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The claim is R(c)=40c+41 for every c≥2, reduced to three finite facts: the base value R(2)=121, and the unsatisfiability of a 321-position and a 521-position spoke template. The reduction is Lean-checked and holds for every D≥1; the two unsatisfiability results carry DRAT proofs.
This completes the partial entry for the same conjecture, which proved it for roughly two thirds of integers via a sc…
Four results, and the first is partly a refutation. Jakimiuk conjectured cp=μp−1 is optimal for every p≥3; the paper proves that for p≥4 and gives a counterexample for every 2<p<4, so the conjecture is false as posed and the corrected range is p≥4. The witness is the two-coordinate vector S2=(ε1+ε2)/2.
The Baranski-Murawski-Nayar-Oleszkiewicz flat-po…
Every purely-maximal ideal of a commutative ring is purely-prime, and the converse holds for several important classes of rings; Tarizadeh conjectured (Conjecture 5.8 of his earlier published paper) that in a commutative ring every purely-prime ideal is purely-maximal. False: there is a commutative ring with a purely-prime ideal that is not purely-maximal.
A generalization of Boppana's entropy inequality, of the kind used in union-closed-sets arguments, proved and formalized: the sharp form with the extremal constant characterized via the unique positive solution of an explicit equation.
The general principle is the interesting part: every semialgebraic property of a bounded fixed-dimensional mean parameter is eventually almost surely predictable. Against merely integrable matrix laws it fails from dimension two.
Curto et al. (Advances in Applied Mathematics, 2024) conjectured that every stable fixed point of a threshold-linear network is minimal. Disproved: an explicit competitive 3-neuron TLN has a stable fixed point whose support strictly contains another's, and 3 neurons is proven smallest possible.
Nathanson asked which subsets of N can occur as product intersection sets of a family of semigroup subsets, for arbitrary and for decreasing families (his Problems 10 and 11). Both are solved by complete classifications.
Can the minimum edge-outerplanarity of a finite loopless planar graph, minimized over all planar embeddings, be computed in polynomial time? Asked by Bentz in 2009.
Gill introduced the probabilistic automatic complexity AP(w) of a string: the least number of states of a probabilistic finite automaton for which w is the unique most probably accepted string of its length. He asked whether AP is unbounded, no string with AP>3 being known. The paper proves AP(w)≤3 for every string over every finite alphabet, with an explicit three-state witness.
Ramachandra and Natarajan conjectured a bound on the pairwise independent correlation gap in their 2025 Operations Research Letters paper. An explicit counterexample refutes it.
For an irreducible crystallographic root system of rank r with Coxeter number h, the paper proves that Au's normalized Witten zeta function has a simple pole at 2/h and evaluates its residue in closed form in terms of the Cartan determinant, the Weyl group order and the invariant degrees.
Does a general pencil of plane cubics over C have exactly 12 common flex lines? Ciliberto, Miranda and Roé asked this in Remark 5.3 of their paper; the answer is yes.
Ross introduced S-perfect numbers, integers expressible as 1+∑λjdj over their proper divisors with coefficients in S, and conjectured that they have the same density as the nondeficient numbers, plus a second conjecture relating odd nondeficient numbers to S-perfection. Both are false.
Coble and Barg introduced binary Coxeter codes, the span of indicators of standard cosets of fixed rank in a finite Coxeter system, generalizing Reed-Muller codes, and proposed a conjectural value for the minimum distance of a general Coxeter code. The conjecture is true, and it yields a decoding consequence.
An existence question settled by exhibiting an object, not a general theorem: one model in the family has no exponential degeneracies for generic couplings, and nothing here says which others do.
The route is worth recording because it is not the one anyone was looking down. The author had tried and failed to find such a model directly. It surfaced instead from an unrelated classification of medium-range spin chain…
subDL satisfies the signed depth relevance property, answering an open question posed by Øgaard (2026). More precisely, every valid inference in subDL contains a propositional variable that occurs in both the premises and conclusion with matching sign and at matching implicational depth.
22 conjectures of Cohen about cyclic numbers (integers with gcd(n,φ(n))=1) settled at once - 16 proved, 6 disproved - together with a complete resolution of a related OEIS problem on sequences whose running averages are Fibonacci numbers (Fried's Conjecture 2).
Does every nontrivial finite simple graph have noninteger Sombor energy? If ρ1,…,ρn are the eigenvalues of the Sombor matrix of a graph G, its Sombor energy is
ESO(G)=i=1∑n∣ρi∣.
The conjecture asserted that ESO(G)∈/Z for every nontrivial graph. A connected graph on nine vertices is exhibited with ESO(G)=64, disproving the conject…
A problem from Fajtlowicz's Graffiti program, studied by Erdős and Staton, on the Havel-Hakimi residue of common-divisor graphs. The paper resolves the problem and extends it, determining the residue's first-order scale and its nontrivial constant from the degree sequence.
Casalaina-Martin and Zhjeqi proved that the first Chern class of every torsion-free coherent quotient of a tensor power of the logarithmic cotangent sheaf is pseudo-effective, noting in Remark 4.5 that torsion-freeness was imposed only for technical reasons. Can it be dropped? Yes.
Is the exact nonreal spectral region of the four-cycle family of row-stochastic nonnegative matrices determined by the Karpelevich constraint, as Ran and Teng conjectured in 2024?
For a simple 3-polytope with at least three faces of size at least 7, must p6≥2039+2p3−4p5−∑k≥7pk? Five minimal ten-face counterexamples refute the printed inequality.
For every connected graph, is the variance of its positive adjacency eigenvalues at most its order divided by its average distance? Exact dumbbell-graph certificates refute the bound under both conventions for average distance.
Nineteen individual exact values, each decided by SAT certificate: unsatisfiable at the claimed n, witnessed satisfiable at n−1. They close cells in DD26's Tables 3-13 but settle no infinite family - the sibling entries do that for the K column. The three overlap cells are Rdih(P4alt,K6)=16, Rdih(P3alt,K9)=17 and Rdih(P9alt,K3)=17, each an instance of a sibling theorem; the remain…
The evenly-divided reading of WOWII Conjecture 72 holds:
⌈(A+L)/3⌉≤t,
where t= tree(G) (order of a largest induced tree), A= average eccentricity and L= maximum neighbourhood independence number.
A stronger reading that divides only L by three is false. The argument rests on two elementary observations (a diametral path is chordless and therefore induces a tre…
The Formal Conjectures pull request flipping this from open to solved is still open rather than merged, so the canonical repository has not yet accepted it.
Must every connected graph satisfy the proposed upper bound on its independence number in terms of residue and largest induced-bipartite-subgraph order? The family K2r+1∨(Kr⊔Kr) violates it for every r≥3.
The directed five-dimensional torus D5(m) has a Hamilton decomposition for every odd m≥3, extending the decomposition program for directed tori beyond the three-dimensional case.
For a semistable one-parameter family of complex projective varieties with smooth nearby fiber Xt and monodromy T, is the map H1(X,Z)→H1(Xt,Z)T surjective? True in degree one, although the integral statement fails in higher degree.
If a finite graph has girth at least five, must its minimum dual degree satisfy δ∗(G)≤−∂n(G), where ∂n(G) is the smallest eigenvalue of its distance matrix? The Hoffman-Singleton graph violates it: dual degree 7 against eigenvalue bound 4.
For a finite connected graph G, let Ls(G) be the maximum number of leaves in a spanning tree and ℓ(G) the average local independence number. Must Ls(G)≥2(ℓ(G)−1)?
VibeMathed records this result as “Written on the Wall II, Graph Conjecture 217.” The registry entry and named primary source contain the available statement and scope.