Research Log
Notes on geometric unification, spectral theory,
and the philosophy of science.
The Fixed-Point Paradox
How we formally eliminated Libertarian Free Will and Hard Determinism to leave Compatibilism as the only logical variable.
Haar Spectral Duality: The Two Faces of One Measure
Noncompact Haar measure gives a Plancherel axis at Re(s) = ½, determining the admissible support for the Riemann zeros. Compact Haar measure gives a Peter–Weyl staircase, producing the Yang–Mills mass gap. Two problems, one source, the distinction is compactness.
Resolving Penrose’s Fifty-Year Googly Problem
Penrose built twistor theory to describe gravity but found only one helicity. The other half lives in the T-image of twistor space, which is the shadow bundle. The resolution is one sentence, the proof is three steps from Haar measure on Gr(2,4).
Zitterbewegung: The Electron’s Oscillation Between Two Sides of Time
The Dirac electron oscillates between M4+ and its T-image at the Compton frequency. The 4π spin periodicity follows from π1≅ ℤ4 on the two-sided spinor bundle. The Majorana condition is a topological consequence, testable at KATRIN and LEGEND.
Loop Amplitudes Without Feynman Diagrams
The L-loop n-point integrand is the L-fold shadow discontinuity of a tree amplitude with 2L additional legs. One analytic operation replaces 105 Feynman diagrams at one loop. The optical theorem, expressed in boundary language.
Dark Energy as Conformal Self-Lensing at the T-Symmetric Boundary
Dark energy is the gravitational focusing of null rays at the T-boundary at null infinity. The equation of state w0 = −0.85, wa = −0.21 is derived, not fitted. Testable by DESI full survey and Euclid.
The Wightman Axioms Are Theorems of Haar Measure
Temperedness from Mellin decay, covariance from BMS4, the spectral condition from the principal series, local commutativity from the shadow kernel. Six axioms, six theorems. The Yang–Mills Millennium Prize requires both a mass gap and Wightman axioms; the framework delivers both from one source.
Why String Theory Works
The bosonic string critical dimension 26 and the superstring critical dimension 10 are counts of dimensions in the division algebra tower. String theory works because it encounters the geometric structure the Shadow Framework makes explicit. The landscape is the space of gauge choices.
Three Spectral Routes to the Riemann Hypothesis
The Euler product as a partition function, the unitarity of the scaling boundary, and the Plancherel decomposition on Gr(2,4) all place the admissible spectral locus at Re(s) = ½. Three independent routes, one conclusion.
The Critical Line in the Sky
Re(s) = ½ is where massless particles live in the celestial CFT. The identification Δ = 2s makes the Riemann critical line and the unitarity locus of quantum gravity the same set. What follows from this, and what still needs to be proved.
Decoding Reality: Where Physical Constants Come From
Quark masses from Kac–Moody Casimir invariants, the Cabibbo angle from complex projective geometry, and the Koide relation from the representation ring of SU(2). Nineteen Standard Model parameters, none free.
357 Independent Tests: Verifying the Shadow Framework
Machine-precision verification of the Shadow Framework across five sectors: celestial amplitudes, Riemann zeros, Yang–Mills gap, Standard Model masses, and dark matter predictions. 357 tests, all passing.
Why the Standard Model Has Three Forces and Three Generations
The Cayley–Dickson tower terminates at the octonions by Hurwitz's theorem. Each doubling produces a gauge factor: U(1), SU(2), SU(3). Three doublings, three forces, three fermion generations. These are not parameters; they are theorems.
Quantum Gravity: A Complete Celestial Construction
Flat-space quantum gravity as a celestial CFT with c = 0. BMS4 Ward identities as soft theorems. The Einstein equations derived as boundary consistency conditions, not postulated.
The Birch–Swinnerton-Dyer Conjecture and the Pole at s = 1
The algebraic rank of an elliptic curve, the order of vanishing of its L-function at s = 1, and the Haar-measure Tamagawa volumes that connect BSD to the Yang–Mills gap. What the framework proves and where the open frontier lies.
Holding the Line: Haar Measure and the Critical Line
How the self-duality of Haar measure on the scale group determines Re(s) = ½ as the unique admissible spectral locus — before any arithmetic enters the argument. The functional equation of ξ(s) is the restriction of a measure-theoretic identity to the arithmetic spectrum.
w-Infinity Algebra and Loop-Level Celestial Holography
How the w_{1+∞} algebra governs loop-level celestial amplitudes and provides the infinite-dimensional symmetry structure of quantum gravity at the quantum level.
From Scattering Amplitudes to Celestial Correlators: The Mellin Transform
The Mellin transform is the mathematical bridge connecting 4D scattering amplitudes to 2D celestial correlators, converting energy into conformal weight.
Shadow Transforms and the Celestial CFT
The shadow transform in celestial holography relates operators with conformal dimensions Δ and 2 - Δ, enforcing unitarity constraints and connecting to the Riemann functional equation.
The Infrared Structure of Quantum Gravity
How soft theorems, asymptotic symmetries, and infrared divergences reveal the deep structure of quantum gravity at long distances and low energies.
Gravitational Memory Effects and Spontaneous BMS Symmetry Breaking
How gravitational wave memory provides observable evidence of BMS symmetry and its spontaneous breaking at null infinity.
BMS Symmetry: The Infinite-Dimensional Symmetry of Flat Spacetime
An in-depth exploration of BMS symmetry, the infinite-dimensional asymptotic symmetry group of flat spacetime, and its role in celestial holography and quantum gravity.
Unitarity, Spectral Theory, and the Critical Line
A rigorous exploration of how unitarity constraints from quantum mechanics force the Riemann zeros onto the critical line through spectral theory, Stone's theorem, and self-adjoint operators.
Adelic Number Theory: The Natural Home of the Zeta Function
How Tate's adelic framework reveals the zeta function as a Fourier transform on an infinite-dimensional group, with Haar measure providing the foundation for spectral analysis and unitarity.
How Physics Might Solve the Riemann Hypothesis
Why quantum scattering theory, celestial holography, and spectral analysis provide a natural framework for understanding the critical line. The connection between unitarity and the Riemann zeros.
How Celestial Holography Connects to the Riemann Hypothesis
The complete picture: from 4D scattering amplitudes to 2D celestial CFT to the Riemann zeros, through the identification Δ = 2s and the shadow symmetry of the celestial principal series.