alignment
Repetend Rigidity at the Golden Scale
An old golden-ratio crossing, revisited where continuous patterns meet the constraints of finite arithmetic.
boundary
The Secondary Term of the Cubic Law
The cubic law gives the leading growth of digit-collision energy. Its secondary term identifies the first systematic correction and the arithmetic that preserves it.
July 1, 2026 · 14 min
boundary
Digit Collisions and the Cubic Law
The collision energy is the cube of the base, split one third to two thirds between diagonal and off-diagonal poles. The cubic constant is unity.
June 4, 2026 · 24 min
boundary
The Orbit's Edge
The orbit is multiplicative. The boundary is additive. The Jacobi sum is the bridge. Two laws from the time of Gauss show up because the boundary forced the character to look at -2.
May 30, 2026 · 25 min
boundary
Carry Boundaries and Bernoulli Spectra
The collision invariant turned out to be a special case. This paper finds the source underneath it. The floor provides the weight. The boundary provides the geometry. The spectrum is their product.
May 29, 2026 · 17 min
essay
The Structure That Survives
The collision invariant is the part of the arithmetic that remains when everything else has been allowed to move. This essay explains the name and the analogy to noble gases.
April 15, 2026 · 9 min
preprint
The Collision Spectrum
The Fourier coefficients factor through Bernoulli numbers and L-function values at s = 1.
March 31, 2026 · 19 min
preprint
The Collision Transform
The collision periodic table, centered and Fourier-transformed. It cancels at s = 1.
March 30, 2026 · 17 min
preprint
The Collision Invariant
A finite signed table for every prime, built from the digit function. The collision invariant.
March 29, 2026 · 21 min
avoidance
The Analytic Collision Transform
The same finite diagonal geometry now carries an exact identity at every s in the critical strip. The analytic factors move. The collision content does not.
January 24, 2026 · 10 min
avoidance
The Collision Spectrum and the L-Function Landscape
The Fourier coefficients factor through Bernoulli numbers and diagonal character sums. The collision weight encodes L-function values at s=1. The digit function meets the critical strip.
March 10, 2025 · 19 min
avoidance
The Spectral Repulsion
Forty percent of the expected overlap between collision weights and prime character sums is missing. The ratio is stable across every prime base from 3 to 37.
January 9, 2025 · 16 min
collision
The Double Transversality
The collision invariant and the prime distribution avoid each other across the strip. The avoidance persists at every lag, across every base tested.
October 3, 2024 · 13 min
collision
The General Neutrality Theorem
Neutrality holds at every odd prime, not just 3. The same reflection identity, the same vanishing, at every scale. The anti-correlation between collision weights and prime sums is universal.
July 21, 2024 · 21 min
collision
The Neutrality Theorem
The mod-3 component of the collision transform vanishes. Remove it and the sum explodes. Neutrality is not decorative. It is structural.
April 23, 2024 · 24 min
collision
The Collision Transform and the Critical Strip
The centered sum converges at s=1. Push it below. The penetration depth measures how far into the critical strip the collision signal survives.
January 13, 2024 · 16 min
collision
The Collision Periodic Table
Forty integers, determined by the last two digits of every prime past 100. Every complement pair sums to exactly -1. At base 12, the table sorts musical intervals by tension.
December 2, 2023 · 20 min
collision
The Centered Collision Sum
Subtract the family bias and the divergence vanishes. The centered sum converges at s=1, and the rate of convergence is controlled by the classical zero-free region.
October 6, 2023 · 14 min
collision
The Collision Fluctuation Sum
The collision deviations, summed over primes, drift downward at the Mertens rate. The drift has a sign, a constant, and a structural origin in the digit function.
April 28, 2023 · 20 min
collision
Silent Primes
At seven primes in base 10, the collision count is exactly zero. The recipe that finds them involves the bin partition and a specific floor-function identity.
January 21, 2023 · 35 min
collision
The Character Structure of the Collision Fluctuation
The collision fluctuation decomposes over Dirichlet characters. Only the odd characters survive, forced there by the complement involution. This is where the algebra begins.
October 14, 2022 · 16 min
spectral
Bin Derangements and the Gate Width Theorem
Nine multipliers produce zero collisions at every prime in base 10. The count is exactly b-1, independent of the prime. The proof identifies the deranging set explicitly.
July 4, 2022 · 11 min
spectral
Phase-Filtered Ramanujan Sums and the Spectral Gate
The last digit of a prime controls which spectral modes survive. A phase filter built from Ramanujan sums explains the gate, and the gate width is universal across primes.
April 17, 2022 · 13 min
spectral
The Autocorrelation Formula
The cross-spectral function had resisted a closed form. It turned out to factor through the digit function evaluated at shifted arguments. The formula closes the spectral chain.
January 26, 2022 · 13 min
spectral
The Spectral Power of the Digit Function
Two neighboring remainders share a digit. Sliding the bins and adding arrows gives two exact ways to read that agreement.
October 26, 2021 · 16 min
spectral
The Spectral Structure of Fractional Fields
Three pairs of binary fractions cover the same digit slots. Those visible cancellations explain two missing directions in the matrix.
July 13, 2021 · 14 min
spectral
The Cross-Alignment Matrix
Compare every fraction to every other. The full grid locates the matches that an average conceals, and its spectrum records their arrangement.
April 11, 2021 · 18 min
spectral
The Coherence Decomposition
The reference score and the average over every pair count agreement differently. An exact relation between them explains the split at twelve.
January 12, 2021 · 15 min
preprint
Primes and the Major Scale
The primes that organize long division also organize musical pitch. Deficit ratios approach the Pythagorean comma in base 12 and the just major scale in base 30.
October 31, 2020 · 17 min
alignment
The Three-Tier Theorem
The alignments at 6 and 12 have exact rational values. No denominator fits between them. Pairing complementary fractions explains the gap.
October 16, 2020 · 15 min
alignment
The Alignment Limit for All Primes
In decimal, the prime 53 has four repeating cycles and two alignment limits. Counting the digits shared by different remainders explains the split.
July 7, 2020 · 15 min
alignment
Digit-Partitioning Primes and the Alignment Formula
Single digits, full cycles, complement pairs. Three kinds of repeating decimal share one boundary and one alignment formula.
April 3, 2020 · 15 min
alignment
Three and the Golden Ratio
The alignment count at denominator twelve leads to a cubic with a golden-ratio factor at three.
January 19, 2020 · 11 min