EulerSolve Research Papers

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

2026-09-05T00:00:00+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

2026-09-05T00:00:00+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

2026-09-05T00:00:00+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

2026-09-05T00:00:00+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

2026-09-05T00:00:00+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

2026-09-05T00:00:00+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

2026-09-05T00:00:00+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

2026-09-05T00:00:00+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

2026-09-05T00:00:00+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

2026-09-05T00:00:00+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

2026-09-05T00:00:00+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

2026-09-05T00:00:00+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

2026-09-05T00:00:00+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

2026-09-05T00:00:00+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

2026-09-05T00:00:00+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

2026-09-05T00:00:00+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

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

2026-09-05T00:00:00+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

2026-09-05T00:00:00+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

2026-09-05T00:00:00+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

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