{ 2, 3, 5, 7, 11 }

Statistical Mechanics from Ten Terms

The partition function IS the Chinese Remainder Theorem. Z factors exactly over five prime channels. Fermi-Dirac uses +sigma, Bose-Einstein uses mirror. All scaling relations sum to D=2.

Part of the Decality — one ring (Z/970200Z), 108 lattice structures.

The Partition Function Factorization Theorem

Z(β) = Z₂ · Z₃ · Z₅ · Z₇ · Z₁₁

The partition function of Z/2310Z factors EXACTLY over CRT channels.
Each channel thermalizes independently. Free energy, entropy, heat capacity — all additive.

Drag the temperature slider to see the factorization live:

    T = 1.0  
    Ztotal = ...  
    ΠZp = ...  
    ratio = ...
  

Why does this work? The eigenvalue decomposes: λ(n) = Σ f_p(n mod p). So exp(-βλ) = Π exp(-βf_p). The sum over Z/NZ = product of sums over each Z/pZ, by the CRT bijection. Number theory IS thermodynamics.

Channel Partition Functions

Click a channel card to see its exact partition function formula.

The E-channel is golden. Z₅(β) = e^-2β + 2e^-β/φ + 2e^βφ, where φ = (1+√5)/2 is the golden ratio. The observer's thermodynamics lives on the golden ratio — connecting DNA redundancy (1.608 ~ φ), depth quadratic zeros, and Fibonacci invisible primes.

Entropy Decomposition

S = Σ S_p. Each channel carries its own entropy. At high T, S → ln(N) = Σ ln(p_i).

T = 2.0 | S = ... | max = ln(2310) = 7.745

Maximum entropy = sum of channel entropies. ln(2310) = ln(2) + ln(3) + ln(5) + ln(7) + ln(11). Each CRT channel is an independent reservoir. The ring's entropy is the SUM of five independent entropies.

Distribution Duality — σ vs Mirror

Fermi-Dirac: 1/(e^x + σ) Bose-Einstein: 1/(e^x - σ)

Matter = +σ (ground state). Force = +mirror (-1). The ONLY difference between fermions and bosons.
At E=0: FD = σ/D = 1/2 (half-filling). BE diverges (condensation).

The classical limit is 1/D. When T → ∞, both FD and BE approach 1/(1+1) = 1/D = 1/2. The bridge prime divides the classical world in half. The quantum signatures (+σ vs +mirror) are what lift this degeneracy.

Critical Exponents — Scaling = D

α + 2β + γ = D = 2

Rushbrooke's relation. ALWAYS sums to D=2 — in 2D Ising, mean field, 3D, EVERYWHERE.
The bridge prime controls all scaling relations universally.

2D ISING (Onsager exact)

Exp	Value	Axiom	Meaning
α	0 (log)	0	Specific heat
β	1/8	σ/D³	Magnetization
γ	7/4	b/D²	Susceptibility
δ	15	K*E	Critical isotherm
ν	1	σ	Correlation length
η	1/4	σ/D²	Correlation decay

MEAN FIELD (d ≥ D² = 4)

Exp	Value	Axiom	Meaning
α	0	0	Specific heat
β	1/2	σ/D	Magnetization
γ	1	σ	Susceptibility
δ	3	K	Critical isotherm
ν	1/2	σ/D	Correlation length
η	0	0	Correlation decay

D=2 defines the boundaries. d_lower = D = 2: below this, no spontaneous order (Mermin-Wagner). d_upper = D² = 4: above this, mean field is exact. The bridge prime IS the boundary of phase transition physics.

Independent Fluctuations

var(E) = Σ_p var(E_p)

Energy fluctuations are INDEPENDENT across CRT channels.
= Block-diagonal backprop = independent neural channels = Liouville conservation.

Three fields, one theorem. Statistical mechanics (independent fluctuations), machine learning (block-diagonal Jacobians), and number theory (CRT independence) are the SAME structure seen from three angles.

Quantum Statistics Constants

Click a card to explore.

All 31 Quantities

#	Quantity	Expression	Primes

Verify partition factorization in .ax Verify Rushbrooke = D in .ax Verify golden Z₅ in .ax

Number Theory Thread

The partition function Z = ∏Z_p factors over CRT channels — just as the number-theoretic partition function p(n) factors through Ramanujan’s congruences mod E=5, b=7, L=11. Both partition functions respect the same primes. The golden ratio in Z_E (the E=5 channel) connects to D-chain class numbers: h(−47)/h(−23) = E/K = 5/3, h(−95)/h(−47) = D³/E = 8/5 = golden complement. Scaling relations (Rushbrooke = D) mirror the shadow function s(p) = (p−1)/2 = D·predecessor.

D-chain class numbers → Partitions & the gate → Modular forms →

Paradigm Contrast

Partition function: complex sum

Z = product of 5 channel Z_p

FD/BE: quantum postulate

FD = +σ, BE = +mirror

Scaling: empirical universality

Rushbrooke = D = 2 (structural)

Independent channels: coincidence

CRT = thermodynamic independence

Critical dimension: d=4 magic

d_upper = D² (bridge squared)

Entropy maximum: ln(N)

= Σ ln(p_i) (CRT decomposed)

Fluctuation-dissipation

= block-diagonal backprop = CRT