delta_sq_sw_bind.py evaluates Section 7's proved statement \(T_N=\sum_{t=1}^N\Delta(t)^2+\sum_{t=1}^{N-1}[\Delta(N)-\Delta(t)]^2\ll_H N^3L^{-2H}\) (\(L=\log N\)) under an explicit, labeled convention -- assumed implied constant \(c(H)=1\) -- because the proof, resting on Siegel-Walfisz, never names \(c(H)\) and never says whether it is effective. That script found the convention binds at \(H=1,2\) and is violated at \(H=4,8\) against the measured \(S(N)=\sum_{t\le N}\Delta(t)^2\) in results_delta_sq.json. This note asks the question that script leaves open: is there a real, computable value in place of the convention, from a known effective version of Siegel-Walfisz -- or does this hit the same ineffectivity wall tb_bind.py already hit for Theorem B's \(\Omega\) constant?
Short answer: no, this does not recur, for the specific line Section 7 actually uses. That line only invokes Siegel-Walfisz at \(q=1\), and the \(q=1\) case of Siegel-Walfisz is not really Siegel-Walfisz at all -- it is the classical prime number theorem remainder derived from the zero-free region for \(\zeta(s)\) alone, which has been effective since de la Vallée Poussin (1899), decades before Siegel's theorem (1935) or Siegel-Walfisz (1936) existed, and for a structural reason that has nothing to do with how much has been proved since: no real Dirichlet character enters at \(q=1\), so there is no Landau-Siegel zero to fail to exclude. An effective \(c_1,K,x_0\) with \(|\Delta(t)|\le Kt\exp(-c_1\sqrt{\log t})\) for \(t\ge x_0\) is known to exist in the literature. What this attempt could not do, for lack of network or library access in this sandbox, is pull a citation-checked numeric value for \(c_1\); delta_sq_sw_effective_bind.py instead computes, from the measured data, the threshold \(c_1\) would have to clear to be violated, and that threshold is loose enough that any plausible published value very likely clears it. Section 4 states precisely what remains unverified and why that is a bounded lookup, not a mathematical wall. Nothing here bears on RH.
1. What Section 7 actually invokes, re-read closely
Section 1 item 3 states (SW) in its general form: for fixed \(B,H>0\), uniformly for \(q\le L^B\), \((b,q)=1\), and \(0\le t\le N\), \(\psi(t;q,b)=t/\phi(q)+O_{B,H}(NL^{-H})\). This general statement, uniform over a modulus \(q\) that grows with \(N\) up to any fixed power of \(\log N\), is the genuine Siegel-Walfisz theorem, and it is the textbook example of an ineffective result: proving it uniformly in \(q\) requires excluding, for every \(q\) in the growing range, the possibility of an exceptional real zero (a Landau-Siegel zero) of \(L(s,\chi)\) for a real primitive character \(\chi\) mod \(q\), and the only known tool for that exclusion, Siegel's theorem, is proved by comparing two hypothetical exceptional characters against each other and produces no computable bound on how close to 1 such a zero could be -- if one existed. This is the same shape of ineffectivity tb_bind.py and THEOREM_B_SEQUENCE.md describe for \(\Theta_\chi\): an existence statement with no extractable constant.
But Section 7 does not use (SW) at a growing \(q\). It uses it once, at \(q=1\): "Using (SW) with \(q=1\) gives, for every fixed \(H\), \(\max_{t\le N}|\Delta(t)|\ll_H NL^{-H}\)." At \(q=1\), \(\phi(q)=1\), the only reduced residue is \(b=1\), and \(\psi(t;1,1)\) is just \(\psi(t)\). There is exactly one Dirichlet character mod 1 (the principal character), and no real primitive character of modulus greater than 1 is involved at any point. The statement being invoked is
\[ \psi(t)=t+O_H(NL^{-H})\quad\text{for every fixed }H, \]
which is a corollary of the classical, effective estimate
\[ \psi(x)=x+O\bigl(x\exp(-c_1\sqrt{\log x})\bigr) \tag{$*$} \]
(or the sharper Vinogradov–Korobov-type saving \(\exp(-c(\log x)^{3/5}/(\log\log x)^{1/5})\), which decays even faster and implies $(*)$), since \(\exp(-c_1\sqrt{\log x})\) beats every fixed power of \(\log x\) as \(x\to\infty\): for any fixed \(H\) there is an explicit, computable \(x_0(H)\) past which \(\exp(-c_1\sqrt{\log x})\le L^{-H}\), and below \(x_0(H)\) the maximum of \(|\Delta(t)|\) is controlled by the trivial, equally effective Chebyshev-type bound \(\psi(t)\le Kt\) for an absolute, explicit \(K\) (e.g. the classical \(\psi(t)<1.04\,t\) for all \(t>0\), a completely elementary, long-published constant with no zero-free-region input at all). So the corollary Section 7 states is not merely "eventually true for some unnamed constant" -- it follows, term by term, from a chain of classical statements each of which has always carried an explicit, computable constant.
2. Why $(*)$ is effective and the general (SW) is not: the actual dividing line
The dividing line is not "Siegel-Walfisz versus something else" -- it is whether a real primitive Dirichlet character is anywhere in the argument.
\(\zeta(s)\) is \(L(s,\chi_0)\) for the principal character mod 1, and the classical zero-free region \(\sigma>1-c_0/\log(|t|+2)\) for \(\zeta(s)\) is proved by the elementary "3-4-1" inequality \(3\log|\zeta(\sigma)|+4\,\mathrm{Re}\log\zeta(\sigma+it)+\mathrm{Re}\log \zeta(\sigma+2it)\ge0\) (\(\sigma>1\)), applied to \(\zeta\) alone -- no auxiliary \(L\)-function, no comparison between two hypothetical characters, and consequently no case split on whether an exceptional zero exists. This argument, first given by de la Vallée Poussin, produces an explicit \(c_0\) directly, and turning a zero-free region into a prime number theorem remainder of the shape $(*)$ by a contour shift past that region is equally elementary and equally explicit; this chain is standard material predating Siegel-Walfisz by decades (see e.g. Ingham, The Distribution of Prime Numbers, ch. III; Titchmarsh, The Theory of the Riemann Zeta-Function, ch. III; or the treatment in Montgomery and Vaughan, Multiplicative Number Theory I: Classical Theory, already the source this hunt cites elsewhere for the large sieve and Vaughan's identity, ch. 6 and ch. 8 for the zero-free region and PNT error term). Numerically explicit versions of \(c_0\) and hence of $(*)$'s \(c_1\) have been published and progressively sharpened for over a century; named examples this attempt can identify (without a verified citation-checked digit from any of them, see Section 4) include Rosser and Schoenfeld (1962), McCurley (1984), Kadiri (2005), Mossinghoff and Trudgian (2015), Platt and Trudgian (2021), and Broadbent, Kadiri, Lumley, Ng and Wilk (2021), the last group of which targets the PNT error term itself rather than only the zero-free region.
Siegel-Walfisz proper, for \(q>1\) in a range growing with \(N\), needs the non-vanishing of \(L(s,\chi)\) near \(\sigma=1\) for every real primitive \(\chi\) mod every such \(q\) simultaneously, and the only known unconditional tool for that is Siegel's theorem: for every \(\epsilon>0\) there is a \(c(\epsilon)>0\) with \(L(1,\chi)>c(\epsilon)q^{-\epsilon}\) for every real primitive \(\chi\) mod \(q\). Siegel's proof is a case split on whether a second, hypothetical exceptional character exists and compares the two; the constant \(c(\epsilon)\) it produces is not extractable from the proof, because one of the two cases the proof splits on is never known to hold or fail. This is a comparison between two objects that might not exist, not an "3-4-1"-style direct estimate, and it is the entire reason Siegel-Walfisz is the textbook example of an ineffective theorem (see also Tatuzawa's 1951 refinement, which makes \(c(\epsilon)\) effective for all but at most one real primitive character of each conductor size -- still not effective for a specific, unidentified exceptional modulus, which is exactly the case a uniform-in-\(q\) statement cannot exclude). At \(q=1\) this case split never arises: there is no real primitive character mod 1 other than the (trivially non-vanishing, non-exceptional) principal character, so nothing about Siegel's theorem is invoked, whether or not the citation attached to (SW) in UPPER_BOUND.md's item 3 names it that way.
3. What this changes for T_N, and what it does not change
Squaring $(*)$'s corollary and summing across \(\sim N\) terms the way (29) does gives, on the same "convention constant" basis delta_sq_sw_bind.py already used (leading constant \(K=1\), labeled, not derived from a named theorem):
\[ T_N \lesssim N^3\exp(-2c_1\sqrt{\log N}), \tag{$**$} \]
which is strictly stronger than \(N^3L^{-2H}\) for any fixed \(H\), for the same reason $(*)$ is stronger than its own \(L^{-H}\) corollary. delta_sq_sw_effective_bind.py evaluates $(**)$ against the measured \(S(N)\) in results_delta_sq.json (the same leading, non-negative half of \(T_N\) delta_sq_sw_bind.py used, with the same one-directional \(S(N)\le T_N\) caveat: a violated bound against \(S(N)\) is also violated against \(T_N\); a bound exceeding \(S(N)\) does not by itself show it would still exceed \(T_N\)).
What this note does not change is Section 7's own assessment of what this component is worth to the larger unconditional upper-bound attempt. Section 7 already says, in the paragraph introducing this line: "This is where the unconditional attempt still loses a power of \(N\) on this component." \(N^3L^{-2H}\) -- or its sharper effective cousin \(N^3\exp(-2c_1\sqrt{\log N})\) -- is a full power of \(N\) above the unproved target (31), \(N^{2+\epsilon}\), for every fixed \(H\) or \(c_1\): the ratio \(N^3\exp(-2c_1\sqrt{\log N})/N^{2+\epsilon}=N^{1-\epsilon} \exp(-2c_1\sqrt{\log N})\to\infty\). Knowing \(c_1\) explicitly makes the \(N^3\)-scale statement a genuine number instead of an unnamed-constant placeholder; it does not shrink the exponent of \(N\), and it does not discharge (31) or move (23)'s other unresolved pieces (\(U_{2\le q\le R_0}\), \(Z_{q\le R_0}\), the fourth-moment excess), which Section 7 already lists as untouched by this estimate. Effectiveness and sufficiency are different questions; this note and its script answer only the first, and only for the \(q=1\) term.
It also does not extend to the general (SW) statement as used elsewhere in this hunt. UPPER_BOUND.md Section 5's major-arc argument and SW_MOMENT_SPLICE.md's splice both work with \(q\) ranging up to \(L^B\) for \(q>1\), which is exactly the regime where the Siegel-zero case split of Section 2 above is unavoidable and no effective constant is known. This note resolves only the degenerate \(q=1\) instance Section 7 happens to use for \(T_N\); it says nothing about, and does not weaken, the ineffectivity of (SW) at \(q>1\) growing with \(N\).
4. What is not verified here, and why that is a bounded gap, not a wall
This attempt's tools are /opt/zeta-venv/bin/python (mpmath, numpy, scipy, sympy, python-flint, cypari2) and local text search (grep, rg, ls, cat, head, tail, wc, find) over this worktree -- no network access and no cached copies of Rosser–Schoenfeld (1962), McCurley (1984), Kadiri (2005), Mossinghoff–Trudgian (2015), Platt–Trudgian (2021), or Broadbent–Kadiri–Lumley–Ng–Wilk (2021) are present in this worktree to cat or grep. So while this attempt is confident, from standard analytic number theory, that some explicit \((c_1,K,x_0)\) triple for $(*)$ is published in the literature above -- the existence of an effective constant here is not in serious doubt, unlike \(\Theta_\chi\)'s exact value or Theorem B's Omega constant in tb_bind.py, which are not just unlooked-up but genuinely open or non-constructive -- this attempt cannot quote a citation-checked numeric digit for \(c_1\) (or the sharper Vinogradov–Korobov exponent triple) without risking a misremembered constant presented as sourced fact, which this hunt's conventions do not allow.
delta_sq_sw_effective_bind.py sidesteps that by solving the comparison the other way: at each ladder \(N\), it computes the threshold \(c_1^(N)=(3\log N-\log S(N))/(2\sqrt{\log N})\) at which the convention-\(K=1\) bound $()$ would exactly equal the measured \(S(N)\); $()$ exceeds (binds against) \(S(N)\) at that \(N\) iff the true \(c_1\) is below \(c_1^(N)\). Across the eight ladder points (\(N=10^5\) to \(10^7\)), \(c_1^*(N)\) ranges from about 2.26 (at \(N=10^5\), the binding constraint across the whole ladder) up to about 2.50 (at \(N=10^7\)). Any true effective \(c_1\) below about 2.26 makes $(**)$ exceed measured \(S(N)\) at every ladder cutoff, in the same "binds, not vacuous" sense delta_sq_sw_bind.py used for \(H=1,2\). Published explicit zero-free-region constants, as far as this attempt can recall without a checked citation, sit well below that -- on the order of a few tenths up to roughly 1 in this normalization, not above 2 -- so it is likely, though not confirmed in this run, that the true effective bound binds comfortably everywhere \(S(N)\) has been measured, unlike the convention-1 \(L^{-2H}\) bound at \(H=4,8\) in results_delta_sq_sw_bind.json, which was violated there. The script also reports an illustrative sweep of \(c_1\) over \([0.05,3.0]\) for the same comparison without relying on the recalled range being exactly right.
The residual task is narrow and concrete: retrieve one of the papers named in Section 1 (or a survey citing its constant, e.g. via a session with network or a local mathematics library this sandbox does not have), extract its explicit \((c_1,K,x_0)\) or Vinogradov–Korobov-type \((c,\kappa,\lambda,K,x_0)\), and rerun delta_sq_sw_effective_bind.py with that value in place of the swept range, replacing "likely, not confirmed" with a checked verdict. That is a literature lookup and an arithmetic substitution, not a new estimate and not an open problem -- a materially different kind of gap from Theorem B's missing \(\Theta_\chi\), missing implied constant, and missing witness sequence in tb_bind.py, none of which exist in the literature to be looked up at all.