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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.
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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.
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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.
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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.
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The Collision Spectrum
The Fourier coefficients factor through Bernoulli numbers and L-function values at s = 1.
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The Collision Transform
The collision periodic table, centered and Fourier-transformed. It cancels at s = 1.
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The Collision Invariant
A finite signed table for every prime, built from the digit function. The collision invariant.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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The Spectral Power of the Digit Function
The squared bin sizes of the digit function control the alignment limit, the spectral structure, and the collision count. One object, three roles.
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The Spectral Structure of Fractional Fields
The eigenvalues of the cross-alignment matrix are determined by the cyclic autocorrelation of the digit function. One formula gives the entire spectrum.
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The Cross-Alignment Matrix
Compare every fraction to every other, digit by digit. The result is a symmetric matrix whose eigenvalues encode the internal structure of the fractional field.
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The Coherence Decomposition
Alignment splits into two parts: a focused component from the repetend orbit, and a pairwise component from cross-matches. The decomposition explains why some integers are more coherent than others.
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Primes and the Major Scale
The primes that organize long division are the same primes that organize musical pitch. In base 30, the alignment deficit lattice produces the just major scale. In base 12, the Pythagorean comma.
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The Three-Tier Theorem
Every positive integer falls into one of three tiers. The mirror bound from the nines complement forces everything with rough part past 7 below the golden line. The classification is unconditional.
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The Alignment Limit for All Primes
Past the digit-partitioning boundary, the alignment splits into lanes. Different smooth factors choose different lanes, and the limit may not exist as a single number. But no prime past 3 reaches the golden threshold.
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Digit-Partitioning Primes and the Alignment Formula
In base 10, exactly three primes produce total digit separation: 3, 7, and 11. Three different mechanisms, one shared condition, one universal formula.
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Three and the Golden Ratio
A cubic equation has a root in (0,1) for every prime. The golden ratio's minimal polynomial divides it at exactly one. The remainder names the prime 3.