Papers

31 research papers from the collision program, in the order the mathematics revealed itself. Each paper is the companion to a blog post that introduces the same material for a general audience.

arXiv preprints

The Collision InvariantMarch 2026
ReadarXivBlog
The Collision TransformMarch 2026
ReadarXivBlog
The Collision SpectrumMarch 2026
ReadarXivBlog

The collision program

Why the Golden Ratio Selects the Prime ThreeJanuary 2020
ReadZenodoBlog
Digit-Partitioning Primes and the Alignment FormulaApril 2020
ReadZenodoBlog
The Alignment Limit for All PrimesJuly 2020
ReadZenodoBlog
The Three-Tier TheoremOctober 2020
ReadZenodoBlog
The Alignment Deficit Lattice and Musical TuningOctober 2020
ReadZenodoBlog
The Coherence DecompositionJanuary 2021
ReadZenodoBlog
The Cross-Alignment MatrixApril 2021
ReadZenodoBlog
The Spectral Structure of Fractional FieldsJuly 2021
ReadZenodoBlog
The Spectral Power of the Digit FunctionOctober 2021
ReadZenodoBlog
The Autocorrelation FormulaJanuary 2022
ReadZenodoBlog
Phase-Filtered Ramanujan Sums and the Spectral GateApril 2022
ReadZenodoBlog
Bin Derangements and the Gate Width TheoremJuly 2022
ReadZenodoBlog
The Character Structure of the Collision FluctuationOctober 2022
ReadZenodoBlog
Silent Primes and the Variance of the Collision CountJanuary 2023
ReadZenodoBlog
The Collision Fluctuation SumApril 2023
ReadZenodoBlog
The Centered Collision SumOctober 2023
ReadZenodoBlog
The Collision Periodic TableDecember 2023
ReadZenodoBlog
The Collision Transform and the Critical StripJanuary 2024
ReadZenodoBlog
The Neutrality TheoremApril 2024
ReadZenodoBlog
The General Neutrality TheoremJuly 2024
ReadZenodoBlog
The Double TransversalityOctober 2024
ReadZenodoBlog
The Spectral RepulsionJanuary 2025
ReadZenodoBlog
The Collision Spectrum and the L-Function LandscapeMarch 2025
ReadZenodoBlog
The Analytic Collision TransformMay 2025
ReadZenodoBlog
The Collision InvariantMarch 2026
ReadarXivZenodoBlog
The Collision TransformMarch 2026
ReadarXivZenodoBlog
The Collision SpectrumMarch 2026
ReadarXivZenodoBlog
The Structure That SurvivesApril 2026
ReadZenodoBlog
Carry Boundaries and Bernoulli Spectra in Long DivisionMay 2026
ReadZenodoBlog
Additive Boundaries and Conductor Energies of Periodic Remainder OrbitsMay 2026
ReadZenodoBlog
The Cubic Law for Digit-Collision EnergyJune 2026
ReadZenodoBlog
Repetend Rigidity at the Golden ScaleSeptember 2026
ReadZenodoBlog
The Secondary Term in Digit-Collision EnergyJuly 1, 2026
ReadZenodoBlog
Magnitude Decorrelation in the Collision SpectrumJuly 22, 2026
ReadZenodoBlog
Conservation and Profinite Dynamics in Digit-Collision EnergyJuly 31, 2026
ReadZenodoBlog
The Stationary-to-Diagonal Transition in Collision CapacityAugust 9, 2026
ReadZenodoBlog
Finite Generation of Collision CapacityAugust 18, 2026
ReadZenodoBlog