Conditional refinements of OpenAI's October 2026 math release

This research draft proposes a stronger coefficient in the released van der Waerden lower bound, together with an elementary geometric lemma and a parameter-transfer argument. It also gives an abstract obstruction for a selected set of matrix-multiplication profile inequalities.

Status: AI-generated mathematical research prepared with Codex. The main refinements are conditional on their stated upstream inputs. They have not received independent human mathematical review or formal verification; the AI-assisted source reviews recorded for them are not a substitute for either. This repository does not claim a verified world record or a new matrix-multiplication algorithm. It is an independent follow-up, not an official OpenAI publication. The rank-split, digit-rank, and short-period refinements were prepared with assistance from Claude (Anthropic).

The conditional review and selected Lean proofs record the review scope and imported assumptions. They do not constitute end-to-end formal verification.

Read the full argument in research-note.tex. Exact checks, parameter choices, and experimental records are included below.

Main proposed refinement

The original manuscript establishes a bound of the form

$$W_r(k)>k^{c k\lfloor\log_2 r\rfloor}$$

with $c=1/100000$, for sufficiently large $k$ and every integer $r\ge2$. Our conditional parameter-transfer argument admits $c=1/9$, with the threshold independent of $r$, and every fixed $c\lt1/8$ in the short-period region below.

That is a $100000/9\approx11{,}111$-fold increase in the coefficient inside the exponent (and $25/18\approx1.39$ times the previous explicit choice $2/25$). The earlier $2/25$, $1/16$, $1/25$, $1/193$, and $1/200$ choices remain included for comparison. No explicit finite coloring record or runtime improvement follows. The threshold may be very large and is not made explicit here.

The earlier repeated-key certificate cover admits $c=1/19$, a factor $100000/19\approx5263$ over the source.

Progress over time

The data behind the chart are in progress/progress.json and the chart is regenerated by progress/progress_chart.py. Plain-language summaries of each step are in progress/.

Why the argument changes

The source bounds the multiplicity of a geometric key by four. Its affinity and separation properties place points of one key on an affine real line, at mutual distances at least one, with squared norms in a half-open interval $[m,m+1)$.

In fact there are at most two such points. Parameterize the line by signed arclength $t$ from its nearest point to the origin, so squared norm is $t^2+C$. Among three parameters, two lie on the same half-line. Their absolute values satisfy $v-u\ge1$, hence

$$v^2-u^2=(v-u)(v+u)\ge1,$$

contradicting membership in a common half-open interval of width one. This elementary lemma is proved directly; applying it to the original construction still requires the source's hypotheses.

Using this multiplicity bound, write $D=\lceil k^a\rceil$, $M=\lceil k^m\rceil$, and $p=k^{-b}$. The note derives the sufficient region

$$a,b,c>0,\quad 3a\lt m\lt 1,\quad b\lt a,\quad 3a\lt 1-b,$$ $$0<\delta<1,\quad 0<\gamma\le\delta/4,\quad \gamma<(1-\delta)/8,\quad 2c<\gamma b/2.$$

Choice	$a$	$m$	$b$	$\delta$	$\gamma$	$c$	Coefficient ratio
Original dimension scales	1/10	1/2	1/20	1/10	1/40	1/4000	25
Retuned parameters	1/5	3/4	1/6	1/10	1/40	1/1000	100
Earlier 500-fold choice	1/4	4/5	247/1000	1/3	41/500	1/200	500
New proposed choice	1/4	4/5	2499/10000	1/3	83/1000	1/193	100000/193

The new choice has strictly positive exact margins and stays below the retained region's endpoint $1/192$. Its full transfer argument and margin table are in near-limit-proof.md.

For the earlier 500-fold row, $\gamma b/2-2c=127/1000000$ and both affine entropy power gaps equal $3/1000$. These strict margins are checked with exact rational arithmetic in parameters.json and audit_math.py.

The same sufficient region admits every fixed positive $c&lt;1/192$. The endpoint is not asserted, and this is only a supremum within the retained estimates. The full transfer proof and its dependencies are in the note; passing the scripts does not establish that transfer as a theorem.

Breaking the $1/192$ ceiling: the sharpened region

The $1/192$ supremum comes from two fixed choices inside the source argument, not from its geometry. First, the outer coloring is only asked to place one quarter of each bit on every tested set of labels, which caps $\gamma$ at $1/12$. Second, the short-period cutoff $h_0=D^2$ makes the global label entropy $k^{3a}$, which caps $b$ at $1/4$. The note's new Section 4 changes three things, with the source's geometric lemmas, anchored pattern count, affinity lemma, prime bound, and digit product untouched:

1. Shorter dilation. Take $h_0=D\lceil\log k\rceil^2$ and $\eta=\varepsilon_{\rm r}/(8D)$. The dilation lemma still holds, the nonregular fraction becomes $\le\varepsilon_{\rm r}$, and the global entropy drops to $O(k^{2a}\log^3k)$.
2. Sharper balance. Ask the local lemma for a $(1/2-\varepsilon)$ fraction of each bit instead of one quarter. The Hoeffding-type tail $2e^{-2\varepsilon^2 l}$ and weights $e^{-\varepsilon^2 l}$ satisfy the asymmetric local-lemma criterion for every fixed $\varepsilon$, since all tested sizes are unbounded.
3. Balance the light positions outside heavy blocks. Those positions are a function of the label word, so they get their own first-family event. Heavy blocks and these light positions then cover all but $h+\delta k=o(k)+\delta k$ positions.

The resulting sufficient region is

$$a,b,c>0,\quad 2a\lt m\lt 1,\quad b\lt a,\quad 2a\lt 1-b,$$ $$0<\varepsilon<\tfrac12,\quad 0<\varepsilon_{\rm r}\le\tfrac12,\quad 0<\delta,\quad \gamma<(\tfrac12-\varepsilon)(1-\varepsilon_{\rm r})(1-\delta),\quad 2c<\gamma b/2,$$

so $b\lt1/3$, $\gamma\lt1/2$, and $c\lt\gamma b/4\lt1/24$; every fixed $c\lt1/24$ is admitted.

Choice	$a$	$m$	$b$	$\delta$	$\varepsilon$	$\varepsilon_{\rm r}$	$\gamma$	$c$	Coefficient ratio
Sharpened explicit choice	333/1000	4/5	331/1000	1/200	1/200	1/200	97/200	1/25	4000

Its decisive margins are $\gamma b/2-2c=107/400000$, $1-b-2a=3/1000$, $a-b=1/500$, and $(1/2-\varepsilon)(1-\varepsilon_{\rm r})(1-\delta)-\gamma=40499/8000000$. The full argument and margin table are in sharpened-region-proof.md; audit_math.py checks them exactly and rejects five boundary controls, including $c=1/24$ itself. Those controls test the earlier condition $2c\lt\gamma b/2$; the new certificate-cover region below is not yet checked by audit_math.py.

Repeated-key certificate cover: conditional 1/19

The separate repeated-key certificate note adds a new counting and probability argument to the sharpened construction above. It bounds literal sets of repeated keys with leading entropy $q^D$, then covers rich monochromatic events by certificates requiring only their selected repeated keys to flip. Different full prescriptions remain different events; the proof uses containment of their union.

The sufficient rich-event inequalities become

$$2c\lt b(\gamma-\tau),\qquad c\lt b\tau,\qquad 0\lt\tau\lt\gamma.$$

Choosing $\tau=\gamma/3$ gives $c\lt\gamma b/3$. With the unchanged sharpened outer input, the note derives $c=1/19$ and admits every fixed $0\lt c\lt1/18$, for sufficiently large $k$ with a threshold independent of integer $r\ge2$. This remains conditional on the precisely stated global outer-color dichotomy. The note includes the full new geometry, certificate count, event cover, exact rational parameter choice, affine count, and digit transfer; finite tests are not premises of the proof.

The initial argument is attributed to a ChatGPT Pro research response by GPT-6 Pro (internal run label "O1"), captured on October 10, 2026; the publication draft was assembled with Codex. Prior source authors and this repository's existing refinements retain their attribution. No literature-priority, formal-verification, explicit coloring, or practical threshold claim is made. The earlier progress chart and parameter tables above document their original stages and have not been regenerated for this separate proof note.

Why $1/24$ is a ceiling of this architecture

Every argument here, and the source manuscript, follows one template: a label-based outer coloring that balances every non-affine progression, keys with at most $\mu$ occurrences per progression carrying Bernoulli($k^{-b}$) flips, a union bound over the $N^2$ progressions for the balanced case, and a union bound over affine signatures for the affine case. barrier-note.md proves that within this template with $\mu=2$, affine key-word entropy $(6+o(1))\log q$, and label entropy $k^{2a+o(1)}$, every admissible coefficient satisfies $c\lt1/24$: the rich bound gives $c\lt\gamma b/4$, balance gives $\gamma\le1/2$, and the affine bound forces $b\lt a$ and $2a+b\le1$, so $3b\lt2a+b\le1$ and $b\lt1/3$.

The note then records, feature by feature, why the three hypotheses resist change: the two-point multiplicity is the reflection symmetry of any quadratic band along a line, and distinguishing the pair costs about $D$ functionals worth $2\log q$ each; the $6\log q$ is tight for floor-words of quadratics; the $k^{2a}$ comes from $2D\log\lambda$ with $\log\lambda\gtrsim h_0\gtrsim D$ forced by the nonregular fraction; and a local-lemma pass over the flips reproduces the $N^2$ union bound because a key's fiber has about $Nk^{-2D}$ elements. AND-ing two flip layers, modular or widened bands, deterministic band parities, and coloring bands in the outer layer are each shown to fail. These explanations are not theorems. audit_math.py confirms the ceiling $1/24$ exactly (template_ceiling). Going further needs a different construction, not different parameters.

Past $1/24$: a rank-split cover of rich events

The sharpened region's last binding inequality, $2c\lt\gamma b/2$, comes from a union bound over all $q^{2D}$ progressions in which each repeated key contributes only one flip. The merged repeated-key proposal, rohanarun/van-der-waerden-refinement#4, observed that repeated keys can be certified with entropy about $q^D$. That gives $c\lt\gamma b/3$ and every $c\lt1/18$. rank-certificate-proof.md restates its repeated-key facts with credit, then splits rich events by the carry rank $\rho$ of the residue classes that contain repeated keys:

1. Start-free certificate. Every key in those classes, singles included, is fixed by $3+\rho$ real projections. These key sets therefore number $q^{2\rho+o(D)}$, and also $q^{D+o(D)}$ by the anchored count.
2. Rank count. Each such class pins an integer linear form in the base-$q$ digits of the step. Progressions of rank $\rho$ therefore number at most $q^{2D-\rho+o(D)}$.

Rich events are then split by whether at least $\theta k$ flipped keys lie in those classes. The worst rank is $\rho=D/2$, and the rich condition becomes

$$5c\lt2\gamma b,$$

with every other inequality of the sharpened region unchanged. Hence every fixed $c\lt2\gamma b/5\lt1/15$ is admitted.

Choice	$a$	$m$	$b$	$\delta$	$\varepsilon$	$\varepsilon_{\rm r}$	$\gamma$	$c$	Coefficient ratio
Rank-split explicit choice	333/1000	4/5	331/1000	1/200	1/200	1/200	97/200	1/16	6250

The exponents and balance constants are those of the $1/25$ choice; only the coefficient changes. The decisive margin is $2\gamma b/5-c=857/500000$. audit_math.py checks:

• every margin, in exact fractions;
• the worst rank $\rho=D/2$, where the two costs sum to $5/2$ (in units of $ck\log k$);
• the two group margins on a grid of ranks;
• the new identities, symbolically;
• the digit pinning, on 2,000 exact instances.

It also rejects five boundary controls, including $c=1/15$. The value $1/16$ lies outside both earlier rich conditions, so it is not a reparametrization of them.

Repeated keys cost nothing: the digit rank

The rank-split argument paid $2\log q$ per direction of the carry span $V$ in its certificate, but saved only $\log q$ per direction in its progression count. digit-rank-proof.md observes that both sides are made of base-$q$ digits in the same pattern:

• Start side. $q^2\langle Y_{a_0},v\rangle$ is fixed, up to a bounded tail, by two integer forms in the start point's digits.
• Step side. Each repeated pair fixes the same two forms in the step's digits: the first exactly, the second up to nine values.

Both counts are therefore governed by one digit rank $R=\dim(S_1V+S_2V)$. The certificate costs $R\log q$ and the progression count $(2D-R)\log q$, so their sum is $2D$ for every $R$. The rich condition becomes

$$2c\lt\gamma b,$$

which is what the union bound gives when no key repeats at all. Every other inequality of the sharpened region is unchanged, so every fixed $c\lt\gamma b/2\lt1/12$ is admitted.

Choice	$a$	$m$	$b$	$\delta$	$\varepsilon$	$\varepsilon_{\rm r}$	$\gamma$	$c$	Coefficient ratio
Digit-rank explicit choice	333/1000	4/5	331/1000	1/200	1/200	1/200	97/200	2/25	8000

These are again the exponents and balance constants of the $1/25$ choice. The decisive margin is $\gamma b/2-c=107/400000$, the same margin that certified $1/25$ under $2c\lt\gamma b/2$; the coefficient doubles. audit_math.py checks:

• every margin, in exact fractions;
• $\kappa+\nu=2$ for every digit rank;
• both group margins on a grid of ranks;
• the digit decompositions, on 3,000 exact instances;
• $\rho\le R\le\min(2\rho,D)$, on random carry spans.

It also rejects five boundary controls, including $c=1/12$. On the rich side, $2c\lt\gamma b$ is also what progressions with no repeated key give on their own. Going past $1/12$ would therefore need $\gamma$ above $1/2$, $b$ above $1/3$, or a new idea for those progressions.

Past $1/12$: short-period runs

The digit-rank region still carries the sharpened region's affine inequality $2a\lt1-b$, which caps $b$ at $1/3$. It comes from paying for the full label word, $\log T_{\rm glob}=O(k^{2a}\log^3k)$, in the estimate for progressions that are not $\gamma$-rich. short-period-runs-proof.md observes that those progressions are short-period, so their color word is much cheaper to pin down:

1. Runs. In a short-period progression, the occurrences of each full label form one run: one residue class modulo the period, consecutive within it.
2. A third local-lemma family. Only $e^{O(D\log k)}$ short-period label words pass through a given label, so the local lemma can also balance their light positions (multiplicity at most $k/M_s$, with $M_s=D\lceil\log k\rceil^2$).
3. Two cases. If light positions fill at least $\delta_sk$ of the progression, about half of them need a flip, and the event is covered by the key word alone. Otherwise at most $M_s$ heavy runs fix the color word, at a cost of $e^{O(D\log^3k)}$.

No full label word is counted, so $2a\lt1-b$ becomes $a+b\lt1$, which already follows from $b\lt a\lt1/2$. The binding constraints are now $2a\lt m\lt1$ and $b\lt a$, so $b\lt1/2$. With the rich condition $2c\lt\gamma b$, every fixed $c\lt\gamma b/2\lt1/8$ is admitted.

Choice	$a$	$m$	$b$	$\delta$	$\varepsilon$	$\varepsilon_{\rm r}$	$\gamma$	$c$	Coefficient ratio
Short-period explicit choice	49/100	99/100	97/200	1/200	1/200	1/200	97/200	1/9	100000/9

The light-run threshold is $\delta_s=1/2$. The decisive margin is $\gamma b/2-c=4681/720000$, and $m-2a$, $1-m$, $a-b$, $1-a-b$ are $1/100$, $1/100$, $1/200$, $1/40$. The old affine gap $1-b-2a=-93/200$ fails on purpose, so this choice lies outside the digit-rank region. audit_math.py checks:

• every margin, in exact fractions;
• that the old affine gap fails;
• the supremum $b\lt1/2$, by linear programming.

It also rejects five boundary controls, including $c=1/8$.

Original work and attribution

All comparisons refer to OpenAI's manuscripts at commit fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb, not to a claim about the latest literature.

• OpenAI, Quantitative Superexponential Bounds for van der Waerden Numbers, September 23, 2026. This is the original construction on which the main refinement depends.
• OpenAI, An Upper Bound of 9/4 for the Matrix Multiplication Exponent, October 2, 2026. Source of the profile constraints considered here.
• OpenAI, A counterexample to Sidorenko's conjecture, September 23, 2026. Source of the finite certificate reproduced for exploratory searches.
• OpenAI, A power saving for planar unit distances, September 23, 2026. Inspected for possible quantitative improvements; none is claimed.

The original authors retain credit for their constructions, lemmas, and results. The finite Sidorenko baseline is derived from the original manuscript. source-manifest.json records the pinned revision and SHA-256 hashes of the inspected source files. Upstream source trees are not bundled; the fetch command below retrieves and verifies them from the original repository. Citation entries supplied by the authors are available in the linked original directories.

Companion result and unsuccessful searches

For the selected scalar matrix-profile constraints, let $u=\min(a,b)$ and $v=\max(a,b)$. The explicit profile

$$P_*(a,b)=\left(\frac{3u-1}{2}\right)^{1/3}\left(v+\frac{u-1}{2}\right)$$

is symmetric, positive, nondecreasing, and separately concave, with $P_*(1,b)=b$. It satisfies shifted tripling and the rank envelope at $t=3/4$, while its diagonal grows as $\Theta(a^{4/3})$. Thus those listed constraints alone cannot exclude $t=3/4$. This is not an actual tensor character or a lower bound on the matrix-multiplication exponent. The note gives an analytic proof; finite LP experiments are exploratory.

The Sidorenko search found no smaller certificate: 56,984 labeled candidates in the contraction/flip run, and 28,281 in the weighted-cut run, of which 2,518 passed the discrete and degree screen. These nonexhaustive searches may overlap and include isomorphic duplicates. Floating-point LP failures are not infeasibility certificates. The original 35-vertex, 66-edge finite certificate passes the independent combinatorial checker; this does not verify the full analytic counterexample proof.

No improved unit-distance exponent was obtained.

Reproduce the checks

Use Python 3.11 or 3.12 in a local checkout:

python3 -m venv .venv
. .venv/bin/activate
python -m pip install -r requirements.txt
python -B audit_math.py
python -B check_complex.py experiments/complex-contractions/baseline.json --output experiments/baseline-independent-check.json

Do not run with -O or PYTHONOPTIMIZE: the verification scripts use assertions. GitHub Actions runs these same checks on pushes and pull requests.

Check	Evidence	Limitation
Four parameter proposals	Exact rational margins for four choices; three invalid boundary controls rejected	Validates inequalities, not the imported lemmas
Sharpened region, $c=1/25$	Exact rational margins for the explicit choice; five boundary controls rejected, including $c=1/24$	Validates inequalities, not the modified local-lemma transfer
Rank-split region, $c=1/16$	Exact margins; worst rank $\rho=D/2$; symbolic identities; 2,000 exact digit-pinning instances; five boundary controls rejected, including $c=1/15$	Validates inequalities and identities, not the counting lemmas or the imported structure
Digit-rank region, $c=2/25$	Exact margins; $\kappa+\nu=2$ for every digit rank; 3,000 exact digit-decomposition instances; five boundary controls rejected, including $c=1/12$	Validates inequalities and identities, not the counting lemmas or the imported structure
Short-period region, $c=1/9$	Exact margins; the old affine gap $1-b-2a$ fails as intended; supremum $b\lt1/2$ by linear programming; five boundary controls rejected, including $c=1/8$	Validates inequalities, not the run lemmas, the third local-lemma family, or the imported structure
Template ceiling $1/24$	Linear program for $\sup b$ and exact product $\tfrac12\cdot\tfrac13\cdot\tfrac14$	A statement about the proof template, not about van der Waerden numbers
Norm-band lemma	Displayed proof plus 26,180 finite rational cases	Finite cases supplement the proof
Abstract matrix profile	Symbolic identities plus 10,000 exact cube comparisons	No tensor realizability claim
Released Sidorenko baseline	Independent reconstruction; duplicate-face and incomplete-exposure controls rejected	Finite combinatorial hypotheses only

Saved outputs are exact-audit.json and baseline-independent-check.json. Earlier LaTeX revisions were compiled; compilation of those revisions does not certify the merged mathematical argument. No compiled PDF is included.

To retrieve the exact upstream source files (optional for the checks above):

python -B fetch_sources.py --output upstream

Exploratory scripts provide --help: search_complex.py, weighted_cuts.py, and profile_lp.py. Searches require explicit seeds, time budgets, and output paths. Their wall-clock-limited candidate counts are machine-dependent; historical summaries are retained under experiments. Use a new output directory when rerunning a search to preserve those records.

What still needs review

The main outstanding work is independent verification of the complete van der Waerden parameter transfer, including uniform constants and all imported hypotheses. The sharpened region additionally relies on the three modifications in Section 4 of the note (shorter dilation, $(1/2-\varepsilon)$-balance in the local lemma, and the separate balancing of light positions outside heavy blocks); each is argued in sharpened-region-proof.md but has not been independently reviewed by a human. The rank-split region further relies on Lemmas A–C of rank-certificate-proof.md, and on the repeated-key structure of the merged proposal #4, which is restated there. Neither has been independently reviewed by a human. The digit-rank region replaces those lemmas with Lemmas A* and C* of digit-rank-proof.md, also not independently reviewed by a human. The short-period region further relies on Lemmas 1 and 2 and the third local-lemma family of short-period-runs-proof.md, likewise not independently reviewed by a human. Pushing past $1/8$ would need $\gamma$ above $1/2$, $b$ above $1/2$ (which $b\lt a$ and $2a\lt m\lt1$ forbid for these keys and this outer coloring), or a new idea for rich progressions without repeated keys. Literature priority has not been established. For matrix multiplication, progress requires additional valid tensor constraints or a different construction. Any smaller Sidorenko finite object would still need the full analytic transfer before it could be called a new counterexample.

The barrier note applies to the earlier full-progression rich-event template. The later certificate and short-period arguments change that template and are consistent with its stated barrier.