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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Explicit massive and nonzero-null free complete integral

Example

Assume ${A C}_{ω}$ for the cotangent-shell supplier. Work with spatial position $q \in R^{3}$ in metres and momentum parameter $α \in R^{3}$ in kg m/s, with the Euclidean dot product and norm $∣ α ∣ = \sqrt{α \cdot α}$ ; at fixed time the spatial phase space has canonical form $ω = \sum_{i = 1}^{3} d q^{i} \land d p_{i}$ . Here $t \in R$ is measured in seconds, $m \geq 0$ is in kg, and $c > 0$ is in m/s. The parameter domain is all $R_{α}^{3}$ when $m > 0$ , and $R_{α}^{3} ∖ {0}$ when $m = 0$ . For a free massive particle with $m > 0$ , or a nonzero massless particle with $m = 0$ and $α \neq 0$ , define

$S (t, q, α) = q \cdot α - t E (α), E (α) = \sqrt{m^{2} c^{4} + c^{2} ∣ α ∣^{2}} .$

Then $S_{t} + E (S_{q}) = 0$ , $S_{q α} = I$ , and its integration coordinates satisfy $β = S_{α} = q - t c^{2} α / E$ , equivalently $q = β + t c^{2} α / E$ . The mixed Hessian is everywhere nonsingular on the stated domain, so this is a complete integral there. $[S] = J s$ , $[α] = k g m / s$ , and $[β] = m$ .

Facts & Assumptions

Given: Flat inertial coordinates, $m \geq 0$ , $c > 0$ , and for $m = 0$ the open parameter domain $α \neq 0$ .

[A1]

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

[L1]

Complete integrals and covariant versus time Hamilton-Jacobi: a smooth HJ family with invertible mixed Hessian defines a local canonical map with new position $β = S_{α}$ , new momentum $α$ , and $ω_{n e w} = \sum_{i} d β^{i} \land d α_{i}$ ; $β$ is constant along characteristics, and the construction is local to the nonsingular domain. For each fixed $α$ here, the initial graph at $t = 0$ lies in the full massive Hamiltonian domain when $m > 0$ and in the open nonzero-momentum domain when $m = 0$ and $α \neq 0$ .

[L2]

Free shell in inertial time has the square-root Hamiltonian: the current H8 draft states the positive-energy Hamiltonian $E = \sqrt{m^{2} c^{4} + c^{2} ∣ p ∣^{2}}$ on the fixed inertial-time branch. The supplier is a mathematical deduction within its stated chart, positive-root and nonzero-null hypotheses; it introduces no physical action adoption.

Verification

technique · substitution and derivative verification

Given: The formulas and domain in the statement, together with [L1]–[L2].

1.1L1L2A1algebra

Differentiation gives $S_{q} = α$ , $S_{t} = - E (α)$ , and hence $S_{t} + E (S_{q}) = 0$ . Also $S_{q_{i} α_{j}} = δ_{i j}$ , so its determinant is one. By [L1] this nonsingular family supplies local canonical integration coordinates on the stated domain.

1.2L1L2A1algebra

Differentiating $E$ gives $\partial_{α_{i}} E = c^{2} α_{i} / E$ wherever $E > 0$ . Thus $β = S_{α} = q - t c^{2} α / E$ and rearranging yields the stated trajectory formula. For $m = 0$ , $E = c ∣ α ∣ > 0$ and differentiability requires $α \neq 0$ , which is exactly the given domain.

2.1givenalgebra∎

The products $q \cdot α$ and $t E$ have action units, while $α$ is momentum and $\partial_{α} S$ is length. Hence the generating function and canonical coordinates have the claimed physical dimensions.

Explicit massive and nonzero-null free complete integral

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources