The Character Structure of the Collision Fluctuation
Abstract
Fix a base, a positive lag, and a finite prime cutoff. The collision deviation at each prime gives a real signal, and the primes carrying that signal split into the unit residue classes of the base. Dirichlet characters give the exact Fourier coordinates of those residue streams. We prove the forward and inverse transforms, identify the total fluctuation with the principal component, establish conjugate pairing, and obtain a Parseval identity. No information is lost. A finite base-ten calculation performed with nfield [4] illustrates the decomposition. The transform exposes every finite residue channel. Understanding those channels requires both the finite structure of the collision coefficients and their behavior across primes.
Four residue streams
In base ten, every prime greater than five ends in 1, 3, 7, or 9. A sum over those primes is therefore four sums before it is one. The four streams may carry the same signal, different signals, or signals that cancel only after they are added.
Dirichlet characters are made for this split. They are multiplicative weights on the unit residue classes, and they form a complete orthogonal basis for functions on those classes. In base ten there are four unit classes and four characters. Passing from the residue streams to character components loses no information. It changes the coordinates in which the finite signal is read.
The collision coefficient used below comes entirely from a finite partition. The character identities are exact at every fixed cutoff.
The finite collision signal
Fix an integer base b\geq2. Let p>b+1 be prime, and let [u]_p denote the least positive representative of a nonzero residue modulo p. For 1\leq r<p, define \delta_{p,b}(r)=\left\lfloor\frac{br}{p}\right\rfloor . The values of \delta_{p,b} lie in \{0,\ldots,b-1\} and partition the nonzero residues into b digit bins.
For g\in\mathbb F_p^\times, define the collision count C_{p,b}(g) = \#\left\{ r\in\{1,\ldots,p-1\}\ \middle|\ \delta_{p,b}(r)=\delta_{p,b}([gr]_p) \right\}. A nonidentity multiplier is constructive when its collision count is positive.
The reference level for the fluctuation is the mean collision count over the constructive multipliers. Its exact value comes from the finite gate structure in Bin Derangements and the Gate Width Theorem [1]. We derive it here because it centers the collision signal before the character transform.
Lemma 1 (Constructive mean). Write p-1=bQ+R, \qquad 0\leq R<b. There are p-b-1=b(Q-1)+R constructive multipliers, and their mean collision count is \overline C_{p,b} = \frac{Q\bigl(b(Q-1)+2R\bigr)} {b(Q-1)+R}.
Proof. Multiplication by b turns the digit bins into residue classes modulo b. Indeed, if x=[br]_p, then br=p\delta_{p,b}(r)+x, and reduction modulo b shows that two residues have the same digit exactly when their corresponding values of x agree modulo b. Consequently C_{p,b}(g) = \#\{x\in\{1,\ldots,p-1\}\mid x\equiv[gx]_p\pmod b\}.
For g\neq1, put c(g)=[b(1-g)^{-1}]_p. This gives a bijection from \mathbb F_p^\times\setminus\{1\} onto \mathbb F_p^\times\setminus\{b\}. We claim that C_{p,b}(g)=0 exactly when 1\leq c(g)<b.
Suppose a collision exists. Write y=[gx]_p=x+mb. The relation c(g)(1-g)\equiv b\pmod p gives x+mc(g)\equiv0\pmod p. If 1\leq c(g)<b and m\geq0, then 0<x+mc(g)=y-m\bigl(b-c(g)\bigr)<p. If m=-n<0, then 0<x-nc(g)=y+n\bigl(b-c(g)\bigr)<p. Either case produces a positive multiple of p smaller than p, which is impossible. Conversely, if b<c(g)<p, take x=p-c(g) and y=x+b. Both lie between 1 and p-1, x\equiv y\pmod b, and y=[gx]_p. Thus a collision exists. There are exactly b-1 nonidentity multipliers with zero collision count. The remaining (p-2)-(b-1)=p-b-1 nonidentity multipliers are constructive.
Let n_d be the size of the dth digit bin. The interval description in (1) shows that every n_d is Q or Q+1. Since the bin sizes sum to p-1, exactly R bins have size Q+1, and the other b-R have size Q. Counting ordered pairs in a common bin, first by the pair and then by their quotient, gives \sum_{g\in\mathbb F_p^\times}C_{p,b}(g) = \sum_{d=0}^{b-1}n_d^2 = bQ^2+R(2Q+1). The identity multiplier contributes p-1, and the zero-collision multipliers contribute zero. The total constructive collision mass is therefore bQ^2+R(2Q+1)-(p-1) = Q\bigl(b(Q-1)+2R\bigr). Division by the number of constructive multipliers proves (3). ◻
Fix a positive integer lag \ell. The collision deviation at p is \Delta_{p,b}(\ell) = C_{p,b}([b^\ell]_p)-\overline C_{p,b}. This remains defined when b^\ell\equiv1\pmod p. In that case the multiplier is the identity and its collision count is p-1. No primitive-root hypothesis is needed because the collision count is defined on the full multiplier group. The prime p=b+1, when it is prime, is excluded because there are no constructive multipliers and the reference mean is undefined.
Characters on the unit classes
Let U_b=(\mathbb Z/b\mathbb Z)^\times and let \widehat U_b be its character group. Its elements are the Dirichlet characters modulo b, restricted to the unit classes [2, 3]. There are \varphi(b) such characters.
For a real cutoff x, a complex parameter s, and r\in U_b, define the residue stream F_{r,x}(s,\ell) = \sum_{\substack{b+1<p\leq x\\p\ {\rm prime}\\p\equiv r\pmod b}} \frac{\Delta_{p,b}(\ell)}{p^s}. For \chi\in\widehat U_b, define the character component \widehat\Phi_{\chi,x}(s,\ell) = \sum_{\substack{b+1<p\leq x\\p\ {\rm prime}}} \frac{\chi(p)\Delta_{p,b}(\ell)}{p^s}. The total finite fluctuation is \Phi_x(s,\ell) = \sum_{\substack{b+1<p\leq x\\p\ {\rm prime}}} \frac{\Delta_{p,b}(\ell)}{p^s}. Here p^{-s} is defined using the real logarithm. These sums are finite, so no convergence hypothesis on s is needed.
Exact reconstruction
Theorem 2 (Finite character decomposition). For every cutoff x, every s\in\mathbb C, and every positive lag \ell, the residue streams and character components satisfy \begin{aligned} \widehat\Phi_{\chi,x}(s,\ell) &= \sum_{r\in U_b}\chi(r)F_{r,x}(s,\ell), \\ F_{r,x}(s,\ell) &= \frac1{\varphi(b)} \sum_{\chi\in\widehat U_b} \overline{\chi(r)}\,\widehat\Phi_{\chi,x}(s,\ell). \end{aligned} If \chi_0 is the principal character, then \Phi_x(s,\ell) = \sum_{r\in U_b}F_{r,x}(s,\ell) = \widehat\Phi_{\chi_0,x}(s,\ell). The transform also satisfies \sum_{\chi\in\widehat U_b} \left|\widehat\Phi_{\chi,x}(s,\ell)\right|^2 = \varphi(b)\sum_{r\in U_b}|F_{r,x}(s,\ell)|^2.
Proof. Partitioning the prime sum by residue class gives \widehat\Phi_{\chi,x} = \sum_{r\in U_b}\chi(r)F_{r,x}, which proves the forward transform. Character orthogonality gives \frac1{\varphi(b)} \sum_{\chi\in\widehat U_b} \chi(t)\overline{\chi(r)} = \begin{cases} 1,&t=r,\\ 0,&t\neq r \end{cases} \qquad (r,t\in U_b). Substituting the forward transform and applying this identity leaves only F_{r,x}, which proves the inverse transform.
Every prime in the sums is coprime to b, so the principal character equals one on every term. This proves (15). Finally, expand the left side of (16), use the forward transform, and apply \sum_{\chi\in\widehat U_b} \chi(r)\overline{\chi(t)} = \varphi(b)\,\mathbf1_{\{r=t\}}. Only the diagonal terms remain, which proves Parseval’s identity. ◻
Corollary 3 (Conjugate pairing). For every s\in\mathbb C, \widehat\Phi_{\overline\chi,x}(\overline s,\ell) = \overline{\widehat\Phi_{\chi,x}(s,\ell)}. In particular, when s is real, conjugate characters have equal component magnitudes.
Proof. The deviations are real. Conjugating the finite sum in (9) replaces \chi by \overline\chi and s by \overline s. ◻
A base-ten snapshot
The group U_{10}=\{1,3,7,9\} is cyclic with generator 3. Label its characters by \chi_j(3)=i^j, \qquad 0\leq j\leq3. Thus \chi_0 is principal, \chi_3=\overline{\chi_1}, and \chi_2 is real.
Table 1 uses lag \ell=1 and exponent s=1. The nfield repository [4] provides the finite character-transform surface used for this calculation and for inspecting the underlying residue streams. Every prime in the range 11<p\leq x is included. For each prime, nfield forms all p-1 digit values, evaluates the collision count for multiplication by 10, subtracts the exact constructive mean (3), and adds the four character weights. The column headed primes counts the eligible primes, not all primes below the cutoff.
| x | primes | \widehat\Phi_{\chi_0,x} | \operatorname{Re}\widehat\Phi_{\chi_1,x} | \operatorname{Im}\widehat\Phi_{\chi_1,x} | \widehat\Phi_{\chi_2,x} |
|---|---|---|---|---|---|
| 100 | 20 | -0.851414 | -0.012410 | -0.168966 | 0.300620 |
| 1000 | 163 | -1.149104 | -0.208657 | -0.217315 | 0.329630 |
| 5000 | 664 | -1.318994 | -0.288890 | -0.234378 | 0.354014 |
| 10000 | 1224 | -1.392595 | -0.316180 | -0.241931 | 0.350235 |
The same nfield calculation reconstructs every residue stream from the four components. In long-double arithmetic, the largest reconstruction error and the largest Parseval residual in these rows are both below 10^{-17}. These finite values verify the calculation and show the four distinct character coordinates, including the conjugate pair; they do not determine cutoff asymptotics.
Character Channels Across Primes
The character components are finite prime sums with collision coefficients. Each coefficient has an exact finite structure at its prime. Across primes, those coefficients form a character channel whose scale is unknown.
Question 4. For fixed b, \ell, and \chi, what is the behavior of \widehat\Phi_{\chi,x}(1,\ell) as x tends to infinity? Does it remain bounded, oscillate without a limit, or admit an unbounded leading scale?
At every finite cutoff, the residue streams determine the character components and the character components reconstruct the residue streams. The principal component alone cannot determine the nonprincipal ones. The arithmetic problem begins where the finite decomposition ends.
References
[1]A. S. Petty, Bin Derangements and the Gate Width Theorem, research note, July 2022, revised August 2026, \href{https://doi.org/10.5281/zenodo.21850917} {doi:10.5281/zenodo.21850917}.
[2]H. Davenport, Multiplicative Number Theory, 3rd ed., Springer, 2000.
[3]H. Iwaniec and E. Kowalski, Analytic Number Theory, American Mathematical Society, 2004.
[4]A. S. Petty, nfield, software repository. https://github.com/alexspetty/nfield