About

This site is the public record of a research program I have been developing since 2009. It starts with one observation about long division.

The floor quotient $\lfloor br/p \rfloor$ that produces repeating decimals also produces, through its collision table, exact $L$-function special values at $s = 1$. I call it the digit function in this program. It is the mechanism by which arithmetic renders its internal structure into the digits we see on the page, and it connects long division to the distribution of the primes.

The digit function is the simplest object in this entire program. It is the rule everyone learns in elementary school for converting fractions into decimals. It is trivially defined and universally familiar. And yet, when you follow it far enough, it connects long division to some of the oldest unanswered questions in number theory. The whole program is the story of how that bridge gets built.

Each article on this site is an introductory companion to a research note linked at the bottom. The articles are written for anyone curious and patient. Someone encountering the material for the first time should be able to follow the story, while a specialist should find nothing to object to. Most of analytic number theory is written for specialists, and from the outside it can feel arcane. There is real beauty in this mathematics, and I want it to be visible to readers who do not normally meet it.

The program

The digit function does two things at every step. It produces a digit and it produces a remainder. The digit tells you where you are. The remainder tells you where you go next. The program follows both.

Alignment. Every fraction has a repeating decimal. When two fractions share a denominator, their digits agree at some positions and disagree at others. The fraction of positions where they agree is the alignment. For fractions built from a prime $p$ and a base-supported multiplier $m$, the alignment has an exact formula. The golden ratio appears as a threshold. The alignment exceeds $1/\varphi$ if and only if $p = 3$. No other odd prime crosses it. The three-tier theorem makes this classification complete, with a gap in the alignment spectrum that no integer can occupy.

Musical tuning. The alignment formula produces a deficit from its limiting value at each prime. The ratios of these deficits, across the base-supported integers, recover the ratios of musical intervals. In base 12, the deficit lattice reproduces the Pythagorean comma. In base 30, it reproduces the just major scale. The primes that organize long division are the same primes that organize musical pitch.

Collisions. The digit function sorts remainders into bins, one bin per digit. When two remainders land in the same bin, that is a collision. The collision count for a prime depends only on the prime's last two digits. Forty possible fingerprints in base 10. Every prime carries one, exactly determined. The fingerprints come in opposing pairs that cancel each other, and the average across all of them is exactly $-1/2$. This is not approximate. It is a theorem.

The gate. Exactly $b - 1$ multipliers produce zero collisions at every prime, independent of which prime. They form a rational family determined by the base alone. The primes where a specific non-gate multiplier also produces zero collisions have a finite recipe. In base 10 at lag one, there are exactly seven such silent primes.

Frequency analysis. Any pattern that repeats can be broken into pure tones, the way a sound can be broken into frequencies. The collision pattern decomposes the same way. Each frequency component corresponds to a Dirichlet character, a periodic filter that detects specific symmetries in the primes. Not all characters appear. The collision geometry kills even characters and characters that factor through the base, leaving only the odd fine-scale characters active. This selection is a theorem, not a heuristic.

Neutrality. Every residue class mod every prime carries its own collision signal. Each one is independently centered-neutral. The centered collision sum over primes in any arithmetic progression converges at $s = 1$. No two progressions lean on each other for cancellation. The neutrality holds simultaneously across all squarefree moduli.

Energy and drift. The collision coefficient at each active character factors exactly into a generalized Bernoulli number times a finite diagonal sum. One identity captures 97.5% of the total energy. The remaining fraction shrinks as the base grows. The collision sum over all primes up to $x$ drifts at the universal rate $-(b-1)/b \log\log x$. In base 10, the coefficient is exactly $-9/10$.

Carry boundaries. Long division occasionally produces a carry. This happens at specific remainders where the digit function jumps. These carry points divide the circle of remainders into arcs. Every collision observable reduces to a carry-boundary observable on a finite cell support. The carry-boundary factorization separates a universal Bernoulli potential from a signed boundary flux. The two factors are independent.

The cubic law. The total collision energy across all characters grows as $b^3$, splitting in the exact ratio $1 : 2$ between diagonal and off-diagonal contributions. The proof connects the energy to classical Dedekind sums and uses three-term reciprocity to separate the leading and secondary terms.

Orbits. The remainder at each step of long division traces a repeating path, an orbit. Some primes produce one long orbit that visits every remainder. Others produce several shorter orbits running in parallel. The carry boundaries cut across these orbits, and the collision energy decomposes into nonnegative conductor-level contributions. No proper-conductor response can hide by cancellation against the top level. The structure that survives cancellation across all primes lives at this intersection.

The analytic bridge. The finite collision table lifts to an analytic family in the critical strip. At every active character, the analytic collision transform has exactly the same zeros, with the same multiplicities, as the Dirichlet $L$-function it encodes. The finite diagonal sum is the switch. When it is nonzero, the $L$-function zeros are visible through the collision lens. When it vanishes, that character channel is silent. This is the connection from long division to the distribution of the primes.

All results are supported by nfield, an open-source analysis engine written in C.

About me

I'm Alex Petty. I live in Virginia.

I am founder and CEO of Singularics (formerly Boston Agile Labs). I help large organizations redesign how they operate so that AI delivers real outcomes, not just new tooling. Two decades of organizational transformation, first in lean and agile delivery, now in AI-native operating design and what I call continuous governance. I also build the tooling. Intent is the delivery system.

I have several patents pending on formal models of identity and recursion for artificial intelligence systems.