ExampleConstruction: AI-generatedVerification: 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.

Bosonic Low Lying Spectrum

Physical scope

A conditional level-zero and first-excited-level calculation in the standard flat critical open/closed bosonic spectra; no stable-vacuum or empirical claim.

Example

Under $D = 26$ and $a = \tilde{a} = 1$ , the open $N = 0$ state has alpha-prime $M^{2} = - 1$ and the $N = 1$ level is massless; the closed $N = \tilde{N} = 0$ state has alpha-prime $M^{2} = - 4$ , while $N = \tilde{N} = 1$ is massless and decomposes into graviton, antisymmetric tensor, and scalar polarizations.

Example references

David Tong, Lectures on String Theory, arXiv:0908.0333v3, full 218-page lecture notes, §§2.3–2.3.3, 3.1–3.1.1 and 5.4–5.4.3 (PDF file pp.47–52, 60–61 and 127–131; printed pp.40–45, 53–54 and 120–124); printed pp.40–45, 53–54 and 120–124); printed pp.40–45, 53–54 and 120–124); printed pp.40–45, 53–54 and 120–124); printed pp.40–45, 53–54 and 120–124); printed pp.40–45, 53–54 and 120–124); printed pp.40–45, 53–54 and 121–124); spectrum conventions; each level and polarization count is checked locally.

Facts & Assumptions

[F1]

Open Bosonic Lightcone Spectrum gives $α^{'} M^{2} = N - 1$ for nonnegative integer open weighted level $N = \sum_{n > 0, i} α_{- n}^{i} α_{n}^{i}$ under $D = 26$ , $a = 1$ .

[F2]

Closed Bosonic Lightcone Spectrum gives $α^{'} M^{2} / 4 = N - 1 = \tilde{N} - 1$ at equal nonnegative weighted closed levels $N = \sum_{n > 0, i} α_{- n}^{i} α_{n}^{i}$ and $\tilde{N} = \sum_{n > 0, i} {\tilde{α}}_{- n}^{i} {\tilde{α}}_{n}^{i}$ .

[A1]

At the first closed excited level, the polarization space is the tensor product of the left and right 24-dimensional transverse vector spaces with the ordinary Euclidean product.

Given: Work only under the two critical flat-spectrum interfaces [F1]–[F2].

Proof

1.1givenF1algebra

Substitute $N = 0$ and $N = 1$ in [F1]. The open ground state has $α^{'} M^{2} = - 1$ , while the first excited level has $α^{'} M^{2} = 0$ .

1.2givenF2A1algebra

Substitute $N = \tilde{N} = 0$ in [F2] to obtain $α^{'} M^{2} / 4 = - 1$ , hence $α^{'} M^{2} = - 4$ . Substituting $N = \tilde{N} = 1$ gives $M^{2} = 0$ . The first closed massless polarization space is $R^{24} \otimes R^{24}$ by [A1].

2.1step 1.2algebra∎

Identify that tensor product with real $24 \times 24$ matrices. Every matrix $A$ decomposes uniquely as $\frac{1}{2} (A + A^{T}) + \frac{1}{2} (A - A^{T})$ into symmetric and antisymmetric parts; the symmetric part decomposes uniquely into its traceless part plus $\frac{tr A}{24} I$ . Their dimensions are $24 \cdot 25 / 2 - 1 = 299$ , $24 \cdot 23 / 2 = 276$ , and $1$ , whose sum is $576 = 24^{2}$ . These are the symmetric-traceless tensor, antisymmetric tensor, and scalar polarizations.

Remarks

The negative values of $M^{2}$ at level zero are tachyonic mass-squared values in this model; they do not state that a stable vacuum exists.

Depends on

Dependency tree · 2 levels

Sources

David Tong, Lectures on String Theory, arXiv:0908.0333v3, full 218-page lecture notes