I want a way to map and measure everything as simply as light does.
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.
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.
28 papers on Zenodo, 30 proof repositories, 2,017 theorems checked by Lean's kernel.
Papers
Newest version first. Each DOI opens the latest version of the paper on Zenodo.
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Five-Dimensional Non-Normal Stability Operator with Gradient-Driven Interface Coupling for Zero-Precursor Filamentary Fractures
scripts in the Zenodo deposit; filament-fracture-lean · 14 Lean theorems
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Certified Obstructions and Exact Forcing: Thickness Bounds, Nine-Colorings, and Integer Dual Certificates for Two Open Benchmarks
earth-moon-kakeya-certificates · certificates and checkers; arithmetic-kakeya-finite-bridges · 127 Lean theorems
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Non-Normality Is Connection Energy
connection-energy-lean · 20 Lean theorems
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Universal Projector Geometry from Electrical Measurements
upg-lean · 104 Lean theorems
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The Operator-First Relational Atlas
relational-atlas-lean · 37 Lean theorems
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Foundational Operator Field Theory on Stratified Manifolds
foft-lean · 11 Lean theorems
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Spectral Stress on a Conservative Matrix Field
spectral-stress-lean · 20 Lean theorems
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The Companion-Matrix Projector Diagnostic
ramanujan-normality-diagnostic · diagnostic code
No paper matches that search. Clear the box to see all of them.
Machine-checked proofs
Every check builds the proofs, replays them in Lean's independent kernel checker, and confirms nothing rests on sorry or extra axioms.
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offset-lean156
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.
See the latest check -
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.
See the latest check -
For What Russell Saw: energy conservation for every oscillator motion (Physlib), which the paper reads as Russell's paired-motion axiom; his
See the latest checkv ∝ 1/aas an inverse-cube pull (exponent arithmetic from the similarity scaling), the apsidal angle diverging as the pull approaches inverse cube, the marginally stable orbit atr = 6Mderived from the Schwarzschild effective potential, Maxwell stress and Ampère's sign, the tide, the vortex and the cone. 43 theorems. -
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.
See the latest check -
For Non-Normality Is Connection Energy:
See the latest check[H, H†] = −2[D, A], the energy identity⅛‖[H, H†]‖² = Σ_edges ‖Dᵢ − AᵢⱼDⱼAᵢⱼ†‖², andHnormal exactly whenDis parallel; the induced transport is a connection and a parallel section commutes with the holonomy around every loop. 20 theorems. -
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.
See the latest check -
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.
See the latest check -
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
See the latest checkTE + ET = (tᵢ + tⱼ)E, and interlacing (exactly one eigenvalue in every gap), and the four-body binary-wall theorems. 63 theorems. -
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
See the latest check(1+z)⁴), gravitational closure withα_Ginvariant, span and offset invariance of eigenspaces, and span invariance of the relative departure from normality. 17 theorems. -
For The Three-Body Problem, Operator-First: the Gram operator's exact spectrum
See the latest check{cos² ϑ, 1}, rank drop at syzygy against gap closing at the poleϑ = 0(quadratically), the 120° symmetry forcing isotropy, the fold, reciprocity of the transport operator, the shifted ledger, and the collinear kernel. 22 theorems. -
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.
See the latest check -
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.
See the latest check -
For Spectral Stress on a Conservative Matrix Field: the exact sum rule, gauge invariance, the factorisation
See the latest checkt = (ρ_b − ρ_c)β_bc β_cb, the amplitudeA = −5√2/16with the vanishing middle band, and theℚ(√2)arithmetic of the limiting spectrum. 20 theorems. -
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.
See the latest check -
For The Permutation Coboundary Constant … Is k/3: every witness for
See the latest checkk = 4, …, 8checked 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), givingN/D = k/3for everyk = 4, …, 8. Also the non-abelian mechanism atk = 4, and the displacement inner products as the cycle-graph Laplacian2I − A. 25 theorems. -
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
See the latest check1/εon the circle|z|² = ε|μ| + ε², the identityε_crit · σ_max = b², transient growth of the closed-form gainM(t)iff|μ| > 2b, and the linear against square-root coalescence paths. 14 theorems. -
For Spectral Stability Across Obstruction Interfaces: the transport operator
See the latest checkαJ ⊗ Δ₁is symmetric (Hermitian), the obstruction field is symmetric, the commutator starts at first order inε, the bulge scaling, and the second-difference error; the leading coupling blockMJ + 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. -
For Electron Screening in Metal Deuterides: the fitting bias points upward (Jensen, for an increasing convex rate), the rate's sensitivity to
See the latest checkU_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. -
For Intrinsic Quantum Geometric Tensor on the Grassmannian: the metric control of Berry curvature
See the latest check|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. -
For Beyond the Event Horizon: the Bures lifts of four-body shape space, the exact
See the latest check1/(6w₃)metric blow-up at the coplanar wall, the discriminant determinant16w₁w₂w₃giving a real spectrum on the regular stratum, the wall block that is a genuine Jordan block whenA₁₂ ≠ 0, and the ledger that survives it. 17 theorems. -
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
See the latest checkTr(Π₁Π₂) = ‖Π₂Π₁‖²_Fas a squared norm (cos²θfor two lines). 11 theorems. -
For The Price of a Direction: directions as channels (
See the latest checkcos²θoverlap,sin²θseparation), exchange symmetry of nonzero spectra, trackability loss at a defective point, the metric–curvature inequality, and the exact ratio-gate onset. 17 theorems. -
upg-lean104
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.
See the latest check -
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.
See the latest check -
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.
See the latest check -
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.
See the latest check -
The rank-two Coxeter group
See the latest checkI₂(m+2)is the dihedral group, against Mathlib's own objects; exact periods2, 3, 4, 6for a representative of each finite rank-two case, drift and no period for the affine representative. 44 theorems. -
Algebraic certificates for the small Diophantine surfaces
See the latest checkz² + (y²+a)z + x³ + bx + c = 0, and the integrality of an explicit rational section. 16 theorems. -
For What Transport Keeps: conserved pairings under dual transport, the Catalan cancellation
See the latest checkA U₀⁻ᵀ f = 450 (G, 1)(Gandlog 2as 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. -
For VOTKP: The Vector of Traits Kept under Peeling: peel shape as the Smith reflection
See the latest checkΓ = 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.
Submitted upstream to Mathlib:
Videos
Each video explains one paper. They also live on the Atlas Memoriae channel on YouTube.
Tools
- 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.
Also
- rhythm-game-one a four-key rhythm game in plain HTML, Canvas and Web Audio.
- Oceans-Journey a small ocean adventure game.