Boltzmann Weighting from Finite Completed-Event Reservoirs: The Completed-Record–Multiplicity Bridge, Signed Finite-Bath Corrections, and a Seven-Face Thermal Consistency Ledger

Abstract

When a finite reservoir becomes a Gibbs weight For a closed system coupled to a finite reservoir, the exact subsystem law is \[ p_a= \frac{g_a\,\Omega_R(E_{\rm tot}-E_a)} {\sum_b g_b\,\Omega_R(E_{\rm tot}-E_b)}. \] The familiar Boltzmann factor is the tangent limit of this finite count, \[ p_a\longrightarrow \frac{g_a e^{-E_a/(k_BT)}}{Z}, \qquad \beta=\frac{\partial\ln\Omega_R}{\partial E}=\frac{1}{k_BT}. \] Version 6.0 fixes the sign of the first finite-bath departure. For a reservoir with positive heat capacity, \[ \ln\!\frac{\Omega_R(E_{\rm tot}-E_a)} {\Omega_R(E_{\rm tot})e^{-\beta_RE_a}} =- \frac{E_a^2}{2k_BT(\xi_a)^2C_R(\xi_a)}\le0, \] so the unnormalised high-energy weight is suppressed relative to the tangent exponential. Negative heat capacity reverses the sign. The release also proves the completed-record/multiplicity bridge: \[ 0\le\ln W_n \le\sum_{e=1}^{n}\ln d_e \le N_{\rm rec}\ln d_{\max}. \] Thus a history of completed records constrains admissible Boltzmann multiplicity without identifying \(N_{\rm rec}\) with \(\ln W\). Under an injective extension history, \[ q_{n+1}=\ln\frac{W_{n+1}}{W_n}\ge0, \qquad \ln W_n=\ln W_0+\sum_e q_e. \] The local-alphabet premise, microcanonical equiprobability, and uniqueness of the source-side per-record valuation remain visible gates. The seven thermal faces are treated as a consistency ledger rather than independent empirical tests. Reader doorway: The Equation on the Tombstone Main Book: 10.5281/zenodo.17527179 Website: quantumtraction.org