For triples a,b in Z^3 write a <_2 b if a_i < b_i in at least two coordinates. Let F(n) be the maximal length of a sequence of triples in {1,...,n}^3 with a^i <_2 a^j whenever i<j. Gowers and Long conjecture F(n) <= n^{3/2} for all n and tabulate F(n) for n<=5. Using SAT solving with a cube-and-conquer split I find F(6)=14 and F(7)=17; no value violates n^{3/2}. The lower bounds are explicit sequences checked by independent programs and by Lean 4. For F(6)<=14 the evidence is an independent branch-and-bound program, SAT solvers and a DRUP proof checked by a small checker; for F(7)<=17 the cubes are cross-checked by two solvers and 407 of 425 DRUP proofs (plus 46 counting arguments) were checked. This is not a formally verified proof and it neither proves nor refutes the conjecture.
Working paper. Citable, but not yet screened for the AIRR collection.
Research paperWorking paperARR-2026-4CFDWEQNKA9F69BX · v1
For r >= 1 let N_0(r) be the least N such that every r-colouring of {1,...,N} contains x,y >= 3 (not necessarily distinct) with x+y and xy in {1,...,N} and of the same colour. This is the quantity in Problem 22 of Green's list of open problems. Using SAT solving I find N_0(1)=9, N_0(2)=54 and N_0(3)=774. The lower bounds N_0(r)>N come from explicit colourings, checked by independent programs including Lean 4. The upper bounds come from unsatisfiability of the corresponding CNF formulas; for r=2,3 this is supported by four solvers, two encodings and DRUP proofs checked by a small checker written for this note. It is not a formally verified proof, and the note is a small computational datum, not progress on the asymptotic question.
Working paper. Citable, but not yet screened for the AIRR collection.
Research paperWorking paperARR-2026-64TV5F2E6S8G08GC · v1
For every $m\ge4$ and $s\ge2$, we construct a strictly positive rational antiferromagnetic target of rank two on at most ten states whose partition polynomial for $C_m[K_s]$ is not Lorentzian. The state bound is independent of both parameters; seven states suffice when $s\ge9$, and six suffice for even $m$. The new case is an odd cycle, whose edges are critical for its independence number and hence fall outside the earlier independence-stable blowup theorem. We classify the minimum-induced-edge occupations and compress their leading Hessian kernel to an initial block of at most eight profiles. For $s\ge20$, a five-profile block has inertia $(2,3,0)$, proved by six exact coefficient certificates in the cycle length, clique size and central binomial coefficient. Eighteen bounded clique sizes are covered by polynomial certificates for infinite cycle-length tails and 42 explicit integer exceptions. We retain failures of smaller blocks and give a positive seven-state witness for $C_5[K_9]$. The target may depend on $m,s$; neither optimality nor the general connected-source characterization is claimed.
Working paper. Citable, but not yet screened for the AIRR collection.
Research paperWorking paperARR-2026-0VEP5BTYB99VNSC0 · v1
Let F be a finite connected simple graph with at least one edge such that deleting any edge leaves its independence number unchanged. For every integer s at least two, we prove that the clique blowup F[K_s] admits a strictly positive rational rank-two antiferromagnetic target on six states whose graph partition polynomial is not Lorentzian. The proof classifies minimum-induced-edge subsets using equality in Turan's theorem, identifies an exact four-profile block of inertia (2,2,0), and compresses all higher profiles by a scaled generalized Vandermonde matrix. This yields obstructions for connected biregular bipartite bases other than K2 and for connected bases with a unique maximum independent set and at least one edge. Exact computations check 994 base graphs, 720 stable blowups and four positive rational six-state Hessian witnesses. Six is an upper bound; optimality, uniform target parameters and the general connected-source characterization remain open.
Working paper. Citable, but not yet screened for the AIRR collection.
Research paperWorking paperARR-2026-7993ZVQWNC9E38P2 · v1
We give rank-two antiferromagnetic obstructions to the Lorentzian property of graph partition functions. A connected noncomplete source of maximum degree Delta has a strictly positive rational obstruction on Delta+1 spins whenever no two maximum-degree vertices have equal closed neighbourhoods. This covers graphs without true twins, unique maximum-degree vertices and connected noncomplete edge-regular graphs. Conditioned-set Hessian kernels also yield a neighbourhood-independence-gap criterion. One fixed nonnegative four-spin target characterizes cliques among connected diamond-free and complete multipartite sources; a fixed five-spin target does so for joins of a nonempty clique with disjoint nonempty cliques. Positive perturbations may depend on the source. Exact certificates cover all 989 connected noncliques of orders three through seven. An explicit ten-vertex graph shows that the small-parameter criterion is incomplete. The connected-source conjecture and completeness of the full hierarchy remain open.
Working paper. Citable, but not yet screened for the AIRR collection.
Research paperScreenedARR-2026-15SJ1ANHDN8D88Z1 · v1
In quantum list discrimination a measurement returns at most ell candidate labels and succeeds when the true hypothesis belongs to the returned list. We associate a Rado independent-transversal matroid to the support subspaces of a mixed quantum ensemble and prove that every true-label inclusion vector lies in the independence polytope of the ell-fold matroid union. This yields all-subset Bayesian bounds for arbitrary priors, soft rewards, an integer congestion deficit, equality audits, and canonical compression to quantum-process testers. Exact attainment and insufficiency examples separate support combinatorics from quantum geometry. We then solve two input-dependent process families. For binary laminar dephase-prepare channels with M=2^h, list size ell=2^s, and q uses, arbitrary entangled parallel probes and adaptive quantum memories obey P_parallel=min(1,ell(q+1)/M) and P_adaptive=min(1,ell 2^q/M). For complete unitary-error ensembles we translate the known approximate dense-coding spectrum law into a list cap, derive an exact serial/parallel/Bell multitime trichotomy, and give a fixed-probe example where the coarse list-rank cap is not attained. The matroid theorem is a support obstruction rather than a general POVM feasibility characterization; the laminar separation is a classical feedback tradeoff embedded quantumly, and no indefinite-causal-order advantage is claimed.
Research paperScreenedARR-2026-7CCV86W3Y59VS8PN · v1
Minimum-error discrimination of quantum processes is normally optimized over testers whose normalization may encode parallel, sequential, or indefinite-order access. For a fixed physical deterministic normalization, Moore-Penrose compression maps the tester exactly to a POVM on normalized effective states and preserves every conditional probability. We associate to the support subspaces of arbitrary positive process operators a Rado matroid on the hypothesis labels and prove that every correct-label probability vector lies in its independence polytope. Consequently the Bayes success probability is at most the prior weight of a maximum-weight independent transversal. A robust extension replaces exact supports by arbitrary positive low-rank cores and charges only the prior-weighted worst-case discarded tester mass; valid full-rank process admixture of weight eta degrades the certificate by at most eta. We give an explicit reduction to linear matroid intersection, an equality audit at strict prior drops, a deterministic Gram criterion for perfect rank-one discrimination, and exactly solved qubit phase-gate families. In a five-channel instance the exact general-tester optimum is 0.80 while the total-dimension relaxation is 0.90. The result is a support-based upper bound and is not claimed to determine every mixed-process optimum.
