Which Bound Sets the Gaussian-Moat Threshold?

In a fixed-parameter reconstruction of the Gaussian-moat argument, the density estimate kicks in at a real scale of roughly $9.6\times10^{1912}$. Even at the smallest parameter pair, the bound that sets the construction’s starting point comes from a different part of the argument.

The underlying OpenAI Math result concerns walks on Gaussian primes, which lie on the integer lattice in the complex plane. The question is whether a walk with a fixed upper bound on its step lengths can reach arbitrarily far. The source audit asks a narrower question. If we carry one set of parameters through the proof’s final caller, which of its prescribed sufficient bounds on the integer input $m$ sets the starting point? Pinned OpenAI Math source

Three conditions share a starting point

The density premise supplies enough well-separated groups of primes at every real scale $X\geq X_{\mathrm{res}}$. The caller starts at $X=\exp(m)$ and uses a rounded sufficient guard on $m$ to ensure that $\exp(m)$ reaches both $X_{\mathrm{res}}$ and $5000\log 2$. It takes the maximum of that guard, the bounds for the finite family of batch certificates and the bound for the smoothing estimate.

In the case $D=d=1$, the radius-one lattice ball contains just $0$, $\pm1$ and $\pm i$. Its five points give the $\log 5$ in the window-width formula. Write $r$ for the fixed positive rate of the construction and $W$ for the number of windows:

The 28 October scale calculation puts this width at about $7.3\times10^{2533}$ for this parameter pair. The 5 November calculation keeps the ceiling and the final $+1$ and compares the thresholds using the exact $W$.

The term that wins

Set $B=100^W$. Every window index $w$ is an integer below $W$, so $100^w$ is at most $B/100$. At $D=d=1$, the source-level calculation bounds all ten batch conditions by $WB$, including their ceilings, and the caller’s density guard by $B$. Within the smoothing requirement, one candidate is exactly $6WB$ and every other candidate is smaller.

Since $6WB$ is already an integer, rounding up leaves it as it is. The complete sufficient threshold for this construction at the specified pair is therefore

The construction still depends on the separate density derivation for every real scale from $X_{\mathrm{res}}$ onwards. That premise comes from replaying the source argument in ordinary mathematics. The identity evaluates the maximum of the prescribed sufficient bounds and leaves the least admissible input open.

The number is far beyond a practical search scale. The comparison shows where the size comes from. Lowering the caller’s density guard alone would leave the maximum at $6WB$. The broader parameter range brings the interval bound into the comparison. 6 November full-domain calculation

Evidence and review

The 5 November derivation defines the rate $r$ in full and records the term-by-term comparison. The 26 October density replay and caller map supply the premise and the formula being evaluated. The 28 October scale calculation gives the earlier estimates. An independent AI challenge of the corrected source packet checked the guard comparisons and ceilings and reported no missing condition. The calculation remains a source-level derivation. Formalising the density premise and elaborating an alternative caller would be separate mathematical and Lean work.

OpenAI GPT models in Codex assisted with the source checks, drafting and editing of this article. The mathematical challenge above was also AI-assisted. Independent human specialist review has yet to take place.