mobius-constant

Exact irrational constants — sqrt(2)**2 == 2, by construction.

>>> from mobius_constant import Sqrt2, Pi, Phi
>>> Sqrt2 ** 2 == 2
True
>>> Pi + Pi == Pi * 2
True
>>> Phi ** 2 == Phi + 1
True

IEEE 754 gets all three wrong. MöbiusConstant gets them right.

The Problem

import math
math.sqrt(2) ** 2 == 2        # False  →  2.0000000000000004
math.sqrt(3) ** 2 == 3        # False
phi = (1 + math.sqrt(5)) / 2
phi ** 2 == phi + 1            # False  →  violates defining identity

Forty-one years of IEEE 754 and irrational constants still can't satisfy their own identities.

The Fix

Every constant is stored as two strands:

Strand	Role	Example (√2)
Binary	float64 — hardware-fast	1.4142135623730951
Truth	Verified digit string — 100 digits, no loss	1.41421356237309504880168872420969807856...

The float gives you speed. The digits give you truth. Comparison uses truth.

The digits are pre-verified, not computed on demand:

• π and α cross-check via the SECS equation: α⁻¹ + S·α = 4π³ + π² + π
• √2, √3, φ are verified by their minimal polynomials
• e is verified by Taylor series convergence
• Round-trip π → α → π recovers all 100 digits exactly

Install

pip install mobius-constant

Pure Python. One dependency: mpmath (for non-integer exponentiation only).

Constants

Constant	Symbol	Value	Identity
Pi	π	3.14159265...	Cross-checked via α
Euler	e	2.71828182...	Taylor convergence
Sqrt2	√2	1.41421356...	x² − 2 = 0
Sqrt3	√3	1.73205080...	x² − 3 = 0
Phi	φ	1.61803398...	x² − x − 1 = 0
Ln2	ln 2	0.69314718...	—
Alpha_inv	α⁻¹	137.035999...	SECS equation
Alpha	α	0.00729735...	1/α⁻¹

Compute Your Own

from mobius_constant import MC, sqrt

# Any square root, exact to 100 digits
s5 = sqrt(MC(5))
assert s5 ** 2 == 5

# Arithmetic propagates both strands
x = MC("1.41421356237309504880168872420969807856967187537694")
y = x * x  # truth strand: exact multiplication of all digits

The α ↔ π Relationship

The fine-structure constant α and π are not circularly defined. Both are independently real:

• π — computable by Chudnovsky, Machin, BBP, dozens of algorithms
• α — measurable from electron g-2, cesium recoil, rubidium recoil

The equation α⁻¹ + S·α = 4π³ + π² + π is the auditor, not the definition. It validates that the digits of both constants are correct. If the round-trip fails at digit N, either your π or your α is wrong — the equation tells you where.

See Also

• mobius-number — Complementary residue arithmetic for rationals (0.1 + 0.2 == 0.3)
• mobius-integer — Dual-strand integer: machine i64 + exact BigInt (Rust)
• mobius-units — Fundamental constants from the eigenvalue tower — one measurement derives them all

Same pattern. Same anatomy. Same fix. Different domain.

License

MIT — Jay Carpenter, 2026