EulerSolve Research Papers

An Exact Fixed-Parameter Algorithm for Extremal Unions of Residue Classes

2026-09-02T00:00:00+03:00

Let A = {n₁ < ⋯ < nᵣ} be a finite set of positive integers. Choose one residue class aᵢ mod nᵢ for each modulus and maximize the natural density of their union. This is the unsettled maximum-density half of Erdős Problem 278. We give an exact uniform characterization whose state space depends on r, not on the magnitudes of the moduli. To a residue tuple we attach the graph in which vertices i and j are adjacent precisely when aᵢ ≡ aⱼ (mod gcd(nᵢ,nⱼ)). Inclusion–exclusion makes the covered density a clique-weighted function of this graph. For every forced edge set, a finite-abelian-group kernel calculation counts compatible tuples by a Smith-normal-form lattice index. Boolean Möbius inversion then counts the tuples with each exact graph. Maximizing over the graphs with positive count gives the exact extremal density. The resulting factoring-free algorithm uses 2^{O(r²)} poly(B) bit operations, where B is the binary input length. We also give an independent prime-power layer formula for the kernel counts, a reduction to gcd kernels, and a factorization over the connected components of the non-coprimality graph. The construction is compatible with known arithmetic-coloring and abelian-arrangement machinery; the contribution is its exact-stratum composition with the Erdős–Graham density objective.

Complex Sectional Curvature Blow-Up under Circle Cheeger Collapse

2026-09-02T00:00:00+03:00

An AIM problem asks whether complex sectional curvatures remain uniformly bounded below in a collapsing circle Cheeger deformation. At a fixed point whose normal circle representation contains rotation blocks of speeds a,b > 0, we derive an exact fixed-point formula. A totally isotropic complex two-plane has K_C(g_ε) = K_C(g) − ab/ε². Thus every fixed component of real codimension at least four forces complex sectional curvature to diverge to −∞. For the standard weight-(1,1) action on the unit round S⁴, the value at either fixed pole is 1 − ε⁻². With only one rotation block, the singular curvature operator is instead rank one and positive semidefinite. The AIM workshop report records the corresponding negative curvature-operator phenomenon but does not identify a complex decomposable direction; the observation here is that its two real negative directions combine into a decomposable totally isotropic bivector. The note is unrefereed and makes no priority claim for this elementary calculation.

A Four-Point Counterexample to Excision for Directed Cubical Homology of Closure Spaces

2026-09-02T00:00:00+03:00

Bubenik and Milićević defined cubical singular homology theories of Čech closure spaces from an interval and either the product or inductive product. Excision is known for the product theories and was left open for the three inductive theories. We give a four-point counterexample for the directed interval J₊. For X = J₊ ⊡ J₊, an interior cover {U,V} has H_1^{(J₊,⊡)}(V,U ∩ V; ℤ) ≅ ℤ, generated by the difference of the two coordinate edges. Under inclusion into (X,U), this class is the boundary of the identity square. Hence the excision map is not injective. The calculation is integral, exact, and accompanied by two independently implemented finite-chain checks. This is an unrefereed note and makes no absolute priority claim.

Maximal Generic Iterated Galois Images Do Not Determine p-Adic Julia Sets

2026-09-02T00:00:00+03:00

An AIM problem asks whether the arithmetic or geometric Galois group of the generic iterated-preimage tower of a rational map over a p-adic field determines its Julia set. We give a uniform negative answer. For every prime p, the quadratic polynomials z² + 1 and z² − p⁻⁶ over ℚ_p have arithmetic and geometric generic iterated Galois images equal to the full group Aut(T₂). The first map has good reduction: its Berkovich Julia set is the Gauss point and its classical Julia set is empty. The second has a type-I Cantor Julia set; moreover, its two contracting inverse branches are defined over ℚ_p, so the classical Cantor set already lies in P¹(ℚ_p). Thus even the full rooted-tree action and the marked normal pair of arithmetic and geometric images fail to determine the cardinality or topological type of the Julia set. The result combines Pink's maximality theorem with a base-field inverse-branch construction. This note is unrefereed and makes no absolute priority claim.

Power-Saving Lower Bounds for Additive Bases of Polynomial Sequences

2026-09-02T00:00:00+03:00

Let h ≥ 2 be fixed, let f be a natural-valued polynomial of degree d ≥ 2, and suppose that A ⊆ ℕ satisfies f(ℕ) ⊆ hA. We prove |A ∩ [0,X]| ≥ X^{4/(3hd)−o(1)} when h is even, and |A ∩ [0,X]| ≥ X^{4/((3h+1)d)−o(1)} when h is odd. Both exponents are strictly larger than the elementary exponent 1/(hd). This gives an affirmative qualitative answer, for every fixed h ≥ 2, to Problem 2.8 from the 2004 AIM problem list on recent trends in additive combinatorics. The argument combines a self-contained graph-path form of an Erdős–Newman two-basis estimate with a divisor bound for repeated differences of polynomial values. For odd h, one summand is first frozen on a large fiber and the remaining summands are then split into two equal blocks. No optimality of the displayed exponents is asserted.

Ramification Portraits of Rigid Lattès Maps

2026-09-02T00:00:00+03:00

We classify the abstract weighted ramification portraits of rigid complex Lattès maps. For a Lattès map induced by an affine torus endomorphism of degree d, the complete portrait is determined by its action φ on the finite branch-value set and by a uniform fiber formula. Reducing the affine map on the four possible Euclidean orbifolds yields nine affine functional graphs for signature (2,2,2,2), four graphs for (3,3,3), four Gaussian parity cases for (2,4,4), and four Eisenstein divisibility cases for (2,3,6). We give the critical-leaf multiplicities in every case and prove realizability. We also answer the overlap question from AIM Problem 6.4: every flexible Lattès portrait, in every possible square degree, occurs for a rigid Lattès map of the same degree. The result classifies weighted directed graphs; it does not classify maps up to Möbius conjugacy or retain cross-ratios and multiplier data.

A Compactness Obstruction to Linear Growth Along Null Geodesics

2026-09-02T00:00:00+03:00

Let (M,g) be a compact semi-Riemannian manifold of indefinite signature whose null geodesics are complete. We prove that no C¹ one-form η can have the property that, along every nonconstant affinely parametrized null geodesic γ: ℝ → M, the function η(γ̇) is affine with nonzero slope. This gives a negative answer to Question 9.2.1 in the Burns–Matveev list of open problems about geodesics. The proof normalizes the null cone by an auxiliary Riemannian metric. Compactness then gives a uniform lower bound for the quadratic form (∇_vη)(v) on normalized null vectors, while homogeneity forces the speed of any fixed null geodesic to remain bounded. Compactness also bounds the norm of η, contradicting the asserted nonzero linear growth. Only null completeness, rather than full geodesic completeness, is used.

Lacunary Counterexamples to a Distinct-Summand Freiman Container Problem

2026-09-02T00:00:00+03:00

Fix K > 1. We construct finite nonempty sets A,B ⊆ ℤ with |A| > |B| and |A+B| < K|A| for which B is not contained in any generalized arithmetic progression of bounded rank and size O_K(|A|). More strongly, for every prescribed rank bound d and size constant C, one counterexample defeats all progressions of rank at most d and cardinality at most C|A|. The construction is B = {0,1,N,…,N^{r−1}} and A = kB. Uniqueness of base-N digits gives exact polynomial growth |sB| = (s+r choose r) for s < N, whereas every rank-d generalized arithmetic progression has h-fold growth at most h^d. This answers Problem 2.5 from the 2004 AIM list on recent trends in additive combinatorics in the negative for every K > 1.

Parallel Nilpotent Endomorphisms Without Parallel Null Vectors

2026-09-02T00:00:00+03:00

We construct a closed complete flat pseudo-Riemannian manifold of dimension 16 and signature (8,8) carrying a nonzero parallel self-adjoint endomorphism N with N² = 0, although neither the manifold nor any connected double cover admits a nonzero parallel vector field. This gives a negative answer, as stated, to Question 12.0.3 in the Burns–Matveev list of open problems about geodesics. The construction starts with a compact flat Riemannian manifold whose holonomy has odd order and zero fixed space, doubles its Euclidean representation, equips the double with the split metric, and sets N(u,v) = (0,u). Odd-order holonomy survives every index-two subgroup. An explicit input is the free (ℤ/3ℤ)² quotient of an Eisenstein four-torus constructed by Bauer and Gleissner. We also show that when the top image of a parallel nilpotent has rank one, the manifold or its orientation double cover does carry a parallel null vector; in particular, the original question is affirmative in Lorentzian and co-Lorentzian signature.

A Positive-Rank Elliptic Curve with No Dense Prime

2026-09-02T00:00:00+03:00

An AIM problem asks whether every positive-rank elliptic curve over ℚ has a prime p for which its rational points are dense in its p-adic points. We give a negative answer. For E: y² = x³ − 1516563 and P = (6403/9, 511280/27), an unconditional full 2-descent and saturation certify E(ℚ) = ℤP. A rational 3-isogeny places the reduction of every rational point in a subgroup of index 3 at every good prime, while separate exact arguments exclude density at the three bad primes 2, 3, and 79. Thus the dense-prime set is empty. We also record an exact finite local criterion: density is equivalent to surjectivity on a fixed finite quotient of the Néron filtration, and, at a good odd prime, on 𝓔(ℤ/p²ℤ).

Forcing Absoluteness of Minimal and Maximal C∗-Tensor Products

2026-09-02T00:00:00+03:00

We answer an AIM question about whether the minimal or maximal C*-tensor product commutes with forcing. We use the standard convention that a ground-model C*-algebra is replaced in the extension by the metric completion of its old normed *-algebra. Under this convention, both ⊗_min and ⊗_max commute canonically with every set-forcing extension, for arbitrary ground-model factors; no separability, density-character, or cardinal-preservation hypothesis is needed. The minimal case follows by extension of faithful spatial representations. For the maximal case, any putative larger new norm would give finitely satisfiable old semialgebraic C*-seminorm constraints. Quantifier elimination for real closed fields and compactness inside the ground model then produce an old C*-seminorm exceeding the old universal norm, a contradiction.

A Free-Uniform-Spanning-Forest Proof of Finite-Index Multiplicativity

2026-09-02T00:00:00+03:00

An explicit question of Abért asks for a free-spanning-forest proof that the first L²-Betti number is multiplicative under passage to a finite-index subgroup. We give such a proof using the free uniform spanning forest (FUSF). For H ≤ Γ of index k, lift a spanning tree of the finite Schreier quotient to a forest W in a labeled Cayley multigraph G, and contract its finite components to obtain a Cayley multigraph X of H. The closed finite-cycle space of G is the graph of a bounded H-equivariant operator over that of X. Equivariant dimensions therefore give the FUSF intensity identity I_H(G) = (k−1) + I_H(X). Combining this with the FUSF formula I_K = 1 + β₁⁽²⁾(K) yields β₁⁽²⁾(H) = kβ₁⁽²⁾(Γ) without importing finite-index multiplicativity of von Neumann dimension.

A Slice–Riesz–Bochner Characterization of Joint Brown Determinant Functions

2026-09-02T00:00:00+03:00

We characterize the functions arising as Fuglede–Kadison determinant transforms of bounded commuting tuples in a fixed II₁ factor. After recovering the log-modulus formula intended by an AIM range question, we show that such functions are exactly the logarithmic potentials of compact probability measures on ℂⁿ. Intrinsically, each complex line must give a normalized subharmonic logarithmic potential with uniformly bounded Riesz support, and the unit-frequency transforms of the slice measures must form a continuous positive-definite function on ℂⁿ. Complex scaling of all slice measures is forced automatically by the single global function; Bochner's theorem then reconstructs the unique joint measure. Every compact probability measure is realized by a commuting normal tuple in any prescribed II₁ factor. In one variable, the representing measure is simply the Riesz measure of one transformed function.

The Spectrum of the Hilbert Matrix on Power-Weighted ℓ² Spaces

2026-09-02T00:00:00+03:00

For α ∈ ℝ, let ℓ²_α be the sequence space with squared norm ∑_{n≥0}|x_n|²(n+1)^α. We determine the spectrum of the classical Hilbert matrix H = ((m+n+1)⁻¹) on these spaces, answering a problem from the 2024 AIM workshop on Riemann–Hilbert problems and Toeplitz matrices. The matrix is bounded exactly when |α| < 1. In that range, its spectrum is the closed lens whose boundary is traced by π/cos(π|α|/2 + iπt), t ∈ ℝ, together with 0. The boundary is the Fredholm essential and continuous spectrum. The lens interior is simple point spectrum when α < 0 and residual spectrum of defect one when α > 0; the interior Fredholm index is −sgn(α). The proof compares the diagonally conjugated matrix, modulo a Hilbert–Schmidt operator, with a half-line Wiener–Hopf operator whose symbol is π/cos(πα/2 − iπt). Hill's complete classification of the latent eigenvectors of the Hilbert matrix then fixes the point and residual parts. This manuscript is unrefereed and makes no absolute priority claim.

Free-by-Cyclic Groups with Unboundedly Many BNS Component Orbits

2026-09-02T00:00:00+03:00

An AIM problem asks for families of free-by-cyclic groups with first Betti number greater than two and many connected components of the Bieri–Neumann–Strebel invariant, even after quotienting by the outer automorphism group. For every m ≥ 2 we construct a linearly growing UPG automorphism of F_{2m+1} whose mapping torus G_m has b₁(G_m) = m+2 and whose BNS invariant is the complement of the m+1 character hyperplanes x+iy = 0 for 0 ≤ i ≤ m. It consequently has exactly 2m+2 connected components. For each component C, we minimize the rank of the kernel of a primitive integral character in C. This minimum is invariant under the full Out(G_m) action. Its m+1 distinct values prove that the components form at least m+1 outer-automorphism orbits. The construction therefore answers the AIM request in the quantitatively unbounded sense. This note is unrefereed and makes no absolute priority claim.

Angle-Data Reconstruction for Triangulated Polyhedral Surfaces and a Genus Deficit

2026-09-02T00:00:00+03:00

An AIM problem asks for a characterization of the face-angle and dihedral-angle data of a triangulated polyhedral surface in ℝ³ and conjectures that the realizable data have dimension E−1 in every genus. We use the realization convention later made explicit by Hempel: a labeled, simplexwise-linear, simplexwise-injective map of a consistently oriented triangulated closed surface, modulo similarities. The combined face-angle and oriented-dihedral map is injective and has a local real-analytic left inverse at every nondegenerate realization. Its actual image therefore has dimension 3V−7 = E−1−6g. Thus the AIM formula is correct for the sphere and fails in every positive genus. The embedded Császár torus gives the concrete count 14 rather than 20. We also give a finite necessary-and-sufficient realizability test: intrinsic sine-law compatibility followed by vertex and non-tree-hinge closure in a dual-spanning-tree development. At generic realizations, Fogelsanger rigidity shows that face angles alone already have rank E−1−6g, identifying the missing 6g as global extrinsic closure codimension. This note is unrefereed and makes no absolute priority claim.

Variable Critical Exponents on a Fixed Free-Deck Regular Cover

2026-09-02T00:00:00+03:00

An AIM problem asks for an infinitely generated Fuchsian group of the first kind whose critical exponent is nonconstant on its quasiconformal Teichmüller space. We give an explicit affirmative construction. For a closed surface S_g, let K be the kernel of the epimorphism π₁(S_g) → F_g that kills a cut system and maps its dual curves to a free basis. For every marked hyperbolic metric m, the group Γ_m = ρ_m(K) is infinitely generated and of the first kind. Its free deck group is nonamenable, so Brooks's theorem gives δ(Γ_m) < 1. When all curves of the killed cut system are pinched to length ℓ, a compactly supported cutoff in one lifted cell gives λ₀(ℍ²/Γ_m) ≤ [g sinh(1)/(π(g−1))]ℓ and 1−δ(Γ_m) ≤ [2g sinh(1)/(π(g−1))]ℓ. Thus the exponents tend to one while remaining strictly below one at every finite metric. All structures lie in the same quasiconformal Teichmüller space. This note is unrefereed and makes no absolute priority claim.

Adjoining a Haar-Generic Matrix to a Parabolic-Free Subgroup of SL₂(Qₚ)

2026-09-02T00:00:00+03:00

Let Γ be a countable subgroup of SL₂(ℚ_p) containing no noncentral element of trace 2 or −2. We prove that, outside a Haar-null set of g ∈ SL₂(ℚ_p), the generated group ⟨Γ,g⟩ has the same property. Thus adjoining one random element preserves the absence of parabolics almost surely, answering Question 4 in Miklós Abért's 2010 list. The key elementary lemma classifies the problematic one-variable generalized words: if all constants lie in a parabolic-free subgroup of SL₂ and the trace of the word is identically 2 or −2, then the word map is identically I or −I. The proof evaluates the word on I+tN over the nilpotent cone. Its two highest coefficients force a central endpoint product and cancellation of the outer exponents, reducing the word by conjugation and induction. This manuscript is unrefereed and makes no absolute priority claim.

Maximal Finite-Set Stabilizers in Thompson's Group T

2026-09-05T00:00:00+03:00

Let D = Z[1/2]/Z be the dyadic circle and let T be Thompson's orientation-preserving circle group. We prove that, for every nonempty finite set A contained in D, its setwise stabilizer is a maximal proper subgroup of countably infinite index in T. If |A| = k, then Stab_T(A) is isomorphic to F^k semidirect C_k, equivalently the regular wreath product F wr C_k, where the cyclic group permutes the factors. Stabilizers with the same cardinality are conjugate, whereas those with distinct cardinalities are pairwise nonisomorphic. The proof establishes a general circular-order criterion: a group with the finite circular extension property acts primitively on the k-element subsets for every k. This yields a countably infinite family of maximal infinite-index subgroups and supplies examples requested in the 2024 AIM problem list on groups of dynamical origin. The theorem is apparently unrecorded in the literature checked; no absolute priority claim is made.

Nonuniformity of Free-Gradient Heat Semigroups under Finite Fisher Information

2026-09-05T00:00:00+03:00

Let X_1,...,X_m be a nonempty finite bounded selfadjoint tuple with finite joint nonmicrostates free Fisher information, and let Delta be the generator of the closed polynomial free-gradient form. We prove that its heat semigroup cannot converge uniformly to the identity in L2 on the operator-norm unit ball of the generated von Neumann algebra. For each tuple we obtain a strictly positive lower bound valid at every positive time, with no restriction excluding one or two variables. The witnesses are coordinate exponentials. A direct proof of the classical marginal L3-density implication, the conjugate-variable adjoint formula, and a one-sided Fejer/Riesz estimate control their generator norm, while their energy grows linearly. Resolvent duality yields the nonuniformity bound. Existing rigidity results then imply non-L2-rigidity and, for at least two variables, primeness. The result addresses the 2006 AIM Free Analysis uniformity question. This preprint is unrefereed and makes no absolute priority claim.

A Seventeen-Dimensional Component of the Hilbert Scheme of Eighteen Points on an Integral Curve

2026-09-05T00:00:00+03:00

We give an explicit integral projective curve whose Hilbert scheme of eighteen points has a rational component of dimension seventeen. This answers affirmatively Problem 20 in the 2010 AIM workshop list Components of Hilbert Schemes, even with the curve required to be integral and projective. The curve comes from the known numerical semigroup ⟨13,14,15,16,17,18,21,23⟩ of Herzog–Kumashiro–Stamate. A five-generator ideal of colength eighteen is a canonical module; its full Hilbert tangent space is the seventeen-dimensional normalization quotient. A flat family of truncated unit translates identifies an open subset of the Hilbert scheme with affine 17-space. Two independent finite syzygy certificates verify the tangent calculation over every field. In characteristic zero, a separate application of Kass's theorem gives a component of dimension d−1 for every d≥18 on the same curve. The semigroup, canonical-module identities and general moduli theorem are established prior work; the point is their explicit Hilbert-scheme application. No absolute priority or minimal-length claim is made.

A Generically Nonreduced Component for Hilbert Function (1,4,10,10)

2026-09-05T00:00:00+03:00

We give a computer-assisted proof that the very-compressed locus with Hilbert function (1,4,10,10) is the reduced support of a generically nonreduced irreducible component of the Hilbert scheme of 25 points on affine four-space over an algebraically closed field of characteristic zero. The support is A^4 times Gr(10,20) and has dimension 104. This supplies the case a=10 omitted from Jelisiejew's published theorem for (1,4,10,a), a=6,7,8,9, and identifies the generic component throughout the interval in AIM Problem 31. The new computation gives 244 primary-obstruction quadrics in 46 normal variables over F_3. Exact F4 rounds produce a positive pure leading power of every variable, proving that the normal obstruction algebra is zero-dimensional. A properness argument transfers this certificate to characteristic zero, where the published Bialynicki-Birula criterion applies. The general obstruction framework and the four earlier cases are established prior work. We do not compute the generic nilpotent local algebra or claim absolute priority.

Smooth Hypersurfaces Beyond the Chevalley-Warning Range over Prime Fields

2026-09-06T00:00:00+03:00

For every prime p, integer n >= 2, and degree d >= n+1, we construct a geometrically smooth hypersurface X in projective n-space over F_p of degree d such that #X(F_p) is not congruent to 1 modulo p. The construction stays over the specified prime field in every characteristic and degree. A finite-field moment polynomial supplies nonzero coefficients with the required point count. In odd characteristic not dividing the degree, a sharper individual-degree bound permits simultaneous avoidance of the singular parameter. When the characteristic divides the degree, positive exponent compositions make a triangular family smooth for all nonzero coefficients, apart from one explicitly handled boundary case. A separate uniform formula treats characteristic two in odd degree. The coefficient selection is a finite deterministic procedure; no efficient complexity bound is asserted.

Quantum Symmetric Algebras on Multiple Standard Copies: Structure and Categorical Non-Equivalence

2026-09-07T00:00:00+03:00

Let E be the standard object in the generic polynomial Hecke category over C(q), and let U be an ordinary finite-dimensional multiplicity space. We determine every homogeneous component of the Berenstein-Zwicknagl quantum symmetric algebra A = S_sigma(E tensor U): in degree n >= 2, only the one-row and one-column representations survive, with multiplicities Sym^n U and exterior^n U. We identify A as a fiber product of two diagonal algebras over its square-zero degree-one truncation. For dim U >= 2, its full category of internal polynomial right modules is not equivalent to the corresponding classical module category, even as an abstract abelian category. The obstruction intrinsically identifies two simple objects whose projective covers have a zero Hom space quantumly and a nonzero Hom space classically. The proof uses an explicit two-dimensional Hecke braid defect.

An Explicit Rational Homotopy from the Faran Map to a Linear Map in Target Dimension Four

2026-09-08T00:00:00+03:00

We construct a homotopy through proper rational holomorphic maps from the unit ball in C^2 to the unit ball in C^4, joining (z,w) -> (z^3,sqrt(3)zw,w^3,0) to (z,w) -> (z,w,0,0). The construction answers the explicit target-four question in the AIM list on the Cauchy-Riemann equations. A mixed-term polynomial family of coefficient rank four joins a unitary transform of the Faran map to a partially tensored quadratic map. An explicit rational segment then degenerates to a Whitney map, which is joined to the linear embedding. The full family is jointly continuous on the closed source ball, and each slice extends holomorphically past that ball. All slices have rational degree at most three. We prove a uniform estimate at the degenerating endpoint and explain why this particular path is not continuous in the C1 boundary topology. No classification of arbitrary ball-map homotopies is asserted.

A Semi-Abelian Group of Order 768 That Is Not an M-Group

2026-09-27T00:00:00+03:00

M. Kida conjectured that every finite semi-abelian group is monomial (J. Group Theory, 2025; Kourovka Notebook, Problem 21.68). We show that the conjecture is false. If a finite group W acts on a finite abelian group N, a linear character of N has stabiliser T in W, and T has an irreducible character that is not monomial, then N ⋊ W is not an M-group. We apply this to the augmentation submodule of the permutation module F₂[W/T], where W = E ⋊ A₄ is the index-two subgroup of C₂ ≀ A₄ and T ≅ SL(2,3) is the binary tetrahedral group acting on the eight quaternion units. The result is a semi-abelian group of order 768 = 2⁸·3 with a non-monomial irreducible character of degree 8. The proof uses only Clifford theory. The example was also checked by exact computation, including every subgroup of index 8. This is an unrefereed note; minimality of the order is not claimed.

Dirac Graphs Without a Spanning Near-Square of an Odd Cycle: A Negative Answer to a Question of Heinig

2026-09-28T00:00:00+03:00

At the 2014 Oberwolfach workshop on combinatorics, P. Heinig asked whether, for every odd n ≥ 7, every n-vertex graph with minimum degree at least ⌈n/2⌉ contains a spanning copy of the graph obtained from the square of an n-cycle by deleting every other edge on the periphery until exactly three consecutive vertices of degree 4 remain. A positive answer would have given a structural reason why the Hamilton cycles of such graphs generate their cycle space. We show that the answer is negative for every odd n ≥ 7, under both natural readings of “periphery”, and for n = 9 under every reading. The counterexample is the complete bipartite graph K_{(n+1)/2,(n−1)/2} with a perfect matching, or a matching and one path with two edges, added inside the larger side; for n = 7 it is the graph that Heinig himself used as a positive example for the cycle-space question. The proof classifies the independent sets of size (n−1)/2 in Heinig’s graph. The cycle-space question itself has since been settled for all large odd n by Hou and Yin, and it is not affected. This is an unrefereed note.

Explicit Closed Elementary Nets That Cannot Be Completed, Including Fields of Transcendence Degree One

2026-09-28T00:00:00+03:00

An elementary net (carpet) of order n over a field K is closed if its elementary net group contains no new elementary transvections, and it is completable if its diagonal can be supplemented to a full net. Completable nets are closed. Koibaev gave closed nets that are not completable over fields of characteristic 0 and 2 and asked for such nets in odd characteristic (Kourovka Notebook, Problem 21.76). Nuzhin (2026) has answered this question, with examples in every characteristic, using closures of finitely generated subgroups over a rational function field in two variables. We give explicit examples with short proofs. An elementary lifting lemma turns a closed pair of additive subgroups of a quotient ring R/J into a closed elementary net of every order n ≥ 3. With F[x,y] → F[x] and a degree argument it gives, for every field F, the net with Fx + yF[x,y] in positions (1,2) and (2,1) and yF[x,y] elsewhere. Lifting the two exceptional pairs of Levchuk’s refinement of Dickson’s theorem along F₃[t] → F₉ and F₂[t] → F₄ gives closed nets over F₃(t) and F₂(t) that are not completable. These one-variable nets also satisfy the hypotheses of Kourovka Problem 19.48. Together with a theorem of Koibaev and Nuzhin on algebraic extensions, it follows that in characteristics 2 and 3 a field carries such nets if and only if it is not algebraic over its prime field. For p ≥ 5 we do not know whether examples exist over F_p(t); by Levchuk’s theorem, lifting from a finite field cannot produce them. This is an unrefereed note.

A Counterexample to a Conjecture of Sportiello on Coloured Permutations in Convex Shapes

2026-09-28T00:00:00+03:00

At the 2018 Oberwolfach workshop on enumerative combinatorics, A. Sportiello compared two families of objects attached to a digitally convex shape λ (a convex polyomino): the permutations whose graph lies in λ and that avoid a 123-pattern whose two corner cells also lie in λ, counted by A_λ, and the red–blue coloured permutations in λ that avoid two coloured versions of the 12-pattern, counted by B_λ. For the full square these numbers are the Catalan number and the central binomial coefficient, and he conjectured that B_λ ≥ A_λ for every digitally convex shape. We show that the conjecture is false. For the band λ = {(x,y) ∈ [7]² : −3 ≤ y−x ≤ 4} one has A_λ = 536 and B_λ = 515. An exhaustive computation shows that every digitally convex shape of side at most 6 satisfies the inequality, and that among the 5,693,968 shapes of side 7 exactly this band and its transpose violate it. For bands of side 10 the ratio B_λ/A_λ drops to about 0.18. This is an unrefereed note.

Small Counterexamples to the All-n Form of the Exact Codegree Conjecture for Tight Hamiltonian Cycles

2026-09-28T00:00:00+03:00

Rödl, Ruciński and Szemerédi stated the following conjecture, which they attribute to Katona and Kierstead: every k-uniform hypergraph on n ≥ k+1 ≥ 4 vertices in which every (k−1)-set lies in at least ⌊(n−k+3)/2⌋ edges has a tight Hamiltonian cycle. They proved it for k = 3 and all sufficiently large n. We observe that the statement, as written for all n ≥ k+1, fails for small n. The smallest counterexample is a 3-graph on 7 vertices: an apex joined to all pairs of a 6-set, together with the ten faces of the hemi-icosahedron on that set. It has minimum codegree 3 = ⌊7/2⌋ but no tight Hamiltonian cycle, and the proof is two lines. A variant of the construction gives counterexamples for k = 4 and k = 5 on k+4 vertices. A computer search finds two more, on 9 vertices for k = 3 and on 10 vertices for k = 5; the second meets the bound (n−k+3)/2 even without the floor. These are exceptions for small n only, and the asymptotic statement is not affected. This is an unrefereed note.

Differences of Signed Representatives: Sharp Answers to Two Questions of Bernert and Arala Santos

2026-09-28T00:00:00+03:00

Let A ⊂ {−n,…,n}∖{0} contain exactly one of k and −k for every 1 ≤ k ≤ n. At the 2025 Oberwolfach workshop on analytic number theory, C. Bernert and N. Arala Santos asked two questions about such sets. First, must A−A contain (1−o(1))n elements of {1,…,n}? Second, for every B ⊆ {1,…,n} with |B| ≥ εn, must the number of pairs (a,b) ∈ A² with a−b ∈ B be ≫_ε n²? We answer both questions affirmatively, with sharp bounds. At most two elements of {1,…,n} are missing from A−A. For every B ⊆ {1,…,n}, the number of pairs (a,b) ∈ A² with a−b ∈ B is at least ⌊(|B|−1)²/4⌋. Both bounds are attained by sets of the form {1,…,a} ∪ {−(a+1),…,−n}, the second for all |B| ≤ ⌊2n/3⌋+1. The proof is a short double-counting argument. This is an unrefereed note.

Sharp Computability Bounds for Nonforking Sections

2026-09-28T00:00:00+03:00

We study computability of sections of the restriction map S_1(N) to S_1(M) for countable stable models M elementary in N. Types are represented by characteristic functions on formulas with named parameters, and the two elementary diagrams are separately decidable. Classical stable definability gives a continuous section computable from the input type together with the fixed oracle 0'. This bound is sharp, already for the decidable, omega-stable, omega-categorical theory of infinitely many infinite equivalence classes. For a fixed decidable-range elementary inclusion in this theory, an oracle X computes a section exactly when it computes the set of classes of N that meet M. Explicit inclusions realize every computably enumerable degree as this least auxiliary degree. Thus a computable section need not exist, although every individual type in the example is computable. In contrast, a decidable full diagram of the predicate expansion (N,M) gives a computable section for every stable theory. The result addresses a Type-2 formulation of an AIM question whose printed statement leaves the effective presentation unspecified.

Weighted Root Deletions and Coefficientwise Toeplitz Positivity

2026-09-28T00:00:00+03:00

For independent indeterminates a, x_1,...,x_N, y_1,...,y_N, we prove that the sequence b_k = a e_k(X) + sum_i y_i e_k(X without x_i) is coefficientwise totally nonnegative: every minor of its upper Toeplitz matrix has nonnegative integer coefficients. In particular, this holds for the coefficients of (u D_z + v) product_i(1+x_i z), coefficientwise in u,v,X. The proof realizes the sequence as sums of bordered principal minors of a star-shaped Gram pencil. A maximal-weight compression in a Schur module expresses the necessary Schur-complement characters as traces against orthogonal projections. An additional letter-content grading separates the independent root weights and yields explicit squared-norm coefficient certificates. This resolves the derivative-plus-constant branch of an AIM total-positivity question, not its other operator conjectures. The note is unrefereed and makes no absolute priority claim.

A Six-Vertex Counterexample to Vertex-Averaged Ordered Reachability

2026-09-28T00:00:00+03:00

We give two loopless directed graph layers on six vertices, each with constant outdegree two, having no rainbow directed cycle and hence no increasing rainbow cycle. For the fixed label order 1<2, the sizes of the sets reachable by possibly trivial increasing paths are 5,5,5,5,5,4. Their average is 29/6<5=1+delta_1+delta_2. Thus the uniform-start-vertex interpretation of the ordered-reachability conjecture attributed to DeVos in Sullivan's 2006 survey is false. Uniform blow-ups give examples on 6m vertices with average 1+23m/6, below the proposed bound 1+4m for every m>=1. The proof is an explicit neighborhood calculation and does not depend on the numerical search that located the example.

A Degree-24 Obstruction to Hopf Structures on Symmetric-Group Supercharacter Spaces

2026-09-28T00:00:00+03:00

Consider the supercharacter theory of S_n whose superclasses are the S_n-conjugacy classes contained in A_n, together with the single block S_n-A_n. We prove that its superclass-function spaces, summed over n with one-dimensional components in degrees zero and one, admit no connected graded Hopf algebra structure over a field of characteristic zero. The argument uses a known nonnegative Euler-product condition for Hopf Hilbert series. The first negative Euler exponent occurs in degree 24 and equals -1: lower degrees force 795 basis monomials, but the required dimension is 794. The two coarser natural supercharacter families fail the same test in degree 4. These obstructions exclude arbitrary graded Hopf operations, not only quotients of symmetric functions. We state the families explicitly and do not claim a classification of all symmetric-group supercharacter theories.

Strict Likelihood Multimodality and Septic Equations for Two-Component Birkhoff Models

2026-09-28T00:00:00+03:00

We give a positive integer dataset on S_4 whose likelihood on the closed two-component Birkhoff model has at least three distinct strict local maxima in distribution space. The dataset has 2013 observations and all three probability vectors have full support. One component of each displayed mode is a boundary component. An exact nonnegative-decomposition certificate and a compactness argument rule out the possibility that the modes are artifacts of a parameter chart or component-label switching. Separately, for every n >= 4, unequal positive parity-constant counts give at least three distinct global maximizers in the closed two-component model. We also determine the first nonzero homogeneous equation degree of the complex S_4 secant variety: it is seven. A 96-term septic and its 24 symmetry images supply local equations at the displayed smooth point. We do not determine the global defining ideal or assert modes with both component matrices strictly positive.

Polynomial Rigidity and a Sharp Degree Bound for Free Quadratic Variation

2026-09-28T00:00:00+03:00

We isolate an elementary highest-degree obstruction to polynomial changes of variables preserving unit free-Brownian diffusion. For every unital moment functional and every nonconstant noncommutative polynomial of degree d, its free quadratic-variation polynomial has degree exactly 2d-2. The leading coefficients form a positive Gram matrix of last-letter derivatives. At any algebraically free self-adjoint tuple, this classifies self-adjoint polynomial unit-covariance maps as affine coisometries. For a standard semicircular m-tuple, the distance of the covariance from the scalars is at least 1/sqrt(m) times the squared norm of the polynomial's highest Wick component. The constant is sharp, and for quadratic polynomials this is a sharp distance-to-affine estimate. The stochastic application concerns a prescribed pathwise transformation in the same filtration; it does not resolve the more general terminal-law steering question in the AIM Free Analysis problem list.

Compact-Unit Types and Fixed Vectors for Symplectic Similitudes

2026-09-28T00:00:00+03:00

For every prime p and every rank r >= 1, let H_r be the compact subgroup of GSp_{2r}(Q_p) consisting of diag(I_r,aI_r) for p-adic units a. We give an elementary proof that every smooth character of H_r occurs in every irreducible infinite-dimensional smooth complex representation of this group. In particular, such a representation has a nonzero vector fixed by the subgroup congruent to H_r modulo p^m for some m. This answers the varying-level fixed-vector question recorded as Problem 3.2 of the AIM automorphic-forms problem list. The proof isolates a nontrivial character on a compact additive subgroup and enlarges that subgroup until its character stabilizer is small enough to make compact-unit averaging nonzero. One long-root subgroup and elementary symplectic transvections then suffice in every rank. The rank-two result and averaging mechanism are due to Roberts and Schmidt; the argument here requires no genericity, Bessel model or twisted-Jacquet nonvanishing theorem, and includes p=2.

Explicit Stationary Comparisons for Join-the-Shortest-Queue Approximations

2026-09-28T00:00:00+03:00

We compare the stationary occupancy law of a finite join-the-shortest-queue system with its finite-buffer and sampled-routing approximations. For N unit-rate exponential servers, Poisson arrival rate 0

Sharp Zero Regions and Hurwitz Criteria for BMV Trace Polynomials

2026-09-28T00:00:00+03:00

For positive-definite Hermitian matrices A,B, we study the zeros of P_m(z)=Tr((A+zB)^m). Loewner bounds lB<=A<=uB give a sharp dimension-independent zero region: the union of two explicit closed disks. Every point of this region is attained by a commuting two-dimensional pair, while a nonreal boundary zero forces an endpoint block decomposition and commutativity. Optimizing the associated angle gives the optimal universal Hurwitz guarantee u/l=3. For arbitrary two-dimensional pairs, including noncommuting ones, a quadratic factorization yields simplicity, an exact stability criterion in terms of Tr(AB)/(Tr A Tr B), and a count of right-half-plane zeros. We also classify real zeros in every dimension. These are explicit zero-location results motivated by an open-ended AIM problem, not a new proof of the BMV coefficient theorem or a classification for every fixed higher-dimensional pair.

Sharp Induced-Norm Paving for Symmetric Weighing Matrices

2026-09-29T00:00:00+03:00

For real symmetric zero-diagonal weighing matrices W with W^2=dI, we deduce induced-l_p epsilon-paving into at most ceil(36 epsilon^(-q)) parts, where q=min(p,p') and q=1 at the endpoints. The exponent q is optimal uniformly over symmetric conference matrices. The elementary compression estimate ||W[S]||_p <= ||W[S]||_2^(2/q) transfers the published Ravichandran--Srivastava Hilbert multi-paving theorem; a single spectral paving also gives quantitative bounds for every p simultaneously. The matching lower order follows from classical Paley conference matrices. This short, theorem-dependent note proves a precise special case, not the general arbitrary-matrix induced-norm paving question. It is self-audited and unrefereed; the observation may be folklore and no absolute priority is asserted.

A Diagonal-Jet Criterion for Normalized Gaussian Analytic Covariances

2026-09-29T00:00:00+03:00

Let R be a twice continuously differentiable, positive semidefinite complex kernel with unit diagonal on a connected plane domain. We give a necessary and sufficient condition for R to be the normalized covariance of a centered proper Gaussian analytic function with positive variance everywhere. A nonnegative residual formed from diagonal derivatives must vanish, and a real one-form determined by the first diagonal derivative must be exact. On a simply connected domain the second condition is a local closedness test. A sharp Gram inequality propagates the diagonal residual condition without division by any off-diagonal kernel value. The covariance is reconstructed up to a positive constant. A polynomial example has a zero in every base section, and an annular example separates local conditions from the global period obstruction. This elementary note concerns the full complex normalized covariance, not arbitrary zero-process correlations. It uses classical positive-kernel and Gaussian-series methods and makes no absolute priority claim.

Low-Degree Curves and Smooth Hilbert Loci on the Fermat Fourfold

2026-09-29T00:00:00+03:00

For the degree-d Fermat fourfold in complex projective 5-space, and for delta>=2 with d>10delta-6, we describe the integral degree-delta Hilbert locus as a disjoint union of 15d^3 smooth open plane-form spaces. The point classification applies Salberger's diagonal-curve inequality after a minimal-support descent already present in the frozen research aid. The scheme-level extension follows from a three-term Wronskian lemma: the normal bundle of a standard plane has no sections on any integral degree-delta plane curve when d>delta+3, including singular curves. Reznick's published nonstandard conic on X_14 makes the conic cutoff d>=15 sharp. Combining the published degree and genus inequalities also gives a stronger necessary genus bound for nonstandard curves.

B-Spline Laws and Nonreal Zeros for Weighted-Cycle Internal DLA

2026-09-29T00:00:00+03:00

We give an exact finite-time distribution for single-source internal diffusion limited aggregation driven by an arbitrary positive nearest-neighbour chain on the integers. The probability of each occupied interval is a B-spline value on the harmonic-scale knots of the chain. Lifting a cycle until its penultimate occupation yields a last-site law and a divided-difference formula for arbitrary positive edge weights. For the four-cycle we characterize the full attainable probability region and every reversible realizing resistance vector up to scale. The shifted last-site polynomial has only negative real zeros for every positive nearest-neighbour chain on three or four vertices, but a five-cycle with resistances (10,10,1,1,20) has a nonreal conjugate pair. Thus five is the smallest possible cycle size for this obstruction.

A Negative Answer to Melnikov's Valency-Variety Problem

2026-09-29T00:00:00+03:00

The valency-variety w(G) of a graph G is the number of distinct vertex degrees of G. In a problem recorded by Vizing in 1968 and later listed by Jensen and Toft, Melnikov conjectured that every graph G with n ≥ 2 vertices satisfies χ(G) > ⌈⌊w(G)/2⌋/(n − w(G))⌉. We show that the conjecture is false. The inequality is equivalent to (2χ(G) − 1)(n − w(G)) ≥ n − 1. For every k ≥ 3 we construct explicit connected k-chromatic graphs that attain equality in this form, a k-chromatic counterexample on 14k − 5 vertices (it has one isolated vertex), and a connected k-chromatic counterexample on 22k − 9 vertices. A blow-up construction gives connected k-chromatic graphs with n − w = 16n/(32k − 13) < n/(2k − 1), so the inequality fails by an amount that grows linearly in n. On the positive side, the inequality holds for all bipartite graphs, and a counting argument based on Turán's theorem shows n − w ≥ μ_k n − O(1) for every K_{k+1}-free graph, where μ_3 = (3 − √5)/4 ≈ 0.191 (Melnikov's bound asks for 1/5, and our constructions give 16/83 ≈ 0.193). The same argument, evaluated exactly by computer and combined with the bipartite case, shows that every graph with at most 36 vertices satisfies Melnikov's inequality. So the smallest counterexample has 37 vertices. For 3 ≤ k ≤ 60, the smallest k-chromatic counterexample has 14k − 5 vertices, and the smallest one without isolated vertices (in particular, the smallest connected one) has 22k − 9 vertices. This is an unrefereed note.

A Negative Answer to Bäumler's Question on the Number of Spin-Glass Ground States

2026-09-29T00:00:00+03:00

Consider the Edwards–Anderson spin glass at zero temperature on an infinite, connected, locally finite graph, with i.i.d. absolutely continuous couplings, and let G(J) be its set of ground states. Bäumler proved that on every locally finite tree, for coupling laws of linear growth, |G(J)| is almost surely 2 or ∞, and that |G(J)| = 2 if and only if simple random walk on the tree is recurrent. He asked whether |G(J)| ∈ {2, ∞} holds for all graphs and all distributions of linear growth. We show that it does not. Join each pair of consecutive integers k, k+1 by m_k internally disjoint paths of length two. For couplings uniform on (−1, 1), the resulting graph has |G(J)| = 4 almost surely if Σ_k m_k^(−1/2) < ∞, and |G(J)| = 2 almost surely otherwise; the first conclusion holds for every absolutely continuous coupling law. We also give a planar graph of maximum degree 4 on which simple random walk is recurrent and |G(J)| = 4 almost surely for couplings uniform on (0, 1), a recurrent example with couplings of both signs, and a recurrent graph on which |G(J)| is a non-degenerate random variable. The last example answers the other half of the question in the form Bäumler posed it in 2019. The examples show that several implications of the tree theorem, between uniqueness, vanishing maximal flow and recurrence, fail on general graphs. The mechanism, a unique cheapest finite domain wall on a two-ended graph, is elementary. The question remains open for bounded-degree graphs with a symmetric coupling law and for quasi-transitive graphs such as ℤ^d. This is an unrefereed note.

When Is the Set of Non-Completable Partial Probability Tensors Convex?

2026-09-29T00:00:00+03:00

In 2017 Kahle, Kubjas, Kummer and Rosen asked whether, for every pattern of observed entries, the set of nonnegative partial tensors that cannot be completed to the joint distribution of independent discrete random variables is convex. The answer is no, and a negative answer is already implicit in examples of Kubjas and Rosen. We determine exactly when the answer is yes. For tensors of format d_1 × ⋯ × d_n with n ≥ 2 and all d_j ≥ 2, the set is convex if and only if no observed entry is pinned, that is, if through every observed entry there is a maximal slice containing no other observed entry. In that case the completable region is the sublevel set {G_E ≤ 0} of an explicit concave function G_E. Otherwise convexity fails, and under the standing assumptions of Kahle et al. it fails even inside the simplex {Σ_e x_e ≤ 1}. For matrices the condition says that every component of the bipartite graph of observed entries is a star. Under these standing assumptions, to which the question refers, the answer is yes for all patterns only in the formats 2 × m and m × 2; under the two conditions stated with the question alone, only in the format 2 × 2. For n ≥ 3 the convex cases under the standing assumptions are exactly the ∏_j d_j "corner" patterns, which include the running example of Kahle et al. For corner patterns we give a one-variable completability criterion, extending the known cubic criterion for 2 × 2 × 2 tensors, and we show that the irreducible boundary hypersurface has degree at most n(n − 1). This is an unrefereed note.

A Proof of the Nichols–Stolz Conjecture on the Almost-Sure Spectrum of the Bernoulli Displacement Model

2026-09-29T00:00:00+03:00

The Bernoulli displacement model is the random Schrödinger operator h_{ω,λ} = h_0 + V_ω on ℓ²(Z) in which each cell of two neighbouring sites carries one single-site potential λ ≠ 0. The potential sits on the left or on the right site of the cell according to an independent Bernoulli variable ω_k. Nichols and Stolz determined the almost-sure spectrum Σ_λ for 0 < |λ| ≤ 2. They conjectured (Oberwolfach Report 55/2009; J. Spectral Theory 1 (2011), Conjecture 6.2) that Σ_λ = σ(h_{ω*,λ}) ∪ σ(h_{ω¹,λ}) for every λ ≠ 0, where the configurations ω* and ω¹ give potentials of period four and two. For |λ| > 2 this means that Σ_λ consists of exactly six bands. We prove the conjecture. In fact, the inclusion σ(h_{ω,λ}) ⊂ σ(h_{ω*,λ}) ∪ σ(h_{ω¹,λ}) = {E ∈ R : |E(E − λ) − 2| ∈ [0, 2] ∪ [|λ|, √(λ² + 4)]} holds for every configuration ω, not only almost surely. The proof uses the reduction of Nichols and Stolz, under which h_{ω,λ}(h_{ω,λ} − λ) − 2 becomes a direct sum of two discrete Schrödinger operators with potentials ±s, where s_k = λ(ω_k − ω_{k+1}). It combines this reduction with an elementary Schur-complement argument. The argument works because s never takes the same nonzero value at two neighbouring sites. This is an unrefereed note.

The Exact Complexity of ε-Dense Steiner Tree

2026-09-29T00:00:00+03:00

In the ε-Dense Steiner Tree problem of Karpinski and Zelikovsky, every terminal is adjacent to at least an ε-fraction of the non-terminals, and a Steiner tree with the fewest edges is sought. For every fixed ε > 0 the problem has a polynomial-time approximation scheme. At an Oberwolfach problem session in 2004, Hauptmann asked for hardness results and noted that it was not even known whether the exact problem is NP-hard; the question was still described as open in 2015 and in 2020. We show that for every fixed ε ∈ (0,1] the problem can be solved exactly in time n^{O(log n/ε)}. The main step is a structural lemma: if H is any set of non-terminals that are all adjacent to terminals, and G[S ∪ H] has r components, then every optimal tree has at most |H| + 2r − 2 Steiner vertices adjacent to terminals. Consequently the problem is not NP-hard, even under Turing reductions, unless NP ⊆ DTIME(2^{O(log² n)}). Conversely, for every fixed ε ∈ (0,1), a reduction from 3-SAT in the style of Megiddo and Vishkin, combined with a dense covering gadget over F_q^d, shows that the problem has no N^{o(log N)}-time algorithm unless the Exponential Time Hypothesis (ETH) fails, and that it is not in P unless FPT = W[2]. Hence, assuming ETH, exact ε-Dense Steiner Tree is neither in P nor NP-hard. Of the two halves of this statement, "not NP-hard" needs only NP ⊄ QP, while "not in P" needs ETH or FPT ≠ W[2]. This is an unrefereed note.

The Critical Window for High-Dimensional Limits of Hyperbolic Poisson k-Plane Processes

2026-09-29T00:00:00+03:00

For 2k > d + 1, the centred and normalized total k-volume of a stationary Poisson process of k-planes in hyperbolic space H^d, observed in a growing ball, converges to a non-Gaussian infinitely divisible law Z_{d,k}. Bühler and Hug asked how the standardized law Z*_{d,k} = Z_{d,k}/(Var Z_{d,k})^{1/2} behaves as d → ∞. Bühler, Hug and Thäle proved that, when k/d → 1/2, Z*_{d,k} tends to the standard Gaussian law if d^{−1}(2k − d − 1)^{d/k} stays asymptotically below eπ and to 0 if it stays above eπ, and they left the critical rate open. We complete the picture. Put m = 2k − d − 1, m_c = (2πe(k − 1))^{1/2} and y = (m − m_c)/m_c^{1/2}. Along every sequence of admissible pairs with d → ∞, the Lévy distance between Z*_{d,k} and the centred Gaussian law with variance Φ(−y/√2) tends to 0, where Φ is the standard normal distribution function. Hence Z*_{d,k} converges in distribution if and only if y converges in [−∞, ∞], every limit law is a centred, possibly degenerate, Gaussian N(0, τ²) with τ² ∈ [0, 1] (where N(0, 0) = δ₀), and every τ² ∈ [0, 1] occurs. At the critical rate k = d/2 + ½(eπd)^{1/2} + O(1) the limit is N(0, 1/2); a shift by c·d^{1/4} gives N(0, Φ(−√2·c/(eπ)^{1/4})). The proof combines an exact Beta representation of the Kolmogorov measure, a reduction lemma for Kolmogorov measures that split between 0 and ∞, an anti-concentration bound, and a central limit theorem for the logarithm of a Beta variable. This is an unrefereed note.

Essential Self-Adjointness of the Laplace–Beltrami Operator for a Family of Non-Regular Almost-Riemannian Structures

2026-09-29T00:00:00+03:00

Oberwolfach Report 47/2019 records the following question, posed in the abstract of L. Rizzi's talk (joint work with V. Franceschi, D. Prandi and M. Seri): is the Laplace–Beltrami operator of the almost-Riemannian structure on R² with orthonormal frame X₁ = ∂ₓ, X₂ = x(x^{2ℓ} + z²)∂_z essentially self-adjoint on the regular region for every ℓ ≥ 1? The case ℓ = 1, and the cases ℓ ≤ n/2 of an n-dimensional version, had been settled by Prandi, Rizzi and Seri in the complete (torus) setting of their Example 7.7; their effective-potential criterion does not apply for larger ℓ. We answer the question affirmatively for every ℓ in the complete setting assumed in the report: the structures X₁ = ∂ₓ, X₂ = c(z) x(x^{2ℓ} + ρ(z))∂_z with c > 0, ρ ≥ 0 and c, cρ bounded, all of which are complete, have an essentially self-adjoint Laplace–Beltrami operator. They include structures that coincide with the model on a strip around the non-regular point; the same holds for the n-dimensional examples of Prandi, Rizzi and Seri for all ℓ and n. We do not prove a localisation theorem for arbitrary complete structures that agree with the model near that point. The proof combines a one-dimensional Hardy inequality obtained with the multiplier δ (the distance from the singular set), whose weight −δΔδ is at least 1 for this family, with a standard Agmon-type argument; the effective potential, in contrast, is not bounded below by 3/(4δ²) when ℓ ≥ 2. The same argument shows that the effective-potential hypothesis in the criteria of Prandi–Rizzi–Seri and Franceschi–Prandi–Rizzi can be replaced by −Δ_ω δ ≥ 1/δ − κ, which is an alternative sufficient condition rather than a strengthening. Taken literally on all of R², the model is incomplete, and its Laplace–Beltrami operator is not essentially self-adjoint for reasons unrelated to the singular set. The more general real-analytic conjecture of the report remains open; we give a real-analytic example without tangency points for which the weak Hardy inequality underlying all these criteria fails. This is an unrefereed note.

Shadowing and Generalized Hyperbolicity for Dissipative Composition Operators without Bounded Distortion

2026-09-29T00:00:00+03:00

Let T_f φ = φ∘f be a dissipative composition operator on L^p(X, B, μ), 1 ≤ p < ∞, induced by a bimeasurable bijection f such that μ∘f and μ∘f⁻¹ are bounded by a multiple of μ. D'Aniello, Darji and Maiuriello proved that if f has bounded distortion, then T_f has the shadowing property if and only if it is generalized hyperbolic, if and only if the masses μ(f^k(W)) of the iterates of a wandering set satisfy one of three growth conditions. They asked, and Maiuriello asked again in Oberwolfach Report 19/2024, whether this remains true without bounded distortion. We show that shadowing and generalized hyperbolicity are equivalent for every dissipative composition operator, for all 1 ≤ p < ∞, for real or complex scalars and without separability assumptions. Both are equivalent to the surjectivity of I − T_f, and to a uniform exponential "tent" condition on the Radon–Nikodym densities d(μ∘f^k)/dμ on the wandering set. The proof represents T_f as a field of weighted shifts and rests on a deterministic lemma about a single fibre. For p = 2, complex scalars and separable L²(μ) the equivalence also follows from a recent theorem of Pituk on separable Hilbert spaces. The characterization by masses fails without bounded distortion in both directions: Bernardes, D'Aniello and Maiuriello recently gave an example satisfying the contraction condition (HC) without shadowing, and we record an elementary hyperbolic example that satisfies none of the three conditions. This is an unrefereed note.

Full Linear Homomesy Spaces for Rowmotion on Rooted Forests

2026-09-29T00:00:00+03:00

For every finite rooted forest with minimal roots, we compute the full spaces of linear order-ideal and antichain statistics that have zero average on every rowmotion orbit. The algorithm retains the orbit through the empty ideal and two affine hulls, each represented by at most n+1 vectors, instead of enumerating the orbits. A deletion lemma for Cartesian products yields scalar recursions for both zero-mesy dimensions and an exact rational basis algorithm using O(n^4) field operations, apart from integer gcd/lcm operations. The two full homomesy dimensions agree; the zero-mesy dimensions can differ by one. A known acyclic-toggle theorem transfers the ideal-indicator result to Coxeter toggle actions.

Exponential Rarity of Multiple Likelihood Modes in Bivariate Seemingly Unrelated Regression

2026-09-29T00:00:00+03:00

The likelihood of the crossed bivariate Gaussian seemingly unrelated regression model can have several local modes. Its almost-sure eventual uniqueness is known. We give an explicit finite-sample bound: for fixed noncollinear regressor vectors of correlation r, with d=1-|r| and k=n-2>=2, the probability of more than one stationary point is at most exp[-k d^3/(16(1+|r|))]+2 exp[-k/16]. The bound is uniform over slopes and all positive-definite error covariances. A determinant normal form gives a directly checkable unimodality certificate, which combines with elementary Gaussian tails. Common regressors are allowed by replacing k with n-p-2 after projection. For Gaussian random predictors of population correlation r0, an explicit bound is 11 exp[-(n-2)(1-|r0|)^3/512]. A change-of-measure argument shows that the probability of multiple modes is exp[-Theta(n)] at each fixed nonsingular random-design parameter. Constants are not claimed optimal. A classical algebraic MANOVA certificate distinguishes the two parts of the motivating AIM question.

A Finite Gap Bound for Local Progression-Free Density

2026-09-29T00:00:00+03:00

Fix s>=3 and consider increasing integer sequences whose every s consecutive terms contain no nontrivial three-term arithmetic progression. We show that every admissible gap sequence can be decreased coordinatewise to an admissible sequence with all gaps at most B_s=1+2 binomial(s,3). Consequently the unrestricted maximum density is the reciprocal of the minimum cycle mean in an explicitly defined finite graph. In particular, it is rational, is computable for each s, and is attained periodically with integer period at most B_s^(s-1). A nonnegative defect derived from this graph characterizes all extremizers, including those with unbounded gaps, for upper, lower, natural and upper Banach density. A short analytic potential proves the sharp value 4/9 for 5<=s<=8.

Logarithmic Equivalence Covers of Powers of Cycles

2026-09-29T00:00:00+03:00

An equivalence graph is a vertex-disjoint union of cliques. We give an explicit cover of every noncomplete cycle power by a logarithmic number of equivalence subgraphs. More precisely, for integers k>=1 and n>=2k+2, ceil(log2(2k+2)) <= eq(C_n^k) <= 4 ceil(log2(k+1))+1. The upper bound partitions the cycle into short clique blocks and uses binary encodings for threshold adjacency between blocks. The lower bound is an application of Alon's multilinear rank method. Consequently the equivalence covering number has order log(k+1) uniformly in n, giving a negative answer to the linear-growth conjecture recorded in Open Problem Garden, including its intended large-n regime. We credit earlier logarithmic co-chain encodings and record a related 2010 conference abstract whose numerical bounds were not available in the located text. No absolute priority claim is made.

Bidirectional Inference Counts on Forests and Cycles

2026-09-29T00:00:00+03:00

For a binary restricted Boltzmann machine on a fixed labeled bipartite graph, consider the pair of conditional maximum-a-posteriori maps in both directions, with the same weights used in the two maps and no ties. We count such pairs exactly on forests and on even cycles. On a forest, local threshold functions can share symmetric weights if and only if their monotonicity directions agree on every input edge essential at both endpoints. Positive rescaling gives a constructive proof and an incidence partition function evaluable by a tree recurrence. For paths, the counts satisfy p_1=2, p_2=14, and p_n=10p_(n-1)+6p_(n-2). For even cycles of length n>=4, the count is (5+sqrt(31))^n+(5-sqrt(31))^n-2^(n+1); the subtraction removes exactly two cyclic families of impossible magnitude comparisons. Two trees with identical degree lists in each bipartition class have different bidirectional counts. This invariant is distinct from the classical one-way inference count, which factors over output degrees.

Finite Bipartite Graphs as Induced Subgraphs of Subfactor Principal Graphs

2026-09-30T00:00:00+03:00

Every finite bipartite simple graph occurs as a vertex-induced unrooted subgraph of the principal graph of an irreducible finite-depth inclusion of hyperfinite type II1 factors. For part sizes m and n, an explicit neighborhood-multiplicity parameter q gives index q3^n, with q at most m+1, and depth at most four. The selected vertices have depths two and three. The complete depth ranks and adjacency spectrum are computed. A separate amplification realizes every finite bipartite multigraph as an ordinary subgraph, with edge deletion allowed.

A Negative Answer to a Question of Basu and Perrucci on Two Multi-Affine Polynomials

2026-09-30T00:00:00+03:00

A real polynomial is multi-affine if it has degree at most one in each variable. Basu and Perrucci proved that the real zero set of one multi-affine polynomial of degree d in R^n has at most 2^(d-1) connected components, independently of n. They asked whether the number of connected components of the common real zero set of two multi-affine polynomials of degree at most d is bounded in terms of d alone; the question was posed again in Oberwolfach Report 9/2025. We show that the answer is no. For m >= 2, explicit multi-affine polynomials of degrees 2 and 4 in 2m+1 variables have a common real zero set homeomorphic to a product of m hyperbolas, with exactly 2^m connected components. A sum-of-squares identity gives a short exact certificate. The construction also combines multi-affine polynomials in disjoint variable sets into a pair whose zero set is homeomorphic to the product of their zero sets. The negative answer persists inside boxes and, as a lower bound, inside any set with nonempty interior.

A Negative Answer to Harrach's Question on the Potential Map in the Unified Eddy-Current Formulation

2026-09-30T00:00:00+03:00

In the unified variational formulation of the parabolic-elliptic eddy-current equations due to Arnold and Harrach, the electric field is written as E = A + ∇φ_A, where A is divergence free and φ_A solves div(σ∇φ_A) = −div(σA). To solve this equation, Arnold and Harrach assume a conductivity σ that is bounded below on its support, and a support made of finitely many Lipschitz domains with disjoint closures. Harrach asked whether the map A ↦ φ_A can be defined for general nonnegative σ ∈ L^∞(ℝ³). We show that it cannot. Let σ be the indicator function of a cyclic necklace of n ≥ 3 closed balls in which consecutive balls touch at one point, or a smooth function that is positive exactly on the open balls. Then there is a smooth, compactly supported, divergence-free field A* for which no φ ∈ H¹_loc(ℝ³) solves div(σ(A* + ∇φ)) = 0. A solid torus whose conductivity is positive almost everywhere but vanishes at least linearly across one cross-section (σ ≤ C|sin(θ/2)|, with θ the azimuth) gives the same conclusion. So neither hypothesis can simply be dropped, although we do not claim that either is necessary. In both examples the eddy-current equation with zero initial data and a smooth divergence-free source that vanishes near the conductor has no solution in L²(0,T;W(curl)). On the positive side, the product σ(A + ∇φ_A), which is what the unified formulation uses, can be defined for every σ ≥ 0 by a weighted orthogonal projection. With this definition the unified formulation stays uniquely solvable and uniformly coercive, and it controls every solution of the eddy-current equation. The eddy-current equation is solvable exactly when, for almost every t, the potential equation for the solution A(t) of the unified formulation has a solution φ(t) ∈ H¹_loc(ℝ³), with ∇φ(·) ∈ L²(0,T;L²_ρ). The ingredients are classical: points have zero capacity, and the energy between touching balls diverges logarithmically. This is an unrefereed note.

The Ford-Circle Packing Has Maximum Area: An Answer to a Question of Propp and Kenyon

2026-09-30T00:00:00+03:00

Two disks of radius 1 centred at (±1, 1) and the x-axis enclose a curvilinear triangle. In the open problem session of the 2015 Oberwolfach workshop on discrete differential geometry, Propp and Kenyon asked whether, among all packings of this triangle by disks that touch the x-axis, the greedy packing has the largest total area. The greedy packing places each new disk in an interstice so that it touches the two disks bounding the interstice. It is a rescaled copy of the Ford circles, and its area is π(ζ(3)/ζ(4) − 1) ≈ 0.34754. We prove that the answer is yes. More generally, let A and B be tangent disks resting on a line, and let G be the greedy packing of the gap between them. For every packing P of this gap by disks resting on the line, and for every α > 1, we show that Σ_{D∈P} r_D^α ≤ Σ_{D∈G} r_D^α, where r_D is the radius of D. The main step is the same inequality for the weight 1/(e^{1/√r} − 1), for which the greedy value of the gap is the product of the weights of A and B. Powers of the radius are superpositions of rescaled copies of this weight. An optimal finite packing contains a chain of tangent disks from A to B, and we bound the value of such a chain by moving two consecutive disks at a time. Along such a move the value is, up to an additive constant, a product of two log-convex functions. The log-convexity reduces to the positivity of an explicit function of two variables whose double power series has non-negative coefficients. We do not discuss uniqueness of the maximizer. This is an unrefereed note.

Euler Numbers Modulo Powers of Two and Arnold's Sequence: Proof of Two Conjectures of Ramassamy

2026-09-30T00:00:00+03:00

Let E_n be the Euler up/down numbers, Σ_n E_n x^n/n! = sec x + tan x, and let e_{n,i} be the Entringer numbers, which form the Seidel–Entringer–Arnold triangle. Arnold observed, without proof, that the least 2-adic valuation m_i of the entries on the i-th diagonal of this triangle is weakly increasing in i, and he introduced the sequence u_k = max{i : m_i < k}. In 2017 Ramassamy conjectured that the sequence (E_n mod 2^k)_{n≥0} is periodic from the index u_k on but not from any earlier index, that its least period is 2^k for k ≠ 2 and 2 for k = 2, and that (u_k) is the f-transform of (2,4,4,4) for an explicit doubling map f. We prove these two conjectures. With h(j) = 2j − 2 − v_2(j), the 2-adic valuation of the tangent number E_{2j−1}, we show that m_i = min_{j≥⌈i/2⌉} h(j) for every i, which proves Arnold's observation, and that u_k = 2 max{j : h(j) < k}. The statements about the period, and the value 2 max{j : h(j) < k} of the preperiod, follow quickly from Stern's classical congruence for the Euler numbers and from the valuation of the tangent numbers. The new ingredients are the determination of m_i and u_k and the proof of the f-transform identity, which rests on the self-similarity h(2^{a−1} + r) = h(r) + 2^a for 1 ≤ r < 2^{a−1}. The proofs are elementary, and computations serve only as consistency checks. This is an unrefereed note.

Excursion Lengths in a Uniform Eulerian Circuit of the Complete Graph: Aldous's Conjecture Holds if and only if i = o(n^(3/2))

2026-09-30T00:00:00+03:00

Let N_i be the number of excursions of length i from a fixed vertex in a uniformly random Eulerian circuit of the complete graph K_n in which every edge is replaced by two opposite arcs. Aldous and Yu (Open Problems in Mathematics, 2014) stated as the natural conjecture that E N_i is asymptotic to e^(−i/n). We prove that for every fixed K, uniformly in 2 ≤ i ≤ K n^(3/2), E N_i = exp(−i/n − i²/(2n³))(1 + O(n^(−1/2) log n)), and that E N_i e^(i/n) → 0 when i/n^(3/2) → ∞. Consequently E N_i ∼ e^(−i/n) holds if and only if i = o(n^(3/2)). This range contains fixed i, the scale i ∼ xn and essentially all excursions, and the length of a uniformly chosen excursion, divided by n, converges in distribution to the standard exponential law. At the scale i ∼ y n^(3/2) the ratio E N_i/e^(−i/n) tends to e^(−y²/2), so the conjecture read literally for all i is false. The proof is elementary. It combines the construction of a uniform Eulerian circuit from a uniform spanning tree (the BEST theorem, in the form used by Kandel, Matias, Unger and Winkler) with a hazard representation of the first excursion and a convexity bound. Exact computations for n ≤ 6 confirm the hazard representation and the closed forms for E N_2 and E N_3, and Monte Carlo simulations up to n = 6400 are consistent with the asymptotic results. This is an unrefereed note.

A Counterexample to a Corner-Peeling Conjecture for Subcomplexes of Z^3

2026-09-30T00:00:00+03:00

Let G be a finite connected subgraph of the grid ℤ³, not necessarily induced, and let X(G) be the cube complex whose cells are the squares and 3-cubes of ℤ³ all of whose vertices and edges lie in G. A corner is a vertex that lies in exactly one maximal cell, and a corner peeling is an ordering v₁, …, vₙ of the vertices in which each vᵢ is a corner of the subgraph induced by v₁, …, vᵢ. In the problem session of the 2026 Oberwolfach workshop Median Geometry and Applications, V. Chepoi, from discussions with J. Chalopin and M. Kokkou, conjectured that G admits a corner peeling whenever X(G) is simply connected and every connected component of every section of X(G) by a coordinate plane is simply connected. We show that the conjecture is false. We give a subgraph G₆₂ of ℤ³ with 62 vertices in [0,3]³ and an induced subgraph H₇₃ with 73 vertices in [0,4]³ such that the cube complex and all its sections are collapsible, but there is no corner at all. Both examples are invariant under a cyclic group of order 6, and the disproof can be checked by hand; we also give explicit sequences of elementary collapses as machine-checkable certificates. A subdivision construction turns every finite subgraph into an induced subgraph whose pieces and corners are dilates of the original ones, so the conjecture for partial subgraphs and the conjecture for induced subgraphs are equivalent. Uncertified solver computations indicate that no counterexample fits into a 3 × 3 × 3 box of lattice points. We also record a smaller counterexample, with 49 vertices, found in an exploratory solver search during an independent verification of this note and checked by computer. This is an unrefereed note.

Hypercyclic Subspaces on ω and a Partial Answer to a Question of Petersson

2026-09-30T00:00:00+03:00

In the report of the 2006 Oberwolfach mini-workshop on hypercyclicity, H. Petersson asked two questions about a (hereditarily) hypercyclic operator T on a separable Fréchet space X. First, does HC(T) ∪ {0} contain the range of an injective operator S ∈ L(X)? Second, for such an S, is every linearly independent n-tuple in Im S hypercyclic for T ⊕ ⋯ ⊕ T? We give a partial answer. Our main result concerns the space ω = K^ℕ. For every hypercyclic operator T on ω there is an injective S ∈ L(ω) such that (Sx_1, …, Sx_n) is hypercyclic for the n-fold direct sum of T whenever x_1, …, x_n are linearly independent. Its range is closed, so every hypercyclic operator on ω has a hypercyclic subspace. The proof uses a special case of a lemma of Shkarin (2011, Lemma 1.5). The hypercyclic-subspace consequence also follows quickly from that lemma combined with a criterion of Menet (2013, Theorem 4.8), and it answers, for an operator on ω and its iterates, a question raised by Menet. On separable Banach spaces, a folklore argument with the entire functional calculus answers the first question positively for every hypercyclic operator. For weakly mixing operators on spaces with a continuous norm, the source itself answers it. In its universal reading, for every such S, the second question has a negative answer: in all these settings there is an S as in the first question whose range contains a pair (x, Tx), and such a pair is never hypercyclic for T ⊕ T. In its existential reading, for some such S, it has a positive answer exactly for the weakly mixing operators, on ω and on spaces with a continuous norm. We leave the first question open for hypercyclic operators that are not weakly mixing on non-normable Fréchet spaces with a continuous norm, and for Fréchet spaces without a continuous norm other than ω. This is an unrefereed note.

Multistationarity of the Altan-Bonnet–Germain Kinetic-Proofreading Network: A Partial Answer to a Question of Rendall

2026-09-30T00:00:00+03:00

In an Oberwolfach abstract of 2017, Rendall remarked that the kinetic-proofreading module of the Altan-Bonnet–Germain model of T-cell activation appears to have deficiency one even under the strongest simplifying assumptions, noted that it is unknown whether it admits multiple steady states, and asked about the asymptotics of solutions of the model. We give a partial answer for the module as published in SBML form (57 species, 158 irreversible reactions, deficiency 34), with mass-action kinetics and independent rate constants. Among 36 natural simplifications, those of deficiency one are exactly the four without CD8, with constant Lck and a single ZAP-70 docking level. For these, every positive stoichiometric class contains exactly one steady state, for all rate constants; the proof combines a flux-balance formula for the bound fraction with injectivity determinants. With two or three docking levels and constant kinases, and for the complete module, there are rate constants with several positive steady states. Exact rational certificates give, for each of these four reduced networks, a class with exactly three steady states, two locally exponentially stable and one unstable, and a class of the complete module with two locally exponentially stable steady states and at least one more. All 36 networks are persistent. The multistationary rate constants are reaction-specific, whereas Altan-Bonnet and Germain use one binding and one unbinding constant for all receptor states. Under that constraint the reduced networks with collapsed Michaelis–Menten steps are monostationary at every docking level; multistationarity of the complete module under this constraint, and global convergence in the deficiency-one case, remain open. This is an unrefereed note.

Partial Results on the Degree of the Algebraic Boundary in Rank-One Tensor Completion

2026-09-30T00:00:00+03:00

Kahle, Kubjas, Kummer and Rosen showed that for a set E of observed entries of a d_1 × ⋯ × d_n tensor with |E| = Σ_j (d_j − 1) that meets every maximal slice and has a full-dimensional completable region (we call such E admissible), the algebraic boundary of the set of partial tensors that are restrictions of joint distributions of independent random variables consists of coordinate hyperplanes and one irreducible hypersurface H. They asked for deg H as a function of n, d_1, …, d_n and E. We give a partial answer. For the corner patterns, the cells that differ from a fixed cell in exactly one coordinate, H is defined by an explicit irreducible discriminant and deg H = n(n − 1) in every format. In general deg H ≤ m!/∏_j (d_j − 1)! with m = |E|, with equality for some patterns. We call E saturated if the maximal minors of its incidence matrix are coprime; this is weaker than unimodularity of that matrix. If E is saturated, the complex completions of a generic partial tensor are the W roots of one Laurent polynomial equation Φ_x(τ) = 1, and we prove q·deg H = W* + deg Den, where W* ≤ W counts the critical values of Φ_x that are not identically zero, q ≥ 1, and Den is a constant multiple of the q-th power of the lowest-degree form of the equation of H. For W = 2 this gives an explicit formula; for matrices deg H = 2 + 2p, where p is the number of observed entries that share their row and their column with other observed entries. For saturated patterns in binary formats we obtain deg H ≤ W + Σ_e μ_e with explicit tropical multiplicities. With exact computer certificates this determines deg H for all admissible patterns in the six formats 2 × 2, 3 × 3, 3 × 4, 2 × 2 × 2, 2 × 2 × 3 and 2 × 2 × 2 × 2. In a modular census of 2343 patterns, which is not a proof, all saturated patterns have q = 1 and W* = W, and the tropical bound is sharp; we do not prove this in general. For patterns that are not saturated we have only the general bound and one certified example. This is an unrefereed note.

A Reciprocal-Block Counterexample to the Forsgård–Shapiro Coefficient-Parity Bound

2026-09-30T00:00:00+03:00

The coefficient-parity conjecture of Forsgård and Shapiro uses the indices at which a_k^2 - a_{k-1}a_{k+1} is nonnegative to bound the number of real zeros of a polynomial with positive coefficients. Katkova, Shapiro, and Vishnyakova already disproved this bound in 2024. We give an alternative explicit degree-77 polynomial with positive rational coefficients for which the selected indices change parity only once, but which has at least three distinct negative real zeros. The construction joins a degree-38 block to its reciprocal. A weighted sum-of-squares identity proves the required root crossing; all coefficients and signs can also be checked by rational arithmetic.

A Positive-Coefficient Counterexample to the Weighted Forsgård–Shapiro Parity Bound

2026-09-30T00:00:00+03:00

Forsgård and Shapiro proposed bounding the number of real zeros of a positive-coefficient polynomial by the parity changes among all indices at which (k+1)a_k^2 - k a_(k-1)a_(k+1) is positive. Katkova, Shapiro, and Vishnyakova already disproved this bound in 2024. We give an alternative counterexample with positive rational coefficients, parity count one, and at least three distinct negative real zeros. An even-degree shift of a degree-77 reciprocal-block seed makes the undesired weighted local quantities negative. A sufficiently small, strictly log-convex factorial prefix fills every missing coefficient without introducing a parity change or destroying the three roots. The explicit witness has degree 1,000,077; no optimal-degree claim is made. Its coefficients have a compact exact formula.

A Finite Moment-Cone Criterion for Derivative-Root Configurations

2026-09-30T00:00:00+03:00

We give a necessary and sufficient finite criterion for prescribed numerical positions of all derivative roots of a real-rooted polynomial-like function. Polynomial-like degree n means that the nth derivative is nowhere zero; the function need not be an ordinary polynomial of degree n. Eliminating integration constants produces m=n(n-1)/2 explicit piecewise polynomial Peano kernels. Realizability is equivalent to zero being interior to their essential-trace convex hull, and to representing an explicit baseline vector by at most m nonnegative trace terms. This gives a finite semialgebraic criterion and a terminating decision procedure for algebraic coordinates. Permitted cross-order coincidences are included. Every feasible array also has an ordinary-polynomial realization of unspecified larger degree.

The Dimension of Trivariate C¹ Splines on Generic Bipyramid Cells

2026-09-30T00:00:00+03:00

A bipyramid cell is a tetrahedral partition with one interior vertex v₀, n boundary vertices that are coplanar with v₀ and surround it, and two further boundary vertices on opposite sides of this base plane. Colvin, DiMatteo and Sorokina determined the dimension of the space S¹_d(Δ) of C¹ splines of degree at most d on such a cell Δ in their collinear and coplanar cases. In the generic case they gave lower and upper bounds and conjectured that the dimension equals the upper bound whenever the two bounds differ. We prove an exact formula for every generic bipyramid cell. Let m be the number of distinct lines spanned by the n interior edges in the base plane. Then dim S¹_d(Δ) = 1, 4, 11 for d = 0, 1, 2, and for d ≥ 3 it equals 2n·C(d,3) + 6d − 1 + (4 − m)₊ if m ≥ 3, and 8·C(d,3) + 8d − 3 − (4 − d)₊ if m = 2. For d ≥ 3 these are exactly the upper bounds as printed in the Oberwolfach abstract of Colvin, DiMatteo and Sorokina, and for d = 2 the upper bound is attained unless m = 3. So the conjecture holds in every case except m = 3, d = 2, where the dimension is 11 and the printed upper bound is 12; the smallest example has a triangular base. We could not access the journal version of their paper; if its bounds for d ≥ 3 are the printed ones and it states this one case differently, the conjecture holds as stated there, and in any case the exact formula answers the question. An elementary cofactor argument shows directly that dim S¹_2(Δ) = 11 for every generic cell. The spline space splits into homogeneous components; the dimensions of those of degree at least 5 also follow from a general formula of Alfeld, Neamtu and Schumaker, so the new part of the formula is the determination of the components of degrees 3 and 4. The proof reduces the problem to bivariate splines on the planar fan in the base plane: each half of the cell projects onto this fan, and gluing the two halves forces the trace on the base plane to be C² across every ray. The same reduction gives a uniform proof of their formulas in the collinear and coplanar cases. The formula agrees with Macaulay2 values of DiPasquale and Villamizar for n = m = 5 and with exact rational computations. This is an unrefereed note.

The Amoeba Dimension of an Arbitrary Loopless Matroid

2026-09-30T00:00:00+03:00

Let M be a loopless matroid on a finite set E with rank function r, and let Σ(M) ⊆ ℝ^E be the support of its matroid fan. Draisma, Eggleston, Pendavingh, Rau and Yuen defined adim(M) as the minimum of 2 dim(Σ(M) + R) − dim R over rational subspaces R ⊆ ℝ^E; for the matroid of a complex linear space this is the dimension of the amoeba of the space. They proved that adim(M) = min Σ_i (2r(P_i) − 1), the minimum over all partitions {P_1, …, P_k} of E, when M is realizable over ℂ, and asked whether this holds for every loopless matroid. The question appears in an Oberwolfach report, in a BIRS problem session and as Conjecture 1.4.1 of their paper. We show that the answer is yes, and that both numbers equal dim(Σ(M) + Σ(M)); the minimum is attained by a rational subspace in the braid arrangement. The lower bound dim(Σ(M) + Σ(M)) ≥ min Σ_i (2r(P_i) − 1) is a theorem of Bernstein, valid for all loopless matroids. The only new ingredient is the inequality 2 dim(Φ + R) − dim R ≥ dim(Φ + Φ) for every finite union of cones Φ and every subspace R, which follows from the Grassmann formula. Consequently S ↦ adim(M|S) is a matroid rank function, which answers one of the sub-questions of the BIRS problem. The others are answered by earlier results: the partition minimum for M|S is a matroid rank function of S (Draisma et al.), and by Bernstein's theorem it equals dim(Σ(M|S) + Σ(M|S)). The inequality can be strict for other fans. Exact computations on connected non-realizable matroids are consistent with the theorem. This is an unrefereed note.

A Telescoping Formula for Evolute Areas in Elliptic Billiards

2026-09-30T00:00:00+03:00

For a noncircular elliptic billiard with a nondegenerate confocal elliptic caustic, we derive closed formulas for the signed area ratios of the caustic contact polygon and the boundary tangent polygon to their discrete evolutes. The formulas hold for every physical periodic orbit of least period greater than four, including star trajectories. Unit complex contact parameters satisfy a biquadratic relation. The consecutive circumcenters have a rational expression whose local area term differs from a constant multiple of the contact-area term by an explicit rational coboundary. A polynomial certificate and cyclic summation prove the inner formula. Conic polarity and a fixed-conic circumcenter transformation yield the outer formula. The exceptional coefficients force least periods three or four, which proves that the stated quotients have nonzero denominators. Exact examples also show why repeating primitive four-cycles does not extend the theorem to a list-length interpretation of period. The results address invariants k703 and k702 of Reznik, Garcia and Koiller in one manuscript, corresponding to AMR-050-0041 and AMR-050-0040 in the frozen ulamai/UnsolvedMath v1.6.0 dataset.

Counterexamples to Conforti's Subtree Conjecture for Mixed-Integer Bipartite Covers

2026-09-30T00:00:00+03:00

For a bipartite graph G = (U ∪ V, E), a set I ⊆ U ∪ V and rationals b_ij, let S(G,I) = {x ∈ R^(U∪V) : x_i + x_j ≥ b_ij (ij ∈ E), x_i ∈ Z (i ∈ I)}, and let k be the least positive integer with kb integral. In the 2008 Oberwolfach report on combinatorial optimization, Conforti conjectured that conv S(G,I) is the intersection of the hulls conv S(T, I ∩ V(T)) over the subtrees T of G whose integral vertices are exactly their leaves. This would place the membership problem for conv S(G,I) in coNP. He noted that the case k = 2 follows from work of Conforti, Gerards and Zambelli, and that the conjecture was open for every k ≥ 3. We show that it fails for every k ≥ 3. For each such k we give two unicyclic counterexamples. One has six vertices. The other has seven, and in it every integral vertex is a pendant vertex with a continuous neighbour, so the standard normalisation of such sets (splitting integral vertices) does not remove it. In both, an explicit point lies in conv S(T, I ∩ V(T)) for every subtree T of G, but violates a facet-defining inequality of conv S(G,I) by (k − 2)/(2k − 3). The proofs are short and by hand. We also describe exact validity certificates based on an extended formulation, and use them to certify a further normalised counterexample for k = 3. The complexity of the membership problem remains open. This is an unrefereed note.

Counterexamples over F̄_p to a Generic Equivalence Problem of Kraft and Russell

2026-09-30T00:00:00+03:00

In Oberwolfach Report 01/2007, Kraft and Russell stated that two morphisms of varieties over an algebraically closed field of infinite transcendence degree, whose fibres over all closed points are isomorphic, become isomorphic after a dominant étale base change, and asked whether this holds over every algebraically closed field, or for counterexamples over F̄_p or Q̄. In 2014 they proved the statement for affine morphisms, with a dominant base change of finite degree. The étale form already fails over every algebraically closed field of positive characteristic, because of Russell's classical purely inseparable forms of the affine line; we record this, and note that it immediately answers a question on positive characteristic raised in a remark of Kaliman. We show that over F̄_p even the finite-degree form fails. For every prime p we give a pair of smooth affine families of threefolds over an open subset Y of the affine line, defined over F_p, whose fibres over each closed point of Y are isomorphic as F̄_p-varieties, but which do not become isomorphic after any base change U → Y whose image contains the generic point, whether it is étale, of finite degree, or neither. The fibres are affine modifications of G_m² × A¹ at two points, and their isomorphism classes are governed by GL₂(Z)-orbits. Over F̄_p every point of G_m² has finite order, and the Frobenius twist stays in the orbit at every closed point but not at the generic point. For p ≡ 1 (mod 4) we give smooth projective families with the same properties: blow-ups of E × E at two points, where E is the curve y² = x³ − x with complex multiplication by Z[i], parametrised by E × E minus the origin or by a curve in it. The case of Q̄ remains open. This is an unrefereed note.

Star Transforms Without Type 2 Singular Directions and Conflitti's Conjecture on Elementary Symmetric Polynomials

2026-09-30T00:00:00+03:00

The star transform integrates a function on the plane along m rays with a common vertex, with directions γ_1, …, γ_m and nonzero weights c_1, …, c_m. In the inversion formula of Ambartsoumian and Latifi, the directions ψ at which Σ_j c_j Π_{i≠j} ⟨ψ, γ_i⟩ vanishes, called singular directions of Type 2, cause instability. They showed that every star with an even number of rays has such directions, and that for m = 3 every choice of weights admits ray directions without them. For odd m ≥ 5 this was left as a conjecture, which the Oberwolfach Report 21/2023 records as unproven. We prove the conjecture for every odd m and all nonzero real weights. The proof reduces the problem, by nearly parallel rays, to a rational function of one variable without real zeros, which we build by nesting explicit three-pole clusters. The same report recalls a conjecture of Conflitti (2006): for even r, the real zero set of the elementary symmetric polynomial e_r in n variables contains no linear subspace of dimension r. Previously the cases r = 2 (Conflitti) and n = r + 1 (Ambartsoumian and Latifi; this case also follows from the irreducibility of e_{n−1}) were known. We prove the conjecture for all even r and all n, using Descartes' rule of signs and a parity argument. Hence the largest subspace in the zero set has dimension min(n, r − 1). For even r ≥ 4 we also show that the only (r − 1)-dimensional subspaces in the zero set are coordinate subspaces. Exact computer certificates for explicit stars with at most 21 rays illustrate the construction; the proofs do not depend on them. This is an unrefereed note.

A Proof of Paták's kb+1 Conjecture for Constrained Stars, with Improved Bounds for Complete Graphs

2026-09-30T00:00:00+03:00

In an Oberwolfach report from 2020, Paták considered closure operators on topological spaces whose closures have at most b path-connected components. He noted that C(b+1,2)(k−1)+b+1 points suffice for a constrained drawing of the star K_{1,k}, conjectured that kb+1 points suffice, and asked two further questions: whether the bounds for complete graphs K_n can be improved, and whether the method extends to higher homology or homotopy. In the journal version (J. Graph Theory, 2025) the conjecture is stated for b-iatlon graphs and proved there for b ≤ 2. We prove the conjecture for all k and b. Every b-iatlon graph with kb+1 vertices contains a constrained copy of K_{1,k}. For every closure operator as above, every set of kb+1 points admits a constrained drawing of K_{1,k}. The bound kb+1 is sharp in both settings. The proof rests on a colouring lemma for families of graphs indexed by the subsets of a finite set and growing with the subset; it extends the bound χ ≤ Δ+1. As a consequence, 1+b+⋯+b^{n−1} points force a constrained copy of K_n, improving the previous bound O(b^{2n−3}). Combined with Paták's topological arguments, this lowers his bounds on Radon and Helly numbers in R^d from O(b^{2d+3}) to O(b^{d+2}), and his bound on Radon numbers on a fixed closed surface from O(b^6) to O(b^3). In the b-iatlon setting, Ramsey numbers give lower bounds, which show that the exponent n−1 is optimal for n = 3, 4. Exact values for complete graphs remain open, and the question on higher homology and homotopy is not addressed. This is an unrefereed note.

The Conformal Parametrization of Nonnegatively Curved Surfaces Is Not a Homeomorphism at Integer Smoothness

2026-09-30T00:00:00+03:00

Let M_0 = ℂ and M_1 = S², with complete metrics g_κ of constant curvature κ ∈ {0, 1}. Every complete smooth metric of nonnegative curvature on M_κ can be written uniquely as φ*(e^{−2u} g_κ), where φ is an orientation-preserving diffeomorphism fixing 0 and 1 (and ∞ if κ = 1) and u is a smooth function. Work of Belegradek, Hu and Banakh shows that the map (u, φ) ↦ φ*(e^{−2u} g_κ) is a homeomorphism when metrics and functions carry the topology of C^{k+α} convergence on compact sets and diffeomorphisms that of C^{k+1+α} convergence, with 0 < α < 1. Belegradek asked whether this remains true for α = 0, and expected that it does not. We show that it fails for every integer k ≥ 0 and on both surfaces: the map is a continuous bijection whose inverse is discontinuous at every point, also on the subspace of positively curved metrics. For k = 0 this follows from an elementary spiral construction. For k ≥ 1 we use diffeomorphisms φ_ε = z + ε z^{k+1} H_ε(|z|²) whose Beltrami coefficient is approximately ε z^{k+1}/z̄, regularized at the scale e^{−1/ε}. They converge to the identity in C^k, while their (k+1)-st derivative at 0 tends to −2(k+1)!. The main work is to choose the conformal factors so that the metrics converge in C^k and keep nonnegative curvature; for k = 1 this needs an additional C¹-small conformal correction. The conformal factors then fail to converge in C^k as well. This is an unrefereed note.

A Negative Answer to the Strong Geography Question of Alfieri and Binns

2026-09-30T00:00:00+03:00

Alfieri and Binns say that an F[U]-module M satisfies the strong geography restriction if it has a direct summand F[U]/U^ℓ ⊕ F[U]/U^(ℓ−1) ⊕ ⋯ ⊕ F[U]/U, where ℓ is the least integer with U^ℓ M_red = 0. They showed that HF⁻(Y) satisfies it when Y is surgery on a knot in S³ or large surgery on a link, and asked whether it holds for every rational homology sphere Y. We show that the answer is no. For the Brieskorn sphere Y = Σ(30,47,83), with either orientation, the reduced Heegaard Floer homology is T_16 ⊕ T_14^10 ⊕ T_13^16 ⊕ ⋯ ⊕ T_1^92, where T_k = F[U]/U^k. So ℓ = 16, but F[U]/U^15 is not a direct summand of HF⁻(Y). The proof combines the Ozsváth–Szabó description of HF⁺ for plumbed manifolds, Némethi's reduction to the graded root of an explicit function τ, and an exact computer calculation. We give a certificate, the 1707 turning points of τ, from which the summand lengths can be rechecked by a short program. The manifold Y is an irreducible integral homology sphere, is not an L-space, and satisfies Lin's weaker restriction. It also shows that the word "large" cannot be removed from the link-surgery theorem of Alfieri and Binns. Computer searches with two independent programs show that among Seifert fibred integral homology spheres Σ(a_1,…,a_n), the counterexamples with the smallest product a_1⋯a_n are Σ(30,47,83) and three spheres with four singular fibres, all of product 117030. This is an unrefereed note.

The Klee–Novik Complexes B(i,d) Triangulate S^i × B^(d−i−1)

2026-09-30T00:00:00+03:00

For 0 ≤ i ≤ d − 2, Klee and Novik defined B(i,d) as the subcomplex of the boundary of the d-dimensional cross-polytope generated by the facets whose xy-words have at most i switches, and asked whether B(i,d) is a combinatorial triangulation of S^i × B^(d−i−1). We show that it is, for all 0 ≤ i ≤ d − 2. Klee and Novik observed that B(i,d) collapses onto the boundary of the (i+1)-dimensional cross-polytope and noted that it is therefore a disc bundle over S^i. We show that |B(i,d)| is in fact a product. That sphere is a join factor of the boundary of the d-dimensional cross-polytope, so it has a regular neighbourhood that is a product, and |B(i,d)| together with an outer collar is another regular neighbourhood of it; uniqueness of regular neighbourhoods gives the result. Consequently ∂B(i,d) is PL homeomorphic to S^i × S^(d−i−2), and a conjecture of Cohen, Klee and Pannell holds. This is an unrefereed note.

The Harrell–Stubbe Gap Inequality for Dirichlet Eigenvalues Is Never Saturated

2026-09-30T00:00:00+03:00

Let E_1 ≤ E_2 ≤ ⋯ be the eigenvalues of the Dirichlet Laplacian on a bounded open set Ω ⊂ R^n, and let M_p(J) = ((n+2p)/n · (1/J) Σ_{j≤J} E_j^p)^{1/p}. Harrell and Stubbe proved that M_1(J)^2 − M_2(J)^2 ≥ ¼(E_{J+1} − E_J)^2, and in a problem list of the 2009 Oberwolfach workshop on low eigenvalues of Laplace and Schrödinger operators they asked whether some Ω and J saturate this inequality. We show that the answer is no: the inequality is strict for every bounded open set and every J. Written in terms of the mean and the variance of E_1, …, E_J, the inequality is a gap estimate of Cheng and Yang, so that estimate is strict as well. The proof analyses the case of equality in the Harrell–Stubbe trace identity behind H. C. Yang's inequality. Equality would put each function x_k u_1, where u_1 ≥ 0 is a first eigenfunction, into a finite sum of eigenspaces. Then every partial derivative of u_1 would lie in H^1_0(Ω), so ∫_Ω Δu_1 = 0, which is impossible because Δu_1 = −E_1 u_1. The same argument shows that Yang's first inequality is strict whenever E_{J+1} > E_1, and hence for every J when Ω is connected; in 2002 Ashbaugh left the strictness of this inequality undecided. By contrast, for the harmonic oscillator and on spheres the analogous bounds are equalities at every spectral gap, that is, for every J with E_J < E_{J+1}. This is an unrefereed note.

Kernels of Signature Operators Twisted by Almost Flat Bundles Are Not Homotopy Invariant

2026-09-30T00:00:00+03:00

At an Oberwolfach problem session in 2006, T. Schick asked the following. Fix closed Riemannian manifolds and a homotopy equivalence f. Do the signature operators twisted by a bundle E and by f*E have isomorphic kernels once the curvature of E is small enough? By a theorem of Hilsum and Skandalis, their indices agree. Sauer and Schick later asked the same for the twisted Betti numbers. We show that the answer is no, by an elementary example. On the flat square torus let f(x,y) = (x + (c/2π) sin 2πy, y), a diffeomorphism isotopic to the identity. Let E_ε be the trivial Hermitian line bundle with the connection d + iε cos(2πy) dx, whose curvature has norm 2π|ε|. The twisted signature operator D = d_∇ + d_∇* has a four-dimensional kernel for E_ε, for every ε, and a zero kernel for f*E_ε whenever εc ∉ 4πℤ; for its chiral half D⁺ the dimensions are 2 and 0. The pull-back moves the harmonic part of the connection form by the curvature flux εc/2. Products give examples in every dimension ≥ 2, for instance kernels of dimensions 16 and 0 on the 4-torus (8 and 0 for D⁺). The answer stays negative for the underlying Euclidean rank-two bundle, for the spin Dirac operator, and for the natural readings of degree-wise twisted Betti numbers; for non-flat bundles the kernel is in general graded only by parity, not by degree. By contrast, the number of eigenvalues near zero is stable: if r = |ε|(1 + c²)^{1/2} < π, both operators have exactly four eigenvalues in [−r, r]. This is an unrefereed note.

A Proof of the Below–Krummeck–Richter-Gebert Conjecture on Cyclic Determinant Quotients

2026-09-30T00:00:00+03:00

For points A_1, …, A_n, A'_1, …, A'_n in K², where K is a field, write [XY] for the 2 × 2 determinant and let α(π) be the cyclic quotient ∏_k [A_{π_k} A_{π_{k+1}}] / ∏_k [A'_{π_k} A'_{π_{k+1}}], with indices read cyclically. Below, Krummeck and Richter-Gebert conjectured in 2003 that Σ sgn(π) α(π) = 0, where the sum runs over the permutations π of {1, …, n} with π_1 = 1. Richter-Gebert posed the problem again at Oberwolfach in 2009. It had been proved for n = 5 and checked by computer algebra for n = 6, and it follows from a symmetry argument when n ≡ 0, 3 (mod 4). We prove the identity for every n ≥ 3 and every field, assuming only that [A'_iA'_j] ≠ 0 for i ≠ j. Equivalently, the associated bracket polynomial vanishes identically. The proof writes the numerator as the trace of a product of traceless 2 × 2 matrices. The reciprocals of the denominators are Parke–Taylor factors. They satisfy a shuffle identity, known from the Kleiss–Kuijf relations of gauge theory, and hence are the coefficients of a Lie polynomial. Mapping this Lie polynomial into 2 × 2 matrices over an exterior algebra, where the commutators of the generators are central, gives zero. The same argument shows that Σ_σ sgn(σ) M_{σ_2} ⋯ M_{σ_n} / ([A'_1 A'_{σ_2}] ⋯ [A'_{σ_n} A'_1]) = 0 for all traceless 2 × 2 matrices M_2, …, M_n when n ≥ 4, and it recovers the vanishing of the same sum without the matrices. This is an unrefereed note.

Algebraic Relations Among Mean Value Coordinates: A Complete Answer for Quadrilaterals and Partial Results for Larger Polygons

2026-09-30T00:00:00+03:00

Mean value coordinates of a planar polygon P with n vertices define a map from P to P^(n−1). At an Oberwolfach mini-workshop in 2022, F. Sottile asked for the homogeneous equations of the Zariski closure S_P of its image. We answer this question for quadrilaterals and give partial results for larger polygons. For every quadrilateral with no three vertices collinear, the ideal of S_P is generated by one explicit irreducible polynomial F_P of degree 14. It is obtained by clearing the square roots in the relation λ_1|x − v_1| − λ_2|x − v_2| + λ_3|x − v_3| − λ_4|x − v_4| = 0, which follows at once from Floater's formula; for the unit square, F_P/16 has 116 integer terms. For every n ≥ 4, S_P is an irreducible surface, and the projection w ↦ Σ w_i v_i / Σ w_i restricts to a map of degree 2^(n−1) from S_P to the plane. Over a general point x, a point w with Σ w_i ≠ 0 and Σ w_i (v_i − x) = 0 lies on S_P exactly when the closed polygon with edge vectors w_i J(v_i − x), J a quarter turn, is circumscribed about a circle. The passage from coordinates to circumscribed polygons is the classical tangent-length picture behind Floater's construction; its converse is what produces equations. For n = 5, 6, 7, computations modulo primes for sample polygons give relations of lowest degree 10, 6, 7, and deg S_P = 52 for n = 5; for a single hexagon they give deg S_P = 152. These values are consistent with a conjectured value 2^(n−3)(6n − 17). A generating set for n ≥ 5 remains open. This is an unrefereed note.

Gasull's Finite Moment Problem for Polynomials with Few Monomials

2026-09-30T00:00:00+03:00

Let f ∈ C[x] have at most k monomials and put M_n = ∫_0^1 f(x)^n dx. A. Gasull asked whether there is a number N(k) such that M_1 = ⋯ = M_{N(k)} = 0 forces f = 0, and, if so, for its value or a good upper bound. We show that N(k) exists for every k. The proof treats the exponents as parameters, applies Hilbert's basis theorem to the moments with their denominators cleared, and uses the known fact, proved by Pakovich and by Françoise, Pakovich, Yomdin and Zhao, that the moments ∫_0^1 f^n dx, n ≥ 1, of a nonzero polynomial cannot all vanish. It gives no explicit bound. We also show that N(k) ≥ k for every k, that N(1) = 1 and N(2) = 2, and, by an exact computer-assisted certificate, that N(3) = 3: after a linear change of variables, a resultant that controls the case k = 3 has only nonnegative coefficients. The statement for k = 3 holds for all real exponents greater than −1/3. For every k, the first k moments force f = 0 when the exponents are sufficiently lacunary. For k = 4 and k = 5 we report exact certificates for all sets of integer exponents with largest exponent at most 50 and at most 16, respectively; these are finite-range computations. Numerically, the real-exponent version of the case k = 4 fails for some exponents with e_1 < 0. The value of N(k), and any explicit upper bound for it, remain open for k ≥ 4. This is an unrefereed note.

Combinatorial Spheres inside Spheres on the Same Vertex Set: Partial Results on a Question of Santos

2026-09-30T00:00:00+03:00

F. Santos asked in an Oberwolfach problem session (2019) whether every combinatorial d-sphere S with n ≥ d+3 vertices is a subcomplex of a combinatorial (d+1)-sphere with the same n vertices. We give partial results. If S has an edge vw with lk(v) ∩ lk(w) = lk(vw), equivalently an edge that lies in no missing face with at least three vertices, then the union of the cones v*ast(v) and w*ast(w) over the two antistars is such a sphere. This extends Datta's construction for flag spheres and recovers his results for joins and for spheres with a vertex of degree d+1. If S has no missing face of dimension d and some vertex link is a stacked sphere, the answer is also positive. We add reformulations in terms of completable antistars, one-point suspensions and balls, closure under connected sums and stellar subdivisions, and restrictions on a smallest counterexample in dimension 3. With computer certificates that are checked by an independent program, every combinatorial 3-sphere with 6 ≤ n ≤ 10 vertices lies in a 4-sphere on the same vertex set: 1320 of the 247,882 ten-vertex 3-spheres have no edge of the above kind, and each of them has an explicit extension. All 337 combinatorial 4-spheres with 9 vertices that we found have such an edge; the completeness of this list is a solver result without proof certificates, and we know no published count to compare it with. A Z_11-invariant neighborly 3-sphere without such an edge and two Z_13-invariant ones have extensions, but none in which every facet meets a fixed pair of vertices. Of nine Z_14-invariant neighborly 3-spheres without such an edge, six extend and three are undecided. These spheres are known from the enumerations of Kühnel and Lassmann and of Köhler and Lutz. The general question remains open. This is an unrefereed note.

Isolated Real Zeros of Sums of Squares: Quaternary Forms and Quinary Quartics

2026-09-30T00:00:00+03:00

For even n let B′_{n,m} be the supremum of the number of real projective zeros of a sum of squares of real forms of degree n in m variables, taken over those with finitely many real zeros. Fröberg, Lundqvist, Oneto and Shapiro conjectured that B′_{n,m} = (n/2)^{m−1}; Choi, Lam and Reznick had proved the lower bound and the case m = 3. A related conjecture of Ottaviani and Shapiro states that a sum of squares of real polynomials of degree at most k in l variables has at most k^l isolated real zeros; it was known for l ≤ 2. Both conjectures follow from the statement that the real points of the base locus of a real linear system of forms of degree k on ℙ^N have at most k^N isolated points. We prove this statement for N = 3 and every k, and for k = 2, N = 4. Consequently B′_{2k,4} = k^3 for every k and B′_{4,5} = 16, and the Ottaviani–Shapiro bound holds for l = 3 and every k and for (k,l) = (2,4). We also give a short proof of B′_{4,4} = 8, a value that Choi, Lam and Reznick expected and that Dressler stated without proof. The main tools are a weighted Bézout inequality, a Jacobian argument along the curve components of the base locus, and an integral closure estimate for the system of partial derivatives. Both conjectures remain open in general, for instance for sextics in five variables and for quartics in six variables. This is an unrefereed note.

A Focal Inversion Area Ratio for Four-Periodic Elliptic Billiards

2026-09-30T00:00:00+03:00

The experimental invariant list of Reznik, Garcia and Koiller prints the value 2 for the focal-inverse area of the outer tangent polygon divided by that of a four-periodic elliptic billiard orbit. We give a rational counterexample to this value and a self-contained area calculation showing that the correct constant is 1/2. The result concerns vertexwise inversion of both polygons about the same focus and signed shoelace area. It applies to primitive four-periodic orbits with a confocal elliptic caustic. This is a reciprocal correction to a tabulated constant, not a refutation of the underlying constancy or a claim about the corresponding general-period conjecture.

Focal Antipedal Centroids of Rectangles Circumscribed about an Ellipse

2026-09-30T00:00:00+03:00

Let a rectangle be circumscribed about a noncircular nondegenerate ellipse with center O and a focus F. We show that its antipedal quadrilateral with respect to F has vertex centroid O and area centroid O+(F-O)/3, independently of the rectangle's orientation. The quadrilateral is always finite and strictly convex. Applied to the known outer tangent rectangles of simple four-periodic elliptic billiards with a confocal elliptic caustic, this calculation proves both constancy assertions in the experimental invariant k406,b of Reznik, Garcia and Koiller and identifies their values. The proof uses an orthonormal support frame and exact polygon first moments; it does not assert the corresponding general-even-period invariant k407. Source record: AMR-050-0025 (raw ID 5100025, k406,b), Hugging Face dataset ulamai/UnsolvedMath. Self-audited, AI-assisted, unrefereed preprint; no independent human review or formalization is claimed. The source and novelty review credits the known four-periodic outer-rectangle lemma and makes no absolute priority claim. The source ZIP includes the English LaTeX manuscript and a portable standard-library exact rational checker.

Stationary Focal Antipedal Centroids for Even Elliptic Billiard Periods

2026-09-30T00:00:00+03:00

Fix a noncircular ellipse and a nondegenerate confocal elliptic caustic supporting a family of billiard orbits of effective even period. Form the outer polygon by intersecting consecutive boundary tangents, then take its antipedal with respect to either focus. We prove that its vertex centroid is fixed throughout the family and lies on the major axis. An exact opposite-edge calculation reduces this centroid to a bilinear trace of a Poncelet polygon inscribed in a circle and circumscribed about a concentric ellipse. We supply a compact-curve proof of trace constancy, adapting the pole cancellation method of Akopyan, Schwartz and Tabachnikov. The calculation proves the constancy assertion k407 of Reznik, Garcia and Koiller, with the effective-period convention stated explicitly. The constant need not be the center of the ellipse. We make no signed-area-centroid or hyperbolic-caustic claim. Self-audited, unrefereed preprint prepared with AI assistance. Established trace methods are credited; no independent human review, formal proof-assistant verification or absolute-priority claim is made. The source identity is AMR-050-0026 in the frozen ulamai/UnsolvedMath v1.6.0 dataset. This manuscript treats the single literal invariant k407, not every invariant in the source list. PDF, English LaTeX source and standard-library reproducibility checkers are included. Author: Alper Ferudun, Mercury Software GmbH.

On a Question of Freslon, Gerontogiannis and Skalski about Quantum Isometries of Cuntz–Krieger Algebras

2026-10-01T00:00:00+03:00

For a primitive 0–1 matrix A, Freslon, Gerontogiannis and Skalski computed the quantum isometry group G_A^∞ of the log-Laplacian spectral triple on the Cuntz–Krieger algebra O_A. They showed that the quantum automorphism group QAut(A) is a quantum subgroup of a larger quantum group G_A^1 and asked if it is one of G_A^∞ when QAut(A) ≠ Aut(A). We show that the answer is negative in general. For the Cuntz algebras O_N and for A = J_N − I_N, with N ≥ 4, the quantum permutation group S_N^+ = QAut(A) is not a quantum subgroup of G_A^∞. For every primitive A with QAut(A) ≠ Aut(A), the canonical inclusion of QAut(A) in G_A^1 does not extend to G_A^∞, and the natural action of QAut(A) on O_A is not isometric for this spectral triple. Quantum groups such as S_M^+, M ≥ 4, cannot sit in G_A^∞ through an embedding that fixes a vertex. On the other hand, for two explicit primitive 4 × 4 matrices, the non-classical QAut(A) (the dual of the infinite dihedral group, and the hyperoctahedral quantum group H_2^+) is a quantum subgroup of G_A^∞. For general A it remains open whether S_M^+, M ≥ 4, can sit in G_A^∞ through an embedding that fixes no vertex. This is an unrefereed note.

Components and Cohomology of Fock's Baby Teichmüller Space

2026-10-01T00:00:00+03:00

At an Oberwolfach problem session in 2006, V. Fock asked for the cohomology and the number of components of the space B_n of maps f: ℤ/n → ℝP¹ with f(i) ≠ f(i+1) and at least three values, modulo SL(2,ℝ). We answer both questions for all n ≥ 3. The space B_n is a real-analytic (n−3)-manifold with exactly n−1 components B_{n,k}, indexed by the winding number k. For k ≠ n/2 they are Hausdorff and contractible, and for k = 1 and k = n−1 they are the Teichmüller space of the ideal n-gon. For even n the middle component is not Hausdorff: two distinct points have no disjoint neighbourhoods exactly when one is represented by a map constant on the even positions and the other by a map constant on the odd positions, and both loci are spheres S^{n/2−2}. The middle component is weakly homotopy equivalent to S^{n−3}. Hence H⁰(B_n; ℤ) ≅ ℤ^{n−1}, H^{n−3}(B_n; ℤ) ≅ ℤ for even n, and all other singular cohomology vanishes; the cohomology of the constant sheaf agrees, and so does its Čech cohomology on the Hausdorff components and, on the middle component, in degrees at most 2. For n = 4 the de Rham cohomology of smooth forms differs, and B_{4,2} is weakly homotopy equivalent to a circle but not homotopy equivalent to it. Modulo PGL(2,ℝ) there are ⌊n/2⌋ components, and for even n the middle one is weakly homotopy equivalent to ℝP^{n−3}; for odd n the space of real tame frieze patterns of width n−3 has (n−1)/2 components, all contractible. This is an unrefereed note.

Extremal Modular Forms Modulo p and a Conjecture of Bannai, Koike, Shinohara and Tagami

2026-10-01T00:00:00+03:00

For even k ≥ 4 let f_k = 1 + O(q^{dim M_k}) be the extremal modular form of weight k for SL_2(ℤ). Bannai, Koike, Shinohara and Tagami conjectured that if k = 12μ and f_k mod p is a nonconstant power series in q^p (their Case (2)), then f_k(τ) ≡ g(p^r τ) (mod p) for an extremal modular form g of smaller weight and some r ≥ 1. We observe that this is a corollary of the classical theory of modular forms modulo p: the key input is the equality w(h^p) = p·w(h) for the filtration of a p-th power, which follows from Swinnerton-Dyer's structure theorem and underlies the theorem of Serre and Katz on the kernel of θ. For every even k ≥ 4 and every prime p in Case (2) we get f_k(τ) ≡ f_{k'}(pτ) (mod p) with k' = w(f_k mod p)/p, where 4 ≤ k' ≤ k/p and k' ≡ k (mod p − 1). So the conjecture holds with r = 1, and by iteration also with the largest possible r. For k = 12μ every prime in Case (2) satisfies 11 ≤ p ≤ 3μ and p ≠ 13, and Case (2) is decided by a finite test. Exact computations for all weights 12μ ≤ 7200 and all even k ≤ 2400, covering 9433 pairs (k, p) in Case (2), agree with these statements. This is an unrefereed note.

The Quantum Symmetriser of a Yang–Baxter Solution Need Not Be a Quasi-Isomorphism

2026-10-01T00:00:00+03:00

For a set-theoretic solution (X, σ) of the Yang–Baxter equation, Farinati and García Galofre showed that a comparison map from braided (co)homology to the Hochschild (co)homology of the structure algebra A = kM(X, σ) can be chosen to be the quantum symmetriser of −σ, and that it factors through a complex A ⊗ B ⊗ A, where B is the Nichols algebra of −σ. They asked whether this complex is a resolution of A, and Lebed asked in a 2019 Oberwolfach report whether the quantum symmetriser is bijective for general solutions; the answer is known to be positive for involutive (characteristic zero) and idempotent solutions. We show that it is negative in general. For the permutation rack X = Z/4, x ◁ y = x + 1, a bijective non-degenerate solution, we give an explicit 2-cycle of weight 3 in A ⊗ B ⊗ A that is not a boundary; it comes from a cubic quantum Serre relation in B. With coefficients in the module on which X acts by 0, the image of the quantum symmetriser misses part of HH₃(A; k) for every quotient of the braided complex. We show that A ⊗ B ⊗ A is a resolution if and only if A is Koszul and B is the Koszul dual coalgebra of A, and that in characteristic zero every finite permutation rack whose permutation has a cycle of length divisible by 4 (resp. 3) fails in weight 3 (resp. 4). For finite non-degenerate solutions that are not involutive and have a finite-dimensional Nichols algebra, such as the dihedral quandle of order 3, a negative answer already follows from a remark of Farinati and García Galofre and a theorem of Jespers, Kubat and Van Antwerpen. By exact computation, 13 of the 29 isomorphism classes of bijective solutions on three points fail. This is an unrefereed note.

All Incorrect Equilibria of the Planar Four-Agent Formation Law Are Unstable

2026-10-01T00:00:00+03:00

Four agents in the plane move by the standard distance-based gradient law ṗᵢ = Σ_{j≠i} e_ij (p_j − pᵢ), e_ij = ‖pᵢ − p_j‖² − d̄_ij², where all six desired distances come from one planar target. In an Oberwolfach report of 2015, B. D. O. Anderson asked whether, as for rectangular targets, all incorrect equilibria are saddle points when the target is a general quadrilateral or a triangle with an interior agent. We show that the answer is yes for every planar target: at every incorrect equilibrium the Hessian of V = ¼ Σ_{i 0, where λ is the affine dependency of the agents, and the smallest Hessian eigenvalue is at most −κ‖λ‖²/2 ≤ −√(8V/3); both constants are sharp. The collinear case is known. For the spanning case we combine the classical self-stress of four planar points, a Schur-complement argument as in recent work of Criscitiello on the s-stress, and an inequality between two 2×2 matrices depending only on λ, proved by explicit identities in the elementary symmetric functions of λ. Consequently almost every initial condition converges to a formation congruent to the target, while collinear initial conditions show that literal global convergence fails. This is an unrefereed note.

A Negative Answer to a Question of Santos on the Dual Diameter of Surfaces with Boundary

2026-10-01T00:00:00+03:00

Let f̄(2,n) be the maximum diameter of the dual graph of a triangulated surface with boundary on n vertices. At a 2012 Oberwolfach workshop, Santos asked whether f̄(2,n) ≤ n − 3. We show that the answer is no. A triangulation M₇ of the real projective plane minus two disjoint open discs has 7 vertices, 10 triangles and dual diameter 5. The same complex appears in work of Holmes on Stanley–Reisner rings with Serre's property (S₂), where it is not identified as a surface. Gluing copies of M₇ along boundary edges gives f̄(2,n) ≥ n − 3 + ⌊(n − 2)/5⌋ for all n ≥ 3, so the excess over n − 3 is unbounded. An orientable surface of genus 2 with 10 vertices and dual diameter 8 gives the bound n − 3 + ⌊(n − 2)/8⌋ for orientable surfaces. A short layering argument shows f̄(2,n) ≤ max(n − 3, 2n − 8), and a theorem of Holmes gives max(2n − 10, n − 2). Computer searches with DRAT-certified unsatisfiability proofs, confirmed by an exhaustive enumeration of the surfaces with at most 10 vertices, show that f̄(2,n) = n − 3 for n ≤ 6 and f̄(2,n) = n − 2 for 7 ≤ n ≤ 10 (for 6 ≤ n ≤ 9 these values also follow from results of Holmes), that M₇ is the only counterexample with at most 7 vertices, and that orientable surfaces with at most 9 vertices satisfy the bound. This is an unrefereed note.

Equal Focal Inverted Pedal Areas Can Vanish in Elliptic Billiards

2026-10-01T00:00:00+03:00

For an elliptic billiard with a nondegenerate confocal elliptic caustic and effective even period, we prove that the pedals of the two unit-focus-inverted orbit polygons are congruent and have equal ordered signed areas. Equality does not imply nonvanishing. We construct a continuous family of genuine ten-periodic billiards of winding three and certify opposite signs of the common area at two rational major semiaxes. Exact rational interval arithmetic and the intermediate value theorem give a member for which both areas are zero. The construction satisfies the reflection law, has ten distinct vertices and a nondegenerate confocal elliptic caustic. As a contrast, we give an explicit positive formula for the counterclockwise four-periodic case. The focal area ratio remains one wherever its denominator is nonzero; the common-zero example obstructs an everywhere-defined literal quotient, not that ratio identity on its natural domain. Self-audited, unrefereed preprint prepared with AI assistance. Classical symmetry is credited; application novelty is undetermined after bounded literature searches. No independent human review, formal proof-assistant verification or absolute-priority claim is made. The source identity is AMR-050-0068 (raw ID 5100068), invariant k908,a in the frozen ulamai/UnsolvedMath v1.6.0 dataset. This is not a hyperbolic-caustic or whole-source-list closure. PDF, English LaTeX source and standard-library exact reproducibility checkers are included. Author: Alper Ferudun, Mercury Software GmbH.

Isogonal Conjugacy and Equal Focal Inverted Pedal Areas

2026-10-01T00:00:00+03:00

Invert the vertices of a triangle about a point and form the pedal triangle of its inverse with respect to that same point. We derive its ordered signed area in terms of the original circumcircle power and the three vertex distances. The formula proves that every finite isogonal-conjugate pair off the sidelines produces equal, nonzero inverse-pedal areas. For a genuine three-periodic elliptic billiard, the optical property of the ellipse and the billiard reflection law make its two foci such a pair. This gives an elementary proof of invariant k908,b: the literal focal area quotient is one for every nondegenerate elliptic-caustic triangular orbit. Two exact orbits in the same ellipse and caustic show that the individual area can nevertheless vary. Classical triangle identities are explicitly separated from the application; no absolute novelty or priority is claimed. AMR-050-0069 (raw ID 5100069, frozen ulamai/UnsolvedMath v1.6.0): COMPLETE_PROOF of the exact N=3 invariant k908,b. Finite isogonal conjugates off triangle sidelines have equal nonzero ordered signed inverted-pedal areas. The actual foci of a genuine three-periodic billiard triangle with a nondegenerate confocal elliptic caustic satisfy those hypotheses, so the denominator is nonzero and the quotient equals 1 throughout the family. The equality was already stated as Observation 1 by Reznik, Garcia and Helman (arXiv:2012.03020v2, Section 3); this note supplies an explicit proof and nonvanishing argument, not a claim to discovering the observation. No other-period or neighboring-invariant closure is claimed. The PDF, LaTeX source and dependency-free exact Python checkers are included. This self-audited, AI-assisted preprint is unrefereed. No independent human review, formal proof-assistant verification or absolute priority is claimed. The bounded literature audit retains access gaps for the final subscription journal version and full book chapters.

Four-Periodic Counterexamples to Two Elliptic Billiard Invariants

2026-10-01T00:00:00+03:00

We give two exact counterexamples using one connected family of convex, primitive four-periodic billiards in a noncircular ellipse with a fixed confocal elliptic caustic. Write A for the orbit's ordered signed area, A' for the area of its consecutive tangent-intersection polygon, A'' for the area of its caustic-contact polygon, and theta_i for the internal angles of the original orbit. The quantity (A'/A) product_i sin(theta_i/2), listed as k107 for N congruent to 0 modulo 4, varies on this family. So does the product A'A'', listed as old k111 for even N in arXiv:2004.12497v11. The first assertion survives in the final 2021 journal table; the old second row does not. The renumbered final-journal k111 is a different, odd-period identity and is not contradicted. With s=a^2+b^2, D=a^2 b^2 and d(t)=(b^2-a^2) sin(t) cos(t), the displayed family has A'A''=16D(D+d(t)^2)/s^2 and (A'/A) product_i sin(theta_i/2)=2(D+d(t)^2)^2/(D s^2). Both vary whenever a>b>0. For a=4, b=3, an axis diamond and an axis-aligned rectangle share the caustic with semiaxes 16/5 and 9/5. Their area products are 331776/625 and 576, and their angle-area quantities are 288/625 and 625/1152. Boundary membership, reflection, interior segment tangency, primitive period and positive signed areas are established explicitly. The contradiction does not rely on self-intersection, hyperbolic caustics or numerical inference. The standard four-periodic geometry and Garcia and Reznik's Proposition 4.9 area identities are credited. The old product row's omission is documented without speculating about its reason. This is one version-qualified corrective note covering frozen dataset records AMR-050-0001 and AMR-050-0005 (raw IDs 5100001 and 5100005), not two deposits or a solution of the full source list. Bounded literature checks do not establish absolute novelty or priority. The manuscript is AI-assisted and unrefereed; no independent human review or formal proof-assistant verification is claimed. Author affiliation and contact: Mercury Software GmbH; alper@mercurycodelab.com; https://github.com/AlperTheKing.

Nonconstant Antipedal Area Ratios in Four-Periodic Elliptic Billiards

2026-10-01T00:00:00+03:00

For a periodic elliptic billiard, let A' be the area of the polygon formed by consecutive tangents to the ellipse, and let A*_M be the antipedal area of the original orbit with respect to a fixed point M. We give an exact counterexample to the claimed constancy of A'/A*_M for periods divisible by four, listed as k402 in Table 5 of Reznik, Garcia and Koiller's 2021 article Fifty New Invariants of N-Periodics in the Elliptic Billiard. The denominator is unprimed and the printed point condition is All; taking M to be the center O is allowed. On a standard connected family of convex primitive four-periodic orbits in a fixed noncircular ellipse with one fixed confocal elliptic caustic, put s=a^2+b^2, D=a^2 b^2 and d(t)=(b^2-a^2) sin(t) cos(t). We derive A'/A*_O=D(D+d(t)^2)/(D^2+s^2 d(t)^2), with sharp range [D s^2/(s^2-2D)^2, 1] on the displayed family. For a=4,b=3, two orbits share the caustic with semiaxes 16/5 and 9/5 and give the ratios 1 and 90000/113569. The analytic continuous-family certificate proves reflection and strictly interior caustic tangency; all witness polygons are finite, convex and positively oriented. The four-periodic family and caustic are known geometry and are explicitly credited. This note refutes the literal k402 assertion and frozen dataset record AMR-050-0019 (raw ID 5100019), not every source invariant. A bounded primary-source search did not identify a correction of this exact row but does not certify novelty or absolute priority. Two standalone exact rational checkers accompany the analytic proof. This AI-assisted manuscript is unrefereed; no independent human review or proof-assistant verification is claimed. Alper Ferudun, Mercury Software GmbH. Contact: alper@mercurycodelab.com. GitHub: https://github.com/AlperTheKing.

Focal Pedal-Antipedal Area Products in Elliptic Billiards

2026-10-01T00:00:00+03:00

For a genuine elliptic billiard orbit of least period divisible by four, we prove that the signed area of its pedal polygon at either focus times the signed area of its antipedal polygon at the same focus is constant throughout the fixed-ellipse, fixed-confocal-caustic family. Convex and coprime star windings are included. We derive explicit factors relating focal pedal area to outer tangent area and focal antipedal area to original area, using a velocity-product telescope, an actual Jacobi quarter-period permutation, and direct signed-area algebra. The known even-period original/outer area-product theorem then gives the requested invariant. Zero signed antipedal area is allowed. The four-periodic value is 16 a^4 b^4/(a^2+b^2)^2. We explicitly qualify the least-period convention: repeating a primitive six-periodic orbit to obtain twelve listed vertices does not meet it, and exact controls show that the unrestricted repeated-walk extension is false. Hyperbolic or degenerate caustics and other source assertions are not covered. Source record: AMR-050-0021 (raw ID 5100021, k403,b), Hugging Face dataset ulamai/UnsolvedMath. Self-audited, AI-assisted, unrefereed preprint; no independent human review or proof-assistant verification is claimed. Established elliptic billiard geometry, Stachel's canonical parametrization and Chavez-Caliz's even area-product theorem are credited. Novelty remains undetermined after a bounded primary search, and no absolute priority claim is made. The source ZIP contains the English LaTeX manuscript, authored proof and source-hypothesis notes, and portable standard-library exact controls with their actual execution outputs. Rounded higher-period observations are corroboration, not the proof.

Center-Pedal Area Products in Elliptic Billiards

2026-10-01T00:00:00+03:00

We prove constancy of the product of the signed area of an elliptic billiard orbit and the signed area of its pedal polygon at the common center, for a fixed nondegenerate confocal elliptic caustic and least period odd or divisible by four. Primitive star trajectories are included. A reflection-coordinate telescope expresses the centered pedal area as a fixed multiple of a Jacobi trace. After complexifying the billiard parameter, the original area and this trace have complementary simple-pole sets; odd pairing and reversed-edge pairing remove every pole of their product. This establishes invariant k203,b under an explicitly stated genuine-period convention. Source record: AMR-050-0012 (raw ID 5100012, k203,b), Hugging Face dataset ulamai/UnsolvedMath. Exact standard-library controls verify the centered-pedal prefactor and actual low-period geometry. Primitive six-periodic controls show why an unrestricted repeated-list interpretation would be false. Hyperbolic and degenerate caustics are excluded. This is a self-audited, AI-assisted, unrefereed preprint; no independent human review, formal proof-assistant verification or absolute priority claim is made. Prior geometry, Jacobi parametrization and the established complex-pole method are credited, and novelty remains undetermined after a bounded primary-literature search.

Outer-Pedal Area Products in Elliptic Billiards

2026-10-01T00:00:00+03:00

For a fixed confocal elliptic-caustic billiard family with least period congruent to two modulo four, we prove that the signed area of the outer tangent polygon times the signed area of its pedal polygon with respect to any fixed point is constant. Coprime star trajectories are included. The tangent-intersection polygon is an explicit affine image of a half-step shift of the billiard orbit. After complexification, its area and the outer-pedal area have disjoint possible simple-pole sets. Central pairing separates the pedal map into two area traces; two reversal symmetries provide the complementary zeros that make their product entire and elliptic. This proves invariant k303,a under an explicit least-period convention. For the center pedal, the same argument also covers every odd period, proving k303,b for all traversal lengths not divisible by four, including repeated traversals. We evaluate the six-periodic constant and give exact counterexamples to the unrestricted arbitrary-point repeated-list interpretation and to adding period four in the centered case. No absolute priority claim is made. Source records: AMR-050-0014 (raw ID 5100014, k303,a) and AMR-050-0015 (raw ID 5100015, k303,b), ulamai/UnsolvedMath v1.6.0. Version 1.1 extends the existing manuscript, whose prior version DOI is 10.5281/zenodo.23087221 and concept DOI is 10.5281/zenodo.23087220. No separate manuscript is added. The k303,a arbitrary-fixed-point theorem retains its explicit genuine least-period condition N congruent to two modulo four. The additional k303,b center-pedal theorem covers every closed traversal length not divisible by four, including repeated traversals and all admitted signed star windings. Both use a fixed noncircular ellipse and fixed nondegenerate confocal elliptic caustic, boundary-tangent feet and signed shoelace areas. The centered N=4 products 8*a^2*b^2 and 2*(a^2+b^2)^2 differ by 2*(a^2-b^2)^2; this refutes an all-period extension, not k303,b. No hyperbolic, degenerate or whole invariant-list extension is asserted. This is a self-audited, AI-assisted, unrefereed preprint. No independent human review, formal proof-assistant verification, first-proof or absolute priority certification is claimed. Novelty remains undetermined after bounded primary literature searches. Credited confocal geometry, Jacobi parametrization and the established meromorphic pole-cancellation method are prior mathematics.

Steiner-Centroid Pedal Area Ratios in Elliptic Billiards

2026-10-01T00:00:00+03:00

For an odd-periodic billiard in an ellipse with a fixed confocal elliptic caustic, let C be the Steiner curvature centroid: the weighted average of the vertices with weights sin(2 theta_i), where theta_i is the internal reflection angle. Let A be the signed area of the original chord polygon and B_C the signed area of its pedal polygon with respect to C. We prove that C is always defined, that B_C is nonzero, and that A/B_C is invariant throughout the family. The result covers every coprime odd winding and odd repeated traversal, with an explicit signed-star convention; circular billiards are treated separately. This resolves invariant k501 in Table 6 of Reznik, Garcia and Koiller's Fifty New Invariants of N-Periodics in the Elliptic Billiard. The proof combines the classical pedal-area quadratic with reflection trace identities, complementary elliptic-function divisors, a character argument on an odd quotient torus, and a strict odd-grid theta inequality. The quotient-torus argument also identifies the moving centroid's axis-aligned elliptic locus. Exact standard-library checks accompany the proof but are not used as a substitute for the general analytic argument. This is an unrefereed, AI-assisted preprint; no absolute priority, independent human review, or formal proof-assistant verification is asserted.

Focal Distance Products of Outer Elliptic Billiard Polygons

2026-10-02T00:00:00+03:00

We prove an explicit positive, phase-independent formula for the product of Euclidean distances from either focus of an ellipse to the consecutive boundary-tangent intersections of an elliptic billiard. The theorem assumes a nondegenerate confocal elliptic caustic and a least period divisible by four. It includes all admitted coprime star windings and repetitions of those genuine periods; concentric circular billiards are treated directly. A factorization of the actual squared focal distance, Jacobi quarter-period pairing and complete cyclic divisor cancellation give the general proof. Standard parametrization and Jacobi identities are credited, and the already known four-periodic case is not presented as new. An unrestricted divisible-by-four listed length is a different assertion: four traversals of an exact primitive triangle give two unequal distance products in the same fixed ellipse and caustic. Thus this is a complete proof of an explicit least-period theorem and a counterexample to its stronger repeated-list extension, not an unqualified closure of every interpretation of invariant k115 or frozen record AMR-050-0007 (5100007) in ulamai/UnsolvedMath v1.6.0. No hyperbolic or degenerate caustic is included. An unchanged standard-library exact checker and an optional, separately identified mpmath geometry diagnostic accompany the analytic proof. This English preprint is AI-assisted, self-audited and unrefereed. Novelty remains undetermined after a bounded primary-source review; no independent human review, proof-assistant verification or absolute-priority certification is asserted.

Original Antipedal Centroids in Elliptic Billiards

2026-10-02T00:00:00+03:00

We establish explicit, phase-independent vertex-centroid formulas for antipedals of the original polygon in elliptic billiards with a nondegenerate confocal elliptic caustic and even least period. The pole is the ellipse center or either focus, and all consecutive antipedal intersections are finite. The result covers admissible coprime star trajectories and repetitions of even primitive orbits; the circle is treated separately. A paired-chord identity and perimeter stationarity yield the focal coefficient in terms of semiaxes, caustic parameter and mean side length. An even number of listed vertices is not sufficient: exact twice-repeated primitive triangles show that the unrestricted even-list extension fails. This note gives a precisely scoped interpretation of invariant k405 in Reznik-Garcia-Koiller's Table 5, rather than declaring every interpretation of the abbreviated dataset record AMR-050-0023 resolved. Source record 5100023 belongs to the frozen Hugging Face dataset ulamai/UnsolvedMath v1.6.0. The construction is the original-polygon antipedal and its unweighted vertex mean, not an outer-polygon antipedal or an area centroid. No hyperbolic or degenerate-caustic extension is claimed. Portable exact symbolic and rational verification files accompany the analytic proof. This is an AI-assisted, self-audited, unrefereed preprint. Classical ingredients are credited; novelty remains undetermined, and no independent human review, proof-assistant verification or absolute-priority certification is asserted.

Inner Steiner Pedal Ratios and Exceptional Zero-Area Families

2026-10-01T00:00:00+03:00

For an odd-periodic elliptic billiard with a fixed nondegenerate confocal elliptic caustic, consider the polygon of caustic contact points and its own Steiner curvature centroid. We prove that this centroid is finite at every real phase and that the signed area of its pedal polygon is a phase-independent real multiple R of the contact polygon's area. Thus the reciprocal area ratio proposed as invariant k503 is constant whenever defined. Its denominator need not be nonzero: we construct a genuine convex five-periodic family in which the stationary pedal area vanishes identically, specifying its nondegenerate ellipse and caustic by a uniquely isolated algebraic root of degree 18. This is a correction to universal definedness, not phase variation of a defined ratio. The result includes odd coprime star windings under an explicit angle convention, odd repetitions, and a separate circle case. The analytic proof uses the pedal-area quadratic, elliptic-function poles and characters, and an odd quotient torus. Portable exact quadratic-field, symbolic-identity, and rational Bernstein certificates accompany the exceptional family. This is an unrefereed, AI-assisted preprint; no absolute priority, independent human review, or formal proof-assistant verification is asserted.

A Six-Periodic Zero of Focal Outer-Antipedal Area

2026-10-02T00:00:00+03:00

For a periodic elliptic billiard, form the polygon of intersections of consecutive boundary tangents, then its antipedal with respect to either focus. Classical central symmetry implies equality of the two signed antipedal areas for effective even periods. Their quotient is therefore one where defined, but it need not be defined everywhere. We give an explicit convex billiard of least period six on an ellipse of aspect ratio 1+sqrt(3) for which both areas vanish, while all antipedal vertices are finite, all edges are nonzero and the vertices are not collinear. A direct coordinate calculation gives the signed area of an axial six-periodic family and isolates the zero exactly. This is a denominator obstruction, not a counterexample to equality on the quotient's natural domain. The earlier original-polygon antipedal zero at aspect ratio 2 is distinguished and credited. Source record: AMR-050-0036 (raw ID 5100036), invariant k608 in the frozen Hugging Face dataset ulamai/UnsolvedMath v1.6.0. Supporting lines and ordered signed area are used. The effective-even scope includes admitted coprime stars and repetitions of even-primitive orbits; no all-phase zero, odd-primitive repeated-list extension, hyperbolic-caustic extension or unqualified whole-source resolution is claimed. A portable Python standard-library exact certificate accompanies the analytic proof. This is a self-audited, AI-assisted, unrefereed preprint. Classical symmetry and the prior original-polygon zero are credited; novelty remains undetermined after a bounded search. No independent human review, proof-assistant verification or absolute-priority certification is asserted.

Focal Inversion Area Products in Elliptic Billiards

2026-10-02T00:00:00+03:00

For an elliptic billiard with a fixed nondegenerate confocal elliptic caustic, we prove that the original signed shoelace area times the signed area of its unit focal inversion is a positive phase-independent constant when the genuine least period is divisible by four. Both foci give the same constant. The theorem includes all admitted coprime star windings and repetitions of those primitive cycles; concentric circles are treated directly. The general proof derives the focal denominator identity, enumerates every possible complex singularity and cancels all poles on a common Jacobi torus. Isotropic tangency and cyclic reversal reduce the apparent higher-order inverse-area poles to simple ones, and the two areas have the required complementary zeros. Classical parametrization and Jacobi methods are credited, as is the already published simple four-periodic value 4. An unrestricted divisible-by-four listed length is a stronger, false assertion: two exact primitive six-periodic phases in one fixed ellipse and caustic have unequal products after being traversed twice into lists of length twelve. A separate repeated-triangle control confirms the same convention boundary. This is a complete proof of an explicit genuine-period theorem and a counterexample to its unrestricted repeated-list extension, not an unqualified closure of every interpretation of source invariant k804,a or frozen record AMR-050-0044 (5100044) in ulamai/UnsolvedMath v1.6.0. Hyperbolic and degenerate caustics are not included. Three standard-library exact controls and a separately identified 33-family numerical diagnostic support, but do not replace, the analytic proof. This English preprint is AI-assisted, self-audited and unrefereed. Novelty remains undetermined after a bounded primary-source review; no independent human review, proof-assistant verification or absolute-priority certification is asserted.

Focal Inversion Area Ratios in Elliptic Billiards

2026-10-02T00:00:00+03:00

For every nondegenerate confocal elliptic-caustic billiard of genuine least period n >= 6 with n congruent to 2 modulo 4, we prove that the original ordered signed area divided by the signed area after unit inversion about either boundary-ellipse focus is a positive phase-independent constant, identical at the two foci. The proof includes all admitted coprime star windings and establishes finite inverse vertices and a nonzero denominator. In confocal Jacobi coordinates, the two area traces have the same two possible simple poles on a reduced quotient torus. Residue cancellation and imaginary antiperiodicity force proportionality; real geometry proves positivity. Repetitions of admitted primitive cycles preserve the ratio, and the circular case is treated directly. The target is k805 in arXiv:2004.12497v11, corresponding to frozen record AMR-050-0046 / 5100046. Garcia and Reznik's 2022 Proposition 4.16 already gives the simple-six-periodic formula, which is expressly credited. Exact controls reproduce the ratio 32/27 at two six-periodic phases. Two primitive triangles in one fixed family, each traversed twice to give listed length six, instead have ratios (10584 + 72 sqrt(105))/25 and (10584 - 72 sqrt(105))/25; this excludes only the stronger unrestricted listed-length interpretation. A separately executed 85-decimal diagnostic covers 27 target families and 18 odd-period controls, five phases each; these finite checks are not the general proof. No hyperbolic or degenerate caustic extension or unqualified whole-source closure is claimed. This English preprint is AI-assisted, originating-researcher self-audited and unrefereed. Novelty remains undetermined; no independent human review, proof-assistant verification, guaranteed indexing or absolute-priority claim is made.

Products of the Two Focal Inversion Areas in Elliptic Billiards

2026-10-02T00:00:00+03:00

For every odd-periodic real billiard family in an ellipse, we prove that the product of the trajectory-order signed areas of the two polygons obtained by unit inversion of the original vertices about the two original boundary foci is a positive phase-independent constant. The result includes every primitive odd period, admitted coprime star winding, orientation reversal and fixed odd repeated traversal; an r-fold traversal multiplies the product by r^2. Circular boundaries are treated directly. A real-caustic parity lemma excludes an omitted nondegenerate odd hyperbolic or degenerate family. In confocal Jacobi coordinates, each inverse-area trace has at most simple poles. For odd periods the two focal pole sets are disjoint, and reversal forces the opposite trace to vanish at each possible pole. Their product is therefore holomorphic on a compact quotient torus and constant. Real geometry establishes positive areas and finite inversion vertices. The target is invariant k903,a in arXiv:2004.12497v11, corresponding to frozen record AMR-050-0062 / 5100062. Reznik, Garcia and Helman's published N=3 formula is expressly credited, together with the classical Jacobi parametrization and meromorphic method. Exact controls verify the common product 1/324 for two non-equivalent triangular phases in one fixed family, odd-repeat scaling and a separate circle case. Genuine period-four phases instead give products 1 and 25/16, showing that the odd hypothesis cannot be removed. A separately executed 85-decimal diagnostic samples 42 odd-period families, including 27 stars, at five phases each; these finite checks are not the general proof. The quantity is signed area in trajectory order, not the unsigned area of a self-crossed polygon's bounded faces. This English preprint is AI-assisted, originating-researcher self-audited and unrefereed. Novelty remains undetermined; no independent human review, proof-assistant verification, guaranteed indexing or absolute-priority claim is made.

A Pentagonal Counterexample to a Poncelet Squared-Side Criterion

2026-10-02T00:00:00+03:00

We construct an explicit fixed circle and strictly nested noncircular ellipse admitting a Poncelet family of primitive convex pentagons with constant sum of squared side lengths, even though the circle center is neither the ellipse center nor either focus. If t is the unique root in (1/10,1/5) of 25t^3-50t^2-40t+8=0, the invariant value is 175/18-(625/48)t. All tangencies occur on the actual edge segments. This gives an exact counterexample to the necessity direction of the fixed-family center-or-focus criterion proposed in Murad 2026, Section 6, Conjecture 1, without contradicting its proved triangle results. The construction starts from a bicentric circle pair. For the polygon obtained by unit inversion of the bicentric vertices about the inner center, we express the squared-side sum as a two-step normal trace. Jacobi's bicentric parametrization and the published MI-II cyclic identity of Khare, Lakshminarayan and Sukhatme show that this trace is constant. A nonsingular projectivity agrees with vertex inversion on the outer circle and carries the complete nested Poncelet configuration to the required circle/ellipse pair. An isolated cubic root then supplies a strictly convex primitive five-cycle, and Poncelet closure supplies the whole family. The investigation originated with frozen record AMR-050-0053, invariant k811. That source uses incompatible descriptions of its dual construction. We prove the squared-side invariance of two precise interpretations separately, crediting the published bicentric cosine theorem and MI-II identity, but do not silently repair or claim unqualified closure of k811. Two standard-library exact checkers certify the displayed algebraic pentagon and distinguish the reciprocal constructions on exact fixtures; the all-family theorem rests on the analytic proof, not finite sampling. This English preprint is AI-assisted, originating-researcher self-audited, and unrefereed. Novelty remains undetermined; no independent human review, proof-assistant verification, guaranteed indexing, or absolute-priority claim is made.

Exact Heisenberg Determinant Limits and Tower-Dependent Homological Torsion

2026-10-02T00:00:00+03:00

We compute the exact finite determinant limit for Kammeyer's element f=1-2a+2b in the integral group ring of the discrete Heisenberg group. Along the congruence tower modulo powers of any fixed odd prime p, the normalized Fuglede-Kadison determinant converges to 2^(1/(p+1)), whereas the infinite determinant is 2. An exact central-character factorization and an integral-lattice tail bound justify the ordinary limit without passing an unbounded logarithm through weak spectral convergence. Attaching a two-sphere and a three-cell to the Heisenberg nilmanifold realizes the same discrepancy in integral homology. The resulting finite connected three-dimensional CW complex has determinant-L2-acyclic universal cover and L2-torsion log 2, but its normalized alternating integral homology torsion tends to log 2/(p+1). This limit therefore depends on the residual tower of a single space and refutes the unrestricted finite-CW formulation of modified homological torsion approximation in Hughes and Luck, arXiv:2510.20959v2, Conjecture 1.2. The space is neither aspherical nor a closed manifold; no aspherical-manifold conjecture or original AIM 11.1 closure is claimed. Kammeyer's underlying counterexample and infinite determinant, Deninger's determinant formula, Luck's nilmanifold vanishing theorem, and Boschheidgen's representation-theoretic ingredients are explicitly credited. Two unchanged standard-library exact checkers support the displayed finite calculations; the general conclusions rest on the analytic proof, not finite sampling. This English preprint is AI-assisted, originating-researcher self-audited, and unrefereed. Novelty remains undetermined. No independent human review, proof-assistant verification, guaranteed indexing, or absolute-priority claim is made.

A Negative Answer to Wolansky's Question on Mutually Dominating Multiphase Measure Spaces

2026-10-02T00:00:00+03:00

In an Oberwolfach report of 2017, Wolansky introduced a domination order between vector-valued measure spaces (X, σ) and (Y, η) with k components, defined by one Markov kernel that transports every σ_i to η_i, and asked whether two atomless spaces that dominate each other must admit deterministic maps T : X → Y and S : Y → X with T_#σ_i = η_i and S_#η_i = σ_i. We show that the answer is no as posed, already for k = 2 and compact atomless spaces. Mutual domination is equality of the colour laws (Blackwell equivalence), and a deterministic transport has to respect colours, so it must carry colour fibres onto colour fibres. This gives explicit examples on [0,1] and [0,1]² in which only one of the two maps exists, and an example on [0,1] in which neither exists. On the positive side, for standard Borel spaces with equal colour laws, T exists whenever the colour fibres of X are atomless, for instance for atomless spaces with k = 1 or with finitely many colours. We give necessary conditions in general and an exact classification for finite spaces. The results are consequences of the classical comparison of experiments (Blackwell, Le Cam, Torgersen), and we claim no novelty beyond answering this question with explicit atomless examples and the atomless-fibre positive case. This is an unrefereed note.

The Sharp Hexagonal Constant in Steinerberger's Geometric Uncertainty Principle on the Square

2026-10-02T00:00:00+03:00

For a partition of [0,1]² into N measurable pieces Ω_i, let F = Σ_i |Ω_i| A(Ω_i) + Σ_i (|Ω_i| − min_j |Ω_j|), where A is the Fraenkel asymmetry with respect to discs. Steinerberger proved F ≥ 1/60000 for large N and asked, in Oberwolfach Report 49/2013, whether the extremal configuration is a hexagonal tiling, which would give the optimal constant c ≈ 0.07. We show that this is so on the square, in an asymptotic sense, with the constant c_H = 0.07446575455…, the asymmetry of the regular hexagon. Every partition of [0,1]², into any number of pieces, satisfies F > c_H + 0.06 m, where m is the smallest measure of a piece, while for every N a honeycomb partition with a boundary correction has F ≤ c_H + O(N^(−1/2)). Hence the infimum I_N over partitions into N pieces lies between c_H + 0.03/N and c_H + O(N^(−1/2)), I_N tends to c_H as N → ∞, and the value c_H is attained by no partition. We do not show that near-minimisers are geometrically close to a honeycomb, and we do not treat other domains. The proof bounds F below by a covering functional of discs of unequal sizes and uses Laguerre cells, a moment lemma of Fejes Tóth for the indicator of the complement of a disc (proved in an appendix), Euler's formula, and a tangent inequality in the number of sides with explicit constants. The same proof answers, in the same asymptotic sense, a weighted version asked by Burchard: the limit of the infimum is min{α c_H, 1}, so honeycomb partitions stop being asymptotically extremal exactly at the weight α = 1/c_H = 13.4289…. This is an unrefereed note.