Introduction

Division at a fixed base and lag produces a finite collision table. After centering, that table has no principal component and no component inherited from the base alone. The surviving character coefficients record the exact finite geometry. Truncated prime-character sums record how primes meet the same characters up to a declared cutoff.

Two directions can then be separated. Within one base, the question is whether large collision coefficients are assigned to large or small prime-character sums as the lag increases. Across coprime bases, the question is whether the centered tables retain independent directions on a common finite space.

The second question has an exact answer. The Chinese remainder theorem places the lifted base signals on orthogonal coordinate axes. The first question is measured at finite cutoffs. The higher-lag data show a persistent negative assignment after two small base-three cases, but they do not supply an asymptotic theorem.

The distinction matters. Orthogonality on the complete finite state space does not imply cancellation along the primes. A finite correlation does not imply convergence. The exact identities and the numerical observations will be kept separate throughout.

The Finite Character Pairing

Fix a base b\geq 2 and a lag \ell\geq 1. Put m=b^{\ell+1}, \qquad U_m=(\mathbb Z/m\mathbb Z)^\times, and define the diagonal digit set G_{b,\ell}= \left\{0\leq n<m\ \middle| \left\lfloor\frac{n}{b^\ell}\right\rfloor=n\bmod b \right\}. The first and last base-b digits agree on this set, and |G_{b,\ell}|=b^\ell.

For a\in U_m, define the exact collision table T_{b,\ell}(a)= -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). No prime representative is needed to construct this table.

Reduction from U_m to U_b partitions the table into fibers A_u=\{a\in U_m\mid a\equiv u\pmod b\}, \qquad u\in U_b. Every fiber has b^\ell elements. Put \mu_{b,\ell}(u)= \frac1{b^\ell}\sum_{a\in A_u}T_{b,\ell}(a), \qquad f_{b,\ell}(a)=T_{b,\ell}(a)-\mu_{b,\ell}(a\bmod b).

Call a character active when its coefficient in (5) is nonzero. Every active character is odd and does not descend to the base.

For a finite cutoff X>m and a real exponent \sigma>0, put \begin{aligned} P_X(\sigma,\chi)&= \sum_{m<p<X}\frac{\chi(p)}{p^\sigma}, \\ F_{b,\ell,X}(\sigma)&= \sum_{m<p<X}\frac{f_{b,\ell}(p\bmod m)}{p^\sigma}. \end{aligned}

The cutoff in (6) is part of the definition. No infinite value at \sigma=1/2 is assumed.

The Anti-Correlation at Higher Lags

The finite calculations were performed with nfield [1]. Every collision table was built directly from (1). The calculation checked reflection, every fiber sum, and the equality between the direct prime sum and the character decomposition. The largest decomposition error in the primary table was less than 2.4\mathbin{\cdot}10^{-14}. Runs with one and eight threads produced identical output.

The tested bases are three, five, and seven. The corresponding lag ranges are one through seven, one through four, and one through three. The primary cutoff is X=2{,}000{,}000 and \sigma=1/2. A coefficient was retained when its magnitude exceeded 10^{-10}. In every row, the only omitted odd characters were those proved inactive by Lemma 1.

For each retained character, the signed contribution is \operatorname{Re}\bigl( \widehat f_{b,\ell}(\chi)P_X(1/2,\chi) \bigr). The Pearson coefficient \rho compares the two retained lists \bigl|\widehat f_{b,\ell}(\chi)\bigr| \qquad\hbox{and}\qquad \bigl|P_X(1/2,\chi)\bigr|. It is reported only when both lists have nonzero variance. Conjugate characters are counted separately because they are separate terms in the full Fourier sum. Keeping one representative from each conjugate pair gives the same Pearson coefficient.

There are thirteen defined correlations. Twelve are negative. The one positive value occurs at base three and lag two. The lag-one base-three row has two conjugate active characters with equal coefficient magnitudes, so its Pearson coefficient is undefined rather than zero.

The three deepest tested rows were also followed through five increasing cutoffs. In each row the correlation becomes more negative throughout the declared range.

Every primary-cutoff net in Table 1 is positive. That sign is not stable under a change of cutoff. At base three and lag seven, the finite net changes sign twice across the five cutoffs. A positive value at one cutoff therefore carries no conclusion about convergence.

Coprime Bases

The cross-base direction is exact. Let \mathcal B be a finite set of pairwise coprime bases. Choose a lag \ell_b\geq1 for each b\in\mathcal B and put m_b=b^{\ell_b+1}, \qquad M=\prod_{b\in\mathcal B}m_b. The Chinese remainder theorem identifies U_M\cong\prod_{b\in\mathcal B}U_{m_b}. Lift the centered signal from base b to U_M by \mathcal F_b(a)=f_{b,\ell_b}(a\bmod m_b). Use normalized counting inner products on every unit group, so \langle h,k\rangle_{U_q}= \frac1{\varphi(q)}\sum_{a\in U_q}h(a)\overline{k(a)}.

Uniform counting on U_M makes its Chinese-remainder coordinates independent and uniform. The random variables \mathcal F_b are therefore independent, not only orthogonal. Conditional averaging shows that their weighted aggregate preserves every component having nonzero weight.

Pairwise coprimality gives the cleanest sufficient condition. There is an exact formula when two moduli share a factor.

Global centering removes the principal common mode. It does not remove a nonconstant mode on U_d. Coprime moduli have a trivial common quotient, so their global zero means are sufficient for orthogonality. More generally, orthogonality follows whenever either conditional average in (12) vanishes identically.

For the ten prime bases 3,5,7,11,13,17,19,23,29,31, the theorem applies at lag one or at any independently chosen lags. The weight w_b=1/b^2 gives one possible combined readout, but the orthogonality does not depend on that choice. It holds for every set of weights.

An exact nfield audit built all ten lag-one tables and checked their 45 pairwise inner products in factored Chinese-remainder coordinates. Every cross term was exactly zero. The weighted Pythagorean identity for w_b=1/b^2 also held exactly. These checks support the implementation in nfield [1]. The proof of Theorem 3 does not depend on them.

The theorem concerns uniform counting on the complete joint unit group. Primes trace one arithmetic sequence through that product space. Their coordinates are not independent samples, so (9) does not assert cancellation of a prime sum.

The Double Transversality

The word double refers to two different directions.

Within a base, the exact character pairing joins the collision spectrum to finite prime-character sums. Twelve of the thirteen defined higher-lag correlations are negative at the primary cutoff. The deepest tested row in each base becomes more negatively correlated across the five declared cutoffs. This is finite evidence about the coefficient-to-character assignment.

Across pairwise coprime bases, the centered collision signals are exactly orthogonal. Their character supports occupy disjoint coordinate axes, and their energies add without cross terms. This is a theorem about the finite collision geometry.

The two statements do not prove one another. The computed within-base pattern uses primes and a cutoff. The proved cross-base pattern uses the complete finite state space and no primes. Their common content is that direction matters. Within a base, the coefficient assignment matters. Across bases, the Chinese-remainder coordinate matters.

Exact Orthogonality and Finite Data

The floor formula, reflection law, centered gate, finite character pairing, and coprime-base orthogonality are exact. None depends on a searched prime representative or a numerical fit.

The higher-lag correlations and signed nets are finite calculations. They use declared moduli, prime intervals, cutoffs, and retention thresholds. nfield checked every residue class in the fourteen collision tables and reproduced the complete ledger [1]. These calculations do not establish a limiting correlation.

Tail windows across bases depend on their starting point. Removing the initial prime segment can change the signs of the individual base contributions. At \sigma=1/2, nfield [1] evaluated the ten natural sums from b^2<p<X at cutoffs from 250{,}000 through 5{,}000{,}000. All fifty values were positive. Mixed signs appeared only after the initial segment through 997 was removed. This is finite evidence, not a positivity theorem. It shows why the exact orthogonality theorem must replace, rather than justify, a claim of prime-sum cancellation across bases.

The remaining problem is cutoff-uniform control of the higher-lag assignment and its effect on the surviving prime-character sum. Until such control is proved, there is no conclusion about convergence below one and no conclusion about zero locations.

References