Thermodynamics — GFT = first law

Statement

The GFT identity (fundamental gap theorem):

is an exact algebraic identity (verified to $< 10^{-15}$ bits at $m = 210$). It reads as the first law of sieve thermodynamics:

The sieve can neither gain nor lose $\log_2 m$ units during its processing: it can only redistribute them between an ordered part ($D_{KL}$) and a disordered part ($H$).

Why it matters

This identification is one of the strongest in PT: the entire sieve thermodynamics (first law, second law, arrow of time, KMS conditions, Bekenstein bound) is derived from the single GFT identity and the Gibbs structure of the sieve — without importing any thermodynamic concept.

• First law: GFT itself (this card).
• Second law ($D_{KL}$ strictly decreasing, $H$ strictly increasing): consequence of T4 convergence and Pinsker’s inequality.
• Arrow of time: differentiation of GFT at fixed $m$.
• Bekenstein: geometric entropy bound on sieve surfaces.

Three partition functions $Z_F = Z_R = Z_P$ (Fisher, Riemann, prime) coincide — anchoring the thermodynamic identification in the strong sense.

Proof — outline

GFT identity. By definition, $D_{KL}(P \| U_m) = \sum_i p_i \log_2(m\, p_i) = \log_2 m + \sum_i p_i \log_2 p_i = \log_2 m - H(P)$.

Hence $\log_2 m = D_{KL} + H$, an exact algebraic identity on every distribution $P$ and every uniform $U_m$.

Thermodynamic identification (BRIDGE role): $U \leftrightarrow \log_2 m$ (total capacity fixed by the modulus), $F \leftrightarrow D_{KL}$ (persistent information = extractable work), $S \leftrightarrow H$ (residual entropy), $T = 1$ (natural temperature in bits/nat). Under this identification, $U = F + TS$ is the first law.

Epiplectic refinement (Finzi et al. 2026): for an observer time-bounded by $T$, GFT becomes $S_T(X) + H_T(X) \leq n + c$ — epistemic structural/entropic separation. The PT identity is the $T \to \infty$ limit.

See also

• GFT — Fundamental gap theorem — source identity
• G1 — Uniqueness of $D_{KL}$ — pins the informational content
• L0 — Unique geometric distribution — maximum entropy
• All theorems