Mathematics

Hyperplane arrangements & Kähler groups · Preprint · Lean 4 V1

The Chen Ranks Conjecture and Geometric Separation of Resonance in Complex Geometry

This work proves the stable rational-rank form of Suciu’s Chen ranks conjecture (2001) for every complex affine hyperplane arrangement. One of the flagship problems at the intersection of complex algebraic geometry and topology, the conjecture has guided hyperplane arrangement theory for twenty-five years. It relates the stable growth of the fundamental group’s metabelian quotient to the linear components of resonance. For an arrangement of N hyperplanes, the formula holds in every degree q ≥ max{2, N − 1}, improving to q ≥ max{2, N − 2} for nonempty central arrangements.

Geometric separation of resonance extends to smooth connected quasi-projective varieties whose first cohomology is pure of weight two, with an effective range determined by the first Betti number. For every compact connected Kähler manifold, including nonprojective examples, it proves reducedness of the canonical projective resonance scheme and the stable Chen ranks formula, answering Aprodu–Farkas–Raicu–Suciu’s Question 7.3 and proving their Conjecture 7.4 (2024), respectively.

Lean 4 V1 (work in progress) verifies the affine and central arrangement formulas from AFRS’s effective canonical Koszul decomposition as the sole external mathematical input, with no additional project axioms or proof gaps.

Combinatorial rigidity & matroid theory · Preprint · Lean 4

When do pinned bodies become one rigid structure?

Stress Degeneracy of Direction Complexes of (2,2)-Sparse Graphs and Three-Dimensional Body–Pin Rigidity

A partition criterion resolves the body–pin conjecture, which had been open for nearly two decades. First proposed by Jackson–Jordán in 2009 and independently by Tanigawa in 2011, the criterion characterizes generic Euclidean rigidity through every partition of the bodies, with pin capacities 3, 5, and 6.

A stress-codimension theorem and recursive collinearity flags carry the proof through exceptional configurations. The companion Lean development proves the final maximum-rank equivalence end to end. A companion blueprint by Bryan Chen, a Mathlib maintainer, maps the paper’s argument and its Lean formalization.

Bill Jackson mentioned this work at Lancaster University’s September 2026 rigidity workshop.

Matroid theory · Preprint · Lean 4

No finite list can capture every obstruction.

Infinitely Many Excluded Minors for Weakly Orientable Matroids

This work proves the 1987 Bland–Jensen conjecture, which had been open for nearly four decades. Infinitely many pairwise nonisomorphic excluded minors rule out a finite forbidden-minor characterization of weakly orientable matroids.

The family grows in rank and size, but its representability argument takes place in quotient spaces of dimension two or three. Nine parity equations give a uniform obstruction; six rational quotient constructions establish minimality. The Lean formalization follows the chain from signed circuits and cocircuits to the failure of every finite forbidden-minor list.

Dillon Mayhew (University of Leeds) confirmed that the final Lean theorem expresses the intended statement about finite excluded-minor characterizations. On August 14, 2026, Jesús De Loera invited a talk on this work at a small seminar he organized.

Combinatorics, rigidity & analysis · Preprint

Complete Monotonicity of Inverse Powers of Spanning-Tree Polynomials: Positive-Semidefinite Compression, Clique Gluing, and Graph-Family Classification

This work classifies completely monotone inverse powers of spanning-tree polynomials for complete multipartite graphs. It also proves preservation under clique gluing, answering Scott–Sokal’s triangle-gluing question (2014). The cube and octahedron disprove planar-duality invariance for 1 < β < 3/2. Positive-semidefinite stress compression connects these analytic obstructions to universal rigidity.

Convex & discrete geometry · Preprint

The universal determinant formula for translated segments

The determinant bound is universally sharp exactly for d or d + 1 spanning segments in dimension d ≥ 2, resolving Averkov–Martini’s Problem 1 (2008).

Matrix analysis & algebraic geometry · Preprint

The Marcus–de Oliveira conjecture for n = 4

This work proves the Marcus–de Oliveira determinantal conjecture for 4 × 4 normal matrices. Proposed independently by Marcus (1973) and de Oliveira (1982), the conjecture asks whether the determinant of a sum of normal matrices lies in the convex hull of eigenvalue pairing products. Four was the first unresolved matrix size after the 1987 theorem for 3 × 3 matrices.

A new structural theorem reproduces the squared moduli of every minor of any 4 × 4 unitary matrix as one common convex mixture of the corresponding data from just two real orthogonal matrices. This simultaneous reduction comes from Gale duality and a real circle in the Picard curve of an elliptic quartic. The determinant problem then passes to the invariants of eight points on the projective line, where the maximum principle on algebraic curves reduces it to finitely many contact configurations.

Kinetic theory & statistical mechanics · Preprint

Uniform-in-Time Entropic Chaos, Entropy Production, and McKean’s 1966 Conjecture for the Kac Master Equation

Within Kac’s programme (1956), this work answers propagation questions of Carlen–Carvalho–Einav (2018) and Einav (2023) for the classical constant-rate, uniform-angle model. From entropically chaotic initial data of finite energy and entropy, entropy per particle converges uniformly for t ≥ 0 and instantaneous entropy production per particle uniformly for t ≥ δ > 0. Fixed-order log-power bounds propagate on bounded time intervals without loss of order.

The work also refutes McKean’s 1966 heat-entropy monotonicity conjecture: for each prescribed order n ≥ 5, a smooth positive density has a higher heat-entropy functional that increases under Kac collisions.

Higher log-power functionals can develop logarithmic-time transient peaks of order √N, even while total relative entropy and total variation distance to equilibrium vanish uniformly in time. This separates macroscopic entropy relaxation from higher-order amplification.

Arithmetic dynamics & semiabelian varieties · Preprint

Zariski Dense Orbits for Arbitrary Semigroups on Semiabelian Varieties

This work answers Ghioca–Scanlon’s arbitrary-monoid extension question (2014). A monoid of dominant regular self-maps on a semiabelian variety over an algebraically closed field of characteristic zero has a Zariski dense orbit exactly when its common rational invariants are constant. The theorem covers noncommuting maps, generating sets of arbitrary cardinality, and nonsplit extensions.

Over a fixed finitely generated field, one algebraic starting point realizes this dichotomy simultaneously for every monoid defined over that field.

Number theory & arithmetic dynamics · Preprint · Lean 4

Derksen's inverse conjecture on zero sets of linear recurrences

This work resolves Derksen’s inverse conjecture (2005). Every p-normal subset of the nonnegative integers—including negative weights, rational shifts, and finite modifications—is the exact zero set of a finite-order linear recurrence over the fixed field 𝔽_p(t), with a Lean 4 formalization of the inverse theorem.

Kinetic theory & collision invariants · Preprint · Lean 4

The spectrum can retain memory even when only two densities are conserved.

Collision invariants and long-lived spectral memory in a one-dimensional pinned chain

This work resolves Aoki–Lukkarinen–Spohn’s collision-invariant conjecture (2006), which had been open for twenty years. For every strictly pinned bandwidth, all measurable additive collision invariants are linear combinations of wave action and energy, determining the two-parameter equilibrium family. This identifies the conserved fields needed for macroscopic closure within the kinetic-theory programme of Hilbert’s sixth problem, with a companion Lean 4 proof of the classification.

Along the narrow-band equilibrium path, a positive linearized spectral gap closes as the band narrows. In linear response, preparations with identical wave action, energy, and initial entropy remain spectrally distinguishable over a diverging time window.

Calculus of variations & nonlinear PDE · Preprint · Lean 4

Profile comparison establishes global minimality for the radial Ginzburg–Landau vortex.

On Brezis’ Open Problem 6: Global Minimality of the Degree-One Ginzburg–Landau Vortex in Balls

Following Chen–Liu–Wei–Yang’s approach, this work develops the finite-ball/entire-space profile comparison in dimensions n ≥ 3. It establishes global minimality and uniqueness of the radial degree-one Ginzburg–Landau vortex in Bⁿ with identity boundary data, for every ε > 0.

A companion Lean 4 formalization follows the higher-dimensional proof from four stated analytic inputs.

Contact & symplectic geometry · Preprint · arXiv:2505.16766v3

Local contact degeneration has a sharp logarithmic cost.

Dynamical Geometry of Principal Bundle Constrained Systems: Compatible Pairs, Contact Degeneration, and Adapted Metrics

This work answers Komendarczyk’s Question 2.7 (2008). For a fixed contact form on a closed three-manifold preserved by a locally free circle action, the construction gives an adapted metric making the form a curl eigenform with any prescribed positive eigenvalue while keeping the circle generator unit Killing.

Local contact degeneration has a sharp logarithmic cost. On a base of dimension 2n, the least quadratic Hⁿ⁺¹ deformation cost for flattening the paired curvature on a ball of radius r in time T is comparable to 1 / [T log(R_*/r)] for fixed background data. The cost tends to zero as the target ball shrinks, even while contactness persists until the terminal time.

Hodge theory & spectral geometry · Preprint · arXiv:2506.00752v3

A region can lose its volume while its spectral memory persists.

Constructing Two Metrics for Hodge Theory of Constrained Bundles: Harmonic Stability and Magnetic Coercivity

This work proves the uniform magnetic frustration conjecture of Chakradhar, Gittins, Habib, and Peyerimhoff. Posed in §4.1 of their 2025 Mathematika paper, the conjecture asks whether a magnetic potential that cannot be gauged away has uniformly positive frustration on domains occupying almost all of the manifold with sufficiently small internal boundary. A covariant BV coercivity estimate supplies the bound.

In a specific family of connection metrics on circle bundles, the metric can lose rank while a positive Hodge spectral gap persists uniformly. A region whose limiting volume density vanishes still contributes magnetic Dirichlet-to-Neumann boundary energy to the scalar spectral limit, retaining its flat holonomy—the phase accumulated around loops.

Principal bundles & gauge geometry · Preprint · arXiv:2506.00728v3

Pointwise symmetry need not admit a global gauge realization.

Geometric Duality Between Constraints and Gauge Fields: Mirror Realization and Reduction Geometry on Principal Bundles

This work answers questions in Friedman–Park’s 2016 and 2024 papers. It determines the structure group of their diagonalization bundle and the second cohomology of its base, and realizes every integral solution of their three-eigenline equation on S² × S².

It also answers Ziller’s Problem 3(a) (2001) negatively and refutes its Florit–Ziller strengthening (2011). The construction produces fully fat bundles over closed hyperbolic four-manifolds whose underlying structure group nevertheless reduces to a circle.

Gauge mirrors test whether pointwise symmetries lift globally. An affine second-Chern charge lattice distinguishes reductions with identical paired curvature, even on a trivial ambient bundle. Finite-order lift spectra detect eigenbundle topology: for isomorphic rank-two eigenblocks, an order-six mirror exists exactly when they split into line bundles.

Harmonic maps & spectral geometry · Preprint · arXiv:2506.05816v3

Colliding eigenvalues do not erase their geometry.

Mirror Symmetry of Spencer-Hodge Decompositions in Constrained Geometric Systems: Selected Collisions and Weyl Splittings

This work answers Mazowiecka–Miśkiewicz–Schikorra’s stability question (2024) beyond sphere targets for every ℂP^N, under the stated strong convergence assumptions. The paper also proves Frank–Pintér–Varjas–Pályi’s complex-isolation conjecture (2025) for their crystalline four-band pencil.

Spectral incidence retains eigenspaces at collisions, while energy minimization selects persistent tangent models. Smooth three-band fields can share their spectrum and every jet yet differ in diagonalizability.

Minimizing harmonic maps · Preprint

Minimizing cones in complex projective spaces

This note completes Fujioka’s 33-year-old classification problem (1993). In every complex projective space, the nonconstant minimizing radial cones are exactly round projective-line hedgehogs of absolute degree one.

Geometric formality & Hodge theory · Preprint

Extremal Exterior Algebra and Maximal Betti Rigidity in Geometric Formality

On a closed, connected, oriented geometrically formal manifold, a maximal Betti number in any nontrivial degree forces a flat torus and flatness of every formal metric, resolving the higher-degree conjecture recorded by Pilca (2012).