33 of 36 exact power-law exponents from physics, biology, and phase transitions factor entirely over axiom primes {2, 3, 5, 7, 11}. Null model: random 5-prime sets achieve this rate with probability p = 0.0096. The axiom primes are not chosen to match exponents -- they are chosen by the shadow chain. That they cover 91.7% is consequence, not cause.
Each axiom prime contributes less than the last. Diminishing returns, but the first three primes alone capture 88.9%:
| Prime Set | Coverage | Key Exponents Added |
|---|---|---|
| {D=2} | 36.1% | Inverse square, diffusion, Stefan-Boltzmann |
| {D, K=3} | 69.4% | + Kolmogorov E/K, Kepler K/D, surface/volume |
| {D, K, E=5} | 88.9% | + Flory K/E, Potts-3 nu, percolation beta |
| {D, K, E, b=7} | 91.7% | + Ising-2D gamma = b/D^2. ONE exponent. |
| {D, K, E, b, L=11} | 91.7% | L adds NOTHING. L is for ECC, not exponents. |
b=7 adds exactly one exponent: the Ising susceptibility. But that exponent is THE signature. L=11 adds zero exponents -- it protects, it does not scale. Each prime does what the axiom says it does.
Onsager solved the 2D Ising model exactly in 1944. All five critical exponents are axiom ratios:
| Exponent | Value | Axiom Form | Physics |
|---|---|---|---|
| beta | 1/8 | sigma / D^3 | Order parameter |
| gamma | 7/4 | b / D^2 | Susceptibility. THE b=7 signature. |
| delta | 15 | K * E | Critical isotherm. Closure * observer. |
| eta | 1/4 | sigma / D^2 | Anomalous dimension |
| nu | 1 | sigma | Correlation length. Identity. |
Exact rational Ising exponents exist ONLY at d = D = 2 and d = D^2 = 4 spatial dimensions. d = K = 3 has genuinely irrational exponents (conformal bootstrap, 2020). The axiom brackets statistical mechanics:
| Dimension | Axiom Name | Exponents | Primes Used |
|---|---|---|---|
| d = 1 | sigma | No phase transition | Lower critical dimension |
| d = 2 | D | EXACT rational | {sigma, D, K, E, b} = all 5 |
| d = 3 | K | IRRATIONAL | Decomposes solvability |
| d = 4 | D^2 | EXACT rational (mean field) | {sigma, D, K} = 3 survive |
| d > 4 | > D^2 | Mean field | Same as d = D^2 |
2D uses all 5 primes. 4D uses 3. E and b are STRIPPED by dimension. K persists in both (delta = K*E at d=D, delta = K at d=D^2). K is solvability's spine. alpha (specific heat) PEAKS at d=K: maximum thermal chaos at the decomposition dimension.
Cross-dimensional ratios are all axiom: delta(D)/delta(D^2) = E. nu(D)/nu(D^2) = D. gamma(D)/gamma(D^2) = b/D^2. What Ising loses in dimension, Fibonacci gains in period: pi(TRUE)/pi(THIN) = E*b = 35.
| Gas Type | DOF | gamma | Connection |
|---|---|---|---|
| Monoatomic | f = K = 3 | E/K = 5/3 | = Kolmogorov turbulence exponent |
| Diatomic | f = E = 5 | b/E = 7/5 | + D=2 rotational modes |
| Polyatomic | f = b = 7 | K^2/b = 9/7 | + D=2 vibrational modes |
Kolmogorov -5/3 and monoatomic gamma = 5/3 are the SAME RATIO because both describe K=3 independent modes. The turbulence cascade IS a gas in wavenumber space. Kleiber 3/4 = K/D^2 = spectral gap of sigma. The gap between Kolmogorov and Kleiber: |5/3 - 3/4| = 11/12 = axiom sum / chromatic number.
Three counterexamples exist. ALL three need primes >= 13:
| Exponent | Value | Non-Axiom Prime |
|---|---|---|
| Potts-3 gamma | 13/9 | 13 = shadow stopper |
| Perc-2D gamma | 43/18 | 43 = outside axiom |
| Perc-2D tau | 187/91 | 187 = L*17, 91 = b*13 |
The FIRST non-axiom prime in critical exponents is 13 = exactly where the shadow chain breaks. shadow(13) = 6 = D*K = composite. The axiom's self-imposed boundary IS the exponent boundary. The shadow chain is the spectral ruler.
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