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 labels on every cut. 127 theorems.
what-russell-saw-lean	For What Russell Saw: Russell's paired-motion axiom as energy conservation for every oscillator motion (Physlib), why his v ∝ 1/a means an inverse-cube pull and spiral orbits, the apsidal angle and the ISCO at r = 6M derived from the Schwarzschild potential, Maxwell stress and Ampère's sign, the tide, the vortex and the cone. 43 theorems.
dirac-time-lean	For The Dirac Time of the Gigantefermion: every exact finite result the paper states: 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 with unitary invariance of entropy proved via Physlib, the moving-frame identity, the equal-endpoint witness, friction on projector rotations and the torus-knot silence criterion. 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 parallel sections commute with holonomy along every edge. 20 theorems.
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 differentiability alone. 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 spectrum, and interlacing (exactly one eigenvalue in every gap), and the four-body binary-wall theorems. 63 theorems.
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, and span and offset invariance of eigenspaces and of the relative departure from normality. 17 theorems.
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 poles (quadratically), the 120° symmetry forcing isotropy, the fold, reciprocity of the transport operator, the shifted ledger, and the collinear kernel. 22 theorems.
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 exact Henrici dissipation rate, 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 transient growth in one operator, and 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.
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, including an exhaustive gauge minimum and a proved-sound branch-and-bound search, which gives N/D = k/3 at both powers of two. Also the non-abelian mechanism at k = 4 and the displacement Gram as half the cycle Laplacian. 25 theorems.
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 exact disk pseudospectrum and ε_crit = σ_min, transient growth iff `	μ
spectral-stability-lean	For Spectral Stability Across Obstruction Interfaces: the transport operator αJ ⊗ Δ₁ is symmetric with real spectrum, 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.
electron-screening-lean	For Electron Screening in Metal Deuterides: the fitting bias points upward (Jensen), the rate's sensitivity to U_e, the ratio's cancellation of every energy-independent factor, and the checked arithmetic of the neutron deficit, the error budget, the discrimination and the drift residuals. 11 theorems.
iqgt-grassmannian-lean	For Intrinsic Quantum Geometric Tensor on the Grassmannian: the metric control of Berry curvature `	Ω
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 proving a real spectrum on the regular stratum, the Jordan block on the wall, and the ledger that survives it. 17 theorems.
emergent-transport-lean	For Intrinsic Emergent Transport Geometry (Directional Transport Geometry): first-order cancellation of the ledger, commutators with a projector are off-diagonal, the projector–metric correspondence in its daggered form, gauge invariance, and persistence Tr(Π₁Π₂) = Σ cos²θ as a squared norm. 11 theorems.
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.
upg-lean	For Universal Projector Geometry: a retained sector runs on its own exactly when its coupling vanishes, and every finite diagnostic (commutator, redistribution, zero-time memory, cross-block metric) agrees. 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: gauge-cutoff certificates, the boundary-matrix Schur certificate, and the longest finite chain from projector geometry to an allocation floor. Finite algebra only; no mass-gap claim. 77 theorems.
sqrt-lower-bound-lean	A complete formalization of the lower square-root bound on operator entanglement in an integrable brickwork circuit (the theorem of Pozsgay and Vona): 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 the finite rank-two cases, drift and no period for the affine one. 44 theorems.
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.
what-transport-keeps-lean	For What Transport Keeps: conserved pairings under dual transport, the Catalan cancellation A U₀⁻ᵀ f = 450 (G, 1) with its retention and dual certificates, the reconstructed Catalan matrix field (column convention, plaquette identity, factored determinant, curvature certificates), and the stress, projector-metric and boundary-response identities. 49 theorems.
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 wall (2n − 1)/(n² + (n − 1)²), the thin-film echo and the exact circle that decides whether a film rises before it falls, and the finite peel theorems. 57 theorems.

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 (submitted): #43703, the rank-two Coxeter group is the dihedral group · #43951, a trace identity for differences of idempotents.

All 28 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.

--- | :--- | | Certified Obstructions and Exact Forcing: Earth–Moon Graph Coloring and Arithmetic Kakeya · DOI 10.5281/zenodo.22812167 | earth-moon-kakeya-certificates: manuscript, certificate producers, independent checkers, re-verified on every push | | The Offset Belongs to the Boundary · DOI 10.5281/zenodo.22181748 | offset-lean: the Lean proofs of its finite algebra | | What Russell Saw: Walter Russell's Universe of Paired Motion, Read Against a Century of Physics · DOI 10.5281/zenodo.22986656 | what-russell-saw-lean: 43 Lean proofs, three built on Physlib; paper, mixer, ledger and scripts in the Zenodo deposit | | The Dirac Time of the Gigantefermion · DOI 10.5281/zenodo.22978630 | dirac-time-lean: Lean proofs of every exact finite result in the paper | | Non-Normality Is Connection Energy · DOI 10.5281/zenodo.22803572 | connection-energy-lean: Lean proofs of the identity and the normality criterion | | Light Keeps the Ledger · DOI 10.5281/zenodo.22123115 | light-ledger-lean: Lean proofs | | The Ledger Outlives the Metric · DOI 10.5281/zenodo.22023237 | gravity-ledger-lean: Lean proofs | | The Square Case: Gram Reduction and Spectral Transport for N = d + 1 Bodies · DOI 10.5281/zenodo.21855590 | square-case-lean: Lean proofs | | 5D Spectral Anatomy of Four-Body Obstructions · DOI 10.5281/zenodo.20818168 | square-case-lean: Lean proofs of the binary-wall algebra | | Universal Projector Geometry from Electrical Measurements · DOI 10.5281/zenodo.21305024 | upg-lean: Lean 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.