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

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