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Trace-Gibbs predictions for coercive finite regulators

Physical scope

Conditional finite-mode, coercive polynomial trace-Gibbs model at positive temperature and finite-form-energy normal competitors. The three QSM batch-36 carriers remain drafts; their repaired mathematical inputs now have exact canonical carriers, with native readiness separate. No data, thermalization, regulator removal, or continuum interacting QFT claim is made.

Statement

Assume the Axiom of Choice and the finite regulated model of Finite coercive regulated polynomial model, the current draft thermal preparation interface Canonical quantum thermal preparation, the current draft normal-state interface Normal-state quantum measurement interpretation, and the current draft mathematical theorem Self-adjoint trace-Gibbs normalization and form-energy optimization. Its inspected exact interface is: for a nonzero separable complex Hilbert space and a densely defined self-adjoint $A$ bounded below, if $e^{- β A}$ is trace class for every $β$ in an open interval $I \subset (0, \infty)$ , then $Z_{β} \in (0, \infty)$ and $ρ_{β} = e^{- β A} / Z_{β}$ is a faithful normal density; every normal density of finite absolute spectral form energy satisfies $E_{A} (ρ) - T S (ρ) \geq F_{β}$ , with equality exactly at $ρ = ρ_{β}$ . The theorem assumes AC. Under these conditions, for each $β > 0$ J $^{- 1}$ the model state is $ρ_{β} = e^{- β E_{*} h} / Z_{β}$ , with $0 < Z_{β} < \infty$ , and the stated free-energy inequality holds for every normal density $ρ$ whose spectral law $μ_{ρ} (C) = Tr (ρ E_{H} (C))$ has finite first absolute moment $\int ∣ λ ∣ d μ_{ρ} (λ) < \infty$ . The energy $E_{H} (ρ) = \int λ d μ_{ρ} (λ)$ and free energy are in joules, entropy $S (ρ)$ is in joules per kelvin, and $T > 0$ is in kelvin. The conditional result is for the selected fixed finite-mode form realization. It asserts no cutoff removal or empirical realization.

Facts & Assumptions

Given: AC, Countable Choice, finite $r \geq 1$ , coercive polynomial data and selected form realization as in the regulated-model definition, energy scale $E_{*} > 0$ J, and temperature parameter $β > 0$ J $^{- 1}$ .

[F1]

Finite-regulator Gibbs trace and bounded expectations: for the self-adjoint lower-bounded finite polynomial form realization, $e^{- b h}$ is positive trace class with finite positive trace for every $b > 0$ .

[F2]

Quantum spectral entropy and relative-entropy domains: for a trace-class density with positive eigenvalues $p_{i}$ , spectral entropy is $S (ρ) = - k_{B} \sum_{i} p_{i} \log p_{i}$ , possibly infinite; the logarithm is natural and acts on dimensionless density eigenvalues.

[A1]

Canonical quantum thermal preparation (current batch-36 draft): for $T > 0$ , $β = (k_{B} T)^{- 1}$ and the canonical trace-Gibbs density is adopted when the stated open-interval trace hypotheses hold. This is a preparation assumption, not thermalization.

[A2]

Self-adjoint trace-Gibbs normalization and form-energy optimization (current batch-36 draft): assume AC, a nonzero separable Hilbert space, self-adjoint lower-bounded $H$ and trace class $e^{- β H}$ for $β$ in an open positive interval. Then each normal density of finite absolute spectral form energy has finite entropy and satisfies $E_{H} (ρ) - T S (ρ) \geq F_{β}$ , with equality exactly at the normalized Gibbs density.

[A3]

Normal-state quantum measurement interpretation (current batch-36 draft): a prepared normal state is a trace-class density on a specified separable Hilbert space, and a sharp self-adjoint measurement with spectral PVM $E_{A}$ assigns $P_{ρ} {A \in C} = Tr (ρ E_{A} (C))$ . Algebraic states are not identified with normal states by this postulate.

[A4]

The Axiom of Choice: assume AC, as required by [A2].

Proof

1.1givenF1algebra

The selected form realization in the model is self-adjoint and bounded below; write its lower bound as $h \geq - C_{0} I$ . Multiplication by $E_{*} > 0$ preserves self-adjointness and gives $H = E_{*} h \geq - E_{*} C_{0} I$ . Theorem [F1] applies at $b = β E_{*} > 0$ , yielding $0 < Z_{β} = Tr (e^{- β H}) < \infty$ and a positive trace-class numerator. Dividing by $Z_{β}$ produces a positive trace-class operator of trace one.

2.1givenA2A4F1F2step 1.1

Since the same trace finiteness holds for every $b > 0$ by [F1] and AC is assumed explicitly by [A4], the map $e^{- β H}$ is trace class for every $β > 0$ ; in particular it is trace class on an open interval around the chosen positive $β$ . The entropy notation for normal densities is fixed by [F2]. The model, units, and this normal density meet the hypotheses in [A2] under its stated AC and finite-form-energy competitor conditions. Substitution into that conditional variational theorem gives the free-energy inequality and uniqueness on exactly that competitor domain.

3.1A1A2A3step 1.1∎

Under the preparation assumptions [A1] and [A3], the density obtained in step 1.1 supplies the stipulated normal Born predictions for each declared sharp measurement with spectral PVM. The batch-36 postulates remain adopted draft physical premises; this conditional argument does not establish native readiness or empirical validity for them. Neither [A1] nor [A2] asserts a physical process that drives states to the Gibbs density.

Supplier adoption status

The three batch-36 interfaces match the conditional uses above. The trace-Gibbs theorem now cites the exact canonical fixed-temperature trace-Gibbs model and summable canonical differentiation carriers, replacing the historical M33/M16 research placeholders. This supplies its mathematical normalization, discrete spectral and local moment-envelope clauses without extending the finite-form-energy competitor domain. The preparation and normal Born interpretations remain explicit physical assumptions. Native review, input rebinding and readiness are separate obligations; this source adoption is not acceptance.

Trace-Gibbs predictions for coercive finite regulators

Statement

Facts & Assumptions

Proof

Supplier adoption status

Depends on

Dependency tree · 2 levels

Sources