Openai 2026 renewal changes law critical honeycomb walks

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theorem_3_3: Claimed exponent-3/4 law for the uniform critical bridge of one exact length on the honeycomb lattice, obtained from a local renewal lower bound by exponential tilting and Fourier inversion; unverified here.

theorem_6_4: Claimed exponent-3/4 law, with moments, for the full-plane honeycomb self-avoiding walk weighted by (rho e^{-1/N})^n at every large N, proved through a recoverable lower bound for the partition function and a two-end estimate at an internal minimum; unverified here.

theorem_8_1: Claimed moment laws n^{3p/4+o(1)}, an all-length upper bound on the maximum radius of uniform endpoint-free honeycomb walks, the half-plane lower law on a density-one set of lengths, and the half-plane thermal and fixed-height laws; unverified here.

theorem_8_2: Claimed honeycomb analogue of the planar exponent-3/4 law for the uniform n-step self-avoiding walk, proved by inserting a long bridge at a minimum and bounding the inverse multiplicity; the lower law holds along a density-one set of lengths, not at every length; unverified here.

OpenAI, Renewal and changes of law for critical honeycomb walks, OpenAI Math Release preprint, September 26, 2026. Released under the Apache License 2.0 at https://github.com/openai/math (revision adc7f1241), folder preprints/Renewal-and-changes-of-law-for-critical-honeycomb-walks-September-26-2026; the held PDF, main.pdf in the release, is retained as openai_2026_renewal_changes_law_critical_honeycomb_walks.pdf, and the release's TeX bundle sits in the same folder.

@misc{OAI:Renewal-and-changes-of-law-for-critical-honeycomb-walks-September-26-2026,
  author = {{OpenAI}},
  title = {{Renewal and changes of law for critical honeycomb walks}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Renewal-and-changes-of-law-for-critical-honeycomb-walks-September-26-2026/main.pdf}{OAI:Renewal-and-changes-of-law-for-critical-honeycomb-walks-September-26-2026}},
  year = {2026}
}

The release README describes the collection as manuscripts "produced by an internal OpenAI model", says that it "includes results at different stages of verification", that not all have Lean formalizations, and that "Some of the unformalized results could have issues". The manuscript's own README carries only the title, the author line "OpenAI", the date and the citation block; it adds no statement about human assistance or review. These are the source's own attestations, recorded here as history and not as this corpus's review. No refereed publication, arXiv version or independent review of the manuscript is recorded here and nothing on this card is independently reviewed.

The release's lean/formalization.yaml does not name this manuscript. Its Lean page for the family lists it, with Polynomial vacuum representations and bridge mass for honeycomb walks and Critical strip-crossing mass on the honeycomb lattice, as an accompanying paper of the family, and names three comparator statement files: HoneycombBridgeFiniteness.lean (finiteness of the critical bridge mass and first length moment at every strip height, also under a corridor confinement of width $h(\log h)^2$), HoneycombFreeEnergy.lean (existence of the free-energy limit of the critical partition function under a force) and CriticalStripMass.lean (strip-crossing mass comparable to $N^{-1/4}$ and first horizontal-displacement moment of return paths comparable to $N^{3/4}$). The page itself says that the $3/4$ spatial and moment laws, the small-force exponent and the near-critical correlation scale lie outside these statements. This listing is read statically from the release's catalogue; the build of the three statements here is recorded below. None of the three files is a formalization of any Erdős problem.

Formal verification here: this corpus's verification built OAI.HoneycombForce.logPartition_tendsto, OAI.PolynomialVacuumBridge.Corridor.finite_bridge_sums and OAI.CriticalStrip.critical_strip_mass at the release's revision adc7f1241b42e322a6451854ab7e4b4c146bf78a (2026-10-06) with toolchain leanprover/lean4:v4.34.1 on 2026-10-08. The axioms of each are exactly propext, Classical.choice and Quot.sound, no sorry appears, and each declaration's fingerprint is identical to its comparator challenge, HoneycombFreeEnergy.lean, HoneycombBridgeFiniteness.lean and CriticalStripMass.lean. Checked clause by clause, logPartition_tendsto certifies only the existence part of Theorem 9.1: for every root vertex $o$, every direction $e$ and every real $s$, the limit of $n^{-1}\log\sum_\gamma\rho^n e^{s\,e\cdot(\gamma_n-o)}$ over $n$-step self-avoiding walks from $o$ exists and is finite. It does not certify $f_e(0)=0$, the small-force law $f_e(s)=s^{4/3+o(1)}$, the thresholds $m_e(t),m_{\mathrm{rad}}(t)=t^{3/4+o(1)}$ or any $3/4$ spatial law. finite_bridge_sums certifies only that, at every strip height $h\ge1$, the critical mass of strict bridges and their first-length mass are finite (its corridor-confined clauses follow from these); it certifies no exponent, so the bound $M_h\le Ch^{13/12}$ of Theorem 2.1 is not certified. critical_strip_mass certifies Theorem 1.1 of the strip-crossing manuscript, the input $B_h\asymp(1+h)^{-1/4}$ of Section 12, as recorded on that card. Beyond these parts no result of this manuscript is certified, and its prose proofs remain unreviewed.

The manuscript cites nine companions from its family for its finite inputs: Uniform marked-polygon estimates and sharp finite bridge moments (the bounded-factor bridge estimates of Theorem 2.1), Radial transfer estimates and polygon length laws for honeycomb walks (slab, arch and polygon inputs of Section 5), Cylinder amplitudes and logarithmic bridge-length windows on the honeycomb lattice (the calibrated inputs of Sections 4 and 7), Polynomial vacuum representations and bridge mass for honeycomb walks and Disk transfer representations and confined bridge mass (Section 10), Signed cylinder propagation and marked polygons on the honeycomb lattice (Section 11), and, for Section 12, Critical strip-crossing mass on the honeycomb lattice, Cylinder loop weights and planar nesting and Mass and covering exponents for fixed-length honeycomb walks. The last of these is the family's principal fixed-length diameter law; the present manuscript supplies the renewal and change-of-law transfers and consumes the companions' estimates as stated inputs. The three other family members, Critical honeycomb chords with prescribed boundary endpoints, Marked polygon correlations and one-arc bounds and Cap-selected amplitudes and triangle chords for honeycomb walks, are not cited by the manuscript.

Read status: claims checked for Theorems 2.1, 2.7, 2.8, 3.3, 3.4, 4.3, 6.4, 7.1, 8.1, 8.2, 9.1, 9.5 and 11.1, Propositions 2.4, 4.1, 5.12, 5.13, 8.4, 12.1 and 12.3 and the finite-input lists of Sections 5, 10, 11 and 12, read clause by clause in the TeX source (main.tex and the twelve section files under sections/ of the release's TeX bundle, from introduction.tex to geometric-transfers.tex) on 2026-10-07, with theorem numbers and pages taken from the held PDF (98 pages); the proofs were read for their structure only and no step was checked; nothing here is independently reviewed.

Contents

Conventions (Section 1, pp. 2--5). Honeycomb vertices are the centers of the triangles of a side-one equilateral tiling, $\rho=(2+\sqrt2)^{-1/2}$ is the critical weight, and a path with $L$ visited vertices has weight $\rho^L$. Ports are midpoints of tiling edges lying on a domain boundary, here a horizontal row boundary; a strict bridge of height $h$ joins a fixed port to a port on the row boundary $h$ rows higher, with every visited vertex strictly between the two. The row boundaries that a bridge crosses exactly once cut it into irreducible bridges, whose critical weights add up to one and so form the law $p$; independent $p$-samples concatenate with no further avoidance condition. $A$ is endpoint distance, $D$ diameter. The introduction lists the principal claims and cites Nienhuis for the predicted exponent $\nu=3/4$, Kesten, Madras--Slade, Lawler--Schramm--Werner and Dyhr et al. for the renewal construction on the integer lattice, Duminil-Copin--Smirnov for $\rho$, Beaton et al. and Krachun--Panagiotis for earlier honeycomb bridge-mass results, and the companions for the finite estimates.

• Section 2, One probability space for the different laws (pp. 6--10). Theorem 2.1 (finite bridge estimates, cited from Uniform marked-polygon estimates, Theorem 1.1): $B_h\asymp h^{-1/4}$, first-length mass $M_h\le Ch^{13/12}$, mass at least $ch^{-1/4}$ on $L\ge ch^{4/3}$, diameter tail $B_h\{D>x\}\le Cx^{-1/4}$. Proposition 2.2 (renewal factorization, $\sum p=1$, $b(s,0)\asymp s^{-3/4}$), Lemma 2.3 (marked-piece identity), Proposition 2.4 (joint increment estimates for one irreducible: $1-\mathbb E_pe^{-sH}\asymp s^{3/4}$, $p(H>x)+p(D>x)\le Cx^{-3/4}$, $p(L>n)\le Cn^{-9/16}$, $1-\mathbb E_pe^{-uL}\asymp u^{9/16}$), Remark 2.5 (exponent-only variants), Lemma 2.6 (Laplace deficit to growth of independent sums), Theorem 2.7 (infinite irreducible-bridge law: $|\gamma_n-\gamma_1|=n^{3/4+o(1)}$ and $R^{4/3+o(1)}$ vertices within distance $R$, almost surely), Theorem 2.8 ($a_{n+2}/a_n\to\rho^{-2}$ for half-plane counts by a hexagon-flip double count; under the uniform half-plane law of length $n$, each fixed initial path has a probability tending to its probability under the infinite law).
• Section 3, Conditioning on one exact length (pp. 10--14). Lemma 3.1 and Theorem 3.2 (local renewal mass $V_m\ge c_2m^{a-1}$ for an integer variable of span one with power tails and two-sided Laplace deficit, by exponential tilting and Fourier inversion; no regular variation assumed). Theorem 3.3 (uniform strict bridges of $n$ visited vertices: $n^{3/4-\xi}\le A\le D\le n^{3/4+\xi}$ with probability tending to one along every sufficiently large even $n$). Theorem 3.4 (critical bridges at fixed height $h$: $L=h^{4/3+o(1)}$, $D=h^{1+o(1)}$ in probability).
• Section 4, Conditioning on length from logarithmic finite estimates (pp. 14--18). A second, self-contained route from weaker inputs. Proposition 4.1 (finite calibrated input, cited from Cylinder amplitudes, Theorems 1.1--1.2 and Proposition 11.2: $B_h\asymp (1+h)^{-1/4}$, confined-bridge mass and first-length bounds, exponential diameter cutoff in strips, arch mass $A_g\asymp g^{-5/4}$, a logarithmic length window). Lemma 4.2 (length deficit with logarithmic loss). Theorem 4.3 ($u_n\ge n^{-7/16-o(1)}$ and $u_{n+2}/u_n\to1$ on even $n$; under the uniform length-$n$ bridge law, the height, the endpoint distance and the maximum radius each grow as $n^{3/4+o(1)}$ in probability; local convergence to the independent-irreducible law), with the transfer to vertex bridges.
• Section 5, Geometric estimates before a change of law (pp. 18--33). Finite inputs cited from Radial transfer estimates (Theorems 1.1, 12.1, 12.2, Proposition 3.1): slab masses, corridor lower bounds, arch mass $a_b\asymp b^{-5/4}$, polygon tails and second moment, thin-cylinder span. Lemmas 5.1--5.4 (increment tails, spatial atoms $Ck^{-8/3}$ and $Ck^{-4/3}$, polygon sewing, adjacent-prefix avoidance $k^{-1+\delta}$), Proposition 5.5 and Corollary 5.6 (marked pairs; $q(t)=t^{-9/16+o(1)}$ from spatial and polygon inputs alone), Propositions 5.7--5.8 (fixed height and two-port arch laws), Lemma 5.9 and Proposition 5.10 (turning-chain estimate with reserved independent groups), Corollary 5.11 (two ends with a reserved height variable), Proposition 5.12 (absolute critical mass $a_n\le C_\delta e^{n^\delta}$, by Hammersley--Welsh unfolding), Proposition 5.13 (absolute fast-path estimate: critical mass of $n$-vertex walks with $D\ge n^{3/4+\epsilon}$ at most $e^{-n^{c}}$).
• Section 6, From renewal pieces to the unrestricted thermal law (pp. 33--39). Lemma 6.1 (half-plane factorization $H_N=U_N/q_N$; as $N\to\infty$ the law of the initial pieces under the fugacity weight tends to that of independent critical irreducibles), Lemma 6.2 (recoverable lower bound $Z_N\ge U_N^2/(2q_N)$ for the plane partition function), Lemma 6.3 (two-end estimate for small endpoint-height difference), Theorem 6.4 (thermal plane walk with weight $(\rho e^{-1/N})^n$: $L=N^{1+o(1)}$, $A,D=N^{3/4+o(1)}$ in probability and all positive moments, at every large $N$).
• Section 7, Censored arms and absolute spatial estimates (pp. 39--49). An independent route from the calibrated inputs: Theorem 7.1 (absolute suppression of rapid travel, $\le C\exp(-R^c)$), Lemmas 7.2--7.4 (atoms, short bridges at a height, censored two-arm survival $f(g)\le C_\eta(1+g)^{-3/4+\eta}$), the kernel calculus of Section 7.3, Proposition 7.5 (critical mass of walks of diameter at most $R$ is at most $\exp(C_\nu(1+R)^\nu)$).
• Section 8, Uniform endpoint-free laws and moments (pp. 49--58). Theorem 8.1 (moments $n^{3p/4+o(1)}$ for the infinite walk and uniform bridges; for uniform endpoint-free walks, planar or half-planar, the maximum radius exceeds $n^{3/4+\epsilon}$ with probability $O(n^{-B})$ for every fixed $B$; half-plane lower law on a density-one set; half-plane thermal and fixed-height laws). Theorem 8.2 (full-plane uniform walks: endpoint distance and maximum radius $n^{3/4+o(1)}$ in probability and in every positive moment along a set of lengths of natural density one; full-plane thermal law as $t\downarrow0$, with no exceptional $t$). Lemma 8.3 (recoverable insertion labels), Proposition 8.4 (favorable exact lengths in $[R^{1-\epsilon},R]$ number $R^{1+o(1)}$; the limsup of every positive moment exponent is $3/4$ times its order; if the exponent exists at all lengths it equals $3/4$).
• Section 9, Correlation lengths, pulling, and prescribed spatial endpoints (pp. 58--67). Theorem 9.1 (exponential thresholds $m_e(t), m_{\mathrm{rad}}(t)=t^{3/4+o(1)}$ uniformly in direction; the limit defining the fixed-length free energy $f_e(s)$ exists, and $f_e(s)=s^{4/3+o(1)}$ as $s\downarrow0$), Corollary 9.2 (one-sided derivatives $s^{1/3+o(1)}$, matching Pincus's relation), Remark 9.3, Proposition 9.4 (lattice local lower bound $cH^{-5/4}$ for a prescribed port displacement with controlled length and diameter, by subsequential Poisson-type limits and Fourier inversion), Theorem 9.5 (near-critical rates $\underline m(N),\overline m(N)=N^{-3/4+o(1)}$; thermal law conditioned to end at $b_N$ with $|b_N-o|=N^{3/4+o(1)}$ has $\ell=N^{1+o(1)}$, $D=N^{3/4+o(1)}$ with moments), Corollary 9.6.
• Section 10, Confined renewal rewards (pp. 67--71). Inputs from Polynomial vacuum representations and Disk transfer representations; Proposition 10.1, Corollary 10.2 (weaker-input consequences, including $|\gamma_n-\gamma_0|\le n^{17/18+o(1)}$), Proposition 10.3 (completed prefix mass $R^{4/3+o(1)}$ in a fixed-proportion ball).
• Section 11, The irreducible length tail from signed cylinder estimates (pp. 71--88). Inputs from Signed cylinder propagation (boundary balance with phase $e^{3iT/8}$, $W_R=R^{-1/4+o(1)}$, cylinder and hexagon visit masses, marked-polygon second moment). Theorem 11.1 ($p(L\ge n)=n^{-9/16+o(1)}$ from these inputs, with the infinite, fixed-height and length-tilted bridge laws), proved through Lemmas 11.2--11.12 and Corollary 11.13 (strip moments, slit bounds, two ordered renewal arms, folding around a fixed irreducible, recoverable cuts, chord second moment).
• Section 12, Geometric transfers to infinite and thermal paths (pp. 88--96). Inputs: $B_h\asymp(1+h)^{-1/4}$ from Critical strip-crossing mass (Theorem 1.1), the first-length bound from Cylinder loop weights (Proposition 11.5), a localization bound and three geometric results (Lemma 4.1, Lemma 5.1, Proposition 3.1, Corollary 6.6 and the diameter clause of Theorem 1.1) from Mass and covering exponents. Proposition 12.1 (stopped renewal paths; the infinite law again), Lemma 12.2 (insertion multiplicities), Proposition 12.3 (thermal stretching by insertion: $\ell=T^{1+o(1)}$, $D$ and endpoint distance $T^{3/4+o(1)}$ in probability).
• References (pp. 96--98): Beaton et al. 2014, Borgs--Chayes--King--Madras 2000, Caravenna--Doney 2019, Duminil-Copin--Smirnov 2012, Dyhr et al. 2011, Garsia--Lamperti 1962, Hammersley--Welsh 1962, Ioffe 1998, Ioffe--Velenik 2008 and 2010, Kesten 1963, Krachun--Panagiotis 2026, Lawler--Schramm--Werner 2004, Madras--Slade 1993, Nienhuis 1982, Pincus 1976, Schilling 2016, and the nine family companions.

External inputs. Every quantitative finite estimate (bridge masses, first length moments, arch masses, polygon tails, cylinder spans, boundary flux identities) is taken from the companions at statement level; the only classical inputs used in proofs are the honeycomb connective constant of Duminil-Copin--Smirnov (for $a_n^{1/n}\to\rho^{-1}$ and $c_m\ge1$), the Hammersley--Welsh unfolding and Schilling's lecture notes, cited for the general Poisson construction of pure-jump infinitely divisible laws (in the proof of Proposition 9.4); the renewal normalization $\sum p=1$ is proved in Proposition 2.2, with Kesten cited in the introduction as background. The manuscript flags nothing as numerical, computer-assisted or conditional beyond this dependence on the companions; it does state its own limits: the full-plane endpoint lower law at fixed length is proved only on a density-one set of lengths, the rate statement of Theorem 9.5 sends the distance to infinity at fixed discount before sending the discount to zero and does not show that the two orders of limits agree, and Corollary 9.6 does not identify higher cumulants. The release folder holds no verification/ directory for this manuscript. The manuscript names no Erdős problem.

Bears on

• Problem 529: the honeycomb analogue of the first question. The page asks, for the uniform $n$-step self-avoiding walk on $\mathbb Z^2$, whether the expected endpoint distance $d_2(n)$ satisfies $d_2(n)/n^{1/2}\to\infty$. The manuscript claims, for the honeycomb lattice and critical weight, that the expected endpoint distance of the uniform $n$-step walk is $n^{3/4+o(1)}$ along a set of lengths of natural density one (Theorem 8.2, $p=1$), that the endpoint distance is $n^{3/4+o(1)}$ in probability at every large even length $n$ for strict bridges (Theorem 3.3), and that it is $N^{3/4+o(1)}$, in probability and in mean, at every large discount scale $N$ for the thermal law (Theorem 6.4); Theorem 8.1 states for uniform walks at every large $n$ only that the maximum radius exceeds $n^{3/4+\epsilon}$ with probability smaller than every fixed inverse power of $n$, from which the upper bound $n^{3/4+o(1)}$ on the expected endpoint distance at every large $n$ is a deduction made here, not a statement of the manuscript. It proves nothing on $\mathbb Z^2$ and claims no universality step, so it does not resolve or partially answer the question as posed; it is a comparison and background. The second question, $d_k(n)\ll n^{1/2}$ for $k\ge3$, is untouched. The claims are unverified here, their finite inputs are consumed from companion manuscripts that are themselves unverified apart from the formally verified strip-crossing mass theorem, and the page's status rests on acceptance evidence.