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Fix a base and a lag. Coordinate agreement between two long-division positions gives a centered integer label. For every denominator coprime to the base, that label is determined by one table modulo a fixed power of the base. A prime dividing the base has label zero. This elementary finite label has an exact spectral consequence.
For any N primes, form the exponential affinity kernel from differences of their labels. If exactly q labels occur, the squared normalized Laplacian has eigenvalue 1 with multiplicity exactly N-q. Its remaining q eigenvalues are simple and lie in [0,1), with one equal to zero. Thus a matrix growing with the prime list has a nontrivial spectrum bounded solely by the base and lag. Dirichlet’s theorem shows that the bound eventually becomes exact. Finite determination supplies a uniform bound on the quotient dimension; the label populations determine its entries.
A growing prime-indexed matrix can have an exactly finite nontrivial spectrum. Fix a base and a lag, and let long division assign one centered collision label to each prime. If q labels occur among N primes, the squared normalized Laplacian of their exponential label kernel has N-q eigenvalues exactly equal to 1. Its remaining q modes are simple and lie strictly below 1. Their number is bounded by the fixed label set while the prime list grows, and becomes constant once every label has appeared.
Coordinate agreement under cyclic shift is the unnormalized Hamming correlation of sequence theory [1]. When the base generates the unit group modulo a prime, the digits of its reciprocal form one orbit and the count below is such a cyclic-shift agreement. Kak and Chatterjee studied Hamming distance and numerical autocorrelation for prime-reciprocal digit sequences [2]. The finite table proved here is evaluated over the complete nonzero residue ensemble. It holds for every denominator coprime to the base and requires neither primality nor a primitive-root hypothesis.
The spectral construction uses these labels to index an arbitrarily large prime matrix. Its quotient dimension is bounded before the prime list begins to grow.
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 map sends a remainder to the next digit.
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.
The collision count records agreement between individual digit bins. Finite determination places its centered values in one table. That finite label set gives the spectral quotient. Reflection separately controls the character support of the 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. ◻
Finite determination holds for every N\geq2 coprime to b. Small primes sit in the same table. In N=mt+a, the quotient t is simply allowed to be zero.
Complementary patterns in rational expansions are classical [3, 4]. The table has the following exact centered mirror law.
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 labels have opposite deviations from -1/2. 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 parity projection of an odd function on a finite unit group is standard finite Fourier analysis [5]. Its content here is the preceding arithmetic fact that the centered collision table is odd. No character factorization is needed for the spectral quotient below.
Watson constructs kernel operators on prime sets from pairwise arithmetic divergences and studies their heat traces and spectral compression [6]. The kernel here has a different and more rigid input. Each entry depends only on the difference between two values in the finite collision table.
Normalized graph Laplacians are standard [9], as is spectral reduction along an equitable partition [8]. The positive definiteness of the exponential distance kernel belongs to classical positive-definite function theory [10]. Perron–Frobenius theory and the simple spectrum of an irreducible symmetric tridiagonal matrix are standard matrix analysis [11]. The theorem below gives a self-contained proof for this kernel. Its arithmetic content is that the partition comes from a fixed residue table, which bounds the quotient dimension uniformly over prime lists.
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 [9].
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 full N-prime matrix carries its entire spectral content in 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 the exact heat trace of a rank-one normalized kernel. Corollary 6 places the collision profile within a fixed distance of it while N grows. Finite compression forces this relation. Any asymptotic universality law beyond the quotient remains open.
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 [7]. 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 [13] checked every unit class in the three tables by exact integer arithmetic, computed all 900 prime labels directly, confirmed the base-prime zero cases, and recovered q=12 and N-q=288 in each system. At \tau=1, its numerical block-action and eigenvalue checks use tolerance 5\times10^{-10}. The exact counts corroborate the displayed examples, while Theorem 5 supplies the proof for every finite prime list.
Twelve is the exact label count in these three systems. The invariant statement 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.
In kinetic theory, a collision invariant is a function of velocity whose sum over the colliding particles is unchanged by a collision. Momentum and kinetic energy give familiar examples [12]. Such identities lead to conservation laws in the macroscopic description. The arithmetic construction uses persistence under local rearrangement as an analogy; its reflection law is not a kinetic conservation law.
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.
The analogy bears no mathematical weight. The exact statement is the finite table and its quotient spectrum. Local remainder motion is compressed first to label populations and then to the q quotient modes.
The mechanism can now be stated without analogy. Finite determination places every prime away from the base in a periodic label table, while the finitely many base primes have label zero. Repeated labels make the exponential kernel block constant. The standard quotient mechanism then becomes exact, while the arithmetic table gives a fixed upper bound on the quotient dimension as new primes are added.
The arithmetic input is the exact long-division table. For the specified exponential kernel, the first N primes contribute their populations among finitely many collision states. Once every state has appeared, each increase in the matrix dimension adds to the multiplicity of the eigenvalue 1. The quotient eigenvalues still depend on the label populations.
The finite table bounds the number of nontrivial modes; the populations determine the quotient. That finite description is the structure that survives.
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