What a Lean Source Audit Can Establish

A formal theorem can have a long chain of prerequisites. One useful question is whether the mathematical conditions supplied by earlier parts of the chain actually fit the conditions its final caller needs. A source audit can make that relationship explicit. It cannot, by itself, show that a replacement theorem has been accepted by Lean.

This distinction matters in a recent bounded review of the Gaussian-moat theorem in OpenAI Math's Family 028. The review worked from pinned public source revisions and asked whether one fixed set of parameters, together with an explicit onset, could satisfy the symbolic caller conditions in Main.lean. A separate earlier record reports a bounded offline build and endpoint/axiom check of the existing theorem. This source audit neither repeats nor extends that check; it examines a possible alternative route through the source-level caller conditions.

Why make the tuple explicit?

The source's existing Main obtains its parameters and density onset existentially. The audit instead carried a fixed tuple through the mathematical construction: $a=I/192$, $\delta=I/48$, $K=5000$, and $t=I/(160000\log 4)$, with an explicit starting threshold $X_{\mathrm{res}}=\max(4001,15L_0/(2I))$. Here $I=(1427/120030000)\,1431!\,2567!/3999!$ and $L_0=(369/200)\,2^{2567}$ are the constants in the reviewed mass estimate.

At the reviewed source level, the mass and thinning argument supplies the required finite density witness for every real $X\geq X_{\mathrm{res}}$. A separate caller audit then maps symbolic bounds for the finite batch range, smoothing term and common onset to the obligations in Main. The cutoffs remain formulas in the input parameters; they are not evaluated numerical cutoffs in that caller map.

The fixed-tuple scale is explicit, but enormous

A later calculation evaluated two scales for this same sufficient construction. The pointwise density onset from the separate mass replay is approximately $9.59610236\times10^{1912}$. This value is derived from pinned-source identities and explicit ordinary-mathematics estimates; it is not a numeric constant defined or certified by the Lean files.

The prescribed finite-window width has a similarly severe lower scale. For every permitted geometric parameter $D\geq1$, the Gaussian lattice ball contains $0,\pm1,\pm i$, so its logarithmic size is at least $\log 5$. Substitution into the fixed-tuple window rule gives a lower bound of about $7.29708934\times10^{2533}$, or at least 2,534 decimal digits, before the other components of the common onset are evaluated. An independent AI arithmetic challenge found no error in these scale calculations; it treated the separate pointwise density-onset derivation as a premise.

This quantifies a severe scale limitation of this particular explicit sufficient construction. It does not show the bounds are sharp, refute the theorem, establish universal practical impossibility, or rule out another proof. The complete finite Main bound remains unevaluated, and no selector, executable cost/batch calculation, alternative theorem or Lean result follows.

The value of this exercise is traceability. A reader can see which fixed witness is being carried, where the density statement applies, and how the caller's symbolic conditions are covered. The reviewers found no mismatch in the inspected cutoff, ceiling, quantifier order or caller conditions at this bounded scope.

A map is not an implemented proof

The result is a source-level alternative proof map. It is conditional on the separately replayed mass and finite-set thinning contract, and it describes how an alternative caller could use that contract. It does not show that unchanged Main.lean chooses this tuple or onset. No alternative Main theorem has been written or elaborated.

No new Lean or Lake run was performed for this result. The cutoffs were not numerically evaluated or certified; no selector or executable cost and batch calculation was produced. The review did not establish source-to-cache correspondence, kernel trust, complete imported-proof closure or theorem-wide assurance. It makes no claim about novelty, a practical application or the physical validity of the underlying result. The alternative construction remains unwritten, and the wider assurance review remains incomplete.

That boundary is useful beyond this example. “The source obligations appear to fit under a stated mathematical contract” and “the proposed formal proof has compiled and been checked” are different claims. A clear audit should say which one its evidence supports.

Sources and review limits

The underlying theorem and Lean source are in the pinned OpenAI Math revision. The fixed-tuple mass replay and symbolic caller map are summarised in the public Research Artifact Ledger's 26 October source-audit update. The 28 October cutoff-scale update records the conditional scale calculation and its limits. The component source review also uses the pinned Mathlib revision.

AI assisted with drafting and editorial development from the cited public source record. The source reviews were AI-assisted and bounded; reviewer runtime identities were not available for readback. No independent human mathematical specialist review is claimed. The featured image is AI-generated conceptual artwork, not data or a depiction of the theorem. The author remains responsible for the account and its limitations.