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
Digit boundaries choose the weights of an L-function family. In base five, twenty entries give its exact fourth moment.
March 31, 2026 · 15 min
preprint
The Collision Transform
A centered digit table gives a convergent sum over primes and an exact cancellation condition at the zeros of its active L-functions.
March 30, 2026 · 15 min
preprint
The Collision Invariant
The collision count grows with the denominator. Its deviation is fixed by a handful of final digits.
March 29, 2026 · 16 min
avoidance
The Analytic Collision Transform
A small digit table supplies the fixed part of a changing function, and the original collision coefficient can be recovered exactly.
January 24, 2026 · 14 min
avoidance
The Collision Spectrum and the L-Function Landscape
In base five, a finite table of digit collisions determines an exact sum of L-function values raised to the fourth power.
March 10, 2025 · 14 min
avoidance
The Spectral Repulsion
The collision and prime spectra overlap less than a fixed-weight shuffle predicts. An exact reflection law reveals a separate additive pattern.
January 9, 2025 · 15 min
collision
The Double Transversality
Centered collision signals from coprime bases remain recoverable when combined. Within each base, higher-lag correlations raise a separate question.
October 3, 2024 · 14 min
collision
The General Neutrality Theorem
Neutrality survives simultaneous prime remainder conditions. In the finite character sums, positive and negative contributions remain.
July 21, 2024 · 15 min
collision
The Neutrality Theorem
The collision table balances exactly. Does that balance survive when we sort the primes by their remainder on division by three?
April 23, 2024 · 15 min
collision
The Collision Transform and the Critical Strip
Give the large primes more weight. The finite digit table selects the L-functions whose zeros can obstruct the sum.
January 13, 2024 · 16 min
collision
The Collision Periodic Table
Forty decimal endings hold the collision deviations. Complementary cells add to minus one, fixing the mean and the negative half of the table.
December 2, 2023 · 13 min
collision
The Centered Collision Sum
Centering a finite table removes the drift and reveals which arithmetic patterns remain.
October 6, 2023 · 16 min
collision
The Collision Fluctuation Sum
A slow drift across the primes leads back to forty integers and a reflection that leaves every pair one short of zero.
April 28, 2023 · 16 min
collision
Silent Primes
Slide a repeating decimal by one place and count the matches. In base ten, seven primes above ten give zero. Their finite recipe opens a question about the spread of positive collision counts.
January 21, 2023 · 18 min
collision
The Character Structure of the Collision Fluctuation
The collision sum hides differences between primes with different last digits. Dirichlet characters separate them, and every stream can be recovered.
October 14, 2022 · 15 min
spectral
Bin Derangements and the Gate Width Theorem
For every prime greater than ten, exactly nine multipliers change every first digit. The same nine fractions identify them all.
July 4, 2022 · 15 min
spectral
Phase-Filtered Ramanujan Sums and the Spectral Gate
Primes can share a bin pattern yet admit different digit matches. Ramanujan’s sum reads the difference from the spectrum.
April 17, 2022 · 15 min
spectral
The Autocorrelation Formula
A repeating decimal can be compared with every rotation of itself. One frequency table recovers the exact match counts.
January 26, 2022 · 15 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