Novel findings.
20 findings
Albertson’s Conjecture
Proved the conjecture through 28 colors and reduced the 29-color frontier to five explicit cases.
Beyond the published r ≤ 26 frontier. The five order-57/58 cases remain open; the extension is AI-reviewed.
Charney–Davis Inequality
Proved γ₃ ≥ 0 for every 17-vertex flag generalized homology 5-sphere, closing this finite case.
Every possible counterexample reduces to one degree profile, which is eliminated. Parts of the argument are Lean checked.
Planar Firey Lᵖ Equality
Classified every equality case for symmetric planar bodies, across every finite p > 1.
Exactly the parallelograms positioned with the origin at a vertex attain equality.
Ramsey Number R(5,5)
Forced new constraints on every hypothetical 43-vertex solution, excluding large symmetry, switching, and catalog-derived families.
Both colors must be 18-connected with binary rank ≥ 8. The asymmetric search remains open: 43 ≤ R(5,5) ≤ 46.
Hadwiger–Nelson Problem
Certified a five-chromatic 552-vertex unit-distance graph and compressed the sub-509 deletion search to 60 effective selectors.
Every ≤508-vertex subgraph of the fixed H514 support is four-colorable. No five-chromatic graph on 509 or fewer vertices was found.
Hypercube Square Saturation
Raised the asymptotic lower constant for square saturation from 3/2, past 7/4, to 5183640/2874791.
The 7/4 theorem is independently AI-reviewed. The strongest Q₆-derived lift still awaits external review.
Grassmannian Identity Inflation
Proved the identity-inflation inequality for every Grassmannian shape, extending a rectangular case to all shapes and scales.
Every integer scale k ≥ 1, with an exact classification of equality.
Sidorenko Stability
Proved sharp quantitative stability for complete bipartite graphs: near equality forces a tightly controlled structure.
All Kₛ,ₜ with s, t ≥ 2, including rank-one rigidity and a sharp fourth-root forcing exponent. The general conjecture remains open.
Strong Seymour Classification
Classified the relevant six-quotient transitive blow-ups into one quotient and twelve Hall cones.
A complete description within this family of tournaments, rather than a solution of the general Seymour conjecture.
Cyclic Fuglede Prime Descent
Extended the prime-descent obstruction through cardinality 2p − 1, pushing the boundary of the spectral-to-tiling search.
A structural result for finite cyclic groups. The full spectral-to-tiling frontier remains open.
Majority C-Coloring Hamming Cores
Classified every Hamming core with 2h + 1 vertices and minimum degree h, exposing the shapes behind majority colorings.
For h ≥ 2: outside a coordinate line, only two line-union families and one exceptional grid occur.
Degree-57 Moore Graph Boundary
Reduced the sharp 398-coclique equality boundary to four exact systems of 24 branches.
A finite reduction inside the hypothetical degree-57 Moore graph. Its existence remains unresolved.