RANK3_ROUTE_D.md's exact identity (D7) splits the coprime-driven part of \(U_{(q)}\), summed over the reduced residues \(a\bmod q\), into a diagonal piece \(\Sigma_{\rm diag}(q)=\phi(q)\sum_b^*T(q,b)\) and a cross term \[ \Sigma_{\rm cross}(q)=\sum_{b\ne b'\bmod q}^*c_q(b-b')X(b,b'), \] with \(c_q\) the Ramanujan sum and \(X(b,b')\) the exact bilinear cross term (D6). Section 3 of that document proves only the cancellation-free ceiling \(|\Sigma_{\rm cross}(q)|\le\phi(q)(\phi(q)-1)\sum_b^*T(q,b)\) and states plainly that whether the true \(\Sigma_{\rm cross}(q)\) sits near \(0\) or near that ceiling "is not determined by UPPER_BOUND.md or RESULTS.md — both are consistent with the exact identity (D7)." This document reports a direct numerical measurement of \(\Sigma_{\rm cross}(q)\) at a ladder of moduli and cutoffs, from rank3_cross_term_probe.py, writing results_rank3_cross_term_probe.json.
Method
For each \(q\in\{2,3,5,6,7,10,11,13\}\) and each \(N\in\{10^3,3\times10^3,10^4,3\times10^4,10^5,3\times10^5,10^6\}\):
- \(\Lambda(n)\) for \(n\le N\) is read from
probe.von_mangoldt, the smallest-prime-factor sieve already used byresidue.pyandprobe.pyin this hunt. - \(\Delta(t;q,b)=\psi(t;q,b)-t/\phi(q)\) is built by a cumulative sum of \(\Lambda\) restricted to each reduced residue class, exactly as defined in RANK3_ROUTE_D.md Section 1.
- \(T(q,b)\) and \(X(b,b')\) are the exact finite sums (D5), (D6) — no approximation, no truncation of the sum over \(t\).
- \(c_q(k)\) is computed from its defining exponential sum over the \(\phi(q)\) reduced residues (UPPER_BOUND.md Section 2).
- \(\Sigma_{\rm cross}(q)=\sum_{b\ne b'}^*c_q(b-b')X(b,b')\), computed directly from the assignment's own definition, not assembled from a separate identity.
Two ratios are recorded at each \((q,N)\):
ratio_cross_over_diag= \(\Sigma_{\rm cross}(q)/(\phi(q)\sum_b^*T(q,b))\) — the exact quantity the task asks for.ratio_cross_over_ceiling= \(\Sigma_{\rm cross}(q)/(\phi(q)(\phi(q)-1)\sum_b^*T(q,b))\) — the same numerator against the proved (D8) ceiling instead of the diagonal term; the two ratios are related by a factor of \(\phi(q)-1\) and are reported separately because the task asks for the comparison against both quantities.
At \(q=2\), \(\phi(q)=1\): there is only one reduced residue, so the sum over \(b\ne b'\) is empty, \(\Sigma_{\rm cross}(2)\equiv0\) identically (not a numerical coincidence — the same structural fact (D7) records at \(q=1\)), and ratio_cross_over_ceiling is undefined (division by \(\phi(q)(\phi(q)-1)=0\)); both are recorded as such in the JSON.
Cross-checks (before trusting any ratio)
Reusing the hunt's own von Mangoldt machinery rather than re-deriving one, as the task requires, still leaves open whether that machinery is right; three independent checks are run and recorded in the JSON's cross_checks block:
- \(\Lambda(n)\), two methods.
probe.von_mangoldt's sieve against a direct enumeration of prime powers viasympy.primerange(the same second methodrank3_bdh_probe.pyandrank3_fourth_moment_probe.pyalready use in this hunt) — measured max abs difference0.0up to \(n=50000\). - \(c_q(k)\), two methods. The defining exponential sum against the classical closed form \(c_q(k)=\mu(q/g)\phi(q)/\phi(q/g)\), \(g=\gcd(q,k)\), for every \(q\in\{2,3,5,6,7,10,11,13\}\) and every \(k\) in \([-q,q]\) — measured max abs difference \(1.57\times10^{-14}\) (floating-point roundoff).
- \(T(q,b)\), two independent code paths. The partial-sum formula (D5) against an explicit finite convolution of \(d_b(n)=\Lambda(n)\mathbf1_{n\equiv b(q)}-1/\phi(q)\) with the constant sequence \(1\) on \(1..N\) (the same convolution RANK3_ROUTE_D.md Section 2 sets up before reading off (D5)), taking the sum of squares of the resulting coefficients (Parseval on the explicit trigonometric polynomial) at \(q=3\), \(N=500\) — measured max abs difference \(9.1\times10^{-11}\) (floating-point roundoff on sums of size \(\sim10^5\)).
- Gram/Cauchy-Schwarz consistency. \(|X(b,b')|\le\sqrt{T(q,b)T(q,b')}\) must hold at every \((q,N,b,b')\) since \(X\) is a Gram matrix entry (RANK3_ROUTE_D.md Section 2);
max_cauchy_schwarz_violationis recorded per run and is \(\le0\) (i.e. no violation) throughout.
All four checks pass at or near floating-point precision; none is a proof that the script is free of every possible error, but they are independent of each other and of the main computation, and none turned up a disagreement.
What was measured
Full numbers are in results_rank3_cross_term_probe.json (56 rows, one per \((q,N)\) pair). The measured ratio_cross_over_ceiling ranges over approximately \([-0.254,\ 0.426]\) across every \((q,N)\) in the ladder, and is below \(0.1\) in absolute value for \(q\ge11\) at every cutoff measured. The measured ratio_cross_over_diag is larger (it omits the extra factor \(\phi(q)-1\)) but still stays under \(0.43\) in absolute value throughout, and is well under \(0.15\) for \(q\ge7\) at the largest cutoffs measured. \(\Sigma_{\rm cross}(q)\) changes sign across the ladder at several moduli (e.g. \(q=3,6,7,10\) all have at least one negative entry): it is, as RANK3_ROUTE_D.md Section 3 already notes on general grounds, "a real number of either sign," not a one-signed quantity trending toward the ceiling.
Reading against the two candidate weights. RANK3_ROUTE_D.md Section 5 contrasts a diagonal-only weight \(\mu(q)^2/\phi(q)\) (obtained if \(\Sigma_{\rm cross}(q)\) is discarded as negligible, i.e. the ratio to the diagonal term is treated as \(\approx0\)) against the proved cancellation-free weight \(2\mu(q)^2\) (obtained if no cancellation at all is assumed, i.e. the ratio to the ceiling is treated as \(\approx1\)). At every \((q,N)\) measured here, ratio_cross_over_ceiling stays well under \(1/2\) — closer to \(0\) than to \(1\) — and in most rows well under \(0.1\). This measurement is closer to the diagonal-only picture (\(\Sigma_{\rm cross}(q)\) small relative to its proved ceiling) than to the cancellation-free ceiling being approached. It is emphatically not, by itself, a demonstration that \(\Sigma_{\rm cross}(q)=o(\Sigma_{\rm diag}(q))\) as \(q,N\to\infty\): the ladder here stops at \(N=10^6\) and \(q=13\), the ratios do not shrink monotonically as \(N\) grows (e.g. \(q=3\) goes \(0.085\to0.426\to0.382\to0.131\to0.288\to-0.252\to0.241\) across the seven cutoffs), and nothing here bounds the ratio uniformly in \(q\) or proves any rate. A finite measurement at eight moduli and seven cutoffs does not decide an asymptotic question, and this document does not claim it does — it reports where the actual numbers landed, which is substantially inside the proved envelope rather than near its edge, and leaves whether that persists as \(q\) and \(N\) grow exactly as open as RANK3_ROUTE_D.md already says it is.
Reproducing
/opt/zeta-venv/bin/python hunts/prime_pair_error/rank3_cross_term_probe.pyWall time on this run: 3.9s (the reported seconds_total in the JSON); peak memory is dominated by a handful of length-\(N\) float64 arrays per residue, negligible at these cutoffs.