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

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Conditional free Poincare and background Killing charges

Physical scope

Under the exact action and background symmetry, adopted mechanical charge observables are conserved; time-dependent boosts include partial-time cancellation.

Statement

Assume ${A C}_{ω}$ for the moment-map supplier. Adopt either the massive free test-particle action or the distinct massless shell action of Free massive test-particle action and Separate massless shell action. Conditional on those model choices and the corresponding future free-shell equations, the affine Poincaré charges are conserved on flat spacetime; in a fixed-time Hamiltonian description the boost charge includes its explicit partial-time derivative. On a prescribed curved background, every mechanical scalar $p_{a} ξ^{a}$ for a Killing field is conserved along the adopted free shell. These are model deductions. They do not establish the empirical adequacy of either action, and they do not extend to arbitrary interacting potentials.

Facts & Assumptions

Given: A flat inertial Minkowski background or a prescribed smooth Lorentzian background with a Killing field, and one of the explicitly adopted free-particle models.

[A1]

Free massive test-particle action: the massive model adopts fixed primitive mass $m > 0$ and the stationary action $- m c \int \sqrt{- g (x^{'}, x^{'})} d λ$ on future timelike curves with the stated endpoint and regularity conditions. The H5 carrier remains a draft adopted postulate with no pipeline_run binding; its current Step 1 record is ready/hash-current, which is input readiness and not empirical evidence or independent proof approval.

[A2]

Separate massless shell action: the massless model is a separate constrained first-order action with nonzero future null momentum and positive multiplier; it is not obtained by putting $m = 0$ in the massive square-root action. This H5 carrier remains a draft adopted postulate with no pipeline_run binding; its current Step 1 record is ready/hash-current, not an empirical finding or independent proof approval.

[L1]

Poincare cotangent moment map and full generator algebra: the current assigned A draft states shell charge descent and the time-dependent boost identity partial_t B_i+{B_i,E}=0. It is a batch-26 draft with H0/H8 inputs still open.

[L2]

Killing scalar conservation in cotangent shell form: for a Killing vector, ${p_{a} ξ^{a}, C_{m}} = 0$ identically, so the scalar is conserved along the constrained free-shell flow; the proof is conditional on H0/H7.

[A3]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): assume countable choice ${A C}_{ω}$ only; no full axiom of choice is used.

[L3]

First-order shell action yields massive and null geodesics: the current H7 draft assumes a prescribed smooth time-oriented Lorentzian background, a compact chart interval, smooth $x, p, e$ , $e > 0$ , and either the future timelike $m > 0$ branch or the separate nonzero future-null $m = 0$ branch; its stationary equations are $x'^{a} = e g^{a b} p_{b}$ , $p'_{a} = - \frac{e}{2} \partial_{a} g^{b c} p_{b} p_{c}$ , $C_{m} = 0$ . It relates the massive square-root functional to the shell equations; the null branch remains separately specified. This is a mathematical variational supplier for explicit functionals and selected future branches; adoption of particle dynamics is a separate physical premise.

Proof

technique · apply the shell first-integral identities to the adopted models

Given: [A1] or [A2] and the actual corresponding constrained shell trajectories.

1.1A1A2A3L1L3algebra

Under [A1] or [A2], the model is restricted to its stated massive future-timelike or nonzero future-null branch. On flat spacetime the affine generators are Killing fields; their phase-space charges are the components of [L1]. The shell brackets vanish, so each time-independent translation or Lorentz component is constant along its shell characteristic. For a fixed-time boost, the derivative is instead $\partial_{t} B_{i} + {B_{i}, E}$ , which vanishes by [L1].

2.1A1A2A3L2L3algebra∎

On a prescribed curved background, let $ξ$ be Killing. By [L2] the bracket with the mass-shell constraint vanishes; along the multiplier equations selected by either adopted action the derivative is $e {p_{a} ξ^{a}, C_{m}} = 0$ . This is a conditional model prediction for the specified background symmetry.

Conditional free Poincare and background Killing charges

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources