevAtomCoefficient_eq_deconvolved_interpolate, sum_Ioo_lagrangeRvachevAtomCoefficient_mul_shifted_rvachevUp, and sum_lagrangeRvachevDecoder_eq_one. The degree bounds and polynomial reconstruction need no distinct-node hypothesis; componentwise Kronecker biorthogonality requires distinct nodes and evaluation inside [-1,1], while the row-sum theorem additionally requires a nonempty node set. This closes the reusable generic finite-node synthesis loop, not a geometric Gaussian closed-form decoder, bundled matrix/right-inverse wrapper, or optimal/minimum-variation decoder theorem.
Finite Appell--Hasse calculus and the geometric Gaussian decoder	FabiusFunction.RvachevAppellHasse	Exhaustive public surface: one definition, Appell.polynomialTransform, and fourteen theorems, Appell.polynomialTransform_apply, Appell.polynomialTransform_monomial, Appell.polynomialTransform_eq_sum_hasseDeriv_of_natDegree_lt, Appell.polynomialTransform_eq_sum_hasseDeriv, rvachevReciprocalMomentRat_odd, rvachevDeconvolutionLinearMap_eq_appellPolynomialTransform, rvachevDeconvolvedPolynomial_eq_sum_even_hasseDeriv, eval_hasseDeriv_prod_X_sub_C_eq_elementarySymmetricEval, eval_rvachevDeconvolvedPolynomial_prod_X_sub_C, eval_rvachevDeconvolvedPolynomial_qFallingPower, lagrangeBasis_eq_nodalWeight_mul_prod_X_sub_C, lagrangeRvachevDecoder_eq_nodalWeight_mul_sum, geometric_nodalWeight_eq_geometricQPochhammer, and geometric_lagrangeRvachevDecoder_eq. Over a commutative semiring the transform replaces monomials by an arbitrary Appell family and equals a finite Hasse-derivative sum; the Rvachev specialization proves odd reciprocal moments vanish and reduces deconvolution to even Hasse derivatives. Taylor/Vieta then give the complementary elementary-symmetric formula for arbitrary root products and the exact q-falling formula. The field-level Lagrange factorization and real geometric specialization give the full Gaussian q-Pochhammer prefactor times the finite even reciprocal-moment sum. These algebraic formulas are total even at zero or colliding nodes, where inverse-defined Lagrange bases are not cardinals; the manuscript's cardinal use retains c>0 and 0<q<1. Composed with normalized_sum_Ioo_rvachevDeconvolvedPolynomial_mul_shifted_rvachevUp and the generic Lagrange synthesis theorem, they make both gq:prop:q-Appell-falling and gq:thm:gaussian-Appell-decoder Exact. Here the coefficients rvachevReciprocalMomentRat are the formal reciprocal-moment sequence: no analytic reciprocal-MGF convergence is claimed. Atom reconstruction remains supplied by the separate synthesis API, and no larger matrix right-inverse or decoder-optimality assertion is added.
Nodes-only Lagrange--Rvachev amplitudes (cor:lag-nodes-only)	FabiusFunction.RvachevLagrangeNodesOnly	Exhaustive public surface: one definition, rvachevDeconvolvedPolynomialRat, and fourteen theorems, map_rvachevDeconvolvedPolynomialRat, rvachevDeconvolvedPolynomial_eq_sum_appell, eval_rvachevDeconvolvedPolynomial_eq_sum_even_iterateDerivative, rvachevDeconvolvedPolynomial_prod_X_sub_C_eq_sum_appell, eval_rvachevDeconvolvedPolynomial_lagrangeBasis_eq_sum_even_iterateDerivative, eval_rvachevDeconvolvedPolynomial_lagrangeBasis_eq_nodalWeight_mul_sum_appell, lagrangeRvachevDecoder_eq_nodalWeight_mul_sum_appell, map_lagrangeBasis_ratCast, map_rvachevDeconvolvedPolynomialRat_lagrangeBasis, lagrangeRvachevDecoder_eq_ratCast, rvachevRawMomentRat_eq_centeredRvachevFullMoment, momentCumulant_rvachevRawMomentRat_eq_centeredRvachevFullCumulant, momentCumulant_rvachevRawMomentRat_even_eq_bernoulliMersenne, and rvachevReciprocalMomentRat_eq_completeBellPolynomial_neg_centeredCumulant. By composition these give cor:lag-nodes-only an Exact/Complete counterpart: the ordinary-derivative formula assumes Set.InjOn v s, the raw omitted-node elementary-symmetric/Appell form uses Lagrange.nodalWeight, and rational nodes give a polynomial over ℚ whose real cast is the decoder polynomial. Value rationality is asserted only at rational evaluation points, including k/M, not arbitrary irrational reals; the lattice equality is total at M=0, while reconstruction retains its separate nonzero/admissible-mesh hypotheses. The Bell identity is formal coefficient algebra, not analytic reciprocal-MGF convergence; odd-cumulant vanishing comes from the pre-existing parity theorem. No single wrapper theorem is claimed. Independently, the two exact synthesis declarations in LagrangeRvachevSynthesis make thm:lag-cardinal Exact/Complete by assembly; the larger compound thm:lag-right-inverse, global atom synthesis, and decoder optimality remain unpromoted.
Typed finite Lagrange--Rvachev Matrix/Markov layer and overlap sign obstruction	FabiusFunction.LagrangeRvachevMatrix	Exhaustive public surface: four definitions/abbreviations, rvachevAtomIndexSet, RvachevAtomIndex, lagrangeRvachevEncoderMatrix, and lagrangeRvachevDecoderMatrix; and six theorems, lagrangeRvachevEncoderMatrix_nonneg, sum_lagrangeRvachevEncoderMatrix_row_eq_one, sum_lagrangeRvachevDecoderMatrix_row_eq_one, lagrangeRvachevEncoderMatrix_mul_decoderMatrix, exists_neg_entry_of_rightInverse_of_row_overlap, and exists_lagrangeRvachevDecoderMatrix_entry_neg_of_row_overlap. The atom type is the exact integer block Ioo (-(2*M)) (2*M). The normalized encoder has entries M⁻¹ * rvachevUp F (v i - k/M), is entrywise nonnegative without distinctness or M ≠ 0, and has unit row sums for IsFabius F, M ≠ 0, and in-range nodes. The sampled decoder has unit row sums for a nonempty distinct node family, and encoder * decoder = 1 when the nodes are distinct and in [-1,1], M ≠ 0, and s.card - 1 ≤ padicValNat 2 M. Generically, a nonnegative matrix with a row-unital right inverse has a negative decoder entry whenever one column is strictly positive in two distinct rows; the Rvachev specialization remains conditional on such an overlap, and encoder row normalization is not used in the generic obstruction. No decoder * encoder projector range/kernel, rank, trace, characteristic-polynomial, Cauchy--Binet, geometric closed-form, or decoder-optimality theorem is asserted.
Sharp universal composite-mesh exactness and least natural meshes	FabiusFunction.CompositeMeshSharpness	Exhaustive public surface: exists_shift_tsum_shifted_monomial_ne_integral_nat_real, rvachevCombExactThrough, rvachevCombExactThrough_iff_padicValNat, rvachevCombExactThrough_iff_pow_two_dvd, rvachevCombExactThrough_two_pow, two_pow_le_of_rvachevCombExactThrough, isLeast_rvachevCombExactThrough, isLeast_rvachevCombExactThrough_even. The IsLeast results quantify over meshes exact for the whole real polynomial space through the stated degree; they do not assert minimality for an individual Legendre polynomial, a fixed Legendre partial sum, or a target-adapted mesh.
Universal endpoint-transfer polynomials and their formal exponential series	FabiusFunction.EndpointTransferPolynomials	endpointTransferPolynomial_succ, endpointTransferPolynomial_eq_partitionExpSum, endpointTransferSeries_eq_exp_subst, aeval_endpointTransferPolynomial, map_endpointTransferSeries
Finite base-b layer regrouping in multiplicative and additive form	FabiusFunction.BaseLayerRegrouping	filter_dvd_eq_image, prod_multiples_eq_prod_filter, sum_multiples_eq_sum_filter, prod_layers_eq_prod_pow_card, sum_layers_eq_sum_nsmul_card, card_filter_pow_dvd, prod_layers_eq_prod_pow_multiplicity, sum_layers_eq_sum_nsmul_multiplicity
Complete homogeneous evaluations, finite formal generating series, Bell/power-sum conversion, fixed-degree asymptotic bounds, denominator-free geometric principal specialization, and a second proof of Gaussian symmetry	FabiusFunction.CompleteHomogeneous, FabiusFunction.CompleteHomogeneousGenerating, FabiusFunction.CompleteHomogeneousBell, FabiusFunction.CompleteHomogeneousAsymptotics, FabiusFunction.GeometricCompleteHomogeneous	CompleteHomogeneousBell exhaustively exports completeHomogeneousPowerSum, completeHomogeneousBellInput, completeHomogeneousEvalOn_insert_eq_sum, bellComplete_completeHomogeneousBellInput, factorialNormalize_completeBellPolynomial_completeHomogeneousBellInput, and completeHomogeneousEvalOn_eq_factorialNormalize_completeBellPolynomial. Its backbone is division-free over every commutative semiring: Bell.complete κ n = n! * h_n; only the normalized h_n = B_n/n! form uses a commutative ℚ-algebra. Empty alphabets, repeated or zero entries, zero divisors, positive characteristic, and the zero ring are included. GeometricCompleteHomogeneous exhaustively exports six theorems: completeHomogeneousEval_geometric, completeHomogeneousEval_scaled_geometric, completeHomogeneousEvalOn_range_pow_eq_gaussianBinomial, completeHomogeneousEvalOn_range_pow_eq_gaussianBinomial_degree, gaussianBinomial_add_symm, and gaussianBinomial_symm_via_completeHomogeneous. The principal specializations and both symmetry proofs need no distinctness, division, cancellation, ordering, topology, or convergence assumptions; the separate generating identities are purely formal, while the asymptotic theorem transfers coordinatewise Big-O through every fixed homogeneous degree.
Fixed-column Stirling series and complete-homogeneous coefficients	FabiusFunction.StirlingCompleteHomogeneous	Focused verification passed. The exhaustive zero-definition/eight-theorem surface is stirlingColumnOGF_eq_completeHomogeneousGeneratingSeriesOn, stirlingSecond_add_eq_completeHomogeneousEvalOn, stirlingSecond_eq_completeHomogeneousEvalOn_of_le, stirlingSecond_add_eq_completeHomogeneousEval, stirlingSecond_add_eq_eval_hsymm, stirlingSecond_add_eq_sum_finsuppAntidiag, pow_mul_descPochhammer_eval_inv_eq_prod_one_sub_natCast_mul, and prod_inv_one_sub_natCast_mul_eq_inv_pow_mul_descPochhammer_eval_inv. The formal-series identity is over commutative rings, coefficient forms are over commutative semirings, and the falling-factorial normalization is over fields with x ≠ 0. The API includes k = 0, requires k ≤ n for the n-k spelling, uses Lean's totalized inverse in the reciprocal theorem, and asserts no analytic convergence.
Infinite products at summable scales	FabiusFunction.ScaledInfiniteProducts	summable_norm_scaled_sub_one, hasProdUniformlyOn_scaled, multipliableUniformlyOn_scaled, hasProdLocallyUniformly_scaled, multipliableLocallyUniformly_scaled, continuous_tprod_scaled, differentiable_tprod_scaled, differentiable_tprod_scaled_of_eq_one, tprod_scaled_ne_zero, tprod_scaled_eq_zero_iff; pointwise deviation summability allows an arbitrary normed-ring target, the compact-uniform API assumes a continuous factor and a complete commutative normed-ring target with a normed unit, local uniformity adds local compactness, holomorphy uses a complete normed complex-algebra target, and zero detection adds a multiplicative norm but needs neither continuity nor local compactness
Geometric reciprocal-Gamma products and the dyadic Rvachev bridge	FabiusFunction.GeometricReciprocalGamma	Exhaustive public surface (six definitions and 23 theorems, 29 declarations): shiftedReciprocalGamma, shiftedReciprocalGamma_zero, shiftedReciprocalGamma_differentiable, shiftedReciprocalGamma_sub_one_isBigO, shiftedReciprocalGamma_eq_zero_iff, shiftedReciprocalGamma_mul_neg, summable_norm_qpow, geometricReciprocalGamma, geometricReciprocalGammaFactors_multipliable, geometricReciprocalGamma_differentiable, geometricReciprocalGamma_zero, geometricReciprocalGamma_mahler, geometricReciprocalGamma_eq_zero_iff, geometricGamma, geometricGamma_meromorphic, geometricGamma_mahler, geometricSincProduct, hasProdLocallyUniformly_geometricSincProduct, geometricSincProductFactors_multipliable, hasProd_geometricSincProduct, geometricReciprocalGamma_mul_neg, geometricSincProduct_differentiable, dyadicReciprocalGamma, dyadicGamma, dyadicReciprocalGamma_differentiable, dyadicReciprocalGamma_zero, geometricSincProduct_inv_two, dyadicReciprocalGamma_mul_neg, rvachevFourierProduct_eq_one_div_dyadicGamma_mul. For complex q with ‖q‖ < 1, including q = 0, the sinc factors now have a named locally uniform product, pointwise Multipliable and HasProd forms, and an entire geometricSincProduct; the reciprocal-Gamma product retains its Mahler, zero, meromorphic-inverse, reflection, and dyadic bridge laws. Normalization at zero is unconditional. geometricGamma and dyadicGamma are totalized pointwise inverses, not proved raw Gamma tprods away from poles.
Complex infinite q-Pochhammer convergence and global geometric-sinc spectral factorization	FabiusFunction.RvachevPochhammerFactorization	Exhaustive public surface (one definition and ten theorems): complexQPochhammerInf; complexQPochhammerInf_eq_tprod, complexQPochhammerInf_eq_qPochhammerInfIn, multipliable_one_sub_mul_pow_complex, hasProd_complexQPochhammerInf, tendsto_finiteQPochhammerIn_complex, summable_norm_sineTerm_qpow_pair, geometricSincProduct_eq_tprod_pair, geometricSincProduct_eq_tprod_complexQPochhammerInf, rvachevFourierProduct_eq_tprod_complexQPochhammerInf, and rvachevFourier_eq_tprod_complexQPochhammerInf. The bridge to the generic symbol is unconditional. Product convergence requires exactly ‖q‖ < 1; the global spectral factorization includes q = 0 and individual zero factors. The factorization module alone supplies no centered characteristic/MGF wrapper or reciprocal outside-disk theory; outer-product normal convergence is the separate downstream API in GeometricPochhammerNormalConvergence.
Entire complex infinite q-Pochhammer products and simple factor zeros	FabiusFunction.QPochhammerEntire	Exhaustive public surface: zero definitions and exactly five theorems, hasProdLocallyUniformly_complexQPochhammerInf, complexQPochhammerInf_differentiable, complexQPochhammerInf_eq_zero_iff, complexQPochhammerInf_eq_zero_iff_eq_inv_pow, analyticOrderAt_complexQPochhammerInf_of_eq_zero. For every fixed strict complex contraction ‖q‖ < 1, the defining factors converge locally uniformly in a, the limit is entire, and its zeros are exactly the displayed factor zeros. If q ≠ 0, these are exactly a = (q^j)⁻¹; without that hypothesis the division-free form includes q = 0, whose sole zero is a = 1. Every zero has analytic order one; the generic-name analytic-order theorem is canonically owned by QPochhammerInfinite. This leaf does not assert joint analyticity in (a,q), reciprocal-nome continuation for `
Finite q-Pochhammer residue-class dissections	FabiusFunction.QPochhammerDissection	Exhaustive public surface: zero definitions and exactly two theorems, finiteQPochhammerIn_dissection and finiteQPochhammerIn_dissection_remainder. Both hold over an arbitrary [CommRing R], including the zero ring, for arbitrary a, q, r, and n, with no division, cancellation, nonvanishing, contraction, or positivity hypothesis. The exact-multiple identity (a;q)_(r*n) = product_(s<r) (a*q^s;q^r)_n is total even at r = 0. The remainder identity assumes exactly u <= r and gives the first u residue classes length n+1 and the remaining classes length n; it includes both endpoints u = 0 and u = r.
General infinite q-Pochhammer convergence, dissection, zero set, parameter regularity, and simple-zero derivatives	FabiusFunction.QPochhammerInfinite	Exhaustive public surface: one definition, qPochhammerInfIn, and exactly 29 theorems, qPochhammerInfIn_eq_tprod, summable_norm_mul_pow, one_sub_ne_zero_of_norm_lt_one, norm_mul_pow_self_lt_one, finiteQPochhammerIn_self_ne_zero, multipliable_one_sub_mul_pow_of_norm_lt_one, hasProd_qPochhammerInfIn, tendsto_finiteQPochhammerIn_qPochhammerInfIn, qPochhammerInfIn_eq_finite_mul_shift, qPochhammerInfIn_succ_shift, qPochhammerInfIn_eq_factor_mul, qPochhammerInfIn_dissection, qPochhammerInfIn_ne_zero, qPochhammerInfIn_eq_zero_iff, qPochhammerInfIn_self_ne_zero, qPochhammerInfIn_eq_tprod_smul, summable_norm_pow_of_norm_lt_one, isBigO_one_sub_sub_one, differentiable_finiteQPochhammerIn, qPochhammerInfIn_eq_zero_iff_exists_inv_pow, hasProdLocallyUniformly_qPochhammerInfIn, continuous_qPochhammerInfIn, pow_sq_mul_finiteQPochhammerIn_inv_pow_self, differentiable_qPochhammerInfIn, hasDerivAt_qPochhammerInfIn_of_mul_pow_eq_one, hasDerivAt_qPochhammerInfIn_inv_pow, deriv_qPochhammerInfIn_inv_pow_ne_zero, deriv_qPochhammerInfIn_ne_zero_of_mul_pow_eq_one, and analyticOrderAt_qPochhammerInfIn_of_eq_zero. The total definition and its defining-product equality require only [CommRing R] [TopologicalSpace R]; outside the convergence hypotheses the tprod retains Mathlib's junk-value convention. The three elementary norm lemmas use exactly [NormedRing R] [NormOneClass R] and the displayed strict norm bound, while finite self-product nonvanishing uses [NormedCommRing R] [NormOneClass R] [NoZeroDivisors R], ‖q‖ < 1, and no completeness. The seven convergence and factorization theorems use [NormedCommRing R] [NormOneClass R] [CompleteSpace R] and ‖q‖ < 1; only the infinite residue dissection additionally assumes 0 < r. The three zero-set theorems add exactly [NormMulClass R]; qPochhammerInfIn_ne_zero assumes forall j, a*q^j != 1, while the exact zero iff and self-nonvanishing need no further side condition. Over [NormedField 𝕜], the scaled-product equality and factor Big-O estimate are unconditional and scale summability assumes ‖q‖ < 1; finite-symbol differentiability needs exactly [NontriviallyNormedField 𝕜]. Over [NormedField 𝕜] [CompleteSpace 𝕜], the inverse-power zero classification assumes q != 0 and ‖q‖ < 1, whereas local uniform convergence and continuity assume ‖q‖ < 1 plus [LocallyCompactSpace 𝕜]. The finite cofactor identity needs only [Field 𝕜] and q != 0. Over ℂ, global differentiability assumes only ‖q‖ < 1; the factor-zero derivative and its nonvanishing additionally assume exactly a0*q^j = 1, including the q = 0 boundary; the inverse-power derivative formula and coefficient nonvanishing additionally assume q != 0; and every zero has analytic order one under the strict contraction. No joint regularity in (a,q), reciprocal-nome continuation, or meromorphic reciprocal theorem is asserted.
Normal convergence of the outer geometric Pochhammer factorization	FabiusFunction.GeometricPochhammerNormalConvergence	Exhaustive zero-definition/three-theorem surface: hasProdLocallyUniformly_geometricSincProduct_complexQPochhammerInf, hasProdLocallyUniformly_rvachevFourierProduct_complexQPochhammerInf, and hasProdLocallyUniformly_rvachevFourier_complexQPochhammerInf. The first assumes exactly a complex strict contraction ‖q‖ < 1 and gives locally uniform convergence on all of ℂ of the outer factors (z²/(k+1)²;q²)_∞ to geometricSincProduct q, including q = 0. The second is the unconditional dyadic nome-1/4 specialization; the third assumes exactly F : BoundedFabius and IsFabius F to identify the limit with its Fourier transform. No joint normality in (q,z), uniformity up to ‖q‖ = 1, outside-disk continuation, or centered MGF wrapper is asserted.
Exact dyadic reciprocal-Gamma zeros and meromorphic pole orders	FabiusFunction.DyadicGammaOrder	dyadicReciprocalGamma_eq_zero_iff, dyadicReciprocalGamma_int_ne_zero_of_nonneg, dyadicReciprocalGamma_nat_ne_zero, dyadicGamma_meromorphic, analyticOrderAt_dyadicReciprocalGamma_int_of_neg, analyticOrderAt_dyadicReciprocalGamma_neg_nat, meromorphicOrderAt_dyadicGamma_int_of_neg, meromorphicOrderAt_dyadicGamma_neg_nat; integer order statements assume a negative center, natural wrappers assume a nonzero index, and negative meromorphic order is Mathlib's encoding of a pole
Elementary evaluations, weighted Pascal, and elementary--complete orthogonality	FabiusFunction.SymmetricFunctionOrthogonality	Exhaustive public surface: elementarySymmetricEval, elementarySymmetricEval_eq_eval_esymm, elementarySymmetricEval_zero, elementarySymmetricEval_comp_equiv, elementarySymmetricEval_option_succ, elementarySymmetricEval_fin_succ, sum_elementarySymmetricEval_mul_completeHomogeneousEval. The definition, Mathlib identification, zero-degree law, reindexing law, and both adjoining-variable recurrences hold over every commutative semiring. Together with the existing completeHomogeneousEval_option_succ, the option recurrence is the exact weighted-Pascal pair. The final alternating elementary--complete convolution is the Kronecker delta over every commutative ring, for every degree and every finite family, including degree zero and the empty family; no division, characteristic, domain, or nonvanishing assumption is used.
Generic finite lower-triangular transforms	FabiusFunction.FiniteTriangularTransform	Exhaustive public surface (one definition and one theorem): lowerTriangularTransform, lowerTriangularTransform_comp. Over [Semiring R] [AddCommMonoid M] [Module R M], with no commutativity assumption on R, a kernel acts by the finite interval sum over Icc 0 n; a total ordered-convolution identity on every pair n,j gives equality of the composite transform with the original sequence as functions. When n < j the required interval is empty. This is the generic finite engine reused by both q-binomial and symmetric-function inversion; it uses no subtraction, topology, or infinite sum.
Weighted elementary--complete transforms and inversion	FabiusFunction.SymmetricFunctionTransform	Exhaustive public surface (four definitions and five theorems): completeHomogeneousKernel, signedElementaryKernel, completeHomogeneousKernel_left_orthogonality, completeHomogeneousKernel_right_orthogonality, completeHomogeneousTransform, signedElementaryTransform, signedElementaryTransform_completeHomogeneousTransform, completeHomogeneousTransform_signedElementaryTransform, weightedSymmetricFunction_inversion. The complete kernel and transform are defined over a commutative semiring; the signed kernel, both total-Icc convolution identities, both whole-sequence inverse equalities, and the inversion iff use [CommRing R] [AddCommMonoid M] [Module R M] and an arbitrary finite weight family. Both kernels are zero-extended above the diagonal and the compositions reuse the generic finite triangular theorem. The sums are finite, so no division, nonvanishing, characteristic, domain, topology, or convergence hypothesis is present.
Formal generating series for finite symmetric alphabets and the reciprocal finite q-binomial theorem	FabiusFunction.SymmetricFunctionGenerating	Exhaustive public surface (two definitions and six theorems): elementarySymmetricGeneratingSeries, completeHomogeneousGeneratingSeries, coeff_elementarySymmetricGeneratingSeries, coeff_completeHomogeneousGeneratingSeries, elementarySymmetricGeneratingSeries_eq_prod, elementarySymmetricGeneratingSeries_neg_mul_completeHomogeneousGeneratingSeries, completeHomogeneousGeneratingSeries_eq_invOfUnit_elementarySymmetricGeneratingSeries_neg, prod_one_sub_qPow_X_mul_gaussianBinomialGeneratingSeries. Both definitions, both coefficient results, and the elementary-product theorem hold over every commutative semiring; reciprocity, the canonical PowerSeries.invOfUnit identification, and the geometric-alphabet Gaussian reciprocal identity hold over every commutative ring, including the empty alphabet, n = 0, singular values of q, positive characteristic, and zero divisors. These are formal-power-series identities only: no analytic evaluation, radius of convergence, or complex convergence claim is made.
Every residual moment of finite interpolation and geometric Richardson rows	FabiusFunction.LagrangeResidualMoments, FabiusFunction.GeometricResidualMoments	sum_weight_mul_pow_card_add, sum_lagrangeEvalWeight_mul_pow_card_add, sum_lagrangeEvalWeight_mul_pow_card_add_zero; the exhaustive zero-definition/nine-theorem GeometricResidualMoments surface is sum_weight_mul_geometric_pow_succ_add, sum_weight_mul_geometric_pow_of_pos, sum_weight_mul_scaled_geometric_pow_succ_add, sum_weight_mul_scaled_geometric_pow_of_pos, sum_geometricLagrangeWeight_mul_pow_succ_add, sum_geometricLagrangeWeight_mul_pow_of_pos, sum_geometricLagrangeWeight_mul_scaled_geometric_pow_of_pos, sum_geometricLagrangeWeight_mul_eval_scaled_geometric, and sum_geometricLagrangeWeight_mul_shifted_pow_of_pos. The polynomial theorem works over every field under exactly Set.InjOn (fun k : ℕ ↦ q ^ k) (Finset.range (p + 1)), reproduces P.eval 0 whenever P.natDegree ≤ p, and allows arbitrary c, including c = 0, strengthening the manuscript's c ≠ 0 hypothesis. Together with the positive-power scaled-moment theorem it makes cor:scaled-geometric-moments Exact by composition. The separate zero-target residual theorem assumes a nonempty distinct field-valued node family, exactly excluding the exceptional 0^0 empty-row case, and gives the negative signed nodal product times the complete homogeneous function.
Arbitrary finite-node and geometric formal-power-series filters	FabiusFunction.FinitePowerSeriesFilter, FabiusFunction.GeometricPowerSeriesFilter	finitePowerSeriesFilter, coeff_finitePowerSeriesFilter, finitePowerSeriesFilter_rescale, map_finitePowerSeriesFilter, coeff_finitePowerSeriesFilter_of_exact_of_le, geometricSeriesFilter, coeff_geometricSeriesFilter_of_exact, geometricSeriesFilter_eq_residual_mk, geometricLagrangeSeriesFilter_eq_residual_mk; these are coefficientwise algebraic identities and assert no analytic convergence or remainder estimate
Unconditionally summable finite-node and geometric analytic-series filters	FabiusFunction.AnalyticSeriesFilter	finiteAnalyticSeriesFilter, summable_finiteAnalyticSeriesFilter_diagonal, finiteAnalyticSeriesFilter_eq_tsum, finiteAnalyticSeriesFilter_eq_head_add_tail_of_exact, finiteAnalyticSeriesFilter_eq_constant_add_tail_of_exact_zero, geometricAnalyticSeriesFilter, geometricAnalyticSeriesFilter_eq_constant_add_gaussian_tsum, geometricLagrangeAnalyticSeriesFilter_eq_constant_add_gaussian_tsum, geometricLagrangeAnalyticSeriesFilter_shifted; the arbitrary finite filter works over a commutative semiring acting on a normed additive group and splits into an exact finite head plus an exact infinite tail under unconditional summability of the weighted sampled series, so a zero-weight node imposes no convergence condition; the geometric and Lagrange wrappers require summability only at nonzero-weight nodes and give the exact denominator-free Gaussian tsum tail; conditionally-only convergent boundary series, a formal-power-series evaluation bridge, a sinc-product instantiation, radius-of-convergence or uniform-convergence theorems, norm/sign/error bounds, asymptotic acceleration, positivity, and Fabius-specific acceleration are not asserted
Named even Fourier-moment modes and their genuine series sum	FabiusFunction.AnalyticMoments	Exhaustive new surface: rvachevFourierMomentTerm, rvachevFourierMomentTerm_zero, rvachevFourierMomentTerm_scale, hasSum_rvachevFourierMomentTerm; for every bounded Fabius solution and every complex frequency, the named even modes have constant term one, scale diagonally by (c^2)^n, and HasSum to the actual Rvachev Fourier transform
Exact Gaussian q-filter of the Rvachev sinc product	FabiusFunction.RvachevQBinomialFilter	Exhaustive public surface: hasSum_rvachevFourierMomentTerm_product, rvachevFourierMomentTerm_pow_scale, geometricLagrange_rvachevFourierProduct_eq_gaussian_tsum, quarterLagrange_rvachevFourierProduct_eq_gaussian_tsum, geometricLagrange_rvachevFourier_eq_gaussian_tsum. The generic entire identity takes arbitrary c,z : ℂ and p : ℕ, uses Gaussian base q = c^2, and assumes only Set.InjOn (fun j ↦ (c^2)^j) (Finset.range (p+1)); it needs no contraction, realness, positivity, or global nonvanishing hypothesis. The quarter specialization c = 1/2, q = 1/4 is assumption-free, and the final theorem holds for every bounded F satisfying IsFabius. This closes the infinite-product coefficient bridge, not the frontier report's finite prefixes P_(b,n), quotient or Bell-coefficient formulas, conditionally convergent boundary cases, analytic error signs or bounds, uniform convergence, derivative estimates, or asymptotics.
Denominator-free finite q-binomial algebra	FabiusFunction.FiniteQBinomialCore	map_gaussianBinomial, gaussianBinomial_succ_succ, gaussianBinomial_succ_succ_alt, gaussianBinomial_symm, finiteQPochhammerIn_add, finiteQPochhammerIn_self_add, finiteQPochhammerIn_self_eq_mul_mul_gaussianBinomial, finite_qBinomial_theorem; the reusable signed-index extension is gaussianBinomialInt, with gaussianBinomialInt_ofNat, gaussianBinomialInt_eq_zero_of_neg, gaussianBinomialInt_eq_zero_of_lt, and total row reflection gaussianBinomialInt_symm. All identities avoid quotient and cancellation hypotheses.
Central Gaussian reduction at squared base	FabiusFunction.CentralQBinomialReduction	Exhaustive zero-definition/six-theorem surface: finiteQPochhammerIn_mul_neg, finiteQPochhammerIn_two_mul, finiteQPochhammerIn_map_ringHom, central_gaussianBinomial_sq_mul_int, central_gaussianBinomial_sq_mul, and central_gaussianBinomial_sq_div. The commutative-ring layer proves sign-pairing, even/odd dissection, naturality, the integral-polynomial certificate, and the division-free identity [2k,k]_(q²)(q²;q²)_k=(q;q²)_k(-q;q)_(2k); the field quotient form assumes exactly its two displayed denominators are nonzero.
Regular central q-binomial sum	FabiusFunction.RegularCentralQBinomialSum	Exhaustive public surface: two definitions, qNumberC and regularCentralQBinomialTerm, and one theorem, hasSum_regularCentralQBinomial. For real 0<q<1 and complex alpha, the theorem gives the actual HasSum evaluation by qGammaC (q^2) (3/2) * qGammaC (q^2) ((alpha+1)/2) / qGammaC (q^2) ((alpha+2)/2), assuming exactly qPochhammerInfIn (q^(alpha+1)) (q^2) ≠ 0. That hypothesis packages nonvanishing of every generalized q-number in the summand and does not exclude even negative integral alpha; there the totalized q-Gamma quotient and the equivalent product value are zero. This makes the q-monograph's thm:regular-central-sum Exact, but proves no classical q→1⁻ limit or classical hypergeometric corollary.
Cyclotomic factorization of finite q-products and Gaussian coefficients	FabiusFunction.CyclotomicFactorization	Exhaustive zero-definition/seven-theorem surface: div_add_div_le_div, div_le_div_add_div_add_one, mem_range_and_mem_divisors_iff, finiteQPochhammerIn_X_eq_prod_cyclotomic, finiteQPochhammerIn_X_eq_gaussianBinomial_mul, prod_cyclotomic_pow_div_extend, and gaussianBinomial_X_eq_prod_cyclotomic. The first two bounds show every Gaussian cyclotomic exponent is zero or one; the commutative-ring layer factors (X;X)_n and supplies the factorial/product-extension identities, while the final exact Gaussian factorization assumes an integral domain.
Complete Gaussian block at a primitive root of unity	FabiusFunction.PrimitiveRootBlock	Exhaustive zero-definition/three-theorem surface: gaussianBinomial_isPrimitiveRoot_eq_zero, neg_one_pow_mul_pow_choose_two, and finiteQPochhammerIn_isPrimitiveRoot. In a commutative integral domain, a primitive d-th root ζ makes [d,k]_ζ = 0 for 0 < k < d; when 0 < d, the phase is (-1)^d ζ^(choose d 2) = -1 and the complete block is (y;ζ)_d = 1-y^d.
The q-Lucas theorem at a primitive root	FabiusFunction.QLucas	Exhaustive zero-definition/seven-theorem surface: add_mul_add_sub_one, choose_two_add, coeff_finiteQPochhammerIn_neg_X, finiteQPochhammerIn_neg_X_block, coeff_block_pow_mul, pow_choose_two_add_mul_eq, and gaussianBinomial_q_lucas. The first two are natural-number quadratic identities; the remaining coefficient, block, and phase lemmas prove [a*d+b,r*d+s]_ζ = choose(a,r) * [b,s]_ζ when 0 < d, ζ is a primitive d-th root in a commutative integral domain, and b,s < d. Its same-shaped two_mul_choose_two helper is private; the unique public Fabius.two_mul_choose_two is owned by QChuVandermonde.
Cyclotomic carry criterion and multiple-index primitive-root value	FabiusFunction.CyclotomicDivisibility	Exhaustive zero-definition/three-theorem surface: cyclotomic_exponent_eq_one_iff, cyclotomic_dvd_gaussianBinomial_iff, and gaussianBinomial_mul_isPrimitiveRoot. For k ≤ n and 0 < d, the Gaussian cyclotomic exponent is one exactly when n % d < k % d; over ℚ[X] this is exactly the divisibility criterion for Φ_d. In a commutative integral domain, a primitive n-th root with 0 < n gives [a*n,b*n]_ζ = choose(a,b).
MacMahon's integral q-Catalan polynomial	FabiusFunction.QCatalan	Exhaustive one-definition/eleven-theorem surface: qCatalan; map_qInt, qInt_X_monic, qInt_X_natDegree, X_sub_one_mul_qInt, qInt_X_eq_prod_cyclotomic, qInt_X_dvd_gaussianBinomial_rat, qInt_X_dvd_gaussianBinomial_int, qInt_X_mul_qCatalan, qCatalan_natDegree, qCatalan_eval_one_mul, and qCatalan_eval_one. The semiring naturality and commutative-ring q-integer identities lead to [n+1]_X ∣ [2*n,n]_X over both ℚ[X] and ℤ[X]; the integral quotient has degree n*(n-1), satisfies (n+1) C_n(1) = choose(2*n,n), and evaluates to the ordinary Catalan number.
Newton interpolation and geometric-grid divided differences	FabiusFunction.NewtonInterpolation	Exhaustive three-definition/nineteen-theorem field-valued surface. Definitions: newtonCoeff, nodeNewtonPoly, and compatibility alias newtonInterpolant. Theorems: newtonCoeff_eq, newtonCoeff_zero, newtonCoeff_mul_prod, nodeNewtonPoly_succ, eval_nodeNewtonPoly, degree_nodeNewtonPoly_lt, nodeNewtonPoly_eq_interpolate, eq_nodeNewtonPoly_of_eval_eq, coeff_nodeNewtonPoly_self, newtonCoeff_eq_sum, nodal_range_pow, prod_erase_pow_sub_pow, newtonCoeff_pow_eq_sum, plus compatibility forms newtonPoly_succ, eval_newtonPoly, degree_newtonPoly_lt, newtonPoly_eq_interpolate, eq_newtonPoly_of_eval_eq, and coeff_newtonPoly_self. It proves triangular reconstruction, interpolation and uniqueness at distinct finite nodes, the divided-difference formula, and its geometric-grid specialization; the evaluation, injectivity, nonzero-product, q ≠ 0, and index hypotheses remain explicit. The node-qualified family avoids the existing scalar newtonPoly; the compatibility family is definitionally identical.
Jackson q-beta integral	FabiusFunction.QBetaIntegral	Exhaustive one-definition/eight-theorem real surface. Definition: qBeta. Theorems: qNumber_pos, qBeta_term_eq, qBeta_eq_prod, qBeta_eq_qGamma, qBeta_comm, qBeta_pos, qBeta_add_one_left, qBeta_add_one_right. For 0 < q < 1 it identifies the Jackson integral with its infinite-product and q-Gamma formulas and derives positivity, symmetry, and both recurrences under the displayed positive-argument hypotheses.
Generic Gaussian-polynomial degree, palindromicity, mean, and linear coefficient	FabiusFunction.GaussianBinomialPalindromic	Exhaustive public surface: zero definitions and fourteen theorems, reflect_add_of_natDegree_le, reflect_one', gaussianBinomial_natDegree_le, gaussianBinomial_zero_left, gaussianBinomial_diag', reflect_gaussianBinomial, coeff_gaussianBinomial_reflect, coeff_gaussianBinomial_zero, coeff_gaussianBinomial_top, gaussianBinomial_natDegree, gaussianBinomial_monic, two_mul_derivative_gaussianBinomial_eval_one, coeff_gaussianBinomial_one_of_pos_of_lt, and coeff_gaussianBinomial_one. Over every commutative semiring it supplies the reflection helpers, the degree bound, zero/diagonal values, reflection in degree k * (n - k), constant and top coefficients one, bounded-index coefficient reversal, and the division-free mean identity. The new interior theorem gives coefficient one under exactly 0 < k and k < n; the total theorem gives if 0 < k ∧ k < n then 1 else 0, explicitly covering k = 0, k = n, and n < k, with no nontriviality hypothesis. Exact degree and monicity require only that the semiring be nontrivial.
Universal Gaussian-polynomial degree and palindromicity	FabiusFunction.GaussianBinomialPolynomialStructure	Exhaustive public surface: zero definitions and five theorems, natDegree_gaussianBinomial_universal, gaussianBinomial_universal_monic, coeff_zero_gaussianBinomial_universal, gaussianBinomial_universal_reflect, and coeff_gaussianBinomial_universal_symm. For k ≤ n, the universal natural-coefficient polynomial gaussianBinomial (X : ℕ[X]) n k has exact natural degree k * (n - k), is monic, has constant coefficient one, and is fixed by reflection in that degree; consequently its coefficients at d and k * (n - k) - d agree whenever d ≤ k * (n - k). The adjacent general-semiring theorem coeff_gaussianBinomial_one supplies the complete linear-coefficient classifier.
Gaussian coefficients and finite q-Pochhammer products at q = -1	FabiusFunction.GaussianBinomialAtNegOne	Exhaustive public surface: zero definitions and five theorems, gaussianBinomial_neg_one_even_even, gaussianBinomial_neg_one_odd_even, gaussianBinomial_neg_one_odd_odd, finiteQPochhammerIn_neg_one_even, and finiteQPochhammerIn_neg_one_odd. All five are total over every [CommRing R], with no domain, characteristic, cancellation, division, or nonvanishing assumption. The three Gaussian identities include columns above the row, where both sides are zero; together with the pre-existing gaussianBinomial_neg_one_even_odd_eq_zero from QBinomialReciprocity, they give the complete four-parity table. The two product identities are (z;-1)_(2a) = (1-z^2)^a and (z;-1)_(2a+1) = (1-z^2)^a(1-z). The derivative and root-multiplicity layer is listed separately in GaussianBinomialAtNegOneDerivative.
First Gaussian-polynomial jets and simple alternating roots at q = -1	FabiusFunction.GaussianBinomialAtNegOneDerivative	Exhaustive public surface: zero definitions and five theorems, gaussianBinomial_X_eval, gaussianBinomial_derivative_eval_neg_one_of_even_degree, gaussianBinomial_derivative_eval_neg_one_even_odd, gaussianBinomial_even_odd_rootMultiplicity_int, and gaussianBinomial_even_odd_rootMultiplicity. The first evaluates the universal Gaussian polynomial [n,k]_X at every point of every commutative semiring. Over every [CommRing R], the first jet theorem gives G'_(n,k)(-1) = -(k*(n-k)/2) G_(n,k)(-1) whenever k*(n-k) is even, including degree zero and columns above the row, while the exceptional theorem gives G'_(2a,2b+1)(-1) = (a-b) * choose(a,b) for all natural a,b, with natural subtraction and binomial zero extension making it total. If b < a, the last two theorems prove that this alternating zero has root multiplicity exactly one, first over ℤ and then over every [CommRing K] [CharZero K]. No simplicity claim is made in arbitrary characteristic, outside the admissible range, or for all cyclotomic zeros.
Simple dyadic roots of the half-base q-binomial polynomial	FabiusFunction.HalfQBinomialRootSimplicity	Exhaustive public surface: zero definitions and one theorem, halfQBinomial_sum_rootMultiplicity_two_pow. For every j < n, the exact rational coefficient polynomial ∑_{k≤n} (-1)^k (1/2)^(k.choose 2) halfQBinomial n k · X^k has root multiplicity one at 2^j. Together with the existing complete root classification, this says that all roots in the half-base locus are simple. It makes no arbitrary-base, arbitrary-field, or general cyclotomic simplicity claim.
Elementary finite q-Pochhammer reversal, termination, and adjacent Gaussian ratios	FabiusFunction.QPochhammerElementaryIdentities	Exhaustive public surface (13 theorems): finiteQPochhammerIn_base_reversal_units, finiteQPochhammerIn_inv_base_reversal_units, finiteQPochhammerIn_base_reversal, finiteQPochhammerIn_inv_base_reversal, prod_pow_sub_pow_eq_finiteQPochhammerIn, pow_mul_finiteQPochhammerIn_inv_pow_eq, finiteQPochhammerIn_inv_pow_eq_self_div, finiteQPochhammerIn_inv_pow_eq_zero_of_lt, one_sub_mul_gaussianBinomial_one, gaussianBinomial_adjacent_mul, gaussianBinomial_row_adjacent_mul, gaussianBinomial_adjacent_div, gaussianBinomial_row_adjacent_div. The two unit reversals hold in every commutative ring. The root-safe terminating numerator, first-column clearer, and both adjacent cross-multiplied identities also hold over every commutative ring, including roots of unity; the two cross identities are total in all n,k, with zero extension making both sides vanish on and above the row boundary. The field reversal wrappers require exactly a != 0 and q != 0; the cleared terminating formula and the k > N zero theorem require q != 0, while its displayed quotient additionally requires (q;q)_(N-k) != 0. The two adjacent quotient theorems remain restricted to k < n and require exactly their displayed Gaussian and linear-factor denominators to be nonzero; they do not require q != 0.
Finite q-Cauchy convolutions and the q-Bernstein partition of unity	FabiusFunction.QBinomialCauchy	Exhaustive public surface: finite_qCauchy_identity, its compatibility spelling finiteQPochhammerIn_mul_eq_sum_gaussianBinomial, finite_qCauchy_identity_reflected, qBernsteinBasis, sum_qBernsteinBasis, and finite_qCauchy_second_identity (one definition and five theorems). The primary identity is (uv;q)_n = sum_(k=0)^n [n choose k]_q (u;q)_k v^k (v;q)_(n-k); its reflected strengthening uses (v;q)_k v^(n-k) (u;q)_(n-k). The specialization u = 0 makes the denominator-free q-Bernstein row sum to one, and the second identity evaluates the two-product Cauchy convolution. All parameters and degrees are arbitrary in every commutative ring, including n = 0, q = 0, roots of unity, positive characteristic, and zero divisors; no quotient, cancellation, or nonvanishing hypothesis is present.
Terminating two-phi-one reversal and the two q-Chu--Vandermonde sums	FabiusFunction.TwoPhiOneReversal, FabiusFunction.QChuVandermonde	Exhaustive public surfaces: TwoPhiOneReversal has two definitions and twelve theorems, including the reflected parameter map, involution, finite/actual-tsum reversals, double reversal, and the degree-zero bridge; QChuVandermonde has ten theorems and no definitions, including both denominator-cleared sums, finite evaluations, actual-tsum wrappers, and the separately named by-reversal route. lem:2phi1-reversal and both formulas of cor:q-chu are exact under their displayed rational/nonvanishing hypotheses. prop:qchu2-by-reversal remains partial: that route additionally assumes C ≠ 0 and (A;q)_n ≠ 0, while the full-domain second formula uses a direct denominator-cleared q-Cauchy proof; rational continuation and a cleared commutative-ring extension remain unformalized.
Bit-position and weighted-subset forms of Gauss's finite theorem	FabiusFunction.BitPositionQBinomial	Exhaustive public surface: prod_one_add_mul_pow_eq_gaussianBinomial, prod_one_add_pow_eq_sum_gaussianBinomial, sum_powersetCard_two_pow, sum_pow_bitPositionSum_filter_eq_gaussianBinomial, sum_pow_bitPositionSum_filter_eq_gaussianBinomial', sum_pow_sum_powersetCard_eq_gaussianBinomial, sum_pow_sum_powersetCard_Icc_eq_gaussianBinomial, gaussianBinomial_one_eq_choose. The last two subset identities give respectively the zero-based range N weight q^(choose r 2) and the literal one-based interval Icc 1 N weight q^(choose (r+1) 2); they are total in N,r over every commutative ring.
Reciprocal symmetry and the q = -1 parity zero of Gaussian coefficients	FabiusFunction.QBinomialReciprocity	Exhaustive public surface (four theorems): gaussianBinomial_reciprocity_units, gaussianBinomial_reciprocity, gaussianBinomial_neg_one_eq_zero_of_odd_degree, and gaussianBinomial_neg_one_even_odd_eq_zero. The total identity q^(k*(n-k)) * [n choose k]_(q⁻¹) = [n choose k]_q holds for a unit in every commutative semiring, including zero divisors, and has a nonzero-parameter wrapper over every semifield. Over every commutative ring, odd polynomial degree forces vanishing at q = -1; hence every odd column of an even row vanishes there, including the above-diagonal zero-extension cases. No quotient formula, cancellation of finite q-Pochhammer factors, domain hypothesis, or characteristic restriction is used.
Gaussian reciprocity and dimension-dominant bounds	FabiusFunction.GaussianBinomialBounds	Exhaustive zero-definition/six-theorem surface: gaussianBinomial_inv, one_le_gaussianBinomial, finiteQPochhammerIn_pow_le_one, gaussianBinomial_le_inv_qPochhammerInfIn, pow_le_gaussianBinomial_of_one_lt, and gaussianBinomial_le_pow_div_of_one_lt. The module reuses the stronger ordered-field theorem finiteQPochhammerIn_self_pos from GeneralQConditionNumber. Its field reciprocity theorem assumes q != 0 and k ≤ n. Over ordered fields, nonnegative q gives the lower bound one; over the reals, 0 ≤ q < 1 gives the uniform upper bound by (q;q)_∞⁻¹, and reciprocity transfers these to Q > 1 as Q^(k*(n-k)) ≤ [n choose k]_Q ≤ Q^(k*(n-k))/(Q⁻¹;Q⁻¹)_∞. The index and order hypotheses remain explicit.
Division-free Gaussian chains, alternating rows, and mutually inverse scalar kernels	FabiusFunction.QBinomialInversion	Exhaustive public surface: gaussianBinomial_mul, sum_gaussianBinomial_alternating_mul_pow, sum_gaussianBinomial_alternating, gaussianBinomialKernel, gaussianBinomialInverseKernel, scaledGaussianBinomialKernel, scaledGaussianBinomialInverseKernel, gaussianBinomialKernel_left_orthogonality, gaussianBinomialKernel_right_orthogonality, scaledGaussianBinomialKernel_left_orthogonality, scaledGaussianBinomialKernel_right_orthogonality. The chain identity holds over every commutative semiring; the alternating sums and all four total-Icc orthogonality theorems hold over every commutative ring. The scale s is independent of the Gaussian base and is arbitrary: neither it nor q is assumed nonzero or invertible.
Scaled and classical q-binomial transforms of module-valued sequences	FabiusFunction.QBinomialTransform	Exhaustive public surface (four definitions and four theorems): scaledGaussianBinomialTransform, scaledGaussianBinomialInverseTransform, scaledGaussianBinomialInverseTransform_transform, scaledGaussianBinomialTransform_inverseTransform, scaledGaussianBinomial_inversion, gaussianBinomialTransform, gaussianBinomialInverseTransform, gaussianBinomial_inversion. The forward definitions need only [Semiring R] [AddCommMonoid M] [Module R M], and the inverse definitions need only [Ring R] [AddCommMonoid M] [Module R M]. For [CommRing R] [AddCommMonoid M] [Module R M], both composition theorems are equalities of whole sequence functions and the two triangular relations are equivalent as whole-sequence equalities. The proofs reuse lowerTriangularTransform_comp; the Gaussian coefficient zero-extends each scalar kernel above its row. These finite algebraic maps require no topology, convergence, division, or invertibility; the unscaled theorem is classical q-binomial inversion.
Scaled Gaussian characteristic polynomials and exact q-difference annihilation	FabiusFunction.QDifferenceAnnihilation	Exhaustive public surface (four theorems): sum_scaledGaussianBinomialInverseKernel_mul_pow, sum_gaussianBinomialInverseKernel_mul_geometric_pow, qDifference_sum_eval₂_eq_map_coeff_mul, qDifference_sum_eval₂_eq_zero_of_degree_lt. Over every commutative ring, sum_(k=0)^n (-s)^(n-k) q^(choose (n-k) 2) [n choose k]_q z^k = prod_(j<n) (z-s q^j). At s = 1 and z = q^d, this gives every monomial moment prod_(j<n) (q^d-q^j), hence zero for d < n. More strongly, after any scalar extension φ : A →+* R from a semiring, the row applied to a polynomial of degree at most n is φ(p.coeff n) * prod_(j<n) (q^n-q^j), and it annihilates every polynomial of degree strictly below n. The statements include n = 0 and the zero polynomial; nodes may collide and the surviving product may vanish. No division, node distinctness, nonzero/invertible base, domain, characteristic, topology, or convergence hypothesis is used.
Exact q-Gaussian inversion specializations	FabiusFunction.QBinomialInversionSpecializations	Exhaustive public surface (two definitions and four theorems): qGaussianResidualCoeff, qGaussianReconstructionCoeff, qGaussianResidualCoeff_eq, qGaussianReconstructionCoeff_eq, qGaussianReconstructionCoeff_residualCoeff_delta, qGaussianResidualCoeff_reconstructionCoeff_delta. The two definitions and their pointwise closed-form theorems require only [Ring R], allowing a noncommutative coefficient ring. At Gaussian base q^2 and scale -q, the residual coefficient is (-q)^(n-k) [n choose k]_(q^2) and its reconstruction coefficient is q^((n-k)^2) [n choose k]_(q^2). Exactly the two total-Icc convolution-delta theorems require [CommRing R].
Denominator-free q-Vandermonde and central convolutions	FabiusFunction.QBinomialVandermonde	Exhaustive public surface: gaussianBinomial_add_vandermonde, gaussianBinomial_add_vandermonde', gaussianBinomial_add_central, gaussianBinomial_add_central_min, gaussianBinomial_two_mul_add_shifted_central, gaussianBinomial_two_mul_sub_shifted_central, gaussianBinomial_two_mul_sub_shifted_central_Icc, gaussianBinomial_two_mul_int_shifted_central, gaussianBinomial_two_mul_int_shifted_central_finsum. All nine hold over an arbitrary commutative semiring without division, cancellation, or a restriction on q; the first seven are the natural-index forms, while the last two prove the report's single formula for every k : ℤ, first on the finite natural range 0,…,N and then literally as a finite-support sum over ℤ.
Geometric Richardson filters, Gaussian coefficients, all residual moments, and finite conditioning	FabiusFunction.GeometricQBinomialLagrange, FabiusFunction.GeometricRichardson, FabiusFunction.GeometricLagrangeWeights, FabiusFunction.GeometricLagrangeQBinomial, FabiusFunction.GeometricLagrangeQMoments	geometricQBinomialWeightNumerator_eq_scaledGaussianBinomialInverseKernel, reversed_finite_qBinomial_theorem, sum_geometricLagrangeWeight_mul_pow_eq_gaussianBinomial, geometricLagrangeWeightPolynomial_eq_forwardGeometricRichardsonPolynomial, geometricLagrangeQMoment_eq_residual_qBinomial, sum_abs_geometricLagrangeWeight_eq_prod. The first theorem, now owned by GeometricQBinomialLagrange, globally identifies the denominator-free geometric numerator with the inverse kernel at base and scale q, including indices above the diagonal and without k ≤ n; it is the s = q specialization of the scaled characteristic polynomial. This does not weaken the separate Field/finite-node-InjOn/in-range assumptions of normalized quotient formulas. The remaining rational closed forms use their stated nonzero-base and nonvanishing finite-denominator hypotheses, while sign and variation assume 0 < q < 1.
Quotient-defined rational geometric moments and exact finite conditioning	FabiusFunction.GeometricLagrangeQMoments	Exhaustive public surface (one definition and 37 theorems): geometricLagrangeQMoment; geometricLagrangeQMoment_eq_weightPolynomial_eval, geometricLagrangeQMoment_eq_forwardRichardson_eval, geometricRootPolynomial_inv_eval_pow_mul_signedPowers, geometricRootPolynomial_inv_eval_pow_mul_triangular, geometricRootPolynomial_inv_eval_one_mul_triangular, geometricLagrangeQMoment_eq_qPochhammer, geometricLagrangeQMoment_zero, geometricLagrangeQMoment_eq_zero, geometricRootPolynomial_inv_eval_pow_eq_qPochhammer_of_le, geometricLagrangeQMoment_eq_residual_qPochhammer, qPochhammer_self_add, qPochhammer_self_pos_of_pos_of_lt_one, qBinomial_pos_of_pos_of_lt_one, gaussianBinomial_eq_qBinomial_of_pos_of_lt_one, qPochhammer_pow_pos_of_pos_of_lt_one, qPochhammer_tail_div_self_eq_qBinomial, geometricLagrangeQMoment_eq_residual_qBinomial, geometricLagrangeQMoment_firstUncancelled, negOnePow_mul_geometricLagrangeQMoment_eq_positiveResidual, negOnePow_mul_geometricLagrangeQMoment_pos, qPochhammer_self_succ, qBinomial_succ_succ_of_pos_of_lt_one', qBinomial_succ_succ_of_pos_of_lt_one, qBinomial_theorem_of_pos_of_lt_one, sum_qBinomial_triangular_succ_eq_neg_qPochhammer, abs_geometricLagrangeWeight_eq_qBinomial, abs_geometricLagrangeWeight_eq_sign_mul, abs_geometricLagrangeWeight_complement_eq_qBinomial, sum_abs_geometricLagrangeWeight_eq_qPochhammer_ratio, neg_qPochhammer_div_self_eq_prod, sum_abs_geometricLagrangeWeight_eq_prod, quarterGeometricLagrangeQMoment_eq_qPochhammer, quarterGeometricLagrangeQMoment_eq_zero, quarterGeometricLagrangeQMoment_eq_residual_qPochhammer, quarterGeometricLagrangeQMoment_eq_residual_qBinomial, quarterGeometricLagrangeQMoment_firstUncancelled, and sum_abs_quarterGeometricLagrangeWeight_eq_qPochhammer_ratio. These are finite rational identities. The quotient and injectivity forms retain their explicit nonzero-denominator hypotheses; positivity, sign, and absolute-value formulas retain 0 < q < 1; no analytic convergence or error estimate is asserted.
Euler generating function of the geometric Richardson triangle	FabiusFunction.GeometricRichardsonGenerating	Exhaustive public surface (three definitions and seven theorems): geometricRichardsonKernel, qPochhammerNormalizedDataSeries, geometricRichardsonTransform; coeff_rescale_qPochhammerSeries_eq_geometricRichardsonKernel, coeff_qPochhammerNormalizedDataSeries, geometricRichardsonTransform_generating, geometricRichardsonTransform_eq_sum_lagrange, geometricLagrangeRichardson_generating, hasSum_geometricRichardsonTransform_mul_pow, and hasSum_geometricLagrangeRichardson_mul_pow. The formal convolution and factorization hold over every commutative ring, without topology or a regularity assumption on q, using Ring.inverse at possibly nonunit finite q-Pochhammer factors. Over a field, q ≠ 0 identifies the convolution with the canonical totalized Lagrange row and proves the report's gq:thm:richardson-generating claim exactly. No root-of-unity exclusion is needed for that algebraic equality, but colliding nodes are not thereby genuine interpolation weights; q = 0 is excluded from the Lagrange identification and its closed formula already fails once repeated nodes occur. The two analytic theorems assume a complete normed field, ‖q‖ < 1, and norm-summability of the normalized data at z; the report-facing Lagrange form additionally assumes q ≠ 0 and concludes HasSum, not a general evaluation theorem for formal power series.
Report-facing geometric complete-homogeneous bridges	FabiusFunction.GeometricLagrangeCompleteHomogeneous	Exhaustive five-theorem surface: completeHomogeneousEvalOn_geometric_range, sum_geometricLagrangeWeight_mul_pow_succ_add_eq_gaussianBinomial, geometricLagrangeQMoment_eq_residual_gaussianBinomial, completeHomogeneousEvalOn_geometric_range_eq_qBinomial, and geometricLagrangeQMoment_eq_residual_qBinomial_via_completeHomogeneous. The semiring principal-specialization alias is denominator-free; the field residual uses finite-node injectivity; the rational quotient bridges retain their stated nonzero-Pochhammer or 0 < q < 1 hypotheses.
Exact finite polynomial filters	FabiusFunction.FinitePolynomialFilterExactness	Exhaustive five-theorem surface: polynomialFilter_response_eq, polynomialFilter_exact, normalizedGeometricRootPolynomial_filter_exact, forwardGeometricRichardsonPolynomial_filter_exact, and forwardGeometricRichardsonPolynomial_filter_firstUncancelled. The first two are arbitrary commutative-semiring response and mass-one/root-cancellation laws. The geometric field specializations preserve the baseline, cancel the prescribed inverse or forward modes, and evaluate the first surviving forward mode as (-1)^n q^(choose (n+1) 2) under their explicit nonzero-base and normalization-denominator hypotheses.
Formal geometric Richardson filters and the quarter Catalan--Gaussian specialization	FabiusFunction.QuarterCatalanRichardson	Exhaustive public surface (three definitions and 15 theorems): finiteRescaleFilter, geometricRichardsonPowerSeriesFilter, quarterCatalanRichardsonFilter; finiteRescaleFilter_coeff, geometricRichardsonPowerSeriesFilter_coeff, geometricRichardsonPowerSeriesFilter_coeff_zero, geometricRichardsonPowerSeriesFilter_coeff_eq_zero, geometricRichardsonPowerSeriesFilter_coeff_eq_qPochhammer, geometricRichardsonPowerSeriesFilter_coeff_eq_qBinomial, geometricRichardsonPowerSeriesFilter_firstUncancelled_coeff_of_nonzero, geometricRichardsonPowerSeriesFilter_firstUncancelled_coeff, quarterCatalanRichardsonFilter_coeff, quarterCatalanRichardsonFilter_coeff_zero, quarterCatalanRichardsonFilter_coeff_eq_zero, quarterCatalanRichardsonFilter_coeff_eq_zero_of_le, quarterCatalanRichardsonFilter_coeff_eq_qBinomial, quarterCatalanRichardsonFilter_coeff_succ_eq_qBinomial, and quarterCatalanRichardsonFilter_firstUncancelled_coeff. These are coefficientwise formal-power-series identities: the generic row diagonalizes rescaling, preserves degree zero, cancels degrees 1,…,p, and exposes every residual and the first survivor; the quarter specialization multiplies those factors by the exact Catalan coefficients. No convergence, real-function error sign, remainder bound, or analytic acceleration is asserted.
Exact lower-Lambert phase locking, reciprocal-grid Richardson moments, Bell/generalized-harmonic residuals, fixed-order growing-row bounds, and analytic extraction of the periodic Fabius endpoint term	FabiusFunction.LambertPhaseLockedRichardson, FabiusFunction.LambertPhaseLockedBell, FabiusFunction.LambertReciprocalAsymptotics, FabiusFunction.FabiusLambertPhaseLockedPullback, FabiusFunction.FabiusLambertPhaseExtraction, FabiusFunction.FabiusLambertPhaseExtractionBell	The phase-locking and analytic chain exposes fabiusLambertPhase_phaseLockedNode, Periodic.apply_fabiusLambertPhase_phaseLockedNode, shiftedReciprocalLagrangeWeight_eq_choose, sum_shiftedReciprocalLagrangeWeight_mul_periodicPhaseLocked, sum_shiftedReciprocalLagrangeWeight_mul_invPow_card_add, sum_shiftedReciprocalLagrangeWeight_mul_invPow_eq_completeHomogeneous, sum_shiftedReciprocalLagrangeWeight_residual, shiftedReciprocalLagrangeWeight_mul_invPow_isBigO_atTop, tendsto_lambertPhaseLockedNode_smallArgument, log_fabius_phaseLockedNode_sub_WikipediaLambertExpansion_isBigO, fabiusPhaseLockedPeriodicEstimator_sub_residual_isBigO, fabiusPhaseLockedPeriodicEstimator_sub_firstOmitted_isBigO, and fabiusPhaseLockedPeriodicEstimator_tendsto_periodicAlong. The Bell leaf exhaustively adds shiftedReciprocalPowerSum, shiftedReciprocalBellInput, shiftedReciprocalPowerSum_eq_sum, shiftedReciprocalBellInput_zero, shiftedReciprocalBellInput_succ, completeHomogeneousEvalOn_shiftedReciprocal_eq_bell, sum_shiftedReciprocalLagrangeWeight_mul_invPow_card_add_eq_bell, sum_shiftedReciprocalLagrangeWeight_mul_invPow_eq_bell, fabiusPhaseLockedResidualTerm_eq_bell, and fabiusPhaseLockedResidualPartialSum_eq_sum_bell. The normalized Bell specialization is stated over a field with rational-algebra structure, and the weighted moment forms assume characteristic zero. Within that setting the definitions are total and require no positivity or nonzero-shift hypothesis. For every fixed row order r and residual depth S, subtracting the first S exact residual terms leaves O(lambda⁻¹^(r+1+S)), and integer phase rays converge to the corresponding value of negativeLaplacePsi. No convergence of an infinite residual series, exponentiated Bell relative-error hierarchy, higher derivative extractor, residual sign/bracketing theorem, or uniformity for growing r or S is asserted.
Infinite q-Pochhammer symbols and the limiting general-q row condition number	FabiusFunction.LimitConditionNumber	qPochhammerInf, multipliable_one_sub_mul_pow, tendsto_finiteQPochhammerIn, qPochhammerInf_self_pos, qConditionNumberLimit, tendsto_sum_abs_qToeplitzWeight, one_div_one_sub_le_qConditionNumberLimit, tendsto_qConditionNumberLimit_atTop_at_one_left, one_lt_qConditionNumberLimit, qConditionNumberLimit_zero, thousand_le_qConditionNumberLimit; for 0 ≤ q < 1 the finite row variation converges to (-q;q)_∞ / (q;q)_∞, whose denominator is strictly positive and whose value is at least 1 / (1-q); consequently the limit tends to +∞ as q → 1⁻, so no uniform-in-q bound exists
Closed real Lambert branches, endpoint continuity, and the principal small-argument equivalent	FabiusFunction.LowerLambertW, FabiusFunction.PrincipalLambertW	The new lower-branch continuity surface is lowerLambertW_continuousWithinAt_branchPoint, lowerLambertW_continuousAt, lowerLambertW_continuousOn, and lowerLambertW_continuousOn_Ico: it gives continuity on the full natural domain [-exp(-1),0), including the finite branch point but not zero. The principal companion exposes principalLambertArg, principalLambertW, principalLambertW_mul_exp, neg_one_le_principalLambertW, principalLambertW_unique, principalLambertW_branchPoint, principalLambertW_zero, principalLambertW_exp_one, mul_exp_strictMonoOn, principalLambertW_strictMonoOn, principalLambertW_nonpos, neg_one_lt_principalLambertW, principalLambertW_image_Ioi, principalLambertW_image_Icc, principalLambertW_continuousWithinAt_branchPoint, principalLambertW_continuousAt, principalLambertW_continuousOn, principalLambertW_continuousOn_Ici, principalLambertW_hasDerivAt, deriv_principalLambertW_pos, deriv_principalLambertW_zero, and principalLambertW_isEquivalent_zero. Thus W₀ is continuous on [-exp(-1),∞), its derivative claims are only above the branch point, and W₀(z) ~ z at zero. The raw branch-point derivative/secant limits and leading square-root laws are recorded in the dedicated modules below; no finite endpoint derivative is asserted.
Exact real-branch pairing, gap bijection, and symmetric identities	FabiusFunction.LambertWBranchPairing, FabiusFunction.LambertWGapBijection, FabiusFunction.LambertWBranchSymmetry	Exhaustive module counts are respectively 0+7, 4+16, and 0+9, hence four definitions and 32 theorems (36 public declarations) in their union. LambertWBranchPairing exports principalLambertW_sub_lowerLambertW_pos, lowerLambertW_eq_principalLambertW_mul_exp_gap, principalLambertW_eq_neg_gap_div, lowerLambertW_eq_neg_gap_mul_exp_div, lowerLambertW_eq_neg_gap_div_one_sub_exp_neg, eq_neg_gap_div_mul_exp, and principalLambertW_lowerLambertW_eq_of_exp_gap. LambertWGapBijection defines gapPrincipal, gapLower, gapArg, and branchGap, and proves gap_denominator_pos, gapPrincipal_mem_Ioo, gapLower_eq_mul_exp, gapLower_eq_sub, gapLower_lt_neg_one, gapLower_mul_exp, gapArg_mem_Ioo, principalLambertW_gapArg, lowerLambertW_gapArg, branchGap_gapArg, gapArg_branchGap, branchGap_invOn, branchGap_bijOn, principalLambertW_gapArg_log, lowerLambertW_gapArg_log, and gapArg_log. LambertWBranchSymmetry exports lowerLambertW_div_principalLambertW_eq_exp_branchGap, principalLambertW_add_lowerLambertW_eq_exp_branchGap, principalLambertW_add_lowerLambertW_eq_cosh_div_sinh_branchGap, principalLambertW_mul_lowerLambertW_eq_exp_branchGap, principalLambertW_mul_lowerLambertW_eq_sinh_sq_branchGap, principalLambertW_add_lowerLambertW_lt_neg_two, principalLambertW_mul_lowerLambertW_pos, principalLambertW_mul_lowerLambertW_lt_one, and principalLambertW_mul_lowerLambertW_mem_Ioo. On the strict common domain x ∈ (-exp(-1),0), the positive gap Δ=W₀(x)-W₋₁(x) gives both branches and their argument by exact exponential-rational formulas; conversely gapArg and branchGap are two-sided inverses between that domain and Δ ∈ (0,∞). For t>1, the three explicit t=exp Δ formulas recover W₀, W₋₁, and the common argument. The symmetric leaf proves the exact ratio, exponential and hyperbolic sum/product forms, W₀+W₋₁<-2, and 0<W₀W₋₁<1. All endpoints are deliberately excluded. These three finite exact modules neither supply a Bernoulli-number expansion of the gap formulas nor assert convergence, remainders, or branch-point/small-input asymptotics.
Actual Bernoulli series for the real Lambert branch gap and its exact complex convergence radius	FabiusFunction.LambertWBranchGapBernoulli	Exhaustive public surface (zero definitions and five theorems): summable_norm_bernoulli_mul_pow_div_factorial, summable_bernoulli_mul_pow_div_factorial_iff, hasSum_bernoulli_mul_pow_div_factorial, hasSum_bernoulli_mul_pow_div_factorial_complex_iff, and principalLambertW_lowerLambertW_eq_bernoulliSeries. For real z with `
Exact second derivatives and curvature of the real Lambert branches	FabiusFunction.LambertWCurvature	Exhaustive public surface: deriv_principalLambertW, deriv_principalLambertW_hasDerivAt, deriv_deriv_principalLambertW, deriv_deriv_principalLambertW_zero, deriv_deriv_principalLambertW_neg, strictConcaveOn_principalLambertW, deriv_lowerLambertW_hasDerivAt, deriv_deriv_lowerLambertW, deriv_deriv_lowerLambertW_pos_iff, deriv_deriv_lowerLambertW_neg_iff, deriv_deriv_lowerLambertW_eq_zero_iff, strictConvexOn_lowerLambertW_left, and strictConcaveOn_lowerLambertW_right. The nonsingular formula W'' = -exp(-2W)(W+2)/(W+1)^3 is proved for the principal branch exactly when z > -exp(-1) and for the lower branch exactly when -exp(-1) < z < 0; in particular W₀''(0) = -2. The principal branch is strictly concave on its full closed domain. The lower branch has its unique smooth-domain inflection at -2 exp(-2), is strictly convex from the branch point through that input, and strictly concave thereafter toward zero. The shape package includes the finite branch point and the shared inflection endpoint, while the lower natural domain remains open at zero; it does not assert derivative limits at the branch point or a Puiseux law.
One-sided vertical tangents and failure of finite branch-point differentiability	FabiusFunction.LambertWBranchPointGeometry	Exhaustive public surface (eight theorems): tendsto_deriv_principalLambertW_branchPoint_atTop, tendsto_deriv_lowerLambertW_branchPoint_atBot, tendsto_principalLambertW_secantSlope_branchPoint_atTop, tendsto_lowerLambertW_secantSlope_branchPoint_atBot, principalLambertW_not_differentiableWithinAt_branchPoint, lowerLambertW_not_differentiableWithinAt_branchPoint, principalLambertW_not_differentiableAt_branchPoint, and lowerLambertW_not_differentiableAt_branchPoint. As z approaches -exp(-1) from the right, the principal derivative and endpoint secant slope (W₀(z)+1)/(z+exp(-1)) tend to +∞, while their lower-branch counterparts tend to -∞. Consequently neither branch has a finite right derivative there, and neither totalized branch is differentiable at the branch point. These statements require no square-root or Puiseux expansion.
Leading signed square-root asymptotics at the real branch point	FabiusFunction.LambertWBranchPointAsymptotics	Exhaustive public surface (one definition and eight theorems): lambertWBranchPointScale, lambertWBranchPointScale_pos, lambertWBranchPointScale_sq, tendsto_principalLambertW_add_one_sq_div_branchPoint, tendsto_lowerLambertW_add_one_sq_div_branchPoint, principalLambertW_add_one_sq_isEquivalent_branchPoint, lowerLambertW_add_one_sq_isEquivalent_branchPoint, principalLambertW_add_one_isEquivalent_branchPoint, and lowerLambertW_add_one_isEquivalent_branchPoint. The scale is sqrt(2 exp(1) (z+exp(-1))), is positive strictly to the right of the branch point, and has square exactly 2 exp(1) (z+exp(-1)). For each branch, (W(z)+1)^2/(z+exp(-1)) → 2 exp(1), equivalently the squared displacement is asymptotic to that linear normalization. The signed leading laws are W₀(z)+1 ~ scale(z) and W₋₁(z)+1 ~ -scale(z) from the right. No O(z+exp(-1)) remainder, convergent Puiseux expansion, or higher coefficient is asserted, and named generic/Fabius phase wrappers for these raw endpoint laws remain open.
Two real Lambert inverses of scaled power--exponential saddles, exact root classification, full phase continuity, and small-input asymptotics	FabiusFunction.PowerExponentialLambert, FabiusFunction.PowerExponentialLambertCalculus, FabiusFunction.PowerExponentialLambertInverse, FabiusFunction.PowerExponentialLambertAsymptotics, FabiusFunction.PowerExponentialLambertFabius	powerExponentialSaddle, powerExponentialPeak, powerExponentialLambertArgument, principalPowerExponentialPhase, lowerPowerExponentialPhase, the three powerExponentialLambertArgument_mem_* domain theorems, both branch solve laws and endpoint values, principalPowerExponentialPhase_mem_Icc, lowerPowerExponentialPhase_mem_Ici, powerExponentialLambertArgument_strictAntiOn, principalPowerExponentialPhase_strictMonoOn, lowerPowerExponentialPhase_strictAntiOn, both branch HasDerivAt/deriv/derivative-sign and interior-continuity pairs, principalPowerExponentialPhase_continuousOn_Icc, lowerPowerExponentialPhase_continuousOn_Ioc, powerExponentialLambertArgument_image_Icc, powerExponentialLambertArgument_image_Ioc, principalPowerExponentialPhase_image_Icc, lowerPowerExponentialPhase_image_Ioc, both branch LeftInvOn/RightInvOn/InvOn packages, powerExponentialSaddle_eq_iff_eq_principal_or_eq_lower, and principalPowerExponentialPhase_ne_lowerPowerExponentialPhase. All of these generic branch results assume m ≠ 0, A > 0, and beta > 0; the root iff additionally assumes x ∈ (0,peak] and lambda ≥ 0 (the restriction is essential for even m), while distinctness holds only for x ∈ (0,peak). The principal phase is continuous on [0,peak], the lower phase on (0,peak], and derivatives remain restricted to (0,peak). The small-input surface is powerExponentialLambertEpsilon, powerExponentialLambertArgument_eq_neg_epsilon, powerExponentialLambertEpsilon_pos, tendsto_powerExponentialLambertEpsilon_nhdsGT_zero, principalPowerExponentialPhase_isEquivalent_rpow, tendsto_lowerPowerExponentialPhase_nhdsGT_zero_atTop, lowerPowerExponentialPhaseIntrinsicMain, and lowerPowerExponentialPhase_sub_intrinsicMain_tendsto_zero: along x ↓ 0, the principal root is equivalent to (x/A)^(1/m), while the lower root diverges and has only the proved intrinsic epsilon-coordinate two-term remainder. The bridges are lowerPowerExponentialPhase_rate_one, generalizedLambertCoordinate_argument_mem_Ioo, generalizedLambertCoordinate_argument_mem_Ico, generalizedLambertCoordinate_solves_saddle_of_mem, powerExponentialSaddle_one_one_log_two, powerExponentialPeak_one_one_log_two, lowerPowerExponentialPhase_one_one_log_two, fabiusPrincipalLambertPhase, fabiusSaddle_eq_iff_eq_principal_or_eq_lower, fabiusPrincipalLambertPhase_ne_fabiusLambertPhase, fabiusPrincipalLambertPhase_continuousOn_Icc, fabiusLambertPhase_continuousOn_Ioc, and fabiusPrincipalLambertPhase_isEquivalent_id; their root/continuity domains specialize respectively to (0,exp(-1)/log 2], (0,exp(-1)/log 2), [0,exp(-1)/log 2], and (0,exp(-1)/log 2]. No cleaned L = log(A/x) form, branch-point Puiseux law, or generic complete asymptotic series is claimed.
Second derivatives of scaled Lambert phases and exact Fabius phase curvature	FabiusFunction.PowerExponentialLambertCurvature, FabiusFunction.PowerExponentialLambertFabiusCurvature	The generic module exposes deriv_principalPowerExponentialPhase_hasDerivAt, deriv_deriv_principalPowerExponentialPhase, deriv_lowerPowerExponentialPhase_hasDerivAt, and deriv_deriv_lowerPowerExponentialPhase: for m : ℕ with m ≠ 0, A > 0, and beta > 0, both branches satisfy lambda'' = lambda * (m - (m - beta*lambda)^2) / (x^2 * (m - beta*lambda)^3) on (0,peak). It deliberately does not assert the generic square-root threshold or global shape classification. The Fabius leaf exhaustively adds fabiusLambertInflectionInput, fabiusLambertInflectionInput_mem_Ioo, fabiusLambertPhase_inflectionInput, deriv_deriv_fabiusLambertPhase, deriv_deriv_fabiusLambertPhase_pos_iff, deriv_deriv_fabiusLambertPhase_neg_iff, deriv_deriv_fabiusLambertPhase_eq_zero_iff, deriv_deriv_fabiusLambertPhase_inflectionInput, strictConvexOn_fabiusLambertPhase_left, strictConcaveOn_fabiusLambertPhase_right, fabiusPrincipalLambertPhase_eq_principalLambertW, fabiusPrincipalLambertPhase_continuousOn_Iic, fabiusPrincipalLambertPhase_hasDerivAt, deriv_fabiusPrincipalLambertPhase, deriv_fabiusPrincipalLambertPhase_hasDerivAt, deriv_deriv_fabiusPrincipalLambertPhase, deriv_deriv_fabiusPrincipalLambertPhase_pos, deriv_deriv_fabiusPrincipalLambertPhase_zero, and strictConvexOn_fabiusPrincipalLambertPhase. The lower phase has its unique inflection at x* = 2 exp(-2)/log 2, is strictly convex on (0,x*] and strictly concave on [x*,peak]; the principal phase is strictly convex on the whole closed half-line (-∞,peak], with second derivative 2 log 2 at zero. Neither module proves a branch-point vertical-tangent limit or Puiseux expansion.
Formal power-series logarithms and finite-product additivity	FabiusFunction.LogSeriesMultiplicative	SaddleExpansion.logOf, SaddleExpansion.massSeries_coeff, SaddleExpansion.constantCoeff_logOf, SaddleExpansion.mul_derivative_logOf, SaddleExpansion.logOf_eq_of, SaddleExpansion.logOf_mul, SaddleExpansion.logOf_one, SaddleExpansion.logOf_prod
Generic unit-interval Laplace-moment bounds	FabiusFunction.UnitLaplaceMomentBounds	unitLaplaceMoment_midpoint_sq_le_all, unitLaplaceMoment_le_of_tilt_sub, pow_mul_exp_neg_le_factorial, fabiusLaplaceMoment_midpoint_sq_le_all, fabiusLaplaceMoment_le_of_tilt_sub
Exact dyadic computation and analytic correctness	FabiusFunction.DyadicAnalytic, FabiusFunction.GlobalDyadic	fabiusDyadicValue, evalFabiusDyadic, fabiusDyadicUnit_cast, extendedFabiusDyadicValue_cast
First and second published papers	FabiusFunction.Paper05442, FabiusFunction.Paper06487	the theorem maps in the module docstrings and docs/PAPER_COVERAGE.md
Corrected sharp and all-orders asymptotics	FabiusFunction.PaperFabiusAsymptotic	abs_log_fabius_dyadic_sub_explicitCumulantMain_le, log_fabius_sub_sharpLambertMain_hasAsymptoticExpansion, fabiusSharpLambertExpansion_two
Complex finite Hamming-weight and Thue--Morse exponential block transforms	FabiusFunction.ThueMorseComplexExponential	sum_range_two_pow_binaryWeight_cexp, sum_range_two_pow_binaryWeight_cexp_affine, sum_range_two_pow_thueMorseSign_cexp, sum_range_two_pow_thueMorseSign_cexp_affine; the weight, exponent, and affine shift are arbitrary complex parameters, and the empty product at m = 0 is included; these declarations are the raw finite transforms, while ThueMorseComplexProductBridge supplies the total complex sinc and negative-Laplace normalizations
Denominator-cleared centered sinc shells and their Thue--Morse zero classification	FabiusFunction.CenteredRvachevThueMorseFourier	centeredSincPartialProduct_dyadic_eq_thueMorse, rvachevFourierProduct_dyadic_eq_thueMorse, rvachevFourierProduct_dyadic_eq_zero_iff_thueMorse, rvachevFourierProduct_dyadic_eq_zero_iff_exists
Finite odd-coset DFT traces and odd-coset-filtered convolution (Ramanujan at power-of-two moduli)	FabiusFunction.HalfIntegerOddDFT	sum_odd_powers_eq_root_filter, oddDFT_add_period, oddDFTPowerTrace_eq_ramanujanConvolution, normalizedOddDFTPowerTrace_eq_two_mul_half
Fourier--Legendre expansions, least squares, coefficient energy, exact rational approximants, Fourier/sinc energy identities, finite translate blocks, and uniformly convergent fixed-scale self-reconstruction	FabiusFunction.FabiusTranslatedLegendreSeries, FabiusFunction.FabiusLegendreLeastSquares, FabiusFunction.FabiusLegendreEnergy, FabiusFunction.FabiusLegendreRationalEnergy, FabiusFunction.FabiusSquareEnergyFourier, FabiusFunction.FabiusLegendreTranslateBlocks, FabiusFunction.FabiusLegendreTranslateSeries	hasSum_canonical_rvachevLegendreSeries_formula, rvachevLegendrePartialSum_pythagorean, rvachevLegendreBlock, intervalIntegral_rvachevLegendreBlock_mul, hasSum_rvachevLegendreCoefficient_energy, hasSum_rvachevLegendreCoefficient_energy_tail, rvachevLegendreSquaredError_partialSum_eq_tsum_tail, fabiusSquareEnergy_eq_tsum_legendre, canonicalRvachevLegendreCoefficientRat, fabiusSquareEnergyPartialSumRat, fabiusSquareEnergyPartialSumRat_pos, monotone_fabiusSquareEnergyPartialSumRat, fabiusSquareEnergyPartialSumRat_three, tendsto_fabiusSquareEnergyPartialSumRat_cast, integral_norm_sq_rvachevFourier_eq_two_mul_fabiusSquareEnergy, fabiusSquareEnergy_eq_integral_Ioi_norm_sq_rvachevFourier, fabiusSquareEnergy_eq_integral_Ioi_tprod_sinc_sq, fabiusSquareEnergy_eq_scaled_integral_Ioi_tprod_sinc_sq, rvachevLegendreDeconvolutionPolynomial, natDegree_rvachevLegendreDeconvolutionPolynomial, leadingCoeff_rvachevLegendreDeconvolutionPolynomial, eval_legendrePolynomial_eq_sum_rvachevUp, eval_legendrePolynomial_even_eq_sum_rvachevUp, rvachevLegendreScale, rvachevLegendreIndexSet, rvachevLegendreAtomCoefficient, rvachevLegendreTranslateBlock, rvachevLegendreTranslateBlock_eq_rvachevLegendreBlock, intervalIntegral_rvachevLegendreTranslateBlock_mul, rvachevTranslateGram, sum_rvachevLegendreAtomCoefficient_mul_gram, hasSum_rvachevLegendreTranslateBlock, hasSum_rvachevLegendreTranslateBlock_uniform, rvachevLegendrePartialSumDeconvolutionPolynomial, natDegree_rvachevLegendrePartialSumDeconvolutionPolynomial, leadingCoeff_rvachevLegendrePartialSumDeconvolutionPolynomial, rvachevLegendrePartialSumTranslateBlock, rvachevLegendrePartialSumTranslateBlockOnInterval, rvachevLegendrePartialSumTranslateBlockOnInterval_apply, rvachevLegendrePartialSumTranslateBlockOnInterval_eq_eval_partialSumPolynomial, rvachevLegendrePartialSumTranslateBlockOnInterval_eq_sum, tendsto_rvachevLegendrePartialSumTranslateBlockOnInterval, rvachevLegendrePartialSumTranslateBlock_tendstoUniformlyOn, tendsto_norm_rvachevLegendrePartialSumTranslateBlockOnInterval_sub, tendsto_rvachevLegendrePartialSumTranslateBlock, eval_rvachevLegendrePartialSumPolynomial_eq_sum_rvachevUp
Inverse construction, exact smoothness locus, interior calculus, curvature, and endpoint steepness	FabiusFunction.FabiusInverse	fabiusInv, fabiusReal_fabiusInv, fabiusInv_hasDerivAt, deriv_fabiusInv_eq_inv_two_mul_rvachevUp, deriv_fabiusInv_pos, fabiusInv_contDiffOn_Ioo, fabiusInv_contDiffAt_infty_iff, fabiusInv_differentiableAt_iff, deriv_deriv_fabiusInv, deriv_fabiusInv_half, deriv_deriv_fabiusInv_half, deriv_deriv_fabiusInv_neg_iff, deriv_deriv_fabiusInv_pos_iff, deriv_deriv_fabiusInv_eq_zero_iff, strictConcaveOn_fabiusInv_firstHalf, strictConvexOn_fabiusInv_secondHalf, id_isLittleO_fabiusInv_pow_at_zero_right, one_sub_isLittleO_one_sub_fabiusInv_pow_at_one_left, tendsto_deriv_fabiusInv_atTop_at_zero_right, tendsto_deriv_fabiusInv_atTop_at_one_left
Fixed-length Fabius increments and exact inverse modulus	FabiusFunction.InverseModulus	Exhaustive public surface (one definition and 31 theorems): fabiusIntervalMass; fabiusIntervalMass_reflect, fabiusIntervalMass_eq_zero_of_add_nonpos, fabiusIntervalMass_eq_zero_of_one_le, monotoneOn_fabiusIntervalMass_firstHalf, antitoneOn_fabiusIntervalMass_secondHalf, strictMonoOn_fabiusIntervalMass_firstHalf, strictAntiOn_fabiusIntervalMass_secondHalf, fabiusReal_sub_le_sub, fabiusReal_lt_fabiusIntervalMass_of_mem_Ioo, fabiusIntervalMass_eq_fabiusReal_iff, fabiusReal_sub_lt_sub, fabiusReal_sub_eq_sub_iff, fabiusReal_add_le, fabiusReal_add_eq_iff, fabiusInv_sub_le_sub_of_mem_Icc, fabiusInv_sub_eq_sub_iff_of_mem_Icc, fabiusInv_sub_le_sub_of_le, fabiusInv_sub_le_sub, fabiusInv_add_le, abs_fabiusInv_sub_le, abs_fabiusInv_sub_eq_iff_of_mem_Icc, dist_fabiusInv_le, fabiusInv_min_one, isGreatest_abs_fabiusInv_sub, sSup_abs_fabiusInv_sub_eq, isGreatest_abs_fabiusInv_sub_Icc, sSup_abs_fabiusInv_sub_Icc_eq, abs_fabiusInv_sub_lt_of_abs_sub_lt_fabiusReal, abs_fabiusInv_sub_le_of_abs_sub_le_fabiusReal, fabiusReal_le_abs_sub_of_le_abs_fabiusInv_sub, forall_abs_fabiusInv_sub_lt_iff. Nonnegative increments are reflection-invariant, vanish on the two constant tails, and have global weak shape; for 0 < h <= 1 they are strictly monotone on the maximal branches [-h,(1-h)/2] and [(1-h)/2,1]. Inside [0,1], the endpoint intervals are the only nondegenerate least-mass intervals, which gives the exact equality cases for constrained forward superadditivity and for the ordered and absolute inverse-gap bounds. Those inverse equality classifications deliberately remain unit-interval statements, because clamping creates additional global cases. The clamped inverse also has global gap bounds and subadditivity, absolute and metric self-moduli, saturation at one, attained exact IsGreatest/sSup moduli at every nonnegative radius over both all real inputs and unit-interval inputs, and the final four effective-injectivity statements. This structural module alone does not construct an explicit recursive denominator; that effective layer is supplied by FabiusInverseEffectiveContinuity below.
Effective dyadic continuity of the totalized inverse	FabiusFunction.FabiusInverseEffectiveContinuity	Exhaustive public surface (two definitions and 12 theorems): inverseFabiusFactorialDenominator, inverseFabiusDeltaDenominator; inverseFabiusFactorialDenominator_eq, inverseFabiusFactorialDenominator_primrec, inverseFabiusDeltaDenominator_primrec, fabiusReal_inverse_two_pow_one_term_lower_bound, inv_inverseFabiusFactorialDenominator_le_fabiusReal, inverseFabiusFactorialDenominator_le_deltaDenominator, inv_inverseFabiusDeltaDenominator_le_fabiusReal, abs_fabiusInv_sub_lt_inverse_two_pow_of_lt_factorialDenominator, abs_fabiusInv_sub_le_inverse_two_pow_of_le_factorialDenominator, abs_fabiusInv_sub_lt_inverse_two_pow_of_lt_deltaDenominator, abs_fabiusInv_sub_le_inverse_two_pow_of_le_deltaDenominator, and fabiusInv_effectivelyUniformContinuous. For r > 0, the positive zeroth recurrence term gives 1 / (2^(r.choose 2) * (r+1)! * (2^r-1)) ≤ F(2^-r). Hence the stronger factorial denominator D!(r)=2^((r+1).choose 2)*(r+1)! and the report denominator DDelta(r)=2^((r+1).choose 2)*(r+1)^(r+1) satisfy 1/D!(r) ≤ F(2^-r) and 1/DDelta(r) ≤ F(2^-r), with D!(r) ≤ DDelta(r). Both natural-valued denominators are primitive recursive. For either denominator, a strict input bound gives the strict inverse-output bound < 2^-r, while the corresponding closed input bound gives ≤ 2^-r. Taking r=n certifies EffectivelyUniformContinuous (fabiusInv F hF). This module itself does not construct the downstream sequential realizer or combined IsComputableRealFunction theorem; those are supplied by EffectiveMonotoneInverse and FabiusInverseComputable. It also does not prove the logarithmic selector r(n), an exact least/ceiling denominator, or an input-bit complexity bound.
Logarithmic reciprocal modulus for the totalized inverse	FabiusFunction.FabiusInverseLogarithmicModulus	Exhaustive public surface (three definitions and 15 theorems): inverseFabiusLogarithmicOrder, inverseFabiusLogarithmicFactorialDenominator, inverseFabiusLogarithmicDeltaDenominator; inverseFabiusLogarithmicOrder_eq_succ_log2, inverseFabiusLogarithmicOrder_primrec, inverseFabiusLogarithmicOrder_isLeast, inverseFabiusLogarithmicOrder_le_self, inverseFabiusLogarithmicFactorialDenominator_of_pos, inverseFabiusLogarithmicDeltaDenominator_of_pos, inverseFabiusLogarithmicFactorialDenominator_primrec, inverseFabiusLogarithmicDeltaDenominator_primrec, inverseFabiusLogarithmicFactorialDenominator_le_deltaDenominator, abs_fabiusInv_sub_lt_inv_nat_of_lt_logarithmicFactorialDenominator, abs_fabiusInv_sub_lt_inv_nat_of_le_logarithmicFactorialDenominator, abs_fabiusInv_sub_lt_inv_nat_of_lt_logarithmicDeltaDenominator, abs_fabiusInv_sub_lt_inv_nat_of_le_logarithmicDeltaDenominator, fabiusInv_effectivelyUniformContinuous_logarithmic, and fabiusInv_effectivelyUniformContinuous_logarithmicDelta. The primitive-recursive selector is exactly r(n)=Nat.log2 n+1; for n>0 it is the least natural r satisfying n<2^r and obeys r(n)≤n. Both logarithmic denominators use value 1 at zero and, at positive n, compose their dyadic predecessor with r(n); both are primitive recursive, and the factorial denominator is no larger than the Delta denominator. For either denominator, both strict and closed input thresholds imply the strict reciprocal output bound <1/n, because 2^-r(n)<1/n. Each denominator separately witnesses EffectivelyUniformContinuous (fabiusInv F hF). The tolerant-bisection realizer, SequentiallyComputable inverse, and combined IsComputableRealFunction theorem are supplied downstream by EffectiveMonotoneInverse and FabiusInverseComputable; the exact endpoint-mass ceiling denominator is supplied by the next leaf, while input-bit asymptotics remain outside Lean.
Sharp exact dyadic reciprocal modulus for the totalized inverse	FabiusFunction.FabiusInverseExactDyadicModulus	Exhaustive public surface (two definitions and ten theorems): inverseFabiusExactDyadicDenominator, inverseFabiusExactLogarithmicDenominator; inverseFabiusExactDyadicDenominator_primrec, inverseFabiusExactLogarithmicDenominator_primrec; inverseFabiusExactDyadicDenominator_pos, inv_inverseFabiusExactDyadicDenominator_le_fabiusAtInverseTwoPow, inverseFabiusExactDyadicDenominator_isLeast, abs_fabiusInv_sub_lt_inverse_two_pow_of_lt_exactDyadicDenominator, exists_fabiusInv_gap_of_lt_exactDyadicDenominator, inverseFabiusExactDyadicDenominator_isLeast_strictModulus, inverseFabiusExactLogarithmicDenominator_of_pos, and abs_fabiusInv_sub_lt_inv_nat_of_lt_exactLogarithmicDenominator. At order r, the first definition is the natural ceiling of the reciprocal of the exact rational value F(2^-r). It is positive, primitive recursive, and least among positive integers whose reciprocal is at most that endpoint mass. Its strict inverse modulus is sharp: every smaller positive denominator fails at the endpoint pair 0, F(2^-r), so it is also least among strict integer moduli for the fixed dyadic output target 2^-r. The logarithmic definition is also primitive recursive; it takes value 1 at zero and equals the fixed-target denominator at the least order r(n) for positive n, where its final theorem gives a strict output bound <1/n. That 1/n result is a witness only, not a leastness theorem for the weaker reciprocal target, and the value at zero is convention-only with no modulus conclusion. Thus the recursive-denominator clause is closed, while no input-bit asymptotic is asserted.
Generic tolerant monotone inversion on the unit interval	FabiusFunction.EffectiveMonotoneInverse	Exhaustive public surface (two definitions and six theorems): Fabius.SequentiallyComputableOn, Fabius.unitClamp; Fabius.unitClamp_sequentiallyComputable, Fabius.tolerantDifference_error, Fabius.tolerantDifference_safe_updates, Fabius.tolerantDifference_inconclusive, Fabius.tolerantBisection_correct, and Fabius.effectiveInversionOn_Icc. At requested precision p, the natural-number realizer performs exactly p dyadic halvings, carrying a bracket index and an optional accepted numerator. Exact signed-code comparisons take the left branch when U+4<L, the right branch when L+4<U, and otherwise certify that the midpoint already has inverse error <2^-p; an accepted numerator is doubled through the remaining depths so the final code always has denominator 2^p. The final bracket endpoint covers the no-hit case, yielding a uniform Computable₂ name with error ≤2^-p. effectiveInversionOn_Icc consumes a supplied computable positive reciprocal inverse modulus; the adjacent gap module constructs one from rational gap lower bounds.
Effective inversion from computable positive rational dyadic gaps	FabiusFunction.EffectiveGapInverse	Exhaustive 4+4 public surface: Fabius.EffectivelyUniformContinuousOn, the structure Fabius.ComputablePositiveRationalSequence, Fabius.ComputablePositiveRationalSequence.value, Fabius.ComputablePositiveRationalSequence.reciprocalDenominator, Fabius.ComputablePositiveRationalSequence.reciprocalDenominator_spec, Fabius.inverseModulus_of_positiveRationalGap, Fabius.effectiveInversionOn_Icc_of_computablePositiveRationalGap, and Fabius.clampedEffectiveInversion_of_computablePositiveRationalGap. A sequence packages computable positive natural numerators and denominators; denominator p / numerator p + 1 is computable, positive, and has reciprocal strictly below its encoded rational value. For a strict increasing inverse pair on [0,1], a lower bound α.value p ≤ f (x + 2^-p) - f x for every x ∈ [0,1-2^-p] yields the inverse modulus. Adding a computable dyadic oracle and both interval maps gives sequential computability and effective uniform continuity of g on [0,1]. The total theorem certifies exactly x ↦ g (unitClamp x); it agrees with g on [0,1] but makes no claim about the un-clamped values of g outside that interval. Together with EffectiveMonotoneInverse 2+6 and FabiusInverseComputable 0+1, the effective-inverse union is three modules and seventeen declarations.
Total computability of the inverse Fabius function	FabiusFunction.FabiusInverseComputable	Exhaustive public surface (zero definitions and one theorem): Fabius.fabiusInv_isComputableRealFunction. It instantiates generic tolerant inversion with the centered-spline dyadic oracle for fabiusReal and inverseFabiusDeltaDenominator, uses unitClamp to turn arbitrary input names into unit-interval names without changing the totalized inverse, and combines the resulting total SequentiallyComputable realizer with the logarithmic-Delta EffectivelyUniformContinuous witness. This is a computability certificate, not a practical running-time or input-bit complexity bound.
Elementary functions and non-elementarity	FabiusFunction.ElementaryFunction, FabiusFunction.AlgebraicBranch, FabiusFunction.InverseBranch, FabiusFunction.NotElementary, FabiusFunction.InverseNotElementary	IsElementary, IsElementary.comp, IsElementary.rpow_of_ne_zero, IsElementary.dense_analyticLocus, analyticDenseOn_of_algebraic, canonical_fabius_not_isElementary_on_Ioo, canonical_fabius_not_isElementary, canonical_fabius_not_algebraicBranch_on_Ioo, IsElementaryOrInverse, fabiusInv_not_analyticAt, canonical_fabiusInv_not_isElementary_on_Ioo, canonical_fabiusInv_not_isElementaryOrInverse_on_Ioo
Computable-real-function theorems	FabiusFunction.FabiusComputableSpline	fabiusSplineApproxPR_computable, extendedFabiusSplineApproxPR_computable, fabius_isComputableRealFunction, globalFabius_isComputableRealFunction
Gaussian-polynomial continuity and the finite-product quotient bridge	FabiusFunction.GaussianBinomialContinuity	Retained 0+3 theorem inventory: continuous_gaussianBinomial, tendsto_gaussianBinomial_nhds_one, and gaussianBinomial_eq_finiteQPochhammerIn_div.
Gaussian-polynomial cumulants, second derivative, and cleared moments at q = 1	FabiusFunction.GaussianBinomialCumulants	Exhaustive public surface (two definitions and 24 theorems): meanAtOne, varAtOne; meanAtOne_one, varAtOne_one, meanAtOne_mul, varAtOne_mul, meanAtOne_prod, varAtOne_prod, eval_one_derivative_X_pow, eval_one_derivative_derivative_X_pow, eval_one_qInt_X, eval_one_derivative_qInt_X, eval_one_derivative_derivative_qInt_X, meanAtOne_qInt_X, varAtOne_qInt_X, one_sub_X_pow_succ_eq, gaussianBinomial_X_mul_prod_qInt, eval_one_gaussianBinomial_X, sum_mean_diff, sum_var_diff, meanAtOne_gaussianBinomial_X, varAtOne_gaussianBinomial_X, eval_one_derivative_gaussianBinomial_X, eval_one_derivative_derivative_gaussianBinomial_X, twelve_mul_secondMoment_gaussianBinomial_eval_one, and twelve_mul_varianceNumerator_gaussianBinomial_eval_one. The last three are the new declarations. The explicit second-derivative formula holds over a characteristic-zero field under exactly k ≤ n. The two denominator-cleared raw-second-moment and variance-numerator identities hold over every commutative semiring for all natural n,k, including the above-row case n < k, where zero extension makes both sides vanish; they require no division, nonvanishing, or characteristic hypothesis.
Jacobi triple product and Euler pentagonal sums	FabiusFunction.JacobiTripleProduct	Retained 2-definition/25-theorem inventory: finite triple-product polynomial and field forms, the bilateral Jacobi HasSum identities, and pentagonal and paired-pentagonal HasSum corollaries.
Infinite q-binomial and reciprocal Euler theorems	FabiusFunction.QBinomialTheoremInfinite	Current 1-definition/29-theorem q-facing inventory: the retained comparison, positivity, uniform Gaussian, Euler-product, q-binomial, and reciprocal-Euler results; the strengthened exponential bound norm_finiteQPochhammerIn_pow_sub_one_le_exp_of_norm_le_one; both fixed-column limit names tendsto_gaussianBinomial_add_const_atTop and compatibility alias tendsto_gaussianBinomial_add_atTop; and the effective IsBigO bounds. The shared finite zero-left identity is imported from GaussianBinomialAtOne.
Effective fixed-column Gaussian convergence	FabiusFunction.GaussianBinomialFixedColumnRate	Exhaustive zero-definition/nine-theorem surface: norm_finiteQPochhammerIn_pow_sub_one_le_exp', norm_finiteQPochhammerIn_pow_sub_one_le, norm_finiteQPochhammerIn_self_mul_gaussianBinomial_sub_one_le, norm_gaussianBinomial_sub_inv_finiteQPochhammerIn_le, norm_gaussianBinomial_add_sub_inv_finiteQPochhammerIn_le, gaussianBinomial_fixedColumn_relativeError_isBigO, gaussianBinomial_shifted_fixedColumn_relativeError_isBigO, gaussianBinomial_fixedColumn_error_isBigO, and gaussianBinomial_shifted_fixedColumn_error_isBigO. The unprimed exponential bound and both fixed/shifted limit names are canonically owned by QBinomialTheoremInfinite; the hypotheses and effective geometric-rate conclusions are unchanged, so thm:fixed-column-limit remains Exact.
Gaussian-binomial asymptotics for real q > 1 (cor:qgreaterone)	FabiusFunction.GaussianBinomialGreaterOneAsymptotics	Exhaustive zero-definition/two-theorem surface: gaussianBinomial_gt_one_fixedColumn_relativeError_isBigO and gaussianBinomial_gt_one_central_isEquivalent. For exactly 1 < q, the first proves (q⁻¹;q⁻¹)_k * (q^(k*(n-k)))⁻¹ * [n,k]_q - 1 = O((q⁻¹)^(n-k+1)) at fixed k; natural subtraction is total, while reciprocity is used only eventually when k ≤ n. The second proves [2m,m]_q ~ q^(m*m) * (q⁻¹;q⁻¹)_∞⁻¹. Together with gaussianBinomial_inv, these give cor:qgreaterone an Exact counterpart without claiming a shifted-central formula or a wider nome domain.
Weighted q-Pascal summation	FabiusFunction.QPascalSummation	Retained 0+4 theorem inventory: sum_gaussianBinomial_succ_mul, sum_gaussianBinomial_succ_mul', Commute.gaussianBinomial_left, and Commute.gaussianBinomial_right.
Quantum-plane binomial expansion	FabiusFunction.QuantumBinomial	Retained 0+2 theorem inventory: quantumPlane_mul_pow and quantum_binomial.
Rogers--Szegő recurrences and generating series	FabiusFunction.RogersSzegoPolynomial	Retained 1-definition/9-theorem inventory: the zero, row-sum, successor, dilation, and three-term laws, the Gaussian successor factor identity, summability and Euler antidiagonal convolution, and hasSum_rogersSzego_generating.
Latest q-calculus closure: bounds, Heine/q-Gauss, complex order, basic hypergeometric series, and q-multinomials	FabiusFunction.QPochhammerInfiniteBounds, FabiusFunction.HeineTransformation, FabiusFunction.QGaussSummation, FabiusFunction.QPochhammerComplexOrder, FabiusFunction.BasicHypergeometricSeries, FabiusFunction.QMultinomial	Exhaustive module counts are respectively 0+5, 2+5, 0+2, 1+4, 2+5, and 1+9: six definitions and thirty theorems. The APIs preserve their explicit strict-contraction, nonvanishing, and denominator hypotheses; the q-multinomial recursion and cleared product identity are division-free, while quotient statements remain conditional.
Foundational polynomial q-calculus, finite-product differentiation, Lambert logarithms, and real q-Gamma	FabiusFunction.PolynomialQDerivative, FabiusFunction.PolynomialQLeibniz, FabiusFunction.QPochhammerDerivative, FabiusFunction.LambertSeriesLog, FabiusFunction.QGamma	Exhaustive counts are 2+17, 0+4, 0+3, 0+4, and 2+10: four definitions and thirty-eight theorems, forty-two public declarations. Definitions: qInt, qDerivative, qGamma, qNumber. Theorems: qInt_zero, qInt_one, qInt_succ, qInt_succ', qInt_add, qInt_one_left, one_sub_mul_qInt, qDerivative_apply, qDerivative_monomial, qDerivative_C, qDerivative_X_pow, qDerivative_X, qDerivative_C_mul_X_pow, eval_qDerivative_mul, qDerivative_comp_C_mul_X, qDerivative_mul, qDerivative_mul'; comp_C_mul_X_comp_C_mul_X, qDerivative_C_mul, qDerivative_iterate_comp_C_mul_X, qDerivative_iterate_mul; hasDerivAt_finiteQPochhammerIn, hasDerivAt_finiteQPochhammerIn_of_ne_zero, hasDerivAt_finiteQPochhammerIn_comp; one_le_norm_natCast_add_one, summable_lambert_series, hasSum_lambert_log_complex, exp_neg_tsum_lambert_eq_qPochhammerInfIn; norm_lt_one_of_pos_of_lt_one, qPochhammerInfIn_rpow_pos, qPochhammerInfIn_self_pos, qGamma_pos, qGamma_one, qGamma_add_one, qGamma_nat_succ, qNumber_natCast, qGamma_mul_qGamma_one_sub, and hasSum_theta_qGamma_reflection. The polynomial laws are division-free over their stated semiring/ring classes for arbitrary nome. The finite logarithmic derivative assumes exactly factorwise nonvanishing, its chain rule exactly the supplied HasDerivAt, and the Lambert results use complex ‖a‖<1, ‖q‖<1 apart from their unconditional norm helper. The positive q-Gamma recurrence uses 0<q<1 and the displayed x>0; qNumber_natCast assumes only q≠1; the reflection product uses only real q<1, and its theta HasSum form uses 0<q<1, 0<x<1. No analytic-polynomial derivative, norm-one continuation, product-level principal-log identity, complex q-Gamma continuation, classical limit, pole, log-convexity, uniqueness, or digamma claim is made.
Gaussian palindromicity, q-exponentials, Jackson integration, and theta quasi-periodicity	FabiusFunction.GaussianBinomialPalindromic, FabiusFunction.QExponential, FabiusFunction.JacksonIntegral, FabiusFunction.ThetaQuasiPeriodicity	Exhaustive module counts are 0+14, 3+8, 1+7, and 1+6: five definitions and thirty-five theorems. They give the degree, monicity, coefficient reversal, mean identity, and total linear-coefficient classifier for Gaussian polynomials; the two q-exponentials and their q-derivative laws; Jackson's fundamental theorem and integration by parts; and the bilateral-theta product, quasi-periodicity, and exact zero criterion. The analytic statements retain their explicit strict-contraction, nonzero-variable, convergence, and nonvanishing hypotheses.
Universal Gaussian structure, q-Pochhammer derivatives, and Jacobi's cubic identity	FabiusFunction.GaussianBinomialPolynomialStructure, FabiusFunction.QPochhammerLogDerivative, FabiusFunction.QPochhammerOrderDerivative, FabiusFunction.JacobiCubic	Exhaustive module counts are 0+5, 0+10, 0+3, and 0+2: twenty theorems and no definitions. They give universal Gaussian degree, monicity, constant coefficient, and reflection symmetry over ℕ[X]; derivative and Lambert-series formulas for the infinite q-Pochhammer product on the unit disc; the complex-order derivative under a nonzero nome and ‖a*q^α‖ < 1; and Jacobi's cubic identity for ‖q‖ < 1.
Cyclotomic factorization and central Gaussian reduction	FabiusFunction.CyclotomicFactorization, FabiusFunction.CentralQBinomialReduction	Exhaustive module counts are 0+7 and 0+6: thirteen theorems and no definitions. The first factors (X;X)_n and [n,k]_X into cyclotomic polynomials, with the Gaussian factorization stated over an integral domain. The second proves finite-symbol sign pairing, even--odd dissection, ring-hom naturality, and the division-free central identity [2k,k]_(q²)(q²;q²)_k=(q;q²)_k(-q;q)_(2k) over every commutative ring; its quotient corollary assumes both denominators are nonzero.
Root-of-unity Gaussian arithmetic, q-Lucas, and MacMahon q-Catalan	FabiusFunction.CyclotomicDivisibility, FabiusFunction.PrimitiveRootBlock, FabiusFunction.QCatalan, FabiusFunction.QLucas	Exhaustive counts are 0+3, 0+3, 1+11, and 0+7: one definition and twenty-four theorems. The tranche proves the cyclotomic carry criterion, Gaussian values and complete q-Pochhammer blocks at primitive roots, the q-Lucas theorem over integral domains, and the integral q-Catalan polynomial with degree n(n-1) and Catalan specialization at q=1. QLucas's local two_mul_choose_two helper is private; the public theorem of that name belongs to QChuVandermonde.
Newton interpolation and the Jackson q-beta integral	FabiusFunction.NewtonInterpolation, FabiusFunction.QBetaIntegral	Exhaustive counts are 3+19 and 1+8: four definitions and twenty-seven theorems. The Newton module constructs triangular coefficients and node-qualified interpolants, proves evaluation, uniqueness, divided differences, and the explicit geometric-grid denominator formula, and retains the seven-name newtonInterpolant compatibility API. The q-beta module defines the Jackson integral, proves its infinite-product and q-gamma evaluations for 0<q<1 and positive arguments, and derives symmetry, positivity, and both successor recurrences.
Integer/complex upper Gaussian coefficients and q-Pfaff--Saalschütz	FabiusFunction.GaussianBinomialInteger, FabiusFunction.GaussianBinomialComplexOrder, FabiusFunction.QPfaffSaalschutz	Exhaustive counts are 1+10, 1+5, and 0+3: two definitions and eighteen theorems. The first module extends Gaussian coefficients to integer upper indices, proves both q-Pascal laws and negative-index reflection, and derives the reciprocal finite q-binomial series. The second uses principal complex powers to package complex upper indices and the generalized reciprocal and finite q-binomial series. The third proves the terminating balanced ₃φ₂ summation algebraically over a field. All nonzero-nome, strict-contraction, and displayed denominator hypotheses remain explicit.
Terminating ₂φ₁ reversal	FabiusFunction.TwoPhiOneReversal	Exhaustive count: two definitions and twelve theorems. Definitions: twoPhiOneFinite, twoPhiOneReflection. Theorems: choose_two_add_succ_choose_two, finiteQPochhammerIn_sub_eq, finiteQPochhammerIn_reversal_ne_zero, finiteQPochhammerIn_inv_pow_self, twoPhiOneReflection_involutive, twoPhiOneFinite_reversal, twoPhiOneFinite_reversal_twice, twoPhiOneFinite_eq_sum_twoPhiOneTerm, twoPhiOne_eq_twoPhiOneFinite_inv_pow, twoPhiOne_reversal, twoPhiOne_reversal_twice, twoPhiOne_one_eq_twoPhiOneFinite_zero. The monograph label lem:2phi1-reversal is Exact: the public result is stated for the actual twoPhiOne tsum, the terminating-series bridge needs no analytic convergence bounds, the reflected parameter map is involutive, and the two displayed prefactors cancel on a second application. The reversal itself retains exactly q,a,c,z ≠ 0 and the three displayed finite-product nonvanishing assumptions; its separate n=0 bridge includes q=0.
The two q-Chu--Vandermonde sums	FabiusFunction.QChuVandermonde	Exhaustive count: zero definitions and ten theorems: two_mul_choose_two, mul_sub_one_eq_mul_sub_add, finiteQPochhammerIn_div_eq_sum_chu, q_chu_vandermonde_first, finiteQPochhammerIn_div_eq_sum_chu_second, twoPhiOneFinite_mul_finiteQPochhammerIn_eq_chu_second, q_chu_vandermonde_second, q_chu_vandermonde_second_by_reversal, twoPhiOne_q_chu_vandermonde_first, and twoPhiOne_q_chu_vandermonde_second. The monograph label cor:q-chu is Exact: its two formulas have actual-twoPhiOne wrappers on the full displayed rational domain q ≠ 0, A ≠ 0, (q;q)_n ≠ 0, (C;q)_n ≠ 0; in particular the second formula assumes neither C ≠ 0 nor (A;q)_n ≠ 0. The label prop:qchu2-by-reversal is Partial: q_chu_vandermonde_second_by_reversal records that proof only on the additional locus C ≠ 0 and (A;q)_n ≠ 0, while the stronger full-domain finite and actual-tsum theorems use a direct denominator-cleared q-Cauchy proof. Rational continuation and the cleared commutative-ring extension asserted in the prose remain unformalized.
Jacobi's two-square count and Lambert identities	FabiusFunction.JacobiTwoSquareCount	Exhaustive count: zero definitions and four theorems, sumSqRep_two_eq_four_mul_twoSquareDivisorSum, sumSqRep_two_eq_four_mul_prod, theta_sq_eq_chi4_lambert, and theta_sq_eq_odd_lambert. For every nonzero natural n, the first theorem proves r₂(n)=4 D(n) in ℤ; the product form additionally assumes even valuation at every prime divisor congruent to three modulo four. The last two theorems unconditionally instantiate the reusable TwoSquareTheorem kernels over every complete normed field under ‖q‖<1, giving the χ₄ and alternating-odd Lambert series.
Noncommutative q-multinomial theorem	FabiusFunction.QuantumMultinomial	Exhaustive count: zero definitions and five theorems. Over an arbitrary semiring, pairwise relations x_j*x_i = q*(x_i*x_j) for i<j, together with commutation of q with every x_i, expand a power of the finite sum into ordered monomials weighted by qMultinomial. The supporting API decomposes tuple antidiagonals, transports Gaussian symmetry to semirings, and proves coefficient commutation. The result is finite and division-free; it makes no convergence claim.

The frontier-facing focused imports above expose exact finite or formal algebra, and their names should not be read as stronger analytic conclusions. The quarter-cell theorems concern the finite spline reportFiniteFabiusApproximant, not a constructed finite inverse G_n; the Catalan--Gaussian filter is an identity in ℚ[[Q]], without convergence or an error bound. The centered Thue--Morse shell concerns the standalone sinc-product model, while the odd DFT module is finite character algebra and does not prove a half-integer aliasing formula or alias-error estimate. Likewise, the geometric principal specialization proves finite residual moments, not spectral-tail convergence. The Lambert tranche goes further: it proves Bell/generalized-harmonic closed forms for every finite residual, fixed-order phase extraction, and weighted residual Big-O estimates, but no convergence of the literal infinite residual series or uniformity as the row order or residual depth grows. The full-order centered parity API is coefficientwise formal algebra; its even-index bridges identify it with the already established compressed moment and cumulant families.

Most analytic theorems first appear in a reusable form with arguments (F : BoundedFabius) (hF : IsFabius F). Canonical corollaries specialize these results to fabius and fabius_spec; globalFabius denotes the signed extension and is intentionally different from the bounded, clamped function outside [0,1]. Range facts such as rvachevUp_nonneg, rvachevUp_le_one, and norm_coe_rvachevUp_le_one need no IsFabius hypothesis because they follow directly from the codomain.

The probability API uses “global” in a deliberately different sense from globalFabius: it means that a formula holds for every real argument. If X is the weighted sum of independent uniform coordinates, then fabiusReal F x = P[X ≤ x] and rvachevUp F x = P[X ≤ 1 - |x|] for every x : ℝ. These identities describe the bounded CDF and its folded bump, not the signed extension extendedFabius F; real-valued and native ℝ≥0∞ measure forms are both available.

The naming scheme distinguishes convergence objects and numeric identities: *_hasSum retains the summability witness, while *_eq_tsum gives the corresponding equality; *_cast bridges exact rational formulas to analysis; and *_zpow uses integer exponents so inverse powers remain visible without division side conditions. The module docstrings state endpoint conventions and any corrections to the printed sources.

Computability as a real function

FabiusComputability.lean formalizes the two clauses in the Grzegorczyk definition of a computable real function. A computable real sequence is presented by one recursive fast dyadic name, uniform in the sequence index and precision: at precision p, a signed-natural pair (a,b) denotes (a-b)/2^p with error at most 2^-p. SequentiallyComputable says that every such sequence is mapped to another such sequence. EffectivelyUniformContinuous uses a recursive positive modulus and the source's reciprocal convention for positive precision indices.

The algorithms in FabiusComputableSpline.lean are entirely natural-number and primitive-recursive. Both compute the Thue--Morse bit, evaluate the finite centered uniform spline on the dyadic grid of order p+3, and round to the nearest dyadic of order p. The bounded evaluator clamps its input to [0,1]; the signed-global evaluator instead rounds an unrestricted signed rational spline code, with negative input names collapsing to the exact zero tail. Each has proved evaluator error 5 * 2^(-(p+3)). Propagating the input-name error through the global 2-Lipschitz bound costs another 2 * 2^(-(p+3)), so the output error is within 2^-p. This proves both fabius_sequentiallyComputable and globalFabius_sequentiallyComputable. The primitive-recursive modulus d(n)=2n proves effective uniform continuity; fabius_isComputableRealFunction and globalFabius_isComputableRealFunction package both clauses for the canonical bounded and signed-global functions. These are computability certificates, not practical running-time claims: the unrestricted positive grid numerator controls the length of a finite primitive-recursive fold. The underlying analytic approximation is stronger than the computability application needs: Fabius.abs_fabiusUniformSpline_sub_extendedFabius_le gives the global error 2^-p, and Fabius.fabiusUniformSpline_tendstoUniformly_globalFabius packages uniform convergence on all of ℝ; the diagonal theorem Fabius.fabiusUniformSpline_tendsto_extendedFabius_of_tendsto also allows the evaluation point to vary with the spline order.

EffectiveMonotoneInverse.lean supplies the generic inverse realizer without ever deciding equality of computable reals. For an output precision p, it runs a fixed-depth dyadic bisection for exactly p steps. At each midpoint, forward and target dyadic approximations are compared by natural-number arithmetic with three outcomes: a certified positive difference moves the right endpoint, a certified negative difference moves the left endpoint, and an inconclusive comparison is already a successful inverse approximation by the supplied reciprocal inverse modulus. The optional accepted numerator is doubled through later frozen steps, so both the successful and no-hit paths finish at denominator 2^p with error at most 2^-p.

EffectiveGapInverse.lean closes the generic gap-to-modulus step. It packages computable positive rational lower bounds for every dyadic forward gap, constructs the computable reciprocal denominator, derives the inverse modulus, and proves sequential computability plus effective uniform continuity on the unit interval. Its total result is precisely the clamped extension fun x => g (unitClamp x); it does not assert that an arbitrary inverse g has the same behavior outside [0,1].

FabiusInverseComputable.lean applies the fixed-depth construction to fabiusReal, its centered-spline dyadic oracle, and inverseFabiusDeltaDenominator. Computable unit clamping extends the unit-interval realizer to arbitrary input names, and the logarithmic-Delta continuity theorem supplies the other clause of Fabius.fabiusInv_isComputableRealFunction. The generic Lean theorem assumes the computable positive reciprocal inverse modulus as input. The adjacent generic gap module derives such a modulus from a supplied computable positive rational gap lower-bound sequence; constructing a particular sequence remains an explicit hypothesis of that interface.

Exact dyadic evaluation

The executable evaluator follows a well-known algorithm, which is Proposition 10 of the paper in computational form. It precomputes the values F(2^-k), removes one highest set bit from the numerator at each step, and evaluates the resulting Taylor polynomial in Horner form. Thus it uses roughly O(n^2 + n * binaryWeight(a)) rational operations for a / 2^n, rather than work proportional to the numerator itself. The rational-input wrapper is the preferred front door because Lean's ℚ representation first reduces inputs such as 10/32 to 5/16.

DyadicCorrectness.lean proves termination, clamping, table-prefix stability, refinement invariance, and representation independence. The inverse-power table is connected axiom-cleanly to the executable moment recurrences in MomentPowerSeries.lean; DyadicClosedForm.lean proves the highest-bit Taylor identity, and DyadicAnalytic.lean proves equality with every bounded analytic Fabius function. GlobalDyadic.lean supplies the corresponding proofs for the signed global extension and for equation (32) at every nonnegative dyadic argument m / 2^n, with no restriction that the representation be reduced or that m ≤ 2^n.

#eval Fabius.fabiusDyadicValue 4 5
-- 305857 / 2073600

#eval Fabius.evalFabiusDyadic (5 / 16 : ℚ)
-- some (305857 / 2073600)

#eval Fabius.evalFabiusDyadic (2 / 3 : ℚ)
-- none

Under the bounded convention, nonpositive inputs evaluate to 0 and inputs at least 1 evaluate to 1. The separate global evaluator retains the paper's oscillating continuation, for example F(3) = -1.

Fourier--Legendre expansion of up

The ordinary Legendre polynomials are constructed from Rodrigues' formula in LegendrePolynomial.lean. The development proves their parity, degree, Sturm--Liouville equation, endpoint values, sharp bound |P_n(x)| ≤ 1 on [-1,1], orthogonality, and exact squared norm 2 / (2n+1).

FabiusLegendreCoefficients.lean evaluates the even Fourier--Legendre coefficients of Rvachev's up function in terms of dyadic Fabius values. The result is the exact finite sum

u_n = 4^(-n) (4n+1) sum (k = 0..n),
  (-1)^(n+k) * choose(2n,n+k) * choose(2n+2k,2n)
  * (2k)! * 2^choose(2k+1,2) * F(2^(-2k-1)).

Finally, FabiusLegendreSeries.lean proves up(x) = ∑' n, u_n P_(2n)(x) for every x ∈ [-1,1]. The convergence is absolute and uniform on that closed interval, so the equality includes both endpoints. The primary public results are Fabius.canonical_rvachevLegendreCoefficient_eq_fabius_sum, Fabius.hasSum_canonical_rvachevLegendreSeries_formula, and Fabius.hasSum_canonical_rvachevLegendreSeries_formula_uniform, with tsum forms available for both. No corresponding equality is asserted outside the natural Legendre interval.

FabiusLegendreLeastSquares.lean proves the finite orthogonal-projection property behind this expansion. If

S_N(x) = sum (n = 0..N), u_n * P_(2n)(x),
E(q) = integral (-1..1), (up(x) - q(x))^2 dx,

then for every real polynomial q of degree at most 2N+1,

E(q) = E(S_N) + integral (-1..1), (S_N(x) - q(x))^2 dx.

Consequently S_N is the unique least-squares minimizer. This is stronger than optimality among polynomials of its visible degree at most 2N: the extra degree is available because the coefficient of P_(2N+1) vanishes. The primary public results are Fabius.rvachevLegendrePartialSum_pythagorean, Fabius.rvachevLegendrePartialSum_least_squares, Fabius.rvachevLegendrePartialSum_error_eq_iff, and Fabius.canonical_rvachevLegendrePartialSum_mem_and_isMinOn.

FabiusLegendreEnergy.lean defines the Legendre blocks B_n(x) = u_n * P_(2n)(x), proves their complete orthogonality formula, and closes the coefficient-energy side of this expansion. In the notation above it proves

integral (-1..1), B_m(x) * B_n(x) dx
  = if m = n then 2 * u_n^2 / (4n+1) else 0,
integral (-1..1), up(x)^2 dx
  = sum (n = 0..infinity), 2 * u_n^2 / (4n+1),
E(S_N)
  = sum (n = N+1..infinity), 2 * u_n^2 / (4n+1),
A_2 := integral (0..1), F(t)^2 dt
  = sum (n = 0..infinity), u_n^2 / (4n+1).

The ten public declarations consist of the two definitions Fabius.rvachevLegendreBlock and Fabius.fabiusSquareEnergy, together with the eight theorems Fabius.intervalIntegral_rvachevLegendreBlock_mul, Fabius.integral_sq_eval_rvachevLegendrePartialSumPolynomial, Fabius.hasSum_rvachevLegendreCoefficient_energy, Fabius.hasSum_rvachevLegendreCoefficient_energy_tail, Fabius.rvachevLegendreSquaredError_partialSum_eq_tsum_tail, Fabius.integral_sq_rvachevUp_eq_two_mul_fabiusSquareEnergy, Fabius.hasSum_fabiusSquareEnergy_legendre, and Fabius.fabiusSquareEnergy_eq_tsum_legendre. The block is defined directly in its polynomial form.

At compiled source checkpoint 9d5f41c2c, FabiusLegendreRationalEnergy.lean adds three executable rational definitions: Fabius.canonicalRvachevLegendreCoefficientRat, Fabius.fabiusSquareEnergyTermRat, and Fabius.fabiusSquareEnergyPartialSumRat. Its fifteen theorems are Fabius.canonicalRvachevLegendreCoefficientRat_cast, Fabius.canonicalRvachevLegendreCoefficientRat_zero, Fabius.fabiusSquareEnergyTermRat_cast, Fabius.fabiusSquareEnergyTermRat_nonneg, Fabius.fabiusSquareEnergyTermRat_zero, Fabius.fabiusSquareEnergyPartialSumRat_cast, Fabius.monotone_fabiusSquareEnergyPartialSumRat, Fabius.fabiusSquareEnergyPartialSumRat_pos, Fabius.fabiusSquareEnergyPartialSumRat_zero, Fabius.fabiusSquareEnergyPartialSumRat_one, Fabius.fabiusSquareEnergyPartialSumRat_two, Fabius.fabiusSquareEnergyPartialSumRat_three, Fabius.hasSum_fabiusSquareEnergy_ratCast, Fabius.fabiusSquareEnergy_eq_tsum_ratCast, and Fabius.tendsto_fabiusSquareEnergyPartialSumRat_cast. Thus every Legendre energy partial sum is represented by a positive rational number, the rational partial sums are monotone, and their real casts converge to A_2. The four displayed values are certified exactly: 1/4, 7/18, 3271/8100, and 3246043/8037225 for cutoffs N = 0, 1, 2, 3, respectively.

At compiled source checkpoint b9b240bc0, FabiusSquareEnergyFourier.lean exports no definitions and exactly four theorems: Fabius.integral_norm_sq_rvachevFourier_eq_two_mul_fabiusSquareEnergy, Fabius.fabiusSquareEnergy_eq_integral_Ioi_norm_sq_rvachevFourier, Fabius.fabiusSquareEnergy_eq_integral_Ioi_tprod_sinc_sq, and Fabius.fabiusSquareEnergy_eq_scaled_integral_Ioi_tprod_sinc_sq. They identify the full real-axis squared Fourier mass with twice A_2, the positive-half-line mass with A_2, and the latter with both the unscaled and 1/(2*pi)-scaled dyadic sinc-product integrals in the Self-Reconstruction report.

In the current compiled tree, RvachevMomentAppell.lean exports six public definitions: Fabius.rvachevRawMomentRat, Fabius.rvachevReciprocalMomentRat, Fabius.rvachevAppellPolynomialRat, Fabius.rvachevAppellPolynomial, and Fabius.rvachevDeconvolvedPolynomial, together with the linear-map package Fabius.rvachevDeconvolutionLinearMap. Its thirty-three public theorems are Fabius.rvachevRawMomentRat_zero, Fabius.rvachevRawMomentRat_even, Fabius.rvachevRawMomentRat_odd, Fabius.rvachevReciprocalMomentRat_zero, Fabius.binomialConv_rvachevRawMomentRat_reciprocal, Fabius.rvachevReciprocalMomentRat_eq_completeBellPolynomial, Fabius.monic_rvachevAppellPolynomialRat, Fabius.natDegree_rvachevAppellPolynomialRat, Fabius.rvachevAppellPolynomial_eq_poly_cast, Fabius.monic_rvachevAppellPolynomial, Fabius.natDegree_rvachevAppellPolynomial, Fabius.eval_rvachevAppellPolynomial_add, Fabius.integral_pow_mul_rvachev_eq_rvachevRawMomentRat_cast, Fabius.integral_eval_rvachevAppellPolynomial_add_mul_rvachev, Fabius.rvachevDeconvolutionLinearMap_apply, Fabius.rvachevDeconvolvedPolynomial_zero, Fabius.rvachevDeconvolvedPolynomial_add, Fabius.rvachevDeconvolvedPolynomial_smul, Fabius.rvachevDeconvolvedPolynomial_finsetSum, Fabius.rvachevDeconvolvedPolynomial_C_mul, Fabius.rvachevDeconvolvedPolynomial_monomial, Fabius.rvachevDeconvolvedPolynomial_X_pow, Fabius.coeff_rvachevDeconvolvedPolynomial_natDegree, Fabius.natDegree_rvachevDeconvolvedPolynomial_le, Fabius.natDegree_rvachevDeconvolvedPolynomial, Fabius.leadingCoeff_rvachevDeconvolvedPolynomial, Fabius.rvachevDeconvolvedPolynomial_eq_zero_iff, Fabius.rvachevDeconvolutionLinearMap_injective, Fabius.rvachevDeconvolvedPolynomial_injective, Fabius.integral_eval_rvachevDeconvolvedPolynomial_add_mul_rvachev, Fabius.integral_eval_rvachevDeconvolvedPolynomial_sub_mul_rvachev, Fabius.integral_eval_rvachevAppellPolynomial_sub_mul_rvachev, and Fabius.integral_eval_rvachevAppellPolynomial_mul_rvachev_eq_zero. They package the full rational raw-moment sequence, its formal binomial-convolution reciprocal and complete-Bell description, rational and real monic Appell families of exact degree, and polynomial deconvolution as an explicit real linear map. In particular deconvolution preserves zero, addition, scalar multiplication, finite sums, and multiplication by constant polynomials; it sends a monomial and X^n to the correspondingly scaled and unscaled Rvachev--Appell polynomial. Its triangular top term is unchanged: it preserves the coefficient in the original natDegree, hence preserves exact natDegree and leadingCoeff, has trivial kernel, and is injective both as the packaged linear map and as the underlying raw operation. Smoothing it by up recovers the original polynomial. Reflection invariance of the even Rvachev density also turns this additive smoothing into the centered x - y convolution for every real polynomial; the corresponding centered Appell identity recovers x ^ n, and every positive-degree Appell polynomial has Rvachev mean zero. These statements do not supply an analytic reciprocal MGF or Appell generating series, a literal differential-operator expansion, reciprocal/deconvolution parity, or the displayed low reciprocal coefficients.

In the current compiled tree, RvachevPolynomialSynthesis.lean exports no public definitions and exactly five public theorems: Fabius.tsum_rvachevDeconvolvedPolynomial_mul_shifted_rvachevUp, Fabius.normalized_tsum_rvachevDeconvolvedPolynomial_mul_shifted_rvachevUp, Fabius.sum_Ioo_rvachevDeconvolvedPolynomial_mul_shifted_rvachevUp, Fabius.normalized_sum_Ioo_rvachevDeconvolvedPolynomial_mul_shifted_rvachevUp, and Fabius.normalized_tsum_shifted_rvachevDeconvolvedPolynomial_mul_rvachevUp. For every nonzero natural mesh M and polynomial of degree at most v₂(M), they give both global tsum synthesis and, on [-1,1], its exact finite k ∈ (-2M,2M) form with the 1/M normalization. The fifth theorem holds for arbitrary real phase θ and real x: it samples up at M⁻¹ * (θ + k), evaluates the deconvolved polynomial at x - M⁻¹ * (θ + k), and reconstructs P.eval x. Taking M = 2 ^ N formalizes arbitrary-phase polynomial reproduction through every degree n ≤ N.

The compiled RvachevSuperconvergentSynthesis.lean extension exports exactly one public definition and eight public theorems. Its definition Fabius.IsRvachevSuperconvergentPhase selects 0,1/2 when v₂(M)+1 is odd and 1/4,3/4 when it is even. The theorem Fabius.isRvachevSuperconvergentPhase_two_pow_iff rewrites this at M=2^N as endpoint phases for even N and quarter phases for odd N. The remaining seven theorems are Fabius.tsum_quarter_monomial_eq_integral_of_even_deg, Fabius.tsum_three_quarters_monomial_eq_integral_of_even_deg, Fabius.tsum_shifted_monomial_eq_integral_superconvergent, Fabius.tsum_shifted_polynomial_eq_integral_superconvergent, Fabius.integral_polynomial_mul_rvachevUp_eq_normalized_tsum_superconvergent, Fabius.normalized_tsum_shifted_rvachevDeconvolvedPolynomial_mul_rvachevUp_superconvergent, and Fabius.normalized_tsum_shifted_rvachevAppellPolynomial_mul_rvachevUp_superconvergent. For every nonzero natural mesh they prove exactness through degree v₂(M)+1, including physical-coordinate quadrature, deconvolved-polynomial reconstruction, and the explicit Appell specialization. This arbitrary-M result is stronger than the dyadic-only manuscript claim. The selected phases are exact representatives rather than a modulo-integer wrapper or a classification, and the module proves neither maximality nor positivity or rationality of the quadrature.

The current-tree module LagrangeRvachevSynthesis.lean exports exactly two public definitions, Fabius.lagrangeRvachevDecoder and Fabius.lagrangeRvachevAtomCoefficient, and exactly seven public theorems: Fabius.natDegree_lagrangeBasis_le_card_sub_one, Fabius.natDegree_lagrangeInterpolate_le_card_sub_one, Fabius.normalized_sum_Ioo_lagrangeRvachevDecoder_mul_shifted_rvachevUp, Fabius.normalized_sum_Ioo_lagrangeRvachevDecoder_eval_node, Fabius.lagrangeRvachevAtomCoefficient_eq_deconvolved_interpolate, Fabius.sum_Ioo_lagrangeRvachevAtomCoefficient_mul_shifted_rvachevUp, and Fabius.sum_lagrangeRvachevDecoder_eq_one. For any finite real node family, the first two bound the basis and interpolant degrees by one less than the node count without requiring distinctness. A nonzero mesh whose two-adic valuation reaches that bound gives exact cardinal and full-interpolant synthesis on [-1,1]; distinct nodes turn the cardinal formula into the componentwise Kronecker-delta identity, and distinct nonempty nodes make each fixed lattice-sample decoder row sum to one. Thus the generic finite-node dictionary, its linear data-to-atom coefficients, componentwise biorthogonality, and exact finite interpolation loop are formalized. No theorem here gives the geometric-node Gaussian q-binomial/q-Pochhammer or elementary-symmetric closed form for the decoder entries, packages the componentwise identity as a matrix/right-inverse equation, or proves an optimal or minimum-variation decoder. The first omission is supplied by the separate RvachevAppellHasse.lean leaf inventoried above, and the typed right-inverse packaging by LagrangeRvachevMatrix.lean; decoder optimization remains open.

The focused-build CompositeMeshSharpness.lean module exports one public definition and seven public theorems. The definition Fabius.rvachevCombExactThrough F M d requires M ≠ 0 and shifted-comb exactness at every real shift for every real polynomial of natural degree at most d. Its two classification theorems prove that this is equivalent to M ≠ 0 ∧ d ≤ v₂(M), equivalently M ≠ 0 ∧ 2^d ∣ M. The remaining order theorems put the canonical mesh 2^d in this class, bound every member below by 2^d, prove that it is the least member, and specialize the least mesh for the complete degree-2N space to 4^N. The real first-defect theorem supplies a shift where the monomial of degree v₂(M)+1 fails. These are universal polynomial-space statements: they do not prove that 2^d is minimal for one Legendre polynomial or that 4^N is minimal for one particular partial sum S_N.

At compiled source checkpoint a3854643d, FabiusLegendreTranslateBlocks.lean exports six public definitions: Fabius.rvachevLegendreDeconvolutionPolynomial, Fabius.rvachevLegendreScale, Fabius.rvachevLegendreIndexSet, Fabius.rvachevLegendreAtomCoefficient, Fabius.rvachevLegendreTranslateBlock, and Fabius.rvachevTranslateGram. Its seven public theorems are Fabius.natDegree_rvachevLegendreDeconvolutionPolynomial, Fabius.leadingCoeff_rvachevLegendreDeconvolutionPolynomial, Fabius.eval_legendrePolynomial_eq_sum_rvachevUp, Fabius.eval_legendrePolynomial_even_eq_sum_rvachevUp, Fabius.rvachevLegendreTranslateBlock_eq_rvachevLegendreBlock, Fabius.intervalIntegral_rvachevLegendreTranslateBlock_mul, and Fabius.sum_rvachevLegendreAtomCoefficient_mul_gram. They specialize the finite synthesis to mesh 2^d, then to the even mesh 4^n. For Q_d = D(P_d), they prove exact degree d and the explicit unchanged leading coefficient (1/2)^d * choose (2*d) d; they also identify the literal finite translate block with the existing polynomial block on [-1,1], and prove both its orthogonality and exact finite atom-Gram formula.

At compiled source checkpoint a3854643d, the focused-build module FabiusLegendreTranslateSeries.lean exports five public definitions: Fabius.rvachevLegendrePartialSumDeconvolutionPolynomial, Fabius.rvachevLegendrePartialSumAtomCoefficient, Fabius.rvachevLegendrePartialSumTranslateBlock, Fabius.rvachevLegendreTranslateBlockOnInterval, and Fabius.rvachevLegendrePartialSumTranslateBlockOnInterval. Its twenty-five public theorems are Fabius.natDegree_rvachevLegendrePartialSumDeconvolutionPolynomial, Fabius.leadingCoeff_rvachevLegendrePartialSumDeconvolutionPolynomial, Fabius.summable_norm_rvachevLegendreTranslateBlock, Fabius.summable_rvachevLegendreTranslateBlock, Fabius.hasSum_rvachevLegendreTranslateBlock, Fabius.tsum_rvachevLegendreTranslateBlock, Fabius.rvachevLegendrePartialSumDeconvolutionPolynomial_eq_sum, Fabius.rvachevLegendrePartialSumAtomCoefficient_eq_sum, Fabius.eval_rvachevLegendrePartialSumPolynomial_eq_tsum_rvachevUp, Fabius.eval_rvachevLegendrePartialSumPolynomial_eq_sum_rvachevUp, Fabius.rvachevLegendrePartialSumTranslateBlock_eq_eval_partialSumPolynomial, Fabius.rvachevLegendrePartialSumTranslateBlock_eq_sum_translateBlock, Fabius.rvachevLegendreTranslateBlockOnInterval_apply, Fabius.rvachevLegendreTranslateBlockOnInterval_eq_smul, Fabius.summable_norm_rvachevLegendreTranslateBlockOnInterval, Fabius.summable_rvachevLegendreTranslateBlockOnInterval, Fabius.hasSum_rvachevLegendreTranslateBlock_uniform, Fabius.tsum_rvachevLegendreTranslateBlock_uniform, Fabius.rvachevLegendrePartialSumTranslateBlockOnInterval_apply, Fabius.rvachevLegendrePartialSumTranslateBlockOnInterval_eq_eval_partialSumPolynomial, Fabius.rvachevLegendrePartialSumTranslateBlockOnInterval_eq_sum, Fabius.tendsto_rvachevLegendrePartialSumTranslateBlockOnInterval, Fabius.rvachevLegendrePartialSumTranslateBlock_tendstoUniformlyOn, Fabius.tendsto_norm_rvachevLegendrePartialSumTranslateBlockOnInterval_sub, and Fabius.tendsto_rvachevLegendrePartialSumTranslateBlock. Thus the literal outer translate blocks are absolutely summable pointwise and in the interval supremum norm, have pointwise and uniform HasSum/tsum forms, and sum to rvachevUp. The same module identifies the partial-sum deconvolution polynomial with the finite sum of separately deconvolved modes, expands every common-mesh coefficient, proves global tsum and finite [-1,1] synthesis at mesh 4^N, and identifies the resulting finite train both with the polynomial partial sum and with the sum of its separately scaled blocks. It also proves that C_N = D(S_N) has exactly the natDegree and leadingCoeff of S_N, including every degenerate case in which the visible degree drops. The bundled common-mesh trains converge to rvachevUp in C([-1,1]), equivalently in the interval supremum norm; the module also exports the raw TendstoUniformlyOn form, convergence of the supremum-norm error to zero, and the pointwise corollary on [-1,1].

The nine modules in this tranche have respectively 6/33, 0/5, 1/8, 2/7, 1/14, 1/14, 1/7, 6/7, and 5/25 public definition/theorem inventories, for exactly 143 public declarations in total. The added 1/8, 1/14, and 1/14 inventories are RvachevSuperconvergentSynthesis, RvachevAppellHasse, and RvachevLagrangeNodesOnly. Universal whole-space mesh sharpness is now proved, but target-specific minimality for an individual Legendre polynomial or partial sum is not. The finite Hasse-operator expansion, odd reciprocal-moment vanishing, and geometric elementary-symmetric decoder are now proved. The modules still do not assert analytic convergence of a reciprocal MGF or Appell generating series, the displayed low reciprocal coefficients, or the displayed closed formulas for the deconvolved Legendre family, coefficient rationality for the atom rows, equality of the fixed-scale and separately scaled coefficient vectors, decoder optimality, an unconditional natDegree(S_N) = 2*N theorem or nonvanishing of its top Legendre coefficient, or the later refinement, projector, and asymptotic layers.

FabiusTranslatedLegendreSeries.lean translates this expansion to the signed global Fabius function on [0,2]. It proves

P_(2n)(x-1) = sum (j = 0..2n),
  (-1)^j * 2^(-j) * choose(2n,j) * choose(j+2n,j) * x^j,

and substitutes this finite polynomial into the coefficient formula above. The resulting public HasSum and tsum theorems display the complete nested k- and j-sums, including the integer exponent k - j + 2*k^2 - 2*n. They use globalFabius both for the value at x and for every dyadic value inside the coefficient. This signed/global interpretation is essential on 1 < x ≤ 2; the bounded CDF-style function is clamped to one there. At x = 0, natural powers use the convention 0^0 = 1. The primary endpoints are Fabius.hasSum_globalFabius_translatedLegendre_formula and Fabius.globalFabius_eq_tsum_translatedLegendre_formula.

Sharp global regularity

The order-theoretic and metric shape of the two functions is developed in six modules that sit directly on top of Differential.lean and are independent of the paper-index files.

Differential.lean proves the single differential equation

F'(x) = 2 up(2x - 1)      for every real x,

Fabius.fabius_hasDerivAt. It specializes to the defining equation F'(x) = 2 F(2x) on [0, 1/2], to its reflection F'(x) = 2 F(2 - 2x) on [1/2, 1], and to 0 outside [0, 1], so no later derivative computation needs the case analysis.

Monotonicity.lean collects everything order-theoretic. Besides the monotonicity, positivity, and support statements that used to live inside the arXiv:1702.06487v3 index file, it proves the strict theory: F is strictly increasing on [0,1] (Fabius.strictMonoOn_fabiusReal), hence injective there, and the intermediate value theorem promotes this to a bijection of [0,1] onto itself (Fabius.bijOn_fabiusReal). The support of up is exactly the open interval, Fabius.support_rvachevUp, and up is strictly increasing on [-1,0], strictly decreasing on [0,1], and equal to one only at the origin (Fabius.rvachevUp_eq_one_iff). The positivity statements are also given in iff form.

Regularity.lean proves that both F and up are 2-Lipschitz and that the constant cannot be improved: F'(1/2) = 2 up(0) = 2, so Fabius.isLeast_lipschitzWith_fabiusReal and Fabius.isLeast_lipschitzWith_rvachevUp identify 2 as the least Lipschitz constant of each. The linear majorant F(x) ≤ 2x holds on all of [0, ∞).

Convexity.lean shows that F is convex on (-∞, 1/2] and concave on [1/2, ∞), strictly so on the two halves of the unit interval, so the midpoint is the unique inflection point. It also gives the exact pointwise formula

F''(x) = 8 * (up(4x - 1) - up(4x - 3)),

and proves that F'' is positive exactly on (0, 1/2), negative exactly on (1/2, 1), and zero outside (0,1) and at the midpoint. The entry points are Fabius.deriv_deriv_fabiusReal, Fabius.deriv_deriv_fabiusReal_pos_iff, Fabius.deriv_deriv_fabiusReal_neg_iff, and Fabius.deriv_deriv_fabiusReal_eq_zero_iff.

EffectiveFlatness.lean replaces the qualitative o(x^n) flatness statement by the effective bound

F(x) ≤ 2^C(n+1,2) * x^n      whenever 0 ≤ x and 2^n x ≤ 1,

obtained by iterating the mean value estimate F(x) ≤ 2x F(2x). Rvachev's function inherits it at both ends of its support through up(x) = F(1 - |x|). SharpFlatness.lean runs the same induction through the fundamental theorem of calculus instead of the mean value theorem, which recovers the factorial the pointwise estimate throws away:

F(x) ≤ 2^C(n+1,2) / n! * x^n .

At x = 2^(-n) the exact value is 2^(-C(n,2)) d_n / n! with d_n the half moment, so the remaining overshoot is exactly 1 / d_n.

GlobalBounds.lean proves that the signed global extension is bounded by one in absolute value, Fabius.abs_extendedFabius_le_one — the missing ingredient that turns equation (3) into the sharp uniform derivative bounds

|F^(k)(x)| ≤ 2^C(k+1,2)      and      |up^(n)(x)| ≤ 2^C(n+1,2),

both attained, at 2^(-k) and 2^(-n) - 1 respectively, so Fabius.isGreatest_abs_iteratedDeriv_extendedFabius and Fabius.isGreatest_abs_iteratedDeriv_rvachevUp are exact suprema.

BoundedDerivatives.lean carries all of that back to the bounded, CDF-style function. The two functions have the same germ at every argument below one, so equation (3) holds verbatim for fabiusReal there; above one the bounded function is locally constant, and the single remaining point x = 1 is caught by continuity. The consequences are flatness at the origin (Fabius.iteratedDeriv_fabiusReal_zero), the global bound Fabius.abs_iteratedDeriv_fabiusReal_le, and the exact attained supremum Fabius.isGreatest_abs_iteratedDeriv_fabiusReal. More sharply, on the matched mesh m / 2^k with m < 2^k, the kth derivative vanishes exactly at even numerators; at odd numerators it is the sharp amplitude 2^C(k+1,2) times thueMorseSign (m / 2). Hence every odd grid point is an extremizer, not only the familiar point 2^-k.

MidpointEndpointTransfer.lean identifies the entire midpoint defect with the endpoint profile on the closed half-cell:

F(1/2 + h) = 1/2 + 2h - F(h),
F(1/2 - h) = 1/2 - 2h + F(h)       for 0 ≤ h ≤ 1/2.

Both translated first derivatives equal 2 - F'(h), and every iterated derivative of order at least two vanishes at 1/2. The same pointwise identity gives the all-real oriented centered-integral formula ∫_[1/2-a,1/2+a] F = a, arbitrary-weight right and left defect-transfer identities, and every Cauchy kernel (a-h)^n/n! for anchored repeated primitives. These are exact pointwise and finite-integral theorems, not an analytic-germ assertion or a finite-spline quarter-cell statement.

InverseMidpointDefect.lean then converts that transmutation into an exact implicit equation for the totalized inverse. With

h(δ) = F⁻¹(1/2 + δ) - 1/2,
E(δ) = h(δ) - δ/2,

one has, throughout 0 ≤ δ ≤ 1/2,

δ = 2h(δ) - F(h(δ)),
h(δ) = δ/2 + F(h(δ))/2,
E(δ) = F(h(δ))/2 = F(δ/2 + E(δ))/2,
0 ≤ h(δ) ≤ 1/2,        0 ≤ E(δ) ≤ 1/4.

The endpoints are exact: h(0)=E(0)=0, while h(1/2)=1/2 and E(1/2)=1/4, so both displayed upper bounds are attained. The offset enclosure actually holds for every δ ≥ 0, and both h and E are globally odd, including the clamped inverse tails. The fixed point is an exact algebraic identity; no all-orders defect bound, asymptotic equivalence, logarithmic expansion, or Lambert-W transfer is inferred without the additional quantitative estimates those conclusions require.

NowhereAnalytic.lean transfers the unnumbered non-analyticity corollary from up to F and determines the analytic locus exactly:

AnalyticAt ℝ (fabiusReal F) x  ↔  x ∉ [0, 1].

The signed extension is likewise analytic at no point of the first block [0, 2).

Paper coverage

Paper05442.lean is the public import for the first paper. It includes all seven theorems, Lemma 1, the unnumbered non-analyticity corollary, and the prose probability proposition. In particular, it proves the original existence-and-uniqueness characterization with the initially unknown scale. Although positivity of that scale remains a source-faithful field of IsOriginalFabius, IsOriginalFabius.mk_of_derivative_law derives it from the remaining smoothness, support, positivity, normalization, and derivative hypotheses. Every bounded Fabius solution folds to an original compact-support solution, and conversely every original solution has scale two and is the fold of a unique bounded Fabius solution. More sharply, equality of two folds is equivalent to equality of their bounded candidates on (-∞, 1]; restoring the omitted strict right tail gives an exact fixed-candidate iff. The paper aggregate also proves weak-* convergence of the finite convolution measures, pointwise convergence of the polynomial step approximants, the infinite-product probability model, the differential identities, Poisson summation, moment formulas, and global rationality at dyadic points. Its Schwartz construction also exposes rapid decay of every real-axis derivative of the entire Fourier transform through rvachevFourier_real_iteratedDeriv_rapidDecay, with the transform-only form rvachevFourier_real_rapidDecay as a direct corollary.

Paper06487.lean is the public import for the arithmetic paper. PaperStatements.lean contains all 18 proved numbered results in the v3 PDF: Propositions 1, 2, 3, 4, 6, 8, 10, 15, 18, 19, and 22; Theorems 7, 9, 13, 17, 20, and 21; and Lemma 1. It also formalizes Question 5, Definition 12, and Conjecture 16. Paper06487Supplement.lean proves assertions made in the surrounding prose and inside proofs.

PaperFabiusAsymptotic.lean is the public aggregate for the first local draft. It proves the exact logarithmic delay equation, the elementary log expansions, explicit dyadic bounds, the full-real quadratic leading term, and the coarse O(t * log t) error. It also proves that the draft's proposed sharp main term has a nonzero (log t / t)^2 equation residual and therefore is not O(t^-2). The draft's unsupported periodic-in-t argument is not used. Independently, a negative-Laplace product, Mellin finite-part analysis, and quantitative Bromwich saddle proof establish a corrected sharp formula with error O(1 / (-log x)). Its centered periodic correction is reconstructed as an absolutely summable Gamma--zeta Fourier series and proved nonconstant. At dyadic arguments the cumulant approximation is fully effective from index 224043 onward. The component normalized-moment estimates, the endpoint/Laplace comparison, and the final evaluated-constant bound are all public Lean theorems, headed by abs_log_fabius_dyadic_sub_explicitCumulantMain_le. More strongly, if lambda = fabiusLambertPhase x, then for every N

log F(x) = fabiusSharpLambertMain x
  + sum (j < N), lambda^(-j) * fabiusSaddleLogCoefficient j lambda
  + O(lambda^(-N)).

The zeroth coefficient is zero, and the first explicit correction is fabiusFirstSaddleCorrection lambda / lambda. A separate theorem expands lambda itself to arbitrary order in -log x and log (-log x). The full formula keeps the oscillatory coefficient functions at the exact Lambert phase; it does not silently replace them by a lower-order phase approximation.

The asymptotic aggregate also audits four linked Stack Exchange discussions. The recurrence sequence is exposed directly as fabiusRecurrenceSequence n = halfMoment n / n!, with its displayed recurrence, Bernoulli recurrence, inverse-dyadic bridge, generating series, and product all proved. Substituting the inverse-dyadic bridge back into the sequence recurrence gives the direct formula

F(2^(-n)) = 2^(-choose(n,2)) / (2^n - 1) *
  sum (k < n), 2^(choose(k,2)) / (n-k+1)! * F(2^(-k))

for every n ≥ 1; exact rational, generic bounded, generic signed-global, and canonical signed-global forms are exposed by fabiusAtInverseTwoPow_recurrence_zpow, fabiusFunction_inverse_two_pow_recurrence_zpow, extendedFabius_inverse_two_pow_recurrence, and globalFabius_inverse_two_pow_recurrence, respectively. The restriction is necessary: at n = 0 the displayed denominator vanishes.

FabiusInverseDyadicClosedForm.lean solves this recurrence completely. If (r₁,…,rₘ) ranges over the ordered compositions of n and sⱼ = r₁+⋯+rⱼ, then

F(2^(-n)) = 2^(-choose(n,2)) *
  sum (r₁,…,rₘ) ⊧ n,
    product (j = 1,…,m), 1 / ((2^sⱼ - 1) * (rⱼ + 1)!).

For n = 0, the unique empty composition and empty product give the initial value F(1)=1, so unlike the recurrence this formula holds for every natural n. The proof is a finite weighted-path expansion with a formal last-edge decomposition. Public endpoints include Fabius.fabiusAtInverseTwoPow_eq_composition_formula, the explicitly nested Fabius.fabiusAtInverseTwoPow_eq_composition_formula_by_length, and generic, canonical, and signed-global real corollaries. The self-contained derivation is integrated into the primary exposition, available as LaTeX source and as a rendered PDF. The consolidated frontier volume retains longer alternative derivations and their provenance without weakening this exact, primary-document integration.

The conjectured finite q-binomial formula is proved exactly in its full stated scope: for all natural m,n, its half-shifted Thue--Morse sum is the signed global Fabius value at m / 2^n. No condition m ≤ 2^n or irreducibility of the dyadic representation is needed. When m ≤ 2^n, the same formula is a corollary for every bounded function satisfying IsFabius. The rational expression is independent of the representation of m / 2^n (in particular, it is unchanged by (m,n) ↦ (2m,n+1)) and is invariant under any common translation of its inner powers. The finite translated expressions are constant polynomials over ℚ, so they can be evaluated in every field over ℚ. In particular, for every real or complex q the fully displayed sum with inner power (j - m * 2^k + q)^(n+k) has that same value; generic, real, complex, and Gaussian-rational endpoints are public. Thus the source's +1/2 formula and the centered form agree, while its QPochhammer/QBinomial factors retain notation-faithful definitions at the fixed q-special-function base 1/2. For the inverse-power specialization, dyadic reflection additionally proves for every real or complex q the raw-coordinate formula with inner power (r+q)^(n+k) and denominator (-2)^(n^2). Its fully literal theorem uses the zero-one thueMorseBit; at n = q = 0, the sole inner power is 0^0 and evaluates to one. The centered and raw scalar APIs are Fabius.qBinomialThueMorseTranslatedFormulaIn and Fabius.qBinomialThueMorseRawTranslatedFormulaIn; the arbitrary-numerator version is Fabius.qBinomialThueMorseDyadicTranslatedFormulaIn.

The global binary-reduction series is also formalized. Its correct outer index starts at m = 0, where Floor[2^(m-1)x] is genuinely Floor[x/2]. For every real x ≥ 0, the series converges absolutely to the signed global Fabius extension. This specializes to the bounded Fabius function on 0 ≤ x ≤ 1. More strongly, for N ≥ 1 the finite telescope through scale N has all-real error at most 2 * 2^-N: it is the uniformly bounded residual for x ≥ 0, while both the partial sum and signed extension vanish for x ≤ 0. Thus Fabius.globalBinaryReductionSum_tendstoUniformly_extendedFabius proves uniform convergence on ℝ. The complete finite inner expression is a constant polynomial in its common translation, so the theorem holds not only for rational q, but for every real or complex q. The missing m = 0 term is zero on 0 ≤ x < 1 and equals one at x = 1; this explains both why the former one-indexed formula worked on the half-open interval and why it failed at the right endpoint. The primary public endpoints are Fabius.hasSum_qBinomialFabiusGlobalSummand, Fabius.globalFabius_eq_tsum_qBinomialFabiusGlobalSummand_real, and Fabius.globalFabius_eq_tsum_qBinomialFabiusGlobalSummand_complex.

The parity-power form of the binary-reduction series is proved as well, with one necessary correction: the all-x sum begins at m = 0. For every real x ≥ 0, Fabius.globalFabius_eq_tsum_fabiusParityPower_literal is the fully expanded source-style identity and targets the signed global extension. Its exponent is interpreted in ℤ, so the n = 0 exponent is genuinely -1: Fabius.fabiusParityPowerExponent_eq_choose_sub_one records the exact normalization. On [0,1], Fabius.fabiusReal_eq_tsum_fabiusParityPowerSummand gives the bounded Fabius function. The original sum starting at m = 1 is retained, with its correct domain 0 ≤ x < 1, as Fabius.globalFabius_eq_tsum_fabiusParityPowerSummand_succ.

The generalized Wolfram DiscreteLimit formula is proved as well. For every real x ≥ 0 and every q : ℂ, its finite q-binomial/Thue--Morse approximants converge to the signed global Fabius value; on [0,1] the limit is the ordinary bounded Fabius function. Separate public specializations cover rational shifts, Gaussian-rational shifts, and arbitrary real shifts, including irrational ones. Lean encodes the inner sum safely with length ⌊2^(n+k) x + 1/2⌋₊; Fabius.fabiusDiscreteLimitRangeLength_eq_floor_add_one proves that this is exactly the successor of the inclusive Wolfram upper bound Floor[2^(n+k) x - 1/2], including its empty case. A finite row can genuinely depend on q at a nondyadic x; it is the limit that is independent of every fixed complex q. The proof reindexes each row as a uniformly bounded Toeplitz average of centered finite Thue--Morse splines, proves their global convergence through finite uniform-distribution CDFs, and controls a complex shift by a decaying Taylor bound. Finally, exact telescope and tsum theorems identify the same limit with the binary-reduction series; they do not assert a termwise equality between the two finite approximations. The primary endpoints are Fabius.fabiusDiscreteLimit_literal_complex_tendsto_globalFabius, Fabius.fabiusDiscreteLimitApproximationComplex_tendsto_fabiusReal, and Fabius.fabiusDiscreteLimitApproximationComplex_tendsto_literal_tsum. The exact finite-remainder telescope is exposed both generically as Fabius.extendedFabius_eq_qBinomial_telescope_add_remainder and with every nested sum displayed as Fabius.globalFabius_eq_qBinomialThueMorse_telescope_add_remainder_complex.

The recurrence sequence's fixed-constant heuristic omits the nonconstant periodic correction. The elementary small-x expression from Math Stack Exchange is formalized verbatim and corrected by adding that term at the exact lower-Lambert phase. The uncorrected claimed error is formally disproved. Exponentiating the corrected formula gives a proved asymptotic equivalent for the Fabius function itself. Finally, the proposed quotient-of-exponentials fit is little-o of the true displaced Fabius bump at the endpoint, so it cannot be an asymptotic equivalent despite its good compact-interval plot.

PaperKFoldThueMorse.lean is the public aggregate for the second local draft. It contains the exact prefix-sum, zero-run, convolution, and generating-series identities; the intended real polygonal interpolation; and a proved corrected pointwise approximation scheme. It also exposes the zero-one sequence thueMorseBit and proves the exact identity expressing it through Log2 of the signed binomial-parity sum. The Stirling estimate used by the draft is proved in its precise O(log n) form. The aggregate also exposes formal counterexamples to the literal normalization, the claimed local and global error estimates, the unbounded “maximum” proxy, and the omitted linear term in the subsequent Stirling calculation. Both qualitative decay comparisons are proved: the Fabius function is smaller than every power at zero, while exp (-c/x) is little-o of it for every c > 0. No Lambert-W theorem is used to justify the false proxy chain; instead, the repaired lower branch, its equation-(9) solution, and its standard two-term expansion are proved separately.

The exact source-to-Lean map is in docs/PAPER_COVERAGE.md. The requirement-by-requirement asymptotic evidence is recorded in docs/ASYMPTOTIC_COMPLETION_AUDIT.md.

The two arXiv sources contain a few statements that are not literally correct. The formalization records the mathematically valid versions next to their proofs. Among them:

1. In the first paper, equation (12) must be a finite convolution; the printed infinite upper index is incompatible with its dependence on m.
2. The closed interval indicators in Theorem 2 double-count shared endpoints. halfEndpointIntervalIndicator gives endpoints weight 1/2, preserving the asserted normalization φ_n(0) = 1 and the pointwise limit.
3. Equation (25) omits t from its exponential, equation (26) needs n > 0, and equation (32) has inconsistent scaling. The Poisson-summation module proves the corrected identities. Its rvachev_poisson_support_specialization_unscaled_of_one_half_le and rvachev_poisson_support_specialization_of_one_half_le declarations also show that the paper's upper bound a ≤ 1 is unnecessary: both formulas hold on the sharp support-controlled ray a ≥ 1/2.
4. In the arithmetic paper, Lemma 1 is false for a negative scale and an arbitrary derivative order. Its proof requires 0 ≤ scale + order; the Lean statement includes that hypothesis.
5. Proposition 2's quotient (exp x - 1) / x has a removable singularity; expm1Div 0 is defined to be 1.
6. The exponent in R_n is positive in equation (27), its proof, and its displayed values. The development uses that consistent positive exponent.

Documentation policy

Every mathematical document in this directory is a LaTeX document, and its compiled PDF is committed alongside its source.

• Format. Mathematics is written in *.tex, never in Markdown. Markdown is reserved for repository bookkeeping that contains no displayed mathematics: this README, AGENTS.md, docs/PAPER_COVERAGE.md, and the coordination files in AGENTS/.
• Style. New documents reuse the preamble of Fabius_Function_and_Rvachev_Up.tex verbatim — the same geometry, fonts, colours, hyperref setup, running heads, section formatting, theorem environments, macros, and listing style. Only the title block and the PDF metadata change.
• Layout. One directory per document, named after it, containing the .tex and the .pdf of the same name.
• The PDF is committed in the same commit as the .tex, built with three pdflatex passes so that the cross-references and the table of contents settle; the .aux, .log, .out and .toc files are not committed. A .tex change without a rebuilt .pdf is an incomplete commit.
• Prose is Libertinus. The preamble falls back to Latin Modern silently when the font package is missing, so builders verify the committed PDF with pdffonts, install Libertinus first when it is absent, and — only when installation fails — commit a fallback build together with a README.md beside the PDF requesting a rebuild on a Libertinus-equipped machine. Math stays Computer Modern by decision. AGENTS.md states the full rule.
• Check the rendered PDF. Never write LaTeX through a shell heredoc or a script that round-trips through unicode_escape: both silently destroy backslashes, and LaTeX will not complain — it renders something plausible and wrong.
• Keep the primary exposition formalization-backed. Every mathematical assertion in Fabius_Function_and_Rvachev_Up.tex must match one or more actual proved Lean declarations. This rule covers not only theorem environments, but also displayed formulas, prose deductions, exact numerical values, inequalities, asymptotics, convergence statements, and claims about algorithms. The exposition records exact declaration names and modules so the correspondence can be audited; similarity to a theorem or an informal consequence is not sufficient.
• Put unformalized work on the research frontier. Any mathematical material without an exact proved Lean counterpart — however obvious, standard, or plausible — belongs in a LaTeX/PDF document under docs/semi-formalized-research-frontiers/, not in the primary exposition. Frontier documents label conjectures, heuristics, partial formalizations, refutations, and the precise outstanding Lean obligations rather than presenting them as established results.
• Treat drafts as a temporary inbox. Content under docs/semi-formalized-research-frontiers/drafts/incoming/ is reviewed claim-by-claim. Lean-backed material is integrated organically into the primary exposition without duplication; everything else is relocated to the research-frontier tree with its provenance. Once a draft is fully dispositioned, it is removed, and an empty drafts/ directory is deleted rather than retained as an archive.

AGENTS.md states the same policy with the exact build commands.

Contributing and coordination

The operational entry point is AGENTS.md: its documentation policy, Lean build guidance, and invariants apply to all work in this directory. Multi-agent coordination is switched by the single file AGENTS/STATUS.md (currently OFF) and specified by AGENTS/PROTOCOL.md: claim-free optimistic Lean work with first-landed-wins integration, one standing owner per canonical document with a fast path for small fixes, a lock-file build mutex, a 2-hour integration-latency cap, bookkeeping on a dedicated orphan branch off main, and a built-in overhead assessment with explicit authority to delete rules that stop paying for themselves. The heavier v1 protocol of the 2026-08 campaign and its rationale survive only in git history (the deleted docs/COLLABORATION.md and docs/MULTI_AGENT_COORDINATION_PROPOSAL.md).

Checking

From the repository root:

lake build +FabiusFunction