2026-09-11. Base: ENDPOINT_SHARP.md (commit 367573c, the endpoint checkpoint, unchanged), UPPER_BOUND.md sections 2 to 5, LOCALIZED_MIXED_ENERGY.md section 6, SIEGEL_UNIFORMITY.md (19). Assignment: take equation (22) of LOCALIZED_MIXED_ENERGY.md as the route, retain every cross term including the model remainder, and use the actual arithmetic coefficients to look for cancellation, with the enlarged-\(Z\) corrected target held fixed.
Result, stated first. With the notation of ENDPOINT_SHARP.md section 1, \(Z=\exp(\sqrt\ell)\), \(\ell=\log N\), and \(\gamma\) the constant of its progression input (1\('\)), for every fixed \(c''<\min(2\gamma/3,\ 2c_m)\), \[ \boxed{\ E_{\rm corr}^{(Z)}(N)\ \ll\ N^3\exp\!\big(-c''\sqrt{\log N}\big).\ } \tag{S$'$} \] ENDPOINT_SHARP.md (18\('\)) gives the same shape with the exponent constant \(c'=\tfrac12\min(2\gamma-7\sigma,\ \sigma/6,\ c,\ 2c_m)\) at \(\sigma=\min(\gamma/20,1/20)\), i.e. \(c'=\gamma/240\); the best that budget allows over \(\sigma\) is \(\gamma/43\). Here the constant is \(2\gamma/3\), and three of the four inputs ENDPOINT_SHARP.md section 6 names as capping the exponent are no longer consumed: the divisor approximant (its section 2, both variants), the minor-arc character estimates of ENDPOINT_BOUND.md section 4, and the terms \(N^3R^{-1/6}\), \(N^{14/5}\), \(N^{2+A(N)}e^{\sqrt\ell/2}\) of the budget. The rate \(\sqrt\ell\) is unchanged; this is a better constant in the exponent and a shorter chain, not a power saving.
The mechanism, in one paragraph. Equation (22) splits the corrected residual \(G_{\rm corr}=X+Y+R_{\rm mod}\) along the model: \(X+Y\) is what the model leaves of \(|F|^2\), \(R_{\rm mod}\) is what the model misses of the target. In coefficient space the model cancels identically in the sum (section 1), so the split is not intrinsic, and the previous chain paid for it twice: on the minor arcs it had to bound the model exponential sum \(H\) (which is what the divisor approximant, the Type I estimate and the character cases were for), and it lost the cross term between \(|F|^2\) and \(|H|^2\) there by writing \((|F|^2-|H|^2)^2\le2|F|^4+2|H|^4\). The repair is to split by arcs instead: insert the model only on the major arcs, where \(|F|^2-|H|^2\) is small because the model reproduces the main term of \(|F|^2\) at every rational with denominator \(r\le R\), and on the minor arcs keep \(G_{\rm corr}\) whole, where it is \(|F|^2\) minus the singular-series polynomial and the minor-arc size of that polynomial is governed by the arithmetic coefficients \(\mu(q)^2/\phi(q)^2\) of the Ramanujan expansion of \(\mathfrak S(h)\), through the large sieve. Every cross term of (22) is retained on the minor arcs because nothing is split there. This is the architecture of UPPER_BOUND.md section 5, whose baseline (1) is the classical \(N^3L^{-C}\), moved to the corrected target with the major-arc cutoff \(R=\exp(\sigma\sqrt\ell)\) that the model and its progression input make available.
Grade: derived, one route, finite checks in section 8, independently read. No power saving, no statement about exceptional zeros beyond what the inputs carry, nothing about the zeros of \(\zeta\).
Independent check, 2026-09-12 (attempt a-0078, ARC_SPLIT_BUDGET_REVIEW.md). No defect in any of the five sections. It verified (2) and (3) to floating-point precision on its own toy model, recomputed sections 3 and 5 against the cited equations including the account of \(R^7\) becoming \(R^2\) and both budgets' optima (\(\gamma/43\) for the old one at its own optimum, \(\gamma/240\) as stated, \(2\gamma/3\) here), confirmed that on the minor arcs the document cites UPPER_BOUND.md and nothing of the divisor approximant, and re-derived the quadratic piece of 4.1 in full with its three ranges and the consistency check against SIEGEL_UNIFORMITY.md (23). Two things it checked at the level of orders rather than symbol by symbol, and said so: the auxiliary sieve-tail bound \(\sum_{m>X,\,g\mid m}\phi(m)^{-2}\ll g\,\phi(g)^{-2}X^{-1}\ell^{O(1)}\) in the \(m>\sqrt N\) range, and the constants of the linear pieces. Neither is a binding term, so an error there confined to logarithmic factors would not change (S\('\)); both are named here so that they are redone if 4.1 is ever consumed by something that binds on them.
1. Notation, and the identity that makes (22) non-intrinsic
From UPPER_BOUND.md section 2, with \(y=\lfloor\sqrt N\rfloor\): \(\mathfrak S_y(h)=\sum_{q\le y}\mu(q)^2c_q(h)/\phi(q)^2\), \(V_y(\alpha)=\sum_{q\le y}\mu(q)^2\phi(q)^{-2}\sum_a^*|K_N(\alpha-a/q)|^2\), \(d_N=\sum_{n\le N}\Lambda(n)^2\), \(a_0(N,y)=d_N-N\mathfrak S_y(0)\), \(G_y=|F|^2-V_y-a_0(N,y)\), and the tail \(D_{\rm tail}(N,y)=2\sum_{h\le N}(N-h)^2|\mathfrak S(h)-\mathfrak S_y(h)|^2\ll N^2\) (its (6)). From LOCALIZED_MIXED_ENERGY.md section 6, with the exceptional correction \(C_N(h)=C_{q,\beta,Z}(h)\) of SIEGEL_UNIFORMITY.md (19) at this \(Z\) (zero when no exceptional zero exists): \[ \widehat C_N(\alpha)=2\sum_{h=1}^NC_N(h)\cos(2\pi h\alpha),\qquad G_{\rm corr}=G_y-\widehat C_N=X+Y+R_{\rm mod}, \] \(X=\mathcal C(2\operatorname{Re}\overline HW)\), \(Y=\mathcal C(|W|^2)\), \(R_{\rm mod}=\mathcal C(|H|^2-V_y)-\widehat C_N\), \(\mathcal C\) removing the constant coefficient. Its (24), from UPPER_BOUND.md (4): \[ \big|\sqrt{E_{\rm corr}^{(Z)}}-\|G_{\rm corr}\|2\big|\le\sqrt{D{\rm tail}(N,y)}\ll N, \qquad\text{so}\qquad E_{\rm corr}^{(Z)}\le2\|G_{\rm corr}\|_2^2+O(N^2). \tag{1} \]
Coefficients. For \(1\le h\le N\), with \(\psi_2(N,h)=\sum_{n\le N-h}\Lambda(n)\Lambda(n+h)\) and \(\psi_2^a(N,h)=\sum_{n\le N-h}a(n)a(n+h)\): \[ \begin{aligned} [X+Y]_h&=\psi_2(N,h)-\psi_2^a(N,h),\\ [R_{\rm mod}]_h&=\psi_2^a(N,h)-(N-h)\mathfrak S_y(h)-C_N(h),\\ [G_{\rm corr}]_h&=\psi_2(N,h)-(N-h)\mathfrak S_y(h)-C_N(h). \end{aligned} \tag{2} \] The model appears in the first two lines with opposite signs and not in the third. Whatever \(a(n)\) is, \(X+Y\) and \(R_{\rm mod}\) carry equal and opposite copies of \(\psi_2^a\), of the sieve remainder inside it, and of the part of the truncated singular series the model reproduces. So \(2\operatorname{Re}\langle X+Y,R_{\rm mod}\rangle=\|G_{\rm corr}\|^2-\|X+Y\|^2-\|R_{\rm mod}\|^2\) is not a quantity with independent arithmetic content; it is the accounting correction for having split a model-independent object along a model. A bound that keeps \(G_{\rm corr}\) whole retains it exactly. That is what the minor arcs below do.
The same identity in the other direction is what the major arcs use: pointwise on the whole circle, \[ G_{\rm corr}=\big(|F|^2-|H|^2\big)+R_{\rm mod}-\kappa,\qquad \kappa:=d_N-\sum_{n\le N}a(n)^2, \tag{3} \] since \(\mathcal C(|F|^2)-\mathcal C(|H|^2)=|F|^2-|H|^2-\kappa\) and \(\mathcal C(|F|^2)-\mathcal C(V_y)-\widehat C_N=[\mathcal C(|F|^2)-\mathcal C(|H|^2)]+R_{\rm mod}\). Section 8 item (1) checks (3) numerically, including the sign of \(\kappa\). Here \(|\kappa|\ll N\ell\): \(d_N\le\ell\,\psi(N)\) and \(\sum a(n)^2\le4b^2N\) with \(b\sim e^{\gamma_E}\sqrt\ell\).
2. The arcs
UPPER_BOUND.md (7) with \(Q:=R\): \[ R=\lfloor\exp(\sigma\sqrt\ell)\rfloor,\quad 2R^2<N,\quad I_{r,a}=\{\alpha:\|\alpha-a/r\|\le R/(rN)\},\quad \mathfrak M=\bigcup_{r\le R}\bigcup_a^*I_{r,a},\quad \mathfrak m=\mathbb T\setminus\mathfrak M . \] The arcs are disjoint and \(|\mathfrak M|\le2R^2/N\). On \(\mathfrak m\), Dirichlet approximation with \(\lceil N/R\rceil\) gives a reduced \(a/r\) with \(R<r\le2N/R\) and \(|\alpha-a/r|\le r^{-2}\) (UPPER_BOUND.md section 5). The radius \(R/(rN)\) rather than ENDPOINT_BOUND.md's \(2R/N\) is used below; it costs nothing and it removes one factor of \(R\) from the major arcs. By (1), \[ E_{\rm corr}^{(Z)}\le2\int_{\mathfrak M}|G_{\rm corr}|^2+2\int_{\mathfrak m}|G_{\rm corr}|^2+O(N^2). \tag{4} \]
3. Major arcs: the model, inserted where it cancels
By (3), \(\int_{\mathfrak M}|G_{\rm corr}|^2\le3\int_{\mathfrak M}(|F|^2-|H|^2)^2+3\|R_{\rm mod}\|_2^2+3\kappa^2|\mathfrak M|\).
(a) The intensity difference. \(|F|^2-|H|^2=W\overline F+H\overline W\), so \(||F|^2-|H|^2|\le|W|(|F|+|H|)\). For \(\alpha=a/r+\theta\in I_{r,a}\), split \(W\) into residue classes modulo \(r\); each class is a progression, so (1\('\)) of ENDPOINT_SHARP.md section 4 (uniform over every prefix \(y\le N\), every \(r\le R\), every residue) and partial summation give \[ |W(\alpha)|\le\sum_{b\bmod r}\Big|\sum_{n\equiv b\ (r)}w_ne(n\theta)\Big| \le r\,(1+2\pi N|\theta|)\,C_1Ne^{-\gamma\sqrt\ell} \le(r+2\pi R)\,C_1Ne^{-\gamma\sqrt\ell}\le8C_1RNe^{-\gamma\sqrt\ell}, \tag{5} \] using \(N|\theta|\le R/r\). Then, by Parseval on the whole circle, \[ \int_{\mathfrak M}(|F|^2-|H|^2)^2\le\sup_{\mathfrak M}|W|^2\int_{\mathbb T}(|F|+|H|)^2 \le64C_1^2R^2N^2e^{-2\gamma\sqrt\ell}\cdot2\Big(d_N+\sum a(n)^2\Big) \ll N^3\ell\,R^2e^{-2\gamma\sqrt\ell}. \tag{6} \] ENDPOINT_BOUND.md (9) has \(N^3\ell^2R^7e^{-2\gamma t}\) for the same integral: \(R^3\) from \(|\mathfrak M|\) times the pointwise bound \(|F|+|H|\ll N\ell\), which Parseval replaces by \(\int(|F|+|H|)^2\ll N\ell\), and two more powers of \(R\) from the arc radius \(2R/N\) in place of \(R/(rN)\). Nothing else in (6) differs from that derivation; the prime-model cancellation is retained in the same way.
(b) The model remainder. By (2), \([R_{\rm mod}]_h=[\psi_2^a(N,h)-(N-h)\mathfrak S(h)-C_N(h)]+(N-h)(\mathfrak S(h)-\mathfrak S_y(h))\), and the first bracket is input (2) of ENDPOINT_SHARP.md section 5 at this \(Z\): \(O(Ne^{-c_m\sqrt\ell})\) uniformly in \(h\). So \[ \|R_{\rm mod}\|2^2\le4\sum{h\le N}\big|\psi_2^a-(N-h)\mathfrak S-C_N\big|^2+2D_{\rm tail}(N,y) \ll N^3e^{-2c_m\sqrt\ell}+N^2 . \tag{7} \] This is LOCALIZED_MIXED_ENERGY.md (24), squared. It is integrated here over \(\mathfrak M\) only, but bounded by its whole-circle norm; no gain and no loss.
(c) The constant. \(\kappa^2|\mathfrak M|\ll N^2\ell^2\cdot R^2/N=NR^2\ell^2\).
Altogether \[ \int_{\mathfrak M}|G_{\rm corr}|^2\ll N^3\ell R^2e^{-2\gamma\sqrt\ell}+N^3e^{-2c_m\sqrt\ell}+NR^2\ell^2+N^2 . \tag{8} \]
4. Minor arcs: no model, the singular-series polynomial instead
On \(\mathfrak m\) the arc model \(A_R\) of UPPER_BOUND.md section 3 vanishes, so its identity (10), \(G_y=B_R-H_R-a_0(N,R)-T_{y,R}\) with \(B_R=|F|^2-A_R\), reads \[ G_{\rm corr}\big|_{\mathfrak m}=|F|^2-H_R-a_0(N,R)-T_{y,R}-\widehat C_N , \] where \(H_R=V_R-A_R\ge0\) is the leakage of the \(q\le R\) kernels outside their arcs and \(T_{y,R}=\mathcal C(V_y-V_R)\) is the truncation between \(R\) and \(y\). Hence \[ \int_{\mathfrak m}|G_{\rm corr}|^2\le5\Big[I_R+\int_{\mathfrak m}H_R^2+a_0(N,R)^2|\mathfrak m| +\int_{\mathfrak m}|T_{y,R}|^2+\int_{\mathfrak m}|\widehat C_N|^2\Big], \qquad I_R=\int_{\mathfrak m}|F|^4 . \tag{9} \] Four of the five are UPPER_BOUND.md's, valid for any \(R\) with \(2R^2<N\):
- \(I_R\le\sup_{\mathfrak m}|F|^2\int_{\mathbb T}|F|^2\ll(N^3/R+N^{13/5})L^6\), its (20), from Vaughan's bound on \(\mathfrak m\) where every approximating denominator exceeds \(R\).
- \(\int_{\mathfrak m}H_R^2\le\|H_R\|_2^2\ll N^3R^{-2}\log(2R)\,w(R)^2\ll N^3\ell^3R^{-2}\), its (13) and (15).
- \(a_0(N,R)=N\log(N/R)+O(N)\), so \(a_0(N,R)^2|\mathfrak m|\ll N^2\ell^2\).
- \(\int_{\mathfrak m}|T_{y,R}|^2\le\|T_{y,R}\|_2^2\ll N^3/R^2\), its (9).
The fifth is new and is proved in 4.1: \[ \int_{\mathfrak m}|\widehat C_N|^2\ll N^3\ell^{O(1)}/R, \tag{10} \] with \(\widehat C_N=0\) when no exceptional zero exists. Therefore \[ \int_{\mathfrak m}|G_{\rm corr}|^2\ll N^3\ell^{O(1)}R^{-1}+N^{13/5}\ell^6+N^2\ell^2 . \tag{11} \] Compare ENDPOINT_BOUND.md (17): \(\ell^{O(1)}[N^3R^{-1/6}+N^{14/5}+N^2D_0\sqrt Z]+N^3e^{-c\sqrt\ell}\). The \(R^{-1/6}\) was the model's minor-arc bound (15), the \(N^{14/5}\) came from bounding \(\int_{\mathfrak m}|H|^4\) through a cubic moment, and the \(N^2D_0\sqrt Z\) was the approximant's level. None of the three objects is present in (9).
4.1 The correction polynomial on the minor arcs
By SIEGEL_UNIFORMITY.md (19), (17), with \(\mathcal L_h=(q/\phi(q))^2S_*(h)\), \(S_*(h)=\prod_{p<Z,\,p\nmid q}\alpha_p(h)=\sum_{d\mid P,(d,q)=1}c_d(h)/\phi(d)^2\), \[ C_N(h)=\mathcal L_h\Big[\frac{c_q(h)}qJ_{12}(h)\Big] -1_{q\ \rm odd}\,\mathcal L_h\frac{\mu(q)}q\big[\chi(-h)J_1(h)+\chi(h)J_2(h)\big], \] \(J_1,J_2\ll N^\beta\le N\), \(J_{12}\ll N^{2\beta-1}\le N\), all three nonnegative and nonincreasing in \(h\) on \(1\le h\le N\) (their integrands are; \(J_1(h)\) has \(T=N-h\)), and all zero at \(h=N\). For a nonnegative nonincreasing weight \(J\) write \(\Phi_J(x)=\sum_{0<|h|\le N}J(|h|)e(hx)\); Abel summation gives \(|\Phi_J(x)|\le J(1)/\|x\|\) and \(|\Phi_J(x)|\le2NJ(1)\).
The quadratic piece. LOCALIZED_MIXED_ENERGY.md (5): \(\mathcal L_hc_q(h)/q=\sum_{d\mid P,(d,q)=1}q\,\phi(qd)^{-2}c_{qd}(h)\), so \[ \widehat C_N^{\rm quad}(\alpha)=\sum_{\substack{d\mid P\\(d,q)=1}}\frac q{\phi(qd)^2} \sum_{a\bmod qd}^{*}\Phi_{J_{12}}\big(\alpha-\tfrac a{qd}\big). \] Split the fractions by their denominator \(m=qd\).
- \(R<m\le\sqrt N\). Bound the whole-circle norm in coefficient space: the coefficient at \(h\) is \(J_{12}(h)\) times a sum over fractions with denominators in \((R,\sqrt N]\), which are \(1/N\)-spaced, so the dual large sieve gives \[ \sum_{h\le N}\Big|J_{12}(h)\sum_{R<qd\le\sqrt N}\frac q{\phi(qd)^2}c_{qd}(h)\Big|^2 \le4N^2\cdot2N\sum_{R<qd\le\sqrt N}\phi(qd)\frac{q^2}{\phi(qd)^4} \le8N^3\frac{q^2}{\phi(q)^3}\sum_{d>R/q}\frac{\mu(d)^2}{\phi(d)^3} \ll N^3\ell^{O(1)}\min\Big(\frac1q,\frac q{R^2}\Big)\le\frac{N^3\ell^{O(1)}}R, \] using \(q/\phi(q)\ll\log\log q\le\ell\), \(\sum_{d>D}\mu(d)^2/\phi(d)^3\ll D^{-2}\ell^{O(1)}\) for \(D\ge1\) and \(O(1)\) for \(D<1\). Both cases of the minimum are at most \(1/R\): if \(q\le R\) then \(q/R^2\le1/R\), and if \(q>R\) then \(1/q<1/R\).
- \(m>\sqrt N\). With \(|c_m(h)|\le\gcd(m,h)\) and \(\sum_{m>X,\,g\mid m}\phi(m)^{-2}\ll g\,\phi(g)^{-2}X^{-1}\ell^{O(1)}\), the coefficient at \(h\) is \(\ll N\,q\,\tau(h)\ell^{O(1)}/\sqrt N\), and with \(q<Z=e^{\sqrt\ell}\), \(\sum_{h\le N}\tau(h)^2\ll N\ell^3\), the mean square is \(\ll N^2e^{2\sqrt\ell}\ell^{O(1)}=N^{2+o(1)}\).
- \(m\le R\), present only when \(q\le R\). These centres lie in \(\mathfrak M\); on \(\mathfrak m\), \(\|\alpha-a/m\|>R/(mN)\). The \(\phi(m)\) points \(a/m\) are \(1/m\)-spaced, so \(\sum_a^*\|\alpha-a/m\|^{-1}\le mN/R+2m(1+\log m)\le4mN/R\) for large \(N\), and \[ \sup_{\mathfrak m}\Big|\sum_{qd\le R}\cdots\Big|\le\sum_{d\le R/q}\frac q{\phi(qd)^2}\cdot2N\cdot\frac{4qdN}R \ll\frac{N^2q^2}{\phi(q)^2R}\sum_{d\le R/q}\frac d{\phi(d)^2}\ll\frac{N^2\ell^{O(1)}}R, \] \[ \int_{\mathfrak m}\Big|\sum_{qd\le R}\cdots\Big|\le\sum_{d\le R/q}\frac q{\phi(qd)}\int_{\|x\|>R/(qdN)}\frac{2N}{\|x\|}dx \ll N\ell\frac q{\phi(q)}\sum_{d\le R/q}\frac1{\phi(d)}\ll N\ell^{O(1)}, \] so this part has \(\int_{\mathfrak m}|\cdot|^2\le\sup\cdot\int|\cdot|\ll N^3\ell^{O(1)}/R\).
The linear pieces (odd \(q\) only). For primitive \(\chi\), \(\chi(h)\tau(\overline\chi)=\sum_{a\bmod q}\overline\chi(a)e(ah/q)\) for every \(h\), with \(|\tau(\overline\chi)|=\sqrt q\), so \(\chi(\pm h)S_*(h)\) is a sum over fractions with denominators dividing \(qd\), \(d\mid P\), \((d,q)=1\), with coefficients of modulus at most \(q^{-1/2}\phi(d)^{-2}\); the weight in front is \((q/\phi(q))^2q^{-1}\), and \(J_1,J_2\le2N\). The same three ranges give: for \(R<qd\le\sqrt N\), by the large sieve, \(\ll N^3(q^2/\phi(q)^4)\sum_{d>R/q}d\,\phi(d)^{-4}\ll N^3\ell^{O(1)}q^{-2}\min(1,q^2/R^2)\le N^3\ell^{O(1)}/R^2\); for \(qd>\sqrt N\), \(N^{2+o(1)}\) as above; for \(qd\le R\), the same tail argument with the weight \(q^{1/2}/(\phi(q)^2\phi(d)^2)\), giving \(\ll N^3\ell^{O(1)}/R\). This proves (10). Consistency check: for \(q>R\) every fraction is in the first two ranges and the bound is \(\ll N^3\ell^{O(1)}/q\), which is SIEGEL_UNIFORMITY.md (23)'s whole-circle energy \(N^{4\beta-1}q^2/\phi(q)^3+N^{2\beta+1}q^2/\phi(q)^4\) up to logarithms, as it must be.
5. The budget
From (4), (8), (11): \[ E_{\rm corr}^{(Z)}(N)\ll N^3\ell R^2e^{-2\gamma\sqrt\ell}+N^3e^{-2c_m\sqrt\ell} +N^3\ell^{O(1)}R^{-1}+N^{13/5}\ell^6+NR^2\ell^2+N^2\ell^2 . \tag{12} \] Take \(R=\lfloor\exp(\sigma\sqrt\ell)\rfloor\) with \(\sigma=2\gamma/3\). The constraints are \(2R^2<N\) and \(\sigma\le1/\sqrt2\) (ENDPOINT_SHARP.md section 4 needs \(R\le e^{\sqrt{\ell/2}}\)), both satisfied since \(\gamma<1/2\) gives \(\sigma<1/3\). Then \(R^2e^{-2\gamma\sqrt\ell}\le e^{-(2\gamma/3)\sqrt\ell}\) and \(R^{-1}\le2e^{-(2\gamma/3)\sqrt\ell}\), so \[ E_{\rm corr}^{(Z)}(N)\ll N^3\ell^{O(1)}e^{-(2\gamma/3)\sqrt\ell}+N^3e^{-2c_m\sqrt\ell}, \] which is (S\('\)). Here \(c_m=1/4-o(1)\) (the fundamental lemma at level \(N^{1/4}\) in input (2), ENDPOINT_HALF.md section 2.1), and \(\gamma<1/2\), so \(2c_m>2\gamma/3\) and the model term is not the binding one; the boxed statement keeps it in the minimum so that nothing is assumed about which constant is smaller.
Against the previous budget. Both budgets balance a major-arc term \(N^3R^ke^{-2\gamma\sqrt\ell}\) against a minor-arc term \(N^3R^{-j}\). The previous one had \(k=7\), \(j=1/6\): optimum at \(\sigma=12\gamma/43\), exponent \(\gamma/43\), and the stated \(\sigma=\gamma/20\) gives \(\gamma/240\). This one has \(k=2\), \(j=1\): optimum at \(\sigma=2\gamma/3\), exponent \(2\gamma/3\). The gain in \(k\) is bookkeeping (Parseval, arc radius); the gain in \(j\) is the mechanism: Vaughan's \(R^{-1}\) for \(|F|^4\) replaces the approximant's \(R^{-1/6}\) for \(|H|^2\), because \(H\) is not bounded on the minor arcs at all.
6. What the arc split removes, and what it keeps
Not consumed by (S\('\)): ENDPOINT_BOUND.md section 2 (the Bonferroni approximant \(B_m\), its (4)-(6)), section 4 (Type I (10), the character estimate (12), Cases A and B (13)-(14), the minor-arc model bound (15)), section 5 (the cubic moment (16) and the minor-arc integral (17)); and in ENDPOINT_SHARP.md, section 2 in both variants (the Mertens correction, the retuned cutoff and (A), the \(\beta\)-sieve alternative) and section 3. The level \(D_0\), the function \(A(N)\), the properties (P1)-(P4) and the term \(N^{2+A(N)}e^{\sqrt\ell/2}\) do not appear. Those statements remain correct as statements about the divisor approximant; the complete bound no longer routes through it.
Consumed: input (1\('\)) with (DF), (Pg), (FL) at level \(D_1\) and (Cmp), all from ENDPOINT_SHARP.md section 4; input (2) from its section 5; the arc estimates of UPPER_BOUND.md (7), (9), (13), (15), (20); and 4.1 above.
ENDPOINT_SHARP.md section 6 names four inputs that cap the exponent at \(\sqrt\ell\) and attain it. After this document, the divisor approximation is not one of them for the complete bound. The cap now rests on (1\('\)) (a zero-free region question for the \(L\)-functions of conductor at most \(R\)), on (2) (the fundamental lemma at a level fixed by \(Z\)), and on the arc balance itself: the minor-arc term decays only polynomially in \(R\), because \(|F|^2\) is of size \(N^2/\phi(r)^2\) at every rational with denominator just above \(R\), while the major-arc term decays like \(e^{-2\gamma\sqrt\ell}\) with \(\gamma\) tied to \(\log R\) through the zero-free region. Balancing a power of \(R\) against \(\exp(-c\,\ell/\log R)\) puts \(\log R\) at \(\sqrt\ell\) whatever the constants. That statement is about this architecture.
7. The route through (22), and the refuted sub-candidate
The assignment asked for cancellation in the cross terms of (22). Section 1 records what they are: the cross term \(2\operatorname{Re}\langle X,Y\rangle\) was already retained by ENDPOINT_BOUND.md, which never separates \(X\) from \(Y\) (its \(D=\|X+Y\|^2\)); the cross term \(2\operatorname{Re}\langle X+Y,R_{\rm mod}\rangle\) is the accounting correction for a split along the model, and the minor arcs retain it exactly by not splitting. The model is inserted only on the major arcs, in the form (3), where its purpose is to cancel the main term of \(|F|^2\) at each \(a/r\).
Refuted sub-candidate: cancellation between \(X+Y\) and \(R_{\rm mod}\) improves the exponent. It does not, in either budget. Whatever the sign of \(2\operatorname{Re}\langle X+Y,R_{\rm mod}\rangle\), the term it can remove is at most \(2\|R_{\rm mod}\|^2\ll N^3e^{-2c_m\sqrt\ell}+N^2\) by (7), and \(2c_m\) is not the binding constant: ENDPOINT_SHARP.md (18\('\)) binds at \(\sigma/6=\gamma/120\), and (12) binds at \(2\gamma/3\). Retaining or discarding that cross term changes the constant in front, not the exponent. There is a genuine cancellation inside it (by (2), the part of the truncated singular series that the model reproduces, and the model's own sieve remainder, appear with opposite signs in \(X+Y\) and \(R_{\rm mod}\)), but it is cancellation of a term that was never binding. The candidate is refuted as an improvement, with the witness being the two budgets' binding terms.
What did move the exponent constant is section 4: the model's presence on the minor arcs, not any cross term, was the cost.
Checked against previous attempts. UPPER_BOUND.md section 5 is this argument for the original \(E\) with \(Q=L^B\) and Siegel-Walfisz in place of (1\('\)); its (1) is \(N^3L^{-C}\), and the reason it stops at logarithms is that Siegel-Walfisz caps the major-arc cutoff at a power of \(L\). The model and (1\('\)) lift that cap to \(e^{\sigma\sqrt\ell}\); nothing else changes. UPPER_BOUND.md section 6 with \(Q=\sqrt N/3\) is the rank-3 programme and is not touched. RANK3_QUARTIC_LITERATURE.md and w-bound-raw-arc-quartic-moment record that no bound on \(\int_{\mathfrak m}|F|^4\) below Vaughan's \(N^3L^6/R\) is available to the hunt; (9) uses exactly Vaughan's. The two judged proposals of a-0071, w-corr-longwindow-meansquare (equation (20)) and w-corr-crossterm-cancellation (the two cross terms), are addressed as follows: the second is answered in this section; the first concerns the mean square (20) of the long-window prefix sums \(A(k)=\sum_{n\le k}w_n\), the \(q=1\) positive-kernel majorant of LOCALIZED_MIXED_ENERGY.md. Scope, corrected 2026-09-12. The first version of this paragraph said (12) does not route through that quantity. That was too strong. (12) does use the prefix sums \(A(k)\), and their analogues in every residue class modulo \(r\le R\), through (1\('\)) in section 3(a): the bound (5) on \(|W|\) over a major arc is the supremum of those prefix sums, times \(r(1+2\pi N|\theta|)\). What (12) does not use is their mean square: a bound for (20) of the shape \(N^{3-\eta}\) would not enter (12), and (12) would not improve if (20) were improved, because (5) is a supremum bound and does not see the mean square; and a gain confined to the major arcs would in any case move the balance constant \(\sigma\), not the shape, since at the optimum the major-arc term is balanced against the minor-arc one. The two proposals are therefore not duplicated by this document; they are made irrelevant to this budget, which is a different thing from being answered.
What a stronger bound would establish. A fixed power saving \(E_{\rm corr}^{(Z)}\ll N^{3-\delta}\) would, through the correlation identity of CORRECTED_RH_BRIDGE.md section 4, bound \(\psi(N)-N\) by \(N^{1-\delta/2}\) up to the exceptional term, which is the strength of a zero-free strip for \(\zeta\). Nothing of that kind is claimed or approached here; (S\('\)) is subpolynomial, like everything before it.
8. Finite checks
arc_split_probe.py, results in results_arc_split_probe.json, about three seconds. No exceptional zero exists at these \(N\), so \(C_N=0\) and \(E_{\rm corr}=E\); nothing below tests 4.1 or any asymptotic statement.
(1) The identity (3), with the toy model \(\nu(n)=b\,1_{(n,30)=1}\) (\(Z=7\), \(b=35/8\)) at \(N=2000\) on a grid of \(2^{14}\) points: the maximum of \(|G_y-(|F|^2-|H|^2)-R_{\rm mod}+\kappa|\) is \(1.0\times10^{-10}\) against \(\max|G_y|=8.0\times10^4\), with \(\kappa=d_N-\sum a(n)^2=5483.7\) entering with the minus sign as written.
(2) The arc split at \(N=8000\), grid \(2^{19}\), \(y=89\): \(E=6.198\times10^8\) from the \(h\)-sum, \(J_{\rm ms}(N,y)=6.972\times10^8\) from the \(h\)-sum and from the grid (relative defect \(2\times10^{-15}\)), \(D_{\rm tail}(N,y)=1.0085\,N^2\), and \((\sqrt E-\sqrt{J_{\rm ms}})/\sqrt{D_{\rm tail}}=-0.19\), inside (1). With the arcs of section 2 at \(Q=R\):
| \(R\) | \( | \mathfrak M | /(2R^2/N)\) | \(\int_{\mathfrak M} | G_y | ^2/J\) | \(I_R/(N^3/R)\) | \(I_R\,R^2/N^3\) | \(\int_{\mathfrak m}(V_y+a_0)^2/(N^3/R^2)\) | \(\sup_{\mathfrak m} | F | ^2/(N^2/R)\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3 | 0.72 | 0.037 | 0.354 | 1.06 | 1.09 | 0.74 | ||||||
| 6 | 0.63 | 0.064 | 0.153 | 0.92 | 0.93 | 0.38 | ||||||
| 12 | 0.62 | 0.101 | 0.136 | 1.63 | 1.59 | 0.33 | ||||||
| 24 | 0.62 | 0.207 | 0.122 | 2.92 | 2.51 | 0.38 | ||||||
| 48 | 0.61 | 0.464 | 0.087 | 4.20 | 2.52 | 0.19 |
Read: the minor-arc fourth moment sits between \(N^3/R^2\) and \(N^3/R\) at this \(N\), below Vaughan's \(N^3L^6/R\) by a factor of order \(10^6\); the minor-arc energy of the main-term polynomial \(V_y+a_0(N,y)\), which is \(H_R+a_0(N,R)+T_{y,R}\) there, is \(1\) to \(2.5\) times \(N^3/R^2\), the order (9)'s second and fourth terms predict; at \(N=8000\) the minor arcs carry most of the mean square at every \(R\) shown, which is what one expects while \(e^{-\gamma\sqrt\ell}\) is not yet small. None of these numbers tests (S\('\)).
(3) The tail mean square behind the large-sieve step of 4.1, at \(N=8000\): \(\sum_{h\le N}|\mathfrak S(h)-\mathfrak S_R(h)|^2\) against \(N\,\mathcal T(R)\), \(\mathcal T(R)=\sum_{q>R}\mu(q)^2/\phi(q)^3\), is \(0.99,\ 0.93,\ 0.90,\ 0.79,\ 0.65\) at \(R=5,10,20,40,80\): the tail is at the orthogonality prediction and below the large-sieve bound \(2N\mathcal T(R)\) by a factor \(2\) to \(3\).
9. Scope
One mechanism, attempted to a checked budget. The result is a better constant in the exponent of the same subpolynomial bound, obtained by not using the sieve model on the minor arcs, and a shorter list of inputs. The divisor approximant of the previous chain is not refuted; it is unused. The refuted candidate is the cross term with the model remainder as a lever on the exponent. No fixed power saving, no novelty claim, no statement about exceptional zeros beyond the inputs, nothing about the zeros of \(\zeta\). Sections 3 to 5 have had an independent read (ARC_SPLIT_BUDGET_REVIEW.md); the two order-level steps of 4.1 it names are recorded at the top.