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.
Divide 1 by 7 in base 10. You get the repeating block 142857. The first digit is 1. The last digit is 7. They do not match. Now divide 2 by 7. The block is 285714. First digit 2, last digit 4. Still no match. Keep going through all six remainders. Some first digits match their last digits. Some do not. Count the matches, subtract what chance would predict, and write down the deviation.
Do this for every prime. For every base. The deviations form a table, and the table has a square mass. That square mass is the collision energy.
The energy grows. In base 3 it is small. In base 101 it is large. In base 10007 it is very large. Plot the energy against the base and the curve is unmistakable. It is cubic. $E_b$ grows like $b^3$.
The question is whether you can prove the constant.
A century of tools, waiting
The floor function is the oldest operation in arithmetic. Divide, keep the integer part, discard the rest. Euclid used it. Every child who does long division uses it. It is so elementary that it barely registers as an object of study.
But the floor has a secret life. When $r$ increases by 1 and you compute $\lfloor br/q \rfloor$, the fractional part rises steadily until it passes 1, then drops. The digit ticks up. The fractional part resets. Rise, reset, rise, reset. That waveform is a sawtooth, and Euler knew about it three hundred years ago.
Dedekind studied sums built from that sawtooth in the 1870s. He was working on the foundations of algebraic number theory, on the structure of ideals in number fields, and the sawtooth appeared as a correction term in the transformation law of the Dedekind eta function. Rademacher proved a three-term reciprocity law for those sums in the 1950s. Conrey and others computed their mean values in the 1990s. The theory is deep and well-developed.
The collision energy is a weighted average of Dedekind sums. The connection is exact. The finite spectral object, with its thousands of character terms, compresses into one classical sum over one modulus. The tools were sitting there for a century.
The weight that vanishes
The collision energy starts as a sum over characters. Thousands of terms, one for each harmonic of the unit group. Each term is a product of two factors, the floor's Bernoulli weight and the collision boundary's flux, both squared. That is exact but unwieldy. Thousands of terms is not a proof. It is a phone book.
The compression works like this. The characters are orthogonal. When you sum all the squared terms at once, the characters cancel against each other and the double index collapses into a single index. Instead of summing over characters, you sum over ratios. Each ratio $v$ is the quotient of two residues from the digit window, and the sum at that ratio turns out to be a Dedekind sum. Thousands of character terms become one sum over ratios, each weighted by a Dedekind sum.
But the compression leaves a combinatorial residue. Not every ratio $v$ appears equally often. Some ratios can be produced by many pairs of window entries. Others by only one. The count of how many pairs produce a given ratio is the weight $w_b(v)$. Think of it as a popularity score. Some ratios land in the window often. Others land there once or not at all. The weight tracks who comes back.
This is the kind of combinatorial bookkeeping that makes formulas correct but hard to see through. You are not looking at the Dedekind sums anymore. You are looking at the Dedekind sums through a screen of multiplicities.
Then the weight disappears.
The weight counts pairs $(k, u)$ in the digit window with $kv \equiv u \pmod{b^2}$. Since $k$ is a unit, the congruence determines $v \equiv k^{-1}u$. Instead of summing over ratios $v$ and asking how popular each one is, sum directly over the two window coordinates $k$ and $u$ that produce it. The multiplicity is absorbed into the parametrization. What remains is bare:
Two coordinates, each running from 1 to $b - 1$. Each term is a classical Dedekind sum evaluated at the reduced ratio $u/k$. No auxiliary weight. No multiplicity screen. The collision energy laid open.
Now the structure is visible. When $u$ equals $k$, the ratio is 1. When $u$ does not equal $k$, the ratio is some other fraction. The energy splits into two natural pieces. The diagonal, where the ratio is 1. And everything else.
One third and two thirds
The diagonal piece of the collision energy is closed. Every term is $s(1, b^2)$, and the classical evaluation gives an exact rational number:
One third. The case where the two window coordinates are equal. A clean fraction. A closed form.
The off-diagonal piece is everything else:
Two thirds. The case where the coordinates differ.
One third and two thirds. They sit at $2\pi/3$ and $4\pi/3$ on the unit circle, equally spaced, complementary. In every base, in every modulus, the same two positions. The diagonal carries one pole. The off-diagonal carries two. Together they close the circle. The proportion $1:2$ is not an artifact of the proof. It is the pole structure of the collision energy.
The off-diagonal requires real work. The reduced fractions $a/k$, with $a$ larger than $k$ and no common factor, are the Farey fractions. Each one appears a known number of times. And each one requires Rademacher's three-term reciprocity to evaluate.
Rademacher's identity breaks a Dedekind sum at a large modulus into a rational main term and two Dedekind sums at smaller moduli. Apply it to every off-diagonal term and collect the results.
Four collected pieces. One large. One constant. Two small-modulus remainders.
The constant piece is $O(b^2)$. Too small to affect the cubic constant.
One of the small-modulus remainders vanishes exactly. The Dedekind sum is an odd function. Summing an odd function over a symmetric set gives zero. The entire piece is killed by a symmetry that Dedekind himself would have recognized.
The other small-modulus remainder survives. It does not vanish. It does not simplify. It requires an analytic estimate. More on that later.
The large piece carries the constant. And the constant depends on a single infinite series.
The conservation law
The diagonal third is exact. It comes from a single classical evaluation and asks nothing more. The off-diagonal two thirds is where the proof lives or dies. The Rademacher decomposition opens it into four pieces. Three are dispatched. The fourth is bounded. But the leading constant of the off-diagonal, the reason it is $\frac{2}{3} b^3$ and not some other multiple of $b^3$, depends on a single infinite series.
Strip away the prefactors and the constant reduces to this:
An infinite sum over coprime pairs. It equals 1.
Many series converge to 1. What matters is that this one does. It is the series the collision geometry produced. The Rademacher decomposition forced it. The Farey fractions shaped it. If this particular sum over coprime pairs had been $0.97$ or $1.03$, the off-diagonal would contribute something irrational, or something involving $\pi$, or something that depended on the base in a way that resisted simplification.
The proof passes through Euler sums, a family of infinite series that Euler studied in the 1700s and that Flajolet and Salvy catalogued in the 1990s. The key input is $\sum_{a \geq 1} H_a / a^2 = 2\zeta(3)$, where $H_a$ is the harmonic number $1 + 1/2 + 1/3 + \cdots + 1/a$. Mobius inversion inserts the coprimality condition. After the cancellation, the ratio $\zeta(3)/\zeta(3)$ appears. The numerator and the denominator are the same object. The coprime pairs do not contribute some fraction of the total. They contribute all of it. Nothing leaks out. Nothing is created.
The series is 1. Unity. The off-diagonal contributes exactly $\frac{2}{3}, b^3$. Add the diagonal third. The collision energy is
The $1:2$ pole split is now proved. The diagonal carries one third by a closed evaluation. The off-diagonal carries two thirds because the coprime Euler sum is 1. One channel and its complement. They sum to unity.
In physics, a conservation law says the total is preserved. Energy flows between channels but the sum does not change. Noether proved in 1918 that every conservation law comes from a symmetry, and every symmetry produces a conservation law. Physicists take this for granted. It is the deepest structural principle they have.
Here the conservation is arithmetic. The collision energy is $b^3$. One third at the diagonal pole. Two thirds at the off-diagonal pole. The total is the cube of the base, not a fraction of it, not a multiple, not an approximation. The Euler sum is 1 because the coprime pairs exhaust the available mass. The reflection identity $S(a) + S(m - a) = -1$ locks every table entry to its complement. The channels balance. The total is conserved.
Unity appears in number before it appears in physics. The integers have conservation laws. They do not need a physical substrate to enforce them. The symmetry is already there, built into the coprime structure of the rationals, and the collision energy inherits it the way a vibrating string inherits its overtones from its length.
The surviving logarithm
The cubic constant is proved. But the result says $b^3 + O(b^2 (\log b)^2)$, and that $(\log b)^2$ is interesting. It is the shadow of something the proof can see but not yet fully resolve.
Remember the four pieces from the Rademacher decomposition. One vanishes by symmetry. One is too small to matter. One delivers the $\frac{2}{3}$ through the Euler sum. The fourth piece survives. It is a sum of Dedekind sums at small moduli, and it does not simplify.
To control it, I needed a way to bound a weighted sum without computing every term. The technique is called Abel summation, and the idea is simple. If you are adding up a list of numbers with weights that shrink as you go, the total cannot be worse than the largest weight times the worst running total of the unweighted list. You separate how big the terms are from how much they cancel.
The running totals are partial sums of Dedekind sums. How well do those cancel? This is where the problem connects to one of the deepest threads in number theory. In 1987, Ilan Vardi showed that Dedekind sums are directly related to Kloosterman sums, a family of finite exponential sums that appear throughout the subject. In 1948, Andre Weil proved that individual Kloosterman sums exhibit square-root cancellation, the same phenomenon that governs the distribution of primes among the integers. The collision remainder inherits that cancellation, one modulus at a time, through the Dedekind-Kloosterman bridge.
The bound works. But it loses one logarithm compared to what the computation shows. Run the tables through a few hundred primes and the true remainder is visibly smaller than the proved bound. The gap comes from cancellation across moduli that the proof handles one at a time instead of collectively. Capturing that collective cancellation is the next problem.
The logarithm also has a second, more surprising source. The rational main term itself, the one that produced the $\frac{2}{3} b^3$, carries hidden secondary structure at the same $(\log b)^2$ scale. The generating function behind the main term has a double pole, and double poles produce logarithmic-squared terms the way a bell produces overtones. The floor itself planted the logarithm there. Sharpening the error requires accounting for that secondary term first, before the remainder can be isolated.
The boundary law
The entire proof lives on the additive side of the integers. Dedekind sums walk the number line. Rademacher reciprocity rearranges the steps. The Euler sum counts coprime pairs. But the same collision energy has a second, multiplicative form, a weighted moment of $L$-functions over Dirichlet characters. The same $1:2$ pole split appears there too. Two languages for the same object. They meet at the collision table.
The collision diagonal asks whether the first digit of the expansion equals the last. That is one question. But any question about digits selects a finite set. Every finite set has a signed boundary. Every signed boundary, fed through the floor's universal Bernoulli weight, produces a spectrum. Do three consecutive digits agree? Is the second digit one larger than the first? Does a particular four-digit pattern appear? Each question, same machinery, different boundary.
The cubic law says the total energy of the collision boundary is $b^3$. One third at the diagonal pole. Two thirds at the off-diagonal pole. The diagonal is a closed rational expression. The off-diagonal is pinned by an Euler sum that equals 1.
Dedekind studied his sums in the 1870s. Euler computed his harmonic series in the 1700s, decades before Gauss was born. Rademacher proved his reciprocity law in the 1950s. Weil proved his bound in 1948.
The collision energy is the new object. A digit-matching question that compresses into a Dedekind average, splits into a diagonal and an off-diagonal, and opens under reciprocity into four pieces. Three evaluated. The fourth controlled. The route from the collision table to the Dedekind sum, from the Dedekind sum to the Rademacher decomposition, from the decomposition to the Euler sum, did not exist before this paper. The ingredients belong to Dedekind, Rademacher, Euler, and Weil. The assembly is mine. And the constant that holds the whole thing together is an infinite series, built from harmonic numbers and coprime pairs, that has been equal to 1 since before anyone thought to write it down.
Try it yourself
Print the collision table at base 3. The antisymmetry $S(a) + S(b^2 - a) = -1$ pairs every entry with its complement.
$ ./nfield table --base 3
Now base 10. Twenty complement pairs, each summing to $-1$.
$ ./nfield table --base 10
Check the reflection identity directly.
$ ./nfield verify antisymmetry
Check the Parseval decomposition. The table-side energy should equal the spectral-side moment, term by term.
$ ./nfield verify decomposition
Code: github.com/alexspetty/nfield
Alexander S. Petty
June 2026
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