The Missing Overlap

Two mass distributions may be broad or narrow for reasons that say nothing about which channels are paired. The question here holds both marginal mass lists fixed and changes only their assignment. Squared collision coefficients and squared truncated prime-character sums provide the two lists on the primitive odd characters modulo b^2. Conjugation groups them into equal-mass pairs, and uniform permutation of those pair labels gives an exact neutral baseline.

The Centered Collision Sum [1] isolates the finite centered table. The Double Transversality [2] examines its pairing with finite prime-character sums. Holding the two mass lists fixed now separates their concentration from their assignment to the same characters.

When the base generates the units modulo a prime, the underlying collision count is periodic Hamming agreement on a reciprocal digit word. Lempel and Greenberger give the general Hamming-correlation setting [3]. For decimal reciprocals of primes, Kak and Chatterjee study Hamming distance from cyclic shifts [4]. Girstmair and Murty and Thangadurai study neighboring digit-value statistics on full or possibly proper multiplicative orbits, with Dedekind sums, class numbers, and generalized Bernoulli numbers attached to odd characters [5, 6]. The signal below is instead an equality indicator on the complete nonzero residue system.

Finite Fourier analysis and convolution are classical [7], as are permutation sums of the form used for the null model [8]. The collision-specific work begins with the exact centered table. Reflection and fiber cancellation force its multiplicative transform into the primitive odd sector. The relabeling theorem then separates marginal concentration from channel assignment. All sixty nontrivial rows in the declared ledger lie below its mean, but their moving values give finite evidence only. The term spectral repulsion refers to this measured assignment deficit, not to random-matrix eigenvalue repulsion and not to an asymptotic theorem.

The same centered table carries an independent additive law. Its self-convolution has zero total mass, a negative well at the reflection class, and a nonpositive additive spectrum. For suitable even targets this becomes an exact sign law for centered collision-weighted prime pairs. It is a second consequence of the table, not an explanation of the multiplicative underlap.

The Finite Collision Table

Fix a base b\geq2, put m=b^2, and write U_m=(\mathbb Z/m\mathbb Z)^\times. For an integer d>m coprime to b, define \delta_{d,b}(r)=\left\lfloor\frac{br}{d}\right\rfloor, \qquad 1\leq r<d. For d\nmid x, let [x]_d denote the representative of x\bmod d in \{1,\ldots,d-1\}. Put \begin{aligned} C_b(d)&= \#\left\{1\leq r<d\mid \delta_{d,b}(r)=\delta_{d,b}([br]_d)\right\}, \\ S_b(d)&=C_b(d)-\left\lfloor\frac{d-1}{b}\right\rfloor. \end{aligned}

Represent each a\in U_m by its unique integer in \{1,\ldots,m-1\}. The lag-one diagonal set is G_b=\{r(b+1):0\leq r\leq b-1\}. Define the raw collision table by T_b(a)= -1-\left\lfloor\frac ab\right\rfloor +\sum_{n\in G_b} \left( \left\lfloor\frac{(n+1)a}{m}\right\rfloor -\left\lfloor\frac{na}{m}\right\rfloor \right).

Reduction modulo b gives fibers A_u=\{a\in U_m:a\equiv u\pmod b\}, \qquad u\in U_b. Each fiber contains b residues. Put \mu_b(u)=\frac1b\sum_{a\in A_u}T_b(a) and define the centered table f_b(a)=T_b(a)-\mu_b(a\bmod b).

The floor formula is intrinsic. It constructs every table entry without searching for a prime representative of the residue class.

The Primitive Odd Sector

Now let b be an odd prime. For a Dirichlet character \chi modulo m, put \widehat f_b(\chi)= \frac1{\varphi(m)}\sum_{a\in U_m}f_b(a)\overline{\chi(a)}.

Let \mathcal O_b denote the odd characters modulo b^2 and let \mathcal A_b denote the primitive odd characters. Their cardinalities are |\mathcal O_b|=\frac{b(b-1)}2, \qquad |\mathcal A_b|=\frac{(b-1)^2}2. The first count is half of \varphi(b^2). The odd characters induced from modulus b number (b-1)/2, which proves the second count.

The sector \mathcal A_b has no self-conjugate characters. The kernel of U_{b^2}\to U_b has odd order b, so every character of order at most two is trivial on that kernel and factors through modulus b. Thus \mathcal A_b splits into k_b=\frac{(b-1)^2}{4} conjugate pairs.

For a finite cutoff X>b^2, define P_X(\chi)= \sum_{b^2<p<X}\frac{\chi(p)}{\sqrt p}, where p runs over primes. The cutoff is part of the definition. No infinite value at exponent one half is assumed.

Assume the two denominators below are nonzero. On \mathcal A_b, define the probability weights p_\chi= \frac{|\widehat f_b(\chi)|^2} {\sum_{\psi\in\mathcal A_b}|\widehat f_b(\psi)|^2}, \qquad q_{\chi,X}= \frac{|P_X(\chi)|^2} {\sum_{\psi\in\mathcal A_b}|P_X(\psi)|^2}. Their primitive-sector overlap and scaled overlap are \Omega_{b,X}=\sum_{\chi\in\mathcal A_b}p_\chi q_{\chi,X}, \qquad \rho_{b,X}=|\mathcal A_b|\,\Omega_{b,X}.

The full odd sector contains a separate occupancy effect. Put q^{\mathcal O}_{\chi,X}= \frac{|P_X(\chi)|^2} {\sum_{\psi\in\mathcal O_b}|P_X(\psi)|^2} and extend p_\chi by zero from \mathcal A_b to \mathcal O_b. Define \eta_{b,X}=\sum_{\chi\in\mathcal A_b}q^{\mathcal O}_{\chi,X}, \qquad \Omega^{\mathcal O}_{b,X}= \sum_{\chi\in\mathcal O_b}p_\chi q^{\mathcal O}_{\chi,X}, \qquad \rho^{\mathcal O}_{b,X}=|\mathcal O_b|\Omega^{\mathcal O}_{b,X}.

At base 3, the primitive odd sector has one conjugate pair. Its primitive-sector scaled overlap is identically one. Any full-sector deficit at that base is an occupancy effect rather than a within-sector assignment effect.

Random Relabeling

Reality of f_b and of the prime weights gives |\widehat f_b(\chi)|=|\widehat f_b(\overline\chi)| and |P_X(\chi)|=|P_X(\overline\chi)|. Permutation sums of the resulting pair masses belong to the classical combinatorial setting of Hoeffding [8]. A direct calculation gives the exact moments needed here. Choose one representative from each conjugate pair and define pair masses x_r=p_\chi+p_{\overline\chi}, \qquad y_r=q_{\chi,X}+q_{\overline\chi,X}, \qquad 1\leq r\leq k_b. Both lists sum to one. Since the two magnitudes agree within each conjugate pair, \rho_{b,X}=k_b\sum_{r=1}^{k_b}x_ry_r.

For a uniform random permutation \pi of the k_b pair labels, put R_\pi=k_b\sum_{r=1}^{k_b}x_ry_{\pi(r)}.

When the variance is positive, the standardized displacement is Z_{b,X}=\frac{\rho_{b,X}-1} {\sqrt{\operatorname{Var}(R_\pi)}}. A negative value records underlap relative to the relabeling model. It is not a probability model for the arithmetic.

The Measured Underlap

For the tested bases, nfield [10] builds every collision value from (3), checks the raw and centered reflection laws and every fiber sum, and verifies conjugate-pair equality.

The tested prime bases are 3,5,7,11,13,17,19,23,29,31,37. The six cutoffs are 250{,}000,\ 500{,}000,\ 1{,}000{,}000,\ 2{,}000{,}000,\ 3{,}000{,}000,\ 5{,}000{,}000. Every prime sum uses exactly the interval in (14).

At base 3, the primitive sector has one pair and \rho_{3,X}=1 at every cutoff. Table 1 gives the ten nontrivial rows at X=2{,}000{,}000.

At this cutoff, the finite ledger has mean primitive-sector ratio 0.5909 and population standard deviation 0.0714 across the ten bases. Table 2 shows how the same statistic moves with X.

Collision and prime energy underlap in every tested nontrivial row of the finite ledger. The magnitude varies with the cutoff, so the limiting assignment remains open.

Effective Support

For a probability vector w=(w_1,\ldots,w_N), let H(w)=-\sum_j w_j\log w_j, \qquad \operatorname{eff}(w)=e^{H(w)}. This is the exponential Shannon entropy [9]. It measures spread, not channel location.

At the cutoff X=2\cdot 10^6, the same finite ledger gives the primitive-sector effective supports from the weights used in Table 1. The collision distribution is more concentrated than the prime distribution in every tested nontrivial base.

Effective support records the two marginal shapes. A narrow distribution and a broad distribution can still place their largest weights on the same channels. The overlap statistic records the assignment.

The Additive Reflection Well

The additive identities do not require a prime base. Let b\geq2 be arbitrary again and put m=b^2.

Extend f_b by zero from U_m to all of \mathbb Z/m\mathbb Z. The extension, still denoted f_b, remains odd. Define its additive convolution by K_b(t)=\sum_{a\bmod m}f_b(a)f_b(t-a).

The zero total in (25) and the negative value in (26) are exact. The nonzero shifts collectively compensate for the reflection well. The well is strictly negative whenever f_b is nonzero, but the placement of the compensating shifts is not determined by the multiplicative overlap statistic.

The Reflected Prime-Pair Sign

For an even integer N, define the ordered centered collision-weighted Goldbach sum H_b(N)= \sum_{\substack{p+q=N\\p,q\ \mathrm{prime}}} f_b(p\bmod m)f_b(q\bmod m). The zero extension assigns weight zero when a prime residue is not a unit.

The sign law is additive and exact. It does not follow from the multiplicative-character underlap. Both statements arise from the same centered collision table, but they use different harmonic structures. For odd prime b, the applicable targets have relative density 1/b^2 among the even integers. At base 10, the relative density is 1/50.

nfield [10] verifies (32) at every applicable even N from 4 through 100{,}000 for bases 3,5,7,10,11,13. This exhausts the stated finite range. Corollary 7 proves the sign law for every admissible target.

The Coefficient Question

The floor table, centered reflection, forced-zero character gate, sector factorization, relabeling moments, additive convolution identities, and reflected prime-pair sign are exact. Finite Fourier analysis, permutation moments, and entropy provide classical machinery. The collision table supplies the particular primitive-sector distribution and the odd additive signal to which that machinery is applied.

The overlap and entropy ledgers have a different status. They cover the displayed bases, prime intervals, and cutoffs. Every one of the sixty nontrivial overlaps lies below the exact relabeling mean, but its magnitude moves with the cutoff. That establishes a finite assignment deficit, not a limiting law.

The additive reflection identity is stronger and separate. It fixes a nonpositive sign for every applicable centered collision-weighted prime pair, without controlling the other additive shifts or explaining the multiplicative deficit. The collision weights are fixed once the base is chosen. The open questions concern their exact distribution over the primitive odd channels and how the prime-side mass moves relative to them as the cutoff grows.

References