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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Massive and null free particles in inertial time

Physical scope

Conditional flat-spacetime kinematic prediction of the two adopted free models; no empirical speed measurement is claimed.

Statement

In the flat inertial-time model of Free shell in inertial time has the square-root Hamiltonian, the coordinate velocity is $\dot{q} = c^{2} p / E$ . For $m > 0$ , its Euclidean magnitude is strictly less than $c$ ; for the separate massless model $m = 0$ with $p \neq 0$ , its magnitude is exactly $c$ . Spatial momentum is constant in both cases. These are deductions from the adopted free actions, not experimental measurements.

Facts & Assumptions

Given: Flat inertial coordinates, $c > 0$ , either the massive model with $m > 0$ or the separately postulated nonzero-null model.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Assume countable choice ACω, meaning every sequence of nonempty sets has a choice function; no full Axiom of Choice is assumed.

[F1]

Free massive test-particle action: the massive square-root action is the adopted free model.

[F2]

Separate massless shell action: the massless branch is separately postulated with nonzero spatial momentum.

[F3]

Free shell in inertial time has the square-root Hamiltonian: $E = \sqrt{m^{2} c^{4} + c^{2} ∣ p ∣^{2}}$ , $\dot{q} = c^{2} p / E$ , and $\dot{p} = 0$ .

Proof

1.1givenA1F3algebra

Carry the inherited bundle-construction hypothesis [A1] when invoking the positive-energy shell supplier [F3]. By [F3], $∣ \dot{q} ∣^{2} = c^{4} ∣ p ∣^{2} / (m^{2} c^{4} + c^{2} ∣ p ∣^{2}) = c^{2} c^{2} ∣ p ∣^{2} / (m^{2} c^{4} + c^{2} ∣ p ∣^{2})$ . For $m > 0$ , the positive term $m^{2} c^{4}$ makes the last fraction strictly less than one, giving speed below $c$ .

1.2givenF2F3algebra

For $m = 0$ with $p \neq 0$ , $E = c ∣ p ∣$ , so [F3] gives $∣ \dot{q} ∣ = c^{2} ∣ p ∣ / (c ∣ p ∣) = c$ . In either case [F3] gives $\dot{p} = 0$ , so momentum is constant.

2.1F1F2step 1.1step 1.2∎

The massive speed statement is a deduction within the [F1] model; the null statement uses its own separate premise [F2]. No step used or asserted an empirical observation.

Massive and null free particles in inertial time

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources