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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BMT orthogonality and conserved spin length

Physical scope

Conditional constraint preservation and restricted-ansatz coefficient matching for the adopted classical BMT polarization model on a prescribed external field. No measured value of the gyromagnetic ratio, quantum-spin derivation, or empirical validation is asserted.

Statement

Assume ${A C}_{ω}$ of The Axiom of Countable Choice ( ${A C}_{ω}$ ) for the canonical tangent-bundle and connection suppliers used by the BMT transport and charged-action equivalence. No full Axiom of Choice is assumed.

Adopt the charged-body quantities and SI tensor convention of Charged and spinning test-body quantities and The electromagnetic two-form, its observer decomposition and the four-current, the polarization transport postulate Classical BMT polarization model, and its translational equation obtained from Stationary charged action and Lorentz force. On a local potential domain, assume the finite curve segments are stationary for the adopted action. Let $z$ be a $C^{2}$ future timelike proper-time curve, $U = d z / d τ$ , $g (U, U) = - c^{2}$ , with prescribed continuous alternating field $F$ along it. Let $m > 0$ , $q \in R$ , $g_{s} \in R$ , and an initial $C^{1}$ spin vector satisfy $g (s, U) = 0$ . Put $α = q / m$ , $G = g_{s} / 2$ , and $κ = g (s, F^{♯} (U))$ . The adopted equations and action result give

$D_{τ} U = α F^{♯} (U), D_{τ} s = α [G F^{♯} (s) + (1 - G) \frac{U}{c^{2}} κ] .$

Then $g (s, U) = 0$ and $g (s, s)$ are constant along every solution. In the restricted ansatz $D_{τ} s = α [a F^{♯} (s) + b U κ / c^{2}]$ , require preservation of orthogonality for every initially orthogonal datum and every allowed alternating field. For $α \neq 0$ this gives $a + b = 1$ . If, in addition, the pointwise projected rest-space matching condition of BMT transport constraint and restricted-ansatz algebra holds with target coefficient $G$ , that lemma gives $a = G$ , and hence $b = 1 - G$ . This coefficient conclusion is only within that ansatz and the stated joint conditions. If $α = 0$ , the rest-space matching condition does not identify $a, b$ .

Facts & Assumptions

[A3]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): assume only countable choice ${A C}_{ω}$ for the canonical tangent-bundle smooth structure and inherited BMT/action-equivalence supplier hypotheses.

Given: The adopted classical BMT model, $c > 0$ , constant $m > 0, q, g_{s}$ , a future timelike proper-time curve, a prescribed alternating field, and an initially orthogonal spin vector.

[A1]

The adopted BMT transport and Lorentz-force translation equations, with $G = g_{s} / 2$ and $κ = g (s, F^{♯} (U))$ . Classical BMT polarization model

[A2]

The trajectory is stationary under the adopted action on a local potential domain, and the initial spin satisfies $g (U, s) = 0$ .

[F1]

The SI field tensor has $F_{0 i} = - E_{i} / c$ and $F_{i j} = ϵ_{i j k} B_{k}$ in the named inertial convention; the charged-body definition uses the same convention. The electromagnetic two-form, its observer decomposition and the four-current; Charged and spinning test-body quantities

[F2]

For the restricted two-term transport ansatz, metric compatibility and skew-adjointness give $D_{τ} g (U, s) = α (1 - a - b) κ$ ; if $a + b = 1$ , orthogonality and spin norm are preserved, and the pointwise projected rest-space matching condition defined in the supplier determines $a = G$ when $α \neq 0$ . BMT transport constraint and restricted-ansatz algebra

[F3]

Under [A3], stationarity of the adopted local charged-particle action implies $m D_{τ} U = q F^{♯} (U)$ on the stated finite fixed-endpoint potential-domain model. Stationary charged action and Lorentz force

Proof

technique · direct

1.1givenA3A1A2F2F3algebra

In the lemma [F2], set $a = G$ and $b = 1 - G$ . The postulate [A1] has exactly this displayed transport equation because $- (G - 1) = 1 - G$ , and [F3] gives the matching translation equation $D_{τ} U = α F^{♯} (U)$ . Thus $a + b = 1$ in the lemma; its derivative identity gives $D_{τ} g (U, s) = 0$ . The initial value is zero by [A2], hence orthogonality is preserved.

2.1givenA3A2F2step 1.1algebra

With $g (U, s) = 0$ from step 1.1 and [A2], the same lemma gives $\frac{1}{2} D_{τ} g (s, s) = g (s, D_{τ} s) = 0$ : the $F^{♯}$ term vanishes by skew-adjointness, and the term proportional to $U$ vanishes by orthogonality. Thus $g (s, s)$ is constant.

3.1givenA3F1F2step 1.1algebra∎

In the general ansatz, [F2] shows that for $α \neq 0$ , preservation for every initially orthogonal datum and every allowed field requires $a + b = 1$ ; an orthogonal rest spin $s = e_{1}$ with an alternating field satisfying $F (e_{1}, e_{0}) \neq 0$ gives an allowed instance with $κ \neq 0$ . The pointwise spatial projection in [F2] gives $a = G$ under the stipulated rest-space matching condition for every $s, B$ ; substituting into $a + b = 1$ gives $b = 1 - G$ . If $α = 0$ , both transport equations vanish and impose no coefficient condition. This establishes uniqueness only under the joint conditions and within the stated ansatz.

Remarks

This is a conditional deduction inside the adopted classical transport model. The original BMT treatment is flat and homogeneous-field; it does not prove the separately adopted curved or varying-field law, and this theorem makes no quantum or empirical claim.

BMT orthogonality and conserved spin length

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources