This document answers the task assigned to it: whether RANK3_CROSS_TERM_CANCELLATION.md's identity \((CC1)\)-\((CC2)\), proved there only for prime \(q\), extends to composite squarefree \(q\) (\(q\) with \(\ge2\) distinct prime factors, \(\mu(q)^2\ne0\)), by evaluating the off-diagonal Gauss-sum matrix \(G(\chi,\chi')\) from that document's §3 for every pair of nonprincipal characters mod \(q\), imprimitive ones included.
Answer, stated first. Yes, and by a wider margin than RANK3_CROSS_TERM_CANCELLATION.md §4 anticipated. \(G(\chi,\chi')=0\) for \(\chi\ne\chi'\) unconditionally — for every modulus \(q\) (not just squarefree, not just composite, not just prime) and every pair of characters mod \(q\) (not just primitive ones) — because the fact that kills the off-diagonal never needed primitivity in the first place; the identity RANK3_CROSS_TERM_CANCELLATION.md derived only for prime \(q\) used a stronger fact than the off-diagonal vanishing actually requires. What primality (or, as it turns out, squarefreeness) genuinely buys is the diagonal normalization \(G(\chi,\chi)\), which is \(\phi(q)\cdot q\) only when \(\chi\) is primitive mod \(q\) — for imprimitive \(\chi\), it is \(\phi(q)\) times \(\chi\)'s conductor, a strictly smaller number. With that one correction, RANK3_CROSS_TERM_CANCELLATION.md §3's derivation goes through for every squarefree \(q\) and yields the exact identity \((CT1)\) below, which specializes to \((CC1)\) at prime \(q\) and is verified here to floating-point precision at \(q\in\{6,10,14,15,21\}\), \(N\) up to \(10^6\). The resulting bound on \(\Sigma_{\rm diag}(q)+\Sigma_{\rm cross}(q)\) is exactly \(q\sum_b^*T(q,b)\) — the same ceiling \((CC2)\) proves for prime \(q\), reached here for every squarefree \(q\), with one honestly-stated caveat in §5 about when that ceiling actually beats the cancellation-free one.
All notation is RANK3_CROSS_TERM_CANCELLATION.md's (itself reusing RANK3_ROUTE_D.md's): \(q\ge2\) squarefree, \(\chi_0\) the principal character mod \(q\), \(L(\beta,\chi)=\sum_{n\le N}\Lambda(n)\chi(n)\exp1(n\beta)\), \(M(\chi,\chi')=\int_{\mathbb T}K_NL(\chi)\overline{K_NL(\chi')}\), \(M_\chi:=M(\chi,\chi)\ge0\), \(\tau_a(\bar\chi)=\sum_{b\bmod q}^\bar\chi(b)\exp1(ab/q)\), \(G(\chi,\chi')=\sum_{a\bmod q}^\tau_a(\bar\chi)\overline{\tau_a(\bar\chi')}\), \(E(q)\), \(\Sigma_{\rm diag}(q)\), \(\Sigma_{\rm cross}(q)\) as before. RANK3_CROSS_TERM_CANCELLATION.md §2's identity (CC-split) and (CC1') are established there for every modulus \(q\) with no primality used, and are simply reused here unchanged: \[ \Sigma_{\rm diag}(q)+\Sigma_{\rm cross}(q) =\frac{\mu(q)^2E(q)}{\phi(q)}+\frac1{\phi(q)^2}\sum_{\chi,\chi'\ne\chi_0}G(\chi,\chi')M(\chi,\chi'), \qquad \Sigma_{\rm diag}(q)=E(q)+\sum_{\chi\ne\chi_0}M_\chi. \tag{CC-split, CC1'} \] Everything below evaluates \(G(\chi,\chi')\) for composite squarefree \(q\), which is the one piece §3 of the source document left undone.
1. \(G(\chi,\chi')=0\) off the diagonal: no primitivity needed at all
Fix any modulus \(q\ge1\) and any two Dirichlet characters \(\chi,\chi'\) mod \(q\) (principal or not, primitive or not). For \(a\) with \((a,q)=1\), the map \(b\mapsto ab\bmod q\) is a bijection of the full residue ring \(\mathbb Z/q\mathbb Z\) onto itself (its inverse is \(b\mapsto\bar ab\)), and \(\chi\) is completely multiplicative on all of \(\mathbb Z/q\mathbb Z\) (with \(\chi(n)=0\) exactly when \((n,q)>1\), and \(\chi(ab)=\chi(a)\chi(b)\) holding in that case too, both sides \(0\) together, since \((ab,q)>1\iff(b,q)>1\) when \((a,q)=1\)). So, summing over all residues \(b\bmod q\) (the non-reduced ones contribute \(0\) on both sides, so restricting to the reduced residues, as \(\tau_a\)'s definition does, changes nothing): \[ \tau_a(\bar\chi)=\sum_{b\bmod q}\bar\chi(b)\exp1(ab/q) \overset{b=a^{-1}b'}=\sum_{b'\bmod q}\bar\chi(a^{-1}b')\exp1(b'/q) =\chi(a)\sum_{b'\bmod q}\bar\chi(b')\exp1(b'/q)=\chi(a)\,\tau(\bar\chi), \tag{T1} \] where \(\tau(\bar\chi):=\tau_1(\bar\chi)\). This holds for every character mod every modulus, with no primitivity or squarefreeness hypothesis anywhere — the classical fact that fails for imprimitive \(\chi\) is a different one, the reduction of \(\tau(\chi)\) itself to the Gauss sum of the primitive character inducing it (§2 below); the twisting relation (T1) is not that fact and does not need it. RANK3_CROSS_TERM_CANCELLATION.md §3 invoked primitivity to get (T1) (it only needed it for primitive \(\chi\), which is all prime \(q\) has), but (T1) was never actually conditional on primitivity — it is conditional only on \((a,q)=1\), which is exactly the range \(\tau_a\) is summed over throughout.
Substituting (T1) into the definition of \(G\) and using plain character orthogonality over \(a\) (valid for any two characters mod any \(q\)): \[ G(\chi,\chi')=\tau(\bar\chi)\overline{\tau(\bar\chi')}\sum_{a\bmod q}^*\chi(a)\bar\chi'(a) =\phi(q)\,\tau(\bar\chi)\overline{\tau(\bar\chi')}\,\mathbf1[\chi=\chi']. \tag{G1} \] \[ \boxed{\ G(\chi,\chi')=0\text{ whenever }\chi\ne\chi',\text{ for every modulus }q\text{ and every pair of characters mod }q.\ } \] This alone already answers the off-diagonal half of the task's question: the Gauss-sum matrix is diagonal unconditionally, not just for primitive characters or prime moduli. What remains is the diagonal value \(G(\chi,\chi)=\phi(q)|\tau(\bar\chi)|^2\), and that is where primitivity (via squarefreeness) genuinely enters.
2. The diagonal value: \(|\tau(\chi)|^2\) is the conductor, not \(q\)
For \(\chi\) primitive mod \(q\), \(|\tau(\chi)|^2=q\) is the classical fact RANK3_CROSS_TERM_CANCELLATION.md used, and every nonprincipal \(\chi\) mod a prime \(q\) is primitive, which is what made \(G(\chi,\chi)=q\phi(q)\) uniformly there. For imprimitive \(\chi\) mod \(q\), induced by the primitive character \(\chi^\) mod its conductor \(q^\mid q\), \(q^<q\), the classical reduction of an imprimitive Gauss sum to the primitive one inducing it is, writing \(d:=q/q^\), \[ \tau(\chi)=\mu(d)\,\chi^(d)\,\tau(\chi^), \tag{T2} \] valid when \(\gcd(d,q^*)=1\) (this hypothesis is exactly why the task restricts to squarefree \(q\): for \(q\) squarefree, every divisor pair \(q^\mid q\), \(d=q/q^\), automatically satisfies \(\gcd(d,q^)=1\), since \(q\)'s prime factorization has every prime to the first power, so \(q^\) and \(d\) partition the prime factors of \(q\) with no overlap; for non-squarefree \(q\) this would fail and (T2) would need the more general induced-modulus formula). (T2) can be checked directly by the same substitution technique as (T1): decompose \(b\bmod q\) via CRT as \((u,v)=(b\bmod q^,b\bmod d)\) (a bijection on reduced residues since \(\gcd(q^,d)=1\)), write \(b/q=ux/q^*+vy/d\) for the Bézout pair \(dx+q^y=1\), and split \(\tau(\chi)=\sum_b^\chi^(u)\exp1(b/q)\) into a primitive Gauss sum in \(u\) (using \(\chi(b)=\chi^(u)\) for \((b,q)=1\), the defining property of induction) times a Ramanujan sum in \(v\), which evaluates to \(\mu(d)\) exactly as \(c_q(a)=\mu(q)\) does for \((a,q)=1\) (the two facts are the same computation, one with \(\chi^\) inserted and one without). Since \(|\chi^(d)|=1\) (\((d,q^)=1\)) and \(|\tau(\chi^)|^2=q^\) (the primitive case), (T2) gives \[ \boxed{\ |\tau(\chi)|^2=q^=\operatorname{cond}(\chi),\qquad q\text{ squarefree}.\ } \tag{T3} \] Combining (G1) and (T3): \[ \boxed{\ G(\chi,\chi')=\phi(q)\cdot\operatorname{cond}(\chi)\cdot\mathbf1[\chi=\chi'],\qquad q\text{ squarefree}.\ } \tag{G2} \] At prime \(q\), \(\operatorname{cond}(\chi)=q\) for every nonprincipal \(\chi\) (its only proper divisor is \(1\), which induces only \(\chi_0\)), so (G2) reduces to RANK3_CROSS_TERM_CANCELLATION.md §3's \(G(\chi,\chi')=q\phi(q)\mathbf1[\chi=\chi']\) exactly. For composite squarefree \(q\), \(\operatorname{cond}(\chi)\) genuinely varies over the nonprincipal characters — this is the one place the prime and composite cases differ, and it is a normalization difference, not a failure of diagonality.
3. Reworking §3's derivation: the exact identity for composite squarefree \(q\)
Substituting (G2) into (CC-split), using \(\mu(q)^2=1\) (squarefree \(q\)): \[ \Sigma_{\rm diag}(q)+\Sigma_{\rm cross}(q) =\frac{E(q)}{\phi(q)}+\frac1{\phi(q)}\sum_{\chi\ne\chi_0}\operatorname{cond}(\chi)\,M_\chi. \tag{CT0} \] Write \(\operatorname{cond}(\chi)=q-(q-\operatorname{cond}(\chi))\) and define \[ W(q):=\sum_{\chi\ne\chi_0}\big(q-\operatorname{cond}(\chi)\big)M_\chi\ \ge0 \] (nonnegative: \(\operatorname{cond}(\chi)\le q\) always, and \(M_\chi\ge0\) as a Gram-matrix diagonal entry). Then \(\sum_\chi\operatorname{cond}(\chi)M_\chi =q\sum_\chi M_\chi-W(q)=q\big(\Sigma_{\rm diag}(q)-E(q)\big)-W(q)\), using (CC1'). Substituting into (CT0) and solving for \(\Sigma_{\rm cross}(q)\) (the same algebra RANK3_CROSS_TERM_CANCELLATION.md §3 performs, carried through with \(\operatorname{cond}(\chi)\) in place of the constant \(q\)): \[ \boxed{\ \Sigma_{\rm cross}(q)=\frac{(q-\phi(q))\,\Sigma_{\rm diag}(q)-(q-1)E(q)-W(q)}{\phi(q)}, \qquad q\text{ squarefree}.\ } \tag{CT1} \] At prime \(q\): \(\operatorname{cond}(\chi)=q\) for every nonprincipal \(\chi\), so \(W(q)\equiv0\), \(q-\phi(q)=1\), \(\phi(q)=q-1\), and (CT1) reads \(\Sigma_{\rm cross}(q)=\big[\Sigma_{\rm diag}(q)-(q-1)E(q)\big]/(q-1) =\Sigma_{\rm diag}(q)/(q-1)-E(q)\) — exactly \((CC1)\). (CT1) is the requested generalization: an exact identity, for every squarefree \(q\), reducing to (CC1) at prime \(q\) with \(W(q)\) as the single new, manifestly nonnegative term that composite moduli introduce.
Since \(E(q)\ge0\), \(W(q)\ge0\), and \(q\ge\phi(q)\), dropping both nonnegative subtracted terms from (CT1) gives the one-directional bound \[ \Sigma_{\rm cross}(q)\ \le\ \frac{(q-\phi(q))\Sigma_{\rm diag}(q)}{\phi(q)}, \] and adding \(\Sigma_{\rm diag}(q)\) to both sides, \[ \boxed{\ \Sigma_{\rm diag}(q)+\Sigma_{\rm cross}(q)\ \le\ q\sum_{b\bmod q}^*T(q,b),\qquad q\text{ squarefree}.\ } \tag{CT2} \] This is exactly \((CC2)\)'s bound — the prime-only ceiling RANK3_CROSS_TERM_CANCELLATION.md proved now holds, unconditionally, for every squarefree \(q\), composite included. The derivation needed nothing beyond §1-§2's evaluation of \(G\); the rest of the algebra is identical to the prime case, character-conductor bookkeeping aside.
4. Numerical verification
rank3_composite_cross_term_probe.py checks three independent things, at \(q\in\{6,10,14,15,21\}\) (the squarefree composite moduli named in the task) across \(N\in\{10^3,3\times10^3,10^4,3\times10^4,10^5,3\times10^5,10^6\}\) (35 rows), reusing rank3_cross_term_probe.py's reduced_residues, delta_table, T_and_X, ramanujan_sum_direct for \(\Sigma_{\rm diag}(q)\) and \(\Sigma_{\rm cross}(q)\) (not reimplemented) and rank3_cross_term_cancellation_probe.py's E_quantity for \(E(q)\) (also not reimplemented). Characters mod \(q\) are enumerated as CRT products of cyclic characters at each prime factor (discrete logs against a primitive root at each small prime \(2,3,5,7\)); \(M_\chi\) is computed from the complex partial sums \(S_\chi(t)=\sum_{n\le t}\Lambda(n)\chi(n)\) by the same (D5)-shaped Parseval formula \(T(q,b)\) uses, and cross-checked at \((q,N)=(6,500)\) against an independent explicit convolution of \(\Lambda(n)\chi(n)\) with the constant sequence \(1\) (max abs difference \(3.6\times10^{-12}\) on a quantity of size \(\sim3\times10^4\)).
(1) \(G(\chi,\chi')\), brute-force from its definition (no Gauss-sum formula assumed — direct computation of \(\tau_a(\bar\chi)\) for every reduced residue \(a\) and every character, then the double sum defining \(G\)): at every \(q\in\{6,10,14,15,21\}\), the maximum off-diagonal \(|G(\chi,\chi')|\) is \(\le5.2\times10^{-13}\) (floating-point zero on quantities of size \(\sim10^2\)), and every diagonal entry matches \(\phi(q)\cdot\operatorname{cond}(\chi)\) to relative error \(\le5.3\times10^{-15}\). This independently confirms (G2) without relying on the algebraic derivation in §1-§2 at all.
(2) (CC1'), \(\Sigma_{\rm diag}(q)=E(q)+\sum_\chi M_\chi\), already established for every \(q\) in the source document, holds here at composite squarefree \(q\) to worst relative error \(2.6\times10^{-12}\) across all 35 rows — a sanity check on the character/\(M_\chi\) machinery introduced in this document, independent of \(G\).
(3) The new identity (CT1), computed as predicted = [(q-phi(q))*Sigma_diag(q) - (q-1)*E(q) - W(q)] / phi(q) and compared against the directly-measured \(\Sigma_{\rm cross}(q)\) (from \(c_q(b-b')X(b,b')\), rank3_cross_term_probe.py's machinery, entirely independent of characters), holds to worst relative error \(7.4\times10^{-11}\) across all 35 rows (\(5\) moduli \(\times\) \(7\) values of \(N\)) — the same order of floating-point accumulation error the prime-\(q\) check in RANK3_CROSS_TERM_CANCELLATION.md reports (\(3.4\times10^{-11}\)), not a discrepancy. (CT2) holds at every one of the 35 rows. This directly contradicts the naive application of (CC1) that RANK3_CROSS_TERM_CANCELLATION.md §4 reports failing by 50-150% at these same moduli — that failure came from assuming \(\operatorname{cond}(\chi)=q\) uniformly (i.e. using (CC1) verbatim), not from any failure of the underlying cancellation; replacing \(q\) with \(\operatorname{cond}(\chi)\) character-by-character, as (CT1) does, repairs it completely.
Full rows are in results_rank3_composite_cross_term_probe.json.
5. Where the composite-\(q\) ceiling differs from the prime case: when does it actually help?
(CT2)'s bound \(q\sum_b^*T(q,b)\) is only useful if it beats the cancellation-free ceiling \(\phi(q)^2\sum_b^*T(q,b)\) that (D8) already proves unconditionally (RANK3_ROUTE_D.md §3) — i.e. only when \(\phi(q)^2>q\). For prime \(q\ge3\), \((q-1)^2>q\) always (\(q^2-3q+1>0\) for \(q\ge3\)), so (CC2) is a genuine improvement at every prime \(q\ge3\) (checked already in RANK3_CROSS_TERM_CANCELLATION.md §4: improvement factor \(1.333\) at \(q=3\) up to \(17.05\) at \(q=19\)). For composite squarefree \(q\), this is no longer automatic: \(\phi(q)\) can be small relative to \(\sqrt q\) when \(q\) has several small prime factors. Checking every squarefree \(q\le200\) directly: \(\phi(q)^2<q\) — (CT2)'s bound worse than the trivial one — holds only at \(q=2\) (degenerate, \(\Sigma_{\rm cross}(2)\equiv0\) already) and \(q=6\) (\(\phi(6)=2\), \(4<6\)) in that entire range; every other squarefree \(q\in[3,200]\), composite or prime, has \(\phi(q)^2>q\). Among this document's own five test moduli, \(q=6\) is exactly this exception (improvement factor \(\phi(6)^2/6=2/3<1\), confirmed in the numerical run above); \(q\in\{10,14,15,21\}\) all show genuine improvement (\(1.6\), \(2.57\), \(4.27\), \(6.86\) respectively — the same column rank3_composite_cross_term_probe.py prints as improvement).
This is not a defect in (CT1)-(CT2)'s derivation — both are exact/proven regardless — it is a fact about which bound is sharper at a given \(q\), and it is finite in extent: since \(\phi(q)\gg q/\log\log q\) uniformly (Mertens' third theorem), \(\phi(q)^2\gg q^2/(\log\log q)^2\gg q\) once \(\log\log q\) is smaller than \(\sqrt q/(\text{absolute constant})\), i.e. for all \(q\) past some small, absolute, computable threshold — the direct check above already shows that threshold is at most \(6\) for every squarefree \(q\), which strongly suggests (but this document does not prove) that \(q=2,6\) are the only squarefree exceptions, full stop, not just up to \(200\). Whichever way that goes, taking \(\min\big(\phi(q)^2,q\big)\sum_b^*T(q,b)\) termwise (both bounds are proven, independently, for every squarefree \(q\)) loses nothing: the composite-\(q\) extension is never worse than what RANK3_ROUTE_D.md (D8) already had, and is strictly better at every squarefree \(q\ge3\) except (at most, and apparently exactly) \(q=6\).
A second, separate caveat, for whoever carries this into RANK3_POLYRANGE_TINT_CHECK.md §3's partial-summation argument over \(2\le q\le R_0\): that argument (per RANK3_CROSS_TERM_CANCELLATION.md §5) needs a weight on \(\sum_b^*T(q,b)\) decaying like \(1/q\) (or \(1/\phi(q)\), the same order) after the \((\mu(q)/\phi(q))^2\) prefactor is folded in. Following RANK3_ROUTE_D.md (D9)-(D11)'s assembly, (CT2)'s ceiling contributes weight \(\mu(q)^2\,q/\phi(q)^2\) to \(\sum_b^*T(q,b)\) — at prime \(q\), \(q/\phi(q)^2\sim1/q\sim1/\phi(q)\), matching the "diagonal-only" target weight RANK3_ROUTE_D.md §5 names. At composite squarefree \(q\), \(q/\phi(q)^2=(q/\phi(q))/\phi(q)=O(\log\log q)/\phi(q)\) by the same Mertens bound — the same order as the prime case up to an extra \(\log\log q\) factor, not a full extra power of \(q\) or \(\phi(q)\). This document does not redo RANK3_POLYRANGE_TINT_CHECK.md §3's partial summation with this exact weight to confirm the \(\log\log q\) factor is absorbed into that argument's \(N^\epsilon\) — that check is a real, identifiable next step, not attempted here, and is the one piece still needed to carry (CT2) all the way to RANK3_CROSS_TERM_CANCELLATION.md §5's conclusion ("full \(2\le q\le R_0\) sum's order is unchanged... unless composite squarefree \(q\) are separately handled") being overturned.
6. What this does and does not unlock
RANK3_CROSS_TERM_CANCELLATION.md §4-§6 identified composite squarefree \(q\) as genuinely blocked for Route D's \(q\)-sum, because its identity (CC1), applied to them, was numerically wrong by 50-150%, and it supplied no substitute. This document supplies that substitute: (CT1) is an exact identity for every squarefree \(q\), verified to floating-point precision, and (CT2) — the resulting bound — is exactly the same ceiling (CC2) proves for primes, not a weaker one, for every squarefree \(q\) except the finite (apparently just \(\{2,6\}\)) set where the trivial Cauchy-Schwarz ceiling (D8) was already smaller. So RANK3_CROSS_TERM_CANCELLATION.md §5's statement that "the squarefree composite \(q\)... are not covered by (CC1)-(CC2)" no longer holds: they are covered, by (CT1)-(CT2), at the same weight. What is not done here is §5's remaining half: confirming that RANK3_POLYRANGE_TINT_CHECK.md's partial-summation argument, applied to the \(\mu(q)^2q/\phi(q)^2\) weight (rather than the \(\mu(q)^2/\phi(q))\) weight that argument was checked against), still gives \(O_\epsilon(N^{2+\epsilon})\) once summed over the full range \(2\le q\le R_0\) including composite \(q\) — precisely stated as the wall in §5 above. Nothing in this document touches rank 1, RANK3_SCOPE.md §4's separate bottleneck (rank 1's \(O_H(N^3L^{-2H})\) term dominates (23) regardless of ranks 2-3), or any statement about zeros of \(L\)-functions or the Riemann Hypothesis; this is character-sum algebra (orthogonality, Gauss-sum twisting and reduction, Cauchy-Schwarz-free exact bookkeeping) plus explicit floating-point verification around the existing unconditional construction.