State of Proof — Paper Watch

The linear instability of Kasner spacetimes

2026-09-20T13:03:54Z

Claim: The abstract says the paper proves linear stability toward the Big Bang modulo an explicit finite-dimensional non-decaying self-similar space, then gives a complete linear description of the expected instability without symmetry assumptions. Because the title foregrounds instability while the first abstract sentence foregrounds stability modulo modes, the exact theorem statement and conventions are a required first check. Why it could matter: Which ripples survive near a spacetime singularity? Kasner spacetimes are exact solutions used to study an extreme mathematical limit of gravity. This paper claims to sort their linear disturbances into decaying behavior and a finite set of persistent self-similar modes. If it holds: Foundational: it would give a sharper linear map for a difficult PDE-and-geometry regime, including the stated Taub-transition mode. If it does not: The exact exceptional modes, stability norm, or linearization may need revision, clarifying where the proposed early-time picture stops applying. Boundary: intake record only; no examination result recorded.

A note on generating polyhedra and quadrangulations

2026-09-19T13:04:00Z

Claim: The abstract says two graph transformations generate every polyhedron other than pyramids from the square pyramid; it separately gives a unique-transformation construction for the specified facial-four-cycle sphere quadrangulations. It is a current constructive classification claim with a bounded combinatorial check surface. Why it could matter: Two moves build many polyhedral graphs The work asks whether complex graph families can grow from one seed through a few reliable moves. Such recipes can make a large mathematical family easier to organize and explore. If it holds: The construction would provide a concise route to generate the stated non-exceptional polyhedra and related quadrangulations. If it does not: An omitted graph family or failed inverse step would locate the limit of the proposed construction. Boundary: intake record only; no examination result recorded.

Boolean Small-Ball Inequalities for Discrepancy Theory

2026-09-19T13:04:00Z

Claim: The abstract states a small-ball inequality for Boolean matrix series under bounded trace and variance conditions, then claims an interlacing-free proof of Kadison–Singer and further discrepancy corollaries. The explicit method and dependencies make the claimed advance an appropriate unexamined proof-method intake. Why it could matter: A new route through matrix discrepancy The paper studies how random plus-or-minus choices can keep a matrix sum controlled. It offers a different proof mechanism for results about balancing many competing effects. If it holds: The method could give mathematicians another reusable way to derive matrix-balancing results and inspect which assumptions carry the work. If it does not: Pinpointing a failed inequality or dependency would clarify which part of the proposed proof route cannot support its advertised consequences. Boundary: intake record only; no examination result recorded.

Nonexistence of a Leech Tree of Order 18: A Computer-Assisted Proof

2026-09-18T14:36:13Z

Claim: The abstract claims that no Leech tree of order 18 exists. It says Lean 4 verifies structural facts reducing a putative example to eight local configurations; conventional mathematics establishes an exact-cover condition and search completeness; exhaustive computation closes the eight cases. The source explicitly calls this a computer-assisted proof rather than an end-to-end Lean formalization. Why it could matter: When a proof has both a kernel and a search Can one weighted tree realize every whole-number distance from 1 through 153 exactly once? This paper says no—and carefully divides its evidence between Lean-checked reductions and an exhaustive search. If it holds: Methods: it would resolve this finite graph puzzle while offering a candid map of what a proof kernel certifies and what remains in the computation-and-checker trust boundary. If it does not: The split record helps locate the problem: a formal reduction, a conventional argument, the search program, its run, or the checker may need correction. Boundary: intake record only; no examination result recorded.

The smallest square tileable by pairwise incomparable integer rectangles

2026-09-18T14:36:13Z

Claim: The abstract claims that for every integer (nleq26) and every number of tiles (kgeq2), the (n× n) square has no tiling by pairwise incomparable integer rectangles, establishing that the known 27-by-27 example is smallest. Its finite stage is reported to search 167,538 candidates using two independently written programs. Why it could matter: Why 26-by-26 may be forever too small Cut a square into integer-sided rectangles, but forbid any rectangle from being at least as wide and tall as another. A 27-by-27 example is known; this paper claims every smaller square is impossible. If it holds: Foundational: it would close a clean finite geometry puzzle and show how structural reasoning can shrink an enormous search to a checkable set of cases. If it does not: A missing tile family, an incomplete reduction, or a mismatched search record would show exactly where the claimed impossibility needs repair. Boundary: intake record only; no examination result recorded.

On the Strong Matroid Secretary Conjecture and Beyond

2026-09-18T14:36:13Z

Claim: The abstract claims a 1/e-competitive ordinal secretary algorithm for every linear matroid, maintaining expected-intersection-dimension bounds while selecting online. It separately claims a 1/2 single-sample prophet algorithm for arbitrary matroids and a black-box 1/64 secretary reduction. This is a newly posted, concrete advance on structured online selection, not a blanket result for all allocation problems. Why it could matter: How do you choose before every offer arrives? Hiring, booking, and bidding all share a cruel timing problem: accept too early and a better option may appear; wait too long and nothing remains. This paper claims a sharp online-selection rule for a structured family of feasible choices. If it holds: Enabling: it would settle the strong secretary guarantee for linear matroids, giving algorithm designers a precise benchmark for online selection under that structure. If it does not: The sharp guarantee or its scope would need revision, revealing which arrival, independence, or representation assumption carries more weight than claimed. Boundary: intake record only; no examination result recorded.

Regularity of asymptotically axisymmetric solutions to the 3D Navier–Stokes equations with analytic forcing

2026-09-18T13:41:39Z

Claim: The abstract and Theorem 1.3 claim regularity for a specified class of forced 3D Navier–Stokes solutions. As a consequence, a singular construction with the named core and anisotropic properties cannot have force that both remains (C²)-bounded to singular time and is locally uniformly real analytic in space. It is a timely mathematical boundary on the September 8 forced Navier–Stokes intake, not a refutation of that claim. Why it could matter: A sharp new boundary around a fluid singularity claim Navier–Stokes singularities are notoriously hard to rule in or out. This paper says a particular proposed route cannot work with a force that remains real analytic in space, narrowing the terrain without resolving every case. If it holds: Foundational: it would impose a concrete regularity constraint on this class of forced singularity constructions and focus scrutiny on the exact behavior of their forcing. If it does not: The proposed restriction would weaken, revealing which symmetry, scale, or analyticity step needs a more careful argument. Boundary: intake record only; no examination result recorded.

Universal completeness of exponentials

2026-09-18T13:41:37Z

Claim: The abstract claims uniformly discrete density-one frequency sets complete in every (L^p(S)) for (|S|<1), density-(v) integer-frequency counterparts for (|S|

A Proof of the Strong Papadimitriou–Ratajczak Conjecture

2026-09-14T13:03:39Z

Claim: The exact declaration constructs a straight-line embedding with strict convex face cycles and a neighboring vertex closer to every distinct destination, under an original-drawing and deletion-connectivity formulation of finite simple 3-connected plane graphs. Why it could matter: A route through every planar network may become greedy In a greedy drawing, each hop toward a destination gets strictly closer. This release claims every sufficiently well-connected planar network can be drawn that way, while keeping the proof’s exact scope and independent review visible. If it holds: It would settle the stated strong graph-drawing conjecture and provide a formal endpoint for studying convex greedy-routing constructions. If it does not: A mismatch between the Lean statement, its assumptions, or the paper’s claimed theorem would identify the boundary needing repair; the conjecture would remain open. Boundary: intake record only; no examination result recorded.

EOC Lean Verification: harmonic discrepancy and cylinder arithmetic

2026-09-13T13:12:00Z

Claim: The current source declares Lean formalizations of finite residue-count and harmonic arithmetic-progression discrepancy bounds, plus cylinder arithmetic. Its documentation explicitly limits these to finite or conditional infrastructure and states that the Collatz conjecture and the project’s Global Occupation Conjecture remain open. Why it could matter: Small Collatz lemmas become machine-checkable Collatz research involves patterns in repeated odd-number transformations. This revision formalizes finite rules for how evenly certain residue classes appear, including a harmonic-weighted version, while explicitly not claiming to solve the famous conjecture. If it holds: It would add reusable machine-checkable building blocks and clearer boundaries between finite arithmetic facts, conditional arguments, and open questions. If it does not: A declaration, dependency, or claimed scope boundary would need repair; the open Collatz problem remains open. Boundary: intake record only; no examination result recorded.

Unfriendly partitions of locally finite Borel graphs

2026-09-11T13:03:55Z

Claim: The abstract says the author constructs a closed, unbounded-degree locally finite Borel graph with no Borel unfriendly partition, answering Thomas's question negatively. It also states positive bounded-degree cases. A named-question counterexample released within the current intake window merits a clearly bounded public examination route. Why it could matter: An infinite network resists a fair-looking split An unfriendly partition puts each vertex with at least as many opposite-side neighbors as same-side neighbors. This paper claims a carefully structured infinite graph where no Borel, or systematically describable, partition can do that. If it holds: It would settle Thomas's Borel-graph question negatively and sharpen the boundary between finite-style graph intuition and measurable infinite structures. If it does not: The proposed graph, its local finiteness, or the measurability obstruction would need repair; the question would remain open. Boundary: intake record only; no examination result recorded.

The generalised semi-Clifford conjecture is false

2026-09-11T13:03:55Z

Claim: The abstract claims a five-qubit gate in the fifth Clifford-hierarchy level that is not generalised semi-Clifford, refuting the Zeng-Chen-Chuang conjecture and showing the hierarchy is not closed under inverses. It is a newly posted concrete counterexample to a named structural conjecture. Why it could matter: A five-qubit gate breaks a tidy hierarchy rule Quantum gates can be sorted into layers of increasing complexity. This paper claims one five-qubit gate in the fifth layer cannot be reshaped into the simple form a long-standing conjecture predicted. If it holds: It would refute the generalized semi-Clifford conjecture and show the Clifford hierarchy is not closed under taking inverses. If it does not: The gate's layer membership or the claimed normal-form obstruction would need correction; the conjecture would remain unsettled. Boundary: intake record only; no examination result recorded.

Nonexistence of a Strongly Regular Graph with Parameters (266,45,0,9): A Certificate-Free Lean Proof

2026-09-09T13:03:41Z

Claim: The paper claims there is no strongly regular graph with parameters ((266,45,0,9)), via a classification-free Lean proof that reduces the remaining case to an impossible projection identity. It is a current example of formal proof construction without external infeasibility certificates. Why it could matter: A stubborn graph pattern may be impossible Strongly regular graphs are highly symmetric networks with exact local rules. This paper claims one long-sought parameter set cannot exist, using a Lean formalization that follows the contradiction through lattice and design arguments rather than an external infeasibility certificate. If it holds: It would close this specific existence question and supply a formally checkable example of a classification-free nonexistence proof. If it does not: The parameter translation, lattice argument, or formal statement may need repair; the graph’s existence question would remain open. Boundary: intake record only; no examination result recorded.

Generalized DBLog: A Verified Contract for Interleaving Database Rows with a Change Log

2026-09-09T13:03:41Z

Claim: The paper states conditions under which a chunked database copy can be interleaved with an active change log without gaps, stale copied state overwriting newer logged updates, or deletion resurrection; it covers multiple DBLog/Debezium/Flink/back-up variants. This is a directly inspectable new proof-method result for committed-state handoff and reconciliation. Why it could matter: Copying live data without losing the plot A database copy can collide with updates still arriving from the live system. This paper claims conditions that prevent missed changes, stale overwrites, and deleted rows returning during that handoff across several change-data-capture designs. If it holds: It would provide a verified foundation for reasoning about copy-to-log handoffs, including watermarking and related capture designs. If it does not: One or more stated conditions or protocol variants may be incomplete, narrowing where the claimed reconstruction guarantee applies. Boundary: intake record only; no examination result recorded.

Explicit Positive-Density Collatz Convergence in Logarithmic Time

2026-09-09T13:03:12Z

Claim: The release claims fixed explicit constants (c>0) and (X₀) such that every (X≥ X₀) has at least (cX) positive starts (n

Finite Time Blowup for Navier–Stokes

2026-09-09T00:00:00Z

Claim: OpenAI claims a finite-time breakdown of smooth, forced, three-dimensional incompressible Navier–Stokes flow from rest with bounded energy, asserting alternatives C and D of the Clay problem. This is the publisher's claim, not a State of Proof validation. Why it could matter: Navier–Stokes: can fluid equations hit a breaking point? Wings, weather and blood flow all involve fluid equations. OpenAI claims that even smooth inputs can drive one idealized model beyond smooth behavior. This concerns the model's limits—not instantly better aircraft or forecasts. If it holds: It would resolve the forced breakdown alternatives of the Clay problem and sharpen research into when smooth fluid models stop applying. If it does not: A gap would identify which assumption or proof step needs repair; ordinary engineering uses would not automatically become invalid. Boundary: intake record only; no examination result recorded.

Stable Singularity of the Euler Equations on R³

2026-09-09T00:00:00Z

Claim: The manuscript presents evidence of a stable finite-time singularity, an approximate profile discovered with a physics-informed neural network, and a framework reducing nonlinear stability to finite quantitative estimates. Why it could matter: AI finds a candidate; completing the proof comes next A neural network finds an approximate pattern that could become a singularity in ideal fluid flow. The manuscript offers evidence and a stability framework, but explicitly lists unfinished proof work. Finding a promising pattern is not yet proving it exists. If it holds: Completing the quantitative certification could turn AI-guided discovery into a rigorous singularity result under the exact stated assumptions. If it does not: An unsuccessful certification would reveal where the approximate pattern or stability estimates need to change. Boundary: intake record only; no examination result recorded.

Extending the Córdoba-Martínez-Zoroa IPM Blow-up to Uniformly Space-Time Smooth Forcing

2026-09-09T00:00:00Z

Claim: The paper claims finite-time density- and velocity-gradient blowup for a periodic incompressible porous-media model with smooth initial density and a force smooth in space and time. Why it could matter: Fluid through a sponge hides a difficult mathematical limit Think of water seeping through a sponge. This idealized porous-flow model asks how sharply density and velocity can vary under smooth inputs. Its value is understanding mathematical limits, not a demonstrated improvement to groundwater prediction. If it holds: It would extend a known singularity construction to forcing smooth in both space and time within the stated periodic model. If it does not: It would expose the step that fails to preserve time smoothness or the claimed gradient growth. Boundary: intake record only; no examination result recorded.

Finite Time Blowup for the Euler Equation

2026-09-09T00:00:00Z

Claim: The paper claims finite-time breakdown for smooth, compactly supported initial flow in the three-dimensional unforced incompressible Euler equations. Why it could matter: Can an ideal fluid break down without an outside push? Take friction and external pushing out of the picture. Can a smooth ideal flow still develop unbounded gradients? This paper claims it can, probing a fundamental limit of the equations used to understand fluid motion. If it holds: It would establish a smooth-data, whole-space breakdown example for unforced Euler, changing the mathematical picture of ideal-fluid regularity. If it does not: The proposed construction would need repair; the general smooth-data question would not be settled by this argument. Boundary: intake record only; no examination result recorded.

Blowup for the Euler Equations with Smooth Forcing

2026-09-09T00:00:00Z

Claim: The paper claims finite-time blowup of vorticity and circulation gradients in three-dimensional incompressible Euler flow with a force smooth in space and time, including at the terminal time. Why it could matter: A smooth push can still produce extreme fluid structure A smoothly applied force need not keep an ideal fluid smooth forever. The authors claim a flow whose twisting and gradients become unbounded. This is about a mathematical limit, not proof that real water reaches infinite speed. If it holds: It would strengthen our understanding of singularity formation under smooth forcing and provide a construction to study related equations. If it does not: A failed estimate or translation would locate what must be repaired before relying on the claimed smooth-forcing result. Boundary: intake record only; no examination result recorded.

Blowup for the Boussinesq Equations with Smooth Forcing

2026-09-09T00:00:00Z

Claim: The paper claims finite-time singularity formation in the two-dimensional inviscid Boussinesq system with smooth forcing, bounded temperature, and unbounded temperature gradient and vorticity. Why it could matter: Warm rises, cold sinks—and the mathematics gets sharper Buoyancy helps warm fluid rise through cooler fluid. This simplified model asks whether smooth inputs can create arbitrarily fine structure while temperature remains bounded. The result could clarify the model's limits, not directly improve tomorrow's forecast. If it holds: It would establish a precise breakdown mechanism for a forced, two-dimensional buoyancy model and support further mathematical study. If it does not: The claimed forcing or stability argument would need repair; it would not invalidate every buoyancy model. Boundary: intake record only; no examination result recorded.

The Erdős-Sós conjecture in dense graphs

2026-09-09T00:00:00Z

Claim: The manuscript claims that for every (gamma), all sufficiently large (n)-vertex graphs with more than ((k-2)n/2) edges contain every (k)-vertex tree whenever (k≥gamma n). It also claims to solve a 51-year-old Erdős-Graham problem on multicolor Ramsey numbers of trees. This is a current, high-consequence partial-regime resolution of a landmark extremal-graph question. Why it could matter: Dense networks must contain every tree shape A dense network cannot avoid a chosen branching pattern forever. This paper claims the exact edge threshold forces every large tree shape to appear, settling the dense regime of a major extremal-graph question. If it holds: It would give combinatorics a sharp dense-network embedding rule and resolve a long-standing multicolor Ramsey question about trees. If it does not: The exact threshold or asymptotic range needs repair, preventing premature use as a universal dense-graph guarantee. Boundary: intake record only; no examination result recorded.

Arithmetic Polyhedra

2026-09-09T00:00:00Z

Claim: The paper claims that every arithmetic reflection group arising from a combinatorial polyhedron is commensurable to one arising from a tetrahedron, square pyramid, or cuboctahedron, proving the Kontorovich-Nakamura conjecture. Its stated intermediate theorem classifies arithmetic ideal right-angled hyperbolic polyhedra as gluings of three seed polyhedra. Why it could matter: Infinite hyperbolic symmetries reduce to three seeds Hyperbolic polyhedra can encode vast families of geometric symmetries. This paper claims every arithmetic case in one major construction descends from just three building blocks, turning a classification puzzle into a finite map. If it holds: It would organize a broad class of arithmetic reflection groups around three seed geometries, enabling sharper classification work. If it does not: The proposed seeds may not cover every case, exposing where arithmeticity or gluing arguments need stronger conditions. Boundary: intake record only; no examination result recorded.

Parking functions, Smirnov words, and noncrossing Chow polynomials

2026-09-09T00:00:00Z

Claim: The paper claims real-rootedness for Chow polynomials of noncrossing partition lattices and proves Conjecture 4.2 of Xiao and Conjecture 11.2 of Ehrenborg-Hetyei-Readdy. It presents interlacing, differential-recurrence, and finite Schur-Szegő-convolution routes, making a current assertion suitable for targeted review. Why it could matter: Hidden polynomial patterns keep their roots orderly Many counting problems produce polynomials whose roots reveal deep regularity. This paper claims two conjectured families have only real roots, using new translations between parking functions, words, and noncrossing structures. If it holds: It would settle two combinatorial real-rootedness conjectures and add reusable interlacing tools for structured counting polynomials. If it does not: The claimed translation or recurrence would need narrowing, preserving caution around predicted root behavior in these families. Boundary: intake record only; no examination result recorded.

The finiteness conjecture for equilibria of electric fields generated by point charges of one sign

2026-09-09T00:00:00Z

Claim: The manuscript claims that finitely many same-sign point charges in three-dimensional space generate an electric field with only finitely many equilibria, answering a 1969 question of Morse and Cairns. It also states an explicit bound for mixed-sign charges away from the zero set of an auxiliary function. A current resolution claim with a bounded algebraic-geometric core warrants prompt, non-validating intake. Why it could matter: A 1969 question about electric balance points may close Point charges create invisible push-and-pull fields. This paper claims that any finite collection with one charge sign has only finitely many balance points, ending a decades-old question and making their global geometry less mysterious. If it holds: Foundational: mathematicians gain a firm finiteness rule for same-sign Coulomb fields and an explicit counting framework for more complicated mixed-sign arrangements. If it does not: The old question remains open, and the failure would reveal where the complex-curve or algebraic counting argument overreaches. Boundary: intake record only; no examination result recorded.

Extending concurrent separation logic to the hardware level to verify the xv6 OS kernel on RISC-V with AI agents

2026-09-09T00:00:00Z

Claim: The authors introduce MachCSL, adapting Iris-style concurrent separation logic to Sail RISC-V sub-instruction semantics, and report an AI-agent-assisted verification of a 6,593-line xv6 kernel implementation that found nine xv6 bugs and one Sail-semantics bug. It squarely tests whether LLM agents can contribute to reviewable low-level formal verification while the source is current. Why it could matter: AI-assisted proofs reach down to computer hardware Operating systems make devices usable, but their deepest rules must survive memory, interrupts, and hardware translation. This paper claims AI agents helped verify a real teaching kernel at that level, while uncovering implementation and specification bugs. If it holds: Methods: it would offer a concrete model for combining formal hardware semantics, human-designed invariants, and machine assistance in reviewable systems verification; reuse elsewhere still requires released artifacts and independent replay. If it does not: The claimed verification scope or bug findings would narrow, showing which model, proof boundary, or AI-produced step needs stronger evidence before such workflows are trusted. Boundary: intake record only; no examination result recorded.

Formalizing Fermat's Last Theorem

2026-09-09T00:00:00Z

Claim: Anthropic reports that Claude agents produced in eleven days the first complete end-to-end, computer-checked Lean proof of Fermat's Last Theorem. This is not a new solution to an open problem—Wiles and Taylor-Wiles proved the theorem in 1995—but a claim that their known mathematical route has been rebuilt as a fully machine-checkable formal artifact at unprecedented scale and speed. Why it could matter: A landmark proof becomes software machines can check Fermat's Last Theorem was solved in 1995. The new achievement is different: turning that enormous human proof into code independent computers can replay. If this scales, AI-generated mathematics can arrive with a checkable receipt—helping researchers find errors and spend time understanding the truth. If it holds: A reproducible end-to-end Lean proof would show that AI can help turn landmark mathematics into independently checkable software, making formal verification a practical companion to human peer review as mathematical output accelerates. If it does not: Fermat's theorem remains proved, but this artifact would not yet demonstrate a reliable new route for machine-checking large AI-assisted proofs; the verification workflow would need repair. Boundary: intake record only; no examination result recorded.

Catalan's constant is irrational

2026-09-09T00:00:00Z

Claim: The manuscript claims that Catalan's constant (G=sumkgeq0(-1)^k/(2k+1)²) is irrational, presenting this as a proof via suitable weights. Irrationality of (G) is a long-standing open problem, so the first primary v1 merits prompt, explicitly non-validating intake. Why it could matter: A famous constant may finally leave mathematical limbo Mathematicians can calculate Catalan's constant to enormous precision but still do not know whether it is a ratio of whole numbers. Settling that basic identity question could reveal new ways to prove that familiar constants are fundamentally non-fractional. If it holds: Foundational: it closes a famous open problem and may supply reusable techniques for proving other constants irrational; it does not imply an immediate new device or speedup. If it does not: The failure identifies where a promising weighted-series argument loses the arithmetic or asymptotic control needed to prove irrationality. Boundary: intake record only; no examination result recorded.

The Truncated Octahedral Graph Has Bondage Number Five

2026-09-09T00:00:00Z

Claim: The paper claims that the planar cubic truncated-octahedral graph has domination number 8 and bondage number 5, giving (5>Delta(T)+1=4) and therefore a counterexample to the stated 1998 planar-graph conjecture. Its finite verification reportedly covers all 58,905 four-edge sets. Why it could matter: One tiny graph may overturn a 28-year-old prediction A 1998 conjecture proposed a limit on how easily coverage can break in certain networks. This paper says a familiar, highly symmetric graph exceeds it. The mathematics can inform coverage models, but practical effects would be indirect. If it holds: Foundational: researchers must replace the conjectured bound and rethink this corner of graph robustness before drawing broader lessons for coverage or fault-tolerance models. If it does not: The 1998 bound survives this test, and the exhaustive check should reveal whether the graph, deletions, or domination count was encoded incorrectly. Boundary: intake record only; no examination result recorded.

Geometry-dependent rank defect in (C¹) cubic spline space

2026-09-09T00:00:00Z

Claim: The manuscript claims a nondegenerate planar 18-triangle complex at (t=1/5) where (dim S¹₃(mathcal T)=34) exceeds Schumaker's predicted lower bound 33 despite no singular interior four-star, refuting the conjectured sufficiency of the local correction (sigma). Why it could matter: Curved digital surfaces have hidden global dependencies Splines are the smooth patches behind computer-aided design, animation, and numerical simulation. This paper says checking each local mesh neighborhood can miss a dependency created by the geometry of the whole surface. If it holds: Enabling: spline software and mathematical models may need global checks, helping prevent silent dimension errors in CAD, surface design, and simulation pipelines. If it does not: The classical local correction may still be sufficient; the unusual extra degree of freedom would trace to a rank or geometry calculation error. Boundary: intake record only; no examination result recorded.

An infinite small-step (ℤ³)-walk with no collinear triple

2026-09-09T00:00:00Z

Claim: The manuscript claims an infinite walk in (ℤ³), using a fixed set of sixteen step vectors, whose vertices contain no collinear triple—answering the Gerver--Ramsey problem popularized as Erdős Problem 193. It is unusually ready for source-to-formal scope mapping because the arXiv record links both a Lean/formal project and a public project surface. Why it could matter: An infinite walk that never lines up three points The construction solves a deceptively simple puzzle: move forever through a 3-D integer grid without ever placing three visited points on one line. It is also a compact test of machine-checked mathematics. If it holds: Foundational: it closes Erdős Problem 193 and supplies a reusable blueprint for formally verified infinite constructions built from finite, checkable rules. If it does not: The puzzle remains open, while the mismatch would teach us where a finite checker or Lean statement failed to cover the infinite walk. Boundary: intake record only; no examination result recorded.

Albertson's Conjecture Holds for r at Most 26

2026-09-09T00:00:00Z

Claim: The paper claims that every graph with chromatic number (r≤ 26) has crossing number at least that of (Kr), extending the previously reported (rleq24) range by resolving the remaining orders for (r=25,26). The paper also states a constrained structural consequence for a hypothetical (r=27) exception. Why it could matter: How much tangling does complex connectivity force? When a network needs many colors to separate conflicting connections, must it also require many crossings when drawn? Settling more cases sharpens the boundary between abstract connectivity and unavoidable geometric congestion. If it holds: Foundational: Albertson's conjecture is established through 26 colors, extending the known frontier by two cases and sharply restricting a possible 27-color counterexample. If it does not: The 25- and 26-color cases remain unproved by this argument; failure would not itself produce a counterexample or show that the conjecture becomes false below 27. Boundary: intake record only; no examination result recorded.

Ten unknowns for Hilbert's tenth problem over the integers

2026-09-09T00:00:00Z

Claim: The manuscript claims that no algorithm decides integer solvability for arbitrary polynomial equations in ten unknowns, improving the previously stated eleven-unknown result. It is a specific advance on a restricted-variable form of Hilbert's tenth problem, not a new resolution of the original 1970 undecidability result. Why it could matter: Some ten-variable equations can defeat every algorithm This is a hard limit on computation, not merely a slow-algorithm result. It says no universal program—not even a future AI—can always decide whether an integer polynomial with ten unknowns has a solution. If it holds: Foundational: it moves undecidability from eleven variables to ten, tightening our map of problems that computation can never solve in full generality. If it does not: The known eleven-variable impossibility remains; the attempted compression identifies where an undecidability encoding needs an extra variable. Boundary: intake record only; no examination result recorded.

A Proof of Fraenkel's Conjecture

2026-09-09T00:00:00Z

Claim: The manuscript claims Fraenkel's conjectured binary density pattern for partitions of the integers into at least three Beatty sequences with distinct moduli. Its proposed route runs through a dimension-free one-third-density statement, Fourier cancellation, and three exact finite verifications, making the proof architecture specific enough to map promptly. Why it could matter: Perfectly interlocking number schedules may have one shape Imagine several repeating schedules that cover every integer exactly once without collision. Fraenkel's conjecture says that, with three or more distinct rhythms, their shares must follow one rigid doubling pattern. If it holds: Foundational: a decades-old classification becomes complete, deepening the mathematics of exact partitions and potentially informing future work on collision-free periodic scheduling. If it does not: Other perfectly balanced patterns may exist, and the failed step would narrow where to search for them. Boundary: intake record only; no examination result recorded.

144 real circles tangent to three conics

2026-09-09T00:00:00Z

Claim: The manuscript exhibits three conics with 144 real tritangent circles, exceeding and thereby contradicting the conjectured maximum of 136. The finite claimed witness makes this a comparatively bounded algebraic-geometry intake target. Why it could matter: Geometry has more tangent circles than we thought Three simple curves can share far more tangent circles than the previous proposed ceiling allowed: 144 instead of 136. That changes the landscape for a classic geometry-counting problem. If it holds: Enabling: geometric solvers and enumerative methods gain a tougher benchmark, improving how researchers count and certify all real solutions to tangency constraints. If it does not: The 136 ceiling may survive; the examination would expose duplicated, non-real, or merely approximate circles. Boundary: intake record only; no examination result recorded.

A Full-Sequence Quantitative Gap Between the Chromatic and Cochromatic Numbers of a Random Graph

2026-09-09T00:00:00Z

Claim: The manuscript claims to resolve Erdős and Gimbel's question by showing, along the full sequence for (Gn ∼ G(n,1/2)), that (χ(Gn)-ζ(Gn)) exceeds an explicit positive multiple of (n/(log n)³) with probability tending to one. It is unusually timely because the author supplies a Lean 4 formalization of the explicit full-sequence lower-bound consequence and a public replay archive, while disclosing AI-assisted development. Why it could matter: Random networks hide a real shortcut in their structure A random network can be grouped more efficiently when groups may be either fully connected or fully disconnected than when only independent groups are allowed. The paper quantifies that advantage at scale. If it holds: Foundational: it resolves an Erdős–Gimbel question and gives researchers a sharper baseline for random-graph partitioning and average-case combinatorial optimization. If it does not: The claimed full-sequence gap is not established; the failure would identify where a probabilistic or formally encoded bound overreaches. Boundary: intake record only; no examination result recorded.

A Counterexample to Belinsky's Conjecture on Cesàro Means at Lebesgue Points

2026-09-09T00:00:00Z

Claim: The paper claims that the Carleson–Trigub–Zagorodniĭ logarithmic-growth condition is not sufficient for Belinsky's 1997 conjecture: it constructs a strictly convex increasing sequence ((am)) with (am≤ 7m⁸) and an (L¹(𝕋)) function for which the specified Cesàro means are unbounded at a Lebesgue point. Why it could matter: A long-standing Fourier averaging test may fail Fourier averaging is a mathematical way to tame unstable wave reconstructions. This counterexample says a long-standing condition still cannot guarantee convergence at a locally well-behaved point; consequences for practical signal and image methods would be indirect and long-term. If it holds: Foundational: analysts must strengthen a proposed convergence test and gain a precise counterexample for building a correct replacement. If it does not: The sufficiency claim may survive; the proposed sequence or function fails one of the required growth, convexity, or local-regularity conditions. Boundary: intake record only; no examination result recorded.

An Explicit Family of Log-Concave Counterexamples to the Gaussian Completely Monotone Conjecture

2026-09-09T00:00:00Z

Claim: The authors claim smooth, strictly log-concave examples in every dimension that violate Gaussian complete monotonicity; in dimension one, the claimed explicit family has a negative signed (m)-th entropy derivative for every sufficiently large (m), persisting for small positive time. The manuscript states that GPT-5.6 Sol Pro developed the proof under author guidance, making the source particularly relevant to the AI-assisted-proof watch lane. Why it could matter: A supposed universal law of entropy breaks Entropy tracks how uncertainty spreads under heat-like smoothing, a core idea in probability and information theory. This AI-assisted proof claims a broad family of beautifully behaved distributions still violates the expected pattern. If it holds: Foundational: researchers lose a proposed universal shortcut for entropy inequalities and gain explicit stress tests for future theorems in information theory and probability. If it does not: The conjecture survives, and the error would pinpoint whether the AI-assisted periodic calculation, real-line transfer, or tensorization went wrong. Boundary: intake record only; no examination result recorded.

Adjoint Closures of Singular Quadratic Pencils and First's Pfister-Type Conjecture

2026-09-09T00:00:00Z

Claim: For a Pfister-type local--global criterion for nonsingular pairs of quadratic forms, the manuscript claims the nonsingularity hypothesis cannot be removed: over every formally real field it constructs a singular pair on (K⁷) whose adjoint closure is its two-dimensional pencil of hyperbolic forms although the pair is not weakly hyperbolic. The new v2 abstract further claims a complete two-dimensional regular-pencil closure dichotomy and minimality of dimension seven. Why it could matter: A hidden exception breaks a local-to-global test Local-to-global principles let mathematicians infer an entire object's behavior from easier local checks. This counterexample says one such shortcut for pairs of quadratic forms breaks when singular cases are allowed. If it holds: Foundational: mathematicians must retain the nonsingularity safeguard or find a replacement, preventing a false local-to-global rule from propagating through quadratic-form research. If it does not: First's broader conjecture remains plausible, and the decomposition examination will show which singular-block argument failed. Boundary: intake record only; no examination result recorded.

Proof of the AGT Conjecture at Generic β

2026-09-09T00:00:00Z

Claim: The authors claim an all-level proof of the four-point SU(2) AGT correspondence with four fundamental hypermultiplets at generic β=-ε₁/ε₂, via generalized-Jack-polynomial Selberg-average factorization and a triangular recursion. Why it could matter: A major piece of a quantum-physics dictionary may be proved The AGT correspondence links calculations in four-dimensional gauge theory to two-dimensional conformal field theory. This paper claims an all-level proof for the four-point SU(2) case at generic beta—not the entire AGT program. If it holds: Foundational: that specific four-point SU(2), generic-beta correspondence becomes a theorem at every expansion level, strengthening one important part of the broader AGT dictionary. If it does not: Finite-level matches may remain, but the claimed universal translation needs repair—likely in a recursion, factorization, parameter, or normalization step. Boundary: intake record only; no examination result recorded.

Square Functions and the Complete Crouzeix Conjecture in Dimension Three

2026-09-09T00:00:00Z

Claim: The manuscript claims to settle the complete Crouzeix conjecture for matrices of order at most three, alongside sharp square-function and spectral-constant results in dimensions two and three. Why it could matter: Small matrix calculations get a stronger safety bound Functions of non-normal matrices can behave far more wildly than their eigenvalues suggest. This result claims a sharp control principle for matrices up to size three, including matrix-valued calculations. If it holds: Enabling: researchers gain firmer error and stability bounds for small-matrix computations, with possible downstream value in numerical analysis, control, and signal processing. If it does not: The complete Crouzeix bound remains unsettled for 3×3 matrices; a failed proof step would not itself establish a counterexample or a breakdown of scalar intuition. Boundary: intake record only; no examination result recorded.

A blueprint for the formalization of norm-variation of multiple ergodic averages for commuting transformations

2026-09-09T00:00:00Z

Claim: The blueprint says its Lean 4 formalization, completed largely automatically with frontier large-language-model assistance, supports norm-variation estimates for multiple ergodic averages of commuting transformations, quantitatively strengthening Tao's norm-convergence theorem and answering an Avigad--Rute question. Why it could matter: A stress test for AI-assisted formal mathematics This mathematics measures whether several interacting processes settle down—and how violently they fluctuate along the way. The bigger story is methodological: a deep modern analysis proof was formalized largely with AI, giving us a rare test of reviewable machine mathematics. If it holds: Methods: it would show AI-assisted formalization can carry a large, current analysis theorem into a proof checker, strengthening the case for faster mathematical work whose logical core remains inspectable. If it does not: A mismatch would expose where the formal theorem, dependencies, or prose diverge—exactly the evidence needed to improve machine-proof workflows before trusting them at scale. Boundary: intake record only; no examination result recorded.

A proof of the Arnold-Givental conjecture

2026-09-09T00:00:00Z

Claim: For a closed symplectic manifold with an anti-symplectic involution and transverse Hamiltonian image of its fixed locus, the authors claim the Arnold-Givental lower bound on the number of intersection points in full generality. Why it could matter: A long-standing rule for the geometry of motion Symplectic geometry is the language of systems whose positions and momenta evolve together. This claimed theorem says certain symmetric shapes cannot be moved through phase space without a minimum number of intersections—a deep rigidity rule for the geometry underlying mechanics. If it holds: Foundational: it would complete a major rigidity principle in symplectic topology and sharpen the geometric toolkit used to understand Hamiltonian motion, without implying an immediate engineering breakthrough. If it does not: A failure would locate a gap in the new localization or Floer-theory machinery, preserve only established partial cases, and prevent the full-generality claim from hardening into lore. Boundary: intake record only; no examination result recorded.

Sequential and distributive dual futile cycle: Hopf bifurcation can occur under parameter-rich kinetics but cannot occur under mass action kinetics

2026-09-09T00:00:00Z

Claim: Claims the sequential/distributive dual futile-cycle ODE system permits Hopf bifurcation under parameter-rich kinetics but not under mass-action kinetics; the author explicitly reports a decisive ChatGPT Sol 5.6 contribution to a nontrivial positivity certificate. Why it could matter: When cell-signaling models can—and cannot—oscillate Cells often control activity by adding and removing chemical tags from proteins. This paper asks when a standard two-site signaling circuit can oscillate, helping researchers avoid attributing rhythmic behavior to a model whose assumptions mathematically forbid it. If it holds: Enabling: it would give systems biologists a firmer rule for choosing kinetic models of multisite phosphorylation and a concrete example of AI producing a checkable polynomial certificate. If it does not: The claimed divide remains unsettled: parameter-rich kinetics may fail to produce the oscillation, mass-action kinetics may fail to exclude it, or both. Boundary: intake record only; no examination result recorded.

Closing the gap and settling the problem of queens on an (n× n) board, each attacking at most one other

2026-09-09T00:00:00Z

Claim: States (q(n)=lfloor4n/3rfloor) for every (nge6), and (q(n)=n) for (nle5), where every placed queen attacks at most one other; it also gives the stated exact-one-attacker formula. This is a new primary-source claim to settle previously conjectural values. Why it could matter: A chessboard puzzle gets an exact answer This gives an exact answer to a deceptively hard chessboard question: how many queens fit when each may attack at most one other? Beyond the puzzle, it is a clean case study in turning clever constructions into universal upper bounds. If it holds: Foundational: it closes the puzzle for every board size and supplies compact constructions and bounds that can benchmark human or machine combinatorial reasoning. If it does not: A missed board size or boundary case would reveal where the proposed universal formula needs an exception or a stronger upper-bound argument. Boundary: intake record only; no examination result recorded.

Refutation of the Non-Cancelling-Intersections Conjecture

2026-09-09T00:00:00Z

Claim: Claims a finite lattice whose top element has no dot-algebra representation, removing the left-linearity restriction from the author's earlier 2608.19414 result and thereby refuting the NCI conjecture as stated. Why it could matter: Some clean formulas have no clean construction The conjecture promised that whenever inclusion–exclusion computes a union without algebraic cancellation, the same result could be built from literal set operations. A counterexample means some tidy numerical identities have no equally tidy structural explanation. If it holds: Foundational: it would block a tempting shortcut in symbolic set reasoning: not every cancellation-free numerical identity can be converted into an equally transparent construction. If it does not: The original structural hope survives, and the finite lattice construction reveals which representation step still needs repair. Boundary: intake record only; no examination result recorded.

The Erdos--Gallai bound for consecutive even cycle lengths

2026-09-09T00:00:00Z

Claim: For sufficiently large (t), claims the sharp Erdős--Gallai edge threshold forces (t) consecutive even cycle lengths, resolving a conjecture of Verstraëte and deriving stated residue-class cycle-threshold consequences. Why it could matter: Dense networks must hide loops of many sizes In a dense network, loops are unavoidable. This theorem claims something much sharper: once a graph crosses an exact density threshold, it must contain loops of many consecutive even sizes, revealing a surprisingly rigid law of network structure. If it holds: Foundational: it would give graph theorists a sharp guarantee about the cycle lengths hidden inside dense networks, strengthening the structural toolkit behind extremal graph algorithms. If it does not: The proposed threshold or exceptional case is incomplete, warning researchers not to use it as a universal guarantee about dense graphs. Boundary: intake record only; no examination result recorded.

Exact-Support Counterexamples to Euclidean-to-Spherical Transfer of Positive Definiteness in Even Dimensions

2026-09-09T00:00:00Z

Claim: For each even dimension and admissible support radius, constructs a smooth radial kernel that is strictly positive definite in Euclidean space but not positive definite on the sphere. Why it could matter: Flat-space models can break on a round world Positive-definite kernels are the similarity and covariance rules behind spatial statistics and many machine-learning methods. This result warns that a kernel valid on flat space can become mathematically invalid on a sphere—crucial for Earth-scale and directional data. If it holds: Enabling: modelers gain exact examples showing when flat-space kernels can produce impossible covariance structures on spherical data, supporting safer geometry-aware choices in geospatial statistics and machine learning. If it does not: The claimed even-dimensional failure narrows or disappears, so some flat-space kernels may transfer more safely than the construction suggests. Boundary: intake record only; no examination result recorded.

Kernel-Checked Exclusions for the Erdős-Selfridge Odd Covering Problem: Any Odd Covering of (ℤ) Has lcm Exceeding 10000

2026-09-09T00:00:00Z

Claim: Lean 4 kernel formalization of the finite exclusion (operatornamelcm>10000) for odd distinct-modulus covers, with 63 public theorems and stated no-sorry/no-nativedecide trust boundary. Why it could matter: A giant number search becomes checkable proof Imagine covering every integer with repeating schedules that use distinct odd periods. The grand puzzle remains open, but this work machine-checks that no such system can have a combined period of 10,000 or less—a milestone for reviewable computational proof. If it holds: Methods: it demonstrates one reproducible route from search-generated certificates to Lean-kernel theorems for this bounded exclusion; broader reuse needs separate evidence. If it does not: A scope or trust-base mismatch would show why compiled code is not enough and protect later searches from inheriting a false foundation. Boundary: intake record only; no examination result recorded.

MathAdv: What Theorem Provers Know, Reason, Formalize, and Generalize

2026-09-09T00:00:00Z

Claim: The authors introduce a 13-domain diagnostic benchmark for natural-language and Lean 4 theorem proving, including reformulation tests, and report that formalization remains a bottleneck and equivalent restatements expose robustness limits. Why it could matter: Can math AI survive a simple rewording? Can an AI prove the same theorem when the wording changes? This benchmark tests theorem provers across 13 areas and finds that many stumble on formalization or equivalent restatements, exposing the difference between genuine robustness and a flattering headline score. If it holds: Methods: it gives developers a better diagnostic map for building math assistants that generalize across subjects and survive harmless rewording, rather than merely excelling on familiar benchmark formats. If it does not: If transformed problems are not truly equivalent or scoring is leaky, model rankings could mislead research; fixing the benchmark still improves evaluation. Boundary: intake record only; no examination result recorded.

A counterexample to Nevanlinna's century-old half-plane problem

2026-09-09T00:00:00Z

Claim: The authors construct a nonconstant meromorphic function on the complex plane whose preimages of 0, 1, and infinity lie on the real axis, while its restriction to the upper half-plane is not in the Nevanlinna class; this is presented as a counterexample to Nevanlinna's half-plane problem. Why it could matter: A century-old shortcut in complex analysis may fail A century-old expectation said that a complex function avoiding three values in half a plane must behave in a controlled way there. This construction says no, forcing analysts to rethink which geometric clues actually guarantee tame growth. If it holds: Foundational: it removes a trusted shortcut in complex analysis and launches the search for stronger conditions that really control meromorphic functions in a half-plane. If it does not: The century-old principle survives; the construction would teach precisely where its claimed preimage or growth property breaks. Boundary: intake record only; no examination result recorded.

Fröberg's Conjecture for Quintics and Septics in Four Variables

2026-09-09T00:00:00Z

Claim: For equal-degree ideals in four variables over characteristic-zero fields, the paper establishes Fröberg's predicted Hilbert series for every generator count when the degree is 5 or 7, using finite Macaulay-multiplication rank certificates; it expressly leaves the unrestricted conjecture outside scope. Why it could matter: Two hard families of polynomial systems become predictable Hilbert series tell algebraists—and computer-algebra software—how many independent polynomial constraints remain at each degree. This paper settles two difficult degree cases, making generic systems of quintic and septic equations more predictable, while leaving the full conjecture open. If it holds: Enabling: the exact rank certificates could give computer-algebra researchers reliable formulas and reproducible test cases for generic polynomial ideals in these two degrees. If it does not: A failed certificate or reduction would limit the covered generator ranges and prevent algebra systems from relying on an overstated formula. Boundary: intake record only; no examination result recorded.

The Multivariable Strong Monodromy Conjecture for Plane Curves

2026-09-09T00:00:00Z

Claim: Claims the topological multivariable Strong Monodromy Conjecture for reduced plane-curve germs, via containment of actual polar hyperplanes in the Bernstein–Sato zero locus. Why it could matter: Two languages for mathematical singularities may finally agree When an algebraic curve crosses or pinches itself, topology and algebra record the damage differently. This theorem claims that, for plane curves, a signal seen in one record must appear in the other—a powerful translation rule for singularities. If it holds: Foundational: it would strengthen the dictionary between geometric, topological, and algebraic descriptions of curve singularities, helping researchers compute and classify these complicated points from whichever representation is tractable. If it does not: The full plane-curve theorem remains unproved; a failed step may reveal a gap or missing hypothesis, but would not by itself show that counterexamples exist. Boundary: intake record only; no examination result recorded.

Generalised Cone Conjecture, I: Beyond Calabi--Yau

2026-09-09T00:00:00Z

Claim: The authors state that this first paper proves the Generalised Cone Conjecture on surfaces. Why it could matter: Turning infinite geometric complexity into a finite map Algebraic surfaces can carry infinitely complicated families of geometric configurations. The cone conjecture says symmetry may compress that infinity into a finite-sided fundamental region, making classification manageable; this paper supplies only part of a two-paper proof. If it holds: Foundational: together with its companion, it would organize broad classes of surfaces into finitely describable symmetry regions and support finiteness results for their minimal geometric models. If it does not: Because this installment treats a companion theorem as a black box, failure there would block the advertised full result while leaving this paper's intermediate theorems potentially intact. Boundary: intake record only; no examination result recorded.

Three omitted values and non-Blaschke point divisors in half-planes

2026-09-09T00:00:00Z

Claim: Constructs a real meromorphic function with the three designated preimage sets real, not of bounded type in either half-plane, and stronger non-Blaschke statements; the authors say the core construction/proof was generated during an autonomous GPT-5.6 Sol Ultra run. Why it could matter: A century-old shortcut may finally be broken The claim says three special output values do not control a complex function's behavior in a half-plane as mathematicians had hoped. Because the core argument was machine-generated, it is also a striking test of autonomous mathematical discovery. If it holds: Foundational: complex analysts lose a century-old shortcut and gain a new example that redraws the boundary between special-value data and global growth. If it does not: The old question stays open, while the failed construction shows exactly where autonomous reasoning lost control of an infinite analytic argument. Boundary: intake record only; no examination result recorded.

A note on normal generation and the first (ℓ²)-Betti number

2026-09-09T00:00:00Z

Claim: For each (n), constructs a countable torsion-free locally free group with first (ℓ²)-Betti number (n) and normal rank one, contradicting the 2011 Osin--Thom conjecture. Why it could matter: One generator may hide unlimited topological complexity A group can be generated in an unexpectedly economical way yet carry arbitrarily large hidden topological complexity. That breaks a proposed bridge used to reason about several major open problems in group theory and topology. If it holds: Foundational: researchers must uncouple normal generation from this complexity measure and revisit consequences tied to the Wiegold, Levin, Kervaire, and Whitehead problems. If it does not: The conjectured bridge survives, and the construction reveals which finiteness or generation condition is doing the real work. Boundary: intake record only; no examination result recorded.

A counterexample to Kanalas' problem of continuously realising types

2026-09-09T00:00:00Z

Claim: Gives a coherent theory, space, and continuous type assignment for which no corresponding sheaf model realizes the assigned types fibrewise, negatively answering a problem of Kristóf Kanalas. Why it could matter: Smooth local choices can still refuse to assemble The paper tests a powerful mathematical instinct: if compatible information changes continuously from place to place, it should combine into one coherent global model. This counterexample says continuity alone is not enough. If it holds: Foundational: theories that build global objects from local data need stronger compatibility conditions, sharpening the logic behind sheaves and local-to-global reasoning. If it does not: A promising local-to-global principle remains alive, and the attempted counterexample exposes which hypothesis actually guarantees assembly. Boundary: intake record only; no examination result recorded.

A Short Proof of a Conjecture Regarding Quadratic Representations of Practical Numbers

2026-09-09T00:00:00Z

Claim: Proves the second Wang–Sun component for odd b and even c; with earlier named work, the author says this settles the full conjecture. Why it could matter: Quadratic recipes keep finding unusually flexible numbers Practical numbers can make every smaller amount from distinct divisor “coins.” This result would finish a conjecture showing that a broad family of quadratic formulas is guaranteed to produce one of these unusually flexible numbers. If it holds: Foundational: it closes the Wang–Sun conjecture and gives number theorists a dependable recipe connecting quadratic expressions with divisor-sum structure. If it does not: The full conjecture remains open, and the failure identifies whether the new proof or its reliance on earlier work needs repair. Boundary: intake record only; no examination result recorded.

Conformal Boundary Deformations under Ricci Lower Bounds: Eigenvalue Counterexamples and Area Obstructions

2026-09-09T00:00:00Z

Claim: Constructs hemisphere metrics with strict Ricci and boundary-convexity inequalities but first boundary Laplace eigenvalue below the proposed bound in every dimension at least three. Why it could matter: Curved spaces can vibrate below the expected floor Eigenvalues encode natural vibration and wave frequencies. The paper says even positively curved spaces with nicely convex boundaries can fall below a proposed frequency floor, showing that those geometric safeguards are not enough. If it holds: Enabling: spectral and geometric models gain a sharper warning about which shape and curvature assumptions can safely predict boundary-wave behavior. If it does not: The proposed spectral floor may survive, and the deformation pinpoints where curvature or boundary conditions prevent the claimed exception. Boundary: intake record only; no examination result recorded.

An elementary counterexample to Escobar's Steklov conjecture on the three-ball

2026-09-09T00:00:00Z

Claim: An explicit polynomial deformation of the unit three-ball has positive Ricci curvature and strictly convex boundary while violating Escobar's proposed first nonzero Steklov-eigenvalue lower bound. Why it could matter: A nearly round ball challenges a spectral-geometry prediction Steklov eigenvalues measure how harmonic behavior inside a shape responds at its boundary. This explicit deformation of a three-dimensional ball appears to violate a curvature-and-boundary prediction, offering a sharp mathematical benchmark rather than a general wave-engineering result. If it holds: Foundational: spectral geometers gain a concrete counterexample showing that positive curvature and convexity do not guarantee the proposed Steklov bound. If it does not: Escobar's proposed bound survives this attack, and the calculation exposes which geometric condition the candidate shape fails to satisfy. Boundary: intake record only; no examination result recorded.

Frankl's Conjecture at Height Four and the Structure of Height-Five Counterexamples

2026-09-09T00:00:00Z

Claim: Proves the union-closed-sets conjecture for empty-set-free families of height at most four (equivalently usual families containing the empty set of height at most five), and constrains possible height-five counterexamples. Why it could matter: A stubborn set puzzle loses another escape route The puzzle asks whether some item must appear in at least half the sets whenever combining any two stays inside the family. This paper settles height four and constrains the smallest height-five counterexamples; taller cases remain open. If it holds: Foundational: it settles height four and narrows the smallest height-five counterexamples, guiding the next search without resolving Frankl's conjecture in full. If it does not: The height-four frontier reopens, but the broken step reveals which structural constraint cannot be trusted in future attacks. Boundary: intake record only; no examination result recorded.

The uniform Littlewood conjecture fails on a set of positive Hausdorff dimension

2026-09-09T00:00:00Z

Claim: Establishes a Hausdorff-dimension-at-least-3/2 counterexample set in a badly-approximable slice and a full-dimension projection statement for the uniform—not classical—Littlewood conjecture. Why it could matter: Number-pair exceptions may form a surprisingly large fractal A uniform number-approximation rule was already known to fail. This paper says the failures occupy a genuinely large fractal family, changing them from isolated oddities into a substantial part of the mathematical landscape. If it holds: Foundational: researchers gain a quantitative map of where uniform simultaneous approximation fails, with new tools for measuring exceptional fractal sets. If it does not: The known counterexamples remain, but claims that they form a large fractal family must be scaled back. Boundary: intake record only; no examination result recorded.

Winning property of counterexamples to Uniform Littlewood's Conjecture

2026-09-09T00:00:00Z

Claim: Shows the BFK25-proposed ULC counterexample set is hyperplane absolute winning, hence full Hausdorff dimension in (ℝ²). Why it could matter: These mathematical exceptions may be remarkably hard to erase The authors claim these counterexamples are not just numerous: they form a robust, full-dimensional fractal set and remain large on broad families of curves and lines. That makes the conjecture's failure structurally durable, not accidental. If it holds: Foundational: the exceptional set becomes robust enough to study with powerful game-based tools, reshaping the geometry of uniform approximation. If it does not: The earlier counterexamples may still stand, but their claimed robustness and full-dimensional structure would remain unproved. Boundary: intake record only; no examination result recorded.

Improved bounds for the smallest 4-chromatic graph of girth six

2026-09-09T00:00:00Z

Claim: Improves the known range to (29 ≤ n₆(4) ≤ 64), supplies an explicit 64-vertex witness, and reports a Lean-checked non-3-colourability certificate. Why it could matter: A smaller impossible-to-three-color network has been found The paper finds a 64-node network with no short loops that still needs four colors, then uses SAT and Lean certificates to check it. It advances both an extremal graph puzzle and reviewable computer-assisted mathematics. If it holds: Methods: it tightens the known size range and demonstrates how search, independent code, certificates, and formal proof can support one verifiable result. If it does not: The failure would expose whether the graph witness, exhaustive search, SAT certificate, or formal checker broke—valuable evidence for better verification pipelines. Boundary: intake record only; no examination result recorded.

The Hodge conjecture for Fermat fourfolds of odd degree at most 199

2026-09-09T00:00:00Z

Claim: A computer-assisted proof for odd-degree Fermat fourfolds through degree 199, combining geometric closure criteria with an exhaustive ((2,2))-orbit census. Why it could matter: A machine-checkable foothold on a million-dollar mystery The Hodge conjecture asks whether certain hidden geometric features always come from actual algebraic shapes. This paper does not solve it generally, but claims a fully checkable proof for a large, precisely bounded family of four-dimensional Fermat varieties. If it holds: Methods: it provides a demanding, replayable case study combining geometry, exhaustive computation, independent enumeration, certificates, and Lean on one bounded family. If it does not: The general conjecture is untouched; the replay would reveal whether the census, geometric closure rules, or code-to-proof bridge failed. Boundary: intake record only; no examination result recorded.

A computer-assisted counterexample to the planar Pompeiu and Schiffer conjectures

2026-09-09T00:00:00Z

Claim: Constructs a bounded simply connected noncircular planar domain yielding counterexamples to the stated Schiffer and Pompeiu formulations through a cubic operator equation and rigorous tail control. Why it could matter: A strange shape may fool two classic tests Some mathematical tests were thought to force a flat shape to be a disk. This paper constructs a noncircular region that appears to pass the same boundary and integral tests, changing what measurements can reveal about shape. If it holds: Enabling: inverse-problem and wave researchers gain a concrete warning that these measurements do not uniquely identify circular geometry, plus an exact benchmark for future methods. If it does not: The conjectures survive, and the failed numerical-to-exact step identifies where an approximate shape stopped being a genuine mathematical counterexample. Boundary: intake record only; no examination result recorded.

Pólya's Conjecture for the Neumann Eigenvalues on Euclidean Balls

2026-09-09T00:00:00Z

Claim: Establishes the Neumann Pólya lower bound for Euclidean balls in every dimension, with exact-rational ancillary verification for a two-parameter estimate in dimensions at least seven. Why it could matter: Round spaces get a guaranteed minimum of wave modes Neumann eigenvalues describe allowed wave patterns in spaces with reflecting boundaries. This paper claims a sharp lower bound—not an exact mode count—for Euclidean balls in every dimension, giving spectral geometry a rigorous benchmark without promising an immediate device. If it holds: Enabling: researchers gain an all-dimensional lower-bound benchmark for wave-mode counts in round domains and a stronger base for estimates on harder shapes. If it does not: The expected lower bound remains unproved for Euclidean balls, so this proposed all-dimensional benchmark cannot yet be treated as established. Boundary: intake record only; no examination result recorded.