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 classical orbit evolution and invariant labels

Physical scope

Selected linear H generates complete group translation on the orbit; m,s labels preserved.

Statement

Assume ${A C}_{ω}$ . Adopt the optional classical spinning-orbit model of Optional adopted classical spinning-orbit model with fixed $m, s > 0$ and selected inertial Hamiltonian $H = c P^{0}$ . Conditional on the KKS orbit and canonical Hamiltonian conventions, its Hamiltonian flow is the one-parameter coadjoint action of the corresponding time-translation subgroup (with the sign fixed by the library convention). It exists for all real evolution parameters and preserves $m$ and $s$ . This is a deduction inside the adopted classical model, not evidence for that model or a derivation of quantum spin.

Facts & Assumptions

Given: The optional model postulate, its fixed orbit, and $H = c P^{0}$ in one inertial frame.

[A1]

Optional adopted classical spinning-orbit model: the model adopts the positive-spin KKS orbit with primitive $m, s > 0$ and chooses $H = c P^{0}$ .

[L1]

Positive classical spin Poincare orbit and stabilizer jump: under ${A C}_{ω}$ , for $m, s, c > 0$ the coadjoint orbit through $P = (m c, 0, 0, 0)$ and $J^{12} = s$ is an eight-dimensional KKS symplectic manifold, with $P^{2} = - m^{2} c^{2}$ , $P$ future, $W \cdot P = 0$ and $W^{2} = m^{2} c^{2} s^{2}$ . Here $W^{a} = \frac{1}{2} ε^{a b c d} P_{b} J_{c d}$ with $ε_{0123} = + 1$ and indices raised by $η = diag (- 1, 1, 1, 1)$ , exactly as in that supplier. No temporal-chart or H10 hypothesis is required. It exists as a batch-26 draft, but is not independently approved.

[F1]

Coadjoint orbits are symplectic manifolds: on a coadjoint orbit the inclusion moment map satisfies $d μ^{ξ} = - ι_{ξ_{O}} ω_{K K S}$ , so linear dual components generate the corresponding infinitesimal coadjoint fields.

[L2]

Canonical Poisson brackets and local Hamiltonian flow fixes the cotangent convention $ι_{X_{H}} ω = d H$ . On the KKS symplectic orbit we define $X_{H}$ by the same intrinsic equation; existence, uniqueness and smoothness follow by inverting the smooth nonsingular matrix of $ω$ in local charts. The cotangent supplier's H25 Step 1 record is ready/hash-current; the carrier remains draft/unbound with ai-altered proof provenance, so that readiness does not independently approve its proof.

[F2]

Poincare semidirect group, dual pairing and moment components: the pure unit $x^{0}$ translation pairs with $p_{0} = - P^{0}$ under the fixed cotangent convention.

[A2]

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

Proof

technique · identify the linear Hamiltonian vector field with a group action

Given: [A1], the KKS moment equation [F1], and the Hamiltonian sign convention [L2].

1.1A1F1F2L1L2A2algebra

Let $ξ_{0}$ be unit translation in the $x^{0}$ direction. The dual pairing of [F2] gives $μ^{ξ_{0}} = p_{0} = - P^{0}$ , so $H = c P^{0} = - c μ^{ξ_{0}}$ . Since $ι_{ξ_{O}} ω = - d μ^{ξ_{0}}$ by [F1] and $ι_{X_{H}} ω = d H$ by [L2], $X_{H} = c ξ_{O}$ . With the library fundamental-field convention, its integral curve is $z (t) = \exp_{G} (- c t ξ_{0}) \cdot z (0)$ .

2.1F1L1A2step 1.1algebra∎

The fundamental field with the library convention integrates as $z (t) = \exp_{G} (- c t ξ_{0}) \cdot z (0)$ for every real $t$ ; differentiating this globally defined action gives $c ξ_{0, O}$ , the field from step 1.1. Thus this particular Hamiltonian flow is complete. The group action preserves the orbit invariants, so $P^{2} = - m^{2} c^{2}$ and $W^{2} = m^{2} c^{2} s^{2}$ remain fixed, preserving $m, s$ .

Conditional classical orbit evolution and invariant labels

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources