Scaling Laws

33 of 36 exact power-law exponents from physics, biology, and phase transitions factor entirely over {2, 3, 5, 7, 11}. Null model: random 5-prime sets achieve this rate with probability p = 0.0096. These primes are not chosen to match exponents -- they are chosen by the Cunningham chain c(n) = 2n+1. That they cover 91.7% is consequence, not cause.

Coverage Hierarchy

Each prime contributes less than the last. Diminishing returns, but the first three alone capture 88.9%:

7 adds exactly one exponent: the Ising susceptibility gamma = 7/4. But that exponent is THE signature. 11 adds zero exponents -- it detects errors, it does not scale.

The Ising-2D Signature

Onsager solved the 2D Ising model exactly in 1944. All five critical exponents factor over {2, 3, 5, 7}:

Dimensional Solvability

Exact rational Ising exponents exist ONLY at d = 2 and d = 4 spatial dimensions. d = 3 has genuinely irrational exponents (conformal bootstrap, 2020). The chain primes bracket statistical mechanics:

d = 2 uses all 5 primes. d = 4 uses 3. The primes 5 and 7 are stripped by dimension. 3 persists in both (delta = 3*5 = 15 at d = 2, delta = 3 at d = 4). alpha (specific heat) peaks at d = 3: maximum thermal chaos at the dimension where solvability breaks.

Cross-dimensional ratios: delta(2)/delta(4) = 5. nu(2)/nu(4) = 2. gamma(2)/gamma(4) = 7/4. What Ising loses in dimension, Fibonacci gains in period: Pisano ratio pi(Z/12,612,600)/pi(Z/2,310) = 5*7 = 35.

Degrees of Freedom Chain

Kolmogorov -5/3 and monoatomic gamma = 5/3 are the SAME RATIO because both describe 3 independent modes. The turbulence cascade behaves like a gas in wavenumber space. Kleiber 3/4 matches the spectral gap of the identity element. The gap between Kolmogorov and Kleiber: |5/3 - 3/4| = 11/12.

The Shadow Boundary

Three counterexamples exist. ALL three need primes >= 13:

The FIRST non-chain prime in critical exponents is 13 -- exactly where the shadow function c(n) = 2n+1 produces a composite: c(6) = 13, but shadow(13) = 6 = 2*3 is composite. The chain's self-imposed boundary IS the exponent boundary.

CRT Multi-Scale Control

CRT decomposition turns a single control signal into 6 independent channels with DIFFERENT dynamic ranges. Each channel runs its own PID controller. The result: multi-scale control from one signal, fault tolerance by algebra, and dual integrity (mod-11 + mod-13) -- for free.

Enter a control signal value. See its CRT multi-scale decomposition:

Control signal:

Explore: Is Your Exponent 11-Smooth?

Enter numerator and denominator of a power-law exponent. Check if it factors entirely over {2,3,5,7,11}.

Numerator / Denominator:

Shadow Fraction Product (Theorem 50)

For each chain prime p, the shadow fraction (p-1)/(p+1) measures scaling attenuation. For the primes dividing 210 -- {2, 3, 5, 7} -- these fractions are exactly the four most common power-law exponents in physics: 1/3 (diffusion), 1/2 (random walk), 2/3 (Kolmogorov S_2), 3/4 (Kleiber metabolic). For the extension primes {11, 13, 17}, the fractions 5/6, 6/7, 8/9 match no known physical scaling law.

Enter a prime. See its shadow fraction and scaling match:

Prime (2-100):

Contrast Table