Mathematics

Contact & symplectic geometry · Preprint · arXiv:2505.16766v3

The sharp cost of losing contact.

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

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

New phenomenon. 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

Zero volume. A lasting spectral memory.

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

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.

New phenomenon. 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

Answers questions in Friedman–Park’s 2016 and 2024 papers. 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.

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

New concepts and phenomena. 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.

Rigidity 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

Resolves the body–pin partition conjecture, 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

Proves the 1987 Bland–Jensen conjecture, 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.