Claude Audits Quantum Gravity

An AI re-derived a published one-loop quantum gravity calculation step by step. It found a sign error and a missing term, and confirmed the physical conclusion.

The paper: E. O. Kahya, Auditing a quantum gravity calculation with an AI model: corrected scalar self-mass and unchanged late-time mode functions during inflation (submitted to arXiv, 2026). PDF · LaTeX source

This repository re-derives, step by step, a published one-loop quantum gravity calculation during inflation: the graviton correction to a massless, minimally coupled scalar on de Sitter space (Paper I, Phys. Rev. D 76, 124005 (2007)), and its use in the scalar's late-time field equation (Paper II, arXiv:0710.5282). The re-derivation was carried out with Claude (Anthropic's AI model, Claude Opus 5.5) as a checking tool. It was published by E. O. Kahya, one of the authors of both papers. Each step, from the Feynman rules (I.6)–(I.8) to the renormalized self-mass (I.167) and the late-time analysis of Paper II, was derived again from the inputs stated in the papers, keeping the dimension Limitations.) Every difference from the printed papers is classified: 4 substantive findings (S1–S4), 6 finite errors absorbed into counterterms (B1–B6) and 23 typographical errors (C1–C23). The complete list of findings gives each one with its evidence.

Bottom line: the main physical conclusion of Paper II survives. The scalar's mode function receives no significant one-loop correction at late times. However, the printed self-mass-squared is

Contents: The question · What was done · Main results · Read it in order · The paper · Complete list of findings · Full derivations · Reproduce the checks · How Claude was used · Papers · Citation

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flowchart LR
    A["Inflation<br/>(de Sitter, a = −1/Hη)"] --> B["A growing sea of<br/>long-wavelength gravitons"]
    B --> C["They couple to the scalar<br/>only through derivatives<br/>(κ h ∂φ ∂φ)"]
    C --> D["One-loop self-mass-squared<br/>M²(x;x′)"]
    D --> E["Quantum-corrected<br/>field equation"]
    E --> F["Mode function Φ(η,k):<br/>does it change at late times?"]

Inflation pulls long-wavelength quanta of massless fields that are not conformally invariant out of the vacuum. Gravitons are such fields, so as inflation goes on the universe fills with an ever-growing sea of infrared gravitons. The question is whether this sea changes how a massless, minimally coupled scalar (a stand-in for a very flat inflaton) propagates. All quantum corrections at one loop are contained in the scalar self-mass-squared

At tree level the plane-wave mode function freezes to a constant once the mode has left the horizon:

Write full self-mass: two one-loop diagrams (a 4-point "seagull" and a 3-point diagram), dimensional regularization, counterterms for a gauge that breaks part of the de Sitter symmetry, and finally the Schwinger–Keldysh (in-in) form of the field equation.

The two papers were split into 18 tasks, which follow the papers' equations in order. They are grouped into nine stages, each with a summary page, a chapter of derivation notes and a folder of code checks.

Paper I: vertices and the 4-point diagram

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flowchart LR
    S1["1 · Feynman rules<br/>(I.6)–(I.8)<br/>Tables I–II"] --> S2["2 · 4-point<br/>contribution<br/>(I.31)–(I.44)"]

Paper I: the 3-point diagram

%%{init: {"flowchart": {"htmlLabels": false, "padding": 18, "nodeSpacing": 35, "rankSpacing": 45}}}%%
flowchart LR
    S3["3 · 3-point<br/>setup<br/>(I.45)–(I.53)"] --> S4["4 · local and<br/>log terms<br/>(I.54)–(I.67)"] --> S5["5 · normal terms<br/>Table III<br/>(I.68)–(I.93)"] --> S6["6 · Tables<br/>IV–V<br/>(I.94)–(I.125)"]

Renormalization (Paper I) and late times (Paper II)

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flowchart LR
    S7["7 · de Sitter<br/>symmetry and<br/>counterterms<br/>(I.126)–(I.145)"] --> S8["8 · renormalized<br/>self-mass<br/>Table X<br/>(I.146)–(I.167)"] --> S9["9 · Schwinger–<br/>Keldysh<br/>late-time mode<br/>function"]

Each stage goes through the same steps: the analytic derivation (index by index, general

Important

Late-time answer: no significant one-loop correction. With the corrected self-mass-squared, and without the simplification (II.44) of Paper II, the solution of the Schwinger–Keldysh mode equation approaches

Table entries are quoted in the paper's sign convention, as printed. S1 then multiplies the whole of Table X by complete list of findings gives every item (Tables A–E) with its evidence, the conventions, a summary for non-experts, and the limitations.

The renormalized self-mass-squared, in the sign fixed by Paper I's own definition (I.168), at

The 23 typographical errors C1–C23 (no effect on any result) #

C1–C19, C22 and C23 are in Paper I; C20 and C21 are in Paper II.

C1 (I.1): the first term has$\partial_\mu\phi,\partial_\mu\phi$ instead of$\partial_\mu\phi,\partial_\nu\phi$ . #

C2 (I.6), left-hand side: the second$\phi$ is missing. #

C3 (I.7), left-hand side:$-\tfrac12\kappa$ should be$+\tfrac12\kappa$ . The vertex itself is correct. #

C4 (I.14): the first coefficient should be$\Gamma(\tfrac D2-1)$ . #

C5 Table II,$U_2$ :$-\tfrac i4$ should be$+\tfrac i4$ . The paper's own (I.33) uses$+\tfrac i4$ . #

C6 (I.39): index placement ($\bar\eta_{\rho\sigma}$ for$\bar\eta^{\rho\sigma}$ ; same matrix). #

C7 The text before (I.54) refers to "the 4th term of (47)"; it should be the 2nd term. #

C8 (I.89): a factor$H^2$ is missing in the second term. Table III is correct. #

C9–C13 Five typos in the appendix "Extracting derivatives": the Table XII header, (A18), Table XIII row$P_{7c}$ , Table XV row$P_{4b}$ and the Table XVIII total. #

C14 (I.128): the list of four-derivative invariants omits$\phi^2R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ and$\phi^2\square R$ ; shift symmetry removes both, so (I.129) is unaffected. #

C15 Table IV,$\gamma_2$ entry: wrong sign. Tables VI–IX use the correct sign, which shift symmetry requires. #

C16 Table VII header: spurious factor$i$ . #

C17, C18 (I.160) and (I.167):$H^2$ is missing next to$\beta_2,\beta_3$ and$\Delta\alpha_2,\Delta\alpha_3$ . #

C19 Table X prints the row$(aa')^2(a^2+a'^2)\nabla^2\square$ twice. #

C20, C21 (II.10) and (II.34):$H^2$ is missing next to$\Delta\alpha_2,\Delta\alpha_3$ (the same omission as in (I.160) and (I.167)). #

C22 (I.170): the same missing$H^2$ next to$\Delta\alpha_2,\Delta\alpha_3$ , and in the result$-\kappa^2a^4\Delta\alpha_3H^2k^2/a^2,\Phi_0$ . #

C23 Appendix, text before (A31): "to$P_1$ and$P_6$ " should read "to$P_1$ and$P_4$ ".

C22 and C23 were noticed while refereeing the write-up and confirmed on the printed pages. Details, with the evidence for each item, are in the complete list of findings.

Each page tells the story of one stage first, then gives the key equations, the result, a comparison table with the paper, and links to the code and to the full derivation.

Feynman rules : expand$\sqrt{-g},g^{\mu\nu}$ to second order in$\kappa$ and get the scalar–graviton vertices (Tables I–II).Tasks 1–2. 2. The 4-point contribution : the seagull diagram, the coincidence limits in dimensional regularization, and where the overall sign comes from.Tasks 3–5. 3. 3-point setup : the effective action, the nine terms after the graviton propagator is inserted, and the mixed derivative of the scalar propagator (I.52).Tasks 6–8. 4. 3-point local and logarithm terms : the local divergences, the de Sitter-breaking$\ln(aa')$ terms, and their exact cancellation against the 4-point terms.Tasks 9–10. 5. Normal terms and Table III : how derivatives are pulled outside, the time-derivative identities, and Table III.Tasks 11–12. 6. Tables IV and V : the ten external operators, the appendix bookkeeping, and the missing$72\gamma_2$ term.Task 13. 7. de Sitter symmetry and counterterms : which isometries the gauge preserves, and why one noninvariant counterterm suffices.Tasks 14–15. 8. Renormalized self-mass and Table X : separating the divergences, the coefficients$\beta_i$ , Table X, and the global pointwise test.Tasks 16–17. 9. Schwinger–Keldysh and late times : Paper II with the corrected self-mass, Tables 2–5, going beyond (II.44), and the late-time mode function.Task 18. 10. Complete list of findings : every result in one place: the substantive findings S1–S4, the absorbable errors B1–B6, the typographical errors C1–C23, what was confirmed as printed, the late-time results, a summary for non-experts, and the limitations.All tasks.

The complete derivation (every step, the index tracking, intermediate results, comparison tables and a calculation record for each task) is in notes/notes.pdf. Its LaTeX sources are in notes/: one file per chapter in notes/chapters/ and the shared macros in notes/macros.tex.

A reader-friendly version of the findings, with the same IDs (S1–S4, B1–B6, C1–C23), is docs/10-complete-list-of-findings.md.

In the notes, each statement carries a label: P (given in the paper), D (derived here), A (assumption or convention) or C (an independent cross-check, by code unless stated otherwise). Each equation states which sign convention it uses: the paper's, or the definition sign of (I.168). To build the PDF yourself: cd notes && tectonic -X compile notes.tex.

git clone https://github.com/eokahya/claude-audits-quantum-gravity.git
cd claude-audits-quantum-gravity
pip install -r code/requirements.txt      # sympy, mpmath

python3 code/run_all.py --quick           # all but the two slowest scripts, about 6 minutes
python3 code/run_all.py                   # everything (38 runs), about 10 minutes
python3 code/run_all.py --folder 08       # one folder

code/run_all.py runs every script from inside its folder and compares its output with the stored expected_output/. At the end it prints a pass/fail table. The stored outputs were produced with Python 3.13.1, SymPy 1.14.0 and mpmath 1.3.0, and they are identical to the outputs recorded during the original calculation. Another SymPy version may print an equivalent expression in a different form; in that case the comparison shows a diff. In the last full run (Apple-silicon Mac) all 38 runs passed in 10 minutes.

You can also run a single check by hand from its folder:

code/README.md lists every script, what it checks and how long it takes. Each folder has its own README, which explains how to read the output.

Method.

Inputs, not targets. Each task took as input only the results of earlier tasks and the definitions and conventions stated in the papers. The printed equation to be checked was not used as an input; it was compared only after the derivation was finished. The exceptions: Tables IV–V were computed from the printed appendix totals after those totals had been checked, and Table V enters the assembly of Table X as printed (its non-analytic terms were confirmed first). Tables 2–5 of Paper II were never used as input. #

General Wherever the paper keeps the spacetime dimension general, so did the derivation. Every step was written out with explicit index tracking.$D$ . #

Code checks are separate from the analysis. Each analytic result was then tested by independent SymPy/mpmath code: exact symbolic identities in general$D$ or at several fixed$D$ , brute-force differentiation compared with the paper's reduced forms, and pointwise evaluation at random points with 20–30 significant digits. Code checks are labelled separately and never replace a derivation. #

Finding the origin of each discrepancy. When a result disagreed with the paper, the cause was traced back. For example, the overall sign comes from the derivative rule in (I.32) and from the step (I.46) → (I.47). Each discrepancy was classified as a typo, absorbable (a finite change absorbed by counterterms), substantive, or a convention. #

Language and provenance. The original working reports, one per task, were written in Turkish (HTML with KaTeX), together with a full log of every script run. The English notes, the docs pages and this README are faithful translations and summaries of those reports, also written with Claude. Each English chapter was then checked against its Turkish source in a separate Claude review pass. The scripts were cleaned up so that each one runs on its own, re-run, and compared with the original logs; the outputs agree.

What a human expert should still review.

• The two sources of the overall sign: Lemma A for the 4-point term (docs/02 ) and the step (I.46) → (I.47)–(I.49) (docs/03 ). Both are short.
• Corrections S2 and S3 and the global test that supports them, check_tableX_global.py . In particular, check how the direct sum is built in its helpercheck_global.py .
• The late-time analysis: the exact $k$ -expansion (kexp.py ) and the solution for$\Phi_1$ .
• The translation from the Turkish reports, if you read Turkish.

Limitations and open points.

• The appendix "Extracting derivatives" of Paper I was audited, not redone from scratch . We checked worked example rows by hand and all ten table totals with the global pointwise test; Tables XII and XIV and (A1)–(A9) were read from the printed pages. (One working report claimed that every row had been reproduced by hand. That claim was an overstatement and is corrected in the notes.)
• A possible extra local term from the appendix's operator reduction is only partly resolved. Any double counting of the local part has exactly the counterterm form$c,(a^D\square-a^{D-2}\nabla^2)\delta^D(x-x')$ , so it is absorbed into$\alpha_2,\alpha_3$ . We have not proved, however, that no other local term arises; that needs a full distributional analysis.
• The global checks are numerical at sample points : two or three random spacelike point pairs, 30 digits, for Table X and for Tables IV–V with the actual propagators. They arenot symbolic identities , and they check only sums. They cannot tell apart equivalent placements of the S3 term (a$-12x$ in the$\delta$ row of Table V, or a changed$\gamma_2$ constant).
• The Schwinger–Keldysh cancellation of the non-falling terms is exact through $k^{10}$ . Beyond that there is a convergent-series argument butno all-orders proof .
• The $a^{-2}$ coefficients of the late-time solution depend on the initial state. As Paper II notes, falling terms cannot be trusted without the corrected initial state.
• Some code checks run only at fixed dimensions: counterterm completeness at $D=4,5,6$ , the gauge-breaking Euler–Lagrange check in$D=3$ , the pointwise check of the nonlocal part of (I.60)–(I.67) at$D=4$ . The general-$D$ statements rest on the analytic derivation.
• The overall sign (S1) rests on the paper's own definition (I.168). Once (I.168) is adopted, the sign of the nonlocal part is physical, because the nonlocal part has no counterterm freedom.
• Assumptions shared with the paper: dimensional regularization preserves the shift symmetry and the residual symmetries of the gauge-fixed action, ultraviolet divergences are polynomial in $H$ , and$D=4$ is set only in terms not multiplied by a pole.

The full list of limitations is at the end of the complete list of findings.

• This work. E. O. Kahya, "Auditing a quantum gravity calculation with an AI model: corrected scalar self-mass and unchanged late-time mode functions during inflation", submitted to arXiv (2026).PDF ,LaTeX source (single self-contained file, as submitted).
• Paper I. E. O. Kahya and R. P. Woodard, "Quantum gravity corrections to the one loop scalar self-mass during inflation",Phys. Rev. D76 , 124005 (2007).doi:10.1103/PhysRevD.76.124005 ,arXiv:0709.0536 .
• Paper II. E. O. Kahya and R. P. Woodard, "Scalar field equations from quantum gravity during inflation",Phys. Rev. D77 , 084012 (2008).doi:10.1103/PhysRevD.77.084012 ,arXiv:0710.5282 .

Equations are cited as (I.n) for Paper I and (II.n) for Paper II. This repository contains no scans or figures from the papers; where a printed equation is compared, it is retyped in LaTeX.

If you use this re-derivation, please cite the paper above (arXiv identifier to be added), the original papers and this repository:

@misc{Kahya2026ClaudeAuditsQG,
  author       = {Kahya, Emre Onur},
  title        = {Claude Audits Quantum Gravity: a step-by-step {AI} re-derivation of a one-loop quantum gravity calculation during inflation},
  year         = {2026},
  howpublished = {\url{https://github.com/eokahya/claude-audits-quantum-gravity}},
  note         = {Step-by-step re-derivation of the one-loop scalar self-mass during inflation
                  (Phys. Rev. D 76, 124005) and of its late-time effect (arXiv:0710.5282),
                  carried out with Claude Opus 5.5}
}

MIT. See LICENSE.