ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)verified 2026-11-04 (gpt-6-luna)

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.

String Worldsheet Genus Coupling Weights

Physical scope

Symbolic substitution of genus values into the formal constant-dilaton topology factor; no amplitude, path-integral measure, or empirical claim.

Example

Assume the adopted local NS–NS model, a constant real dilaton $Φ_{0}$ , and its formal Euclidean topology-weight convention. Put $g_{s} = e^{Φ_{0}} > 0$ . For compact connected Riemann surfaces with compatible Euclidean metrics, supplied Spin structures, and genus $g = 0$ , $1$ , and $2$ , respectively, the factors are $g_{s}^{- 2}$ , $1$ , and $g_{s}^{2}$ . These are exact symbolic topology factors in the adopted model; they are not complete scattering amplitudes, observations, or claims that a path-integral measure has been constructed.

Example references

String Coupling Genus Weight

Timo Weigand, §5.1

David Tong, §6.1.1

Facts & Assumptions

Given: A compact connected Riemann surface with compatible Euclidean metric, supplied Spin structure, genus $g \in {0, 1, 2}$ , constant dilaton $Φ_{0}$ , and the Axiom of Choice.

[F1]

String Coupling Genus Weight: the formal topology factor is $g_{s}^{2 g - 2}$ under its stated Euclidean sign and mathematical hypotheses.

[F2]

Euler characteristic of an orientable compact surface: under full Axiom of Choice, the underlying nonempty compact connected orientable boundaryless surface has $χ = 2 - 2 g$ for its unique genus $g$ .

[A1]

RNS NS Background Sigma Model is the adopted local physical model; this example derives no claim about its empirical validity or measure.

[A2]

The Axiom of Choice is the full choice principle assumed by [F1] and [F2].

Verification

1.1givenA2F1F2algebra

For genus $g = 0$ , [F1] gives exponent $2 g - 2 = - 2$ , so the factor is $g_{s}^{- 2}$ .

1.2givenA2F1F2algebra

For genus $g = 1$ , the exponent is $2 g - 2 = 0$ , so the factor is $g_{s}^{0} = 1$ . This is the formal torus topology factor only.

2.1givenF1F2A1A2algebra∎

For genus $g = 2$ , the exponent is $2 g - 2 = 2$ , so the factor is $g_{s}^{2}$ . All three substitutions use the same definition and the same full-choice premise; no additional string-interaction normalization is inferred.

String Worldsheet Genus Coupling Weights

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources