Physics theoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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Invariance of a regular-shell microcanonical preparation

Physical scope

A classical Hamiltonian flow that is globally defined and invertible on a compact regular energy band with the stated coarea measure; invariance is not ergodicity, mixing, equilibration, or equality of time and ensemble averages.

Statement

Let a physical Hamiltonian model satisfy Regular Hamiltonian shell preparation data on a nonempty compact regular energy band, and adopt its normalized shell law by Microcanonical sampling assumption. In addition to the definition's $C^{2}$ regularity, assume its numerical Hamiltonian $\hat{H}$ is $C^{\infty}$ . Let $ϕ_{t}$ be the physical Hamiltonian flow defined there, with $t$ measured in seconds, and assume it is an invertible smooth map of the whole band for every real time. Then, conditional on the current batch-12 draft mathematical interfaces Hamiltonian volume, recurrence, and ergodic scope and Coarea and invariant measure on compact regular energy shells, every shell probability $ν_{\hat{E}}$ is invariant: $ν_{\hat{E}} (ϕ_{t}^{- 1} (A)) = ν_{\hat{E}} (A)$ for every Borel set $A \subseteq Σ_{\hat{E}}$ and every $t \in R$ . This result asserts no ergodicity, time-average equality, or thermalization.

Facts & Assumptions

Given: A canonical Hamiltonian chart, a nonempty compact regular energy shell, its normalized coarea preparation, and a globally defined invertible flow on the band.

[F1]

Regular Hamiltonian shell preparation data: the dimensionless energy level is regular and compact and the normalized coarea probability $ν_{\hat{E}}$ is defined.

[F2]

Microcanonical sampling assumption: on this admitted regular shell the assigned microcanonical preparation is $ν_{\hat{E}}$ .

[L1]

Hamiltonian volume, recurrence, and ergodic scope (current batch 12 draft): under AC, a smooth Hamiltonian on a symplectic manifold has a local flow preserving symplectic volume wherever defined; completeness or global existence is not inferred. The current Statement and full proof are present and were inspected; Step 1 readiness is missing, so this premise remains conditionally used.

[L2]

Coarea and invariant measure on compact regular energy shells (current batch 12 draft): for $C^{2}$ energy on an open Euclidean domain with compact regular energy bands, the coarea construction has finite positive mass on each nonempty shell; a Hamiltonian flow that is a diffeomorphism of a full compact band and preserves that band preserves each normalized shell measure. The current Statement and full proof are present and were inspected; Step 1 readiness is missing, so this premise remains conditionally used.

[A1]

AC is included as the exact set-theoretic assumption of [L1]; no ergodicity hypothesis is assumed.

[A2]

The numerical Hamiltonian is $C^{\infty}$ , strengthening the shell definition's $C^{2}$ assumption to match [L1]; physical flow time $t$ is measured in seconds, its dimensionless time is $\hat{t} = t / (1 s)$ , and the physical flow is assumed to be a smooth invertible map of the whole band for every real time. This global-flow condition is not inferred from the local Hamiltonian equations.

Proof

Given: The regular-shell model, its adopted shell probability, AC, and an invertible smooth Hamiltonian flow defined on the full band.

1.1F1L1L2A1A2givenalgebra

By [F1] and [A2], the canonical Hamiltonian data, smoothness, and full-band invertible flow match the hypotheses of [L1]–[L2]. In dimensionless time, energy is conserved along each flow line because $\frac{d}{d \hat{t}} \hat{H} ({\hat{ϕ}}_{\hat{t}} (x)) = d \hat{H} (X_{\hat{H}}) = \hat{ω} (X_{\hat{H}}, X_{\hat{H}}) = 0$ ; the global-flow condition is an assumption, not a consequence of local equations.

2.1F2L1L2step 1.1∎

The draft suppliers [L1] and [L2] give volume preservation and shell-measure invariance for the matched model in step 1.1. Normalization by the positive finite shell mass preserves invariance, and [F2] assigns that normalized measure as the preparation. Hence $ν_{\hat{E}} (ϕ_{t}^{- 1} (A)) = ν_{\hat{E}} (A)$ . This deduction remains conditional while both suppliers lack current Step 1 readiness.

Invariance of a regular-shell microcanonical preparation

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources