Two Independent Directions

Center a collision table. Its mean disappears, but its variation remains. Combining such tables from different bases raises a concrete question. Can the combined signal retain the separate patterns? For coprime bases, the Chinese remainder theorem supplies independent coordinates, and averaging over those coordinates recovers every component. Their orthogonality has no cutoff and no prime input.

Within one base, collision coefficients are paired with truncated prime-character sums. Their observed association depends on a finite cutoff. The table and its fiber centering arise in The Centered Collision Sum [1]. The General Neutrality Theorem [2] gives the finite character pairing and the lag-one assignment question. Higher lags and combinations of bases provide two distinct ways to examine that structure.

When the base generates the units modulo a prime, the 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]. The definitions below instead use the complete nonzero residue system and require neither primality nor a primitive-root hypothesis for the finite collision table.

Finite Fourier inversion, product characters, and their orthogonality are classical [5]. The exact cross-base theorem is their application to the fiber-centered collision signals. The within-base contribution is a declared finite ledger at higher lags. All definitions and proofs needed for both directions are included here. The name double transversality records their coexistence without identifying a finite numerical pattern with an exact structural theorem.

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.

Represent each a\in U_m by its unique integer in \{1,\ldots,m-1\}. 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}

Equation (8) is a finite identity. At \sigma=1/2, its value is the value at the declared cutoff X.

The Anti-Correlation at Higher Lags

The finite calculations were performed with nfield [7]. 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 [6] 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. In these odd-prime-base computations, every retained character is nonreal. 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. The net is cutoff-dependent even where the magnitude correlation moves steadily.

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. Orthogonality 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. Theorem 3 supplies the symbolic proof.

The theorem concerns uniform counting on the complete joint unit group. Prime-restricted sums form a different arithmetic object. They follow one arithmetic sequence through the product space instead of averaging uniformly over every coordinate.

Two Meanings of 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 computed within-base pattern uses primes and a cutoff. The proved cross-base pattern uses the complete finite state space and no primes. Across bases, the coordinate split removes every cross term and allows each component to be recovered. Within a base, no comparable theorem fixes the assignment of collision coefficients to prime-character sums.

The Active Sector

The floor formula, reflection law, centered gate, and finite character pairing are exact. The cross-base result is a classical product-space mechanism applied to these particular signals. Fiber centering supplies the zero means, so distinct coordinate lifts are orthogonal, their Fourier supports meet only at the absent principal character, and their energies add.

The higher-lag correlations and signed nets have a narrower status. 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 [7]. The proofs above establish the finite identities independently of that calculation. No row implies a cutoff-uniform or limiting correlation law.

The unresolved question is therefore within one base. Does the negative association persist under a mathematically fixed normalization as the cutoff and lag vary, or does it belong only to the displayed windows? Across pairwise coprime bases, orthogonality is already exact. Within a base, the coefficient-to-prime assignment remains open.

References