Part of the Decality — one ring (Z/970200Z), 108 lattice structures.
A molecule in 3D space gains degrees of freedom in steps of D=2. Translation (K=3) → rotation (+D=2 → E=5) → vibration (+D=2 → b=7). Click each node.
| Gas | Type | γ measured | Axiom | γ axiom | Error |
|---|
The ring Z/970200Z has 48,750 distinct eigenvalue classes. Entropy decomposes additively over the five CRT channels. 48,750 = 2 × 3 × 54 × 13. The shadow-chain stopper 13 appears in the entropy.
Onsager's exact solutions for the 2D Ising model. All five critical exponents factor over {1, 2, 3, 5, 7}. Only γ = 7/4 requires b=7 (depth).
| Law | Exponent | Axiom | Meaning |
|---|---|---|---|
| Stefan-Boltzmann | T4 | TD² | D²=4 spatial DOF each contribute T |
| Debye (low T) | T3 | TK | K=3 dimensions of phonon modes |
| Wien displacement | T1 | Tσ | Linear shift of peak frequency |
| Fick diffusion | t1/2 | tσ/D | Random walk in duality |
| Dulong-Petit | Cv=3R | Cv=K*R | Closure = classical limit per atom |
| Equipartition | U=fkT/2 | U=f*kT/D | D=2 divides energy per DOF |
| Kolmogorov | k-5/3 | k-E/K | = monoatomic γ. Same K=3 modes. |
| Planck prefactor | 2hν3/c2 | D*h*νK/cD | D polarizations, K mode density, D light |
Click or drag to change temperature. The curve shifts as Tσ (Wien) and total power grows as TD² (Stefan-Boltzmann).
Both turbulence and noble gases have K=3 independent modes. A monoatomic gas has 3 translational degrees of freedom → γ = (K+D)/K = E/K = 5/3. Turbulent energy cascades through 3D wavenumber space → E(k) ~ k-E/K. The ratio 5/3 is not a coincidence. It is the same theorem.
The E2=25 channel has floor(25/2)+1 = 13 = GATE distinct eigenvalue levels. 13 is where the Cunningham chain stops (shadow(13)=6=composite). The same boundary that stops the axiom's self-generation appears as a factor in the ring's total entropy: 48,750 = 2 × 3 × 54 × 13.
The D3=8 and K2=9 channels both produce exactly 5 eigenvalue classes (floor(8/2)+1 = floor(9/2)+1 = 5). Despite having different sizes and different algebraic structures, they carry identical entropy: ln(5) = 1.609. Duality cubed = closure squared, informationally.
Kolmogorov 5/3 = E/K appears twice: as the monoatomic gas ratio AND as h(−47)/h(−23) = 5/3 along the D-chain of class numbers. The same ratio rules turbulence, thermodynamics, and algebraic number theory. Partition function factorization Z = ∏Zp echoes the partition function p(n), which stays axiom-smooth for n≤12 and breaks at the GATE p(13)=101. Ising exponent γ=b/D²=7/4: consecutive Fibonacci ratios from the D-chain (13/8).
D-chain class numbers → Partitions & the gate → Modular forms →
Verify the numbers. Explore the ring.
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