Independent researcher in mathematical physics. No lab, no institution: a laptop, published data, and about two and a half years of asking what can be measured and what cannot.
I want a way to map and measure everything as simply as light does.
The work runs through one idea: a projector, the operator that keeps part of a space and discards the rest, carries both the geometry and the bookkeeping of a physical system. Most of my papers follow that idea into a different problem: entanglement, gravity, knots, many-body motion, arithmetic.
Results are graded by what supports them: proved, derived, recognised or measured. The ones that can be machine-checked are checked in Lean.
| Repository | What it proves | Check |
|---|---|---|
| offset-lean | The finite algebra behind The Offset Belongs to the Boundary: the entanglement offset as log-odds of empty versus full, asymmetry = tanh(offset/2), endpoint bounds, a boundary-column degree mechanism. 156 theorems + 7 headline results. |  |
| arithmetic-kakeya-finite-bridges | For the Epoch FrontierMath arithmetic Kakeya problem: rational forcing equals integer forcing, and every completed configuration keeps two independent surviving labels on every nonempty cut of initially unknown sites. 127 theorems. |  |
| what-russell-saw-lean | For What Russell Saw: energy conservation for every oscillator motion (Physlib), which the paper reads as Russell's paired-motion axiom; his v ∝ 1/a as an inverse-cube pull (exponent arithmetic from the similarity scaling), the apsidal angle diverging as the pull approaches inverse cube, the marginally stable orbit at r = 6M derived from the Schwarzschild effective potential, Maxwell stress and Ampère's sign, the tide, the vortex and the cone. 43 theorems. |
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| dirac-time-lean | For The Dirac Time of the Gigantefermion: the predictive quotient (Cayley–Hamilton closure and minimality), projector tangent geometry and its block formulas, the common-clock corollaries, the weak-value bound, path-ledger descent, finite return in discrete and continuous time, the entropy/correlation ledger (its nonnegativity inequalities as hypotheses) with unitary invariance of total entropy proved via Physlib, the moving-frame identity, the equal-endpoint witness, friction on projector rotations and the torus-knot silence criterion; what is not formalized is listed in its LIMITATIONS. 86 theorems + 3 headline results. |  |
| connection-energy-lean | For Non-Normality Is Connection Energy: [H, H†] = −2[D, A], the energy identity ⅛‖[H, H†]‖² = Σ_edges ‖Dᵢ − AᵢⱼDⱼAᵢⱼ†‖², and H normal exactly when D is parallel; the induced transport is a connection and a parallel section commutes with the holonomy around every loop. 20 theorems. |
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| light-ledger-lean | For Light Keeps the Ledger: boundary elimination and the matrix Smith transform, the transition-window census, Gram positivity, optical response as a reweighted metric and the peel, rigidity, completion steps, the Edition 7.2 spectral bounds, upward redistribution and oscillator realization, with the exact counterexamples that bound each claim. 122 theorems. |  |
| gravity-ledger-lean | For The Ledger Outlives the Metric: projector tangents are off-diagonal, the signed metric and reciprocal determinant, the local overlap cap, and the projector-overlap Taylor bridge from first-order differentiability (no second derivative), for idempotents of locally constant trace. 30 theorems. |  |
| square-case-lean | For The Square Case and 5D Spectral Anatomy of Four-Body Obstructions: the Gram invariant, the forced transport coefficient, symmetrisation of the eigenvalue block, the eigenframe relation TE + ET = (tᵢ + tⱼ)E, and interlacing (exactly one eigenvalue in every gap), and the four-body binary-wall theorems. 63 theorems. |
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| matter-at-a-scale-lean | For Matter at a Scale: the Bohr rod and the unique closure of the units dictionary (mass falls as atoms grow; the rejected assignment misses by (1+z)⁴), gravitational closure with α_G invariant, span and offset invariance of eigenspaces, and span invariance of the relative departure from normality. 17 theorems. |
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| three-body-lean | For The Three-Body Problem, Operator-First: the Gram operator's exact spectrum {cos² ϑ, 1}, rank drop at syzygy against gap closing at the pole ϑ = 0 (quadratically), the 120° symmetry forcing isotropy, the fold, reciprocity of the transport operator, the shifted ledger, and the collinear kernel. 22 theorems. |
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| foft-lean | For Foundational Operator Field Theory: the isotropic baseline carries no non-normality, the ledger factorisation with its vanishing first term, chirality killing every odd moment, the trace identities behind the exact Henrici dissipation rate and the norm-conserving dual flow, and degree-three homogeneity of the flow. 11 theorems. |  |
| suns-lean | For the Suns paper: the square-root law at an exceptional point, pair exchange after one loop and restoration after two, the branch-free ledger, modal decay against a positive numerical abscissa (initial transient growth) in one operator, and arithmetic checks of the mode ratio and lane width. 15 theorems. |  |
| spectral-stress-lean | For Spectral Stress on a Conservative Matrix Field: the exact sum rule, gauge invariance, the factorisation t = (ρ_b − ρ_c)β_bc β_cb, the amplitude A = −5√2/16 with the vanishing middle band, and the ℚ(√2) arithmetic of the limiting spectrum. 20 theorems. |
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| relational-atlas-lean | For The Operator-First Relational Atlas: the finite bridges graded THEOREM, including normality as commutation (B2), the traceless off-block change (B31), the critical angle and its complex continuation (B39), the reciprocal reduction to one comparison (B51), paired spectra (B68) and the repelling stratum (B74). 37 theorems. |  |
| coboundary-k3-lean | For The Permutation Coboundary Constant … Is k/3: every witness for k = 4, …, 8 checked in the Lean kernel, with the gauge minimum checked exhaustively (k = 4, 5) or by a proved-sound branch-and-bound search (k = 6, 7, 8), giving N/D = k/3 for every k = 4, …, 8. Also the non-abelian mechanism at k = 4, and the displacement inner products as the cycle-graph Laplacian 2I − A. 25 theorems. |
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| filament-fracture-lean | For the filament operator paper: the invariant interface subspace and Jordan chain inside the five-dimensional operator, the factorised characteristic polynomial, the closed-form resolvent norm equal to 1/ε on the circle |z|² = ε|μ| + ε², the identity ε_crit · σ_max = b², transient growth of the closed-form gain M(t) iff |μ| > 2b, and the linear against square-root coalescence paths. 14 theorems. |
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| spectral-stability-lean | For Spectral Stability Across Obstruction Interfaces: the transport operator αJ ⊗ Δ₁ is symmetric (Hermitian), the obstruction field is symmetric, the commutator starts at first order in ε, the bulge scaling, and the second-difference error; the leading coupling block MJ + JM = tr(M) J, the symplectic Fourier eigenmodes, and the Appendix A norm bound; and, recovered from the June drafts, the cascade's coupling-over-gap theorem and the skew-transport commutator [H, Hᵀ] = 2[A, D]. 44 theorems. |
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| electron-screening-lean | For Electron Screening in Metal Deuterides: the fitting bias points upward (Jensen, for an increasing convex rate), the rate's sensitivity to U_e, the ratio's cancellation of every energy-independent factor, and arithmetic checks of the paper's numbers on its stated inputs (reaction and neutron rates, error budget, discrimination, drift residuals; the discrimination uses the paper's rounded error total 0.024). 11 theorems. |
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| iqgt-grassmannian-lean | For Intrinsic Quantum Geometric Tensor on the Grassmannian: the metric control of Berry curvature |Ω| ≤ 2√(det g), proved from positive semidefiniteness of the Gram matrix in every finite dimension, plus the off-diagonal structure of tangent vectors, rigidity killing curvature, and the two-level saturation identity. 11 theorems. |
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| event-horizon-lean | For Beyond the Event Horizon: the Bures lifts of four-body shape space, the exact 1/(6w₃) metric blow-up at the coplanar wall, the discriminant determinant 16w₁w₂w₃ giving a real spectrum on the regular stratum, the wall block that is a genuine Jordan block when A₁₂ ≠ 0, and the ledger that survives it. 17 theorems. |
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| emergent-transport-lean | For Intrinsic Emergent Transport Geometry (Directional Transport Geometry): first-order cancellation of the ledger at a self-adjoint baseline, commutators with a projector are off-diagonal, the projector–metric correspondence in its daggered form, gauge invariance, and persistence Tr(Π₁Π₂) = ‖Π₂Π₁‖²_F as a squared norm (cos²θ for two lines). 11 theorems. |
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| price-of-a-direction-lean | For The Price of a Direction: directions as channels (cos²θ overlap, sin²θ separation), exchange symmetry of nonzero spectra, trackability loss at a defective point, the metric–curvature inequality, and the exact ratio-gate onset. 17 theorems. |
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| upg-lean | For Universal Projector Geometry: the Hamiltonian commutes with a retained projector exactly when the coupling vanishes; the commutator, redistribution and zero-time memory vanish together, and the cross-block metric equals the coupling energy. 104 theorems. |  |
| elemental-peeling-lean | For Elemental Peeling: what a removal takes away and what later records recover, force recovery from readings, the boundary ledger, and the exclusion certificates. 88 theorems. |  |
| yang-mills-lean | For Retained Geometry and Certified Gauge Cutoffs: exact LDLᵀ certificates for a two-plaquette SU(3) model, the boundary-matrix Schur steps, and three finite lanes (projector commutator → coupling → feedback; resolved penalties → Schur floor; allocation → positive residual), not composed into one implication. Finite algebra only; no mass-gap claim. 79 theorems. |  |
| sqrt-lower-bound-lean | A complete formalization of the lower square-root bound on operator entanglement (the theorem of Pozsgay and Vona) for the finite core to which their Lemma 4.1 reduces the integrable brickwork circuit: 30 files, 661 theorems. |  |
| coxeter-rank2-lean | The rank-two Coxeter group I₂(m+2) is the dihedral group, against Mathlib's own objects; exact periods 2, 3, 4, 6 for a representative of each finite rank-two case, drift and no period for the affine representative. 44 theorems. |
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| small-diophantine-lean | Algebraic certificates for the small Diophantine surfaces z² + (y²+a)z + x³ + bx + c = 0, and the integrality of an explicit rational section. 16 theorems. |
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| what-transport-keeps-lean | For What Transport Keeps: conserved pairings under dual transport, the Catalan cancellation A U₀⁻ᵀ f = 450 (G, 1) (G and log 2 as free symbols) with its retention certificate and the weak-duality lemma, the reconstructed Catalan matrix field (column convention, plaquette identity, factored determinant, Bézout certificates for the given curvature polynomials), and the stress, projector-metric and boundary-response identities. 49 theorems. |
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| votkp-lean | For VOTKP: The Vector of Traits Kept under Peeling: peel shape as the Smith reflection Γ = tanh(½ ln IE_{k+1}/IE_k), peel steps that add like velocities, the shell part (2n − 1)/(n² + (n − 1)²) of a step across a shell wall, the thin-film echo and the circle through the origin that decides whether a film reflects more than its bulk, and the finite peel theorems. 57 theorems. |
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Every check builds the proofs, replays them in Lean's independent kernel checker, and audits that nothing rests on sorry or extra axioms.
Upstream to Mathlib: #43703 (open), the rank-two Coxeter group is the dihedral group · #43951 (closed), a trace identity for differences of idempotents.
All 29 Zenodo records, newest version first. Each DOI always opens the latest version. Every paper has a proof repository; those marked in progress are being formalized paper by paper.
| Paper | Latest | DOI | Proofs / code |
|---|---|---|---|
| What Russell Saw: Walter Russell's Universe of Paired Motion, Read Against a Century of Physics | 2026-09-27 | 10.5281/zenodo.22986656 | what-russell-saw-lean · 43 Lean theorems |
| The Dirac Time of the Gigantefermion: Memory First, Projector Geometry, Retained History, and the Cost of an Arrow | 2026-09-26 | 10.5281/zenodo.22978630 | dirac-time-lean · 86 + 3 Lean theorems |
| VOTKP: The Vector of Traits Kept under Peeling — Elemental Peeling of Atoms and Knots | 2026-09-25 | 10.5281/zenodo.23011140 | votkp-lean · 57 Lean theorems |
| The Ledger Outlives the Metric: Wall Species, the Reciprocity Obstruction, and a Measured Crossing Behind an Operator-Side Reading of Entropic Gravity | 2026-09-25 | 10.5281/zenodo.22023237 | gravity-ledger-lean · 30 Lean theorems |
| Five-Dimensional Non-Normal Stability Operator with Gradient-Driven Interface Coupling for Zero-Precursor Filamentary Fractures | 2026-09-25 | 10.5281/zenodo.21184980 | scripts in the Zenodo deposit; filament-fracture-lean · 14 Lean theorems |
| Certified Obstructions and Exact Forcing: Thickness Bounds, Nine-Colorings, and Integer Dual Certificates for Two Open Benchmarks | 2026-09-17 | 10.5281/zenodo.22812167 | earth-moon-kakeya-certificates · certificates and checkers; arithmetic-kakeya-finite-bridges · 127 Lean theorems |
| Non-Normality Is Connection Energy | 2026-09-17 | 10.5281/zenodo.22803572 | connection-energy-lean · 20 Lean theorems |
| Compound Eye (research tool) | 2026-09-09 | 10.5281/zenodo.22674828 | software deposit |
| The Offset Belongs to the Boundary: the Trace Split of the Entanglement Hamiltonian, a Measured Clausius Region, and an Offset with No Bulk | 2026-09-06 | 10.5281/zenodo.22181748 | offset-lean · Lean proofs |
| What Transport Keeps: Projector Selection, Conserved Pairings, and Spectral Geometry after the Ramanujan Challenge | 2026-09-06 | 10.5281/zenodo.22543486 | what-transport-keeps-lean · 49 Lean theorems · ramanujan-normality-diagnostic · diagnostic code |
| A Subgraph Density Obstruction to Biplanarity, and the Thickness of Inflated Cycles | 2026-09-04 | 10.5281/zenodo.22307183 | see earth-moon-kakeya-certificates |
| Light Keeps the Ledger: Matter, Direction, and the Wall Between Them | 2026-08-27 | 10.5281/zenodo.22123115 | light-ledger-lean · 122 Lean theorems |
| The Permutation Coboundary Constant of the Complete Complex Is k/3 for 4 ≤ k ≤ 8 | 2026-08-25 | 10.5281/zenodo.22090958 | coboundary-k3-lean · 25 Lean theorems |
| The Three-Body Problem, Operator-First: A Complete Spectral Anatomy of Relational Transport on the Shape Sphere | 2026-08-24 | 10.5281/zenodo.20764188 | three-body-lean · 22 Lean theorems |
| Universal Projector Geometry from Electrical Measurements | 2026-08-21 | 10.5281/zenodo.21305024 | upg-lean · 104 Lean theorems |
| Matter at a Scale: Zero, Span, and the Logarithm Between Them | 2026-08-20 | 10.5281/zenodo.22019892 | matter-at-a-scale-lean · 17 Lean theorems |
| Transient Structure at Solar Rigidity Interfaces and the Exceptional Point of the Dynamo Wave | 2026-08-19 | 10.5281/zenodo.21895551 | suns-lean · 15 Lean theorems |
| The Operator-First Relational Atlas | 2026-08-18 | 10.5281/zenodo.21972561 | relational-atlas-lean · 37 Lean theorems |
| Electron Screening in Metal Deuterides: Measurement Systematics, Evolving Target State, and an Experimental Arbitration Protocol | 2026-08-14 | 10.5281/zenodo.21935285 | electron-screening-lean · 11 Lean theorems |
| The Price of a Direction: Boundary Channel Budgets, Directional Capacity, and Kakeya Structure in Operator Transport | 2026-08-13 | 10.5281/zenodo.21918463 | price-of-a-direction-lean · 17 Lean theorems |
| Foundational Operator Field Theory on Stratified Manifolds | 2026-08-09 | 10.5281/zenodo.21254648 | foft-lean · 11 Lean theorems |
| Beyond the Event Horizon: An Operator-First Theory of Persistent Geometry and Black Hole Dynamics | 2026-08-09 | 10.5281/zenodo.21147366 | event-horizon-lean · 17 Lean theorems |
| Spectral Stress on a Conservative Matrix Field | 2026-08-08 | 10.5281/zenodo.21825872 | spectral-stress-lean · 20 Lean theorems |
| The Square Case: Gram Reduction and Spectral Transport for N = d + 1 Bodies | 2026-08-08 | 10.5281/zenodo.21855590 | square-case-lean · 63 Lean theorems |
| Intrinsic Quantum Geometric Tensor on the Grassmannian and Spectral Lower Bounds on Dissipation | 2026-08-05 | 10.5281/zenodo.20768258 | iqgt-grassmannian-lean · 11 Lean theorems |
| The Companion-Matrix Projector Diagnostic | 2026-08-05 | 10.5281/zenodo.21787207 | ramanujan-normality-diagnostic · diagnostic code |
| Intrinsic Emergent Transport Geometry: Emergent Metrics from Spectral Projector Hierarchies | 2026-07-01 | 10.5281/zenodo.21089303 | emergent-transport-lean · 11 Lean theorems |
| Spectral Stability Across Obstruction Interfaces in Geometric Transport Systems | 2026-06-24 | 10.5281/zenodo.20818899 | spectral-stability-lean · 44 Lean theorems |
| The 5D Spectral Anatomy of Four-Body Obstructions: Kinematic Transport Geometry on Stratified Shape Manifolds | 2026-06-23 | 10.5281/zenodo.20818168 | square-case-lean · binary-wall proofs |
- ramanujan-normality-diagnostic: builds the companion matrix of a linear recurrence and splits it into symmetric and skew parts, a quick signal for whether a simple closed form is likely to exist.
- rhythm-game-one: a four-key rhythm game in plain HTML, Canvas and Web Audio.
- Oceans-Journey: a small ocean adventure game.
Copyright (c) 2026 Jeromie Beasley. Code and proofs: MIT. Written text: CC BY 4.0. See LICENSING.md.