2026-09-12. Base: ARC_SPLIT_BUDGET.md (commit a537765; its arcs, its minor-arc bound (11) and its budget (12) are reused unchanged), ENDPOINT_SHARP.md sections 1, 4 and 5 (the model at \(Z=\exp\sqrt\ell\), the exceptional data of TT Definition 2.1 with threshold \(c_0\), the four sieve cases of its section 4, input (2)), UPPER_BOUND.md (7), SIEGEL_UNIFORMITY.md (17), (19), (23). Assignment: attack the binding term of the complete budget with the target \(E_{\rm corr}^{(Z)}\) and every existing control held fixed.
Result, stated first. Let \(c_0\) be the threshold in the target's exceptional data (\(\beta>1-c_0/\log Z\)), \(c\) the constant of the zero-free region (ZF) below, \(b\) Page's constant in the form (Pg\('\)) below, and \(c_P\) the Page constant ENDPOINT_SHARP.md calls \(c\). Under \[ c_0\le\min\big(1/6,\ \sqrt c/2,\ \sqrt b/2,\ c_P\big), \tag{H} \] which TT's "sufficiently small" \(c_0\) satisfies, for every fixed \(c''<c_0\), \[ \boxed{\ E_{\rm corr}^{(Z)}(N)\ \ll\ N^3\exp\!\big(-c''\sqrt{\log N}\big).\ } \tag{S$''$} \] ARC_SPLIT_BUDGET.md (S\('\)) has the constant \(2\gamma/3\) with \(\gamma<c_0/2\) from ENDPOINT_SHARP.md section 4, so at most \(c_0/3\). The rate \(\sqrt\ell\) is unchanged.
What changed, and why it is structural. The major arcs are now bounded by the explicit formula for the character sums \(\sum_n\Lambda(n)\chi(n)e(n\theta)\) on each arc, with the zero-free region evaluated at the arc's own modulus and height rather than at a worst case over all \(r\le R\). Two consequences. First, every zero other than the ones the target already subtracts contributes \(N^{1-c/\log(rT)}\) with \(rT\le R^4\), which is \(N\exp(-(c/(4\sigma))\sqrt\ell)\): far below the balance once \(\sigma\) is small. Second, the one thing that does not shrink is a possible real zero \(\tilde\beta\) of a real character of conductor \(\tilde q\le R\) that Page's theorem allows but the target does not subtract, because \(\tilde\beta\le1-c_0/\sqrt\ell\): it contributes \(N^{\tilde\beta}\le N\exp(-c_0\sqrt\ell)\), with no loss. The binding term of the complete budget is therefore the target's own threshold \(c_0\), and not the progression constant \(\gamma\) of a black-box theorem. Section 5 makes this two-sided: if such a zero exists at \(1-\kappa/\sqrt\ell\) with \(c_0<\kappa\) (in a window), then \(E_{\rm corr}^{(Z)}\gg N^3e^{-2\kappa\sqrt\ell}/\tilde q^2\), so an unconditional exponent above \(2c_0\) would exclude real zeros \(1-\tilde\beta\le c'/\log\tilde q\) for every real character, which is a Siegel-zero statement nobody has. The exponent of this target is pinned to \([c_0,\,2c_0]\) by what is known.
Grade: derived, one route, finite checks in section 7, independently read (sections 2 to 5, even case of section 5). No power saving, nothing about the zeros of \(\zeta\) beyond the classical inputs named, and the lower bound of section 5 is conditional on a zero that may not exist.
Independent check, 2026-09-12 (attempt a-0080, MAJOR_ARC_EXPLICIT_REVIEW.md, judged by a-0081). Four of five items confirmed, one defective: the model term in (4) and (5) dropped the Gauss-sum factor \(\sqrt r\), so it is \(R^{3/2}E_{\rm md}\) and not \(RE_{\rm md}\), the model exponent in (6) is \(2/3-3\sigma\) and not \(2/3-2\sigma\), and (H) needs \(c_0\le1/6\) in place of the printed \(2/9\). Corrected in place, with the correction marked at (4). The conclusion (S\('')\) is unchanged under the corrected (H). The check also found the closing sentence of section 5's corollary too broad and it is restated there. The check verified the two derivative-test bounds numerically with its own script, recomputed both optima and the exponents, and confirmed the Page matching including the case of characters induced by a real primitive character of conductor not dividing \(r\). The parity remark at the end of section 5 was added after the check and has not been independently read.
1. Inputs
The model, arcs and target are those of ARC_SPLIT_BUDGET.md sections 1 and 2: \(R=\lfloor e^{\sigma\sqrt\ell}\rfloor\), arcs \(I_{r,a}\) of radius \(R/(rN)\), \(\mathfrak M\), \(\mathfrak m\). Write, for \(\chi\bmod r\), \[ F_\chi(\theta)=\sum_{n\le N}\Lambda(n)\chi(n)e(n\theta),\qquad H_\chi(\theta)=\sum_{n\le N}a(n)\chi(n)e(n\theta),\qquad I_\rho(\theta)=\int_1^Nt^{\rho-1}e(t\theta)\,dt . \]
(EF) The explicit formula (Davenport, Chapters 17 and 19; for imprimitive \(\chi\) the difference from the primitive inducing character is \(O(\omega(r)\log y)\)): for \(\chi\bmod r\), \(2\le T\le y\), \(r\le y\), \[ \psi(y,\chi)=\delta_\chi y-\sum_{|\gamma_\rho|\le T}\frac{y^\rho}\rho +O\Big(\frac yT\log^2(ry)+y^{1/4}\log y\Big), \] the sum over zeros \(\rho=\beta+i\gamma\) of \(L(s,\chi)\) with \(0<\beta<1\).
(ZF) The zero-free region: there is an absolute \(c>0\) such that for every \(r\ge1\) and \(\chi\bmod r\), \(L(s,\chi)\ne0\) in \(\sigma\ge1-c/\log(r(|t|+2))\), except possibly for one real simple zero when \(\chi\) is real (Davenport, Chapter 14).
(Pg\('\)) Page's theorem, in the form quoted from Drappeau and Fiorilli in ENDPOINT_SHARP.md section 4: there is an absolute \(b>0\) such that for \(Q,T\ge2\) the product \(\prod_{q\le Q}\prod_{\chi\bmod q}L(s,\chi)\) has at most one zero in \(\mathrm{Re}\,s>1-b/\log(QT)\), \(|\mathrm{Im}\,s|\le T\), and it is real and belongs to a unique primitive real \(\tilde\chi\) of conductor \(\tilde q\).
(GS) Gauss sums: for \((a,r)=1\) and any \(\chi\bmod r\) induced by the primitive \(\chi^\bmod r^\), \(\sum_{b\bmod r}\chi(b)e(ab/r)=\overline\chi(a)\tau(\chi)\) with \(\tau(\chi)=\mu(r/r^)\chi^(r/r^)\tau(\chi^)\) (Montgomery and Vaughan, Theorem 9.10), so \(|\sum_b\chi(b)e(ab/r)|\le\sqrt{r^*}\le\sqrt r\).
(Md) The model in progressions: the four cases of ENDPOINT_SHARP.md section 4 with the sieve level \(D_1\), which give, for every \(r\le R\), every \(b\) with \((b,r)=1\) and every prefix \(y\le N\), with \(\chi_e,q_e,\beta_e\) the TT-exceptional data when they exist, \[ \sum_{\substack{n\le y\\n\equiv b\ (r)}}\nu(n)=\frac y{\phi(r)}+O(E_{\rm md}),\qquad \sum_{\substack{n\le y\\n\equiv b\ (r)}}\nu(n)\chi_e(n)n^{\beta_e-1} =1_{q_e\mid r}\,\chi_e(b)\frac{y^{\beta_e}-1}{\beta_e\phi(r)}+O(E_{\rm md}), \] \(E_{\rm md}\ll N(\log\ell)e^{9-\sqrt\ell/2}+\ell^{1/2}e^{\sqrt\ell}N^{1/2}\ll Ne^{-\sqrt\ell/3}\). For \((b,r)>1\) both sums vanish, since every prime factor of \(r\le R<Z\) divides \(P\).
(In2) Input (2): \(\|R_{\rm mod}\|_2^2\ll N^3e^{-2c_m\sqrt\ell}+N^2\), \(c_m=1/4-o(1)\), ARC_SPLIT_BUDGET.md (7).
2. The exponential-sum explicit formula on an arc
Fix \(r\le R\), \((a,r)=1\), \(|\theta|\le R/(rN)\), and \(T\) with \(2\le T\le N\). Integrating \(e(t\theta)\) against \(d\psi(t,\chi)\) and inserting (EF), whose remainder is at most \((N/T)\log^2(rN)+N^{1/4}\log N\) uniformly for \(t\le N\), partial summation against \(e(t\theta)\) (total variation \(2\pi|\theta|N\)) gives \[ F_\chi(\theta)=\delta_\chi K_N(\theta)-\sum_{|\gamma_\rho|\le T}I_\rho(\theta) +O\Big((1+N|\theta|)\Big[\frac NT\ell^2+N^{1/4}\ell\Big]\Big)+O(\ell^2), \tag{1} \] the last term being the prime powers of primes dividing \(r\) that \(\delta_\chi K_N\) counts and \(F_{\chi_0}\) does not.
The integrals. Write \(\phi(t)=\gamma\log t+2\pi\theta t\), so \(I_\rho(\theta)=\int_1^Nt^{\beta-1}e^{i\phi(t)}dt\), \(\phi'(t)=\gamma/t+2\pi\theta\), \(\phi''(t)=-\gamma/t^2\). Three bounds, each on dyadic blocks \([U,2U]\) where \(t^{\beta-1}\le U^{\beta-1}\), summed over \(U\le N\):
- trivially \(|I_\rho|\le N^\beta/\beta\le2N^\beta\);
- if \(|\gamma|\ge4\pi N|\theta|\) and \(|\gamma|\ge1\): \(|\phi'(t)|\ge|\gamma|/(2t)\ge|\gamma|/(4U)\) on the block and \(\phi'\) is monotone, so the first-derivative test gives \(\ll U^{\beta-1}\cdot U/|\gamma|\), and summing, \(|I_\rho|\ll N^\beta/|\gamma|\);
- if \(1\le|\gamma|<4\pi N|\theta|\): \(|\phi''|\ge|\gamma|/(4U^2)\) on the block, so the second-derivative test gives \(\ll U^{\beta-1}\cdot U/\sqrt{|\gamma|}\), and summing, \(|I_\rho|\ll N^\beta/\sqrt{|\gamma|}\).
The zero sum. Let \(\mathcal Z_\chi(T)\) be the zeros of \(L(s,\chi)\) with \(|\gamma|\le T\) other than a possible exceptional real zero, put \(H_0=4\pi N|\theta|\le4\pi R/r\), and let \(\beta_\chi(U)\) be the largest \(\beta\) among them with \(|\gamma|\le U\). By (ZF), \(N^{\beta_\chi(U)}\le N\exp(-c\,\ell/\log(r(U+2)))\). With the zero count \(N(U,\chi)\ll U\log(rU)+\log r\) (Davenport, Chapter 16), \[ \sum_{\rho\in\mathcal Z_\chi(T)}|I_\rho(\theta)| \ll N^{\beta_\chi(H_0)}\Big[\log r+\sqrt{H_0}\log(rH_0)\Big] +\sum_{\substack{U=2^jH_0\\ U\le T}}N^{\beta_\chi(2U)}\log(rU). \tag{2} \] The first bracket is the zeros with \(|\gamma|<1\) (at most \(O(\log r)\) of them, trivial bound) and the stationary range \(1\le|\gamma|<H_0\) (second-derivative bound, \(\sum_{|\gamma|\le H_0}|\gamma|^{-1/2}\ll\sqrt{H_0}\log(rH_0)\)); the sum is the far range, where each dyadic block costs its zero count times \(N^\beta/U\). Since \(rH_0\le4\pi R\), the first term is \(\ll N\exp(-(c/\sigma)\sqrt\ell\,(1+o(1)))\sqrt{R/r}\,\ell\). In the far range \(\beta_\chi(2U)\) increases with \(U\), so the sum is at most \(\ell\cdot N\exp(-c\,\ell/\log(3rT))\log(rT)\).
Choice of \(T\). Take \(T=R^3\). Then \(rT\le R^4\), so the far range is \(\ll N\ell^2\exp(-(c/(4\sigma))\sqrt\ell\,(1+o(1)))\); and the remainder in (1) is \(\ll(1+R/r)(N/R^3)\ell^2\ll N\ell^2R^{-2}\), while \(RN^{1/4}\ell\) is negligible. Altogether, for every \(\chi\bmod r\), with the exceptional real zero \(\beta_\chi\) (if any) kept aside, \[ F_\chi(\theta)=\delta_\chi K_N(\theta)-1_{\beta_\chi}I_{\beta_\chi}(\theta)+O(N\,\Upsilon(r)),\qquad \Upsilon(r):=\ell^2\Big[e^{-(c/\sigma)\sqrt\ell(1+o(1))}\sqrt{R/r}+e^{-(c/(4\sigma))\sqrt\ell(1+o(1))}+R^{-2}\Big]. \tag{3} \]
3. The major-arc lemma
For \(\alpha=a/r+\theta\in I_{r,a}\), splitting into residue classes and expanding the coprime ones in characters, \[ F(\alpha)=\sum_{(b,r)>1}e(ab/r)F_b(\theta)+\frac1{\phi(r)}\sum_{\chi\bmod r} \Big(\sum_{(b,r)=1}\overline\chi(b)e(ab/r)\Big)F_\chi(\theta) =O(\ell^2)+\frac1{\phi(r)}\sum_\chi\chi(a)\tau(\overline\chi)F_\chi(\theta), \] by (GS); the first sum is the prime powers of primes dividing \(r\). The same expansion of \(H\) uses (Md) with partial summation, which costs \((1+2\pi N|\theta|)\le1+2\pi R/r\) per class and \(r\) classes: \[ H(\alpha)=\frac1{\phi(r)}\sum_\chi\chi(a)\tau(\overline\chi)H_\chi(\theta),\qquad H_\chi(\theta)=\delta_\chi K_N(\theta)-1_{\rm exc}1_{q_e\mid r}1_{\chi=\chi_{e,r}}I_{\beta_e}(\theta)+O\big(rE_{\rm md}(1+R/r)\big), \] where \(\chi_{e,r}\) is the character mod \(r\) induced by \(\chi_e\), and the identification of the twisted main term uses \(\chi_e(n)=\chi_e(b)\) on the class \(b\) when \(q_e\mid r\), and its vanishing when \(q_e\nmid r\), exactly as in ENDPOINT_SHARP.md section 4. (For \(\chi_e^2=\chi_0\) on \((n,r)=1\) the twisted sum with \(\chi=\chi_{e,r}\) is the untwisted model sum with the weight \(n^{\beta_e-1}\), whose main term is \(\int_1^Nt^{\beta_e-1}e(t\theta)dt=I_{\beta_e}(\theta)\) by (Md) and partial summation.) Subtracting, with \(|\tau(\overline\chi)|\le\sqrt r\): \[ |W(\alpha)|\le\sqrt r\,\max_{\chi\bmod r}\big|F_\chi(\theta)-H_\chi(\theta)\big|+O(\ell^2) \le\sqrt r\,\max_\chi\big|F_\chi-\delta_\chi K_N+1_{\rm exc}1_{q_e\mid r}1_{\chi=\chi_{e,r}}I_{\beta_e}\big| +8R^{3/2}E_{\rm md}+O(\ell^2). \tag{4} \] Corrected 2026-09-12 (defect found by the independent check a-0080): the first version wrote \(8RE_{\rm md}\) for the model term. The factor \(\sqrt r\) from the Gauss sums multiplies everything inside \(F_\chi-H_\chi\), the model error \(O(rE_{\rm md}(1+R/r))\) included, so the model term is \(\sqrt r\cdot r\,E_{\rm md}(1+R/r)\le8R^{3/2}E_{\rm md}\). The correction propagates to (5), to the squared bound in section 4, and to (H), where \(2/9\) became \(1/6\); it does not touch the \(\Lambda\) side or the conclusion.
Matching the exceptional zeros. Apply (Pg\('\)) with \(Q=R\), \(T=R^3\): at most one zero \(\tilde\beta\) of \(\prod_{r\le R}\prod_\chi L(s,\chi)\) lies in \(\mathrm{Re}\,s>1-b/\log(R^4)=1-b/(4\sigma\sqrt\ell)\), \(|\mathrm{Im}\,s|\le R^3\), and it is a real zero of a unique primitive real \(\tilde\chi\) of conductor \(\tilde q\le R\). Three cases for the exceptional real zero \(\beta_\chi\) of a real \(\chi\bmod r\) in (3), and for the model's \(\beta_e\):
- If \(\chi\) is induced by \(\tilde\chi\) and \(\tilde\chi=\chi_e\) (the TT-exceptional character; this happens exactly when \(\beta_e>1-b/(4\sigma\sqrt\ell)\), which \(\beta_e>1-c_0/\sqrt\ell\) implies under (H), \(c_0\le b/(4c_0)\)): then \(\beta_\chi=\beta_e\), \(1_{q_e\mid r}=1\), and the two terms in (4) cancel.
- If \(\chi\) is induced by \(\tilde\chi\ne\chi_e\): \(\tilde\chi\) is not TT-exceptional at \(Z\) (it has conductor \(\tilde q\le R<Z\), so the failing condition is the threshold), hence \(\tilde\beta\le1-c_0/\sqrt\ell\) and \(|I_{\tilde\beta}(\theta)|\le2N^{\tilde\beta}\le2Ne^{-c_0\sqrt\ell}\). This is the term (4) cannot make smaller. It is present only for \(r\) with \(\tilde q\mid r\).
- Every other zero of every \(L(s,\chi)\), \(\chi\bmod r\le R\), with \(|\gamma|\le T\) lies outside the Page region, hence has \(\beta\le1-b/(4\sigma\sqrt\ell)\); it also lies outside (ZF)'s region. It is in \(\mathcal Z_\chi(T)\) and is covered by (3). Corrected 2026-09-12 (
SHARP_EXPONENT.mdsection 2.1; the checka-0080confirmed the original wording and did not see this): the first version added in parentheses that a real zero of a real character which is not the Page zero is not exceptional for (ZF) either. That does not follow: (ZF) allows one real zero per real \(\chi\) in its own region, Page allows one among all characters in a smaller region, and a real zero in \((1-c/\log2r,\ 1-b/\log R^4]\) is the first and not the second. Such a zero is not in \(\mathcal Z_\chi(T)\); there is at most one per real \(\chi\), it contributes \(|I_\beta|\le2N^\beta\le2Ne^{-(b/(4\sigma))\sqrt\ell}\) to \(F_\chi\), and the term \(e^{-(b/(4\sigma))\sqrt\ell}\) is to be added to \(\Upsilon(r)\) in (3) and (5). In the budget (6) it is \(Re^{-(b/(2\sigma))\sqrt\ell}\), with exponent \(b/(2\sigma)-\sigma\ge\sigma\) at \(\sigma=c_0\) under (H) (\(c_0^2\le b/4\)); nothing else changes.
So, with \(\tilde\chi\) the Page-exceptional character at \((R,R^3)\) when it exists and differs from \(\chi_e\), \[ \boxed{\ \sup_{\alpha\in I_{r,a}}|W(\alpha)|\ \ll\ \sqrt r\,N\,\Upsilon(r) +1_{\tilde q\mid r}\sqrt r\,Ne^{-c_0\sqrt\ell}+R^{3/2}Ne^{-\sqrt\ell/3}.\ } \tag{5} \] Compared with ARC_SPLIT_BUDGET.md (5), which is \(\ll RNe^{-\gamma\sqrt\ell}\) with \(\gamma<c_0/2\): the residue-class factor \(r\) has become \(\sqrt r\) by (GS); the partial-summation factor \(1+2\pi N|\theta|\) is gone on the \(\Lambda\) side, because the explicit formula is applied to the exponential sum itself and the height enters only through the zero-free region; and the loss of the factor \(1/2\) in \(c_0/2\), which came from the small-\(y\) reduction \(\log y\ge\ell/2\) in the proof of (1\('\)), does not occur, because there is no prefix \(y<N\) anywhere in (4).
4. The budget
Squaring (5) and using \(\sqrt r\le\sqrt R\), \(\sqrt{R/r}\cdot\sqrt r=\sqrt R\): \[ \sup_{\mathfrak M}|W|^2\ll N^2\ell^4\Big[Re^{-(2c/\sigma)\sqrt\ell(1+o(1))}+Re^{-(c/(2\sigma))\sqrt\ell(1+o(1))}+R^{-3} +Re^{-2c_0\sqrt\ell}\Big]+R^3N^2e^{-2\sqrt\ell/3}. \] By ARC_SPLIT_BUDGET.md (3) and (6), \(\int_{\mathfrak M}(|F|^2-|H|^2)^2\le\sup_{\mathfrak M}|W|^2\int_{\mathbb T}(|F|+|H|)^2\ll\sup_{\mathfrak M}|W|^2\cdot N\ell\), and its (8), (11) carry the rest unchanged. Hence \[ E_{\rm corr}^{(Z)}(N)\ll N^3\ell^{O(1)}\Big[Re^{-2c_0\sqrt\ell}+Re^{-(2c/\sigma)\sqrt\ell(1+o(1))}+Re^{-(c/(2\sigma))\sqrt\ell(1+o(1))}+R^{-3}+R^3e^{-2\sqrt\ell/3}+R^{-1}\Big] +N^3e^{-2c_m\sqrt\ell}+N^{13/5}\ell^6+N^2\ell^2 . \tag{6} \] (Corrected 2026-09-12, found by the check a-0082 of SHARP_EXPONENT.md: the second term was printed as \(R^2e^{-(2c/\sigma)\sqrt\ell(1+o(1))}\); squaring \(\sqrt r\cdot\sqrt{R/r}=\sqrt R\) gives \(R\). It is dominated by the third term and nothing else changes. The check a-0080 did not see this.) With \(R=\lfloor e^{\sigma\sqrt\ell}\rfloor\) the exponents are \(2c_0-\sigma,\ 2c/\sigma-\sigma,\ c/(2\sigma)-\sigma,\ 3\sigma,\ 2/3-3\sigma,\ \sigma\). The first and last balance at \(\sigma=c_0\), where the others are at least \(\sigma\) provided \(c_0^2\le c\), \(c_0^2\le c/4\), \(c_0\le1/6\) (the last corrected from \(2/9\) with the model term, see (4)); together with \(2c_m=1/2-o(1)>c_0\) and the matching conditions \(c_0^2\le b/4\), \(c_0\le c_P\) of section 3, this is (H). Therefore \[ E_{\rm corr}^{(Z)}(N)\ll N^3\ell^{O(1)}\exp(-c_0\sqrt\ell), \] which is (S\(''\)).
Against (S\('\)). Same architecture, same minor arcs, same (In2). The major-arc term changed from \(N^3\ell R^2e^{-2\gamma\sqrt\ell}\) to \(N^3\ell^{O(1)}Re^{-2c_0\sqrt\ell}\) plus terms that decay faster than any fixed multiple of \(\sqrt\ell\) as \(\sigma\to0\). The balance moved from \(\sigma=2\gamma/3\) to \(\sigma=c_0\), and since \(\gamma<c_0/2\), the exponent constant improved by a factor of more than \(3\).
A repair of (1\('\)) recorded in passing. The factor \(1/2\) in \(\gamma<c_0/2\) came from assuming \(\log y\ge\ell/2\) after discarding \(y\le Ne^{-\gamma\sqrt\ell}\). The discarded range in fact gives \(\log y\ge\ell-\gamma\sqrt\ell\), so \(y^{\tilde\beta-1}\le e^{-c_0(\sqrt\ell-\gamma)}\) and (1\('\)) holds with any \(\gamma<\min(c_0,\delta/\sqrt2,1/2)\); the arc-split budget then gives \(2c_0/3\). That is a local improvement of the checkpoint's constant, not used here, recorded so the comparison is fair: against the repaired checkpoint the present gain is a factor \(3/2\), and the change of binding term is the same.
5. The ceiling: a conditional lower bound
Proposition (conditional). Let \(\tilde\chi\) be a primitive real character of odd conductor \(\tilde q\le e^{\varepsilon\sqrt\ell}\) with \(\tilde\chi(-1)=1\), and suppose \(L(s,\tilde\chi)\) has a real zero \(\tilde\beta=1-\kappa/\sqrt\ell\) with \[ c_0<\kappa<\min(\sqrt c,\sqrt b)-3\varepsilon,\qquad\kappa+3\varepsilon<1/3 . \] Also require \(0<\varepsilon<\kappa\) and \(\kappa+\varepsilon<1/4\). These guards were missing from the displayed window: the first makes the quadratic residual negligible against the linear cross term, and the second supplies the model-error separation used at the end of the proof. They leave a nonempty window near \(\kappa=c_0\) under (H'') of SHARP_EXPONENT.md. This is a conditional implication, not an existence statement for the zero. Then \(\tilde\chi\) is not TT-exceptional at \(Z\), and for large \(N\) \[ E_{\rm corr}^{(Z)}(N)\ \ge\ c_1\,\frac{N^3e^{-2\kappa\sqrt\ell}}{\tilde q^{\,2}}\,(1+o(1)), \tag{7} \] with an absolute \(c_1>0\).
Proof. Take one arc: by ARC_SPLIT_BUDGET.md (1), (3), \(E_{\rm corr}^{(Z)}\ge\tfrac12\int_J|G_{\rm corr}|^2-O(N^2)\) for any \(J\subset\mathbb T\), and \(\int_J|G_{\rm corr}|^2\ge\tfrac12\int_J(|F|^2-|H|^2)^2-2\|R_{\rm mod}\|^2-2\kappa_0^2|J|\). Let \(J\) be the union over \((a,\tilde q)=1\) of \(J_a=\{a/\tilde q+\theta:\ |\theta|\le1/(8N)\}\). On \(J_a\), run sections 2 and 3 with \(r=\tilde q\), \(\sigma\) replaced by the actual height: \(H_0\le\pi/2\), so there is no stationary range; take \(T=\tilde q^{2}e^{\kappa\sqrt\ell}\ell^4\), so the remainder in (1) is \(\ll N\ell^2/T\ll Ne^{-\kappa\sqrt\ell}\tilde q^{-2}\ell^{-2}\) and the far range is \(\ll N\ell^2\exp(-c\ell/\log(3\tilde qT))\ll N\ell^2\exp(-(c/(\kappa+3\varepsilon))\sqrt\ell(1+o(1)))\). Page at \((Q,T)=(\tilde q,T)\): \(\tilde\beta>1-b/\log(\tilde qT)\) since \(\kappa<b/(\kappa+3\varepsilon)\), so \(\tilde\beta\) is the Page zero. Other zeros are either covered by the ordinary zero-free estimate (3), or are a real character's own exceptional zero below the Page threshold. The latter contribute the additional parallel error with \(b\) in place of \(c\). Put \(c_*=\min(c,b)\) to cover both cases. Hence, by (4) with \(r=\tilde q\), and (GS) with \(\tau(\tilde\chi)=\sqrt{\tilde q}\) for even primitive real \(\tilde\chi\), \[ W(a/\tilde q+\theta)=-\frac{\tilde\chi(a)\sqrt{\tilde q}}{\phi(\tilde q)}I_{\tilde\beta}(\theta)+O\big(\sqrt{\tilde q}\,N\ell^2e^{-(c_* /(\kappa+3\varepsilon))\sqrt\ell(1+o(1))}+Ne^{-\kappa\sqrt\ell}\tilde q^{-3/2}\ell^{-2}+\tilde q^{3/2}Ne^{-\sqrt\ell/3}\big), \] and the error is \(o(N^{\tilde\beta}/\sqrt{\tilde q})\) under the stated window: the first term because \(c_* /(\kappa+3\varepsilon)>\kappa+\varepsilon\), the second by the factor \(\ell^{-2}\) (this is why \(T\) carries \(\ell^4\); with \(\ell^2\) the ratio would be \(\phi(\tilde q)/\tilde q^2\), not small for \(\tilde q=3\)), the last because \(\kappa+3\varepsilon<1/3\). If the TT-exceptional \(\chi_e\) existed with \(q_e\mid\tilde q\), it would be a distinct character in the same Page box with \(\beta_e>1-c_0/\sqrt\ell>\tilde\beta\), contradicting uniqueness. Thus this case cannot occur. There is no TT main term on these arcs, so \(H(a/\tilde q+\theta)=(\mu(\tilde q)/\phi(\tilde q))K_N(\theta)(1+o(1))+O(\tilde q^{3/2}Ne^{-\sqrt\ell/3})\). For \(|\theta|\le1/(8N)\): \(|K_N(\theta)|\ge N/2\), \(|I_{\tilde\beta}(\theta)|\ge N^{\tilde\beta}/4\), and the arguments of \(K_N(\theta)\) and \(I_{\tilde\beta}(\theta)\) both lie within \(\pi/8\) of \(\pi(N+1)\theta\), so \(\mathrm{Re}(\overline{K_N}I_{\tilde\beta})\ge\cos(\pi/4)|K_N||I_{\tilde\beta}|\). Since \(\tilde q\) is odd and squarefree, \(\mu(\tilde q)=\pm1\), and \[ |F|^2-|H|^2=2\operatorname{Re}(\overline HW)+|W|^2 =-\frac{2\mu(\tilde q)\tilde\chi(a)\sqrt{\tilde q}}{\phi(\tilde q)^2}\operatorname{Re}\big(\overline{K_N}I_{\tilde\beta}\big)(1+o(1)), \] of modulus \(\ge c_2N^{1+\tilde\beta}\sqrt{\tilde q}/\phi(\tilde q)^2\) on \(J_a\). Integrating over \(|\theta|\le1/(8N)\) and summing over the \(\phi(\tilde q)\) values of \(a\): \(\int_J(|F|^2-|H|^2)^2\ge c_3N^{2\tilde\beta+1}\tilde q/\phi(\tilde q)^3\ge c_3N^{2\tilde\beta+1}/\tilde q^2\). The subtracted terms are \(\ll N^3e^{-2c_m\sqrt\ell}+N^2\ell^2\), and \(N^{2\tilde\beta+1}/\tilde q^2=N^3e^{-2\kappa\sqrt\ell}\tilde q^{-2}\gg N^3e^{-(1/2)\sqrt\ell}\) since \(2\kappa+2\varepsilon<1/2\). This gives (7). \(\square\)
The order \(N^{2\tilde\beta+1}/\tilde q^2\) is the first term of SIEGEL_UNIFORMITY.md (23) for the correction that the target would carry if \(\tilde\chi\) were counted as exceptional; (7) says the target, which does not count it, carries it as error instead.
The other parities (added after the check a-0080, which reviewed the even case only; this part has not been independently read). For odd \(\tilde\chi\) (\(\tilde\chi(-1)=-1\)) with odd \(\tilde q\), \(\tau(\tilde\chi)=i\sqrt{\tilde q}\), so the cross term is \(2\operatorname{Re}(\overline HW)=(2\mu(\tilde q)\tilde\chi(a)\sqrt{\tilde q}/\phi(\tilde q)^2)\operatorname{Im}(\overline{K_N}I_{\tilde\beta})(1+o(1))\). Writing \(M=(N+1)/2\), \(\overline{K_N(\theta)}I_{\tilde\beta}(\theta)=S(\theta)\int_1^Nt^{\tilde\beta-1}e((t-M)\theta)dt\) with \(S=|K_N|\), and the imaginary part is \(S\int_0^{M-1}\big[(M+u)^{\tilde\beta-1}-(M-u)^{\tilde\beta-1}\big]\sin(2\pi u\theta)\,du\). The bracket is \(-(1-\tilde\beta)\,2u\,M^{\tilde\beta-2}(1+O(u/M))\), and \(\sin(2\pi u\theta)\) has one sign on \(0<u<M\) when \(0<\theta\le1/(8N)\), so for \(1/(16N)\le\theta\le1/(8N)\) the imaginary part is \(\asymp(1-\tilde\beta)N^{1+\tilde\beta}=(\kappa/\sqrt\ell)N^{1+\tilde\beta}\) with a fixed sign. The argument above then gives \[ E_{\rm corr}^{(Z)}(N)\ \ge\ c_1'\,\kappa^2\,\frac{N^3e^{-2\kappa\sqrt\ell}}{\tilde q^{\,2}\,\ell}\,(1+o(1)), \] which is (7) with an extra \(\kappa^2/\ell\); the corollary's arithmetic absorbs a power of \(\ell\) in \(e^{(\kappa'-2\kappa)\sqrt\ell}\), so its conclusion holds for odd \(\tilde\chi\) of odd conductor as well. For even conductors (\(4\mid\tilde q\) or \(8\mid\tilde q\)), \(\mu(\tilde q)=0\), the model has no main term at \(a/\tilde q\), the cross term vanishes and only \(|W|^4\) remains: \(\int_J|W|^4\asymp N^{4\tilde\beta-1}/\tilde q^{\,2}\), the second term of SIEGEL_UNIFORMITY.md (23), so \(E_{\rm corr}^{(Z)}\gg N^3e^{-4\kappa\sqrt\ell}/\tilde q^{\,2}\) and the corollary for those conductors starts at \(\kappa'>4c_0\). The ceiling \(2c_0\) is therefore set by the odd conductors, both parities of the character.
Corollary (what an exponent above \(2c_0\) would establish). Suppose \(E_{\rm corr}^{(Z)}(N)\ll N^3\exp(-\kappa'\sqrt\ell)\) unconditionally for some fixed \(\kappa'>2c_0\), and fix \(\varepsilon\) small. We may first reduce \(\kappa'\) to a smaller number still above \(2c_0\), since the assumed bound implies every weaker one. Choose it sufficiently close to \(2c_0\), and then choose \(\varepsilon\), so the midpoint below meets every guard of the corrected Proposition. Let \(\tilde\chi\) be as in the Proposition with a real zero \(\tilde\beta<1\). Choose \(N\) with \(\sqrt{\log N}=(c_0+\kappa'/2)/(2(1-\tilde\beta))\), so that \(\kappa=(1-\tilde\beta)\sqrt\ell=(c_0+\kappa'/2)/2\in(c_0,\kappa'/2)\). If \(\tilde q\le e^{\varepsilon\sqrt\ell}\), i.e. \(1-\tilde\beta\le\varepsilon(c_0+\kappa'/2)/(2\log\tilde q)\), then (7) and the assumed bound contradict each other once \(\tilde q^2<e^{(\kappa'-2\kappa)\sqrt\ell}\), which the same inequality supplies for small \(\varepsilon\). Hence: no even primitive real character of odd conductor \(\tilde q\) has a real zero \(\tilde\beta\) with \(1-\tilde\beta\le c'/\log\tilde q\), \(c'=c'(\kappa',c_0,\varepsilon)>0\), for all \(\tilde q\) beyond an effective bound. Scope of that sentence, corrected after the check a-0080: it is an effective zero-free interval \((1-c'/\log\tilde q,1)\) for the real zeros of even primitive real characters of odd conductor, beyond an effective bound; with the remark below it extends to odd characters of odd conductor. It is not the classical Siegel-zero statement, which has no parity or conductor qualification and no effective threshold. It is a statement of the same shape on a restricted family, and for that family Landau and Page still give only "at most one such zero per range"; excluding it is open there too.
Consequence for this hunt. With the target fixed as it is, the exponent constant in \(N^3\exp(-c''\sqrt\ell)\) lies in \([c_0,2c_0]\): (S\(''\)) gives the lower end unconditionally, and passing the upper end needs the Siegel-zero statement above. The remaining gap, a factor \(2\), is the minor-arc term: (6) balances \(Re^{-2c_0\sqrt\ell}\) against \(R^{-1}\), and the true size of the minor-arc fourth moment is \(N^3/R^2\) (measured in ARC_SPLIT_BUDGET.md section 8 between \(N^3/R^2\) and \(N^3/R\)). With \(R^{-2}\) in place of \(R^{-1}\) the balance would move to \(\sigma=2c_0/3\) and the exponent to \(4c_0/3\); with the true major-arc size (the term \(Re^{-2c_0\sqrt\ell}\) is itself an upper bound for what (7) shows is \(\asymp e^{-2\kappa\sqrt\ell}/\tilde q^2\) on the arcs that carry it) and \(R^{-2}\), to \(2c_0\). Both steps are the rank-3 quartic problem (RANK3_QUARTIC_LITERATURE.md, w-bound-raw-arc-quartic-moment): a bound \(\int_{\mathfrak m}|F|^4\ll N^3\ell^{O(1)}R^{-2+\epsilon}\) for \(R=e^{\sigma\sqrt\ell}\). Nothing in this hunt or its literature record gives it.
6. Routes killed on the way, with the witness
- The shape \(\sqrt\ell\). Every unconditional input here decays like \(\exp(-c\,\ell/\log(rT))\) on an arc of modulus \(r\) and height \(T\), and the minor arcs decay polynomially in \(R\). Balancing a power of \(R\) against \(\exp(-c\ell/\log R)\) puts \(\log R\asymp\sqrt\ell\) whatever the constants. The one input with a better rate, the Vinogradov-Korobov region for \(\zeta\) (rate \(\ell^{3/5}\),
LOCALIZED_MIXED_ENERGY.md(19) for the \(r=1\) arc), has no analogue in the conductor aspect: for \(L(s,\chi)\) the \(\log r\) term of the region is not improved, and \(\log r\) dominates \((\log T)^{2/3}\) for every \(r>e^{(\log T)^{2/3}}\), which at \(T\le R^3\) is every \(r\) beyond \(e^{(3\sigma)^{2/3}\ell^{1/3}}\). So (1) of the assignment is out of reach of the inputs the hunt has. - The minor arcs through the explicit formula. On the Farey arcs that cover \(\mathfrak m\), of radius \(1/(q\sqrt N)\), the heights reach \(N|\theta|\asymp\sqrt N/q\), so \(\log(qT)\asymp\ell/2\) and (ZF) gives \(N^\beta\le Ne^{-2c}\): no decay. The inner parts \(|\theta|\le R'/(qN)\) of those arcs are exactly the major arcs at cutoff \(R'\), so this route is the present one with a different name, and the outer parts are Vaughan's.
- Averaging over moduli instead of the pointwise zero-free region (Bombieri-Vinogradov in Drappeau and Fiorilli's form,
ENDPOINT_SHARP.mdsection 4): its error \(xe^{-\delta\sqrt{\log x}}\) has the \(\sqrt\ell\) rate built in, from the \(x/Q_1\) term at the lower cutoff of its large-sieve range. Summing (5) over \(r\) with it instead of the pointwise bound would trade \(2\gamma\) for \(\gamma+\delta\) in the old budget, a constant, and is superseded by section 3. - Koukoulopoulos's short-interval theorem (
RANK3_Z_COMPONENT.mdinput) as a variance bound on the major arcs through Gallagher's lemma: its \(E(y,h;q)\) is a first moment with a maximum over residues, so it yields \((\log x)^{-A}\) and not a power; recorded in the Ostoyae log of the previous round and not repeated. - The cross terms of (22):
ARC_SPLIT_BUDGET.mdsection 7.
7. Finite checks
major_arc_explicit_probe.py, results in results_major_arc_explicit_probe.json. Zeros of \(\zeta\) from mpmath.zetazero. Nothing here tests an asymptotic statement, (ZF), or (Pg\('\)); no exceptional zero is reachable, so section 5 is untested.
(1) The integral bounds of section 2. For the first \(60\) zeros of \(\zeta\) (\(\gamma\le163.0\)), \(N\in\{10^4,10^6\}\) and \(N\theta\in\{0,1,10,100\}\), \(480\) integrals \(I_\rho(\theta)\) by composite Gauss-Legendre with panels at a quarter oscillation. In the far regime \(|\gamma|\ge4\pi N|\theta|\) the largest value of \(|I_\rho|/(N^\beta/|\gamma|)\) is \(1.010\); in the stationary regime the largest value of \(|I_\rho|/(N^\beta/\sqrt{|\gamma|})\) is \(0.064\). Both bounds hold with constants at most \(1.01\) on this sample; the second has room, as expected of a worst-case test.
(2) The exponential-sum explicit formula (1). At \(N=2000\), with the \(202\) zeros of \(\zeta\) with \(|\gamma|\le400\), the defect \(|F(\theta)-K_N(\theta)+\sum_\rho I_\rho(\theta)|\) is \(1.76,\ 4.48,\ 1.76,\ 1.77\) at \(N\theta=0,\ 0.5,\ 2,\ 5\), against \(|F(\theta)|=1994,\ 1273,\ 14.7,\ 14.8\): between \(0.001\) and \(0.010\) of the displayed remainder scale \((N/T)\log^2N\,(1+N|\theta|)\). At \(N\theta=2\) and \(5\) the main term \(K_N\) is small and the zero sum alone reproduces \(F\) to within \(1.8\). This checks the sign and normalisation of (1); it says nothing about the zero-free region.
(3) The Gauss-sum expansion of section 3. With the toy model \(\nu=b1_{(n,30)=1}\) at \(N=3000\), modulus \(r=7\) and all six characters mod \(7\), the exact identity \(\sum_{(a,7)=1}|W(a/7+\theta)|^2+|W(\theta)|^2=\frac76\sum_\chi|V_\chi(\theta)|^2+7|W_0(\theta)|^2\) (with \(W_0\) the class \(0\bmod7\)) holds to a relative \(4\times10^{-13}\) at \(\theta=3/(7N)\). In the toy the class \(0\) is not small (\(7|W_0|^2=6.6\times10^5\) against \(1.2\times10^5\) for the character side), because \(Z=7\) does not exceed the modulus; in the document's setting \(r<Z\), the class is empty on the model side and carries only prime powers of primes dividing \(r\) on the \(\Lambda\) side.
8. Scope
One mechanism, the explicit formula on the major arcs with the zero-free region at the arc's own modulus and height, attempted to a checked budget. It improves the exponent constant of the same subpolynomial bound and, more usefully, identifies the binding term of the complete budget as the target's own exceptional threshold \(c_0\), with a conditional lower bound that puts the reachable exponent in \([c_0,2c_0]\) short of a Siegel-zero theorem. No power saving; no change to the \(\sqrt\ell\) shape, which section 6 argues is inherent to the inputs; nothing about the zeros of \(\zeta\); the lower bound is conditional on a zero that may not exist; it is proved for even characters of odd conductor, extended to odd characters of odd conductor in a remark that has not been independently read, and gives only the weaker \(4c_0\) for even conductors. Sections 2 to 5 have had an independent read (MAJOR_ARC_EXPLICIT_REVIEW.md); its one defect is corrected in place.