The Structure That Survives
Abstract
Fix a base and a lag. Long division assigns every prime an integer collision label. For primes not dividing the base, the label is determined by one finite table modulo a fixed power of the base. A prime dividing the base has the exact label zero. Complementary unit classes have labels that sum to minus one.
The finite label gives an exact spectral compression. Take any finite list of primes and form the exponential affinity kernel from the differences of their collision labels. If exactly q labels occur, then the normalized kernel reduces to a q-state quotient. The squared normalized Laplacian has eigenvalue 1 with multiplicity exactly N-q. Its remaining q modes are simple, distinct, and strictly below 1. At fixed base and lag, the number of nontrivial modes is therefore bounded independently of the number of primes. This is the precise sense in which the collision structure survives the growing prime sequence.
Long Division
People ask about the name. Why call the object a collision invariant? The echo from physics is intentional, but the name was not chosen for ornament. The mathematics pushed me there.
Fix an integer base b\geq2. For every integer N\geq2 and every 0\leq r<N, put \delta_{b,N}(r)=\left\lfloor\frac{br}{N}\right\rfloor. This is a schoolroom operation. It sends a remainder to the next digit. Nothing in it announces a finite spectral theorem.
Let \ell\geq1 and put B=b^\ell. Multiplication by B moves the remainders modulo N. The lag-\ell collision count is C_{b,\ell}(N) =\#\left\{1\leq r<N\ \middle|\ \delta_{b,N}(r)=\delta_{b,N}([Br]_N)\right\}, where [x]_N is the least nonnegative residue of x modulo N. The centered collision deviation is S_{b,\ell}(N) =C_{b,\ell}(N)-\left\lfloor\frac{N-1}{b}\right\rfloor.
At first the definition looks local. A remainder stays in its digit bin or it does not. Count the ones that stay. That should have been the whole story.
It was not.
The count settles into one finite table. Reflection locks that table into exact pairs. Character decomposition then carries the same finite object into Dirichlet L-values at one [1, 2, 3]. The local motion looks noisy. The aggregate does not.
The All-Prime Table
The finite table can be written without any large-modulus hypothesis. Put m=b^{\ell+1}=bB and define the diagonal set G_{b,\ell} =\left\{d(B+1)+bk\ \middle|\ 0\leq d<b,\ 0\leq k<b^{\ell-1}\right\}. It has B elements and contains both 0 and m-1.
Theorem 1 (Finite table). Let N\geq2 and (N,b)=1. Let a be the least positive residue of N modulo m. Then S_{b,\ell}(N) =-1-\left\lfloor\frac ab\right\rfloor +\sum_{n\in G_{b,\ell}} \left( \left\lfloor\frac{(n+1)a}{m}\right\rfloor -\left\lfloor\frac{na}{m}\right\rfloor \right). In particular, S_{b,\ell}(N) depends only on N modulo m.
Proof. For 1\leq r<N, set n(r)=\left\lfloor\frac{mr}{N}\right\rfloor. Writing n(r)=Bq+s with 0\leq s<B gives \delta_{b,N}(r)=\left\lfloor\frac{n(r)}{B}\right\rfloor. If Br=q'N+y with 1\leq y<N, then n(r)=bq'+\delta_{b,N}(y), \qquad y=[Br]_N. The two digits agree exactly when n(r)\in G_{b,\ell}.
The number of remainders in the nth slice is \left\lfloor\frac{(n+1)N}{m}\right\rfloor -\left\lfloor\frac{nN}{m}\right\rfloor, apart from the terminal endpoint r=N. The terminal slice belongs to G_{b,\ell}, so C_{b,\ell}(N) =-1+\sum_{n\in G_{b,\ell}} \left( \left\lfloor\frac{(n+1)N}{m}\right\rfloor -\left\lfloor\frac{nN}{m}\right\rfloor \right). Write N=mt+a with t\geq0 and 1\leq a<m. Every summand separates into t and the corresponding summand with a. Since |G_{b,\ell}|=B and b\nmid a, \left\lfloor\frac{N-1}{b}\right\rfloor =Bt+\left\lfloor\frac ab\right\rfloor. The B copies of t cancel and give (2). Nothing in the argument requires t to be positive. ◻
For a unit class a modulo m, let T_{b,\ell}(a) denote the right side of (2).
Corollary 2 (All primes). Every prime has a collision label. If p\nmid b, then S_{b,\ell}(p)=T_{b,\ell}(p\bmod m). If p\mid b, then S_{b,\ell}(p)=0.
Proof. The first statement is Theorem 1. If p\mid b, then B\equiv0\pmod p. For 1\leq r<p, one has \delta_{b,p}(r)=\frac bp r>0, \qquad \delta_{b,p}([Br]_p)=\delta_{b,p}(0)=0. Thus the collision count is zero. Also \lfloor(p-1)/b\rfloor=0. ◻
The lower bound sometimes attached to finite determination is therefore unnecessary. Small primes do not sit outside the table. The quotient in N=mt+a is simply allowed to be zero.
Reflection
The table has an exact mirror law [1].
Proposition 3 (Reflection). For every unit class a modulo m, T_{b,\ell}(a)+T_{b,\ell}(m-a)=-1. Consequently the centered table h_{b,\ell}(a)=T_{b,\ell}(a)+\frac12 is odd under reflection.
Proof. For 1\leq n\leq m-2, coprimality gives \left\lfloor\frac{n(m-a)}m\right\rfloor =n-1-\left\lfloor\frac{na}{m}\right\rfloor. The paired floor increments at a and m-a therefore sum to 1 on every interior slice. The two endpoint slices contribute 0 and 2. There are B-2 interior members of G_{b,\ell}, while \left\lfloor\frac ab\right\rfloor +\left\lfloor\frac{m-a}{b}\right\rfloor=B-1. Substitution into (2) gives the constant sum -1. The centered identity follows immediately. ◻
Complementary classes cannot vary independently. Every peak has a valley. Every arithmetic echo has its mirror. For a Dirichlet character \chi modulo m, define \widehat h_{b,\ell}(\chi) =\sum_{a\in(\mathbb{Z}/m\mathbb{Z})^\times} h_{b,\ell}(a)\overline{\chi(a)}. Pairing a with -a gives the following immediate consequence.
Corollary 4 (The parity gate). If \chi(-1)=1, then \widehat h_{b,\ell}(\chi)=0. Only odd characters can survive the centered collision transform.
The full collision transform adds a finer centering along reduction fibers. That removes the induced coarse characters as well. At lag one in an odd prime base, the remaining coefficients are primitive and odd. Their exact factorization contains a finite diagonal character sum and a generalized Bernoulli factor, equivalently a Dirichlet L-value at one [2, 3]. The harmonic structure is not placed on top of the table. It is what reflection permits the table to retain.
The Finite Quotient
The finite collision label can be placed inside a normalized kernel. Kernel operators on prime sets also appear in Watson’s spectral framework [4]. The purpose here is narrower. No spectral dimension or universality claim is needed. The question is exact. How much information can a kernel retain when its input is only the collision label?
Let p_1,\ldots,p_N be distinct primes and write s_i=S_{b,\ell}(p_i). Suppose the distinct labels among the s_i are x_1<\cdots<x_q. Let I_\alpha=\{i\mid s_i=x_\alpha\}, \qquad n_\alpha=|I_\alpha|. For a scale \tau>0, define K_{ij}=\exp\left(-\frac{|s_i-s_j|}{\tau}\right), \qquad d_i=\sum_{j=1}^N K_{ij}. Let D=\operatorname{diag}(d_1,\ldots,d_N) and put A=D^{-1/2}KD^{-1/2}, \qquad L=I-A, \qquad H=L^2. This is the squared normalized Laplacian of the collision-label affinity [6].
Theorem 5 (Finite spectral quotient). With the notation above, define d_\alpha =\sum_{\beta=1}^q n_\beta \exp\left(-\frac{|x_\alpha-x_\beta|}{\tau}\right) and the symmetric q by q matrix Q_{\alpha\beta} =\frac{\sqrt{n_\alpha n_\beta} \exp\left(-|x_\alpha-x_\beta|/\tau\right)} {\sqrt{d_\alpha d_\beta}}. Then \operatorname{spec}(H) =\operatorname{spec}\bigl((I_q-Q)^2\bigr) \mathbin{\uplus}\{1^{[N-q]}\}. The matrix Q has simple eigenvalues 1=\mu_1>\mu_2>\cdots>\mu_q>0. Consequently the q quotient eigenvalues of H are the distinct numbers (1-\mu_1)^2,\ldots,(1-\mu_q)^2 in [0,1). One is zero. The eigenvalue 1 has multiplicity exactly N-q.
Proof. Let W be the space of vectors that are constant on every block I_\alpha. Its orthogonal complement consists of vectors whose entries sum to zero on every block. Since the kernel is constant on I_\alpha\times I_\beta, the normalized kernel A annihilates W^\perp. Hence H is the identity on W^\perp, which has dimension N-q.
For each block put e_\alpha=n_\alpha^{-1/2}\mathbf{1}_{I_\alpha}. The vectors e_1,\ldots,e_q form an orthonormal basis of W. Direct substitution shows that the matrix of A|_W in this basis is precisely Q. This proves the spectral decomposition apart from the assertion that no quotient eigenvalue of H equals 1.
It remains to locate the spectrum of Q. Put E_{\alpha\beta} =\exp\left(-\frac{|x_\alpha-x_\beta|}{\tau}\right). For 1\leq\alpha<q, set r_\alpha =\exp\left(-\frac{x_{\alpha+1}-x_\alpha}{\tau}\right). When q=1, the assertion follows at once from Q=(1). Suppose q>1. Let P be the tridiagonal Hermitian matrix whose quadratic form is y^*Py =|y_1|^2+ \sum_{\alpha=1}^{q-1} \frac{|y_{\alpha+1}-r_\alpha y_\alpha|^2}{1-r_\alpha^2}. The identity E_{\alpha\beta} =\prod_{k=\alpha}^{\beta-1}r_k \qquad(\alpha<\beta) shows by direct multiplication that PE=I. Thus P=E^{-1} and E is positive definite. Every adjacent off-diagonal entry of E^{-1} is nonzero.
Put R=\operatorname{diag} \left(\sqrt{\frac{n_1}{d_1}},\ldots, \sqrt{\frac{n_q}{d_q}}\right). Then Q=RER. It follows that Q is positive definite, while Q^{-1}=R^{-1}E^{-1}R^{-1} is again an irreducible symmetric tridiagonal matrix. An eigenvector of such a matrix is determined by its first entry through the three-term recurrence. Its eigenspaces are one-dimensional. The eigenvalues of Q are therefore simple.
The positive vector with entries \sqrt{n_\alpha d_\alpha} is a Q-eigenvector with eigenvalue 1. Since every entry of Q is positive, the Perron–Frobenius theorem makes 1 its simple largest eigenvalue. Positive definiteness now gives 1=\mu_1>\mu_2>\cdots>\mu_q>0. The map \mu\mapsto(1-\mu)^2 is strictly decreasing on (0,1]. The quotient eigenvalues of H are therefore distinct and strictly below 1. ◻
The theorem is not a numerical pattern. The full N-prime matrix contains no spectral information beyond the q collision states and their populations.
Corollary 6 (Heat trace). For t\geq0, put \Theta_H(t)=\operatorname{tr}(e^{-tH}). Then \Theta_H(t) =(N-q)e^{-t}+\sum_{\alpha=1}^q \exp\bigl(-t(1-\mu_\alpha)^2\bigr). Moreover, 0\leq \Theta_H(t)-\bigl(1+(N-1)e^{-t}\bigr) \leq(q-1)(1-e^{-t}).
Proof. The first identity is the spectral decomposition in Theorem 5. Since \mu_1=1, its contribution is 1. For 2\leq\alpha\leq q, one has e^{-t}\leq \exp\bigl(-t(1-\mu_\alpha)^2\bigr) \leq1. Adding these q-1 inequalities gives the bounds. ◻
The comparison profile 1+(N-1)e^{-t} is not an empirical fit. It is the exact heat trace of a rank-one normalized kernel. Corollary 6 shows why the collision profile remains within a fixed distance of it while N grows. The resemblance is forced by finite compression. It is not evidence for a separate universality law.
Exact Compression
Let q_{b,\ell} be the number of distinct values in the finite unit table, together with the zero label for primes dividing the base if zero is not already present. This number depends on the base and lag, not on how many primes are listed.
Corollary 7 (Fixed spectral complexity). For the first N primes, the number of eigenvalues of H different from 1 is at most q_{b,\ell}. Every unit class modulo m contains infinitely many primes. Consequently every table value eventually occurs, and for all sufficiently large N the multiplicity of 1 is exactly N-q_{b,\ell}. In particular, the proportion of nontrivial modes tends to zero as N\to\infty.
Proof. The first statement is Theorem 5. Dirichlet’s theorem on primes in arithmetic progressions supplies infinitely many primes in every unit class modulo m [5]. Thus every value of T_{b,\ell} occurs among the prime labels, while the finitely many primes dividing b supply the exact zero channel. After all distinct labels have appeared, the quotient has dimension q_{b,\ell} and Theorem 5 gives the stated multiplicity. The final assertion follows by division by N. ◻
At lag one, exact evaluation of every unit class gives the following label sets.
| Base | Distinct labels | Label set |
|---|---|---|
| 7 | 12 | -6,-5,-4,-3,-2,-1,0,1,2,3,4,5 |
| 10 | 12 | -9,-7,-4,-3,-2,-1,0,1,2,3,6,8 |
| 12 | 12 | -11,-7,-5,-3,-2,-1,0,1,2,4,6,10 |
All twelve labels occur among the first 300 primes in each of these bases. The 300 by 300 Hamiltonian therefore has eigenvalue 1 with exact multiplicity 288. Only twelve modes remain, and those twelve are obtained from the quotient matrix. Independent finite evaluation with nfield [8] checked every unit class in the three tables, computed every prime label directly, compared every non-base prime with the finite formula, checked the exact zero label at the base primes, and verified the full block decomposition and quotient spectrum for all three 300-prime systems. These calculations check the displayed examples. They are not used in the proof of Theorem 5.
The number twelve is not asserted to be universal. What is universal is the mechanism. A fixed collision table gives a fixed upper bound on the number of modes that can remain nontrivial while the prime list grows.
The Collision-Invariant Analogy
In kinetic theory, collision invariants are the quantities that remain available after microscopic collisions have been aggregated into the macroscopic description [7]. The arithmetic construction is not a physical gas, but it has the same formal attitude.
Residues move under multiplication. Digit-bin coincidences appear and disappear. The centered count does not wander freely. It collapses to one finite table. Reflection removes half of its character channels. The spectral quotient shows that an arbitrarily long prime list still carries only finitely many collision modes at fixed base and lag.
Prime moduli are especially clean because their nonzero residues form one multiplicative group. If a physical image is useful, they resemble a noble gas more than a reactive medium. The particles in that image are not literal primes, and reflection is not momentum conservation. The useful part of the image is the survival of a small exact structure after many local rearrangements.
I was not looking for a metaphor. I was looking for the structure that remained.
Conclusion
Long division gives a digit partition. Base dilation moves the remainders through that partition. Exact centering produces an integer collision label for every prime. Finite determination places those labels in one table. Reflection locks complementary classes together. The character transform retains only the permitted harmonic channels. The normalized kernel then reduces exactly to the collision-value quotient.
This last reduction makes the word invariant concrete. As the list of primes grows, the ambient matrix grows with it, but the number of nontrivial collision modes does not. Each surviving mode is simple and strictly separated from the bulk eigenvalue. The local arithmetic is allowed to move. The exact structure remains.
References
[1]A. S. Petty, The Collision Invariant, arXiv:2604.00045.
[2]A. S. Petty, The Collision Transform, arXiv:2604.00047.
[3]A. S. Petty, The Collision Spectrum, arXiv:2604.00054.
[4]D. F. Watson, Spectral Geometry of the Primes, arXiv:2604.03351.
[5]H. Davenport, Multiplicative Number Theory, third edition, revised by H. L. Montgomery, Springer, New York, 2000.
[6]F. R. K. Chung, Spectral Graph Theory, American Mathematical Society, Providence, 1997.
[7]C. Cercignani, The Boltzmann Equation and Its Applications, Springer, New York, 1988.
[8]A. S. Petty, nfield, software repository. https://github.com/alexspetty/nfield