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

When a finite reservoir becomes a Gibbs weight For a closed system coupled to a finite reservoir, the exact subsystem law is \[ p_a= {g_a\,Ω_R(Eₜₒₜ-E_a)} {∑_b g_b\,Ω_R(Eₜₒₜ-E_b)}. \] The familiar Boltzmann factor is the tangent limit of this finite count, \[ p_a {g_a e-E_a/(k_BT)}{Z}, β=∂lnΩ_R/∂ E=1/k_BT. \] Version 6.0 fixes the sign of the first finite-bath departure. For a reservoir with positive heat capacity, \[ ln\!{Ω_R(Eₜₒₜ-E_a)} {Ω_R(Eₜₒₜ)e-β_RE_a} =- E_a^2/2k_BT(ξ_a)^2C_R(ξ_a)≤0, \] 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: \[ 0≤ln W_n ≤∑ₑ₌₁ⁿln d_e ≤ Nrecln dₘₐₓ. \] Thus a history of completed records constrains admissible Boltzmann multiplicity without identifying \(Nrec\) with \(ln W\). Under an injective extension history, \[ qₙ₊₁=ln{Wₙ₊₁}{W_n}≥0, ln W_n=ln W_0+∑_e q_e. \] 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