Openai 2026 mass covering exponents fixed length honeycomb walks

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corollary_1_4: The moment form of the manuscript's main claim: the radius of gyration of a uniform n-step honeycomb self-avoiding walk is n^{3/4+o(1)} with the same polynomial failure probability as Theorem 1.1, and for every fixed p > 0 the p-th moments of the diameter and of the radius of gyration are n^{3p/4+o(1)}; the expectation statistic closest to Problem 529's d_2(n), though for the diameter and on the honeycomb lattice.

theorem_1_1: The manuscript's main claim: for every delta, k > 0 and every large n, a uniform n-step honeycomb self-avoiding walk has diameter between n^{3/4-delta} and n^{3/4+delta}, local mass n^{±delta} min(n, s^{4/3}) and covering number n^{±delta}(1 + n s^{-4/3}) at every visited center and every radius 1 ≤ s ≤ n, outside probability C n^{-k}; the honeycomb analogue of the spatial-extent question in Problem 529.

OpenAI, Mass and covering exponents for fixed-length 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/Mass-and-covering-exponents-for-fixed-length-honeycomb-walks-September-26-2026; the held PDF, main.pdf in the release, is retained as openai_2026_mass_covering_exponents_fixed_length_honeycomb_walks.pdf, and the release's TeX bundle sits in the same folder.

@misc{OAI:Mass-and-covering-exponents-for-fixed-length-honeycomb-walks-September-26-2026,
  author = {{OpenAI}},
  title = {{Mass and covering exponents for fixed-length honeycomb walks}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Mass-and-covering-exponents-for-fixed-length-honeycomb-walks-September-26-2026/main.pdf}{OAI:Mass-and-covering-exponents-for-fixed-length-honeycomb-walks-September-26-2026}},
  year = {2026}
}

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

Formalization, as the release lists it. The release's formalization catalogue lean/formalization.yaml at the held revision carries no entry for this manuscript. The release's Lean page for this family names three other manuscripts of the family as its accompanying papers (Polynomial vacuum representations and bridge mass for honeycomb walks, Renewal and changes of law for critical honeycomb walks, Critical strip-crossing mass on the honeycomb lattice) and three comparator statement files, lean/ComparatorChallenges/HoneycombBridgeFiniteness.lean (finiteness of the critical bridge masses and first length moments), lean/ComparatorChallenges/HoneycombFreeEnergy.lean (existence of the free-energy limit) and lean/ComparatorChallenges/CriticalStripMass.lean (strip-crossing mass comparable to $N^{-1/4}$ and displacement moment comparable to $N^{3/4}$). The page's own scope sentences attach to each result separately: it calls the bridge-finiteness result "supporting summability statements"; of the free-energy result it says that it "does not state the paper's small-force exponent, near-critical correlation scale, or the $3/4$ spatial and moment laws"; and the strip result, it says, proves that the first horizontal-displacement moment of return paths is comparable to $N^{3/4}$, which the comparator file states as a $3/4$ moment law in the strip height. None of the three states this manuscript's fixed-length diameter, mass or covering laws. The three comparator results were built at the release's pinned revision with only propext, Classical.choice and Quot.sound, each fingerprint identical to its challenge: finite_bridge_sums certifies only that the bridge and first-length mass sums are finite, with no exponent; logPartition_tendsto only that the forced free energy exists; and critical_strip_mass the strip manuscript's Theorem 1.1, as the critical-strip card records. None of these Lean files is a proof of any Erdős problem.

Companions. The release files this manuscript in a thirteen-member family, "The three-quarter exponent for honeycomb self-avoiding walk", whose description restates this manuscript's abstract. The manuscript itself cites three family members as the source of every analytic input it uses: Critical strip-crossing mass on the honeycomb lattice, whose card is [[discrete_geometry/openai_2026_critical_strip_crossing_mass_honeycomb_lattice/_index|the strip-crossing card]] (it supplies Theorem 2.2 below); Cylinder loop weights and planar nesting (Theorems 2.3 and 2.4 and the cylinder tail bound (4)); and Marked polygon correlations and one-arc bounds (Theorems 2.5 and 2.6). The last two are not held in this library. Renewal and changes of law for critical honeycomb walks, whose card is [[discrete_geometry/openai_2026_renewal_changes_law_critical_honeycomb_walks/_index|the renewal card]], belongs to the same family and is listed on the release's Lean page, but this manuscript does not cite it; its introduction (p. 3) says the renewal process it needs is "developed locally" and that it assumes no change-of-ensemble theorem. The remaining family members (radial transfer estimates and polygon length laws; critical chords with prescribed boundary endpoints; signed cylinder propagation and marked polygons; cylinder amplitudes and logarithmic bridge-length windows; disk-transfer representations and confined bridge mass; polynomial vacuum representations and bridge mass; uniform marked-polygon estimates and sharp finite bridge moments; cap-selected amplitudes and triangle chords) are not cited here and are not held in this library.

Read status: claims checked for Theorem 1.1 and Corollaries 1.2--1.4, read clause by clause in the TeX source (sections/00_introduction.tex lines 17--61) on 2026-10-07, together with the statements of the imported inputs Theorems 2.2--2.6 and the tail bound (4) (sections/01_inputs.tex) and of the intermediate results Proposition 3.1, Proposition 4.10, Theorem 5.2, Theorem 6.2, Theorem 7.4, Theorem 8.1, Propositions 8.6--8.7 and Lemma 9.1; the proofs were read for their structure only and no step was checked; nothing here is independently reviewed.

Contents

The PDF has 66 pages; the TeX source is main.tex with one file per section under sections/ and the bibliography in references.tex. Theorems, lemmas, propositions and corollaries share one counter per section. The manuscript proves its own geometric chain from Section 3 onward; every analytic estimate about critical honeycomb weights is imported from the three companion preprints named above, which are unrefereed manuscripts of the same release.

• Section 1, Introduction (sections/00_introduction.tex, pp. 1--4). Fixes the honeycomb lattice $\mathbb H$ as the planar dual of the unit equilateral triangular tiling, a root vertex $o$, the set $\mathcal W_n$ of $n$-edge self-avoiding paths from $o$ with free endpoint, $c_n$ its size, $\mathbb P_n$ the uniform law, and the statistics $V(\gamma)$ (visited vertices), $D(\gamma)$ (Euclidean diameter), $M_\gamma(z,s)$ (visited vertices in the closed ball of radius $s$ about $z$) and $\mathcal N_\gamma(s)$ (least number of closed radius-$s$ balls covering $V(\gamma)$). States Theorem 1.1: for every $\delta,k>0$ and every integer $n\ge n_0(\delta,k)$, outside $\mathbb P_n$-probability $C_{\delta,k}n^{-k}$, $n^{3/4-\delta}\le D\le n^{3/4+\delta}$, $n^{-\delta}\min\{n,s^{4/3}\}\le M_\gamma(z,s)\le n^{\delta}\min\{n,s^{4/3}\}$ and $n^{-\delta}(1+ns^{-4/3})\le \mathcal N_\gamma(s)\le n^{\delta}(1+ns^{-4/3})$, the last two simultaneously for every $z\in V(\gamma)$ and real $s\in[1,n]$. Corollary 1.2: the projection of $V(\gamma)$ onto each of the three lattice normals has span at least $n^{3/4-\delta}$ with the same probability convention. Corollary 1.3: $\log\mathcal N_\gamma(s)/\log(D/s)\to4/3$ in probability, uniformly over $1\le s\le n^{3/4-\epsilon}$ for fixed $\epsilon\in(0,3/4)$. Corollary 1.4: the radius of gyration is $n^{3/4+o(1)}$ with the same convention, and $\mathbb E_n[D^p]=n^{3p/4+o(1)}$ and $\mathbb E_n[R_g^p]=n^{3p/4+o(1)}$ for every fixed $p>0$. The text (p. 2) states that the results "do not include convergence to a continuum curve, a continuum Hausdorff dimension, a universality theorem on changing lattice, or a fixed-length lower bound on $|\gamma_n-o|$", and that "A lower bound for the diameter does not imply that the two endpoints are far apart." Subsection 1.2 introduces the critical activity $\rho=(2+\sqrt2)^{-1/2}$, port paths, bridges and irreducible bridges, and outlines the three obstacles (confining a modified bridge to a corridor, controlling all subpaths of one walk, controlling repeated visits to one ball) and the exact-length transfer through $c_n\rho^n\ge1$. Subsection 1.3 places the exponent in Nienhuis's Coulomb-gas prediction, the Duminil-Copin--Smirnov connective constant $\sqrt{2+\sqrt2}$ and its bound $c/T\le B_T\le1$ on the strip-crossing mass, the later results of Beaton, Bousquet-Mélou, de Gier, Duminil-Copin and Guttmann ($B_T\to0$), Glazman and Manolescu, and Krachun and Panagiotis (a polynomial bound on $B_T$ and quantitative sub-ballisticity), Kesten's renewal method, the strip-conditioning identity of Dyhr, Gilbert, Kennedy, Lawler and Passon, and the $\mathrm{SLE}_{8/3}$ dimension $4/3$ of Lawler, Schramm and Werner and Beffara; it says a lattice scaling limit is neither input nor output.
• Section 2, Critical masses and the analytic inputs (sections/01_inputs.tex, pp. 4--7). Fixes ports (edge midpoints of the triangular tiling), port paths weighted $\rho^{|\gamma|}$ with $|\gamma|$ the visited-vertex count, horizontal bands of height $d_0=\sqrt3/2$, strict bridges, arches, the bridge kernel $b_h(x)$, $B_h=\sum_xb_h(x)$ with $B_0=1$, the arch masses $K_h(j)$, $A_h$ and $m_h=\sum_{j>0}jK_h(j)$, and the normalized bridge law $\mathbb P^{\mathrm{br}}_h$. Lemma 2.1 (the one input the manuscript proves itself): a weak bridge extends to a strict port bridge at bounded cost in height, length, weight and inverse multiplicity. The imported inputs, stated as theorems with their sources: Theorem 2.2 (strip companion, Theorem 1.1 and Lemma 2.1): the boundary winding identity $\sum_{b\in\partial D}\sum_{\gamma:a\to b}\rho^{|\gamma|}e^{3iW(\gamma)/8}=1$ for every finite simply connected region $D$ made of triangular tiles and bounded by a simple polygon and each boundary source port $a$, convex exit mass at most $1/c$ with $c=\cos(3\pi/8)$, finiteness of all strip sums, $cA_h+B_h=1$, $B_{h+1}\le B_h$, $B_h\asymp(1+h)^{-1/4}$, $m_h\asymp h^{3/4}$ and $m_{h+1}-m_h\asymp B_h$ with constants independent of $h$; Theorem 2.3 (cylinder companion, Lemma 11.4, Theorem 11.2, Corollary 11.3): the chord mass through an internal mid-edge of a convex domain, or of a strip of fixed positive height, is comparable to the sum of the two adjacent face-nesting masses, and the diameter-truncated planar nesting mass is $P_2(r)=r^{1/12+o(1)}$, the middle-face strip nesting mass $S_k=k^{1/12+o(1)}$; Theorem 2.4 (cylinder companion, Proposition 11.5): the unnormalized first-length mass of arches and bridges from one bottom port of the $h$-band strip is at most $h^{13/12+o(1)}$; Theorem 2.5 (marked companion, Theorem 1.1): the length-square mass of unrooted polygons of diameter at most $H$, modulo translation, is at most $H^{2/3+o(1)}$; Theorem 2.6 (marked companion, Theorem 8.3): a uniform bound $A(p,p')\le b^C$ on the unnormalized critical mass of single self-avoiding arcs from one marked port to the other on a physical two-marker cylinder, with the logarithmic parameters $L_X=\tfrac34\log s\ge L_*$, $L_Y=\tfrac34\log b$, $s\le b$ and at least a $1/2-\delta_1$ fraction of upper $X$-sites, with no positive lower bound on $L_X/L_Y$; and display (4) (cylinder companion, Corollary 8.2, from its Theorem 8.1 fugacity bound $Z_N(2\cosh v)\le\exp(C\log N(1+v^2))$ for real $v\ge0$): for every sufficiently large even $N=2m$ and uniformly in integers $a\ge0$, the critical mass $w_N(a)$ of families of $a$ pairwise disjoint polygons separating the two markers of the balanced physical two-marker cylinder of $N$ bands is at most $\exp(C\log N-ca^2/\log N)$.
• Section 3, The critical mass of paths in a box (sections/02_full_box.tex, pp. 7--10). Proposition 3.1 (full-box susceptibility): the critical mass of walks confined to a box of side $H$, over all lengths and endpoints from the worst start, is at most $C_0(1+H)^{C_1}$. The proof imposes a westward-step suffix test whose failures are counted by a Fibonacci bound ($\rho\varphi<1$ with $\varphi$ the golden ratio, using that a honeycomb vertex has one edge in each direction), extracts a cylinder arc eligible for Theorem 2.6 with $b\asymp H$ and bounded or growing $s$, and bounds the unsuccessful prefixes by a strip estimate $U(W)=C\exp(CW^{3/4})$ from Lemma 2.1 and Theorem 2.2.
• Section 4, Localized bridges, recoverable cuts, and length moments (sections/03_sewing.tex, pp. 10--25). Lemma 4.1: convex cuts compare lost and new exit masses (from the real part of the winding identity), and the lateral loss of a height-$h$ strip sum beyond lateral extent $M$ is at most $ChM^{-5/4}$ for $M\ge C_1h$. Lemma 4.2: $\sum_x|b_h(x+1)-b_h(x)|\le C(B_h-B_{h+1})$, by adding two rhombi above adjacent top ports and taking the imaginary part of the winding identity at $\lambda=\pi/8$. Lemma 4.3: seam moment bounds for once-crossed and twice-crossed recovery lines. Lemma 4.4: a prescribed endpoint at slope at most $v$ in a tube of radius $\delta h$ has bridge mass at least $h^{-5/4-o(1)}$. Lemma 4.5: free-endpoint bridges in a widening tube have mass at least $ch^{-1/4}(s/h)^P$. Lemma 4.6: two ports on a line at distance $r$ are joined by arches of diameter $O(r)$ and mass at least $r^{-5/4-o(1)}$. Proposition 4.7: the localized strip second length moment is at most $h^{29/12+o(1)}$, by closing paths into polygons and applying Theorem 2.5. Lemmas 4.8 and 4.9 and Proposition 4.10: for a sufficiently large fixed $C$, the first-length mass of height-$M$ bridges from a fixed source of diameter at most $CM$ is at least $M^{13/12-o(1)}$, by transferring the bulk length of Theorem 2.3 through exterior connectors, whence $\mathbb P^{\mathrm{br}}_M\{|\gamma|\ge M^{4/3-\xi}\}\ge M^{-o(1)}$ for every fixed $\xi>0$ by weighted Cauchy--Schwarz. Lemma 4.11 and Corollary 4.12: a bridge or arch can turn around an obstacle in a corridor with free terminal port at mass at least $cH^{-1/4}$ with no exponent slack, by a second-moment count of its representations.
• Section 5, From localized moments to a typical bridge (sections/04_renewal.tex, pp. 25--27). Derives Kesten's renewal structure in the port convention: $B(z)=(1-I(z))^{-1}$, $\sum_hI_h=1$, $\sum_{h\ge l}I_h\le Cl^{-3/4}$, the half-plane walk as a concatenation of independent irreducible bridges, and the exact conditioning identity that a renewal at height $h$ has probability $B_h$ and the conditioned prefix has the critical height-$h$ bridge law. Lemma 5.1 (overshoots; a trial has probability at least $h^{-o(1)}$ of containing $h^{4/3-\xi}$ vertices). Theorem 5.2 (typical finite-bridge length): for every $\varepsilon>0$, the normalized height-$H$ bridge law gives $|\gamma|$ between $H^{4/3-\varepsilon}$ and $H^{4/3+\varepsilon}$ outside probability $O_\varepsilon(H^{-c_\varepsilon})$, with the diameter between $d_0H$ and $CH^{1+\varepsilon}$; the lower bound by amplifying independent trials and paying for the conditioning, the upper bound by Theorem 2.4 and Markov.
• Section 6, Ordered renewal probes (sections/05_probes.tex, pp. 27--40). Defines the stopped law $\mathcal Q_t$ and the renewal-occurrence measure $\mathcal R$, the ordered test $E(h)$ for two renewal strings from neighboring ports, and the masses $F(h,r,d)$ and $F_{12}(h_1,h_2,r)$ with one or two inspections of a single long irreducible. Lemma 6.1 (jump charges). Theorem 6.2 (ordered probes): for integers $1\le h,h_1,h_2\le r$ with $r$ dyadic and $d\ge0$, $\mathbf P(E(h))\le K_1h^{-3/4}$, $F(h,r,d)\le K_2r^{-3/4}(1+d)^{-5/4}$ and $F_{12}(h_1,h_2,r)\le K_3r^{-3/4}$, proved by strong induction through Lemma 6.3 (hairpin mass bounds, built from Corollary 4.12) and Lemma 6.4 (usable renewals after a regular history). Corollary 6.5: $S_2(h)\le Ch^{-3/4}(1+\log h)^2$ for disjoint first-hit paths from neighboring ports; Corollary 6.6: $S_2(H)\le C_\epsilon H^{-3/4+\epsilon}$.
• Section 7, Amplification against fast travel (sections/06_fast.tex, pp. 40--44). Lemma 7.1 (an integrated short-bridge estimate from Theorem 5.2), Lemma 7.2 (turning-extremum decomposition of a weak bridge into $k=2^d$ up pieces and $k-1$ retreats with recoverable cuts), Lemma 7.3 (avoidance with marked terminal renewals). Theorem 7.4 (fast bridges): for every $\varepsilon>0$ and $A<\infty$, the total mass $\sum_{H\le h\le2H}B_h[\,\cdot\,]$ of bridges of height in $[H,2H]$ and length at most $H^{4/3-\varepsilon}$ is at most $C_{\varepsilon,A}H^{-A}$, by charging each turn an adjacent-arm avoidance factor from Corollary 6.6 and taking the fixed number of turns large.
• Section 8, Length in small regions and repeated crossings (sections/07_slow.tex, pp. 44--58). Works with the unnormalized measure $\mu_H$ of rooted walks of diameter at most $C_0H$. Theorem 8.1 (uniform local length): outside $\mu_H$-mass $C_{\tau,A}H^{-A}$, every vertex subwalk $\sigma$ has $|\sigma|\le H^\tau(1+s(\sigma))^{4/3}$ with $s(\sigma)$ its span in band units, simultaneously in every lattice normal direction. Lemma 8.2 (small returns inside one irreducible, with arbitrary logarithmic gain), Lemma 8.3 (thin excursions), Lemma 8.4 (turning-chain witness bound over random trial trees of depth $\lceil\log H\rceil$, using Theorem 6.2 and Corollary 6.5 at the turns). Lemma 8.5 (many bridges in one slab): for fixed $D$, $a$ strict bridges that cross the same $j\ge1$ bands inside one slab of width $Dj$ and avoid one another, with their starting ports, ending ports and matching prescribed, have critical mass at most $\exp(C_D\log(2+j)-c_Da^2/\log(2+j))$, by reflecting and concatenating them into separating polygons on a cylinder and applying (37), the Section 8 restatement of (4). Proposition 8.6 (uniform local mass): $\#(V(\gamma)\cap\overline B(v,s))\le H^\tau s^{4/3}$ for every visited $v$ and $1\le s\le H$ outside mass $C_{\tau,A}H^{-A}$. Proposition 8.7 (uniform temporal modulus): every $l$-step subwalk has diameter at most $H^\tau(1+l)^{3/4}$ outside mass $C_{\tau,A}H^{-A}$, from Theorem 7.4.
• Section 9, Completion of the fixed-length theorem (sections/09_completion.tex, pp. 58--61). Lemma 9.1: $Z_n=c_n\rho^n\ge1$ for every $n\ge0$, by submultiplicativity against $B_h\asymp h^{-1/4}$. The transfer: with $H=Kn$, a bad set of unnormalized mass $C_AH^{-A}$ has $\mathbb P_n$-probability at most $C_AH^{-A}/Z_n\le C_AH^{-A}$ at every length, with no averaging over lengths; Lemma 9.1's display is (40), and displays (41)--(43) restate Theorem 8.1, Proposition 8.7 and Proposition 8.6 under $\mathbb P_n$. Proofs of Theorem 1.1 (pp. 59--60) and Corollaries 1.2--1.4 (pp. 60--61).
• Appendix A, An alternative avoidance proof using asynchronous renewals (sections/A_asynchronous.tex, pp. 61--65). Lemma A.1 (sewing past an obstacle at mass $H^{-1/4-o(1)}$), Lemma A.2 (irreducible displacement tail $\mathbb P(W>x)\le Cx^{-3/4}$), Theorem A.3: $S_2(H)\le H^{-3/4+o(1)}$, stated as independent of Sections 6 and 7 and weaker than Corollary 6.5.
• References (references.tex, pp. 65--66): ten external entries (Beaton et al. 2014; Beffara 2008; Duminil-Copin and Smirnov 2012; Dyhr, Gilbert, Kennedy, Lawler and Passon 2011; Glazman and Manolescu 2020; Kesten 1963; Krachun and Panagiotis 2026; Lawler, Schramm and Werner 2004; Madras and Slade 1993; Nienhuis 1982) and the three companion preprints.

The manuscript flags nothing as numerical or computer-assisted. Its results are conditional in one sense it states itself: every estimate about critical honeycomb weights (Theorems 2.2--2.6 and display (4)) is imported from the three companion preprints, which are unrefereed manuscripts of the same release, and, in its words (Section 1.2, p. 2), "The analytic proofs belong to the complete companion articles identified there." The release provides no verification/ folder for this manuscript.

Bears on

• Problem 529: comparison and background, not a claimed answer. The page asks about $d_k(n)$, the expected endpoint distance of an $n$-step self-avoiding walk in $\mathbb Z^k$, in particular whether $d_2(n)/n^{1/2}\to\infty$. Theorem 1.1 and Corollary 1.4 claim, on the honeycomb lattice only, that the diameter is $n^{3/4+o(1)}$ with high probability and that $\mathbb E_n[D^p]=n^{3p/4+o(1)}$; the manuscript states that it proves no universality statement across lattices and no fixed-length lower bound on $|\gamma_n-o|$, so it supplies neither the $\mathbb Z^2$ question nor the endpoint statistic the page uses. The claim is unverified here, and the page's status rests on acceptance evidence, not on this card.