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Build a tau count one layer at a time.
IDWT uses levels 23 = 1 + 22.
• A one-level pattern contributes 1 at every depth.
• Vandermonde’s identity therefore reduces the tau count to a sum of 22-level counts at depths 0 through 10.
• Add those eleven layers.
Result: 64,512,240 portions in the assigned tau pattern.
How IDWT checks an orbit’s stability:
1. Solve for the steady locked orbit and its coupled source.
2. Linearize the equations, keeping all 15 source components.
3. Compute the eigenvalues and reduce the numerical step to check their real parts.
For turning ratios L = 2 through 8, the calculation finds no growing mode: the locked orbits are linearly stable.
![Calculated geometric chart of projected inner and outer radii versus viewing angle χ from 0 to 45 degrees for the unit-radius locked four-direction orbit. Starting from r squared equals one half times [1 plus sin(2χ) cos(2Lθ)], the cosine bounds give radii sqrt[(1 minus or plus sin(2χ))/2]. At zero angle both equal 1/sqrt(2). At 45 degrees the outer radius is 1 and the inner radius is zero. Shading marks the interval of projected radii, not density or measurements.](https://glommer.org/uploads/7a/130170-thumb.jpg)
Turn the view; change the ring.
IDWT derives the apparent radii of its locked four-direction orbit:
1. r² = [1 + sin(2χ) cos(2Lθ)]/2.
2. The cosine ranges from −1 to +1.
3. So the inner and outer radii are √[(1 ∓ sin(2χ))/2].
At χ = 0 the radius is constant. At 45° the inner radius reaches zero and petals appear.
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Density × volume = field norm.
IDWT derives the scaling near the frequency threshold:
1. Call the gap s = m₀ − ω. The density scales as s⁹.
2. The spatial size scales as s⁻¹ᐟ², so volume in d bound directions scales as s⁻ᵈᐟ².
3. Multiply the two.
Result: N = c_d s^(9 − d/2). At depth 6, the exponent is 6.
Shrink the orbit; speed up its rhythm.
For a hydrogen-like atom with Z protons, IDWT works through this scaling:
• Net pull: a − b = Z.
• Orbit radius: r = rH/Z.
• Squared orbital rate: Ω² = Z/r³, so it grows as Z⁴.
• Take the square root.
The orbital rate scales as Z²: Z = 2 gives four times hydrogen’s rate.
Energy can leave our view while remaining in the field.
IDWT derives the balance for a packet moving along a hidden spatial direction:
1. Start with local energy conservation.
2. Integrate along the visible direction at our slice.
3. The visible boundary flux vanishes, leaving the hidden-direction flux.
The slice gains and loses energy through that flux; total energy stays constant.
A right-angle turn doubles this geometric count.
IDWT calculates the field count on a round sphere in four spatial directions:
1. At a right-angle tilt, the numerator integral becomes ∫₀^π 1 dθ = π.
2. The untilted normalization is ∫₀^π sin²θ dθ = π/2.
3. Divide: π ÷ (π/2) = 2.
The normalized right-angle count is exactly 2.
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From a mass ratio to three masses.
IDWT uses neutrino counts 2,002 : 11,628 : 65,780.
1. Write each mass as its count × one scale μ.
2. The measured squared-mass gap gives μ²(65,780² − 2,002²) = 2.534 × 10⁻³ eV².
3. Take the square root, then multiply by each count.
Result: 1.533, 8.903 and 50.362 meV.
How much of a freely oriented pattern points along one observed direction?
IDWT derives the average in ten spatial dimensions:
1. A depth-d pattern has d bound directions.
2. Rotational symmetry shares its squared projection equally across all ten axes.
3. The shares sum to d, so each averages d/10.
For depth 6, the mean squared share is 0.6.
One fold becomes two visible crossings.
IDWT follows a bent free line through our observed slice:
1. Near the fold, its hidden coordinate is w = x²/(2ρ) − vt.
2. Set w = 0 to find the crossings: x = ±√(2ρvt).
3. Their separation is 2√(2ρvt).
The pair separates as √t: four times the elapsed time gives twice the gap.
![Calculated geometric diagram for the L=5 locked orbit. Left: five petal tips form a regular pentagon on a unit circle. Right: the end-on path has coordinates x3=cosθ sin(5θ) horizontally and x1=cosθ cos(5θ) vertically. Dashed horizontal lines connect polygon vertex heights to ring-axis crossings at 1, 0.309 (twice) and −0.809 (twice), as listed in Table 2. Steps show tip condition Lθ=πj and shared coordinate x1=cos[πj(L+1)/L].](https://glommer.org/uploads/f9/127481-thumb.jpg)
A polygon hidden in an orbit.
IDWT derives the Frost-circle geometry from its locked electron path:
1. Petal tips occur at Lθ = πj.
2. Their shared coordinate is x₁ = cos[πj(L+1)/L].
3. The end-on orbit crosses its axis at those same heights.
For L = 5: 1, 0.309 (×2), −0.809 (×2)—the five polygon levels.
How does radius set an orbit’s rhythm?
In IDWT’s nearly circular proton orbit:
1. The up-quark pull sets the fourth-direction oscillation.
2. Subtract the down-quark push to get the pull setting revolution.
3. Divide their squared rates to get L².
At width 100 and radius 0.9999 Bohr radii, the ratio is L = 3.
Where does a shell capacity of 32 come from?
IDWT counts angular patterns on a shell in four spatial directions:
• Degree-3 polynomials: 20.
• Subtract the 4 Laplacian constraints: 16 harmonic patterns.
• Two electrons per pattern: 2 × 16 = 32.
The general capacity is 2(m+1)²: 2, 8, 18, 32, 50.
