Introduction

The collision table supplies one distribution. Truncated prime sums supply another. Random relabeling asks what their overlap would be if the channel names carried no information.

Some character channels are removed exactly by centering and reflection. Among the channels that remain, the arithmetic assignment can still place large collision weights beside small prime weights. Those effects must be separated before underlap can be measured.

The separation is exact. The induced odd channels form a forced zero sector. The primitive odd channels form the eligible sector, and conjugation reduces that sector to pairs with identical squared magnitudes. Uniform permutation of those pairs has an elementary mean and variance. The finite prime data can therefore be compared with the relabeling null model rather than with an unqualified constant.

The nfield calculations [2] show a persistent direction. Every nontrivial row in the tested base and cutoff ledger lies below the relabeling mean. They do not show a cutoff-independent ratio near 0.6. The direction survives. The proposed constant does not.

Centered reflection also makes the collision table odd. Convolving that table with itself creates a negative well at the reflection class and zero total mass across all additive shifts. Its additive spectrum is nonpositive. This is the structural Goldbach obstruction. It is logically separate from the multiplicative-character overlap.

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. If [x]_d denotes 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)|. 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

Using nfield [2], we built all collision values from (3), checked both reflection identities and every fiber sum, verified conjugate-pair equality, and obtained identical output with one and eight threads.

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 nfield ledger [2] has mean primitive-sector ratio 0.5909 and population standard deviation 0.0714 across the ten bases. That single-cutoff summary does not define a constant. Table 2 shows the movement with X.

Collision and prime energy underlap in every tested nontrivial row of the nfield ledger [2]. The size of that underlap remains cutoff-sensitive.

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 [1]. It measures spread, not channel location.

At the cutoff X=2{,}000{,}000, nfield [2] computed the primitive-sector effective supports from the same weights used in Table 1. The collision distribution is more concentrated than the prime distribution in every tested nontrivial base.

Effective support does not prove repulsion. A narrow distribution and a broad distribution can still place their largest weights on the same channels. The overlap statistic measures the assignment. Entropy only describes the two marginal shapes.

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 Goldbach Obstruction

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 obstruction 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.

The nfield check [2] verified (32) at every applicable even N from 4 through 100{,}000 and for bases 3,5,7,10,11,13. It found no violation and verifies the implementation over the tested range. The proof does not depend on the check.

Exact Obstructions and Finite Data

The floor table, centered reflection, forced-zero character gate, sector factorization, relabeling mean and variance, additive convolution identities, and Goldbach obstruction are exact.

The overlap and entropy ledgers are finite nfield calculations [2]. They use the listed prime intervals, bases, and cutoffs. nfield [2] reproduces the complete ledger from the direct floor table. The sixty nontrivial overlap rows all lie below the primitive-pair relabeling mean. That is finite evidence of a persistent assignment deficit.

No cutoff-independent repulsion constant is proved. No limiting overlap is defined without a rule coupling the prime cutoff to the growing base. The finite data also do not explain why the collision energy concentrates or why the surviving channels receive the observed prime weights.

Centering alone forces a nonpositive additive spectrum, zero total convolution, and a negative reflection well. It yields a nonpositive Goldbach sum on targets divisible by the collision modulus. It does not control the other additive shifts and does not turn a finite multiplicative underlap into an asymptotic theorem about primes.

References