The Analytic Collision Transform

The collision program has been building toward a door.

On one side is the finite world of long division. Digit bins, collision counts, a periodic table of primes that depends on the last two digits. Everything on this side is proved, exact, and finite. On the other side is the analytic world where the Riemann zeta function and its cousins live. The critical strip, the zeros, the deep structure that governs how primes distribute themselves among the integers. The Collision Spectrum and the L-Function Landscape brought the program to that door and proved an exact identity at the threshold.

This paper walks through.

The same finite diagonal architecture that produced the boundary identity, the same character sum, the same diagonal set, now carries a different kernel. The periodic zeta kernel, slotted into the same slot where the Bernoulli kernel sat, turns the boundary identity into a strip identity. A formula at every point inside the critical strip, with the L-function on one side and the finite collision geometry on the other.

Then it walks back. The strip identity picks up a standard analytic factor from the functional equation on the way through. That is expected. It is the signature that odd characters leave on every L-function identity they touch. Peel that factor off, through a completion that any analytic number theorist would recognize, and the formula returns to the same Bernoulli-diagonal pattern that the spectrum theorem proved at the boundary. The finite architecture went into the strip and came back carrying the same structural fingerprint it left with.

The raw theorem

The diagonal set $G$ comes from the digit function of long division, $\lfloor br/p \rfloor$. It picks out the positions where the leading digit and the trailing digit agree. That set, and the character sum built from it, are the finite objects the collision program contributes. Everything else in the theorem is standard analytic number theory.

The diagonal character sum

is the finite load-bearing term. It is computed entirely from the digit function and the diagonal set. What changes is the kernel. Replacing the Bernoulli slice kernel by periodic zeta gives an exact character-weighted model sum in the strip, and summing that model identity over the diagonal set gives the theorem.

The completion

The stronger statement comes after one more step. The raw strip transform at s = 1 carries a Gauss-sum factor from the odd functional equation. Define the completed transform by removing that standard factor, indexed in the same convention as the spectrum theorem. The boundary value is then

That is the same Bernoulli-times-diagonal coefficient as the spectrum theorem. The same finite diagonal geometry does both jobs: it reaches into the strip, and after completion it returns to the boundary in the spectrum convention.

The part that does not move

The L-function changes as s moves through the strip. The gamma factor changes. The sine and the exponential change. But the diagonal sum $S_G(\chi)$, computed from the digit function $\lfloor br/p \rfloor$, is the same at s = 1/2 as it is at s = 1 as it is at s = 0.01. The finite digit geometry of long division is the invariant part of the analytic collision transform. Everything else moves around it.

The digit function produces a finite structural quantity that is load-bearing at every point in the critical strip.

Fixed conductor, open horizon

This is still fixed conductor. It does not approach the Riemann Hypothesis. It does not give variable-conductor control. But it does show that the same finite diagonal geometry supports both a strip theorem and a boundary recovery in one formal package.

The claim is specific. The kernel replacement, the character decomposition, the Hurwitz connection, the functional equation are all standard. The diagonal set and the digit function that produces it are not. They come from the collision program, and they are the part that survives the round trip.

Paper: The Analytic Collision Transform

Try it yourself

The numerical check is built into the nfield program. Run:

./nfield analytic --base 5

The output shows five rows, one for each value of s tested. The |LHS| column is the analytic collision transform computed directly from the periodic zeta kernel and the diagonal set. The |RHS| column is the theorem's prediction, computed from the gamma factor, the diagonal sum, and the L-function independently. The rel.err column is how far apart they are.

The numbers match to about eight decimal places. The relative error is about 10⁻⁸ at every s in the tested cases. That is not a fit. It is a direct numerical check of the exact identity, one value at a time, across the strip.

The |S_G| line near the top is the diagonal collision sum. That number does not appear in the s column. It is the fixed finite quantity the entire identity is built around.

Try other bases and lags. The first argument is the lag (default 1), and any prime base works:

./nfield analytic --base 3
./nfield analytic --base 7
./nfield analytic 2 --base 5
./nfield analytic 3 --base 5

Lag 2 at base 5 means m = 125 and the diagonal set has 25 elements. Lag 3 means m = 625 with 125 elements. The theorem holds at every lag and every prime base, at the same precision.

To see the collision periodic table that underlies all of this:

./nfield table --base 5

Code: github.com/alexspetty/nfield

Alexander S. Petty
May 2025

.:.