scope and method

How the retuning was found

In August 2026, OpenAI announced a deterministic polynomial-time many-one reduction from 3SAT showing that the Euclidean GapCVP promise problem at approximation factor $n^{1/400}$ is NP-hard. They presented the result as one of ten advances in mathematics and theoretical computer science, and described the collection as resolving or making substantial progress on long-standing problems. I sometimes call it a breakthrough, but that is my characterization, not a quotation from OpenAI. The source is the paper $n^{1/400}$-Hardness of the Euclidean Closest Vector Problem and OpenAI's August announcement.

The published $n^{1/400}$ specialization was not tight within the retained proof architecture. A small retuning of the same construction reaches $n^{1/30}$. I do not know whether OpenAI had already considered that parameter choice, so the precise claim is about the published specialization, not about what its authors realized.

The parameter audit exposes three main knobs: the field-size exponent $Q$, the support exponent $\kappa$, and the moment-horizon exponent $\tau$. OpenAI's fixed settings make the $1/400$ argument work. The retuning method looks for constructions with this kind of coupled resource budget, then:

1. treat the parameter exponents as variables;
2. write the strict inequalities the argument actually uses; and
3. eliminate the internal resource coordinates.

In this case, eliminating $\kappa$ and $\tau$ exposes the condition $(1-28c)Q>9+14c$. When this works, the result can be a new resource lower bound or a sharper frontier for the retained architecture. It is not automatically a lower bound for GapCVP itself or for every possible reduction.

The informal derivation is short because this is parameter retuning: showing that the new values work is largely an audit of the inherited construction and its lemmas, not a new lattice construction. That makes the explanation compact, not automatic; finite constants, interfaces between lemmas, and gap-transfer steps still give the argument many opportunities to fail.

To catch those failures, I put the concrete values back into the large, roughly 130,000-line Lean development. Lean accepted the$c=1/30$ specialization with no sorry or admit in the modified proof files, and the axiom probes reported no sorryAx. They reported only standard Lean axioms. This checks the concrete specialization and its interfaces; it is not an independent recreation of the imported proof and is not external specialist review.

The discovery was partly luck: this construction happened to expose three separable but coupled resource coordinates. It was also the result of applying the same three-step method to dozens of problems over the preceding months. I directed the questions, research decisions, and editorial judgment; most of the mathematical exploration, parameter search, and formalization in this episode were carried out by GPT-5.6. In this account, "I" includes that human direction and AI work.

For provenance, the work used a regular ChatGPT Pro account with GPT-5.6 Sol at xhigh, roughly a quarter of one week's quota, and a large multicore DigitalOcean droplet. Those details explain how the audit was done; they are not evidence for the theorem.

One reduction, several resource coordinates

Let the field size satisfy $q\asymp N^Q$, the large-support threshold be $K=N^\kappa$, and the moment horizon be $T=N^\tau$. The output dimension obeys $M\le 40Nq^2$. A target Euclidean GapCVP exponent $c$is feasible only when all four resources fit at once.

The FPRD move is to keep the coordinates visible. Just as the integer 24 may be presented as $4\cdot3\cdot2$, the same reduction may be presented by the resource factorization that makes its bottleneck inspectable. The representation does not change the theorem; it reveals which constraint is doing the work.

FPRD-T113 · retained-resource frontier

The strict frontier is 1/28

The retained construction requires strict exponent inequalities

The last two give $Q>2+7\kappa$. Substituting the support requirement yields the single necessary bound

If $c\ge1/28$, the left side is nonpositive for $Q\ge0$ while the right side is positive, so no retained tuple exists. If$0\le c<1/28$, choosing Q large enough and then choosing natural-number$\kappa,\tau$ with slack gives a witness. Lean checks both directions.

The number $1/28$ is an unattained supremum for these retained inequalities. It is not a universal barrier for GapCVP reductions, and the general frontier theorem is not yet one uniformly executable end-to-end reduction for every smaller exponent.

FPRD-T114 · concrete checked reduction

At 1/30, one tuple clears every obligation

The three exponent comparisons have explicit slack:

Taking q to be the least power of two at least$N^{242}$ gives$M\le40N^{485}$. Separate quarter-field budgets handle large-support fibers and the anchor/Hankel exceptional set. The exact relation $\ell+1\le N$sharpens completeness to$r^2\le4R\le4Nq$.

The remaining support margin reduces to$16^{15}\cdot40<N^{10}$, already true at $N=100$ and increasingly strong thereafter. The parity lift preserves squared Euclidean distance, so the Euclidean exponent $1/30$becomes binary exponent $1/15$ for nearest-codeword and syndrome decoding.

What was checked

The integer-arithmetic certificate replays every concrete field, support, reconstruction, interpolation, valuation, radius, parity, and gap-transfer condition. The Lean development reaches the concrete Euclidean GapCVP theorem and both binary NP-hardness endpoints. Thirteen axiom probes report only standard Lean axioms, and the modified proof files contain no admitted proof.

This closes the parameter bookkeeping for the retained construction at$c=1/30$. It does not claim a new lattice-hardness method, practical parameter sizes, novelty, or external specialist review.

Evidence and provenance

• GapCVP parameter-retuning paper and publication record
• Algebraic-sewing and frontier note
• Independent Lean verification note
• Complete parameter audit
• Executable integer-arithmetic checker