ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)verified 2026-11-04 (gpt-6.1-sol)

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/brane gas cosmology proposal

Physical scope

A one-circle closed-bosonic-string spectrum and its exact momentum/winding duality in source units. It does not assert a Hagedorn cosmological solution, stabilized compactification, perturbation production, or empirical history.

Example

A string/brane-gas proposal treats strings and specified branes as thermal sources; a brane here is an extended object with a declared spatial worldvolume dimension and wrapping. On a compact circle of dimensionless radius $R$ in the source convention $α^{'} = 1 / 2$ , the integer $n$ labels quantized momentum and the integer $m$ is the signed winding number, counting wraps of the string around the circle. A “Hagedorn regime” here means a proposed thermal phase near a limiting temperature associated with exponential growth of the string-state density. A “quasi-static” phase would require the model to state an interval and tolerance on the scale-factor variation; neither is supplied here. Such proposals also require a specified thermal ensemble, stabilization mechanism, perturbation production and controlled transition; these are not consequences of the one-circle spectrum below.

Consider a closed bosonic string on a target with one compact spatial circle of dimensionless string-frame radius $R > 0$ . In the source convention $α^{'} = 1 / 2$ , $R$ and $M$ in the displayed spectrum are dimensionless numerical values in the corresponding string units; $n, m \in Z$ and the oscillator levels $N_{L}, N_{R} \in N_{0}$ are dimensionless. With the declared convention $N_{L} = N$ , $N_{R} = \tilde{N}$ , an allowed state satisfies the level-matching condition $N_{R} - N_{L} = n m$ . The mass formula is $M^{2} = \frac{n^{2}}{4 R^{2}} + m^{2} R^{2} + N_{L} + N_{R} - 2$ . This one-circle spectrum is a precise closed-string input. No universal Hagedorn temperature is assigned. The gas ensemble, thermodynamic state, stabilization mechanism, quasi-static solution and transition dynamics are missing inputs, as is perturbation generation/transfer. Adding branes requires their species, wrapping and source data; none is specified in this toy spectrum.

Example references

Alternative string-cosmology model data

M. Cicoli, J. P. Conlon, A. Maharana, S. Parameswaran, F. Quevedo and I. Zavala, String Cosmology: from the Early Universe to Today, arXiv:2303.04819v2

Y. Li, Y.-F. Hang, H.-J. He and S. He, Scattering amplitudes of Kaluza-Klein strings and extended massive double-copy, arXiv:2111.12042

Facts & Assumptions

[F1]

Alternative string-cosmology model data: A datum specifies the action and fields, a background branch, sources and transitions, perturbation variables and transfer, control conditions, and observable outputs; missing thermal or transfer data are not inferred.

[A1]

Adopt the source-reported one-circle bosonic closed-string spectrum in $α^{'} = 1 / 2$ string units, with the allowed-state condition $N_{R} - N_{L} = n m$ in the left-right convention $N_{L} = N$ , $N_{R} = \tilde{N}$ of the cited level-matching formula. The dimensionless $R$ and $M$ are the string-frame circle radius and mass in the source's numerical string-unit convention; the duality acts on the momentum/winding integers.

Verification

Given: $R > 0$ , $n, m \in Z$ , and an allowed state in [A1] with fixed $N_{L}, N_{R} \in N_{0}$ and $N_{R} - N_{L} = n m$ .

1.1givenA1algebra

Define $R^{'} = 1 / (2 R)$ , $n^{'} = m$ , and $m^{'} = n$ . Then $\frac{(n^{'})^{2}}{4 (R^{'})^{2}} = m^{2} R^{2}$ and $(m^{'})^{2} (R^{'})^{2} = \frac{n^{2}}{4 R^{2}}$ , so the momentum and winding contributions exchange exactly.

2.1A1step 1.1algebra

The oscillator contribution $N_{L} + N_{R} - 2$ is unchanged, and $n^{'} m^{'} = n m = N_{R} - N_{L}$ , so the allowed-state level-matching condition remains true under the same interchange. Therefore $M^{' 2} = M^{2}$ for every allowed state in this model.

2.2step 1.1algebra

The fixed positive radius satisfies $R = 1 / (2 R)$ , hence $R_{c} = 1 / \sqrt{2}$ . The map identifies radii on opposite sides of $R_{c}$ while exchanging momentum and winding labels.

3.1F1A1step 2.1∎

The source's Hagedorn/string-gas cosmology discussion is a proposal requiring a specified thermal ensemble and stabilized moduli; the spectral identity in steps 1.1–2.2 does not derive a quasi-static cosmological phase, a transition, or observed perturbations.

Remarks

The commutative square summarizes the exact mass-spectrum invariance established in steps 1.1–2.1. Its horizontal arrow is the map $(R, n, m) \mapsto (1 / (2 R), m, n)$ at fixed oscillator levels; the vertical arrows evaluate the source mass formula. The diagram records the calculation but is not a proof premise.

Depends on

Alternative string-cosmology model data