Metadata

bibkey: liu2026quasirhdual
authors: Baiying Liu
year: 2026
title: Complete-dual applications to the quasi-Riemann inverse fourth moment
doi: null
url: https://arxiv.org/abs/2610.12234v2
claim: Source-conditional applications of the pinned Gauss, Poisson, cubic and sextic inputs pay specified complete-dual regions and reduce an original inverse-fourth improvement to a small-product-squarepart head estimate; that remaining estimate is not supplied.
strata_touched: []
license: citation-only
triage: anchor

Complete-dual applications to the quasi-Riemann inverse fourth moment

This companion to the Liu source card records repo-derived applications, not additional theorems attributed to Liu. It reuses that card’s pinned primary manuscript, original coefficients, physical rows, collision zeros and fixed or polylog profile/height contract. The primary Gauss/Poisson formulas, Heath–Brown cubic sieve and original sixth-order sieve remain the analytic suppliers. No new sieve, literature-priority claim or Lean certification is made.

Here $D = U$ , $q_{a}$ denotes ideal norm, $t (a)$ the original fixed multiplicative ray class, and $K_{U, Ω}$ the complete physical-row kernel defined in the actual-order-six correlation. The baseline exponent is $B_{*} = (31 - 106) /6$ . The intrinsic product-squarepart norm $P$ and coefficient-envelope large- $P$ bound are C16–C18.

Reusing Möbius–Gauss cancellation in the complete dual

The squarefree Gauss pairing and its additive-character transport already supply the detector’s eq:signal, pair-Gauss formula and shift to the DDDS convention. They must be reused rather than reconstructed from a rational conductor. In particular, on their stated equal-ray coprime squarefree packet, cancelling the two original Möbius signs in the root factor leaves angular and cubic Gauss coefficients. It does not remove the character of each complete dual ideal.

The joint matrix retains order six. For a good primary squarefree column $x$ , let $χ_{x}$ be the original raw sextic character, including its zeros, and put $κ_{x} = χ_{x}^{2}$ , $λ_{x} = χ_{x}^{3}$ . In precisely the cited additive conventions, the already supplied shift gives

$B_{m, x} := λ_{x} (m) \overline{g_{3} (- 3 m, x)} = \overline{γ_{2} (x)} \overline{χ_{x} (m)} .$

Indeed $\overline{\widetilde g_3(-3m,x)} =\overline{\gamma_2(x)}\kappa_x(m)$ and $λ_{x} κ_{x} = \overline{χ_{x}}$ . If $(m, x) > 1$ , both sides are zero, by the primitive cubic Gauss identity at the colliding prime. For any identical finite row and column sets, therefore, $B = \overline{C} diag (\overline{γ_{2}})$ , where $C_{m, x} = χ_{x} (m)$ . The diagonal is unitary. These two matrices have the same singular values; the same nonnegative separate row and column weights or real zero mask preserve the identity with the conjugated matrix. A complex coupled weight must be transported too. Thus this identity alone supplies no smaller cubic operator norm. It does not exclude extra cancellation for the actual coefficient vector.

Opening the actual coefficient repairs one applicability mismatch. The fixed-twist numerator average in (B2) of the growing-ray supplier cannot simply be evaluated on a diagonal where both its quadratic twist and numerator are determined by $m$ . There is, however, an admissible way to use that supplier for each fixed $m$ : open $A_{D}$ itself.

Consider only the $t_{a} = t_{b} = 1$ , common-column-gcd $h = 1$ , $ℓ = r = 1$ term in the existing complete Poisson majorant, within a fixed equal-ray packet and with fixed bad-prime data. Here $D = U$ , $q_{x}, q_{y} ≍ U$ , the primitive good conductor is $x y$ , and the natural dual range is $q_{m} ≪ U$ . This is a specified term of that majorant, not a positive domination claim for a signed subset of the earlier $E_{6}$ . For a bounded selector $ω_{M}$ on good dual ideals $q_{m} ≍ M$ , $1 \leq M \leq U$ , define this normalized piece explicitly by

$E_{1} (M) = x, y good sf, (x, y) = 1 t (x) = t (y) = τ \sum g_{D} (x) \overline{g_{D} (y)} ϵ_{x, y} \frac{U}{q _{x} q _{y}} \times m good \sum ω_{M} (m) \overline{χ_{x, y} (m)} W (\frac{q _{m} U}{q _{x} q _{y}}) .$

Here $ϵ_{x, y}$ and $χ_{x, y}$ retain the source root convention and original ratio zeros. Only this good-dual, equal-ray term is selected; other original terms are not deleted from the full moment.

Detect $(x, y) = 1$ by $\sum_{e ∣ (x, y)} μ (e)$ . For each squarefree $e$ , allocate $e = e_{1} e_{2}$ in the exact convolution $A_{D} (x, 1) = \sum_{uv = x} V (q_{u} / D) V (q_{v} / D)$ and write $x = e uv$ after allocation. The two remaining lengths are

$D^{'} = D / q_{e_{1}}, N^{'} = D / q_{e_{2}}, D^{'} N^{'} = U / E, E = q_{e} .$

Cubic Gauss CRT leaves a cubic symbol between $u$ and $v$ . For fixed $m$ , its full sextic dual factor is an allowed finite inducing twist on each variable, with extra good modulus $f_{m} = rad (m)$ and $F_{m} = q_{f_{m}} \leq q_{m}$ , including any primes used only for zero extension. The theta numerator is the other convolution variable, with the existing $e$ -shift; its coprimality with $m$ is enforced by the original dual zeros. This avoids the earlier numerator/conductor overlap. Keep every squarefree and finite-class constraint; the original denominator exclusions are retained in the application of (B2).

Apply (B2) directly by Mellin inversion and Cauchy with the actual outer coefficient vector. There is no $ϑ$ weight in this sum, so no bounded-multiplier contraction or deletion of such a weight is used. Both convolution factors have the angular conjugate-cubic Gauss form; choose the smaller one as the outer length. For each allocation the resulting scalar $S_{e, m}$ satisfies

$∣ S_{e, m} ∣^{2} ≪ U^{ε} F_{m} E^{1/2} D^{'} N^{'} min (D^{'}, N^{'}) \leq U^{ε} F_{m} U^{3/2} / E .$

The allocation count is a divisor loss and $\sum_{q_e\ll U}q_e^{-1} \ll\log U $, so t h e i rs u m cos t s$ U^\varepsilon$. Alternatively the existing cubic large sieve gives the summed bound $U^{5/3 + ε}$ , independent of $m$ : sum $D^{'} N^{'} (D^{'} + N^{'} + (D^{'} N^{'})^{2/3})$ over the same allocations. Its first two terms cost $U^{3/2 + ε}$ and the last costs $U^{5/3 + ε}$ .

The Poisson prefactor and the two original $D^{- 1}$ coefficients leave $U^{- 1}$ at $q_{x}, q_{y} ≍ U$ . There are $O (M)$ dual ideals in the dyad and $F_{m} ≪ M$ . Consequently this specified procedure supplies

$∣ E_{1} (M) ∣ ≪ U^{ε} min {U^{1/2} M^{2}, U^{2/3} M} .$

The coupled Poisson profile does not cost a power of $M$ here. Writing $s = M / U \leq 1$ , $ρ_{x} = q_{x} / U$ , $ρ_{y} = q_{y} / U$ and $ρ_{m} = q_{m} / M$ , its scaled factor is $(\rho_x\rho_y)^{-1/2} \mathscr W(s\rho_m/(\rho_x\rho_y))$ on fixed compact annuli. Smoothness of the original radial Fourier transform at zero bounds all required logarithmic derivatives uniformly for $0 < s \leq 1$ ; compact Mellin separation pays the resulting polynomial heights with the same fixed seminorm contract.

The two certificates cross at $M = U^{1/6}$ . The angular application improves the cubic certificate only below that point, where the cubic certificate was already below $U$ . At the full natural length $M ≍ U$ , the better of these certificates is merely $U^{5/3 + ε}$ . In particular the angular application does not enlarge the range $M < U^{B_{*} - 2/3}$ already paid by this cubic certificate. This compares bounds produced by specified procedures, not lower bounds for the actual sum. The other common gcds, original row parameters, finite components and full long-conductor contraction remain outside this restricted calculation. The Gauss transport and analytic suppliers are reused; no improved full inverse-fourth exponent, signed Robin conclusion, literature-priority claim or Lean certification is supplied.

A joint bound for the same good-dual term

For precisely the preceding $E_{1} (M)$ , averaging the raw sextic matrix jointly before taking absolute values improves the paid dual range. This is a repo-derived, source-conditional application of the existing pair-Gauss formula, the sixth-order sieve and the preceding scalar cubic certificate. It retains the original $g_{D}$ , equal-ray packet, good-dual selector and $t_{a} = t_{b} = h = ℓ = r = 1$ restrictions. For $U^{1/3} \leq M \leq U$ , the resulting bound is

$∣ E_{1} (M) ∣ ≪ U^{ε} (U^{17/18} M^{1/6} + U^{5/6} M^{1/3}) . (J1)$

The matrix and the common gcd. Reuse the classwise pair-Gauss formula in its original additive convention: $ϵ_{x, y} = c_{τ} γ_{1} (x) \overline{γ_{1} (y)}$ , $∣ γ_{1} ∣ = 1$ . The class constant includes the original $χ_{y} (- 1)$ sign. Together with $\overline{χ_{x, y} (m)} = \overline{χ_{x} (m)} χ_{y} (m)$ , this is a raw-character Gram kernel on the coprime locus. Extend this product formula algebraically off that locus and impose $(x, y) = 1$ by $\sum_{e ∣ (x, y)} μ (e)$ . This does not extend the primitive ratio formula to noncoprime conductors.

Use the already justified compact Mellin separation of the Poisson profile. For each separated term, write $x = er, y = es$ and $q_{e} ≍ E$ . The column length is $B ≍ U / E$ . The original bound $∣ g_{D} (x) ∣ ≪ U^{- 1/2 + ε}$ implies that the product of the two column-vector $ℓ^{2}$ norms is $≪ U^{ε} / E$ . The $O (E)$ choices of $e$ in its dyad cancel this factor after a triangle inequality; dyads cost $U^{ε}$ . The original squarefree masks $(r, e) = (s, e) = 1$ remain column masks. The shared character factor leaves the row mask $(m, e) = 1$ . A bounded complex selector or Mellin row factor is controlled by Cauchy between the two positive squared norms; it is not deleted from the signed sum by a contraction assertion.

Separate small sixth-power-free cores. Write uniquely $m = n v^{6}$ , where $n$ is sixth-power-free. No condition $(n, v) = 1$ is imposed. Fix $1 \leq L \leq M$ and first select $q_{n} > L$ . For fixed $e, v$ , the collision zeros retain the column masks $(r, v) = (s, v) = 1$ and the row mask $(n, e) = 1$ ; terms with $(v, e) > 1$ vanish. Restricting a positive squared norm preserves the existing sixth-order sieve bound

$F_{6} (T, B) ≪ (TB)^{ε} (T + B + T^{5/6} B^{1/3} + T^{1/3} B^{5/6}), T ≍ M / q_{v}^{6}, B ≍ U / E .$

All column phases, including $γ_{1} (er)$ and the original coefficient, stay in the actual vectors. This uses no growing inducing-character supplier. Sum over $q_{v} ≪ (M / L)^{1/6}$ and then over the common-gcd dyads. Ideal counting gives

$∣ E_{1, q_{n} > L} (M) ∣ ≪ U^{ε} [M + U (M / L)^{1/6} + M^{5/6} U^{1/3} + M^{1/3} U^{5/6}] . (J2)$

For example the $T$ , $T^{5/6}$ and $T^{1/3}$ terms sum respectively $q_{v}^{- 6}$ , $q_{v}^{- 5}$ and $q_{v}^{- 2}$ , all convergently; the $B$ term counts the allowed $v$ . The remaining negative powers of $E$ cannot increase the bound.

For $q_{n} \leq L$ , count the dual ideals first:

$# {m : q_{m} ≍ M, q_{n} \leq L} ≪ M^{1/6} q_{n} \leq L \sum q_{n}^{- 1/6} ≪ M^{1/6} L^{5/6} .$

The preceding per- $m$ cubic certificate is uniform in $m$ and is $U^{2/3 + ε}$ after the same coefficient normalization. Consequently

$∣ E_{1, q_{n} \leq L} (M) ∣ ≪ U^{2/3 + ε} M^{1/6} L^{5/6} . (J3)$

Taking $L = U^{1/3}$ balances (J3) against $U (M / L)^{1/6}$ . For $M \leq U$ , both $M$ and $M^{5/6} U^{1/3}$ are at most $M^{1/3} U^{5/6}$ , proving (J1). For $M < U^{1/3}$ the preceding cubic certificate already costs at most $U^{1 + ε}$ .

Thus this specified term costs $U^{13/12 + ε}$ throughout $M \leq U^{3/4}$ , and has exponent strictly below $B_{*}$ for $M \leq U^{m}$ with fixed $m < 13 - 56 = 0.752551 \dots$ . The joint certificate improves the preceding cubic certificate once $M > U^{1/3}$ ; at $M = U$ it still costs $U^{7/6 + ε}$ . The proof of (J3) above uses the original Möbius convolution; the coefficient interface below analyzes the new Type-II convolution directly. In particular (C31), a whole moment comparison, does not transport a selected dual sum. Other common gcds, original components, the full small- $P$ contraction, signed Robin restoration and RH remain outside (J1). This is an application of the cited suppliers, with no new sieve, priority or Lean-certification claim.

A coefficient interface for the cubic low-core estimate

The cubic certificate, unlike the angular (B2) certificate, accepts arbitrary complex coefficient vectors. This permits a direct application to the actual Type-II factor, without a selected-sum transport theorem. The following repo-derived interface reuses the primary manuscript’s lem:fixed-gauss-phase, its Gauss CRT formula and eq:cubic-large-sieve (Heath–Brown’s cubic large sieve).

Let $b (d)$ be supported on good squarefree ideals, independent of the physical row and dual ideal, with $∣ b (d) ∣ \leq C_{b} τ_{k} (d)$ for fixed $k$ . Keep the original fixed multiplicative ray twist $ξ$ , and define the genuine self-convolution

$h_{b} (a) = D^{- 1} ξ (a) d_{1} d_{2} = a \sum b (d_{1}) b (d_{2}) V (q_{d_{1}} / D) V (q_{d_{2}} / D), D = U . (J4)$

Define $E_{1, b}$ by substituting $h_{b}$ for $g_{D}$ in the same specified good-dual, equal-ray term. To bound one fixed dual ideal, detect $(x, y) = 1$ as before, allocate $e = e_{1} e_{2}$ between the two factors of $x = e uv$ , and put $D^{'} = D / q_{e_{1}}$ , $N^{'} = D / q_{e_{2}}$ . The original squarefree column imposes $u, v$ squarefree, $(uv, e) = 1$ and $(u, v) = 1$ .

Write $κ_{u} (v) = χ_{u} (v)^{2}$ and $θ_{e} (w) = χ_{e} (w) χ_{w} (e)$ . The already supplied CRT and reciprocity formulas give, on this coprime locus,

$γ_{1} (e uv) = γ_{1} (e) θ_{e} (u) θ_{e} (v) R (u, v) κ_{u} (v) γ_{1} (u) γ_{1} (v) . (J5)$

The class factor $R (u, v)$ is fixed after a finite partition of $t (u), t (v)$ , with their product class still prescribed by $t (x)$ . The two coefficient vectors retain $b (e_{1} u)$ , $b (e_{2} v)$ , the respective $V$ factors, $ξ$ , $γ_{1}$ , $θ_{e}$ , norm phases and the original $\overline{χ_{u} (m)}$ , $\overline{χ_{v} (m)}$ zeros. The conditions $(u, e) = (v, e) = 1$ are separate coefficient masks. The only coupled mask $(u, v) = 1$ is exactly the zero extension of $κ_{u} (v)$ . If $(e, m) > 1$ , the whole scalar is zero. The negative-root side is handled by conjugating its entire scalar. Thus these are legitimate arbitrary coefficient vectors for the cubic sieve; no Möbius–Gauss cancellation or removal of a bounded multiplier is required.

For the unnormalized scalar $H_{e, m}^{e_{1}, e_{2}}$ from this allocation, the two squared vector norms are at most $C_{b}^{2} U^{ε} D^{'}$ and $C_{b}^{2} U^{ε} N^{'}$ . Cauchy and the cubic sieve give

$∣ H_{e, m}^{e_{1}, e_{2}} ∣^{2} ≪ C_{b}^{4} U^{ε} D^{'} N^{'} [D^{'} + N^{'} + (D^{'} N^{'})^{2/3}], D^{'} N^{'} = U / q_{e} . (J6)$

The allocation count costs a divisor loss. The first two terms sum $UD\sum_{e_1}q_{e_1}^{-2}\sum_{q_{e_2}\ll D}q_{e_2}^{-1} \ll U^{3/2+\varepsilon}$ and its transpose. The last sums $U^{5/3} \sum_{e} τ (e) q_{e}^{- 5/3} ≪ U^{5/3}$ . Restore the two factors $D^{- 1}$ , and apply Cauchy between the two scalars of each separated rectangular profile. The per- $m$ certificate is therefore $C_{b}^{4} U^{2/3 + ε}$ . The original compact Mellin separation pays all polynomial profile heights. No derivatives of $b$ are taken and no new power of the Type-I cutoff is introduced.

Consequently (J1)–(J3) hold for $E_{1, b}$ , with the factor $C_{b}^{4}$ . In particular they hold for $b = λ_{II, b}$ and $b = r_{A} λ_{II, b}$ , both bounded by $τ_{3}$ on squarefree support. This directly pays their specified $M \leq U^{3/4}$ prefix at $U^{13/12 + ε}$ in the $P = 1$ part of (C20). The angular branch $U^{1/2} M^{2}$ is not asserted for these arbitrary coefficients: its inner special Gauss sequence is a stronger hypothesis than the cubic sieve requires. Neither the full natural dual range nor the other original terms follow from this coefficient interface.

A multiple-Dirichlet-series model of the raw phase

The Gauss numerator shift and (J5) identify a precise model for a specified scalar, before applying a bound. On the squarefree coprime support of (J5), its moving phase, with the fixed- $e$ factors left in the actual coefficient vectors, is

$γ_{1} (uv) \overline{χ_{uv} (m)} = q_{uv}^{- 1/2} z mod uv \sum χ_{uv} (z) e (m z / (uv)) .$

This is the normalized sextic Gauss sum with numerator $m$ . Both sides vanish when $(m, uv) > 1$ . The original $b (e_{1} u)$ , $b (e_{2} v)$ , profiles, fixed-core twists and masks remain present; the identity does not turn them into standard Dirichlet-series coefficients. It reuses the pinned Gauss shift, rather than lowering the order of the original dual character.

An exact function-field model of this raw phase can be read from Sawin–Whitehead, Functional equations of axiomatic multiple Dirichlet series, Weyl groupoids, and quantum algebra, arXiv:2507.08662v2, Lemma 2.2 and Proposition 4.1. This is a repo-derived parameter match using their existing formulas, not an analytic estimate attributed to that paper. Let $q \equiv 1 (mod 12)$ be a prime power, let $χ$ have order six on $F_{q}^{\times}$ , and let $u, v, m$ be monic polynomials, with $uv$ squarefree. The variable order is $(u, v, m)$ . In their notation the required symmetric matrix is

$M = 425245550 (mod 6) .$

Indeed their quadratic character is $ξ = χ^{3}$ , so Lemma 2.2 uses the derivative character $ξ χ = χ^{4}$ . Splitting the derivative of $uv$ gives diagonal exponents $4, 4$ and cross exponent $4 + 4 \equiv 2$ ; the numerator shift gives exponents $- 1 \equiv 5$ against $m$ . Reciprocity has no sign for this choice of $q$ . Proposition 4.1, since $M_{33} = 0$ , permits arbitrary $m$ , including nonsquarefree $m$ . With $a_{M}$ denoting their axiomatic coefficient and $g_{χ}$ their finite-field Gauss sum, the exact normalization is

$q^{- (d e g u + d e g v) /2} g_{χ} (m, uv) = (\frac{g _{χ}}{q})^{d e g u + d e g v} a_{M} (u, v, m) .$

For $(m, uv) = 1$ this follows from Lemma 2.2 and (4.1); on a collision both sides are zero. The remaining factor depends only on degrees and has modulus one, so it is transported by a change of the first two series variables.

The finite-root-system continuation hypothesis fails for this model. In the bicharacter convention of their Section 5, let $ζ$ be a primitive sixth root of unity. The $u, v$ diagonal labels are $- ζ^{4} = ζ$ , and their mutual label is $ζ^{2}$ . The reflection exponent in either direction is therefore

$m_{uv} = m_{vu} = min {k \geq 0 : ζ^{k + 1} = 1 or ζ^{k + 2} = 1} = 4.$

Both directions satisfy Theorem 3.10’s divisibility condition: $g cd (6, 4 + 3) = 1$ divides every incident entry. Lemma 5.16(2) preserves the parameter matrix under each of these two reflections, so the reflections can be composed repeatedly. On the invariant span of the first two basis vectors their matrices and product are

$s_{u} = (- 1 0 41), s_{v} = (14 0 - 1), s_{u} s_{v} = (154 - 4 - 1) .$

The product has eigenvalues $7 \pm 43$ and hence infinite order. Thus the reflection groupoid has infinitely many morphisms, even though these two reflections preserve its parameter matrix. This model does not meet the finite arithmetic-root-system hypothesis of Theorem 5.5. Finiteness of the set of parameter matrices alone would not supply that hypothesis.

This calculation concerns the unweighted function-field model and uses classical Gauss identities and the cited reflection formulas. It supplies no number-field continuation or bound for the actual $b$ -weighted scalar. In particular it does not identify all prime-power coefficients with the original source, transport its angular/ray data, or pay the other original Poisson terms. Continuation on a smaller domain, methods for infinite groupoids and other representations are not excluded. Neither a stronger inverse-fourth estimate nor (J24), signed Robin restoration or RH follows from this model identification.

The reflected star retains pairing and continuation obligations

Under the same $q \equiv 1 (mod 12)$ and order-six $χ$ hypotheses, Sawin–Whitehead, Theorem 4.7 and matrix transform (1.11) reflect the $m$ -Dirichlet coordinate, in order $(u, v, m)$ , to

$M_{⋆} = τ_{m} (M) = 001001110 (mod 6) .$

For $uv$ squarefree, Proposition 4.1 gives $a_{M_{⋆}} (u, v, m) = (u / m)_{χ} (v / m)_{χ}$ for every monic $m$ , including nonsquarefree $m$ , with collision zeros retained. This is a same-sign character pairing. By contrast, Example 5.12’s (5.15) uses a character and its conjugate for an absolute second moment. Theorem 4.7 acts on degree-residue vectors with a scattering matrix; a functional-equation conversion must retain its Gauss/root-number factors and degree shifts and costs. The original $b$ weights have not become canonical MDS coefficients. The separate central-moment route is already assessed.

Popa–Walsh, A decomposition of Weyl group multiple Dirichlet series for symmetrizable Kac-Moody root systems, arXiv:2607.11834v2 provide genuine infinite-type continuation for a fixed symmetrizable Cartan matrix and its Chinta–Gunnells action, with distinct hypotheses. Proposition 3.6 continues the corrected average $D_{re} Δ_{im} Z_{ξ}$ holomorphically to the complexified Tits-cone interior, for a fixed $ξ \in P^{+}$ . Theorem 4.12 requires Chinta–Gunnells invariance and Hypothesis $H (R)$ on $Ω (R)$ , with $R \geq lo g_{q} r$ : $D_{re} Z$ must be holomorphic there and $Z$ must have the prescribed absolutely convergent Taylor expansion. Its decomposition converges normally in the stated polydisk. For their constructed global function-field WMDS, with $q \equiv 1 (mod 2 n)$ , Theorem 5.4 gives normal convergence of $D_{re} Z$ on the explicit Weyl-invariant $X (R_{q, r})_{C}$ , $R_{q, r} = max (1, lo g_{q} r)$ . This supplies meromorphic continuation on that subdomain, rather than unrestricted full-cone continuation or estimates for arbitrary insertions.

In the displayed star basis, the bicharacter labels are $q_{ii} = - 1$ and $q_{u m} q_{m u} = q_{v m} q_{m v} = ζ_{6}$ , a primitive sixth root. The ordinary diagonal Cartan-type relation $q_{ij} q_{ji} = q_{ii}^{a_{ij}}$ cannot hold on either nontrivial edge: no integer power of $- 1$ equals $ζ_{6}$ . This excludes directly importing that conventional Cartan-type realization in this basis; it excludes neither alternative constructions nor other continuations.

This is a repo-derived applicability check using the cited formulas. Identification of the actual coefficients, prime-power and angular/ray data, and a uniform bound with the original $b$ weights over the full natural dual length remain unproved bridges. No analytic saving, natural-boundary theorem, originality claim or RH conclusion follows.

One-factor completion retains a joint coefficient array

This is a repo-derived, source-conditional applicability and cost check for (J4)–(J6), reusing finite Fourier/Gauss identities. Fix an original ray/unit component and its original representative of each ideal, extending coefficients by zero to $O = Z [ω]$ . Put $δ = - 3$ , $e_{K} (z) = exp (2 πi Tr_{K / Q} z)$ ; the codifferent is $δ^{- 1} O$ , so $e_{K} (r x / (δ f))$ pairs $O / (f)$ perfectly. After the finite reciprocity correction, extract from one actual $u$ factor its primitive finite character $χ_{m}$ , of conductor $f = (f)$ and $Q = N f$ . At fixed $m, v$ , write that factor as $S_{m, v} = \sum_{u} a_{v} (u) χ_{m} (u)$ , $H_{m} = \sum_{v} c_{m} (v) S_{m, v}$ , with the original outer weights $c_{m} (v)$ . Set, for $x, r mod f$ ,

$A_{m, v} (x) = u \equiv x (f) \sum a_{v} (u), B_{m, v} (r) = Q^{- 1/2} x mod f \sum A_{m, v} (x) e_{K} (- r x / (δ f)) .$

Fourier inversion and the primitive Gauss identity give exactly

$S_{m, v} = γ_{m} r mod f \sum B_{m, v} (r) \overline{χ_{m} (r)}, γ_{m} = Q^{- 1/2} x mod f \sum χ_{m} (x) e_{K} (x / (δ f)), ∣ γ_{m} ∣ = 1, r \sum ∣ B_{m, v} (r) ∣^{2} = x \sum ∣ A_{m, v} (x) ∣^{2} .$

Gao–Zhao v2, §2.2 (2.3) and Lemma 2.3 give the squarefree Gauss magnitude, coprime shifts and local collision zeros, with $e (z) = e_{K} (z / δ)$ . The primitive identity vanishes at nonunits of $f$ . For imprimitive original constituents use this primitive conductor and retain every extra zero mask in $a_{v}$ ; unit Gauss normalization at the larger original modulus is not asserted. Injective periodization of the support $q_{u} ≪ N_{u}$ gives $∥ B_{m, v} ∥_{2} = ∥ a_{v} ∥_{2}$ ; $N_{u} = o (Q)$ suffices by the diameter bound below. But $a_{v}$ retains $b (e_{1} u)$ , $γ_{1} (u)$ , $θ_{e} (u)$ , $κ_{u} (v)$ , the original profiles, angular/ray data and all zeros. Thus $B$ depends on both $m$ and $v$ ; no common coefficient vector or short support follows. For a nontrivial primitive ray character and the smooth compactly supported radial majorant specified in de Faveri v1, Lemma 3.1, the ideal sum has prefactor $N / Q$ and rapidly decaying dual length $Q / N$ . That smooth-coefficient statement is not applied to this $a_{v}$ .

For $e_{1} = e_{2} = 1$ , $N_{u} = N_{v} = D = U$ and $Q ≍ U$ , rowwise Cauchy over all $Q$ transformed coordinates gives the specified certificate

$∣ H_{m} / D ∣^{2} ≪ C_{b}^{4} U^{ε} Q N_{u} N_{v}^{2} / D^{2} ≍ C_{b}^{4} U^{3/2 + ε} .$

Here $H_{m} = H_{1, m}^{1, 1}$ is the unnormalized (J6) scalar. Its elementary physical-coordinate certificate instead uses $N_{u}^{2} N_{v}^{2} / D^{2}$ ; completion followed by this Cauchy costs $Q / N_{u} ≍ U^{1/2}$ . The already owned (J6) certificate is $C_{b}^{4} U^{2/3 + ε}$ . These are upper certificates for the same normalized scalar, not lower bounds or full-(J24) exponents.

Moving collision zeros test the short-dual premise. At $N_{u} = N$ , $Q ≍ N^{2}$ , take any fixed finite rational centers $Ξ \subset K / O$ . Choose a fixed ideal $H$ with $ξ H \subset O$ for every $ξ \in Ξ$ ; this includes packets from all original fixed bad moduli. Suppose $E = ∥ a_{v} ∥_{2}^{2} > 0$ and a good prime $p ∣ v$ has $P = N p$ and $(p, H) = 1$ . The actual $κ_{u} (v)$ makes $a_{v} (u) = 0$ whenever $p ∣ u$ , including the zero extension outside support. Choose $P$ representatives $h \in H$ modulo $p$ with $N h ≪_{H} P$ , using a fundamental parallelogram of $Hp$ . Require reduction modulo $f$ injective on the support and all its translates by these $h$ . For $q_{u} ≪ N$ , $P = o (N)$ and $Q ≍ N^{2}$ , their squared diameter is $O_{H} (N + P) < Q$ , which verifies this requirement: a nonzero element of $(f)$ has norm at least $Q$ . For each supported $u$ , some $h$ has $u + h \equiv 0 (mod p)$ ; no alias can fill that zero. With $T_{h} A (x) = A (x + h)$ , consequently $\sum_{h} ∥ T_{h} A - A ∥_{2}^{2} \geq E$ .

Let $E_{tail} (Ξ, L)$ be the $B$ energy outside the union of bands $dist (r / f, ξ + O)^{2} \leq L / N$ ; equivalently $∣ r - f ξ ∣^{2} \leq L Q / N$ modulo $f O$ , the ordinary dual norm scale. On these bands $∣ e_{K} (r h / (δ f)) - 1 ∣^{2} ≪_{H} P L / N$ , since $ξ$ annihilates $H$ ; elsewhere it is at most $4$ . Parseval, summed over the $P$ translations, therefore gives

$E \leq C_{Ξ} P^{2} (L / N) E + 4 P E_{tail} (Ξ, L), E_{tail} (Ξ, L) \geq \frac{1 - C _{Ξ} P ^{2} L / N}{4 P} E .$

An actual nonzero instance. Use $b = λ_{II, b}$ , $A_{b} = N^{1/6}$ , $U = N^{2}$ , and a fixed original $V \neq = 0$ . Use the (J5) factor with its original two $V$ profiles and fixed Mellin heights. In a compact interval where $∣ V ∣ > 0$ , choose squarefree $v = pqr$ with $P ≍ lo g N \leq A_{b}$ and $q_{q}, q_{r} ≍ N / P > 2 A_{b}$ . The exact C25/C37–C40 coefficient has only $1, p$ in its $W$ -weighted divisor sum, giving $K_{A_{b}} (v) = 1 - 2 = - 1$ and $b (v) = μ (v) + K_{A_{b}} (v) = - 2$ . Choose $u$ as a distinct rough semiprime of norm in the same nonzero $V$ interval times $N$ , with both primes $≍ N > 4 A_{b}$ ; (C33) gives $b (u) = 2$ . The existing fixed-ray prime-pair counting input supplies these choices: fix prime classes with the prescribed product classes, use disjoint fixed relative norm intervals at the displayed scales, and delete the finitely many collisions. Choose a good prime dual $m$ of norm $≍ N^{2}$ in an allowed class; its primitive conductor has $Q ≍ N^{2}$ , and it meets neither $u$ nor $v$ . Original unit representatives and finite twists are retained, $κ_{u} (v) \neq = 0$ , and all remaining (J5) phases have unit modulus. The specified nonzero $V$ intervals and allowed classes therefore give $a_{v} (u) \neq = 0$ and $E > 0$ for this actual factor. Taking $L = N^{1/2}$ (eventually $2 \leq L \leq N /10$ ) gives $E_{tail} / E \geq 1/ (8 P) ≫ 1/ lo g N$ . This refutes uniform $E_{tail} ≪_{A} L^{- A} E$ for these actual Type-II factors, even after any fixed finite rational packets. It is no lower bound for $H_{m}$ , a row moment or (J24): cross terms can cancel. Growing or $m$ -dependent packets and joint-array estimates remain possible. Completion leaves the weighted $\sum_{v} c_{m} (v) B_{m, v} (r)$ contraction against $\overline{χ_{m} (r)}$ joint in $m, v, r$ . The unpaid quantitative target remains exactly (J24), with each eligible factor replaced by this array, at full natural lengths and with every original component and required profile derivative. No new harmonic-analysis theorem, analytic saving, originality, RH proof or Lean certification is claimed.

A common-gcd region with all original row expansions

The same cubic interface and joint sextic estimate also pay an additional part of the original small- $P$ region. The result here remains an equal-ray, total-squarefree-column statement. For $h_{b}$ in (J4), using the complete original physical row kernel, define

$E_{τ, H}^{sf} (b) = a, c sf, q_{a}, q_{c} ≍ U t (a) = t (c) = τ, q_{(a, c)} ≍ H \sum h_{b} (a) \overline{h_{b} (c)} K_{U, Ω} (a, c) . (J7)$

For $1 ≪ H ≪ U^{1/4}$ , the source-conditional bound is

$∣ E_{τ, H}^{sf} (b) ∣ ≪ C_{b}^{4} U^{ε} [U + U^{7/6} H^{- 1} + U^{10/9} H^{- 5/18} + U^{7/6} H^{- 1/2}] . (J8)$

Keep the common support and row indicators. Write $a = h x, c = h y$ . The ideals $h, x, y$ are squarefree and pairwise coprime, $q_{x}, q_{y} ≍ B = U / H$ , and $t (x) = t (y) = τ / t (h)$ . The primitive ratio conductor is $x y$ , and its zero-only radical is $rad (h S)$ . Expand the sixth-power-free physical-row indicator with $ℓ$ , and its zero-only primes with $r ∣ rad (h S)$ , exactly as in the source Poisson argument. Put $κ = q_{ℓ}^{6} q_{r}$ . Omit empty original row sums with $κ > c_{Ω} U$ before applying Poisson; hence $q_{ℓ} ≪ U^{1/6}$ . All original restrictions, including $(ℓ, h x y S) = 1$ , remain in force.

Poisson contributes $H / κ$ and natural full-dual length $κ U / H^{2}$ . Detect $(x, y) = 1$ by a common divisor $e$ . At $q_{e} ≍ E$ , the normalized column-vector norm product is $≪ C_{b}^{4} U^{ε} / (H E)$ ; $O (E)$ choices of $e$ cancel the $E^{- 1}$ . Summing $O (H)$ choices of $h$ leaves $H / κ$ . All $r$ and finite-class phases remain column factors or collision masks accepted by the positive-norm estimates.

For the scalar low-core bound allocate $h = h_{1} h_{2}$ , $e = e_{1} e_{2}$ between the two original factors. Their remaining lengths obey

$D^{'} = \frac{D}{q _{h_{1}} q _{e_{1}}}, N^{'} = \frac{D}{q _{h_{2}} q _{e_{2}}}, D^{'} N^{'} = \frac{U}{H E}, D^{'}, N^{'} ≪ D .$

The argument of (J5)–(J6), with coefficients $b (h_{1} e_{1} u)$ and $b (h_{2} e_{2} v)$ , gives for each fixed $h$ and full dual ideal, including the Poisson and coefficient normalization,

$\frac{C _{b}^{4} U ^{ε}}{κ} (U^{1/2} + U^{2/3} H^{- 2/3}) ≪ \frac{C _{b}^{4} U ^{2/3 + ε} H ^{- 2/3}}{κ} .$

The last comparison uses $H ≪ U^{1/4}$ ; it is not assumed for larger $H$ . Arbitrary divisor-bounded $b$ preserves these separate vectors.

Retain four high-core terms. First take the dual ideal good and write it as $n v^{6}$ , with $n$ sixth-power-free and no coprimality condition between $n, v$ . For a good-dual dyad $M$ and $q_{n} > L$ , the same sixth-order sieve gives

$\frac{C _{b}^{4} H}{κ} U^{ε} [M + B (M / L)^{1/6} + M^{5/6} B^{1/3} + M^{1/3} B^{5/6}] . (J9)$

The complementary low-core count costs $C_{b}^{4} κ^{- 1} U^{2/3 + ε} H^{1/3} M^{1/6} L^{5/6}$ . For $M \geq B^{1/3}$ , choose $L = B^{1/3}$ to balance it against the second term in (J9). At $M ≪ κ U / H^{2}$ , the four costs are

$C_{b}^{4} U^{ε} [\frac{U}{H}, κ^{- 5/6} U^{10/9} H^{- 5/18}, κ^{- 1/6} U^{7/6} H^{- 1}, κ^{- 2/3} U^{7/6} H^{- 1/2}] .$

Unlike the $ℓ = r = 1$ case, the enlarged dual range can exceed $B$ ; none of these four terms is discarded. Summing $q_{ℓ} ≪ U^{1/6}$ costs respectively $U^{1/6}$ , a constant, $lo g U$ , and a constant. The sums over $r ∣ rad (h S)$ cost $U^{ε}$ ; their powers are respectively $r^{0}, r^{- 5/6}, r^{- 1/6}, r^{- 2/3}$ . For $M < B^{1/3}$ , the scalar cubic certificate and $O (H)$ choices of $h$ instead give $C_{b}^{4} U^{1 + ε} / κ$ , summably.

A general dual ideal is $s m_{0}$ , with $s$ supported on the fixed bad primes $S$ and $m_{0}$ good. Its natural good-dual length is $M ≪ κ U / (H^{2} q_{s})$ . The four endpoint costs gain $q_{s}^{- 1}, q_{s}^{- 1/6}, q_{s}^{- 5/6}, q_{s}^{- 1/3}$ , respectively; all four $S$ -supported sums converge. For the small- $M$ certificate, rapid-decay truncation leaves a polylogarithmic number of $s$ . The original radial transform and its logarithmic derivatives decay arbitrarily fast beyond the natural length, paying all further dyads. No bad-dual component or original auxiliary expansion is deleted. These sums prove (J8).

In particular,

$U^{1/6} \leq H \leq U^{1/4} ⟹ ∣ E_{τ, H}^{sf} (b) ∣ ≪ C_{b}^{4} U^{13/12 + ε} . (J10)$

For total-squarefree columns, $P = H$ and $q_{f} ≍ U^{2} / H^{2}$ . Thus (J10) pays the additional same-class region $U^{1/6} \leq P < U^{1/4}$ , with conductor scales between $U^{3/2}$ and $U^{5/3}$ . It applies to the original Möbius factor and directly to the actual Type-II and rough factors above. It does not pay nonsquarefree columns, smaller $H$ , or the original signed cross-class contraction. A finite positive class partition is not an identification with that signed contraction. The complete inverse-fourth gain, signed Robin restoration and RH remain unproved. All Gauss, Poisson and sieve inputs are reused; the new region is a source-conditional application, without priority or Lean certification.

Extending the paid region to cube-free low-repeat columns

Fix $T \geq 1$ and write actual cube-free columns uniquely as $a = t_{a}^{2} s_{a}$ , $c = t_{c}^{2} s_{c}$ , with squarefree, coprime factors within each column. For $h_{b}$ from (J4), retain the complete original kernel and define

$E_{τ, \leq T} (P_{0}; b) = q_{a}, q_{c} ≍ U, t (a) = t (c) = τ r_{2} (a) \neq = r_{2} (c), q_{t_{a}}, q_{t_{c}} \leq T P_{0} \leq P (a, c) < U^{1/4} \sum h_{b} (a) \overline{h_{b} (c)} K_{U, Ω} (a, c) . (J11)$

Provided $P_{0} / T^{2} ≫ 1$ , the fixed-core application below gives

$∣ E_{τ, \leq T} (P_{0}; b) ∣ ≪ C_{b}^{4} U^{ε} [U T^{2} + U^{7/6} T^{4} P_{0}^{- 1} + U^{10/9} T^{23/9} P_{0}^{- 5/18} + U^{7/6} T^{3} P_{0}^{- 1/2}] . (J12)$

Freeze the repeated support before applying the existing CRT. Put $J = t_{a} t_{c}$ , $C = (t_{a}, t_{c})$ , and record the single-occupancy primes from the opposite repeated support by $d_{a} ∣ t_{c} / C$ , $d_{c} ∣ t_{a} / C$ . Exactly,

$a = A gx, c = B g y, A = t_{a}^{2} d_{a}, B = t_{c}^{2} d_{c},$

where $g, x, y$ are squarefree, pairwise coprime and prime to $J$ . In particular,

$(s_{a}, s_{c}) = g, P (a, c) = q_{t_{a}} q_{t_{c}} q_{g}, (a, c) = (A, B) g .$

Thus the sharp $P$ cut is a scalar restriction after fixing these cores and $g$ ; it does not mask the remaining bulk rectangle. At $q_{g} ≍ G$ , $G ≫ P_{0} / T^{2}$ and $G < U^{1/4}$ . The fixed primitive conductor and full conductor are

$f_{0} = rad (\frac{A}{( A , B )} \frac{B}{( A , B )}), F_{0} = q_{f_{0}}, f = f_{0} x y .$

The support of $f_{0}$ lies in the symmetric difference of $t_{a}, t_{c}$ ; $F_{0} \leq q_{J}$ and $F_{0} / (q_{A} q_{B}) \leq 1$ . The zero-only radical is $R = rad (g A BS) / f_{0}$ . Reuse the all-column finite CRT formula, which already treats reduced valuations $1, 2, - 1, - 2$ . It supplies $Γ (ρ) γ_{1} (x) γ_{- 1} (y)$ , separate fixed-core twists in $x, y$ and a reciprocity phase fixed on bulk classes. The primitive $ρ$ has conductor $f_{0}$ . The remaining $\overline{ρ (m)}$ , including its zeros, is a bounded row multiplier. For $m = n v^{6}$ it becomes $\overline{ρ (n)}$ with the $v$ -collision mask. These are accepted by the positive-norm estimates; no moving-conductor contour estimate or uniformly fixed bad-prime assumption is introduced.

Set $X = U / (q_{A} G)$ , $Y = U / (q_{B} G)$ and $B_{0} = U / G$ , so $X, Y \leq B_{0}$ . For the original $κ = q_{ℓ}^{6} q_{r}$ expansion, Poisson contributes $U / (κ F_{0} X Y)$ and natural dual length

$M ≪ \frac{κ F _{0} X Y}{U} = \frac{κ U F _{0}}{q _{A} q _{B} G ^{2}} \leq \frac{κ U}{G ^{2}} .$

After detecting $(x, y) = 1$ by $e$ , $q_{e} ≍ E$ , the normalized vector-norm product is at most $C_{b}^{4} U^{ε} X Y / (U E)$ . Multiplying the Poisson factor gives at most $C_{b}^{4} U^{ε} / (κ E)$ . The $O (E)$ choices of $e$ and $O (G)$ choices of $g$ therefore give exactly the high-core envelope (J9), with $H = G$ and $B = B_{0}$ .

For the scalar cubic bound, open the two actual factors with

$d_{1} = t_{a} d_{a, 1} g_{1} e_{1} u, d_{2} = t_{a} d_{a, 2} g_{2} e_{2} v,$

where the subscripts allocate $d_{a}, g, e$ between them. The separate vectors retain $b (t_{a} d_{a, 1} g_{1} e_{1} u)$ and $b (t_{a} d_{a, 2} g_{2} e_{2} v)$ ; no multiplicativity or Möbius cancellation is used. Their remaining length product is $X / E$ and each length is at most $D$ . The calculation (J5)–(J6), summed over allocations and $e$ , bounds the unnormalized squared scalar by $C_{b}^{4} U^{ε} (X D + X^{5/3})$ , and likewise for $Y$ . Restoring both normalizations and Poisson gives at most

$\frac{C _{b}^{4} U ^{ε}}{κ F _{0}} (D + X^{2/3}) (D + Y^{2/3}) \leq \frac{C _{b}^{4} U ^{ε}}{κ} (U^{1/2} + U^{2/3} G^{- 2/3}) .$

The second term suffices because $G < U^{1/4}$ . The same low/high-core split as in (J8)–(J9), retaining all four terms and all original $ℓ, r$ and bad-dual sums, now gives for each fixed repeated-core/occupancy pattern

$C_{b}^{4} U^{ε} [U + U^{7/6} G^{- 1} + U^{10/9} G^{- 5/18} + U^{7/6} G^{- 1/2}] .$

The enlarged $R$ costs only a divisor loss; its primes are never deleted from the original zeros. There are $O (T^{2} U^{ε})$ repeated-core pairs and occupancy patterns. Substitute $G ≫ P_{0} / T^{2}$ to obtain (J12). Also $q_{A}, q_{B} \leq T^{3}$ , whence $X, Y ≫ U^{3/4} T^{- 3}$ . For the following parameter range, the bulk cannot have $x = y = 1$ , and actual ratio order six is retained.

In particular, let

$T = U^{θ}, P_{0} = U^{1/6} T^{6}, 0 < θ < 1/72.$

The four terms of (J12) become $U T^{2}$ , $U T^{- 2}$ , $U^{115/108} T^{8/9}$ and $U^{13/12}$ . All are at most $U^{13/12}$ , so

$∣ E_{τ, \leq T} (P_{0}; b) ∣ ≪ C_{b}^{4} U^{13/12 + ε}, U^{1/6 + 6 θ} \leq P < U^{1/4} . (J13)$

This extends the total-squarefree result (J10) to all cube-free low-repeat columns in the stated equal-ray region. It applies directly to the original Möbius, actual Type-II and rough factors of (J4). A strong whole large-repeat tail for the latter two factors is not supplied by this argument.

Truncate before the positive class partition

For the original Möbius source, the existing whole-tail estimate can be combined with (J13) without estimating signed cross-class correlations. First set $g = g_{D} = g_{\leq T} + g_{> T}$ , and only then put $c_{τ, T} = g_{\leq T} 1_{t (a) = τ}$ . For $J_{0} = ∣ T ∣$ and the original physical-row operator $A$ , finite Cauchy and Hilbert-space triangle give

$T (g_{\leq T}) \leq J_{0} τ \sum T (c_{τ, T}), T (g) \leq T (g_{\leq T}) + T (g_{> T}) . (J14)$

No class filter is applied to the old tail bound. The coefficients, physical rows and zeros are unchanged. Apply the cubic-block supplier and (C18) afresh to each $c_{τ, T}$ through its coefficient envelope. Together with (J13), they pay the entire $P \geq P_{0}$ part at $U^{13/12 + ε}$ and the cubic part at $U^{1 + ε}$ . Thus the only additional sufficient input is the aggregate real upper bound

$R_{T} (P_{0}) := Re τ \sum E_{6, P < P_{0}} (c_{τ, T}) ≪ U^{B_{*} - σ + ε}, σ > 0. (J15)$

The sum is the original ordered unequal-parity contraction with both coefficients replaced by $c_{τ, T}$ . Separate absolute bounds for every class are stronger than this sufficient target. Assuming (J15), the owned bound $T (g_{> T}) ≪ U^{B_{*} - θ /25 + ε}$ and (J14) yield

$T (g_{D}) ≪ U^{m a x {B_{*} - θ /25, 13/12, B_{*} - σ} + ε} . (J16)$

Every fixed $0 < ν < min {θ /25, B_{*} - 13/12, σ}$ is consequently a full original inverse-fourth saving, if (J15) is supplied uniformly for the original finite presentations and required profile derivatives. Restore those presentations using their fixed positive norm costs. For example, $θ = 1/1000$ gives $P_{0} = U^{259/1500}$ and tail saving $1/25000$ ; an additional (J15) with $σ = 1/100000$ would suffice for any fixed $ν < 1/100000$ . No such residual estimate is supplied here. The $θ /50$ cost in the earlier moment-difference comparison (C15) is unnecessary for this one-sided upper bound: (J14) combines the new head estimate directly with the tail norm.

The positive partition is a sufficient upper-bound route, not an identification with or estimate of the original signed cross-class $E_{6}$ . Nor is (J15) asserted necessary or equivalent to RH. For a truncated Type-II factor, its packets no longer have the two inverse- $L$ series used in the original Möbius tail proof. The fixed-core sixth-order sieve followed by Minkowski gives

$T (h_{II, > T}) ≪ U^{ε} (U + U^{7/6} T^{- 2/3}) .$

Indeed, at fixed $t$ the squarefree column length is $U / q_{t}^{2}$ , the normalized squared coefficient norm is $≪ U^{ε} / q_{t}^{2}$ , and $χ_{t} (u)^{2}$ is a bounded row multiplier. Apply the existing sixth-order sieve and then Minkowski over $q_{t} > T$ . This particular certificate beats $B_{*}$ only for $θ > (56 - 12) /2 = 0.123724 \dots$ , incompatible with $θ < 1/72$ . The direct density bound below is stronger, but also cannot pay a cutoff this small. Neither certificate replaces the Möbius tail in (J16). The residual (J15), full inverse-fourth improvement, signed Robin restoration and RH remain open. These are source-conditional applications of existing suppliers, not Lean-certified results.

Summing every repeated core with unequal bulk norms

Retaining the core weights and the two bulk lengths separately removes the low-repeat restriction from a further paid region. For $h_{b}$ in (J4), let $E_{τ, P}^{all} (b)$ be the original equal-ray, unequal-parity contraction with $P \leq P (a, c) < 2 P$ , truncated at $P (a, c) < U^{1/4}$ . Keep every cube-free column and all original rows. The same suppliers give

$∣ E_{τ, P}^{all} (b) ∣ ≪ C_{b}^{4} U^{7/6 + ε} P^{- 5/12}, 1 \leq P < U^{1/4} . (J17)$

In particular, summing dyads pays $U^{1/5} \leq P < U^{1/4}$ at $C_{b}^{4} U^{13/12 + ε}$ , with no repeated-core cutoff.

Keep the actual factor cap. Reuse the fixed-core decomposition in (J11)–(J12), writing $t_{a} = c_{0} r_{0}$ , $t_{c} = c_{0} s_{0}$ , with $c_{0}, r_{0}, s_{0}$ pairwise coprime and squarefree. Let $C, R, S, G$ be their norm dyads and that of $g$ . Then $P ≍ C^{2} RSG$ . The occupancy choices satisfy $d_{a} ∣ s_{0}$ , $d_{c} ∣ r_{0}$ , and exactly

$(A, B) = c_{0}^{2} d_{a} d_{c}, f_{0} = r_{0} s_{0}, F_{0} = q_{f_{0}} ≍ RS .$

The actual lengths remain $X = U / (q_{A} G)$ , $Y = U / (q_{B} G)$ , and the natural dual length is $M ≪ κ U F_{0} / (q_{A} q_{B} G^{2})$ . In the scalar allocation each original factor on the $a$ side contains $t_{a}$ , so its remaining length is at most $D / q_{t_{a}}$ . The calculation behind (J6), summing the allocations and the common coprimality ideal $e$ , therefore gives for one fixed dual ideal

$e \sum ∣ H_{a, e} (m) ∣^{2} ≪ C_{b}^{4} U^{ε} X (D / q_{t_{a}} + X^{2/3}) ≪ C_{b}^{4} U^{ε} X^{5/3} .$

Indeed $D / q_{t_{a}} \leq X^{2/3}$ is equivalent, up to fixed annular constants, to $q_{t_{a}} q_{d_{a}}^{2} G^{2} \leq U^{1/2}$ ; its left side is at most $P^{2}$ . The other side is identical with $a, c$ interchanged. No multiplicativity of $b$ is required.

Bound complete positive side norms separately. Apply Cauchy to the full $(e, m)$ sum before choosing its internal core thresholds. For the $X$ side, its sixth-power-free dual core above $L$ costs

$C_{b}^{4} U^{ε} X [M + X (M / L)^{1/6} + M^{5/6} X^{1/3} + M^{1/3} X^{5/6}] .$

Below $L$ , the scalar certificate costs $C_{b}^{4} U^{ε} X^{5/3} M^{1/6} L^{5/6}$ . At $L = X^{1/3}$ these give

$e, m \sim M \sum ∣ H_{a, e} (m) ∣^{2} Q (M, X) ≪ C_{b}^{4} U^{ε} XQ (M, X), = M + M^{1/6} X^{17/18} + M^{5/6} X^{1/3} + M^{1/3} X^{5/6} . (J18)$

If $M < X^{1/3}$ , the all-row scalar bound $M X^{5/3}$ is already bounded by the second term of $XQ (M, X)$ . The $Y$ norm chooses its own threshold $Y^{1/3}$ . These choices need not agree: Cauchy uses two complete positive norms, not a common partition of the signed contraction. Common- $e$ and dual collision masks stay in the norms; fixed-core twists and $\overset{ρ}{ˉ} (m)$ remain bounded coefficient factors or row multipliers throughout.

Poisson and the two original $D^{- 1}$ coefficients now leave $κ^{- 1} F_{0}^{- 1/2} Q (M, X) Q (M, Y)$ per fixed core and $g$ . Counting their ideals in the dyads gives the positive majorant

$\frac{C _{b}^{4} U ^{ε} GC RS}{κ} Q (M, X_{0}) Q (M, Y_{0}), ⎩ ⎨ ⎧ X_{0} = U / (C^{2} R^{2} G), Y_{0} = U / (C^{2} S^{2} G), M ≪ κ U / (C^{4} RS G^{2}) . (J19)$

The occupancy ideals are omitted only from this increasing positive majorant: they decrease $X, Y, M$ , and their count costs divisors. They have not been removed from the CRT identities or original zeros. Also $q_{A} G, q_{B} G ≪ P^{2}$ , so the actual $X, Y ≫ U / P^{2} ≫ U^{1/2}$ ; the bulk principal exception $x = y = 1$ is absent.

Sum the original auxiliary rows and all core dyads. Write $C = U^{c}, R = U^{r}, S = U^{s}, G = U^{g}$ and $p = 2 c + r + s + g$ for the norm scales. Fixed annular factors and dyads are absorbed in $U^{ε}$ . In (J19) the length exponents, before the $κ$ factor, are

$x = 1 - 2 c - 2 r - g, y = 1 - 2 c - 2 s - g, m = 1 - 4 c - r - s - 2 g .$

Expand the two square roots into sixteen positive terms. The four $M$ -powers in $Q$ are $α = (1, 1/6, 5/6, 1/3)$ . A term $i, j$ has $κ$ -power $(α_{i} + α_{j}) /2 - 1$ ; summing the original $q_{ℓ} ≪ U^{1/6}$ adds at most $max (0, (α_{i} + α_{j}) /2 - 5/6)$ to its $U$ exponent. The zero-only $r$ -divisors cost $U^{ε}$ . For a dual part supported on the fixed bad primes, the decay power is $(α_{i} + α_{j}) /2 > 0$ , so every such sum converges. The original transform’s rapid decay pays dyads beyond the natural length. No original auxiliary expansion or bad-dual term is omitted.

Each resulting exponent is affine in $c, r, s, g$ . At $p = 0$ all sixteen exponents are at most $7/6$ . At $p = 1/4$ , the simplex vertices and maxima are

Nonzero core exponent	Maximum over the sixteen terms
$c = 1/8$	$11/12$
$r = 1/4$ or $s = 1/4$	$17/16$
$g = 1/4$	$25/24$

Thus every exponent on $0 \leq p \leq 1/4$ is at most $(1 - 4 p) 7/6 + 4 p (17/16) = 7/6 - 5 p /12$ . This proves (J17). This is an unequal-length and weighted-core application of the existing suppliers; no new sieve or literature-priority claim is made.

An untruncated sufficient target for the actual Type-II factor

Combine (J17) with the coefficient-envelope bound (C18) for $P \geq U^{1/4}$ . For any genuine $h_{b}$ in (J4), first take the positive finite class partition $c_{τ} = h_{b} 1_{t (a) = τ}$ . The entire $P \geq U^{1/5}$ part is paid at $C_{b}^{4} U^{13/12 + ε}$ , and its cubic block at $C_{b}^{4} U^{1 + ε}$ . Consequently the additional sufficient input

$Re τ \sum E_{6, P < U^{1/5}} (c_{τ}) ≪ U^{B_{*} - σ + ε}, σ > 0, (J20)$

would imply $\mathcal T(h_b)\ll U^{\max(13/12,B_*-\sigma)+\varepsilon} $f or f i x e d$ C_b$. The head here has no repeated-core truncation, so no Type-II tail estimate is needed. This supplies a different route from the smaller, truncated original-Möbius target (J15); neither residual is supplied.

For the actual Type-II factor $b = λ_{II, b}$ , use its exact polynomial identity with the paid Type-I part (C30) and apply the $L^{4}$ triangle inequality directly. Assuming (J20) uniformly for its original finite components and required profile derivatives, the original Möbius fourth moment then has the same exponent $max (13/12, B_{*} - σ)$ . Every fixed $0 < ν < min (B_{*} - 13/12, σ)$ would be a saving. The smaller $(B_{*} - 13/12) /4$ transfer allowance in (C31)–(C32) belongs to their moment-difference comparison; this one-sided consumer does not compare those signed differences. No bound for (J20), complete inverse-fourth improvement, signed Robin restoration or RH is proved by the paid sector alone.

A small parity core in the full natural good-dual range

A separate application pays an arithmetic part of the full natural dual range in precisely $E_{1, b} (M)$ from (J1)–(J6). Retain its good-dual selector, equal ray and $t_{a} = t_{c} = h = ℓ = r = 1$ restrictions, and select $q_{r_{2} (m)} \leq A_{0}$ . Then

$∣ E_{1, b, q_{r_{2} (m)} \leq A_{0}} (M) ∣ ≪ C_{b}^{4} U^{ε} U^{8/9} M^{1/6} A_{0}^{5/6}, 1 \leq A_{0} \leq M \leq U . (J21)$

Write uniquely $m = a b^{2} v^{6}$ , where $a = r_{2} (m)$ is squarefree, $b$ is cubefree and $a b^{2}$ is sixth-power-free. No coprimality between $a, b, v$ is imposed. For fixed $a, v$ ,

$\overline{χ_{x} (m)} = \overline{χ_{x} (a)} \overline{κ_{x} (b)} 1_{(x, v) = 1}, κ_{x} = χ_{x}^{2} .$

Thus the $b$ family is genuinely cubic. Its fixed $a$ twist and all $v$ and common- $e$ collisions remain in the separate coefficient vectors or row masks. This is a support restriction, not a global conversion of the sextic matrix to a cubic one.

At norm dyads $q_{a} ≍ A$ , $q_{v} ≍ V$ , the cubic row length is $B ≍ (M / (A V^{6}))^{1/2} \leq U$ . Reuse the cubefree-index cubic supplier $Θ_{3} (B, X) ≪ (BX)^{ε} [B + X + (BX)^{2/3}]$ . Here $X \leq U$ , so it costs at most $U^{1 + ε}$ . The J1 common- $e$ vector norms and $O (A V)$ choices of $a, v$ give $C_{b}^{4} U^{1 + ε} A V$ . Independently, J6 and ideal counting of these same duals give $C_{b}^{4} U^{2/3 + ε} M^{1/2} A^{1/2} V^{- 2}$ . Their crossover is $V_{0} = U^{- 1/9} M^{1/6} A^{- 1/6}$ . Summing the minimum over $V$ , then over $A \leq A_{0}$ , gives (J21). When $V_{0} < 1$ , the scalar branch alone is smaller than that envelope.

In particular $A_{0} \leq U^{1/30}$ pays this selected support at $U^{13/12 + ε}$ through $M ≍ U$ , beyond J1’s uniformly paid prefix $M \leq U^{3/4}$ . The remaining large parity cores, other original auxiliary expansions and bad duals are outside (J21). This restricted result does not supply (J15) or (J20). All results in this note remain source-conditional applications, without analytic Lean certification or an RH proof.

A direct repeat tail and a smaller sufficient head

The support-density argument already used in (C35) applies to a general repeat cutoff, without first fixing each repeated core. Combining it with the joint core constraints in (J19) gives a smaller sufficient head for every genuine $h_{b}$ in (J4), including the actual Type-II factor. These are applications of the same sixth-order and cubic suppliers, not new sieves or literature-priority claims.

Apply the sieve to the whole tail. Write each supported column uniquely as $a = t_{rep} (a)^{2} s$ , with $t_{rep} (a), s$ squarefree and coprime. For $1 \leq T \leq U$ , put $h_{b, > T} = h_{b} 1_{q_{t_{rep} (a)} > T}$ . The original divisor envelope and ideal counting give

$∥ h_{b, > T} ∥_{2}^{2} ≪ C_{b}^{4} U^{- 1 + ε} q_{t} > T \sum # {s : q_{s} ≪ U / q_{t}^{2}} ≪ C_{b}^{4} U^{ε} q_{t} > T \sum q_{t}^{- 2} ≪ C_{b}^{4} U^{ε} T^{- 1} .$

Every such column is cube-free, hence sixth-power-free. Directly apply the owned sixth-order supplier at lengths $U, U$ to this one row-independent coefficient vector:

$T (h_{b, > T}) ≪ C_{b}^{4} U^{7/6 + ε} T^{- 1} . (J22)$

The fixed physical presentations, their twists and the original zeros are transported exactly as in that supplier’s existing family interface. An enlargement is made only within the positive row norm. This is a fresh sieve application to a supported vector, not a contraction of an already estimated moment under a column mask. No inverse- $L$ identity or factorization of the truncated coefficient is required. The same estimate applies to the required finitely many profile derivatives within the original fixed or polylogarithmic height contract.

Set $T = U^{1/12}$ and truncate before the positive ray partition: $c_{τ, T} = h_{b} 1_{q_{t_{rep} (a)} \leq T} 1_{t (a) = τ}$ , where $t (a) = τ$ denotes the existing ray label. The whole discarded tail is now paid at $C_{b}^{4} U^{13/12 + ε}$ .

Retain both repeat caps in the complete-dual estimate. For each dyad $U^{1/6} \leq P < U^{1/4}$ , use (J18)–(J19) on these coefficients, retaining the complete original equal-ray, unequal-parity kernel. The cutoff is constant after fixing its cores; it does not change either actual convolution-factor vector. Then

$∣ E_{6, P \leq P (a, c) < m i n (2 P, U^{1/4})} (c_{τ, T}) ∣ ≪ C_{b}^{4} U^{13/12 + ε}, T = U^{1/12} . (J23)$

Here is the additional joint estimate needed beyond (J17). In its notation put $C = U^{c}, R = U^{r}, S = U^{s}, G = U^{g}$ . The two caps give $c + r \leq 1/12$ , $c + s \leq 1/12$ , while $p = 2 c + r + s + g \geq 1/6$ . Fixed annular constants enter $U^{ε}$ . For $α = (1, 1/6, 5/6, 1/3)$ and $β = (0, 17/18, 1/3, 5/6)$ , each of the sixteen positive terms has exponent

$f_{ij} = g + c + (r + s) /2 + \frac{1}{2} (α_{i} + α_{j}) (1 - 4 c - r - s - 2 g) + \frac{1}{2} β_{i} (1 - 2 c - 2 r - g) + \frac{1}{2} β_{j} (1 - 2 c - 2 s - g) + max {0, (α_{i} + α_{j}) /2 - 5/6} .$

The last term already pays the original $ℓ$ sum, exactly as in (J17). Across the sixteen terms, the largest coefficients of $c, r, s, g$ are respectively $- 14/9, - 1/12, - 1/12, - 5/18$ . Thus each exponent decreases in each coordinate. Scaling all four coordinates down by $1/ (6 p)$ preserves the two caps and can only increase the bound, so it suffices to treat $p = 1/6$ . On that face, $0 \leq r, s \leq 1/12 - c$ and $g = 1/6 - 2 c - r - s$ . Its square sections contract to the apex, giving exactly the following five vertices and maxima:

$(c, r, s, g)$	$max_{i, j} f_{ij}$
$(0, 0, 0, 1/6)$	$13/12$
$(0, 1/12, 0, 1/12)$	$77/72$
$(0, 0, 1/12, 1/12)$	$77/72$
$(0, 1/12, 1/12, 0)$	$13/12$
$(1/12, 0, 0, 0)$	$1$

Affine interpolation proves (J23). Occupancy ideals are omitted only from the increasing positive majorant, as in (J19). All original $ℓ, r$ , common- $e$ masks, fixed-core twists, collision zeros and bad-dual sums remain covered by that calculation. In particular no new principal bulk exception is introduced.

The sufficient original-coefficient input. Use (C18) for $P \geq U^{1/4}$ and (J23) below it. Apply the existing cubic-block bound to each $c_{τ, T}$ afresh through its divisor envelope. The additional sufficient estimate is now

$Re τ \sum E_{6, P < U^{1/6}} (c_{τ, T}) ≪ C_{b}^{4} U^{B_{*} - σ + ε}, T = U^{1/12}, σ > 0. (J24)$

Assuming this uniformly for the original components and required profile derivatives, finite positive partition and the whole-tail norm triangle (J14), now with (J22), imply

$T (h_{b}) ≪ C_{b}^{4} U^{m a x {13/12, B_{*} - σ} + ε} . (J25)$

For actual $b = λ_{II, b}$ , combine with (C30) by the direct $L^{4}$ triangle as in (J20). Every fixed $0 < ν < min {B_{*} - 13/12, σ}$ would then be an original Möbius fourth-moment saving. This remains conditional on (J24). It bounds the aggregate real head, without requiring separate absolute estimates for each ray or subtracting a masked moment from a full one. The small-cutoff original-Möbius route (J15) remains separate.

Two rough prime-overlap shapes remain. For the actual rough factor $b = r_{A} λ_{II, b}$ in (C33), every prime exceeds $2 A = 2 U^{1/12}$ . A nonunit repeated core therefore exceeds $T$ and is already in the paid tail; the retained coefficient is precisely the squarefree part $g_{sf}$ of (C34)–(C35). On two such columns $P = q_{g}$ , where $g$ is their squarefree gcd. If $q_{g} < U^{1/6}$ , then $g$ is either $1$ or one prime of norm $2 U^{1/12} < q_{g} < U^{1/6}$ , since a product of two supported primes exceeds $4 U^{1/6}$ . Thus in this rough specialization (J24) consists only of the coprime squarefree-column contraction and the one-common-prime contraction, with their actual coefficients, phases and zeros. Their combined real upper bound is not supplied. The nonrough Type-II part remains outside this specialization, so a saving for these rough shapes alone does not prove the full (J24), signed Robin restoration or RH.

A simultaneous cubic average in the full weighted small-parity sector

The following repo-derived application uses de Faveri, Optimal large sieve for fixed order characters, arXiv:2610.04045v1, §8, Proposition 8.1, (8.1) on the original two convolution variables and the square part of the complete dual ideal. The other inputs are the Gauss CRT and smooth calculus of the pinned primary manuscript, lem:fixed-gauss-phase, eq:sextic-sieve-pair-gauss, eq:sextic-sieve-pair-poisson, eq:sextic-sieve-transform-bounds and lem:smooth-calculus, with the already matched all-column CRT. This is a paper/source-conditional estimate, with no new sieve, literature-priority claim or Lean certification.

The sector is defined after the original Poisson expansion. Keep $c_{τ, T}$ in (J24), $T = U^{1/12}$ , and the actual $b = λ_{II, b}$ in (J4), with $∣ b ∣ \leq τ_{3}$ . More generally the argument accepts the stated $C_{b} τ_{k}$ envelope. For $1 \leq P < U^{1/6}$ and $H \geq 1$ , let $D_{P, \leq H}$ be the sum of the actual complete-dual terms of $\sum_{τ} E_{6, P \leq P (a, c) < m i n (2 P, U^{1/6})} (c_{τ, T})$ whose good dual part satisfies $q_{r_{2} (m_{good})} \leq H$ . The selector is inserted in each pair’s Poisson series before profile separation and the common- $e$ expansion. This definition retains every original finite ray/angular component, original $ℓ, r$ term, repeated core, occupancy choice, allocation, bad-prime dual part and required derivative profile. All dual dyads are retained, with the original Fourier decay. It defines neither a positive physical moment nor a nonnegative subset of $E_{6}$ . The source-conditional bound is

$∣ D_{P, \leq H} ∣ ≪ C_{b}^{4} U^{1 + ε} (H^{1/2} + P^{1/3} H^{2/3}) . (J26)$

Constants have the same fixed-data and profile-seminorm dependence as (J18)–(J19); their original fixed or polylogarithmic aggregate costs are required. In particular,

$H (P) = U^{1/8} P^{- 1/2} ⟹ ∣ D_{P, \leq H (P)} ∣ ≪ C_{b}^{4} U^{13/12 + ε} . (J27)$

Since $H (P) \geq U^{1/24}$ in this $P$ range, the same estimate, summed over $P$ , pays the entire original expanded sector with $q_{r_{2} (m_{good})} \leq U^{1/24}$ . No repeated-core cutoff beyond the one already present in $c_{τ, T}$ is added.

Actual coefficients in a fixed core. Use exactly the notation of (J18)–(J19): $t_{a} = c_{0} r_{0}$ , $t_{c} = c_{0} s_{0}$ , with pairwise coprime squarefree $c_{0}, r_{0}, s_{0}$ ; $d_{a} ∣ s_{0}$ , $d_{c} ∣ r_{0}$ ; and

$a = A gx, c = B g y, A = t_{a}^{2} d_{a}, B = t_{c}^{2} d_{c}, f_{0} = r_{0} s_{0}, F_{0} = q_{f_{0}}, X = \frac{U}{q _{A} G}, Y = \frac{U}{q _{B} G} .$

The primitive fixed character $ρ$ on $f_{0}$ has local exponents $2 - 1_{p ∣ d_{c}}$ at $p ∣ r_{0}$ , and $- 2 + 1_{p ∣ d_{a}}$ at $p ∣ s_{0}$ , in the original raw sextic symbols. These exponents are $1, 2, - 1, - 2$ , respectively; its original zeros are retained. The remaining zero-only radical is $R_{0} = rad (g A BS) / f_{0}$ . Fix the original $r ∣ R_{0}$ , the allowed $ℓ$ and $κ = q_{ℓ}^{6} q_{r}$ . The row substitution supplies $ρ (r) χ_{x} (r) \overline{χ_{y} (r)}$ and the original $ℓ$ -coprimality masks. Its Möbius signs remain outside the two side norms. The complete CRT root is the cited one, including $Γ (ρ)$ , the fixed bulk-class reciprocity factor and $γ_{- 1} (y) = χ_{y} (- 1) \overline{γ_{1} (y)}$ .

After detecting $(x, y) = 1$ , fix the squarefree $e$ and allocate $d_{a} = d_{a, 1} d_{a, 2}$ , $g = g_{1} g_{2}$ , $e = e_{1} e_{2}$ . On the $a$ side put

$k_{i} = t_{a} d_{a, i} g_{i} e_{i}, x = e uv, d_{1} = k_{1} u, d_{2} = k_{2} v, q_{u} ≍ L_{1}, q_{v} ≍ L_{2} .$

Thus $L_{1} L_{2} ≍ X / q_{e}$ and each $L_{i} ≪ D / q_{t_{a}} \leq D$ . Every original convolution coefficient remains $b (k_{1} u) b (k_{2} v)$ ; in particular the three-slot definition of $λ_{II, b}$ is neither opened into new independent vectors nor replaced by an optimizing vector. Its envelope is used only for squared vector norms. Use $t_{c}, d_{c}$ on the other side. The original caps $q_{t_{a}}, q_{t_{c}} \leq T$ are scalar restrictions on these fixed cores and remain in the outer sum.

Here is an explicit separated presentation, with constants depending only on the fixed data of this application left outside the scalar. Let $δ_{i} (w)$ retain squarefreeness, goodness, $(w, ℓ t_{a} t_{c} g e) = 1$ and its chosen ray class; let $ζ_{i} (w)$ be the chosen factor in the original multiplicative ray/angular expansion, with its original coefficient outside the scalar. For the (J4) component itself $ζ_{i} = 1$ ; factors already included in $ξ$ are not inserted again. Partition the two variable classes far enough that $R (u, v)$ in (J5) is constant. With $θ_{e} (w) = χ_{e} (w) χ_{w} (e)$ , write

$A_{u} B_{v} = b (k_{1} u) V (q_{k_{1}} q_{u} / D) ξ (u) γ_{1} (u) θ_{e} (u) η_{1} (u) q_{u}^{i t_{1}}, = b (k_{2} v) V (q_{k_{2}} q_{v} / D) ξ (v) γ_{1} (v) θ_{e} (v) η_{2} (v) q_{v}^{i t_{2}} . (J28)$

To specify $η_{i}$ , fix a generator $s$ of the bad-prime part of $m$ , together with any original unit component. On the $a$ side its factors are exactly

$η_{i} (w) = δ_{i} (w) ζ_{i} (w) ρ (w) χ_{w} (f_{0} r) \overline{χ_{w} (s)} .$

For the $c$ side conjugate its entire scalar: replace $ρ (w) χ_{w} (f_{0} r)$ in this formula by $\overline{ρ (w)} χ_{w} (- f_{0} r)$ , retain its own original finite factors and profiles, and conjugate the resulting side in the final pairing. The $- 1$ is precisely the retained $γ_{- 1}$ sign. Factors evaluated at $e$ or the allocated fixed cores stay outside; any original collision there makes that term zero. All displayed $η_{i}$ have modulus at most the original component bound. Fixed core characters are coefficient factors, not added to the bad set of the cited theorem. Norm normalizations on compact annuli are included in the separated smooth profile; $t_{1}, t_{2}$ are its real Mellin heights. This describes each original finite component separately, with its original coefficient in the outer sum.

For this fixed component the unnormalized scalar is therefore

$H_{β z^{2}} = u, v \sum A_{u} B_{v} κ_{u} (v) \overline{χ_{u} (β z^{2}) χ_{v} (β z^{2})}, κ_{u} = χ_{u}^{2} . (J29)$

The full dual ideal is $m = s β z^{2}$ , where $m_{good} = β z^{2}$ uniquely, $β = r_{2} (m_{good})$ is squarefree, and $(β, z) > 1$ is allowed. The bounded primitive-core factor $\overline{ρ (m)}$ and the common- $e$ collision mask are still outer row multipliers. Formula (J29) is an algebraic extension of the original coprime bulk formula, used only inside its exact Möbius detection. It asserts no primitive conductor formula for the added noncoprime pairs.

Match the fixed-vector theorem, including its zeros. Multiplicativity and the zero-preserving convention for powers give, for every $u, v, β, z$ ,

$κ_{u} (v) \overline{χ_{u} (β z^{2})} = \overline{χ_{u} (β)} κ_{u} (v z^{2}) . (J30)$

A prime common to $u$ and any of $v, β, z$ makes both sides zero. The $v$ -dual zeros remain in the other factor of (J29). For fixed $β, s, e$ , allocations and separating heights, set $λ_{u} = A_{u} \overline{χ_{u} (β)}$ . This vector is squarefree in $u$ and independent of both $v$ and $z$ .

The character correspondence is exact, rather than an identification by order alone. The primary manuscript defines $κ_{u} (w) = (w / u)_{3}$ on its generators congruent to $1 (mod 3)$ . De Faveri’s Example 2.2 identifies its cubic family with $χ_{u}^{dF} (w) = (u / w)_{3}$ for these same primary generators. The primary manuscript’s precise cubic reciprocity in lem:fixed-gauss-phase gives equality on coprime inputs; both families are zero on collisions, by their respective definitions. Apply that example with its fixed exclusion of the prime above $3$ ; our original extra fixed bad primes are retained as support masks. On added positive-norm rows use the source character family itself; the equality with the raw kernel is needed only on the original good rows. Thus no growing modulus or extra reciprocity weight is inserted in Proposition 8.1. The sextic sign $R$ has already been retained in (J5), and is not an omitted cubic correction.

Cauchy in $v$ gives, before any row enlargement,

$∣ H_{β z^{2}} ∣^{2} \leq (v \sum ∣ B_{v} ∣^{2} ∣ χ_{v} (β z^{2}) ∣^{2}) v original \sum u \sum λ_{u} κ_{u} (v z^{2})^{2} .$

Bound the first factor by $C_{b}^{2} U^{ε} L_{2}$ and enlarge the $v, z$ sets only in the remaining nonnegative squared norm. Proposition 8.1 applies with $(q, a, b) = (u, v, z)$ , $∥ λ ∥_{2}^{2} ≪ C_{b}^{2} U^{ε} L_{1}$ , and gives

$z \sim Z \sum ∣ H_{β z^{2}} ∣^{2} F (A, B; Q) ≪ C_{b}^{4} (U Z)^{ε} L_{1} L_{2} F (L_{2}, Z; L_{1}), = A B + min (A, B)^{2/3} max (A, B)^{1/3} Q + min (A, B)^{1/3} max (A, B)^{2/3} Q^{2/3} . (J31)$

The outer ideals in the cited theorem are arbitrary, so $z$ need not be squarefree or coprime to $β$ . Its only squarefree requirement is the one already met by $u$ . Remark 8.2’s unworked cube-free inner extension is not used. In particular this argument never treats the moving joint coefficient array $B_{m, v}$ as a common inner vector. Common- $e$ , fixed-core and $v$ -dual row masks can be enlarged only at the positive-norm step above; they have not been deleted from the signed contraction.

Sum allocations and parity cores before restoring normalization. For $L_{1}, L_{2} ≪ D$ and every $Z \geq 1$ ,

$F (L_{2}, Z; L_{1}) ≪ D Z + D^{4/3} Z^{2/3} .$

For $Z \leq D$ its two mixed terms are at most $D^{4/3} Z^{2/3}$ ; for $Z \geq D$ they are bounded by $D^{4/3} Z^{2/3}$ and $D Z$ , respectively. Thus this comparison imposes no upper cutoff on $Z$ . In a good-dual dyad $q_{m_{good}} ≍ M$ , $Z ≍ (M / q_{β})^{1/2}$ and only $q_{β} ≪ M$ occur. Ordinary ideal counting yields

$q_{β} \leq H \sum Z ≪ M^{1/2} H^{1/2}, q_{β} \leq H \sum Z^{2/3} ≪ M^{1/3} H^{2/3},$

where empty $Z < 1$ boxes are omitted; the bounds also hold when $H > M$ . Cauchy over the actual allocations costs only divisors. Since $L_{1} L_{2} ≍ X / q_{e}$ and $\sum_{q_{e} ≪ X} τ_{k} (e) / q_{e} ≪ U^{ε}$ , summing $e$ gives

$e, m_{good} \sim M q_{r_{2} (m_{good})} \leq H \sum ∣ H_{a, e} (m) ∣^{2} ≪ C_{b}^{4} (U M)^{ε} X [D M^{1/2} H^{1/2} + D^{4/3} M^{1/3} H^{2/3}] . (J32)$

The $(U M)^{ε}$ notation retains the source theorem’s length loss on the unrestricted dual dyads. The original nonempty-row cutoff bounds $ℓ, r$ and the core scales polynomially in $U$ , so $N_{s} ≪ U^{O (1)}$ below. Rapid decay absorbs the remaining $(1 + M / N_{s})^{ε}$ factor; the final loss is $U^{ε}$ after choosing the input epsilon smaller.

The $Y$ side has its own allocations and the identical bracket. Apply Cauchy between these two complete $(e, m)$ side norms with all original outer masks bounded there. Restore $D^{- 2} = U^{- 1}$ and the Poisson factor $U / (κ F_{0} X Y)$ : per fixed core the bracket in (J32) is multiplied by $1/ (κ F_{0})$ . Counting $c_{0}, r_{0}, s_{0}, g$ in their dyads gives precisely the (J19) prefactor $GC RS / κ$ .

Occupancy ideals remain in every identity and coefficient. In this positive majorant they only decrease $X, Y$ and the natural length; their number is a divisor loss. With the same $X_{0}, Y_{0}$ as (J19), write

$N_{s} = \frac{κ U}{C ^{4} RS G ^{2} q _{s}} .$

This is an upper bound for the actual natural good-dual length, not a truncation. For $α = 1/2, 1/3$ and fixed $A_{1} > 1$ , $\sum_{M \geq 1 dyadic} M^{α} (1 + M / N_{s})^{- A_{1}} ≪ N_{s}^{α}$ , also when $N_{s} < 1$ . On every dual dyad, the original transform and each required logarithmic derivative contribute arbitrary decay in $M / N_{s}$ beyond this scale. Summing all dyads in (J32), including the case $N_{s} < 1$ , replaces $M^{1/2}, M^{1/3}$ by $N_{s}^{1/2}, N_{s}^{1/3}$ in the positive bound. The core envelope is consequently

$C_{b}^{4} U^{1 + ε} [C^{- 1} κ^{- 1/2} q_{s}^{- 1/2} H^{1/2} + G^{1/3} C^{- 1/3} (RS)^{1/6} κ^{- 2/3} q_{s}^{- 1/3} H^{2/3}] . (J33)$

The $ℓ$ sums are $\sum q_{ℓ}^{- 3}$ and $\sum q_{ℓ}^{- 4}$ , both convergent. The original zero-only $r$ sums have powers $q_{r}^{- 1/2}$ and $q_{r}^{- 2/3}$ and cost $U^{ε}$ over divisors of $R_{0}$ . The two fixed- $S$ dual sums converge by the displayed $q_{s}^{- 1/2}$ and $q_{s}^{- 1/3}$ factors. All core dyads and finite partitions cost the original $U^{ε}$ allowance. No principal bulk exception appears: $X, Y ≫ U / P^{2} ≫ U^{1/2}$ , as in (J19).

For completeness, profile separation is made before applying (J31). On fixed annuli its transform factor has argument a fixed monomial in the normalized $u, v, z$ norms and the ratio $M / N_{s}$ . lem:smooth-calculus supplies a common coefficient measure for these rows. Below the natural scale all required logarithmic derivatives are uniformly bounded by smoothness at zero; above it they have arbitrary rapid decay. Differentiating any of the originally required profiles changes its finite seminorm and logarithmic factors, not $b$ , its squarefree support, or the independence of $λ_{u}$ . The arbitrary-vector bound (J31) is uniform in the separating norm phases and bounded multiplicative angular factors. Minkowski therefore restores every original finite component and required derivative with its original fixed or polylogarithmic total cost. No polynomial height budget is renewed for each component.

Finally $P ≍ C^{2} RSG$ , so $C^{- 1} \leq 1$ and $G^{1/3} C^{- 1/3} (RS)^{1/6} ≪ P^{1/3}$ . Summing (J33) proves (J26). At $H = H (P)$ its two terms are respectively $U^{17/16 + ε} P^{- 1/4}$ and $U^{13/12 + ε}$ , which proves (J27).

The complementary target keeps the original signs. In the restricted $P = 1$ , good-dual, $h = ℓ = r = 1$ consumer, (J32) and the original normalization give

$∣ E_{1, b, q_{r_{2} (m)} \leq H} (M) ∣ ≪ C_{b}^{4} U^{ε} [U^{1/2} M^{1/2} H^{1/2} + U^{2/3} M^{1/3} H^{2/3}] .$

For $1 \leq M \leq U$ and $H \geq 1$ , the first term is bounded by the second. At $M = U$ this pays $H \leq U^{1/8}$ , beyond the $U^{1/30}$ range already paid by (J21). The full-parameter assertion is (J27), not an extrapolation from this restricted example. Let $D_{>}$ denote the remaining actual expanded sum in (J24), with $q_{r_{2} (m_{good})} > H (P)$ in its $P$ dyad and with all the same original parameters and weights. Exact Poisson expansion and the complementary selectors give

$Re τ \sum E_{6, P < U^{1/6}} (c_{τ, T}) = Re D_{>} + Re P < U^{1/6} \sum D_{P, \leq H (P)} .$

The second summand is now bounded by $C_{b}^{4} U^{13/12 + ε}$ . A bound $\operatorname{Re}\mathfrak D_{>} \ll C_b^4U^{B_*-\sigma+\varepsilon} $, f orso m e$ \sigma>0$ uniformly under the same original component/derivative contract, would supply (J24) with the resulting exponent $max {13/12, B_{*} - σ} < B_{*}$ . Such a bound is not supplied. The complement includes the natural $M ≍ U$ squarefree good duals; its aggregate real part still requires joint cancellation. This application proves no complete J24 estimate, inverse-fourth saving, signed Robin restoration or RH conclusion.

Retaining the shorter modulus and both actual side lengths

The same sector $D_{P, \leq H}$ defined before (J26) admits an estimate with no $P^{1/3}$ loss. Reuse de Faveri v1 Proposition 8.1, (8.1), and Example 2.2 in exactly the character correspondence of (J28)–(J31), together with the owned Gauss, Poisson and smooth-calculus applications through (J33). For the original $c_{τ, T}$ of (J24), $T = U^{1/12}$ , and its actual $b = λ_{II, b}$ , or the stated $C_{b} τ_{k}$ envelope, the source-conditional strengthening is

$∣ D_{P, \leq H} ∣ ≪ C_{b}^{4} U^{1 + ε} (H^{1/2} + H^{2/3}), 1 \leq P < U^{1/6}, H \geq 1. (J34)$

This is the identical complete weighted sector: the parity selector is still inserted in each original Poisson series, before separation and common- $e$ detection. In particular all repeated cores, occupancies, common- $e$ signs, original $ℓ, r$ expansions, bad duals, finite components and derivative profiles in that definition remain present. The constants and aggregate profile/height costs are those of (J26)–(J33). Uniformly in the same $P$ range,

$H = U^{1/8} ⟹ ∣ D_{P, \leq U^{1/8}} ∣ ≪ C_{b}^{4} U^{13/12 + ε} . (J35)$

This fixed cutoff also pays the sum over all the $P$ dyads with their original signs. It enlarges the tapered paid sector of (J27), whose cutoff is $U^{1/8} P^{- 1/2}$ ; it does not change that earlier estimate.

Orient each actual allocation before applying the supplier. In (J28)–(J29), fix the cores, the original finite component and heights, $s$ , $e$ and its allocations, and the squarefree parity ideal $β$ . Let

$Q = min (L_{1}, L_{2}), L = max (L_{1}, L_{2}), K = Q L ≍ X / q_{e}, W_{a} = D / q_{t_{a}} .$

On nonempty annuli $Q, L \geq 1$ up to fixed annular constants, which are absorbed throughout, and $L ≪ W_{a}$ . Cubic reciprocity in the pinned lem:fixed-gauss-phase gives $κ_{u} (v) = κ_{v} (u)$ on coprime primary inputs. Both sides are zero on collisions, so this equality also holds on the algebraically extended scalar in (J29). If $v$ is the shorter variable, interchange the two entire vectors and their separate masks and heights. The scalar and its fixed outside phases are unchanged. In particular this operation does not replace sextic reciprocity by a sign-free assertion: its original $R$ factor and the negative-root side’s $- 1$ factor remain exactly as in (J28).

The shorter variable is now the squarefree modulus $q$ in Proposition 8.1. Its actual vector is the corresponding vector of (J28) multiplied by $\overline{χ_{q} (β)}$ , as in (J30), and is independent of the other original convolution variable and $z$ . The remaining convolution coefficient, including its dual zeros, is removed only by the same Cauchy inequality used for (J31). Thus (J31) applies with $(A, B; Q) = (L, Z; Q)$ ; it is not an application to a moving vector or a new optimized vector. Example 2.2 and the precise primary cubic reciprocity already checked there identify the source character in either orientation. Its exclusion of the prime above $3$ is fixed; the other original excluded primes and core characters stay coefficient masks or factors, not growing bad data for the supplier. The outer $z$ remains arbitrary, and $(β, z) > 1$ is allowed. No cube-free inner extension from Remark 8.2 is needed.

For the function $F$ in (J31), retaining the actual product gives

$F (L, Z; Q) ≪ L Z + K^{2/3} Z^{2/3}, 1 \leq Q \leq L, Z \geq 1. (J36)$

Indeed, if $Z \leq L$ , the two mixed terms are $Z^{2/3} L^{1/3} Q$ and $Z^{1/3} L^{2/3} Q^{2/3}$ . Their ratios to $K^{2/3} Z^{2/3}$ are respectively $(Q / L)^{1/3}$ and $Z^{- 1/3}$ , both at most one. If $Z \geq L$ , the ratios are $(Q / Z)^{1/3}$ and $L^{- 1/3}$ , again at most one. This comparison has no upper restriction on $Z$ . It preserves $K$ in place of the earlier bound $D^{2}$ and uses the shorter modulus in the first ratio.

Consequently a single allocation satisfies

$z \sim Z \sum ∣ H_{β z^{2}} ∣^{2} ≪ C_{b}^{4} (U Z)^{ε} K [W_{a} Z + K^{2/3} Z^{2/3}] .$

Both original coefficients $b (k_{1} u) b (k_{2} v)$ remain those in (J28). Their separate squared norms supply the factor $C_{b}^{4} K$ ; their fixed allocated divisor factors are absorbed by $U^{ε}$ . No multiplicativity, cancellation or differentiability of $b$ is used. For $m_{good} = β z^{2} \sim M$ , use the same parity counts as in (J32): the sums of $Z$ and $Z^{2/3}$ over $q_{β} \leq H$ are at most $M^{1/2} H^{1/2}$ and $M^{1/3} H^{2/3}$ . Empty boxes are omitted, so these statements hold also for $H > M$ . Cauchy over the actual allocations costs only divisors. Summing the common ideal $e$ uses $\sum_{q_{e} ≪ X} τ_{j} (e) / q_{e} ≪ U^{ε}$ for the first term and the convergent $\sum_{e} τ_{j} (e) / q_{e}^{5/3}$ for the second, with fixed $j$ . Thus the complete positive side norm obeys

$e, m_{good} \sim M q_{r_{2} (m_{good})} \leq H \sum ∣ H_{a, e} (m) ∣^{2} ≪ C_{b}^{4} (U M)^{ε} X [W_{a} M^{1/2} H^{1/2} + X^{2/3} M^{1/3} H^{2/3}] . (J37)$

The other side has its own shorter-variable choice, actual $Y$ and $W_{c} = D / q_{t_{c}}$ . The choices need not agree across the two sides or across allocations: each bounds a complete positive side norm. Cauchy then pairs the two norms on the same original $(e, m)$ set. The common- $e$ Möbius coefficient, $\overline{ρ (m)}$ and all other outer phases are retained in the signed pairing, with their absolute values bounded at this step. No positivity of the selected physical contraction is asserted.

Restore all four unequal-side cross products. Keep $C, R, S, G, X_{0}, Y_{0}$ from (J19). The actual lengths satisfy $X \leq X_{0} = U / (C^{2} R^{2} G)$ and $Y \leq Y_{0} = U / (C^{2} S^{2} G)$ , while $W_{a} ≍ D / (CR)$ and $W_{c} ≍ D / (CS)$ . Restoring the two original $D^{- 1}$ coefficients and the original Poisson factor, then counting the cores in their dyads, gives the positive majorant

$\frac{C _{b}^{4} ( U M ) ^{ε} GC RS}{κ} B_{a} (M, H) B_{c} (M, H), {B_{a} = W_{a} M^{1/2} H^{1/2} + X_{0}^{2/3} M^{1/3} H^{2/3}, B_{c} = W_{c} M^{1/2} H^{1/2} + Y_{0}^{2/3} M^{1/3} H^{2/3} . (J38)$

Occupancy ideals remain in the original coefficients, conductor and zeros. They are omitted only from this increasing positive majorant: they decrease $X, Y$ and the natural dual length and have only a divisor number of choices. In particular they do not increase $W_{a}, W_{c}$ . The original caps $q_{t_{a}}, q_{t_{c}} \leq T$ are retained, although no additional use of them is needed for the bound below.

For each fixed bad-prime dual part $s$ , the natural good-dual scale is bounded above by the same $N_{s} = κ U / (C^{4} RS G^{2} q_{s})$ as before (J33). Expand the square root in (J38) into its four positive cross products before summing dual dyads. Their $M$ powers are $1/2, 5/12, 5/12, 1/3$ . The original transform and required logarithmic derivatives supply arbitrary decay above $N_{s}$ . The dyadic inequality used for (J33) therefore replaces each $M^{α}$ by $N_{s}^{α}$ , including when $N_{s} < 1$ , without truncating the complete dual. With a smaller input epsilon, the source length loss is absorbed exactly as for (J32). The resulting four core costs, in the order first–first, first–second, second–first, second–second, are bounded by

$C_{b}^{4} U^{1 + ε} [C^{- 2} (RS)^{- 1/2} κ^{- 1/2} q_{s}^{- 1/2} H^{1/2} + C^{- 11/6} G^{- 1/6} R^{- 5/12} S^{- 7/12} κ^{- 7/12} q_{s}^{- 5/12} H^{7/12} + C^{- 11/6} G^{- 1/6} R^{- 7/12} S^{- 5/12} κ^{- 7/12} q_{s}^{- 5/12} H^{7/12} + C^{- 5/3} G^{- 1/3} (RS)^{- 1/2} κ^{- 2/3} q_{s}^{- 1/3} H^{2/3}] . (J39)$

All four products are needed for asymmetric $X, Y$ ; no equality of the side lengths or allocations is assumed. Every displayed core power is nonpositive, so there is no $P$ loss when summing their dyads. Since $κ = q_{ℓ}^{6} q_{r}$ , the original $ℓ$ sums are respectively $\sum q_{ℓ}^{- 3}$ , $\sum q_{ℓ}^{- 7/2}$ and $\sum q_{ℓ}^{- 4}$ , all convergent. The original $r ∣ R_{0}$ sums retain powers $q_{r}^{- 1/2}, q_{r}^{- 7/12}, q_{r}^{- 2/3}$ and cost divisors. The bad-prime dual sums have strictly positive decay powers $1/2, 5/12, 1/3$ and converge. Core dyads and allocations cost the original $U^{ε}$ allowance. The bulk lengths still satisfy $X, Y ≫ U / P^{2} ≫ U^{1/2}$ , so no principal $x = y = 1$ exception is added.

Profile separation and restoration are precisely those already justified for (J31)–(J33): separate the whole original coupled profile on annuli before the fixed-vector application, use smoothness at zero and rapid decay at infinity, and restore it by its common coefficient measure. The shorter-variable choice depends only on the allocation’s lengths, not on a moving row. It only exchanges the two original separating phases; the source norm bound is uniform in them. Minkowski restores all original finite ray/angular presentations and required derivatives with the same fixed or polylogarithmic aggregate seminorm/height cost. No derivative falls on $b$ and no height budget is renewed per component. Since $H^{7/12} \leq H^{2/3}$ for $H \geq 1$ , (J39) proves (J34). At $H = U^{1/8}$ the endpoint powers are $U^{17/16 + ε}$ and $U^{13/12 + ε}$ , proving (J35).

The enlarged sector has its own exact signed complement. Let $D_{>}^{unif}$ be the actual original expanded sum in (J24) selected, in every $P < U^{1/6}$ dyad, by $q_{r_{2} (m_{good})} > U^{1/8}$ . Keep every original coefficient, parameter, phase, zero, bad dual and profile on this complementary support. Exact complementary selectors give

$Re τ \sum E_{6, P < U^{1/6}} (c_{τ, T}) = Re D_{>}^{unif} + Re P < U^{1/6} \sum D_{P, \leq U^{1/8}}, T = U^{1/12} . (J40)$

This complement differs from the tapered $D_{>}$ after (J33); the selector $U^{1/8} P^{- 1/2}$ there has not been silently changed. The second term in (J40) is paid at $C_{b}^{4} U^{13/12 + ε}$ by (J35). A uniform aggregate real bound $\operatorname{Re}\mathfrak D_{>}^{\rm unif} \ll C_b^4U^{B_*-\sigma+\varepsilon} $f orso m e$ \sigma>0$, under the same original component and derivative contract, would supply the saving needed in (J24)–(J25), with exponent $max {13/12, B_{*} - σ} < B_{*}$ . That complementary bound is not supplied. In particular the natural $M ≍ U$ squarefree good duals remain in it and still require joint cancellation. The refinement is a source-conditional application of the cited suppliers, with no new sieve, literature-priority or Lean-certification claim. It proves no large-parity complement estimate, full (J24), inverse-fourth saving, signed Robin restoration or RH conclusion.

An every-split support obstruction for the actual Type-II convolution

The standard well-factorable import has a support obstruction for both the actual $b = λ_{II, b}$ on its original inverse annulus and the genuine joint coefficient $h_{b}$ in (J4). This is a repo-derived applicability result, using the existing coefficient identities and fixed-ray prime-ideal supply below. It supplies no strict inverse-fourth improvement and does not settle (J24) or (J40).

The source definition and its domain. The existing owner is the Maynard well-factorable section. In the pinned primary, Definition 1 requires, for every real $Q = Q_{1} Q_{2}$ , $Q_{i} \geq 1$ , a convolution of two 1-bounded sequences, the $i$ th supported on $1 \leq n \leq Q_{i}$ . Definition 2 requires the same for every real $Q = Q_{1} Q_{2} Q_{3}$ , with three separately prescribed supports. One convenient internal factorization or fixed prime block is insufficient. Theorem A and Theorem 1.1 estimate the integer prime-AP discrepancy $π (x; q, a) - π (x) / φ (q)$ , at levels respectively $Q \leq x^{4/7 - ε}$ and $Q \leq x^{3/5 - ε}$ , with fixed $a$ and arbitrary fixed logarithmic saving. They supply no Hecke ideal-coefficient or common- $β$ signed-contraction theorem here; even satisfying their weight definition would leave that transport unproved.

For the support calculation only, define the following ideal-norm analogue, without importing an analytic theorem. For $j = 2$ or $3$ and $Q \geq 1$ , write $S_{j} (a; Q)$ when every real factorization $Q = \prod_{i = 1}^{j} Q_{i}$ , $Q_{i} \geq 1$ , permits

$a (n) = d_{1} \dots d_{j} = n \sum i = 1 \prod j γ_{i} (d_{i}), γ_{i} (d) = 0 if N d > Q_{i} . (J41)$

The convolution is over good integral ideals, each counted once, and the complex $γ_{i}$ may depend on the split. There is no amplitude bound in this definition. Thus failure of $S_{j}$ also rules out the literal ideal-norm version with 1-bounded or divisor-bounded factors. This definition does not aggregate different ideals of equal norm into an integer-modulus weight.

A nonzero semiprime forces a larger level. If $a (n) \neq = 0$ , $n = pq$ for distinct prime ideals, and $p_{0} = min (N p, N q)$ , then

$S_{j} (a; Q) ⟹ Q \geq N n p_{0}^{j - 1}, j = 2, 3. (J42)$

Indeed, any supported convolution first forces $Q \geq N n$ . For $1 \leq r < p_{0}$ , prescribe $Q_{1} = \dots = Q_{j - 1} = r$ and $Q_{j} = Q / r^{j - 1}$ . All prescribed lengths are at least one: for these two values of $j$ , $Q \geq N n \geq p_{0}^{j - 1}$ . Every divisor of $n$ other than the unit has norm at least $p_{0}$ , so the first $j - 1$ factors must all be units. The last must be $n$ , forcing $Q \geq N n r^{j - 1}$ . Let $r ↑ p_{0}$ to obtain (J42).

Apply this to $b$ restricted to the original annulus $q_{d} ≍ D = U$ , with the original $A = A_{b} = U^{1/12}$ . The balanced nonzero semiprimes in C33/C36 already have $b (pq) = 2$ and $N p, N q ≍ U^{1/4}$ . Their identity is reused, not rederived. The fixed-ray prime-ideal supply chooses their product inside a fixed interval where the original $V$ is nonzero. Hence the obstruction also holds with the actual $V (q_{d} / D)$ and unit-modulus ray twist included. Formula (J42) requires levels at least $U^{3/4 + o (1)}$ for binary and $U^{1 + o (1)}$ for triple factorability; neither is available at the original $Q = U^{1/2 + o (1)}$ . These are necessary levels, not existence results at larger levels.

This failure is a common zero of all convolution terms at an offending split. Changing amplitude bounds cannot fill it. A finite signed sum of convolutions with those same prescribed support bounds, or smaller ones, also vanishes there, regardless of its coefficients. A fixed block decomposition that does not admit the offending split is a different hypothesis and is not ruled out by this observation.

The joint $h_{b}$ has its own obstruction. Fix the original nonzero smooth complex annular $V$ of one (J4) component, and choose $t_{0} > 0$ with $v_{0} = V (t_{0}) \neq = 0$ . Fix $c > 2$ . Choose a sufficiently small fixed $δ > 0$ such that $c (1 - δ) > 2$ and

$∣ V (t) - v_{0} ∣ \leq ∣ v_{0} ∣/10 for t_{0} (1 - δ)^{2} \leq t \leq t_{0} (1 + δ)^{2} .$

Using the existing fixed-ray prime-pair supply, choose four distinct good prime ideals with

$N q_{j} \in c U^{1/12} [1 - δ, 1 + δ], N p_{i} \in (t_{0} / c) U^{5/12} [1 - δ, 1 + δ], i, j = 1, 2.$

The cited fixed-ray prime-ideal theorem applies in each of these fixed relative intervals as its length tends to infinity. Delete $S$ and the finitely many coincidences. For the displayed (J4), the inner ideal conditions are goodness and squarefree support, and $ξ$ has modulus one there. If a further original ray selector is retained, qualify the construction to an allowed component as follows: take $t (q_{1}) = t (q_{2}) = r$ and $t (p_{1}) = t (p_{2}) = s$ , with $sr = κ$ an allowed factor class on which the original multiplier is nonzero. A retained column class must then allow $τ = κ^{2}$ . The same supply provides these primes in the fixed classes. No assertion is made for an incompatible selector, a zero component, or an arbitrary varying profile. The fixed unit representatives and finite ray multipliers are retained; four independent ray phases are not substituted for the multiplicative $ξ$ of (J4).

The formula below is for (J4). A compatible further selector keeps all four terms; any additional nonzero multiplier constant on these fixed classes is retained as their common scalar.

Put $w = p_{1} p_{2} q_{1} q_{2}$ and $t_{ij} = N (p_{i} q_{j}) / D$ . All $t_{ij}$ lie in the displayed nonzero neighborhood of $t_{0}$ . Since both primes in each mixed pair exceed $2 A$ , the existing C40 with $b_{0} = 1$ gives $b (p_{i} q_{j}) = 2$ on this original annulus. The only divisors of $w$ in the $D$ annulus are the four mixed pairs: single primes and $q_{1} q_{2}$ are below it, $p_{1} p_{2}$ and every triple are above it. Thus exactly four ordered factorizations contribute to (J4), each with $b (d_{1}) b (d_{2}) = 4$ , and multiplicativity gives

$h_{b} (w) = 8 D^{- 1} ξ (w) [V (t_{11}) V (t_{22}) + V (t_{12}) V (t_{21})] \neq = 0. (J43)$

For completeness, each product in brackets differs from $v_{0}^{2}$ by at most $(21/100) ∣ v_{0} ∣^{2}$ , so the bracket differs from $2 v_{0}^{2}$ by at most $(21/50) ∣ v_{0} ∣^{2} < 2∣ v_{0} ∣^{2}$ . This continuity estimate is valid for complex $V$ and retains interference between both pairings; it uses neither positivity nor an absolute-value replacement of the sum.

Here $Nw ≍ U$ . A divisor of $w$ without a $p_{i}$ has norm $O (U^{1/6})$ ; one containing a $p_{i}$ has norm $≫ U^{5/12}$ . At the natural level $L = U^{1 + o (1)}$ , prescribe the binary split $Q_{1} = U^{1/3}$ , $Q_{2} = L / U^{1/3} = U^{2/3 + o (1)}$ . Both lengths are at least one for large $U$ . The first divisor can contain no $p_{i}$ , so its complement contains both and has norm at least $N p_{1} N p_{2} ≍ U^{5/6} > Q_{2}$ . Every supported convolution is zero at $w$ , contradicting (J43). Thus genuine joint $h_{b}$ , not merely $b$ , fails binary factorability at its natural level.

There are also the following stronger necessary levels for this witness. Write $p_{*} = min_{i} N p_{i}$ . If $S_{2} (h_{b}; L)$ holds, nonvanishing first gives $L \geq Nw$ . Prescribe $Q_{1} = r$ , $1 \leq r < p_{*}$ , approaching $p_{*}$ from below, and $Q_{2} = L / r$ . These lengths are at least one. Again the second divisor must contain both large primes, so $L \geq r N p_{1} N p_{2}$ . For triple factorability the equal split requires that some factor can contain each large prime. Consequently

$S_{2} (h_{b}; L) S_{3} (h_{b}; L) ⟹ L \geq p_{*} N p_{1} N p_{2} ≫_{V, c, δ} U^{5/4}, ⟹ L \geq (i max N p_{i})^{3} ≫_{V, c, δ} U^{5/4} . (J44)$

The equal triple split has all lengths $L^{1/3} \geq 1$ ; below its stated threshold no factor can carry the largest prime. These are necessary bounds only, with no sufficiency or optimality assertion. The same common-zero argument covers arbitrary amplitudes and finite signed sums with the offending support bounds.

Limit of the obstruction. Opening $b$ into its actual prime or cofactor blocks remains legitimate, but cannot certify standard every-split well-factorability at these original levels. General fixed-block multilinear estimates and signed cancellation are not excluded. Nothing here is an energy lower bound, a lower bound or obstruction for the signed (J40) complement, or a joint estimate reaching the sufficient local exponent $1 + B_{*} - σ$ for $σ > 0$ . The exponent $13/6$ is the generic local energy baseline. The original Hecke domain, raw sextic collision zeros, common $β$ , signs and common- $e$ detector, fixed/polylog profile seminorms, and all unit/ray/core/bad-dual aggregation costs remain in the original contract. In particular, (J40) and its unpaid aggregate real complementary bound are unchanged. No new analytic estimate, literature-originality claim, Lean certification, inverse-fourth saving, signed Robin restoration or RH completion follows.

The two-factor predecessors C21–C50 and the explicit reciprocal premise are in the analytic note. The following C51–C76 concern the four-factor exceptional interface of this Fourier note; they retain the original signed J40 scope.

Four inverse factors on the same coprime realization

Retain $K = Q (- 3)$ , $q_{d} = N d$ , the fixed $S$ , and the ideal-counting convention of (C41). Let $D = U^{1/2}$ and $A = U^{α}$ , where $0 < α < 1/4$ is fixed. On the original inverse annulus use exactly $b_{A} = μ + K_{A}$ of (C37), with its original smooth $W$ and collision zeros. The support condition $2 A < z ≍ U^{1/4}$ then holds eventually in (C25). All statements below are paper-level repo-derived deductions; $R_{β}$ always means the explicit principal reciprocal hypothesis (C48), including polynomial vertical growth on smaller closed strips. In particular $R_{7/8}$ is source-conditional on the pinned manuscript, thm:main and lem:logarithmic-control, and is not certified here.

For a smooth complex $Ψ$ supported on a fixed compact rectangle in $(0, \infty)^{4}$ , and squarefree $P$ outside $S$ , put

$Q_{4} (P; Ψ) C_{4} (P) Θ_{4, A} (P) = d_{1} d_{2} d_{3} d_{4} sf (d_{1} d_{2} d_{3} d_{4}, SP) = 1 \sum i = 1 \prod 4 b_{A} (d_{i}) Ψ (q_{d_{1}} / D, \dots, q_{d_{4}} / D), I_{Ψ} = \int Ψ (x) d x, = κ^{4} p \in S \prod (1 - q_{p}^{- 1})^{4} p \in / S \prod (1 + 4 q_{p}^{- 1}) (1 - q_{p}^{- 1})^{4} p ∣ P \prod (1 + 4 q_{p}^{- 1})^{- 1}, = e_{1} e_{2} e_{3} e_{4} sf (e_{1} e_{2} e_{3} e_{4}, SP) = 1 \sum \frac{\prod _{i = 1}^{4} μ ( e _{i} ) 2 ^{ω (e_{i})} W ( q _{e_{i}} / A )}{\prod _{p ∣ e_{1} e_{2} e_{3} e_{4}} ( q _{p} + 4 )}, κ = Res_{s = 1} ζ_{K} (s) . (C51)$

The squarefree product forces all four factors to be pairwise coprime. This is the actual relation obtained from squarefree columns $s = d_{1} d_{2}$ , $t = d_{3} d_{4}$ and common-column gcd $(s, t) = 1$ . Squaring (C41) would allow primes shared across the two columns; even its unforced local factor would be $(1 + 2/ q_{p})^{2}$ , instead of $1 + 4/ q_{p}$ . Thus neither $Θ_{A} (P)^{2}$ nor independent copies of the two-variable mean describe this realization. The old two-variable result remains applicable in its original scope.

Four-factor absolute-error theorem. Suppose $1/2 \leq β < 1$ and $R_{β}$ . For every fixed polynomial range $q_{P} \leq U^{C}$ and $ε > 0$ , a finite derivative order $J$ gives

$Q_{4} (P; Ψ) ∣ Θ_{4, A} (P) ∣ = D^{4} C_{4} (P) I_{Ψ} Θ_{4, A} (P) + O_{ε} (N_{J} (U q_{P})^{ε} [D^{3 + β} + D^{7/2} A^{1/2}]), ≪_{ε} A^{- 4 (1 - β) + ε} q_{P}^{ε}, 0 < C_{4} (P) ≪_{K, S} 1. (C52)$

Here $N_{J}$ is the maximum of the partial derivatives of $Ψ$ up to order $J$ in each variable on that rectangle. Constants depend on $C, α, K, S, W$ , supports, strip margins and the reciprocal growth constants. The displayed $q_{P}^{ε}$ version is retained when summing frequency shells; no condition $q_{P} \leq A^{2}$ is imposed.

Proof. Reuse the divisor identity (C37), the literal Euler-factor method (C44), the deleted-factor estimate (C45) under (C48), and the normal-convergence argument (C47). Expand first the $K_{A}^{4}$ term and fix the four $e_{i}$ . Its series is $\prod_{i} ζ_{K} (s_{i}) G_{e, P} (s)$ . Writing $x_{i} = q_{p}^{- s_{i}}$ , the local factors of $G$ are

$p ∤ SP e_{1} e_{2} e_{3} e_{4} p ∣ e_{j} p ∣ SP (1 + x_{1} + x_{2} + x_{3} + x_{4}) \prod_{i} (1 - x_{i}) x_{j} \prod_{i} (1 - x_{i}) \prod_{i} (1 - x_{i}) . (C53)$

The first factor has no linear term. The literal product therefore converges normally whenever every $ℜ s_{i} > 1/2$ ; one never divides by its potentially vanishing numerator. On smaller closed strips its bound is $\ll_\eta(q_P\prod_iq_{e_i})^\eta \prod_iq_{e_i}^{-\Re s_i} $. A t$ \mathbf s=\mathbf1$, forcing a prime into any one variable gives $q_{p}^{- 1} / (1 + 4 q_{p}^{- 1}) = (q_{p} + 4)^{- 1}$ . Four smooth Mellin inversions and successive contour shifts to $1/2 + δ$ give the residue in (C52). For a subset of $j \geq 1$ shifted variables, summing the forced divisors costs

$N_{J} (U q_{P})^{ε} D^{4 - j /2} A^{j /2}, j = 1, 2, 3, 4. (C54)$

This follows from the same sums $\sum_{q_{e} \leq 2 A} τ (e) q_{e}^{- 1/2 - δ}$ and $\sum_{q_{e} \leq 2 A} τ (e) / q_{e}$ as (C44). Since $A < D$ , the $j = 1$ term dominates. Choose $J$ once to pay four Mellin integrals, their horizontal joins and the ordinary zeta growth.

There are fifteen other terms in $(μ + K_{A})^{4}$ . In each choose one $μ$ variable, fix the other three, and use (C45)/[(C48)](../Analytic/liu2026quasirh.md#the-coupled-euler-product-under-an-explicit-reciprocal-strip) with $r = rad (SP \prod_{i \neq = j} d_{i})$ , so $q_{r} ≪ q_{P} D^{3}$ . That inner sum is $O (N_{J} D^{β + ε} q_{P}^{ε})$ ; each remaining absolute mass is $O (D lo g^{2} (2 A))$ , or $O (D)$ for a $μ$ factor, by (C46). This proves the $D^{3 + β}$ error for their entire sum. Only the principal reciprocal is used: there is no application to nontrivial ray or angular Hecke inverses.

Finally the four-variable Mellin series of $Θ_{4, A}$ is

$p \in / SP \prod (1 - \frac{2 \sum _{i} q _{p}^{- z_{i}}}{q _{p} + 4}) = i \prod ζ_{K} (1 + z_{i})^{- 2} H_{4, P} (z) . (C55)$

After dividing by $\prod_{i} (1 - q_{p}^{- 1 - z_{i}})^{2}$ , the local correction minus one has only terms $O(\sum_iq_p^{-2-\Re z_i} +\sum_{i,j}q_p^{-2-\Re z_i-\Re z_j}) $o n s ma ll ers t r i p s . T h u s$ H_{4,P}$ is holomorphic for every $ℜ z_{i} > - 1/2$ , normally convergent and $≪_{η} q_{P}^{η}$ uniformly in the heights there. Excluded primes have factor $\prod_{i} (1 - q_{p}^{- 1 - z_{i}})^{- 2}$ , costing $q_{P}^{η}$ . Move all four lines to $β - 1 + δ$ . The simple pole of each $W$ at zero is canceled by its corresponding double reciprocal-zeta zero. Rapid Mellin decay and (C48) prove the second line of (C52), with no residue or mixed polar divisor. This completes the proof. Fixed polynomial ranges allow $δ, η$ to be allocated within any final $U^{ε}$ loss.

The constant-ray exceptional component with actual columns

Use Xu–Sang v2, §3.1 (3.3)–(3.4), §3.2 (3.8), and §3.3 (3.11), only as finite Gauss, Poisson and exceptional-frequency identities. Their field is this same $K$ ; columns use the literal raw sextic symbol and its zero extension. Fix the corefree branch, original common- $e = 1$ , and source common-column gcd $C = 1$ . This $C$ is not a repeated-column core. The source moving twist ideal is the unit ideal. No arbitrary-arithmetic-weight version of its Proposition 3.1 or Lemma 5.1 is assumed.

Normalization and the multiplier. In the source conventions set

$e_{K} (z) w (y) G (s, k) = exp (4 πi Im z / 3), d μ (z) = (2/ 3) d Re z d Im z, = \int_{C} w (z) e_{K} (- zy) d μ (z) = w_{rad} (∣ y ∣^{2}), = q_{s}^{- 1/2} x mod s \sum χ_{s} (x) e_{K} (k x / s) = γ_{1} (s) \overline{χ_{s} (k)} (s good sf) . (C56)$

The lattice is self-dual for this measure. The last equality includes collision zeros, since the raw character is primitive at every prime of a good squarefree $s$ . It is the same Gauss convention as the pinned manuscript’s eq:sextic-sieve-pair-gauss and (J5). Consequently an original side scalar $γ_{1} (s) \overline{χ_{s} (m)}$ is $χ_{s} (- 1) G (s, - m)$ . The true change-of-form multiplier is $χ_{s} (- 1)$ , together with the original $ξ (s)$ and finite ray/angular factors; it is not an extra inverse Gauss sum and is not a replacement of $b_{A}$ by $1$ . The original (J5) CRT factors are already inside $γ_{1} (s)$ and must not be inserted a second time.

Fix a common refined total ray $[s] = [t] = σ$ and include its selectors in both original side weights. For one such component write its arithmetic column as

$c_{s} = 1_{s good sf} d_{1} d_{2} = s \sum b_{A} (d_{1}) b_{A} (d_{2}) V_{1} (q_{d_{1}} / D) V_{2} (q_{d_{2}} / D) a (d_{1}, d_{2}), (C57)$

where $a$ retains the original factor-class selectors, $ξ (s)$ , $χ_{s} (- 1)$ and original angular factors in this orientation. Separated norm heights belong to the $V_{i}$ , with their actual polynomial seminorm costs. Distinct side profiles are allowed by polarization. Consider the unnormalized element-row Gauss pairing $\mathcal E=\sum_m w(m/\sqrt U) (\sum_s c_sG(s,-m))\overline{(\sum_t c’_tG(t,-m))}$. This specifies a local Gauss object with actual columns, not the full physical-row moment or a selected positive part of J40. The source normalization is $U^{- 1}$ on each column; taking its column profile to be $t^{1/2} c_{s}$ shows that $E$ is $U^{2}$ times that normalized pairing. This use is an identity before any smooth-column estimate. Equivalently, direct opening gives Poisson multiplier $U / q_{s} q_{t} = (t_{s} t_{t})^{- 1/2}$ , $t_{s} = q_{s} / U$ . The original $D^{- 1}$ side normalizations of (J4), and all physical row/core prefactors, still have to be restored in their own contraction.

On the $C = 1$ branch the finite correlation is exactly $χ_{s} (k) \overline{χ_{t} (- k)} r (s, t)$ . The source’s refined equal-total-ray condition gives $r (s, t) \overline{χ_{t} (- 1)} = 1$ . Its exceptional frequencies have the unique form

$k = ϵ e_{S} (r) a^{6}, ϵ \in O^{\times}, 0 \leq r_{p} \leq 5 (p \in S), e_{S} (r) = p \in S \prod π_{p}^{r_{p}}, P_{a} = rad (a)_{p \in / S} . (C58)$

Here $a$ ranges once over integral ideals, including its possible $S$ factors, with fixed generators; all six units occur. $P_{a}$ is a frequency puncture, not the original product-squarepart $P$ of J40. The characters of $ϵ e_{S} (r)$ are constant on a refined total ray fiber, so they cancel between its two columns. Sixth powers leave precisely the original mask $1_{(s t, a) = 1}$ . There is no coprimality replacement on a different row family.

After that cancellation, let $Ω_{ϵ, r} (d)$ be the remaining actual four-factor finite-ray/angular weight, including both total-class selectors and the original component coefficient. Expand only this given weight under its original presentation contract. Let $η_{ϵ, r}$ be the coefficient with four trivial ray characters and zero angular modes, with normalized counting measure on each ray group and normalized Haar measure on each angular circle. This defines a linear component even when it is zero. For four specified factor classes in a ray group of order $h$ , with no further multiplier, the coefficient is $h^{- 4}$ ; a total-class relation or any additional weight must instead be averaged as its actual table. In particular, one must not assign $h^{- 4}$ to a general total-class selector. All nontrivial characters and angular modes remain in $Ω - η$ . This is a coefficient expansion, not independence of the four actual ideal classes or angles.

Write $z = q_{e_{S} (r)} q_{a}^{6} / U$ and

$Ψ_{z} (x) I_{4} (z) E_{exc} = V_{1} (x_{1}) V_{2} (x_{2}) \overline{V_{3} (x_{3}) V_{4} (x_{4})} (x_{1} x_{2} x_{3} x_{4})^{- 1/2} w_{rad} (\frac{z}{x _{1} x _{2} x _{3} x _{4}}), = \int Ψ_{z} (x) d x, = E_{rad} + R_{tw}, E_{rad} = ϵ, r \sum η_{ϵ, r} a \sum Q_{4} (P_{a}; Ψ_{q_{e_{S} (r)} q_{a}^{6} / U}) . (C59)$

Precisely, $R_{tw}$ is the very same sum over $ϵ, r, a, d_{1}, d_{2}, d_{3}, d_{4}$ , with the squarefree product, $(\prod d_{i}, S P_{a}) = 1$ , the same $\prod b_{A} (d_{i}) Ψ_{z}$ , and weight $Ω_{ϵ, r} (d) - η_{ϵ, r}$ . This is an exact signed residual. No absolute value, new beta row, positivity of the selected frequencies, or monotonicity under selection of physical rows is asserted. The $k = 0$ diagonal is outside (C59).

The proof of (C59) is finite opening of the two Gauss sums, row Poisson with (C56), the $C = 1$ correlation above, and (C58). Squarefreeness of each column and their gcd $1$ give exactly (C51)’s four-way mask. The correlation phases cancel on each original total-class fiber before taking the stated linear coefficient. Thus no unproved arithmetic column-mean theorem is used to identify this component.

Summing the proved component. Assume $R_{β}$ and the same fixed or polylogarithmic aggregate profile/height contract. Applying (C52) to (C59) gives

$E_{rad} ∣ R_{KK} ∣ ∣ E_{rad} ∣ = U^{2} ϵ, r \sum η_{ϵ, r} a \sum C_{4} (P_{a}) Θ_{4, A} (P_{a}) I_{4} (q_{e_{S} (r)} q_{a}^{6} / U) + R_{KK} + R_{μ}, ≪_{ε} N_{J} U^{23/12 + ε} A^{1/2}, ∣ R_{μ} ∣ ≪_{ε} N_{J} U^{(10 + 3 β) /6 + ε}, ≪_{ε} N_{J} U^{ε} [U^{13/6} A^{- 4 (1 - β)} + U^{(10 + 3 β) /6} + U^{23/12} A^{1/2}] . (C60)$

Here $R_{μ}$ is exactly the aggregate of the fifteen $μ$ -containing terms in (C59); $R_{KK}$ is the remainder of the all- $K_{A}$ term after its displayed residue. The finite sum over units and $S$ exponents remains visible in the main term. There is no additional planar area multiplier: $κ^{4}$ already accounts for counting ideals once, and (C56) fixes the Fourier measure.

To justify the full frequency sum, on each shell $z ≍ R \geq 1$ all the required derivatives of $Ψ_{z}$ are $O_{B} (N_{J, B} R^{- B})$ . Ideal counting gives $O ((R U)^{1/6})$ possible $a$ ; the ball $z \leq 1$ has $O (U^{1/6})$ . First retain any fixed polynomial shell range. Its $q_{P_{a}}^{ε}$ factor in (C52) is paid by choosing its internal loss smaller; one cannot discard frequencies merely because $q_{P_{a}} > A^{2}$ . For the remaining shells the original divisor envelope bounds the four columns absolutely, and arbitrary Fourier decay makes their sum smaller than any prescribed power by enlarging the fixed shell cutoff and $B$ . The same argument bounds the residue tail using the absolute divisor mass of $Θ_{4, A}$ . It proves (C60) without truncating the actual frequency identity. In (C59)–(C68), $N_{J}$ denotes the original aggregate of component seminorms against the absolute coefficient measures, including the $∣ η_{ϵ, r} ∣$ costs; it is required to be fixed or polylogarithmic. One derivative order is chosen for all shells, finite components and separated heights. If their growth has degree $h$ , log-Fourier decay of order greater than $h + 5$ is integrable in four variables. Derivatives act only on the external profiles, never on $b_{A}$ , and no budget is restarted per $a$ or ray class.

Joint cutoff certificate and the unpaid estimates

Put $B_{*} = (31 - 106) /6$ and $Δ = 7/6 - B_{*} = (56 - 12) /3 = 0.082482904638630 \dots$ . The unnormalized local energy comparison exponent is $1 + B_{*}$ ; restoring the original $D^{- 2} = U^{- 1}$ coefficient normalization gives the exponent $B_{*}$ used in J40. This comparison does not identify the entire Gauss object (C59) with J40 or pay its omitted parameters. At the unchanged $A = U^{1/12}$ , (C52) and (C60), conditionally on $R_{7/8}$ , give the following explicit certificate:

$Θ_{4, A} (P) ∣ E_{rad} ∣ 17/8 - (1 + B_{*}) 101/48 - (1 + B_{*}) ≪_{ε} U^{- 1/24 + ε} q_{P}^{ε}, ≪_{ε} N_{J} U^{ε} (U^{17/8} + U^{101/48} + U^{47/24}), = 0.040816237971963 \dots, = 0.019982904638630 \dots, 1 + B_{*} = 2.084183762028037 \dots . (C61)$

The $101/48$ term is the fifteen- $μ$ aggregate, not the smaller error that would result from squaring the two-factor mean. These are upper-bound certificate gaps, not lower bounds for any actual term. No full energy improvement follows, and no selected Poisson component has been declared nonnegative.

The same coefficient pays Type I. For $A = U^{α}$ , keep the actual split (C25), the original physical family, all finite ray presentations, original zero extensions and the shared finite-height allowance. Reuse (C29), whose primary suppliers are the pinned manuscript, lem:buffered-bins, lem:detector-dyads, and the zero-prime-slot case of lem:plain. These require the original nonprincipal inducing rows, polynomial-size original deletion masks, bounded annular length ranges, and finite internal Mellin heights whose total lies within the same buffer. There is no new conductor lower bound. The proof of (C29) is uniform in $Q ≪ U^{α}$ , since only the smooth profile’s $Q / A$ ratio changes. It supplies, on the very same bin,

$T_{Q} (δ) f (δ) ≪ U^{ε} min {U^{f (δ)}, U^{1 + 2 α δ}}, = R (δ) + δ = δ + \frac{15 - 16 δ}{15 - 6 δ} (1/50 < δ \leq 5/6), f (δ) = 1 (5/6 \leq δ \leq 1) . (C62)$

The floor separately costs $U^{1 + α /25 + ε}$ . Norm heights and all internal Mellin variables share the original buffer; the Fourier/Mellin tail orders are chosen once after the desired loss. There is no infinite-local-contour extension or differentiation of $b_{A}$ . Dyadic and fixed/polylog aggregate costs are absorbed only within the original common envelope.

Differentiation gives $f^{'} (δ) = 1 - 150/ (15 - 6 δ)^{2}$ and $f^{''} (δ) = - 1800/ (15 - 6 δ)^{3} < 0$ . Hence the unique maximum is

$δ_{*} = (15 - 56) /6 = 0.458758547680685 \dots, f (δ_{*}) = B_{*}, α_{I} = \frac{B _{*} - 1}{2 δ _{*}} = \frac{3 - 6}{6} = 0.091751709536137 \dots . (C63)$

This bin is also inside the range $δ \leq 3/4$ if that restriction is imposed from the source’s $7/8$ Hecke region. For the full bin range, the envelope $max_{δ} min {f (δ), 1 + 2 α δ}$ has a strict gap below $B_{*}$ exactly when $0 < α < α_{I}$ . Indeed at $δ_{*}$ its minimum is $B_{*}$ for $α \geq α_{I}$ . Below that threshold, the increasing line intersects the decreasing side of $f$ at

$δ_{α} = \frac{15 - 10/ ( 1 - 2 α )}{6} > δ_{*}, τ (α) = 1 + 2 α δ_{α} < B_{*} . (C64)$

Before this crossing the minimum is at most the line’s value there; after it, $f$ decreases to $1$ . The floor is smaller since $2 δ_{α} > 1/25$ . Thus $τ$ is an admissible full-range Type-I exponent, and $τ (1/12) = 13/12$ , recovering (C30). This is a limit of these two UPPER BOUND inputs. It does not assert that the critical bin is populated at its counting bound or that any actual Type-I moment attains that exponent.

For $α < α_{I}$ , the same positive $L^{4}$ Minkowski argument as (C31) gives $T_{II} ≪ U^{B_{*} + ε}$ and

$∣ T_{μ} - T_{II} ∣ ≪ U^{(3 B_{*} + τ (α)) /4 + ε} . (C65)$

The original cubic and large-product-squarepart payments add $13/12$ , so the full difference-transfer exponent is $max {13/12, (3 B_{*} + τ (α)) /4}$ , with reserve $min {B_{*} - 13/12, (B_{*} - τ (α)) /4}$ . Here $B_{*} - 13/12 = 0.000850428694703 \dots$ . At $α_{I}$ the interference reserve vanishes. A one-sided triangle transfer also needs a Type-I exponent below $B_{*}$ and does not repair this envelope limit.

The coupled-current-input incompatibility. Making just the four-factor residue upper bound in (C60) smaller than $U^{1 + B_{*}}$ requires the sufficient-estimate inequality $4 α (1 - β) > Δ$ . At $β = 7/8$ this becomes

$α > α_{G} = 2Δ = 0.164965809277260 \dots, whereas the paid Type-I envelope requires α < α_{I} . (C66)$

The windows are disjoint, though both endpoints satisfy the exact-split support limit $α < 1/4$ . As $α ↑ α_{I}$ the residue certificate approaches exponent $13/6-\alpha_I/2 =2.120790811898598\ldots>1+B_*$, while the Type-I reserve tends to zero. Moreover the $R_{7/8}$ bound for $R_{μ}$ in (C60) is $U^{101/48 + ε}$ for every such cutoff, so merely changing $A$ does not pay that error either. The $KK$ contour error alone is below target throughout this Type-I window.

For general $β$ , the main-term and Type-I inequalities can overlap only if $β < 2 - 6 /2 = 0.775255128608411 \dots$ ; at the fixed $α = 1/12$ the main-term inequality is $β < 13 - 56 = 0.752551286084110 \dots$ . The separate errors still require $(10 + 3 β) /6 < 1 + B_{*}$ and $23/12 + α /2 < 1 + B_{*}$ . These are conditional algebraic thresholds for the displayed estimates, not new zero-free regions, necessary RH conditions, actual lower bounds or a global impossibility theorem. A hypothetical squared-two-factor interface has the same formal residue exponent, but (C51) shows why it cannot replace this four-factor theorem or its fifteen-term error.

The next exceptional input has a fixed meaning. Define

$S_{4} (U) = U^{- 1/6} ϵ, r \sum η_{ϵ, r} a \sum C_{4} (P_{a}) Θ_{4, A} (P_{a}) I_{4} (q_{e_{S} (r)} q_{a}^{6} / U) . (C67)$

At the unchanged cutoff a sufficient new frequency-average estimate is $∣ S_{4} (U) ∣ ≪ N_{J} U^{- Δ - σ + ε}$ for some fixed $σ > 0$ . Currently (C52) supplies only $U^{- 1/24 + ε}$ . This target asks for cancellation on the same coupled frequency realization. It must be accompanied by $|\mathcal R_\mu|,|\mathcal R_{\rm tw}| \ll\mathcal N_JU^{1+B_*-\sigma+\varepsilon}$, or a bound of this strength for their same-object signed sum together with the residue. The $KK$ remainder is already smaller. Principal $R_{β}$ supplies neither the ray/angular residual estimate nor a nonprincipal inverse Hecke estimate. All these targets are unproved.

A sufficient local nonexceptional interface, also unproved. The precise gap in Xu–Sang v2, (2.6), (3.6)–(3.7) and Lemma 3.2 concerns arithmetic columns on the same Poisson row. Freeze its $C, b$ , $g = (rad C, k)$ , tensor branch and ray term. Put $X = L^{2} / q$ , $N = L / (q_{C} q_{b})$ , $N_{i} = N / q_{f_{i}}$ , $K_{row} ≍ RX / q_{C}$ and $k = g k^{'}$ . Let $D_{src} = N d$ be the source moving-twist radical norm, distinct from $D = U$ , and let $R_{C, g}$ be its (3.7). The fixed $Π_{i}$ masks are independent of $k^{'}$ . Let $d_{i} (n)$ be the actual compressed column coefficient at $C b f_{i} n$ , including its original side/core allocations; for actual squarefree columns the admissible branch has $C$ squarefree and $f_{i} = 1$ . A sufficient hypothesis $W (λ)$ is

$0 < q_{k^{'}} ≪ K_{row} / q_{g} ψ_{k^{'}}^{*} \neq = 1 \sum n \sum d_{i} (n) ν_{i} (n) χ_{n} (k^{'}) χ_{n} (g d_{e} C^{♭}) 1_{(n, Π_{i}) = 1} V_{i} (q_{n} / N_{i})^{2} ≪ N_{i} (K_{row} / q_{g}) D_{src} R_{C, g} U^{λ + ε} . (C68)$

The symbols $ν_{i}, d_{e}, C^{♭}, ψ_{k^{'}}^{*}$ are exactly those of the frozen source branch (3.6). Quantification includes all its original polynomial-size masks, collision zeros, fixed twists and prescribed bounded profiles, with fixed polynomial height costs and no new conductor lower bound. It is not a claim for optimized or plain columns. Divisor-boundedness gives $\sum ∣ d_{i} (n) ∣^{2} ≪ N_{i} U^{ε}$ ; that mass statement does not prove (C68).

To check local sufficiency, Cauchy on the same $k^{'}$ row gives $w_{C, g} M_{1} M_{2}$ , where $M_{i}$ is the left side of (C68). The source relation $w_{C, g} R_{C, g} / q_{g} \leq q_{C}$ bounds this by $NRX D_{src} U^{λ + ε}$ . On dyads $q_{C} ≍ C_{0}, q_{b} ≍ B_{0}$ , restoring $q / L^{3}$ and the $O (C_{0} B_{0})$ frozen labels gives $R D_{src} U^{λ + ε}$ . Shell decay pays $R$ . In the corefree $q = L = U$ , $D_{src} = 1$ normalization of (C59), restoration of $U^{2}$ therefore gives $U^{2 + λ + ε}$ . Thus $λ < B_{*} - 1$ would suffice locally; $λ = 1/12$ would leave the reserve $B_{*} - 13/12$ above. The original plain-coefficient (2.6) does not supply (C68). Its proof would be a new actual-coefficient estimate. External log-Fourier separation pays its fixed polynomial height cost with sufficiently rapid decay; derivatives still avoid $b_{A}$ .

Even these exceptional and nonexceptional local targets together do not close J40. They must be placed back on its original common beta rows, with all product-squareparts $P < U^{1/6}$ , common- $e$ signed Möbius detector, $ρ$ , nontrivial cores, common-column gcds, unit/ray/ angular terms, bad duals and full natural dual dyads. Moving $D_{src}$ losses and every profile/height coefficient must then be summed under the one original fixed/polylog aggregate budget; they cannot enter an unspecified constant. A physical squarefree-row subset may be restricted after a complete positive energy bound, not by monotonicity of a signed nonexceptional or exceptional component. The actual same-beta signed contraction, signed Robin restoration and RH remain open. No Lean, source-major-proof or priority certification is made.

Exact frequency residue and the averaged coupled theta

This continuation evaluates only the displayed all- $K_{A}$ residue sum in (C60). Retain (C51), (C58)–(C60), their original profiles and finite coefficients, and set $A = U^{1/12}$ . For each fixed $d = q_{e_{S} (r)} > 0$ , put $X = (U / d)^{1/6}$ and $C_{0} = C_{4} (1)$ . All sums below count integral ideals once, including the $S$ factors of $a$ . Define the signed or complex functional and Mellin transforms

$T_{d} (U) = a \sum C_{4} (P_{a}) Θ_{4, A} (P_{a}) I_{4} ((q_{a} / X)^{6}), I (v) = \int_{0}^{\infty} I_{4} (t) t^{v - 1} d t, W (z) = \int_{0}^{\infty} W (t) t^{z - 1} d t . (C69)$

Here $I_{4}$ is smooth at zero and rapidly decreasing with the original profile envelope; $I$ is holomorphic for $ℜ v > 0$ . The original cutoff is $1$ below $1$ and $0$ above $2$ . Omit its four factors temporarily and define

$D (s, z) = a; e_{1} e_{2} e_{3} e_{4} sf (e_{1} e_{2} e_{3} e_{4}, S) = 1, (a, e_{1} e_{2} e_{3} e_{4}) = 1 \sum \frac{C _{4} ( P _{a} ) q _{a}^{- s} \prod _{i} μ ( e _{i} ) 2 ^{ω (e_{i})} q _{e_{i}}^{- z_{i}}}{\prod _{p ∣ e_{1} e_{2} e_{3} e_{4}} ( q _{p} + 4 )} . (C70)$

In (C70), $a$ is unrestricted; only the product of the four $e_{i}$ is squarefree. Absolute convergence holds for $ℜ s > 1$ , $ℜ z_{i} > 0$ . For $q = q_{p}$ , $Q = q^{- s}$ , $Z = \sum_{i} q^{- z_{i}}$ , the normalized local factor of $D / C_{0}$ is $(1 - Q)^{- 1}$ at $p \in S$ , and otherwise

$1 - \frac{2 Z}{q + 4} + \frac{q}{q + 4} \frac{Q}{1 - Q} = \frac{1 - [ 4 Q + 2 ( 1 - Q ) Z ] / ( q + 4 )}{1 - Q} . D = C_{0} ζ_{K} (s) i \prod ζ_{K} (1 + z_{i})^{- 2} H (s, z) . (C71)$

Indeed a prime dividing $a$ contributes $q / (q + 4)$ to $C_{4} (P_{a}) / C_{0}$ and cannot divide any $e_{i}$ ; otherwise it may enter exactly one $e_{i}$ . Outside $S$ , $H_{p}$ is the numerator in (C71) divided by $\prod_{i} (1 - q^{- 1 - z_{i}})^{2}$ ; inside $S$ it is just that inverse product. No numerator is inverted. Write $σ = ℜ s$ , $τ_{i} = ℜ z_{i}$ . For large $q$ , uniformly in all heights on fixed closed real substrips,

$H_{p} - 1 = O (q^{- 1 - σ} + i \sum q^{- 2 - τ_{i}} + i \sum q^{- 1 - σ - τ_{i}} + i, j \sum q^{- 2 - τ_{i} - τ_{j}}), σ > 0, τ_{i} > - 1/2, σ + τ_{i} > 0. (C72)$

Every exponent then exceeds $1$ , proving normal convergence and a uniform bound for $H$ . Its finite denominators do not vanish in this domain. In particular $H$ introduces no mixed polar divisor.

Exact residue. Put $L_{q} = 1 - 4/ [q (q + 4)]$ and $\overset{ˉ}{D}_{q} = (q^{2} + 4 q - 4) / (q - 1) = q + 5 + 1/ (q - 1)$ . At $s = 1$ the numerator in (C71) is $L_{q} (1 - 2 Z / \overset{ˉ}{D}_{q})$ . Consequently

$Res_{s = 1} D (s, z) \overset{ˉ}{C} \overset{ˉ}{F} (z) \overset{ˉ}{Q}_{A} = \overset{ˉ}{C} \overset{ˉ}{F} (z), = κ C_{0} p \in / S \prod L_{q_{p}} = κ^{5} p \in S \prod (1 - q_{p}^{- 1})^{4} p \in / S \prod (1 + 4/ q_{p} - 4/ q_{p}^{2}) (1 - q_{p}^{- 1})^{4}, = p \in / S \prod (1 - 2 \sum_{i} q_{p}^{- z_{i}} / \overset{ˉ}{D}_{q_{p}}) = \prod_{i} ζ_{K} (1 + z_{i})^{- 2} \overset{ˉ}{H} (z), = e_{1} e_{2} e_{3} e_{4} sf (e_{1} e_{2} e_{3} e_{4}, S) = 1 \sum \frac{\prod _{i} μ ( e _{i} ) 2 ^{ω (e_{i})} W ( q _{e_{i}} / A )}{\prod _{p ∣ e_{1} e_{2} e_{3} e_{4}} D ˉ _{q_{p}}} . (C73)$

The Euler product for $\overset{ˉ}{F}$ is literal for $ℜ z_{i} > 0$ ; elsewhere use its displayed factorization. Outside $S$ , $\overset{ˉ}{H}_{p}$ is its numerator divided by $\prod_{i} (1 - q_{p}^{- 1 - z_{i}})^{2}$ , and inside $S$ it is the inverse denominator. Cancellation of linear terms proves normal convergence for every $ℜ z_{i} > - 1/2$ , with uniform bounds on fixed closed real substrips, as in (C55). Thus the $a$ average changes the prime denominator to $q + 5 + 1/ (q - 1)$ , while each of the four inverse-zeta exponents stays $- 2$ . This is a coupled theta, not a square of a two-variable theta.

Conditional absolute-error evaluation. Assume the explicit $R_{β}$ . For fixed $0 < h < min {1 - β, 1/2}$ and $h < σ < 1$ ,

$T_{d} (U) = \frac{C ˉ}{6} X I (1/6) \overset{ˉ}{Q}_{A} + O_{K, S, W, h, σ, R_{β}} (N_{J} X^{σ} A^{- 4 h}), ∣ \overset{ˉ}{Q}_{A} ∣ ≪_{K, S, W, h, R_{β}} A^{- 4 h} . (C74)$

These are absolute bounds, uniform for the specified fixed finite $d$ family and profiles in one common envelope. The moment is signed or complex; it is not a positive mass or a claimed nonzero number.

Proof. Fivefold Mellin inversion initially on $ℜ s > 1$ , $ℜ z_{i} > 0$ gives the integral of $[\widehat I(s/6)/6]X^s\prod_i[\widehat W(z_i)A^{z_i}] \mathcal D(s,\mathbf z) $, w i t hm e a s u re$ (2\pi i)^{-5}ds,d\mathbf z$. Move the four $z_{i}$ lines to $- h$ , then the $s$ line to $σ$ . The condition $σ > h$ keeps all mixed terms inside (C72). Only $s = 1$ contributes a residue, exactly (C73) and its fourfold Mellin integral $\overset{ˉ}{Q}_{A}$ . The factor $1/6$ is from the sixth norm power. Moreover $W (z) = - z^{- 1} \int_{1}^{2} W^{'} (t) t^{z} d t$ has only a simple pole at zero, with residue $1$ . The corresponding double zero of $ζ_{K} (1 + z_{i})^{- 2}$ cancels it throughout this deformation. There are no $z_{i} = 0$ residues. This cancellation says nothing about poles at $z_{i} = ρ - 1$ for zeros $ρ$ of $ζ_{K}$ .

For explicit height control set $δ = 1 - β - h > 0$ . The premise $R_{β}$ bounds each inverse-zeta square by $O ((1 + ∣ℑ z_{i} ∣)^{2 B_{δ}})$ on the shifted strips. Fixed-field meromorphic continuation supplies polynomial zeta growth of some finite degree $L$ on the $s$ strip away from $1$ ; bounded-height joins detour that pole. Choose one finite $J > L + 8 B_{δ} + 20$ , enlarged for the original derivative contract. A sufficient envelope bounds $sup_{t \geq 0} (1 + t)^{J + 3} ∣ I_{4}^{(j)} (t) ∣$ for $0 \leq j \leq J$ , together with the required original-profile and cutoff seminorms. Integration by parts gives rapid Mellin decay on the fixed positive $s /6$ strips and on the $z_{i}$ lines. This pays all five vertical integrals, horizontal joins and tails by absolute integrability, giving the error in (C74). The fourfold argument for $\overset{ˉ}{F}$ proves its second bound. No growing-height or new reciprocal allowance is used. All component/height seminorms are aggregated against absolute coefficient measures within the same fixed or polylogarithmic budget.

There is no extra $\prod_{p \in S} (1 - 1/ q_{p})$ from $a$ : its ideals include all $S$ factors, accounting for the fifth $κ$ . The factor $d^{- 1/6}$ is already in $X$ . The six units remain in the explicit finite sum with $η_{ϵ, r}$ , without an extra factor or division by six. The original $U^{2}$ , $D^{- 1}$ side factors and physical-row/core prefactors of (C59)–(C60) must all be restored in their original places.

Scale and remaining obligation. At source-conditional $R_{7/8}$ , choose $h$ just below $1/8$ and $σ$ just above $h$ , allocating fixed margins to $ε$ . For every $ε > 0$ this yields

$\overset{ˉ}{Q}_{A} T_{d} (U) T_{d} (U) / X ≪_{ε} U^{- 1/24 + ε}, = \frac{C ˉ}{6} d^{- 1/6} U^{1/6} I (1/6) \overset{ˉ}{Q}_{A} + O_{ε} (N_{J} U^{- 1/48 + ε}), = \frac{C ˉ}{6} I (1/6) \overset{ˉ}{Q}_{A} + O_{ε} (N_{J} U^{- 3/16 + ε}) . (C75)$

Thus this isolated $U^{2} T_{d}$ has principal bound $O (N_{J} U^{17/8 + ε})$ and evaluation error $O (N_{J} U^{95/48 + ε})$ , times its original prefactor in modulus. Relative to $U^{13/6}$ the principal saving is still $1/24$ ; the new error saves $3/16 > Δ$ . Since $Δ - 1/24 = 0.040816237971963 \dots$ , the principal bound still misses the sufficient local target $U^{1 + B_{*} - ν}$ . The evaluation error is below that target when $ε + ν < 3/16 - Δ$ .

If the same $I_{4}$ is used throughout the finite sum, its leading scalar is $\widehat I(1/6)\sum_{\epsilon,\mathbf r} \eta_{\epsilon,\mathbf r}d_{\mathbf r}^{-1/6}$. If profiles differ, retain their individual moments inside that sum. An exact vanishing of this scalar would remove this residue, but must be proved from the actual profiles and coefficient tables. Smoothness alone neither forces nor excludes it; no such cancellation is proved here. Without it, one sufficient new hypothesis, with the same uniform cutoff/derivative contract, is

$∣ \overset{ˉ}{Q}_{A} ∣ ≪_{ε} A^{- 12 (Δ + ν) + ε}, 12Δ = 0.989794855663562 \dots, (C76)$

or a sufficient signed bound for the actual finite aggregate. The provided contour gives exponent $1/2$ , not this exponent. Further joint deformation must control the residue itself: $R_{7/8}$ does not exclude zeros at or left of $7/8$ , bound crossed zero residues, or force $\overset{ˉ}{H}$ to vanish there. No new zero-free input or inverse-zeta pole cancellation is assumed. All old (C60) error estimates remain unchanged. The target bounds for the other fifteen $μ$ terms, $R_{tw}$ , (C68), other cores/gcds, bad duals and the original same- $β$ signed J40 aggregation remain unpaid. A selected Poisson component has no positivity or monotonicity under physical-row selection. This reduction is paper-level only and does not strengthen the principal estimate, prove a lower bound or impossibility theorem, certify the manuscript, or close J40, signed Robin restoration or RH.

The joint mixed-Mobius and residue companion proves a source-conditional improvement of the fifteen radial mixed terms, with the original profiles and punctures, and records the actual finite residue interface. Its scope and remaining J40 obligations are explicit.

A small reduced-frequency sector of the actual C68 columns

The reduced-frequency companion retains C77–C86 for the small nonunit sector of the actual bare-J4 columns, with the original and exact redesigned cutoffs. Its original-W0 dominant endpoint retains C87–C92’s sharper $R^{1/6}$ bound on $q_{r} ≍ U$ . The all-annulus original-W0 endpoint extends bare nonunit coverage in C93–C97 to every $r > 1$ , with all sixth-power repetitions, independent physical-row and column scales, one original coupled envelope and the weighted large-shell tail. The source-conditional raw $U^{κ}$ , $D^{- 2}$ -restored $U^{B_{*}}$ endpoint has zero strict C68 reserve; the exceptional original-W0 principal scalar, other branches and full C68/J40/RH remain open. C51–C76 remain in this note.

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