Boltzmann Weighting from Finite Completed-Event Reservoirs: A Conditional QTT Reconstruction of the Canonical Distribution, the Role of k_B, and a Seven-Face Thermal Consistency Ledger

Abstract

When does finite counting become the canonical Boltzmann law? This paper separates the microscopic support, the probability premise, the reservoir approximation, and the laboratory units map. A finite completed-event reservoir gives the exact subsystem marginal \[ p_a= \frac{g_a\,\Omega_R(E_{\rm tot}-E_a)} {\sum_b g_b\,\Omega_R(E_{\rm tot}-E_b)}. \] The familiar Gibbs weight is then recovered as a controlled large-reservoir limit, rather than assumed to be exact for every finite bath: \[ q_a=\frac{g_a e^{-\beta_R E_a}}{Z_R}, \qquad \beta_R=\left.\frac{\partial\ln\Omega_R}{\partial E}\right|_{E_{\rm tot}}. \] The finite-curvature correction is explicit. If \[ s_R(E_{\rm tot}-E_a) =s_R(E_{\rm tot})-\beta_R E_a+r_a, \qquad |r_a|\le\varepsilon_R, \] then \[ e^{-2\varepsilon_R} \le \frac{p_a}{q_a} \le e^{2\varepsilon_R}, \qquad \lVert p-q\rVert_{\rm TV} \le \frac{e^{2\varepsilon_R}-1}{2}. \] This turns "the reservoir is large" into a quantitative statement that can be checked. The count-to-entropy map is also isolated as a theorem. Every zero-preserving, additive, monotone map from dimensionless count entropy to laboratory entropy is linear: \[ S_{\rm lab}=\Phi(\ln W)=k_B\ln W, \qquad k_B=\Phi(1)>0. \] Thus one common scalar must propagate through every thermal face written on the same laboratory entropy and temperature rails. The theorem fixes the role of \(k_B\); the SI decimal remains fixed by the laboratory kelvin convention. The QTT-specific proposal is the microscopic support: a cell-aligned reservoir of completed A5-X events. The paper keeps two additional obligations visible: a finite local physical-label alphabet and microcanonical equiprobability. They are premises, not hidden consequences of A6. A conditional edge cross-lock Combining the standard Unruh relation with the QTT edge definitions \[ a_\ast=\frac{c^2}{\ell_A}, \qquad E_\ast=\frac{\hbar c}{\ell_A} \] gives \[ \boxed{2\pi k_B T_U(a_\ast)=E_\ast}. \] No value of \(G\) is required for this cancellation. The standard Unruh relation retains its established QFT provenance; the equation is a conditional cross-sector lock. Thermal inheritance ledger The release follows the same scalar through entropy, the first law, the canonical limit, thermal angular and cyclic times, blackbody radiation, and the ballistic noise-heat identity. Their dependencies are printed explicitly: they are consistency and propagation faces, not seven independent likelihood factors. Scientific status SIGMA-KB-ADDITIVE-MONOTONE-MAP-CLOSED SIGMA-A5X-CELL-ALIGNED-COUNT-CLOSED SIGMA-FINITE-RESERVOIR-MARGINAL-CLOSED-GIVEN-P2 SIGMA-CANONICAL-LIMIT-ERROR-BOUND-CLOSED SIGMA-LOCAL-ALPHABET-P1-OPEN SIGMA-EQUIPROBABILITY-P2-OPEN OBS-QTT-EXCLUSIVE-THERMAL-DISCRIMINATOR-OPEN Version: 5.0 Concept DOI: 10.5281/zenodo.20322035 Author: Ali Attar Website: quantumtraction.org Main book: Quantum Traction Theory: Main Book v10.01 Related QTT anchors Entropy Production as Anchored Modular Charge Unified Equilibrium Law: modular-charge fifth face Universal Quantum Capacity Laws and Precision QED Corpus Tree entry Derivation Atlas QTT Lexicon Included public files PDF paper, Version 5.0 Full reconstruction and verification package