Metadata

bibkey: corvalan2023interpolation
authors: Álvaro Corvalán
year: 2023
title: Interpolation and Extrapolation Statements equivalent to the Riemann Hypothesis
doi: null
url: https://arxiv.org/abs/2312.00211v3
claim: "The preprint proposes fixed-weight extrapolation closure criteria. Its printed dictionary and quantifier rewrites require calibration; the weighted norm construction admits the existing actual FIB target transport, but no required approximation estimate is supplied."
strata_touched: []
license: citation-only
triage: anchor

Fixed-weight approximation and the actual FIB target

The inspected primary is arXiv:2312.00211v3, revised 1 June 2026. Its PDF has SHA256 28ef4a091c67287cea59a93f9ef447450978257c3b0b6182860ce8bbac08f05e; the author-source archive has SHA256 f756d275cf3a72a42dabe55898ba6f5794e12ae904cbe9d8579df6d81341f71d. The selected definitions, Criterion 21, Theorem 24 and their supporting arguments were read. This is a scoped source-applicability analysis and paper application, without an independent whole-paper proof audit, originality claim or Lean certification. The printed criterion is not adopted without the normalization qualifications below.

What the proposed sufficient condition actually asks for

Let

$E_{θ} = (L^{1} (0, 1), L^{2} (0, 1))_{θ, \infty; K}, ∥ f ∥_{E_{θ}} = t > 0 sup t^{- θ} K (t, f; L^{1}, L^{2}), p (θ) = \frac{2}{2 - θ} .$

Criterion 21, author TeX label N-B-with-extrapol, proposes a positive fixed weight $M$ , tempered at $θ = 1$ and $O (1)$ there, such that

$f \in Φ in f 0 < θ < 1 sup M (θ) ∥ χ - f ∥_{E_{θ}} = 0, χ = 1_{(0, 1)} . (I1)$

Theorem 24, label strong Nyman-Beurling-Milman equivalence, repeats the closure formulation. The weight is chosen once. For each error tolerance, one approximant must serve all interpolation parameters in the supremum; neither the weight nor the approximant may be chosen separately after observing each parameter. The paper supplies no coefficients or truncation-dependent error bound attaining this closure.

The cited Jawerth–Milman input is Theorem 20, label M_delta. It assumes a mutually closed Banach couple, tempered weight and exact interpolation functors with characteristic function $t^{θ}$ . The paper credits Milman’s Extrapolation and Optimal Decompositions with Applications to Analysis, Theorem 21, p.44. This standard interpolation result is reused as a literature input; its upstream proof is not reproved or independently certified here.

Calibration before using the printed criterion

The following distinctions concern the actual displayed source formulas. They do not establish that no corrected weighted criterion can hold.

The source writes its dictionary constraint as $\sum_{a} c_{a} ρ (1/ a) = 0$ with $a \geq 1$ , where $ρ (u) = u - ⌊ u ⌋$ . At $a = 1$ this uses $ρ (1) = 0$ . Consequently the printed dictionary admits $ρ_{1}$ alone and excludes an individual standard balanced probe $ρ_{n} - ρ_{1} / n$ for $n \geq 2$ . It is not literally the standard constraint $\sum_{a} c_{a} / a = 0$ . The definitions in the introduction, Theorem 4 and the later remarks retain this endpoint issue. The existing classical criterion and the balanced natural generators in the coefficient-functional application remain the separate dictionary suppliers.
Immediately after its norm-closure statement, Criterion 21 changes $in f_{f} sup_{θ}$ to $sup_{θ} in f_{f}$ . The same interchange recurs later. These expressions are not generally equal; (I1) is the uniform closure condition used here.
In the reverse proof the bound $M (θ) \leq k$ does not justify the printed lower comparison by $k ∥ χ - f ∥$ . At a fixed $θ$ the valid consequence of a small weighted norm is division by the positive fixed value $M (θ)$ .
Conditions only near $θ = 1$ do not control an arbitrary weight on the rest of $(0, 1)$ . A full-interval supremum needs a controlled extension there, or an explicitly restricted endpoint interval.

In particular, no RH equivalence for the source’s literal printed $Φ$ is imported into the project. The useful object retained below is its weighted norm, applied to an independently specified dictionary.

A controlled weight and an independently specified dictionary

For definiteness take a fixed $a > 0$ and

$M_{a} (θ) = (1 - θ)^{a}, ∥ f ∥_{X_{a}} = 0 < θ < 1 sup (1 - θ)^{a} ∥ f ∥_{E_{θ}} . (I2)$

This weight is positive and bounded throughout the interval, and $M_{a} ((1 + θ) /2) = 2^{- a} M_{a} (θ)$ supplies the endpoint tempering. It is an available choice, not a weight forced by the five FIB modes. No equivalence with an unweighted $L^{2}$ norm is assumed.

On $(0, 1)$ use the standard balanced natural span

$b_{n} (x) = {\frac{1}{n x}} - \frac{1}{n} {\frac{1}{x}}, V = span {b_{n} : n \geq 2} . (I3)$

Extend each function by zero outside $(0, 1)$ . The same formula for $b_{n}$ on the half-line is already zero for $x > 1$ . Write $W_{a} = \overline{V}^{X_{a}}$ ; this closure is defined without asserting that it contains $χ$ or the FIB target. Its definition is not a new RH criterion.

The existing Binet transport acts in this norm

Reuse FIB §391, including its actual coefficients and absolute inverse budget:

$q = φ^{- 2}, β_{d} = lo g (1 - (- q)^{d}), γ * β = β * γ = 1, d \sum ∣ γ_{d} ∣ \leq \frac{1}{β _{1} - T}, T = d \geq 2 \sum ∣ β_{d} ∣ < β_{1} .$

Here $β_{d}$ is a coefficient sequence, distinct from the ATOM $β = ρ (α)$ . For the zero-extended convention define

$U_{d} f (x) = 1_{(0, 1/ d)} (x) f (d x), B_{β} = d \geq 1 \sum β_{d} U_{d}, B_{γ} = d \geq 1 \sum γ_{d} U_{d}, Φ_{β} = B_{β} χ . (I4)$

These are the restrictions of the same support-preserving half-line operators. They are not ordinary unrestricted unit-interval dilations. The endpoint bounds are $∥ U_{d} ∥_{L^{1}} = d^{- 1}$ and $∥ U_{d} ∥_{L^{2}} = d^{- 1/2}$ . Applying the same operator to each term in the defining $K$ -functional decomposition therefore gives

$∥ U_{d} f ∥_{X_{a}} \leq d^{- 1/2} ∥ f ∥_{X_{a}} .$

Thus the existing absolute coefficient and inverse budgets give, in the new specified norm as well,

$∥ B_{β} ∥_{X_{a} \to X_{a}} \leq B_{1/2}, ∥ B_{γ} ∥_{X_{a} \to X_{a}} \leq G_{1/2}, B_{γ} B_{β} = B_{β} B_{γ} = I, (I5)$

where $B_{1/2} = \sum_{d} ∣ β_{d} ∣ d^{- 1/2}$ and $G_{1/2} = \sum_{d} ∣ γ_{d} ∣ d^{- 1/2} \leq (β_{1} - T)^{- 1}$ . No inverse theorem is reproved: this is the existing Dirichlet-inverse transport applied with the common endpoint bound.

The dictionary is compatible with the same operation:

$U_{d} b_{n} = b_{d n} - \frac{1}{n} b_{d}, b_{1} = 0. (I6)$

Consequently each finite coefficient truncation of either transport maps $V$ into $V$ . Operator-norm convergence then preserves $W_{a}$ , and the inverse gives $B_{β} W_{a} = W_{a}$ . In particular the actual-target error budgets satisfy

$\frac{dist _{X_{a}} ( χ , W _{a} )}{G _{1/2}} \leq dist_{X_{a}} (Φ_{β}, W_{a}) \leq B_{1/2} dist_{X_{a}} (χ, W_{a}) . (I7)$

This transports both the target and the permissible closed span. It provides no positive value, zero value or rate for either distance. The earlier $L^{2}$ support classification is not silently transferred to this different norm.

Finite transport tails can be paid independently

For a finite balanced approximant $v \in V$ and an integer $D \geq 1$ , use $B_{β, D} = \sum_{d \leq D} β_{d} U_{d}$ . The already supplied bound $∣ β_{d} ∣ \leq q^{d} / (1 - q)$ gives

$∥ B_{β} v - B_{β, D} v ∥_{X_{a}} \leq \frac{q ^{D + 1}}{( 1 - q ) ^{2} D + 1} ∥ v ∥_{X_{a}} . (I8)$

Hence an actual finite approximant to the transported target can be formed with the explicit error budget

$∥ Φ_{β} - B_{β, D} v ∥_{X_{a}} \leq B_{1/2} ∥ χ - v ∥_{X_{a}} + \frac{q ^{D + 1}}{( 1 - q ) ^{2} D + 1} ∥ v ∥_{X_{a}} . (I9)$

By (I6) the output still has finite balanced natural support. If the input indices are at most $N$ , the output indices are at most $D N$ . The exponentially small second term is a transport cost; the first term is the unproved approximation estimate. A small transport tail alone cannot make that first term small. Nor does (I9) assert that the transported finite coefficients are the fixed natural prefixes $e_{n} = (μ * β)_{n}$ considered in §391.

Remaining consumer

The source opens a norm choice for a future approximation argument. Using it requires an actual balanced family $v_{j}$ , one fixed weight, and a proved common-parameter bound $∥ χ - v_{j} ∥_{X_{a}} \to 0$ , or the corresponding actual-target bound. The displayed transport supplies neither. Any RH conclusion must also use the classical criterion with its correct dictionary and the required embeddings; it does not follow by importing the paper’s literal printed definition.

The five modes encode all natural indices and can organize the finite support in (I6). Their legal seams and interval-length conservation do not estimate this weighted approximation error. The Robin same-integer signed integral remains unpaid. This application reuses standard interpolation and the existing Binet inverse; it is not a new supplier theorem, a proof of RH or a claim that the weighted route outperforms existing approximation methods.

Strength of a local cutoff block

The logarithmic Selberg coefficients are already discussed in Báez-Duarte 2002, Introduction, printed p.2. The classical source Conrey–Myerson, arXiv:math/0002254v1, Theorem 1 and Remark 1 distinguishes uniform convergence of its fractional-part sums from the weighted integral near its moving endpoint. Its sums use ${n u} / n$ , whereas the actual residual below uses ${t / n}$ ; its pointwise estimate is not an upper bound for this residual’s local quadratic energy. The following is a repo-derived paper application of Mellin continuation to the specified coefficients, not a result stated in Corvalán’s paper, a new unconditional estimate, or a Lean-certified implication. No originality of the Mellin obstruction is claimed.

For integers $N \geq 3$ put

$L = lo g N, a_{N, n} = μ (n) \frac{lo g ( N / n )}{L} (1 \leq n \leq N), c_{N} = n \leq N \sum \frac{a _{N, n}}{n} .$

For the actual balanced approximant $v_{N} = - \sum_{2 \leq n \leq N} a_{N, n} b_{n}$ , its residual on $1/ (m + 1) < x < 1/ m$ is

$R_{N} (m) = 1 + c_{N} m - n \leq m i n (N, m) \sum a_{N, n} ⌊ m / n ⌋ .$

Thus, for integer $m \geq N$ , $R_{N} (m) = 1 + \sum_{n \leq N} a_{N, n} {m / n}$ . Define the actual local energy

$E_{N} = m = N \sum N^{2} \frac{∣ R _{N} ( m ) ∣ ^{2}}{m ( m + 1 )} . (I10)$

A proposed supply $E_{N} \leq C (1 + lo g N)^{A}$ , with one finite $A \geq 0$ and one $C$ for all sufficiently large integers $N$ , already entails RH by the argument below. The estimate cannot be treated as an independently paid, weaker precursor to (I1). This implication is a one-way strength analysis. The later classical-error application supplies a paper converse only for the every-epsilon subpower version of the first proportional block; no fixed polylogarithmic supplier is established.

Passing integer cutoffs and cells without losing the actual source

For real $y > 1$ and $t > 0$ retain the same coefficients through

$S_{y} (t) = lo g y + n \leq y \sum μ (n) lo g (y / n) {t / n} .$

This is an auxiliary interpolation: at integer arguments $S_{N} (m) = L R_{N} (m)$ ; within a cell its difference from the balanced step residual is accounted for by (I12). If $N \leq y < N + 1$ , the summation indices agree and, uniformly for every real $t$ ,

$S_{y} (t) - S_{N} (t) = lo g (y / N) (1 + n \leq N \sum μ (n) {t / n}), ∣ S_{y} (t) - S_{N} (t) ∣ \leq 2. (I11)$

For $m \leq t < m + 1$ , all $⌊ t / n ⌋ = ⌊ m / n ⌋$ . Consequently

$S_{N} (t) - S_{N} (m) = (t - m) U_{N}, U_{N} = n \leq N \sum \frac{μ ( n ) lo g ( N / n )}{n}, ∣ U_{N} ∣ \leq L (1 + lo g N) . (I12)$

Using $\int_{m}^{m + 1} t^{- 2} d t = 1/ [m (m + 1)]$ , equations (I11)–(I12) and the assumed estimate yield

$\int_{y}^{2 y} ∣ S_{y} (t) ∣^{2} \frac{d t}{t ^{2}} ≪ L^{A + 2} + \frac{( 1 + L ) ^{4}}{N} . (I13)$

Indeed, all intersecting cells have $N \leq m \leq 2 N + 1 \leq N^{2}$ for $N \geq 3$ ; the cell slope costs $O (∣ U_{N} ∣^{2} / N)$ and the cutoff correction costs $O (1/ N)$ . On changing variables $t = q y$ , (I13) gives, for some fixed finite $B$ ,

$∥ S_{y} (q y) ∥_{L^{2} (1 < q < 2)} ≪ y (1 + lo g y)^{B} . (I14)$

The small bounded range of $y$ is absorbed in the constant. These steps use the actual step-function residual; they do not assume that fractional parts are uniformly Lipschitz across their jumps.

Mellin in the cutoff variable exposes a reciprocal-zeta factor

For every fixed smooth test function $h$ supported in $(1, 2)$ , set

$F_{h} (s) = \int_{1}^{\infty} \int_{1}^{2} h (q) S_{y} (q y) d q y^{- s - 1} d y .$

Equation (I14) and Cauchy–Schwarz make this integral and its $s$ -derivatives absolutely convergent on compact subsets of $ℜ s > 1/2$ . Thus $F_{h}$ is holomorphic there. Initially for $ℜ s > 1$ , absolute interchange and $y = n v$ give

$F_{h} (s) = \frac{\int _{1}^{2} h ( q ) d q}{s ^{2}} + \frac{H _{h} ( s )}{ζ ( s )}, H_{h} (s) = \int_{1}^{2} h (q) H_{q} (s) d q, H_{q} (s) = \int_{1}^{\infty} lo g v {q v} v^{- s - 1} d v . (I15)$

Here $\sum μ (n) n^{- s} = 1/ ζ (s)$ is the classical Dirichlet identity, reused on its absolute-convergence half-plane. The integrals $H_{q}$ are holomorphic for $ℜ s > 0$ and locally uniformly bounded in $q$ and $s$ .

There is no common zero of $H_{q} (s)$ as $q$ runs through $(1, 2)$ . For a fixed $s$ with $ℜ s > 0$ , write

$J_{q} (s) = q^{- s} H_{q} (s) = \int_{q}^{\infty} lo g (t / q) {t} t^{- s - 1} d t .$

Differentiation at $1 < q < 2$ gives

$\frac{d}{d q} (q \frac{d}{d q} J_{q} (s)) = (q - 1) q^{- s - 1} \neq = 0. (I16)$

Hence $H_{q} (s)$ cannot vanish identically on any open subinterval of $(1, 2)$ . If a nontrivial zero $ρ$ had $ℜ ρ > 1/2$ , continuity in $q$ would supply a smooth compactly supported $h$ with $H_{h} (ρ) \neq = 0$ . Multiply (I15) by $(s - 1) ζ (s)$ ; analytic continuation extends the identity to $ℜ s > 1/2$ . Evaluation at $ρ \neq = 1$ forces $H_{h} (ρ) = 0$ , a contradiction. This argument allows arbitrary zero multiplicity. The usual functional-equation symmetry then gives RH.

Only a fixed proportional part of (I10) is needed for this obstruction. If its upper endpoint is $2 N$ , restrict $q$ to $(5/4, 7/4)$ ; for large $N$ every intersecting cell lies in $[N, 2 N]$ , while (I16) still prevents a common zero. Thus discarding the remote part $2 N < m \leq N^{2}$ does not make the proposed uniform polylogarithmic estimate an already available supplier.

For the actual FIB application, (I5)–(I9) still transport any error estimate that is genuinely supplied. They provide no value for (I10). This strength analysis closes neither the required uniform approximation nor the same-integer signed Robin margin.

A paid low-denominator part of the actual residual

Reuse the existing discrete large-sieve input, equation (1), namely Montgomery’s Theorem 3, printed p.559, and the duality in §4, Lemma 2. The following is a paper application to the coefficients and residual of (I10); the classical large sieve is not reproved. This reduction alone supplies no bound for the complete energy.

Write $e (u) = exp (2 πi u)$ and retain the zero-frequency coefficient

$d_{N} = 1 + \frac{1}{2} n \leq N \sum a_{N, n} (1 - 1/ n), T_{N} (m) = R_{N} (m) - d_{N} . (I17)$

The finite Fourier expansion of the periodic sawtooth at integer $m$ is

${m / n} - \frac{n - 1}{2 n} = - \frac{1}{n} h = 1 \sum n - 1 \frac{e ( hm / n )}{1 - e ( - h / n )} .$

It follows by summing $j e (- hj / n)$ over $0 \leq j < n$ ; $n = 1$ has an empty sum. Merge every frequency $h / n = r / q$ into its reduced fraction before applying a separation estimate. This gives, exactly for $m \geq N$ ,

$T_{N} (m) = 2 \leq q \leq N \sum 1 \leq r < q (r, q) = 1 \sum B_{N, q, r} e (m r / q), B_{N, q, r} = - \frac{b _{N, q}}{q ( 1 - e ( - r / q ))}, b_{N, q} = j \leq N / q \sum \frac{a _{N, q j}}{j} . (I18)$

This retains all aliases of the original coefficients, rather than assuming that the unreduced list is separated. Non-squarefree $q$ have $b_{N, q} = 0$ . For squarefree $q$ , the coefficient is the short coprime sum

$b_{N, q} = \frac{μ ( q )}{lo g N} j \leq N / q (j, q) = 1 \sum \frac{μ ( j )}{j} lo g \frac{N}{q j} . (I19)$

This is exactly the existing logarithmic coprime Möbius sum $\overset{m}{ˇ}_{q}^{(1)}$ in the Zuniga-Alterman source note, with $X = N / q$ and the prefactor $μ (q) / lo g N$ . A fixed- $q$ asymptotic alone would not suffice when $q, X$ vary together. Instead, reuse its uniform finite bound, obtained from Lemma 5.1, equation (5.2), the positive convolution (7.6) and Lemma 7.2(a):

$∣ \overset{m}{ˇ}_{q}^{(1)} (X) ∣ \leq 2 q / φ (q) (q \geq 1, X > 0) .$

Here $φ (q)$ denotes Euler’s totient, distinct from the golden ratio used for the FIB geometry. It follows that $∣ b_{N, q} ∣ \leq 2 q / [φ (q) lo g N]$ for every squarefree $q \leq N$ , including the short sums with $q$ close to $N$ . Non-squarefree $q$ still contribute zero. This uses the source’s uniform convolution bound rather than inserting a fixed-parameter limit.

For $1 \leq Q \leq N$ , the coefficient energy therefore obeys

$V_{N} (Q) := 2 \leq q \leq Q \sum 1 \leq r < q (r, q) = 1 \sum ∣ B_{N, q, r} ∣^{2} ≪ \frac{Q}{( lo g N ) ^{2}} . (I20)$

Indeed, the elementary sine bound gives $\sum_{r = 1}^{q - 1} ∣1 - e (- r / q) ∣^{- 2} ≪ q^{2}$ . Reuse the existing finite positive-moment bound with $s = 2$ and $X = Q$ : $\sum_{q \leq Q} (q / φ (q))^{2} \leq Q W (2)$ , where the fixed Euler product $W (2)$ converges. Combining these two inputs proves (I20) with a constant independent of $N, Q$ . Neither input estimates the signed quadratic cross terms of the residual.

Let $I$ be a consecutive integer interval in $[M, 2 M) \cap [N, N^{2}]$ , with integer $M \geq N$ , and use the actual weight

$∥ f ∥_{I}^{2} = m \in I \sum \frac{∣ f ( m ) ∣ ^{2}}{m ( m + 1 )} .$

For $1 \leq Q \leq N$ , split (I18) as $T_{N} = T_{N, \leq Q} + T_{N, > Q}$ . Distinct reduced fractions of denominator at most $Q$ have circular separation at least $Q^{- 2}$ . In the quoted dual large sieve, take these frequencies, the coefficients $B_{N, q, r}$ , and the actual interval $I$ , whose length $H$ is at most $M$ . Since $m (m + 1) \geq M^{2}$ , (I20) gives

$∥ T_{N, \leq Q} ∥_{I}^{2} ≪ \frac{Q ( M + Q ^{2} )}{M ^{2} ( lo g N ) ^{2}} . (I21)$

In particular, if $Q^{2} \leq M$ , the weighted triangle inequality gives

$∣ ∥ R_{N} ∥_{I} - ∥ d_{N} + T_{N, > Q} ∥_{I} ∣ \leq ∥ T_{N, \leq Q} ∥_{I} ≪ \frac{Q / M}{lo g N} \leq \frac{M ^{- 1/4}}{lo g N} . (I22)$

This is a norm error for removing a specified part, not an orthogonal decomposition on the short interval. Take disjoint blocks $I_{j} = [2^{j} N, 2^{j + 1} N) \cap [N, N^{2}] \cap Z$ and $Q_{j} = ⌊ 2^{j} N ⌋$ , keeping the nonempty blocks $0 \leq j \leq ⌊ lo g_{2} N ⌋$ . Define the retained residual $A_{N} (m) = d_{N} + T_{N, > Q_{j}} (m)$ for $m \in I_{j}$ . Applying (I21) on each block and summing the geometric factors $2^{- j /2}$ gives the whole crossing-block comparison

$E_{N} - m = N \sum N^{2} \frac{∣ A _{N} ( m ) ∣ ^{2}}{m ( m + 1 )}^{1/2} ≪ \frac{N ^{- 1/4}}{lo g N} . (I23)$

Thus the discarded low-denominator part has an independently paid norm bound on the complete block. The retained frequencies satisfy $q > M$ on the block at scale $M$ ; their coefficients in (I19) involve only $j < N / M$ . The zero-frequency term remains tied to the same coefficients:

$d_{N} = 1 + \frac{1}{2} (S_{N} - c_{N}), S_{N} = n \leq N \sum a_{N, n} = \frac{1}{lo g N} \int_{1}^{N} \frac{M ( t )}{t} d t, M (t) = n \leq t \sum μ (n) . (I24)$

At scale $M ≍ N$ , applying the same large sieve to all denominators would give the allowance $O (N / lo g^{2} N)$ , not a polylogarithmic bound. Neither (I20) nor (I23) controls the retained signed cross terms or their interaction with $d_{N}$ . Separate bounds for $d_{N}$ and each component are sufficient but need not be necessary, because the terms can cancel on these finite intervals. A uniform polylogarithmic bound for the retained norm squared would, by (I23) and the preceding strength analysis, still entail RH.

Here $q$ is a reduced frequency denominator, not necessarily a prime. The cut $q \leq Q$ is a cut by frequency order, not by numerical frequency magnitude or a single residue observation. Observing modulo $D = lcm (1, \dots, Q)$ retains all orders dividing $D$ , including some greater than $Q$ ; for example $Q = 3$ also retains order $6$ . Thus the paid spectral cut cannot silently replace the earlier modulus-based visible/hidden cut. The five FIB modes can encode the indices $q, j$ ; their separate address statistics do not determine the joint condition $(q, j) = 1$ or the short Möbius sum (I19). Squarefree composite denominators are retained. This application narrows the unestimated arithmetic part while preserving the actual source, mean and weight. It supplies neither the complete FIB approximation rate nor the same-integer Robin sign, and has not been Lean compiled or claimed as an original large-sieve result.

A numerical-frequency cut of the same residual

A frequency’s reduced denominator and its circular distance from an integer are different observations. The same three inputs used in (I20)–(I23) also pay for deleting frequencies away from integers. This is another application to the actual coefficients, not a new large-sieve theorem or a claim that the remaining quadratic form is small. Write $L = lo g N$ and let $0 < η \leq 1/4$ .

The elementary sine bound, including the case $η q < 1$ , gives

$1 \leq r < q ∥ r / q ∥ \geq η \sum ∣1 - e (- r / q) ∣^{- 2} ≪ \frac{q ^{2}}{1 + η q} .$

Here $∥ t ∥$ is distance to the nearest integer. Restricting the sum to coprime numerators only decreases it. Apply the uniform coefficient bound following (I19), then partial summation to the already supplied finite moment $\sum_{q \leq X} (q / φ (q))^{2} \leq X W (2)$ . The angular coefficient energy satisfies

$V_{N, η}^{away} := 2 \leq q \leq N \sum 1 \leq r < q (r, q) = 1 ∥ r / q ∥ \geq η \sum ∣ B_{N, q, r} ∣^{2} ≪ \frac{1}{L ^{2}} q \leq N \sum \frac{( q / φ ( q ) ) ^{2}}{1 + η q} \leq \frac{W ( 2 ) lo g ( 1 + η N )}{η L ^{2}} . (I25)$

For precision, the implied constant in the first inequality also multiplies the final allowance. The final comparison for the weighted positive-moment sum follows from integrating $(1 + η t)^{- 1}$ on $0 \leq t \leq N$ ; it does not use a fixed-moment limiting average in place of a finite bound.

Retain the actual blocks $I_{j}$ , $M_{j} = 2^{j} N$ and $Q_{j}$ of (I23). On each block define the exponential sum

$D_{N, η, j} (m) = Q_{j} < q \leq N \sum 1 \leq r < q (r, q) = 1 ∥ r / q ∥ \geq η \sum B_{N, q, r} e (m r / q) .$

Apply the same large sieve separately on dyadic denominator shells $Q \leq q < 2 Q$ , $Q = 2, 4, \dots$ , truncated at $N$ . On a shell, the coefficient argument above gives the uniform allowance

$V_{Q}^{away} ≪ \frac{1}{η L ^{2}} Q \leq q < 2 Q \sum \frac{( q / φ ( q ) ) ^{2}}{q} ≪ \frac{1}{η L ^{2}} .$

This includes $η Q < 1$ . On each actual block keep the additional $q > Q_{j}$ restriction: it only decreases that shell’s coefficient energy. The shell’s separation cost is $O (Q^{2})$ , not $N^{2}$ . For the shell response $D_{Q}$ defined piecewise on these blocks,

$∥ D_{Q} ∥^{2} ≪ V_{Q}^{away} j \sum \frac{M _{j} + Q ^{2}}{M _{j}^{2}} ≪ \frac{1}{η L ^{2}} (\frac{1}{N} + \frac{Q ^{2}}{N ^{2}}) .$

The full weighted norm is on $N \leq m \leq N^{2}$ . Use its triangle inequality across denominator shells only after this estimate. There are $O (1 + L)$ shells and $\sum_{Q} Q ≪ N$ , so

$(j \sum ∥ D_{N, η, j} ∥_{I_{j}}^{2})^{1/2} ≪ \frac{1}{η L} (\frac{1 + L}{N} + \frac{\sum _{Q} Q}{N}) ≪ \frac{1}{η L} . (I26)$

This uses the original large-sieve separation on every shell. It avoids the extra logarithm that would result from applying the full $N^{2}$ separation cost after merging all shell energies in (I25). No improvement of arbitrary-coefficient separation by angular width is assumed. The last block and endpoint $N^{2}$ remain included. Define the retained response, for $m \in I_{j}$ , by

$C_{N, η} (m) = d_{N} + Q_{j} < q \leq N \sum 1 \leq r < q (r, q) = 1 ∥ r / q ∥ < η \sum B_{N, q, r} e (m r / q) .$

The deleted response is the sum of all low-denominator frequencies and the high-denominator frequencies away from integers. These two sets are disjoint; their functions need not be orthogonal on $I_{j}$ . Combining (I23), (I26) and the weighted triangle inequality yields

$E_{N} - m = N \sum N^{2} \frac{∣ C _{N, η} ( m ) ∣ ^{2}}{m ( m + 1 )}^{1/2} ≪ \frac{N ^{- 1/4}}{L} + \frac{1}{η L} . (I27)$

All constants are independent of $N, η$ . For example, $η_{N} = (4 lo g N)^{- 1}$ lies in $(0, 1/4]$ for $N \geq 3$ , and the right side is $O ((lo g N)^{- 3/4})$ . Thus an actual shrinking angular region, together with the same complete-period mean, retains the weighted norm up to a vanishing error. For the asymptotic choice $η_{N} = (lo g N)^{- a}$ this allowance tends to zero when $0 < a < 2$ ; at $a = 2$ it gives only $O (1)$ . If $η \leq 1/ N$ , all nonzero frequencies are deleted. The displayed angular allowance is then at least $N / L$ ; alternatively the unrestricted coefficient bound (I20) gives the allowance $O (N / L)$ directly. Taking an arbitrarily narrow angular region therefore does not supply a small-error reduction through these estimates.

The remaining frequencies satisfy both $q > M_{j}$ and $∥ r / q ∥ < η_{N}$ . They need not be nearly constant over a block: $M_{j} η_{N}$ can grow. The symmetric angular cut preserves conjugate pairing, so the retained and deleted responses remain real. It leaves all signed correlations within the retained set and with $d_{N}$ unpaid. A bound for an isolated reciprocal-phase Möbius sum would require a further uniform summation and kernel correspondence before it could estimate this complete norm. As with (I23), a uniform polylogarithmic bound for the retained norm squared would still imply RH.

The earlier Mellin strength argument also has a quantitative parameter substitution. For any one fixed $0 \leq ϑ < 1$ , a bound $E_{N} ≪ N^{ϑ} (1 + lo g N)^{A}$ for all sufficiently large integers $N$ , with one finite $A$ , changes (I14) to $∥ S_{y} (q y) ∥_{2} ≪ y^{(1 + ϑ) /2} (1 + lo g y)^{B}$ . The same integral (I15) is then holomorphic for $ℜ s > (1 + ϑ) /2$ , and (I16) excludes zeros in that strict half-plane. Consequently any fixed power saving $E_{N} ≪ N^{1 - δ} (1 + lo g N)^{A}$ , $0 < δ \leq 1$ , would already supply a fixed zero-free half-plane $ℜ s > 1 - δ /2$ at every height. This is a parameter application of the preceding paper argument, not an independently supplied bound. A saving parameter that tends to zero with $N$ cannot be substituted for a fixed $δ$ , and no conclusion on the boundary line is asserted.

This is a numerical-frequency observation of the same periodic response. It is not the FIB composition rotation $C = M J$ , and no arithmetic invariance under that active rotation is asserted. The five-mode encoding can organize the retained indices without supplying their Möbius cancellation. Equations (I25)–(I27) remain paper applications of the cited existing inputs, without a new Lean declaration or an originality claim.

A numerator cut that follows the observation block

The same supplied inputs pay a narrower restriction on the circular numerator $k (r, q) = min (r, q - r)$ , distinct from the angle $k (r, q) / q$ . For every integer $K \geq 1$ , the sine estimate used in (I25) gives

$1 \leq r < q (r, q) = 1 k (r, q) > K \sum ∣ B_{N, q, r} ∣^{2} ≪ \frac{∣ b _{N, q} ∣ ^{2}}{K + 1}, 2 \leq q \leq N \sum 1 \leq r < q (r, q) = 1 k (r, q) > K \sum ∣ B_{N, q, r} ∣^{2} ≪ \frac{N}{( K + 1 ) L ^{2}} . (I28)$

Indeed, $∣1 - e (- r / q) ∣ \geq 4 k (r, q) / q$ , and at most two numerators have a given positive $k$ . Sum the tail of $k^{- 2}$ and use the same uniform $b_{N, q}$ bound and finite totient moment as in (I20). When $K \geq q /2$ the corresponding tail is empty.

Fix $0 < b < 2$ and use the actual blocks and denominator cutoffs of (I23). On block $I_{j}$ , put

$K_{j} = ⌈ \frac{N ^{2}}{M _{j} L ^{b}} ⌉ .$

Let $G_{N, η, b, j}$ be the sum over the frequencies retained in $C_{N, η}$ for which $k (r, q) > K_{j}$ , with their original coefficients and phases. Its coefficient energy is bounded by (I28). The original $N^{- 2}$ separation, the actual weight and $K_{j} + 1 \geq N^{2} / (M_{j} L^{b})$ give

$j \sum ∥ G_{N, η, b, j} ∥_{I_{j}}^{2} ≪ j \sum \frac{N ( M _{j} + N ^{2} )}{M _{j}^{2} ( K _{j} + 1 ) L ^{2}} ≪ L^{b - 2} (\frac{1 + ⌊ lo g _{2} N ⌋}{N} + j \sum \frac{N}{M _{j}}) ≪ L^{b - 2} . (I29)$

The last geometric sum is at most two, and the other term is uniformly bounded for $N \geq 3$ . Empty frequency sets and the final singleton block require no exception.

Define $C_{N, η, b}^{num}$ on $I_{j}$ by retaining exactly $q > Q_{j}$ , $∥ r / q ∥ < η$ , and $k (r, q) \leq K_{j}$ , together with the unchanged $d_{N}$ . The added deletion intersects the previous retained set, so no frequency is subtracted twice. Equations (I27)–(I29) give

$E_{N} - ∥ C_{N, η, b}^{num} ∥ ≪ \frac{N ^{- 1/4}}{L} + \frac{1}{η L} + L^{b /2 - 1} . (I30)$

Here the norm is the same weighted norm on the complete interval $N \leq m \leq N^{2}$ . The numerator deletion itself does not require an angular restriction: dropping that restriction from its definition still gives (I29), and comparison with the denominator-only retained response has error $O (N^{- 1/4} / L + L^{b /2 - 1})$ .

For example, use $η = (4 L^{3/2})^{- 1}$ and $b = 3/2$ . The complete comparison still costs $O (L^{- 1/4})$ , while the remaining circular numerators also obey $k \leq ⌈ N / (2^{j} L^{3/2})⌉$ . For $q ≍ N$ this restricts the angle to $O (2^{- j} L^{- 3/2} + N^{- 1})$ . This is a further paid restriction, including an increasingly small numerator allowance on later blocks; it is not a bound for the retained signed norm.

The original mean cannot be removed by rebinding the response at zero. Although $R_{N} (0) = 1$ and the exact full expansion can be written as $1 + \sum B_{N, q, r} (e (m r / q) - 1)$ , deleting a set in that expression would remove $G (m) - G (0)$ . Estimate (I29) controls $G (m)$ on the actual blocks and does not separately control the constant $G (0)$ . Keeping the original $d_{N}$ preserves the paid estimate.

For $q > N /2$ the actual coefficient is $b_{N, q} = μ (q) lo g (N / q) / L$ . Even after (I30), the earliest block contains $m ≍ N$ , $k = 1$ , $q ≍ N$ , with reciprocal-phase parameter $mk ≍ N$ . Results requiring a faster phase do not cover that endpoint. The coprimality condition, exact Fourier kernel, block-dependent cutoff and mean interaction all remain part of any supplier obligation. This is a further application of the same cited inputs, without a new supplier theorem, originality claim or Lean verification.

Reusing a known centered coefficient family

Ehm’s Gram-matrix and centering source, Section 8.2 and Proposition 8.1, already provides a correction to the Selberg coefficients for centering the relevant moments. Its known construction need not be reinvented. The linked application compares that modified balanced family with the actual family here: the full $L^{2} (0, 1)$ norm cost is $O (1/ lo g N)$ , and the difference of the square roots of the actual local energies is $O (lo g N / N)$ , for any fixed correction constants. The centering property additionally requires the source’s stated linear system. These allowances permit honest transfer of a future complete estimate; the source leaves its signed inversion error unestimated. They do not reuse the original squarefree Fourier support for the modified coefficients, or supply the retained joint bound in (I30).

A complete energy allowance from the existing Mertens input

The Ng source note already records the classical unconditional Vinogradov–Korobov estimate for the actual $M (x) = \sum_{n \leq x} μ (n)$ . Reuse it without proving a new Mertens theorem. For some fixed $c_{0} > 0$ and all sufficiently large $x$ ,

$∣ M (x) ∣ ≪ x e^{- c_{0} Ξ (x)}, Ξ (x) = \frac{( lo g x ) ^{3/5}}{( lo g lo g x ) ^{1/5}} .$

The existing source note also records the Lee–Leong v5 explicit supplier. Only the asymptotic allowance above is used here; no numerical constant or starting threshold is certified in this application. It pays the following complete energy bound for the original coefficients, mean and discrete weight:

$E_{N} ≪ N e^{- c Ξ (N)} for some fixed c > 0. (I31)$

Here and below $N$ is a sufficiently large integer. To check the actual parameters, fix $0 < κ < c_{0} /8$ and put

$Y = N e^{- κ Ξ (N)}, X = N e^{κ Ξ (N)} .$

Eventually $1 < Y < N < X < N^{2}$ . Uniformly for $Y \leq x \leq N$ , $Ξ (x) /Ξ (N) \to 1$ , since $Ξ (N) = o (lo g N)$ . Thus the known Mertens input gives

$Y \leq x \leq N sup ∣ M (x) ∣ ≪ N e^{- (c_{0} /2) Ξ (N)} .$

For the actual integer samples $N \leq m \leq X$ , split the $n$ -sum in $R_{N} (m)$ at $Y$ . Its part $n \leq Y$ is bounded absolutely by $Y$ . On $[Y, N]$ set

$g_{m} (x) = \frac{lo g ( N / x )}{L} {m / x} .$

Its supremum plus total variation is $O (1 + m / Y)$ , including all jumps. On smooth pieces the variation is bounded by $\int_{Y}^{N} [1/ (Lx) + m / x^{2}] d x$ ; there are $O (1 + m / Y)$ jumps of size at most one. Discrete Abel summation uses the actual values $g_{m} (n)$ , so no continuity convention at shared integer jumps is required. The boundary $M (⌊ Y ⌋) = M (Y)$ is also covered by the displayed bound. Consequently

$∣ R_{N} (m) ∣ ≪ 1 + Y + N e^{- (c_{0} /2) Ξ (N)} (1 + m / Y) ≪ N e^{- κ Ξ (N)} (N \leq m \leq X) .$

The last step uses $m / Y \leq e^{2 κ Ξ (N)}$ and $c_{0} /2 > 3 κ$ . The original telescoping weight now gives

$N \leq m \leq X \sum \frac{∣ R _{N} ( m ) ∣ ^{2}}{m ( m + 1 )} ≪ N e^{- 2 κ Ξ (N)} .$

For the entire remaining range $X < m \leq N^{2}$ , use $∣ R_{N} (m) ∣ \leq 1 + N$ and $\sum_{m > X} 1/ [m (m + 1)] \leq 1/ X$ . Its energy is therefore at most $O (N^{2} / X) = O (N e^{- κ Ξ (N)})$ , proving (I31). This argument never separates or optimizes the actual mean and cross terms; it bounds the original residual itself.

For every fixed $A > 0$ , (I31) implies $E_{N} ≪_{A} N / (lo g N)^{A}$ . This is a complete allowance, stronger than any one fixed logarithmic saving, but supplies no fixed power saving: $c Ξ (N) / lo g N = c (lo g N)^{- 2/5} (lo g lo g N)^{- 1/5} \to 0$ . It therefore supplies neither the uniform polylogarithmic energy needed by the earlier strength argument nor a fixed zero-free half-plane. The exact FIB approximation rate and the same-integer signed Robin estimate remain unresolved. Equation (I31) is a paper application of the already owned classical input, without a new Mertens result, originality claim or Lean certification.

Classical RH errors calibrate the first proportional block

For the original coefficients and residual of (I10), distinguish the first proportional block from the complete window:

$E_{N}^{first} := m = N \sum 2 N \frac{∣ R _{N} ( m ) ∣ ^{2}}{m ( m + 1 )} . (I32)$

The classical inputs are the RH consequences $M (t) = \sum_{n \leq t} μ (n) ≪_{δ} t^{1/2 + δ}$ for every fixed $δ > 0$ , and $ψ (t) - t ≪ t lo g^{2} t$ , where $ψ (t) = \sum_{k \leq t} Λ (k)$ . Both are stated in Conrey, Riemann’s Hypothesis, Section 3, printed p.6, author draft of 28 April 2019, PDF SHA256 6a704095b7359d17e27bd54eae3116c2ba621d2499e0b3c61c9c3a66c30619dc. They are reused, without a weak-Mertens mean-square hypothesis, zero simplicity or a reciprocal- $ζ^{'}$ moment condition. The calculation below is a paper application to the specified local residual, not an originality claim for these classical criteria or a Lean certification.

A related discrete-window precedent is Gallardo-Gutiérrez–Seco, arXiv:2206.02654v6, Theorem 3.1 and (15), printed pp.9–10. At its parameter $σ = 1/2$ , (15) uses the same weight $1/ [m (m + 1)]$ on a moving window. Its residual, however, is $1 + \sum_{k \leq N} μ (k) ({m / k} - N {m / N} / k)$ , without the logarithmic Selberg coefficients. Its bound is used along a selected subsequence. Neither identical cell weights nor that subsequence supplies a bound for (I32).

Retain $U_{N}$ from (I12). The classical convolution identity $\sum_{d ∣ k} μ (d) lo g (N / d) = Λ (k) + L 1_{k = 1}$ gives

$L R_{N} (m) = m U_{N} - ψ (m) + T_{N} (m), T_{N} (m) = N < n \leq m \sum μ (n) lo g (N / n), N \leq m \leq 2 N . (I33)$

Indeed, the full divisor sum over $k \leq m$ is $ψ (m) + L$ . Its omitted $n > N$ terms have $⌊ m / n ⌋ = 1$ throughout this block, including $m = 2 N$ ; the logarithmic weight at $n = N$ is zero. Substitution in the exact residual formula gives (I33).

Under RH fix $0 < δ < 1/2$ . The Dirichlet identity for $1/ ζ$ and its derivative at 1 give the classical normalizations $\sum μ (n) / n = 0$ and $\sum μ (n) lo g n / n = - 1$ ; convergence and local differentiation follow from the quoted Mertens bound. Abel summation then gives the exact tails

$U_{N} - 1 T_{N} (m) = n > N \sum \frac{μ ( n ) lo g ( n / N )}{n} = - \int_{N}^{\infty} M (t) \frac{1 - lo g ( t / N )}{t ^{2}} d t ≪_{δ} N^{- 1/2 + δ}, = M (m) lo g (N / m) + \int_{N}^{m} \frac{M ( t )}{t} d t ≪_{δ} N^{1/2 + δ} . (I34)$

The second bound is uniform on $N \leq m \leq 2 N$ . Combining (I33)–(I34) with the quoted Chebyshev error and $\sum_{m = N}^{2 N} 1/ [m (m + 1)] \leq 1/ N$ yields

$E_{N}^{first} ≪_{δ} \frac{N ^{2 δ}}{L ^{2}} + L^{2} . (I35)$

In particular RH implies $E_{N}^{first} ≪_{ε} N^{ε}$ for every fixed $ε > 0$ . Conversely, use the existing (I11)–(I16) argument on $J = (5/4, 7/4)$ . For every fixed positive $ε$ its norm bound is $O_{ε} (y^{(1 + ε) /2} (1 + lo g y)^{B})$ with finite $B$ . For each $ℜ s > 1/2$ choose a fixed $ε < 2ℜ s - 1$ and apply (I15) on a neighborhood of $s$ . Its no-common-zero argument gives RH. Thus the calibrated paper equivalence is

$RH ⟺ for every ε > 0, E_{N}^{first} ≪_{ε} N^{ε} for all sufficiently large integers N . (I36)$

Constants and the lower cutoff may depend on $ε$ ; neither $δ$ nor $ε$ varies with $N$ in the estimates used. This supplies no converse for a fixed polylogarithmic allowance, no RH bound for the entire window $N \leq m \leq N^{2}$ , and no optimal Nyman–Beurling approximation rate.

There is a stronger known supplier with additional assumptions, already cited in the FIB source, Section 391.3: Bettin–Conrey–Farmer, arXiv:1211.5191v1, Theorem 1, printed p.2. It uses these same logarithmic coefficients. Set $F_{N} (x) = 1_{(0, 1)} (x) + \sum_{n \leq N} a_{N, n} {1/ (n x)}$ on the half-line and $D_{N} = ∥ F_{N} ∥_{2}^{2}$ . The classical Mellin–Plancherel dictionary identifies this with that paper’s critical-line integral. On $m \leq t < m + 1$ , $m \geq N$ , the exact relation is $F_{N} (1/ t) = R_{N} (m) + c_{N} (t - m)$ . Thus, for $J_{N} = \int_{1/ (2 N + 1)}^{1/ N} ∣ F_{N} (x) ∣^{2} d x$ ,

$E_{N}^{first} - J_{N} \leq \frac{∣ c _{N} ∣}{N}, E_{N}^{first} \leq 2 D_{N} + \frac{2∣ c _{N} ∣ ^{2}}{N} .$

The lower endpoint is $1/ (2 N + 1)$ because the sample $m = 2 N$ contributes its whole cell. The cited theorem assumes RH and, for some $δ > 0$ , $\sum_{∣ℑ ρ ∣ \leq T} ∣ ζ^{'} (ρ) ∣^{- 2} ≪ T^{3/2 - δ}$ ; the latter includes zero simplicity. It supplies $D_{N} \sim (2 + γ - lo g (4 π)) / lo g N$ , and hence $E_{N}^{first} ≪ 1/ lo g N$ under those complete assumptions. This reuses the known stronger supplier with a paid cell comparison; it does not remove its extra hypothesis or assert a first-block asymptotic with the same constant.

A fixed polylogarithmic first-block allowance also constrains zero multiplicity. If $E_{N}^{first} ≪ L^{A}$ with fixed $A \geq 0$ , the same cell calculation on $J$ gives $∥ S_{y} (y \cdot) ∥_{L^{2} (J)} ≪ y (1 + lo g y)^{A /2 + 1}$ . For a zero $ρ$ on the critical line choose $h \in C_{c}^{\infty} (J)$ with $H_{h} (ρ) \neq = 0$ using (I16). The Mellin integral obeys $F_{h} (ρ + u) = O_{h} (u^{- A /2 - 2})$ as $u ↓ 0$ . The meromorphic right side of (I15) has a pole of order $m_{ρ}$ at $ρ$ , so

$m_{ρ} \leq ⌊ A /2 + 2 ⌋ . (I37)$

This uses meromorphic continuation near $ρ$ , not holomorphy on the boundary. The test function may depend on $ρ$ without changing the exponent. The every- $ε$ condition in (I36) has no controlled $ε$ -dependence from which to deduce (I37). An extra uniform multiplicity consequence does not establish that the fixed polylogarithmic condition is strictly stronger than RH.

Five-mode resolution as a paid cutoff sampling interface

The five modes can specify a sufficiently dense set of actual cutoff parameters. They do not change the Möbius coefficients inside any cutoff. The following application reuses (I11), the canonical finite Zeckendorf interval bound and the legal-word count in the FIB source, Sections 105–106, with canonical decoding owned by the existing golden-window note. It claims neither special Fibonacci cancellation nor an original sampling theorem.

First retain an integer $N \geq 3$ , allow real $y \geq N$ , and put $Δ = y - N$ . Unlike (I11), new dictionary indices may now enter. For every real $t$ the exact difference is

$S_{y} (t) - S_{N} (t) = lo g (y / N) (1 + n \leq N \sum μ (n) {t / n}) + N < n \leq y \sum μ (n) lo g (y / n) {t / n} .$

The first term has magnitude at most $2Δ$ . There are at most $Δ$ new indices, each with logarithmic weight at most $lo g (y / N) \leq Δ/ N$ . Hence

$∣ S_{y} (t) - S_{N} (t) ∣ \leq 2Δ + \frac{Δ ^{2}}{N} \leq 3Δ (0 \leq Δ \leq N), (I38)$

uniformly in $t$ , including across fractional-part jumps.

Let a fixed set $Q \subseteq N_{\geq 3}$ have the left coverage property: for every sufficiently large real $y$ , there is $N \in Q$ with $0 \leq y - N \leq C N (lo g N)^{B}$ , for fixed $C > 0$ and $B \geq 0$ . For $q \in J = (5/4, 7/4)$ , every cell meeting $q y$ then has its integer left endpoint in $[N, 2 N]$ once $N$ is sufficiently large. Use (I12), the actual weight $1/ [m (m + 1)]$ , $y ≍ N$ , and (I38), to obtain

$∥ S_{y} (y \cdot) ∥_{L^{2} (J)}^{2} ≪ N L^{2} E_{N}^{first} + (1 + L)^{4} + Δ^{2} . (I39)$

Consequently the right side of (I36), restricted to all sufficiently large $N \in Q$ for each fixed $ε$ , still implies RH by the same Mellin argument. RH supplies that restricted condition by (I35). This transfers the required inner Mellin window, without claiming to reconstruct every full moving first-block energy from one left sample.

Here is a concrete $Q$ using exactly the five occupancy modes. Use $F_{0} = 0, F_{1} = 1$ , keep the separate unit bit, and group $F_{3}, F_{4}, F_{5} = 2, 3, 5$ and their subsequent triples into windows. For $L \geq 1$ and each integer

$F_{6 L + 3} \leq n < F_{6 L + 9},$

form $Q_{L} (n)$ by replacing the lowest $L$ windows by null, retaining all higher digits and the unit bit. Deletion preserves the nonadjacency rule and seams. The canonical finite interval bound gives

$0 \leq n - Q_{L} (n) < F_{3 L + 3} .$

The leading digit has index at least $6 L + 3$ and is not deleted; therefore $Q_{L} (n)$ stays in the same annulus. The standard Fibonacci square identity yields

$F_{3 L + 3}^{2} = (F_{3 L + 1} + F_{3 L + 2})^{2} \leq 2 (F_{3 L + 1}^{2} + F_{3 L + 2}^{2}) = 2 F_{6 L + 3} \leq 2 Q_{L} (n) .$

Let $Q$ be the union of these images. Every real $y \geq 34$ has $n = ⌊ y ⌋$ in one of these annuli. With $N = Q_{L} (n)$ ,

$0 \leq y - N < 1 + 2 N \leq 2 N . (I40)$

There are $3 L + 6$ remaining high slots, indexed $3 L + 3$ through $6 L + 8$ . The known nonadjacent-word count and at most two choices of unit bit give $∣ Q \cap [F_{6 L + 3}, F_{6 L + 9}) ∣ \leq 2 F_{3 L + 8}$ . Thus each annulus needs only $O (F_{6 L + 3})$ distinct cutoff points; the constant includes its fixed scale ratio. Combining (I35), (I39) and (I40) gives the paper application

$RH ⟺ for every ε > 0, E_{N}^{first} ≪_{ε} N^{ε} for all sufficiently large N \in Q . (I41)$

The truncation creates a different integer cutoff; (I38) pays for that change of source. It is not a lossless coarse observation of the same integer. Each $R_{N}$ still includes every original $n \leq N$ with its original coefficient. A single sequence such as $F_{3 L}$ has gaps of order $N$ and is not justified by this square-root coverage argument. Nor does this replace the separate $⌊ N / d ⌋$ schedule issue for Binet convolution transport in the FIB source, Section 391. The unresolved task is an unconditional, uniform signed energy bound on this fixed set; the digit-count and gap bounds do not supply it. No new Lean declaration or certified RH implication is asserted here.

Bracketing cutoffs by the published derivative-variation bound

A stronger cutoff interface follows by directly applying S. S. Dragomir, Bounds for the deviation of a function from the chord generated by its extremities, Bull. Austral. Math. Soc. 78 (2008), 225–248, Theorem 5.2, (5.5), printed pp.234–235. For an absolutely continuous real function whose derivative has a bounded-variation representative, that theorem bounds the endpoint chord error by

$∣ f (z) - chord_{a, b} f (z) ∣ \leq \frac{( z - a ) ( b - z )}{b - a} Var_{[a, b]} (f^{'}) \leq \frac{b - a}{4} Var_{[a, b]} (f^{'}) .$

The theorem includes piecewise linear functions; its sharpness example is an absolute-value function. The supplier is this published bound, not a new interpolation theorem. Its application here keeps the original coefficients and cutoff family.

Fix integers $3 \leq A < B$ , put $h = B - A$ , and freeze the observation argument $t > 0$ . In the logarithmic cutoff coordinate $x$ ,

$f_{t} (x) = S_{e^{x}} (t) = x + n \leq B \sum μ (n) {t / n} (x - lo g n)_{+}, lo g A \leq x \leq lo g B . (I42)$

This is absolutely continuous and piecewise linear. Its right-slope representative has variation at most $\sum_{A < n \leq B} ∣ μ (n) ∣ {t / n} \leq h$ ; choosing this representative also handles the endpoint hinges. For $A \leq y \leq B$ , define

$η = \frac{lo g ( y / A )}{lo g ( B / A )}, P_{y} (t) = (1 - η) S_{A} (t) + η S_{B} (t) .$

Dragomir’s theorem applies to (I42) and gives, uniformly in $t$ ,

$∣ S_{y} (t) - P_{y} (t) ∣ \leq \frac{h}{4} lo g (B / A) \leq \frac{h ^{2}}{4 A} . (I43)$

The improvement over (I38) uses both endpoint functions. It does not claim that one endpoint alone has this quadratic cutoff error.

Retain $J = (5/4, 7/4)$ . If $B / A < 8/7$ , then $y J$ lies inside both $[A, 2 A]$ and $[B, 2 B]$ . For either $K = A$ or $K = B$ , (I12), the exact cell weights and $y ≍ K$ give

$∥ S_{K} (y \cdot) ∥_{L^{2} (J)}^{2} ≪ K (lo g K)^{2} E_{K}^{first} + (1 + lo g K)^{4} .$

Indeed, the cell containing each $q y$ has its integer left endpoint in $[K, 2 K]$ ; changing from $d q$ to $d t / y$ costs $1/ y$ , while removing the energy weight costs $O (K^{2})$ . The within-cell slope contributes $O (∣ U_{K} ∣^{2})$ , bounded by the fourth logarithmic power. Convexity for the endpoint mixture and (I43) now yield

$∥ S_{y} (y \cdot) ∥_{L^{2} (J)}^{2} ≪ K \in {A, B} \sum K (lo g K)^{2} E_{K}^{first} + (1 + lo g A)^{4} + \frac{h ^{4}}{A ^{2}} . (I44)$

Consequently a fixed unbounded set of integer cutoffs with consecutive gaps $B - A ≪ A^{3/4} (lo g A)^{C}$ , for a fixed $C \geq 0$ , is enough for the every- $ε$ condition: suppose that for each $ε > 0$ there are constants $C_{ε}, N_{ε}$ such that $E_{N}^{first} \leq C_{ε} N^{ε}$ for every sampled $N \geq N_{ε}$ . Apply this at both bracketing cutoffs. Using a smaller fixed $ε$ to absorb logarithms, (I44) gives $∥ S_{y} (y \cdot) ∥_{2}^{2} ≪_{ε} y^{1 + ε}$ for every sufficiently large real $y$ . The existing (I15)–(I16) argument excludes zeros in $ℜ s > 1/2$ by choosing $ε < 2ℜ s - 1$ . This is another paid cutoff application of that argument, not an unconditional estimate for $E_{N}^{first}$ .

A coarser grid using the actual five occupancy modes

The same canonical interval and legal-word bounds cited above supply a concrete bracketing grid. For $L \geq 1$ and each integer

$F_{12 L + 3} \leq n < F_{12 L + 15},$

replace the lowest $3 L$ triple windows by null, keeping all higher digits and the separate unit bit. Write the output as $Q_{L} (n)$ and the union of these images as $Q_{chord}$ . The deleted positions are exactly $F_{3}, \dots, F_{9 L + 2}$ , so the canonical interval bound gives

$0 \leq n - Q_{L} (n) < F_{9 L + 3} .$

The leading digit has index at least $12 L + 3$ , survives deletion, and keeps the output in the original annulus. The elementary Fibonacci bounds $φ^{k - 2} \leq F_{k} \leq φ^{k - 1}$ for $k \geq 2$ imply

$F_{9 L + 3} \leq φ^{5/4} Q_{L} (n)^{3/4} .$

These annuli cover all integers at least $F_{15} = 610$ . Thus every real $y \geq 610$ has a sampled $N \leq y$ with

$0 \leq y - N < (1 + φ^{5/4}) N^{3/4} . (I45)$

The lower endpoint of every annulus is itself retained. For consecutive sampled points $A < B$ , let $y ↑ B$ through $(A, B)$ . Every left sample supplied by (I45) is at most $A$ ; monotonicity of $x + (1 + φ^{5/4}) x^{3/4}$ gives $B - A \leq (1 + φ^{5/4}) A^{3/4}$ . In particular $B / A \to 1$ , as required for (I44); there is no cross-annulus gap exception.

The remaining high slots run from $9 L + 3$ through $12 L + 14$ , a total of $3 L + 12$ slots. The existing nonadjacent-word count and at most two unit choices give

$∣ Q_{chord} \cap [F_{12 L + 3}, F_{12 L + 15}) ∣ \leq 2 F_{3 L + 14} = O (F_{12 L + 3}^{1/4}) . (I46)$

Combining the supplied conditional converse (I35) with (I44)–(I46) therefore gives the paper application

$RH ⟺ for every ε > 0, E_{N}^{first} ≪_{ε} N^{ε} for all sufficiently large N \in Q_{chord} . (I47)$

The gain is a coarser sufficient cutoff grid: $O (N^{1/4})$ points per fixed-ratio annulus instead of the $O (N^{1/2})$ grid used with the one-sided bound. Every sampled residual still uses all original indices $n \leq N$ , with their original signed Möbius coefficients. No digital orthogonality or special Fibonacci cancellation is assumed. Both bracketing energies must be controlled uniformly; an almost-all cutoff estimate cannot be substituted for that condition. The known unconditional allowance (I31) remains insufficient, and no new sufficient unconditional energy estimate or RH proof is supplied. These are source-checked paper applications, without Lean certification.

A paid deletion of original endpoint responses

The stronger all-short-interval Möbius saving recorded in the existing short-interval source controls an actual signed part of the original residual. This is an application of that classical estimate, not a new Möbius theorem. It uses the original centered response atoms from (I17)–(I18), including all their Fourier aliases, rather than just a pair of frequencies.

Set $L = lo g N$ and $Ω (N) = (L / lo g L)^{1/3}$ for sufficiently large integer $N$ . Fix $0 < η < 1/24$ , and take integer $Q, H$ with

$N /2 < Q < Q + H \leq N, H \geq Q^{7/12 + 2 η} .$

The cited estimate with $θ = 7/12 + η$ gives a constant $c = c_{η} > 0$ such that

$0 \leq u \leq H max Q < n \leq Q + u \sum μ (n) ≪_{η} H e^{- c Ω (N)} . (I48)$

Here the maximum can be taken over integer $u$ . For $u \geq Q^{7/12 + η}$ , apply the source directly at the same start $Q$ . For shorter prefixes, the trivial bound is at most $Q^{7/12 + η} \leq H Q^{- η}$ . This power surplus absorbs the exponential allowance since $Ω (N) = o (lo g N)$ and $Q ≍ N$ . The constant $c$ incorporates that scale comparison. Thus (I48) is uniform over all the prefixes required by summation by parts; it is not merely a bound for one full interval.

For the actual integer observations $N \leq m \leq 2 N$ , define the original-index centered band

$G_{N, Q, H} (m) = Q < n \leq Q + H \sum \frac{μ ( n ) lo g ( N / n )}{L} ({m / n} - \frac{n - 1}{2 n}) . (I49)$

To apply (I48), retain the actual real weight

$w_{m} (x) = \frac{lo g ( N / x )}{L} (\frac{m + 1/2}{x} - ⌊ m / x ⌋ - \frac{1}{2}) .$

On $[Q, Q + H]$ , the ratio $m / x$ lies in $[1, 4)$ , so there are only finitely many floor thresholds. On each smooth piece, $∣ w_{m}^{'} (x) ∣ ≪ 1/ (N L)$ ; every jump has magnitude $O (1/ L)$ . The actual values at integer thresholds are the left-limit values, and are included in its bounded variation. Consequently, uniformly in $m, Q, H$ ,

$∥ w_{m} ∥_{\infty} + Var_{[Q, Q + H]} (w_{m}) ≪ L^{- 1} .$

If $Q = N - H$ , then $lo g (N / x) ≪ H / N$ throughout the band. Each jump gains that factor, and integration of the continuous derivative costs $O (H / (N L))$ . Thus the endpoint band has the stronger bound

$∥ w_{m} ∥_{\infty} + Var_{[N - H, N]} (w_{m}) ≪ \frac{H}{N L} . (I50)$

Discrete Abel summation uses the actual values $w_{m} (n)$ , so it does not discard any floor jump or endpoint. Combine (I48)–(I50) with $\sum_{m = N}^{2 N} 1/ [m (m + 1)] < 1/ N$ . In the same weighted first-block norm as $E_{N}^{first}$ , this gives

$∥ G_{N, Q, H} ∥_{first}^{2} ∥ G_{N, N - H, H} ∥_{first}^{2} ≪_{η} \frac{H ^{2}}{N L ^{2}} e^{- 2 c Ω (N)}, ≪_{η} \frac{H ^{4}}{N ^{3} L ^{2}} e^{- 2 c Ω (N)} . (I51)$

For any fixed $0 < λ < c /2$ , take an integer bandwidth $H ≍ N^{3/4} e^{λ Ω (N)}$ . This satisfies the required length and endpoint conditions eventually, because $7/12 + 2 η < 3/4$ and $H = o (N)$ . The second estimate gives

$∥ G_{N, N - H, H} ∥_{first}^{2} ≪_{η} L^{- 2} e^{- (2 c - 4 λ) Ω (N)} = o (1) . (I52)$

By comparison, summing absolute values before using Möbius cancellation only gives the allowance $H^{4} / (N^{3} L^{2})$ , which at this bandwidth is $≍ L^{- 2} e^{4 λ Ω (N)}$ and does not certify vanishing. Thus the signed input pays an additional, quantitatively larger band. Here $H$ is the width of the original dictionary-index band, not the cutoff sampling gap used in (I43)–(I47).

Write $G_{N} = G_{N, N - H, H}$ and $Z_{N} = R_{N} - G_{N}$ on the same first block. The full-period zero-frequency coefficient $d_{N}$ of (I17) is retained: each original centered atom has zero mean on a common multiple of its period. The mean over the first observation block need not be retained. All remaining coefficients keep the original logarithmic cutoff $N$ ; $Z_{N}$ is not identified with a residual reweighted at cutoff $N - H$ . The exact comparison is

$E_{N}^{first} E_{N}^{first} - ∥ Z_{N} ∥_{first} = ∥ Z_{N} ∥_{first}^{2} + 2ℜ ⟨ Z_{N}, G_{N} ⟩_{first} + ∥ G_{N} ∥_{first}^{2}, \leq ∥ G_{N} ∥_{first} = o (1) . (I53)$

A small norm of $G_{N}$ does not assert an $o (1)$ difference of squared energies without a bound for the remaining norm and cross term. Nor does it bound arbitrary Fourier subparts of $G_{N}$ : an original index $n$ includes reducible fractions $h / n$ , so this is not a pure band of reduced Fourier denominators. All those aliases travel together.

This pays one original-response deletion and leaves the other source indices and their interaction with $d_{N}$ in $Z_{N}$ . It gives no improved complete-energy allowance over (I31), no sufficient unconditional RH energy estimate and no same-integer signed Robin estimate. The source application and derivation here are not Lean-certified.

Pay the terminal covariance while retaining the endpoint

The same short-interval source also records the Huxley prime-error estimate needed here. Reuse that input and (I33), rather than replacing the original residual by a separately centered or optimized one. The following application pays the terminal fluctuation in the mixed term; two actual local moments and the endpoint value remain.

Keep the same $N, L, Ω (N)$ and $G_{N}$ as in (I48)–(I53), and put $Q = N - H$ , $K = 2 N$ , $w_{m} = 1/ [m (m + 1)]$ . In this section choose a common constant $c_{⋆} > 0$ , small enough for the Möbius bound at $Q$ , the Möbius and prime-error bounds at $2 Q$ , and their scale comparisons. This can decrease the constant $c$ used in (I48). Fix

$0 < λ < c_{⋆} /2, H ≍ N^{3/4} e^{λ Ω (N)}, H \in N .$

All statements are for sufficiently large integer $N$ . In particular, $H < N /3$ . This is an explicitly restricted parameter family, not a claim for every $λ$ allowed by the unreduced constant in (I52).

For auxiliary sums up to $2 N$ , write $b_{n} = μ (n) lo g (N / n)$ , always with the original cutoff $N$ . Define

$A = Q < n \leq N \sum \frac{b _{n}}{n}, B = Q < n \leq N \sum b_{n}, S_{m} = Q < n \leq m /2 \sum b_{n} .$

The last sum is empty before the terminal interval. For $Q < n \leq N$ and $N \leq m \leq K$ , $1 \leq m / n < 3$ , so $⌊ m / n ⌋ = 1 + 1_{2 n \leq m}$ . The exact centered-atom formula therefore gives

$L G_{N} (m) = (m + \frac{1}{2}) A - \frac{3}{2} B - S_{m} . (I54)$

Let $r = R_{N} (K)$ and keep the actual local moments

$P_{0} = m = N \sum K w_{m} R_{N} (m), P_{1} = m = N \sum K w_{m} (m + \frac{1}{2}) R_{N} (m) .$

No assertion sets either moment to zero. Interchanging finite sums and telescoping the original weight, including the sample $m = 2 n$ , gives

$D := m = N \sum K w_{m} S_{m} = Q < n \leq N \sum b_{n} (\frac{1}{2 n} - \frac{1}{K + 1}) = \frac{A}{2} - \frac{B}{K + 1} . (I55)$

Set $C_{N, H} = ⟨ R_{N}, G_{N} ⟩_{first}$ . These quantities are real, and the exact mixed decomposition is

$C_{N, H} A_{N, H} = A_{N, H} - \frac{1}{L} m = N \sum K w_{m} S_{m} (R_{N} (m) - r), = \frac{1}{L} [A (P_{1} - r /2) - B (\frac{3}{2} P_{0} - \frac{r}{K + 1})] . (I56)$

In particular, the endpoint has been retained in $A_{N, H}$ ; subtracting it inside the terminal fluctuation has not removed it from the original mixed term.

On the support of $S_{m}$ define

$V_{μ} (m) = m < n \leq K \sum b_{n}, V_{Λ} (m) = ψ (K) - ψ (m) - (K - m) .$

Subtract (I33) at $m$ and $K$ , using its original $U_{N}$ . This yields

$L (R_{N} (m) - r) = - (K - m) (U_{N} - 1) + V_{Λ} (m) - V_{μ} (m) . (I57)$

Here the terms $b_{n}$ with $n > N$ occur only in the auxiliary identity (I33); no new dictionary atoms or reweighted cutoff are introduced.

The prefix argument of (I48), applied at $Q$ and $2 Q$ , and discrete Abel summation give

$m sup ∣ S_{m} ∣ ≪_{η} \frac{H ^{2}}{N} e^{- c_{⋆} Ω (N)}, 2 Q \leq m \leq K sup ∣ V_{μ} (m) ∣ + 2 Q \leq m \leq K sup ∣ V_{Λ} (m) ∣ ≪_{η} H e^{- c_{⋆} Ω (N)} .$

For the second assertion the source interval starts at $2 Q$ and has length $2 H$ . Differences of its prefixes give the terminal sums. The weight $lo g (N / n)$ has bounded supremum plus variation on that interval. Short prefixes for $Λ - 1$ have the trivial bound $O (u lo g N)$ ; the fixed power margin between the source exponent $7/12 + η$ and $H \geq N^{7/12 + 2 η}$ absorbs this logarithm and the exponential allowance. Thus an all-prefix prime-error bound is used, not just a bound for the full interval.

The unconditional VK–Mertens input already used in (I31) also pays the exact tail from (I34), with its estimate changed to

$U_{N} - 1 = - \int_{N}^{\infty} M (t) \frac{1 - lo g ( t / N )}{t ^{2}} d t ≪ e^{- b Ξ (N)} for some b > 0.$

Indeed that input makes the Abel tail integral absolutely convergent and gives the normalizations $\sum μ (n) / n = 0$ and $\sum μ (n) lo g n / n = - 1$ by taking the right-hand limits of $1/ ζ (s)$ and its derivative at $s = 1$ . After substituting $u = lo g t$ , absorb the polynomial factor in $\int_{L}^{\infty} (1 + u) e^{- c_{0} Ξ (e^{u})} d u$ by decreasing the positive exponential constant. No RH estimate from (I34) is used.

There are at most $2 H$ samples with $S_{m} \neq = 0$ , their total weight is $O (H / N^{2})$ , and $K - m \leq 2 H$ on this support. Consequently (I56)–(I57) give the quantitative paid remainder

$∣ C_{N, H} - A_{N, H} ∣ ≪_{η} \frac{H ^{4}}{N ^{3} L ^{2}} (e^{- 2 c_{⋆} Ω (N)} + e^{- b Ξ (N) - c_{⋆} Ω (N)}) . (I58)$

This bounds the actual products on their common support; no independence or separate optimizing configuration is assumed. At the stated bandwidth the first allowance tends to zero since $4 λ < 2 c_{⋆}$ , and the second does so because $Ξ (N) /Ω (N) \to \infty$ . Reuse (I52) for the smaller common constant to obtain

$C_{N, H} = A_{N, H} + o (1), E_{N}^{first} - ∥ R_{N} - G_{N} ∥_{first}^{2} = 2 A_{N, H} + o (1) . (I59)$

The second identity retains the mean interaction already present in (I53). It supplies an $o (1)$ error for replacing the terminal covariance by its endpoint contribution, not an $o (1)$ estimate for the complete mixed term or squared-energy difference. The unpaid quantity is $A_{N, H}$ with the displayed jointly realized $A, B, P_{0}, P_{1}, r$ . The old pointwise input for $R_{N}$ still gives only an allowance of order $H^{2} e^{- κ Ξ (N) - c_{⋆} Ω (N)} / (N L)$ for the complete mixed term, which does not certify vanishing at this bandwidth. Full-space moment centering does not set these local moments or endpoint to zero. This is an application of existing short-interval estimates and the owned residual identity, without a new distribution theorem, originality claim, sufficient unconditional RH energy bound, same-integer signed Robin estimate or Lean certification.

Fixed-half-plane inputs sharpen the actual energy benchmark

The already recorded OpenAI source states a zero-free half-plane with boundary $7/8$ . The Liu v2 source, Theorem 27.1, improves that boundary to $B_{new} = 0.874957019420098946 \dots$ . Use these as paper-level inputs, with their strict boundary and principal-pole qualifications retained. Neither external Lean closure has been independently compiled here. The following applies those existing inputs to the actual family (I10); it is not a new zero-free-region or Mertens theorem.

Here is the ordinary-integer analytic interface, including the parameter change needed to use a fixed boundary other than $1/2$ . Suppose $1/2 \leq B < 1$ and $ζ$ has no zero in $ℜ s > B$ . The classical zero-free-half-plane/Perron argument gives, for every fixed $B < β < 1$ ,

$M (x) ≪_{β} x^{β}, ψ (x) - x ≪_{β} x^{β} . (I60)$

This is not obtained by simply renaming the RH exponent in the Conrey quotation. The fixed source snapshot’s logarithmic-control proof, label lem:logarithmic-control, supplies the classical disk argument: Borel–Caratheodory followed by three circles uses a nonvanishing regularized function, polynomial strip growth, and a bounded Euler logarithm on a fixed disk to the right of one. Its stated functions are Hecke functions and regularized $ζ_{F}$ , not the ordinary integer Mertens function. To apply that argument here, use instead $F (s) = (s - 1) ζ (s) / (s + 1)$ , with its removable nonzero value at one. It has those same elementary growth and Euler-logarithm properties and is nonvanishing on $ℜ s > B$ . For any fixed $σ_{0} > B$ and $η > 0$ , the argument gives $1/ ζ (σ + i t) ≪_{σ_{0}, η} (3 + ∣ t ∣)^{η}$ on $σ \geq σ_{0}$ ; Cauchy’s estimate also bounds the logarithmic derivative by $O_{σ_{0}} (lo g (3 + ∣ t ∣))$ on fixed sub-strips away from the pole at one.

For the remaining interface reuse Koukoulopoulos, The Distribution of Prime Numbers, author’s authorized preliminary version, Theorem 7.2, printed p.71. The inspected PDF has SHA256 1445e107bba9c89db6529aeb4bca3ecf01170fc2854a675f5ca6f288c77ea0a8. Its coefficient condition holds with $f (n) = μ (n)$ and $θ = A = 0$ . Choose $B < σ_{0} < β$ , initial line $c = 1 + 1/ lo g x$ , and height $T = x^{2}$ . The shifted vertical integral is $O (x^{σ_{0}} T^{η} lo g T)$ , the horizontal integrals are $O (x T^{η - 1})$ , and that theorem’s truncation error is $O (x lo g x / T + 1)$ . Take $0 < η < (β - σ_{0}) /4$ to obtain the Mertens part of (I60), with no residue at one for $1/ ζ$ . For the prime error directly use the same book’s explicit formula, Theorem 5.1, printed p.57, and its zero-counting Lemma 8.2(a), printed p.84. With $T = x^{2}$ and $ℜ ρ \leq B$ they give $ψ (x) - x ≪ x^{B} lo g^{2} x$ , retaining every multiplicity and the main term $x$ . Absorbing the logarithm gives the second part of (I60). Thus neither Perron nor the prime explicit formula is reproved here. Only their ordinary- $ζ$ parameter interface is made explicit; the source’s Hecke lemma is not quoted as a theorem about ordinary integer Mertens sums.

Apply (I60) first to the exact identity (I33). Since $β < 1$ , the same Abel tails and normalizations used in (I34) converge, now with $U_{N} - 1 ≪_{β} N^{β - 1}$ and $T_{N} (m) ≪_{β} N^{β}$ uniformly on $N \leq m \leq 2 N$ . Thus the actual residual, including its mean and endpoint, satisfies $∣ R_{N} (m) ∣ ≪_{β} N^{β} / L$ there, and $E_{N}^{first} ≪_{β} N^{2 β - 1} / L^{2}$ .

For the complete window reuse the bounded-variation splitting of (I31), without changing any coefficient. Set $X = N^{2 - β}$ . At each actual sample $N \leq m \leq X$ , use $Y = m N^{β}$ , so $1 \leq N^{β} \leq Y \leq N$ . The lower part has absolute bound $Y$ and the upper part uses $sup_{Y \leq t \leq N} ∣ M (t) ∣ ≪_{β} N^{β}$ with the already proved variation allowance $O (1 + m / Y)$ . Consequently

$∣ R_{N} (m) ∣ ≪_{β} 1 + Y + N^{β} (1 + m / Y) ≪_{β} m N^{β} .$

The original weight gives at most $O_{β} (N^{β} lo g N)$ on this range. On the whole remaining range $X < m \leq N^{2}$ , the existing $∣ R_{N} (m) ∣ \leq 1 + N$ and telescoping weights give $O (N^{2} / X) = O (N^{β})$ . Floors and ceilings at $X, Y$ only change fixed constants. Choosing $β > B$ sufficiently close to $B$ and absorbing the logarithm yields, for every fixed $ε > 0$ ,

$E_{N}^{first} ≪_{ε} N^{2 B - 1 + ε}, E_{N} ≪_{ε} N^{B + ε} . (I61)$

For $B = 7/8$ these exponents are $3/4 + ε$ and $7/8 + ε$ . The stated Liu boundary gives respectively $0.749914038840197892 \dots + ε$ and $0.874957019420098946 \dots + ε$ . These fixed power savings improve the benchmark (I31) when its classical VK input is supplemented by the cited fixed-half-plane results. They neither alter what (I31)’s own input supplies nor attain the every- $ε$ subpower first-block estimate (I36). In particular $B_{new} > 1/2$ leaves a positive power in the first-block allowance. Combining it with (I52) also fails to certify vanishing of the full mixed term at the bandwidth of (I59). The actual joint main term $A_{N, H}$ , the same-integer signed Robin estimate, and RH remain unresolved. This is a scoped application of the cited analytic inputs, without a mathematical priority claim or Lean certification.

A coupled Perron kernel for the unpaid joint main term

Retain the actual $A, B, P_{0}, P_{1}, r$ and $A_{N, H}$ of (I56), and the bandwidth and common constant of (I58)–(I59). Reuse the Mellin kernel $H_{q}$ of (I15). Symmetric Mellin/Perron inversion is used only at a fixed integer $N$ here; no zero-sum or contour-shift estimate is assumed. This applies the classical inversion interface to the jointly realized functional, rather than replacing its coefficients or centering its local moments.

Put $a = \frac{Q + 1/2}{N}, h = 1 - a = \frac{H - 1/2}{N}, W_{a} (s) = \int_{a}^{1} x^{s - 1} lo g (1/ x) d x = \frac{1 - a ^{s} ( 1 + s lo g ( 1/ a ))}{s ^{2}},$ with the removable value at zero. Write $\mathscr P_s F=(2\pi i)^{-1}\lim_{T\to\infty} \int_{c-iT}^{c+iT}F(s),ds $f or a f i x e d$ c>1$. The exact band inversion is $B = P_{s} \frac{N ^{s} W _{a} ( s )}{ζ ( s )}, A = P_{s} \frac{N ^{s - 1} W _{a} ( s - 1 )}{ζ ( s )} . (I62)$ The half-integer lower cut selects exactly $Q < n \leq N$ ; the upper weight is zero. Thus no lower-boundary half weight or changed cutoff is hidden in this formula.

The residual inversion needs an additional integer-jump term. Set $J_{m} = n ∣ m n \leq N \sum b_{n} = Λ (m) - 1_{m > N} μ (m) lo g (N / m) .$ The equality holds throughout $N \leq m \leq 2 N$ , including both endpoints: any omitted divisor must be $m$ itself. Then $L R_{N} (m) = L - \frac{J _{m}}{2} + P_{s} \frac{N ^{s} H _{m / N} ( s )}{ζ ( s )} . (I63)$ Indeed, invert $lo g v {q v}$ supported on $v \geq 1$ and use the absolutely convergent Dirichlet series on $ℜ s = c$ . At $n ∣ m$ the fractional part jumps down by one, so symmetric inversion returns its two-sided average and must be corrected by $- b_{n} /2$ . The factor $lo g (N / N) = 0$ removes a separate upper-cutoff jump. For fixed $N$ these inversion limits follow from local bounded variation and integrability in logarithmic coordinates.

For any function specified on the actual grid $m / N$ , define $p_{N} (f) q_{N} (f) p_{N} (v) = m = N \sum 2 N w_{m} (m + \frac{1}{2}) f (m / N) - \frac{1}{2} f (2), = \frac{3}{2} m = N \sum 2 N w_{m} f (m / N) - \frac{f ( 2 )}{2 N + 1}, = p_{N} (H_{∙} (v)), q_{N} (v) = q_{N} (H_{∙} (v)) . (I64)$ Let $Z (m / N) = 1 - J_{m} / (2 L)$ and set $K_{N} (u, v) E_{grid} = W_{a} (u - 1) p_{N} (v) - N W_{a} (u) q_{N} (v), = \frac{A p _{N} ( Z ) - B q _{N} ( Z )}{L} .$ Substitution into (I56) gives the exact jointly signed expression $A_{N, H} = E_{grid} + \frac{1}{L ^{2}} P_{u} P_{v} \frac{N ^{u + v - 1} K _{N} ( u , v )}{ζ ( u ) ζ ( v )} . (I65)$ Here the double symbol means the rectangular limit with independent heights $T_{u}, T_{v} \to \infty$ . Each finite double integral separates into products of one-variable integrals from (I62)–(I64), whose limits exist for this fixed $N$ . This proves that rectangular limit, without asserting absolute convergence, exchanging $N \to \infty$ with it, or authorizing a spectral contour shift. Both actual $H_{2}$ endpoint terms remain present in $p_{N}$ and $q_{N}$ .

There is a paid part of (I65). Since $J_{m} = O (L)$ , $∥ Z ∥_{\infty} = O (1)$ , $∥ p_{N} ∥ = O (1)$ and $∥ q_{N} ∥ = O (N^{- 1})$ . Reuse the short-interval bounds underlying (I58), namely $∣ A ∣ ≪ H^{2} N^{- 2} e^{- c_{⋆} Ω}$ and $∣ B ∣ ≪ H^{2} N^{- 1} e^{- c_{⋆} Ω}$ . They give $E_{grid} ≪ \frac{H ^{2}}{N ^{2} L} e^{- c_{⋆} Ω} = o (1) . (I66)$ This pays the integer-jump and constant terms; it does not pay the remaining double integral or replace $Z$ by a local mean.

The low-frequency kernel has an explicit calibration. Put $Φ (v) = \int_{1}^{2} \frac{q - 3/2}{q ^{2}} H_{q} (v) d q .$ Uniformly for $u$ in a fixed compact subset of $C$ and $v$ in a fixed compact subset of $ℜ v > 0$ , including any fixed order of mixed derivatives, Taylor expansion and Riemann summation give $K_{N} (u, v) = \frac{h ^{2}}{2} Φ (v) + O_{u, v} (h^{3} + h^{2} / N) . (I67)$ In particular $p_{N} (v) - N q_{N} (v) = Φ (v) + O_{v} (1/ N)$ . Its endpoint coefficient is $- 1/2 + N / (2 N + 1) = O (1/ N)$ , so this leading cancellation has not deleted the endpoint from the exact formula.

For fixed simple nontrivial zeros $ρ, σ$ , the corresponding ordered double residue of the meromorphic integrand in (I65), with the factor $L^{- 2}$ included, is consequently $\frac{H ^{2} N ^{ρ + σ - 3}}{2 L ^{2}} (\frac{Φ ( σ )}{ζ ^{'} ( ρ ) ζ ^{'} ( σ )} + o (1)) . (I68)$ This is a calculation for that fixed pair, not an asymptotic for $A_{N, H}$ and not an assumption that all zeros are simple. At a conjugate pair its displayed prefactor has scale $N^{2ℜ ρ - 3/2} e^{2 λ Ω} / L^{2}$ . The coefficient may vanish, other residues may cancel, and no spectral remainder has been paid. Thus existence of a zero with real part greater than $1/2$ alone does not turn this calculation into an obstruction to $A_{N, H} = o (1)$ .

The missing uniformity is visible already in $∣ W_{a} (σ + i t) ∣ ≪_{σ} min (h^{2}, h / (1 + ∣ t ∣))$ , from integration by parts. The compact-set expansion (I67) cannot be inserted over an entire Perron line; heights of order $1/ h ≍ N / H$ and beyond need their own reciprocal-zeta and joint cancellation estimates. Equations (I62)–(I68) supply the actual coupled kernel and a paid grid error, without a bound for that double integral, a new RH criterion, a same-integer signed Robin estimate, a mathematical priority claim or Lean certification.

A paid rectangular truncation on the original Perron lines

Reuse Koukoulopoulos, The Distribution of Prime Numbers, Theorem 7.2, printed p.71, PDF p.82, the same truncated Perron input used for (I60), and the actual short-interval coefficient bounds of (I66). The following application pays a finite-height error in (I65). It does not estimate the integral inside the retained rectangle or a remainder after shifting its lines.

Retain the sufficiently large integer parameter family of (I59), including $H < N /3$ . Put $X = Q + 1/2$ , retain $h = (H - 1/2) / N$ , and choose $κ_{N} = 1 + 1/ lo g X$ . All finite integrals in this section use this same line. Let $P_{s, T}$ denote $(2 πi)^{- 1}$ times integration from $κ_{N} - i T$ to $κ_{N} + i T$ . For $U, V \geq 2$ , define $A_{U} R_{V} (m) = P_{s, U} \frac{N ^{s - 1} W _{a} ( s - 1 )}{ζ ( s )}, = Z (m / N) + \frac{1}{L} P_{s, V} \frac{N ^{s} H _{m / N} ( s )}{ζ ( s )} . B_{U} = P_{s, U} \frac{N ^{s} W _{a} ( s )}{ζ ( s )},$ Here $Z = 1 - J_{m} / (2 L)$ is the exact grid correction of (I63). Uniformly on the original integer grid, $∣ A - A_{U} ∣ ∣ B - B_{U} ∣ N \leq m \leq 2 N max ∣ R_{N} (m) - R_{V} (m) ∣ ≪ h L / U + h / N, ≪ N h L / U + h, ≪ N L / V + 1. (I69)$

To retain the short-band gain in the first two estimates, set $M_{U} (x) = P_{s, U} x^{s} / [s ζ (s)]$ and $f_{τ} (x) = x^{- τ} lo g (N / x)$ , $τ = 0, 1$ . Apply the cited theorem to $μ$ , with $θ = A = 0$ in its notation. Since $κ_{N} \geq 1 + 1/ lo g x$ for $X \leq x \leq N$ , it gives $sup_{X \leq x \leq N} ∣ M (x) - M_{U} (x) ∣ ≪ N L / U + 1$ . The weights decrease to zero at $N$ and satisfy $f_{τ} (X) + \int_{X}^{N} ∣ f_{τ}^{'} (x) ∣ d x = 2 f_{τ} (X) ≪ h N^{- τ} .$ Exact Abel summation gives, respectively for $B, A$ , $- f_{τ} (X) M (X) - \int_{X}^{N} f_{τ}^{'} (x) M (x) d x, τ = 0, 1.$ Replacing $M$ by $M_{U}$ gives exactly $B_{U}, A_{U}$ : exchange the finite integrals and integrate the weights to recover $W_{a} (s)$ and $W_{a} (s - 1)$ . Thus both cutoff terms travel together through the same band. The lower endpoint $X$ is a half integer; the upper weight is zero. Multiplying the uniform Perron error by the displayed variation proves the first two rows of (I69), with no changed cutoff.

For the last row, elementary integration of the fractional-part Mellin transform gives, for $1 \leq q \leq 2$ and $ℜ s > 1$ , $H_{q} (s) = \frac{q}{( s - 1 ) ^{2}} + \frac{d}{d s} (\frac{1 + q ^{s} ( ζ ( s ) - 1 )}{s}) . (I70)$ The floor thresholds $k \geq 2$ supply the factor $ζ (s) - 1$ ; the factor $lo g v$ vanishes at the lower integration endpoint. Writing $m = qN$ , division by $ζ (s)$ gives explicitly $\frac{N ^{s} H _{q} ( s )}{ζ ( s )} = \frac{q N ^{s}}{( s - 1 ) ^{2} ζ ( s )} + \frac{m ^{s} - N ^{s}}{s ^{2} ζ ( s )} + m^{s} (\frac{lo g q}{s} - \frac{1}{s ^{2}}) + \frac{m ^{s}}{s} \frac{ζ ^{'} ( s )}{ζ ( s )} - \frac{m ^{s} lo g q}{s ζ ( s )} .$ Since $∣1/ ζ (κ_{N} + i t) ∣ \leq ζ (κ_{N}) ≪ L$ , the square-denominator pieces have tail $O (N L / V)$ . The remaining pieces are ordinary Perron integrals of $ζ^{'} / ζ$ , $1/ ζ$ and the Dirichlet series $1$ , evaluated at $m = qN$ , with the bounded coefficient $lo g q$ where present. The cited theorem for $Λ$ , with $θ = 0, A = 1$ , gives $O (N L^{2} / V + L)$ ; for $μ$ and the sequence supported at $1$ it gives $O (N L / V + 1)$ . At integer $m$ , replacing full summation by the symmetric Perron limit costs at most $O (L)$ for $Λ$ and $O (1)$ for $μ$ . These costs are retained. Division by $L$ gives the last row of (I69), including $m = N, 2 N$ and their original $J_{m}$ .

Let $S = R_{N} - Z$ , $S_{V} = R_{V} - Z$ , and define the finite rectangle $M_{N} (U, V) = \frac{1}{L ^{2}} P_{u, U} P_{v, V} \frac{N ^{u + v - 1} K _{N} ( u , v )}{ζ ( u ) ζ ( v )} = \frac{A _{U} p _{N} ( S _{V} ) - B _{U} q _{N} ( S _{V} )}{L} . (I71)$ Finite integration and the actual finite grid justify the equality. All endpoint terms of (I64) remain in this expression. The original coefficients satisfy $∣ A ∣ + ∣ B ∣/ N ≪ h^{2} e^{- c_{⋆} Ω}$ , while $∥ S ∥_{\infty} ≪ N / L$ follows from $\sum_{n \leq N} lo g (N / n) ≪ N$ and $∥ Z ∥_{\infty} = O (1)$ . Together with the operator bounds used for (I66), (I69) therefore gives $∣ A_{N, H} - E_{grid} - M_{N} (U, V) ∣ ≪ \frac{N h}{UL} + \frac{h}{L ^{2}} + (h^{2} e^{- c_{⋆} Ω} + \frac{h L}{U} + \frac{h}{N}) (\frac{N}{V} + \frac{1}{L}) . (I72)$ This is a bound for the same jointly realized functional, using its actual coefficients rather than independently optimized values. Set $U = H$ and $V = N$ , and reuse the quantitative bound for $E_{grid}$ in (I66). At the bandwidth of (I59), (I72) yields $A_{N, H} = M_{N} (H, N) + O (\frac{1}{L} + e^{- (c_{⋆} - 2 λ) Ω} + \frac{L}{N}) = M_{N} (H, N) + o (1) . (I73)$ The positive margin is exactly $2 λ < c_{⋆}$ . This baseline does not use the fixed zero-free half-plane of (I60).

If that input is adopted, retain its full premise: fixed $1/2 \leq B < 1$ , no zeta zero in $ℜ s > B$ , and fixed $B < β < 1$ . Then (I60)’s application gives $∥ S ∥_{\infty} ≪_{β} N^{β} / L$ . In the proof of (I72), replace only the first norm bound by this one and take $U = H N^{β - 1}$ , $V = N$ . The same products give $A_{N, H} = M_{N} (H N^{β - 1}, N) + O_{β} (\frac{1}{L} + e^{- (c_{⋆} - 2 λ) Ω} + L N^{1/2 - β}) = M_{N} (H N^{β - 1}, N) + o (1) . (I74)$ Here $β > 1/2$ , and constants may depend on this fixed $β$ and the adopted input. In particular the cited Liu boundary yields $U\asymp N^{0.624957019420098946\ldots+\varepsilon} e^{\lambda\Omega}$ for fixed $0 < ε < 1 - B_{new}$ , with the paper-level source status and uncompiled external Lean boundary of (I60)–(I61) unchanged.

These bounds pay the original-line rectangular truncation. They do not give an absolute tail estimate or a zero-sum truncation after contour deformation. The retained heights still extend beyond $1/ h ≍ N / H$ ; the compact expansion (I67) cannot be substituted over those rectangles. A sufficiently strong estimate inside them, the same-integer signed Robin estimate, and RH remain unresolved. This is an application of the cited Perron theorem, without a mathematical-priority claim or Lean certification.

Transport the actual endpoint into a smooth common test

Retain the original bandwidth and fixed constants of (I58)–(I59), including $0 < 2 λ < c_{⋆}$ and $H < N /3$ . Put $ϵ_{N} = e^{- c_{⋆} Ω}$ and $v_{N} = e^{- b Ξ (N)}$ , where $b > 0$ is the unconditional tail constant used there. All quantities below come from the same integer $N$ , its actual band, and its original endpoint. This applies the already owned uniform short-interval estimates, the cell identity (I12), and the bump bounds cited below; it is not a new cancellation theorem.

The actual coefficients contain a relation lost by separate norm bounds: $δ_{N, H} := A - B / N e_{N, H} := B / (2 N + 1) - A /2 = Q < n \leq N \sum μ (n) lo g (N / n) (1/ n - 1/ N) ≪ h^{3} ϵ_{N}, = - δ_{N, H} /2 - B / [2 N (2 N + 1)] ≪ h^{3} ϵ_{N} . (I75)$ Indeed, the first sum’s nonnegative real weight is decreasing, with supremum plus variation $O (h^{2} / N)$ . The same full-prefix Mobius bound used in (I48) is $O (H ϵ_{N})$ on this band. Abel summation gives the first row; the second uses $∣ B ∣ ≪ N h^{2} ϵ_{N}$ and $N h ≫ 1$ . These bounds do not replace $e_{N, H}$ by a separately optimized value.

Use the existing real-argument interpolation $R_{N} (t) = S_{N} (t) / L$ from (I11)–(I12), and set $I_{j} = \int_{1}^{2} q^{- j} R_{N} (Nq) d q$ , $j = 1, 2$ . This interpolation agrees with $R_{N} (m)$ at integers; the actual balanced residual remains a step function on its original cells. The cell slope is exactly $U_{N} / L$ . Comparing the cell integrals of $t^{- 2}$ and $t^{- 1}$ with the original samples, including $m = 2 N$ , gives $∣ P_{1} - I_{1} ∣ + ∣ N P_{0} - I_{2} ∣ A_{N, H} ≪ N^{- 1} N \leq m \leq 2 N max ∣ R_{N} (m) ∣ + ∣ U_{N} ∣/ L ≪ L^{- 1}, = L^{- 1} (A I_{1} - \frac{3 B}{2 N} I_{2} + e_{N, H} r) + O (h^{2} ϵ_{N} / L^{2}) . (I76)$ For the last bounds reuse $U_{N} = O (1)$ from the unconditional tail and $max ∣ R_{N} ∣ ≪ N / L$ from the original coefficients. The endpoint $r = R_{N} (2 N)$ in (I76) is unchanged.

The same uniform short-interval inputs pay the local real-argument oscillations at both ends: $osc_{[N, N + N h]} R_{N} + osc_{[2 N - N h, 2 N]} R_{N} ≪ \frac{H}{L} (ϵ_{N} + v_{N}) . (I77)$ To see this, (I33) holds throughout $N \leq t \leq 2 N$ with the weak $ψ (t)$ and the actual sum over $N < n \leq t$ . Subtract at two points. The linear term uses $U_{N} - 1 ≪ v_{N}$ ; the other terms use the full-prefix Mobius and $Λ - 1$ bounds underlying (I58). At the lower end apply that same uniform supplier near $N$ ; at the upper end use its terminal prefixes. For all these starts, $Ω (x) - Ω (N) = o (1)$ , so the same chosen common constant works. Short prefixes cost $O (uL)$ and real versus integer endpoints cost $O (L)$ before division by $L$ ; the fixed power surplus in $H$ absorbs both in $H ϵ_{N}$ . Thus no jump is suppressed.

Here is an explicit mass-preserving construction. Fix nonnegative $ρ \in C_{c}^{\infty} (0, 1)$ , supported in $[1/4, 3/4]$ , with integral one, and put $χ (x) = \int_{- \infty}^{x} ρ (u) d u$ , $χ_{h} (q) = χ ((q - 1) / h) χ ((2 - q) / h)$ and $η_{-, h} (q) = h^{- 1} ρ ((q - 1) / h), η_{+, h} (q) = h^{- 1} ρ ((2 - q) / h) .$ Let $β_{j, -}$ and $β_{j, +}$ be the integrals of $(1 - χ_{h} (q)) q^{- j}$ over $[1, 1 + h]$ and $[2 - h, 2]$ , respectively, and define $w_{j, h} (q) = χ_{h} (q) q^{- j} + β_{j, -} η_{-, h} (q) + β_{j, +} η_{+, h} (q), w = A w_{1, h} - \frac{3 B}{2 N} w_{2, h} + e_{N, H} η_{+, h} .$ Each $w_{j, h}$ preserves the original $q^{- j}$ mass separately on the two boundary intervals and equals it away from them. Moreover $β_{j, \pm} = O (h)$ and $∥ w_{j, h} - q^{- j} ∥_{L^{1} (1, 2)} ≪ h$ . The last term of $w$ transports the original endpoint to a normalized average supported in $(2 - h, 2)$ ; it does not delete that endpoint. Equations (I75)–(I77) give $w A_{N, H} \in C_{c}^{\infty} (1, 2), = \frac{1}{L} \int_{1}^{2} w (q) R_{N} (Nq) d q + O (E_{N}), ∥ w^{(k)} ∥_{\infty} E_{N} ≪_{k} h^{2 - k} ϵ_{N}, = \frac{h ^{2} ϵ _{N}}{L ^{2}} + \frac{H ^{4}}{N ^{3} L ^{2}} (ϵ_{N}^{2} + ϵ_{N} v_{N}) = o (1) . (I78)$ For each boundary deformation, subtract the endpoint’s constant value before using (I77); its mass is zero. The resulting error is $O (h H (ϵ_{N} + v_{N}) / L)$ in each $I_{j}$ . The transported endpoint costs $O (∣ e_{N, H} ∣ H (ϵ_{N} + v_{N}) / L^{2})$ in the final functional. These are exactly the allowances in (I78). At the original bandwidth, $4 λ < 2 c_{⋆}$ and $Ξ/Ω \to \infty$ pay them. The common level has not been centered away: $\int w = A lo g 2 - 3 B / (4 N) + e_{N, H}$ need not be zero.

Retain the exact scalar channels before cutting smooth frequencies

Choose the preceding fixed profiles from the standard bump already used in the research volume, Lemma 31.1, whose paper calculation gives $∥ ϕ^{(k)} ∥_{1} \leq 1 2^{k} (k!)^{2}$ for $ϕ (x) = e^{- 1/ (1 - 4 x^{2})} 1_{∣ x ∣ < 1/2}$ . The same cited Cauchy estimate also bounds $∥ ϕ^{(k)} ∥_{\infty}$ by these constants, as required for the profile derivatives. The related Tlas source note records the original bump and its Fourier convention. This reuses the standard Gevrey bound; no new bump or saddle theorem is asserted. One may take $ρ (x) = 2 ϕ (2 x - 1) / \int ϕ$ and $χ$ as above. Leibniz bounds and the analytic coordinate $q = e^{x}$ then give fixed $C > 0$ such that, for $κ = κ_{N}$ and every integer $k \geq 0$ , $∥ \partial_{x}^{k} [e^{(κ + 1) x} w (e^{x})] ∥_{1} + ∥ \partial_{x}^{k} [x e^{(κ + 1) x} w (e^{x})] ∥_{1} \leq C^{k + 1} (k!)^{2} h^{2 - k} ϵ_{N} .$ All integrals are over $0 < x < lo g 2$ with zero extension; both endpoints are flat. Constants are independent of $N, H, k$ . For example, expand $(q \partial_{q})^{k}$ with Stirling numbers of the second kind, and use $S (k, j) (j!)^{2} \leq k^{k} k! \leq e^{k} (k!)^{2}$ ; products with the fixed analytic factors only change $C$ . Optimizing the integration-by-parts order, as for the cited bump, gives fixed $a, C^{'} > 0$ and $g_{w} (s) = \int_{1}^{2} w (q) q^{s} d q, ∣ g_{w} (κ_{N} + i t) ∣ + ∣ g_{w}^{'} (κ_{N} + i t) ∣ \leq C^{'} h^{2} ϵ_{N} e^{- a h ∣ t ∣} . (I79)$ The $h^{- k}$ costs are retained; these are not fixed-complexity tests.

Put $c_{w} = \int_{1}^{2} qw (q) d q$ , $d_{w} = \int_{1}^{2} w (q) d q$ and $B_{all} = \sum_{n \leq N} μ (n) lo g (N / n)$ . Define $C_{N, H} = \frac{N c _{w} U _{N} - d _{w} B _{all}}{L ^{2}}, F_{w} (s) = \frac{ζ ^{'} ( s )}{ζ ( s )} \frac{g _{w} ( s )}{s} - \frac{1}{ζ ( s )} (\frac{g _{w}^{'} ( s )}{s} - \frac{g _{w} ( s )}{s ^{2}}) .$ Fix $N, H$ and hold the resulting $w$ fixed during Mellin inversion. On the original line, the exact smooth-moment identity is $\frac{1}{L} \int_{1}^{2} w (q) R_{N} (Nq) d q = C_{N, H} + \frac{1}{L ^{2}} P_{s} N^{s} F_{w} (s) . (I80)$ The integral on the right is absolutely convergent. For verification, integrate (I70) against $w$ . The full kernel is $\frac{\int w ( q ) H _{q} ( s ) d q}{ζ ( s )} = \frac{c _{w}}{( s - 1 ) ^{2} ζ ( s )} - \frac{d _{w}}{s ^{2} ζ ( s )} + \frac{d}{d s} \frac{g _{w} ( s )}{s} + F_{w} (s) .$ The two square-denominator inversions are exactly $N c_{w} U_{N}$ and $- d_{w} B_{all}$ . Integration by parts gives $P_{s} N^{s} (g_{w} / s)^{'} = - L d_{w}$ ; this cancels the literal $d_{w} / L$ from $R_{N} = 1 + \dots$ . Equivalently, integrate the real-argument (I33); finite Dirichlet summation and Mellin inversion give the same two scalar terms. The $q$ integrals have no half-jump term, since the threshold set has measure zero. The original integer endpoint has already been paid by its transport in (I78).

For $∣ t ∣ \geq 2$ , absolute Euler series on the original line give $∣1/ ζ (κ_{N} + i t) ∣ ≪ L$ and $∣ ζ^{'} / ζ (κ_{N} + i t) ∣ \leq - ζ^{'} (κ_{N}) / ζ (κ_{N}) ≪ L$ . Together with (I79), they prove the absolute tail bound, for $h T \geq 1$ , $\frac{1}{L ^{2}} \int_{∣ t ∣ > T} ∣ N^{κ_{N} + i t} F_{w} (κ_{N} + i t) ∣ d t ≪ \frac{N h ^{2} ϵ _{N}}{L} e^{- a^{'} h T} (I81)$ for fixed $a^{'} > 0$ . Indeed the remaining integral is bounded by a constant times $\int_{h T}^{\infty} e^{- a u} d u / u$ . Choose a sufficiently large fixed $C_{0}$ and $T = C_{0} L^{2} / h$ . Equations (I78)–(I81) yield $A_{N, H} = C_{N, H} + \frac{1}{L ^{2}} P_{s, T} N^{s} F_{w} (s) + o (1), T ≍ \frac{N}{H} (lo g N)^{2} . (I82)$ For example $a^{'} C_{0} > 1$ pays (I81), since $N h^{2} ϵ_{N} ≍ N^{1/2} e^{- (c_{⋆} - 2 λ) Ω}$ . This uses the original $H ≍ N^{3/4} e^{λ Ω}$ family, without adopting the fixed-half-plane input (I60).

The scalar channels in $C_{N, H}$ remain exact. Their sizes and signs have not been paid separately, and they cannot be dropped. Without extracting them, the complete smooth kernel retains only a $1/ t^{2}$ rational tail; smoothness alone would not justify (I82)’s height. The useful change is a paid transport of the same endpoint and a frequency remainder with an absolute bound. The joint bound for $C_{N, H}$ plus the retained integral is still missing. No shifted zero-sum estimate, fixed-complexity nilsequence interface, same-integer signed Robin bound, RH proof, mathematical-priority claim or Lean certification follows from this application.

Two smoothing scales allow a controlled full-residue shift

Retain the original integer family, bandwidth and constants of (I58)–(I59), including $0 < 2 λ < c_{⋆}$ , and the half-integer cut $a = (Q + 1/2) / N$ from (I62). This application uses the standard smooth bump already cited in (I79), the complete kernel (I70), and Inoue’s published good-height supplier, (IH1)–(IH2). It introduces no new good-height theorem. The new interface controls the smoothing and contour errors for the complete joint functional; it does not estimate its interior residue sum.

Fix a nonnegative even $ϕ \in C_{c}^{\infty} (- 1, 1)$ with integral one, obtained by normalizing the cited bump, and put $F_{ℓ} (s) = \int_{- 1}^{1} ϕ (t) e^{ℓ t s} d t, 0 < ℓ \leq 1.$ On every fixed vertical strip, integration by parts gives $F_{ℓ} (σ + i τ) ≪_{ϕ, j} (1 + ℓ ∣ τ ∣)^{- j}$ for every fixed integer $j \geq 0$ . Also $F_{ℓ} (0) = 1$ and $F_{ℓ}^{'} (0) = 0$ . Only these existing smooth-bump properties are needed here.

Hold $a$ fixed while varying the real cutoff $x$ , and define $A (x) η = a x < n \leq x \sum μ (n) lo g (x / n) / n, = H^{- 1}, B (x) A = a x < n \leq x \sum μ (n) lo g (x / n), = \int ϕ (t) A (N e^{η t}) d t, B = \int ϕ (t) B (N e^{η t}) d t .$ Here $A (N) = A$ and $B (N) = B$ with their original integer band. If $∣ x - N ∣ \leq Δ ≪ H$ , variation on the common part and the $O (Δ + 1)$ endpoint integers give $∣ A (x) - A ∣ ∣ A - A ∣ ≪ h (Δ + 1) / N, ≪ N^{- 1}, ∣ B (x) - B ∣ ∣ B - B ∣ ≪ h (Δ + 1), ≪ 1. (I83)$ Indeed the common log weight varies by $O (Δ/ N)$ and the endpoint weights are $O (h)$ . For this smoothing $Δ ≪ N / H$ and $h N / H ≍ 1$ . No extra Mobius cancellation is used in these differences.

For the residual use a different scale, moving both its cutoff and its argument. Write $S_{x} (t) = lo g x + n \leq x \sum μ (n) lo g (x / n) {t / n}, S_{N} (q) = S_{N} (qN),$ so $S_{N} (m / N) = L R_{N} (m)$ at the original integer samples. Set $ν = N^{- 1/2}$ and $S_{N} (q) = \int ϕ (t) S_{N e^{ν t}} (qN e^{ν t}) d t$ . The real-cutoff version of (I33) is $S_{x} (t) = t U_{x} - ψ (t) + x < n \leq t \sum μ (n) lo g (x / n), U_{x} = n \leq x \sum μ (n) lo g (x / n) / n, x \leq t \leq 2 x,$ with the same full-weight $ψ$ at integers. The divisor-convolution proof of (I33) applies unchanged: any omitted divisor in this range must be the integer itself. Between consecutive integers, $(x U_{x})^{'} = U_{x} + \sum_{n \leq x} μ (n) / n = O (L^{2})$ , and $x U_{x}$ is continuous at integer cutoffs. Prime jumps cost $O ((Δ + 1) L)$ when $∣ x - N ∣ \leq Δ$ ; the final Mobius sum changes by $O (Δ + 1)$ . Uniformly for $1 \leq q \leq 2$ , this gives $∥ S_{N} - S_{N} ∥_{\infty} ∥ S_{N} ∥_{\infty} + ∥ S_{N} ∥_{\infty} ≪ N L^{2}, ≪ N . (I84)$ The second row also follows directly from $\sum_{n \leq x} lo g (x / n) ≪ x$ . The $+ 1$ in the variation bound retains the integer jumps, including both original endpoints. The actual balanced step residual has not been replaced by an identical continuous function; (I84) pays the change of samples.

Keep exactly the discrete operators $p_{N}, q_{N}$ of (I64), whose sup-norm operator bounds are $O (1)$ and $O (N^{- 1})$ . Define $A_{N, H} = \frac{A p _{N} ( S _{N} ) - B q _{N} ( S _{N} )}{L ^{2}} .$ Using (I83)–(I84) and the already owned $∣ A ∣ + ∣ B ∣/ N ≪ h^{2} e^{- c_{⋆} Ω}$ yields $∣ A_{N, H} - A_{N, H} ∣ ≪_{ϕ} L^{- 2} + N h^{2} e^{- c_{⋆} Ω} ≪_{ϕ} L^{- 2} + e^{- (c_{⋆} - 2 λ) Ω} = o (1) . (I85)$ For example, coefficient errors act on a residual bounded by $N$ ; residual errors act on the original coefficients. Both smoothings are deterministic linear operations on the same arithmetic data. If the band coefficients were instead smoothed on spatial width $N$ , this argument would leave $h N / L^{2}$ , which grows at the original bandwidth. The distinct scales are essential to this error estimate.

The absolutely convergent smoothed Mellin formulas on $ℜ s = 2$ are $A B S_{N} (q) = \frac{1}{2 πi} \int_{(2)} \frac{N ^{u - 1} W _{a} ( u - 1 ) F _{η} ( u - 1 )}{ζ ( u )} d u, = \frac{1}{2 πi} \int_{(2)} \frac{N ^{u} W _{a} ( u ) F _{η} ( u )}{ζ ( u )} d u, = L + \frac{1}{2 πi} \int_{(2)} \frac{N ^{v} H _{q} ( v ) F _{ν} ( v )}{ζ ( v )} d v . (I86)$ These follow from (I62)–(I63) with the varying real cutoff. Each integer jump occurs at a discrete set of smoothing parameters, of measure zero, so the symmetric and actual values have equal integrals. The literal average of $lo g (N e^{ν t})$ is $L$ by evenness. The two band multipliers are $F_{η} (u - 1)$ and $F_{η} (u)$ ; they cannot be factored into one common multiplier.

Apply the owned good-height theorem with $U = N^{4}$ . It supplies one $T_{N} \in [N^{4}, N^{4} + N^{4/3}]$ such that $- 1 \leq σ \leq 2, ξ \in {- 1, 1} sup ∣ ζ (σ + i ξ T_{N}) ∣^{- 1} ≪_{ε} T_{N}^{ε} .$ No RH, simple-zero hypothesis or negative moment of $ζ^{'}$ is assumed. Let $Z_{N}$ contain each distinct nontrivial zero with $∣ℑ ρ ∣ < T_{N}$ once. Define $a_{N}^{\circ}, b_{N}^{\circ}, z_{N} (q)$ as the sums of the full residues at $ρ \in Z_{N}$ of the three respective integrands in (I86), omitting the factor $(2 πi)^{- 1}$ . Multiple zeros are included through their complete meromorphic residues. Moving the lines from $2$ to $- 1$ gives $A B S_{N} (q) = a_{N}^{\circ} + O_{ϕ} (h^{2} N^{- 2} + N^{- 10}), = b_{N}^{\circ} + O_{ϕ} (h^{2} N^{- 1} + N^{- 10}), = lo g (2 π) + z_{N} (q) + O_{ϕ} (N^{- 1} L^{2} + N^{- 10}), 1 \leq q \leq 2. (I87)$ The last error is uniform in $q$ . Here are the contour costs and all extra residues needed for this shift.

On the horizontal edges, standard polynomial growth on a slightly wider fixed strip and Cauchy’s estimate give $∣ ζ^{'} ∣ ≪ T_{N}^{2}$ , so the good-height bound gives $∣ ζ^{'} / ζ ∣ ≪_{ε} T_{N}^{2 + ε}$ there. It does not give an $O (lo g T_{N})$ horizontal bound. Formula (I70) therefore bounds $H_{q} / ζ$ polynomially, uniformly in $q$ . As $T_{N} / H \geq N^{3}$ and $T_{N} / N \geq N^{7/2}$ , the fixed integration-by-parts order $j = 20$ in $F_{ℓ}$ pays both horizontal edges and the omitted original-line tails by $O_{ϕ} (N^{- 10})$ . For instance, the horizontal integrands are bounded by $O (N^{2} T_{N}^{2 + ε})$ before the smoothing multipliers; each has width three, and their smoothing decay is at least $N^{- 60}$ . The original-line tails can be integrated using the same decay and the absolutely convergent Dirichlet series.

On the entire left line the functional equation, Stirling bounds and the Euler series on $ℜ (1 - s) = 2$ give $∣ ζ (- 1 + i τ) ∣^{- 1} ≪ (1 + ∣ τ ∣)^{- 3/2}, ∣ ζ^{'} / ζ (- 1 + i τ) ∣ ≪ lo g (2 + ∣ τ ∣) .$ The complete identity (I70) and $a > 2/3$ imply $∣ W_{a} (σ + i τ) ∣ 1 \leq q \leq 2 sup ∣ H_{q} (- 1 + i τ) / ζ (- 1 + i τ) ∣ N^{- 1} \int_{R} \frac{lo g ( 2 + ∣ τ ∣ )}{1 + ∣ τ ∣} (1 + ν ∣ τ ∣)^{- 20} d τ ≪ h^{2} (- 2 \leq σ \leq 2), ≪ \frac{lo g ( 2 + ∣ τ ∣ )}{1 + ∣ τ ∣}, ≪ N^{- 1} L^{2} . (I88)$ Thus the full left-line integrals cost respectively $h^{2} N^{- 2}$ , $h^{2} N^{- 1}$ and $N^{- 1} L^{2}$ . Their tails above $T_{N}$ are also $O_{ϕ} (N^{- 10})$ . No absolute bound on interior residues is used.

There is no pole at $s = 1$ : the original integral for $H_{q}$ is holomorphic on $ℜ s > 0$ , and its apparent singularity in (I70) is removable. At zero, (I70) gives $H_{q} (s) = \frac{1}{2 s ^{2}} + O (1), \frac{H _{q} ( s )}{ζ ( s )} = - \frac{1}{s ^{2}} + \frac{lo g ( 2 π )}{s} + O (1) .$ Since $F_{ν}^{'} (0) = 0$ , the extra residue in the last integral of (I86) is $- L + lo g (2 π)$ . It cancels the literal $L$ and retains the common level $lo g (2 π)$ in (I87). The band integrands have no pole at zero, and no trivial zero is crossed before $ℜ s = - 1$ . This proves (I87) with every nonzero contour contribution accounted for. The other application’s formula (CH1b) in the Chirre–Helfgott note requires $y > e^{2} T$ , so it is not used at $y ≍ N, T ≍ N^{4}$ ; only the owned good-height and functional-equation inputs are reused.

Using $p_{N} (v), q_{N} (v)$ from (I64), define the complete smoothed kernel and finite double residue sum $K_{N} (u, v) Σ_{N, H} = [W_{a} (u - 1) F_{η} (u - 1) p_{N} (v) - N W_{a} (u) F_{η} (u) q_{N} (v)] F_{ν} (v), = ρ, σ \in Z_{N} \sum Res_{u = ρ} Res_{v = σ} \frac{N ^{u + v - 1} K _{N} ( u , v )}{ζ ( u ) ζ ( v )} = a_{N}^{\circ} p_{N} (z_{N}) - b_{N}^{\circ} q_{N} (z_{N}) . (I89)$ This is a finite sum of full residues, with no assumption about simple zeros. From (I83)–(I87) and the original physical-space bounds, $∥ z_{N} ∥_{\infty} ≪ N$ and $∣ a_{N}^{\circ} ∣ + ∣ b_{N}^{\circ} ∣/ N ≪ h^{2} e^{- c_{⋆} Ω} + N^{- 1}$ . They pay all cross terms from (I87); the retained constant contributes $O ((h^{2} e^{- c_{⋆} Ω} + N^{- 1}) / L^{2})$ . Consequently $A_{N, H} = \frac{Σ _{N, H}}{L ^{2}} + O_{ϕ} (L^{- 2} + e^{- (c_{⋆} - 2 λ) Ω}) . (I90)$ Both $f (2)$ endpoint terms remain in the original discrete operators. The full $W_{a}, H_{q}$ are used through the growing heights, without substituting the fixed-compact expansion (I67).

The remaining estimate is $∣ Σ_{N, H} ∣ = o (L^{2})$ for each sufficiently large original integer $N$ and its specified band. A fixed zero pair, an average over $N$ , or separately optimized residue sums do not give that estimate. Equations (I83)–(I90) are a repository application of existing analytic tools, with no mathematical-priority claim or Lean certification. They control the errors in a full-residue representation; they do not prove the joint cancellation, the complete same-integer signed Robin bound, or RH.

A larger denominator cut for the actual joint linear functional

Keep the original integer family of (I59), and put $ϵ_{N} = e^{- c_{⋆} Ω}$ and $Δ_{⋆} = c_{⋆} - 2 λ > 0$ . Reuse the complete reduced-frequency expansion (I17)–(I19), including all aliases in $b_{N, q}$ , its uniform coefficient bound, and the finite first positive moment from Weingartner’s existing source note. This application estimates a linear contribution to the original joint functional, rather than the entire residual norm.

For an integer-grid function $f$ , define $J_{N} (f) k_{N, m} = \frac{1}{L} (m = N \sum 2 N k_{N, m} f (m) + e_{N, H} f (2 N)), = \frac{A ( m + \frac{1}{2} ) - \frac{3}{2} B}{m ( m + 1 )}, e_{N, H} = \frac{B}{2 N + 1} - \frac{A}{2} . (I91)$ Thus $J_{N} (R_{N}) = A_{N, H}$ exactly, with both original endpoint contributions combined in their actual coefficient. Let $T_{N, \leq D}$ consist of every frequency in (I18) whose reduced denominator is at most the integer $D$ , where $2 \leq D \leq N$ . In particular its coefficient $b_{N, q}$ still contains every $q j \leq N$ , including $q j > D$ ; this is not a truncation of the original coefficient dictionary.

The finite geometric sum and elementary sine bounds give $m = u \sum v e (m r / q) \leq \frac{2}{∣1 - e ( r / q ) ∣}, r = 1 \sum q - 1 ∣1 - e (r / q) ∣^{- 2} ≪ q^{2}, r = 1 \sum q - 1 ∣1 - e (r / q) ∣^{- 1} ≪ q lo g (2 q) .$ The already owned $∣ b_{N, q} ∣ \leq 2 q / [φ (q) L]$ and finite moment $\sum_{q \leq D} q / φ (q) ≪ D$ therefore imply $u \leq v sup m = u \sum v T_{N, \leq D} (m) m sup ∣ T_{N, \leq D} (m) ∣ ≪ q \leq D \sum ∣ b_{N, q} ∣ q ≪ D^{2} / L, ≪ q \leq D \sum ∣ b_{N, q} ∣ lo g (2 q) ≪ D lo g (2 D) / L . (I92)$ Here $u, v$ are integers. Enlarging the numerator sums to all $1 \leq r < q$ only increases these positive majorants. The second finite moment $\sum q^{2} / φ (q) ≪ D^{2}$ needed in the first row follows at once by multiplying the first moment bound by $D$ ; no new totient distribution theorem is used.

For the continuous extension of the actual weight, $k_{N} (x) = [A (x + 1/2) - 3 B /2] / [x (x + 1)]$ , differentiation gives $N \leq x \leq 2 N sup ∣ k_{N} (x) ∣ + \int_{N}^{2 N} ∣ k_{N}^{'} (x) ∣ d x ≪ \frac{∣ A ∣ + ∣ B ∣/ N}{N} ≪ \frac{h ^{2} ϵ _{N}}{N} .$ Discrete Abel summation with the first row of (I92), including both integer endpoints, controls the sum in (I91). For its last term use the actual relation $e_{N, H} ≪ h^{3} ϵ_{N}$ from (I75) and the point-value bound in (I92). Consequently, uniformly for $2 \leq D \leq N$ , $∣ J_{N} (T_{N, \leq D}) ∣ ≪ \frac{ϵ _{N}}{L ^{2}} (\frac{h ^{2} D ^{2}}{N} + h^{3} D lo g (2 D)) . (I93)$ The same $A, B$ are used in both terms; no independence or separate optimal choice is introduced. The second term retains the endpoint, and its additional factor $h$ uses the same-source relation (I75).

Put $κ_{⋆} = min (c_{⋆} /2, Δ_{⋆}) > 0$ . For any fixed $0 \leq κ \leq κ_{⋆}$ , take $D_{N} (κ) = ⌊(N / h) e^{κ Ω} ⌋$ . At the original bandwidth, $2 \leq D_{N} (κ) \leq N$ eventually and $D_{N} (κ) ≍ N^{3/4} e^{(κ - λ) Ω}$ , since $Ω = o (L)$ . Substitution into (I93) gives $∣ J_{N} (T_{N, \leq D_{N} (κ)}) ∣ ≪ \frac{e ^{- (c_{⋆} - 2 κ) Ω}}{L ^{2}} + \frac{e ^{- (Δ_{⋆} - κ) Ω}}{L} = o (1) . (I94)$ Both exponents are nonnegative throughout the stated closed range; even at its upper endpoint the logarithmic denominators tend to infinity. At $κ = 0$ this recovers the smaller cut $⌊ N / h ⌋$ and the exponentially decaying bound. The positive endpoint $κ_{⋆}$ uses the remaining slack in the same inequality, rather than an additional distribution theorem.

This is a pointwise-in- $N$ estimate for the complete frequency component. The norm cut (I21) alone does not even pay the $κ = 0$ range: weighted Cauchy on the interior sum would give only $O (h^{2} ϵ_{N} D (1 + D^{2} / N) / L^{2})$ , whose allowance there is of order $N^{1/8} e^{- (c_{⋆} - λ /2) Ω} / L^{2}$ and does not tend to zero. The improvement comes from summing the oscillation against the actual weight before taking its absolute value.

For the remaining cut choose $D_{N} = D_{N} (κ_{⋆})$ , retain the original constant $d_{N}$ from (I17), and put $T_{N, > D_{N}} = R_{N} - d_{N} - T_{N, \leq D_{N}}$ . Linearity and (I94) give $A_{N, H} = J_{N} (d_{N} + T_{N, > D_{N}}) + O (\frac{e ^{- (c_{⋆} - 2 κ_{⋆}) Ω}}{L ^{2}} + \frac{e ^{- (Δ_{⋆} - κ_{⋆}) Ω}}{L}) . (I95)$ The retained aliases now have $q > D_{N}$ and $j \leq N / q < N / D_{N}$ (up to the integer endpoint), but their joint coprimality and actual Mobius signs remain. This reduced-denominator cut is not a cut by prime size or the same as observing one modulus.

The constant must remain coupled to those frequencies. Directly from (I64), $p_{N} (1) - N q_{N} (1) = lo g 2 - 3/4 + O (N^{- 1})$ , whose limit is nonzero. Thus $A = B / N + O (h^{3} ϵ_{N})$ does not annihilate the constant channel. The still missing estimate is $J_{N} (d_{N} + T_{N, > D_{N}}) = o (1)$ with its actual coefficients and endpoint. This is a project application of the owned Fourier, coefficient and variation inputs, without a mathematical-priority claim or Lean certification. It does not supply the full joint bound, signed Robin inequality or RH.

A different proposed supplier, vanishing unweighted critical-line mollifier error on the active height intervals, is ruled out by the Radziwill source and exact parameter map. That lower bound neither rules out the weighted Nyman–Beurling norm tending to zero nor supplies a sign for the retained functional in (I95).

Preserve primitive-numerator cancellation at the endpoint

Keep the complete coefficients and actual functional of (I91). For an integer $m$ and $q \geq 2$ , set $S_{q} (m) = {m / q} - \frac{q - 1}{2 q}, Π_{q} (m) = - \frac{1}{q} 1 \leq r < q (r, q) = 1 \sum \frac{e ( m r / q )}{1 - e ( - r / q )},$ and put $S_{1} = Π_{1} = 0$ . Grouping the already owned finite sawtooth expansion by reduced denominator and applying classical Mobius inversion gives $q S_{q} (m) = d ∣ q \sum d Π_{d} (m), Π_{q} (m) = \frac{1}{q} d ∣ q \sum μ (q / d) d S_{d} (m), ∣ Π_{q} (m) ∣ \leq \frac{σ ( q )}{2 q} . (I96)$ Indeed $∣ S_{d} (m) ∣ \leq (d - 1) / (2 d) \leq 1/2$ on the integer grid. The first equality retains every reduced denominator of the original $q$ -periodic response; inversion is applied to this finite divisor identity before taking an absolute value. This reuses standard Fourier grouping and Mobius inversion in the current application.

The existing finite positive-moment bound gives $\sum_{q \leq D} (q / φ (q))^{2} \leq D W (2)$ , with the fixed finite Euler product $W (2)$ defined there. The same source already supplies $σ (q) / q \leq q / φ (q)$ , hence $q \leq D \sum \frac{σ ( q )}{φ ( q )} \leq q \leq D \sum (\frac{q}{φ ( q )})^{2} \leq D W (2) .$ No additional average-order estimate is needed. Since $T_{N, \leq D} (m) = \sum_{2 \leq q \leq D} b_{N, q} Π_{q} (m)$ with the same full aliases as in (I18), its owned coefficient bound and (I96) now give $m \in Z sup ∣ T_{N, \leq D} (m) ∣ ∣ J_{N} (T_{N, \leq D}) ∣ \leq \frac{1}{L} q \leq D \sum \frac{σ ( q )}{φ ( q )} \leq \frac{D W ( 2 )}{L}, ≪ \frac{ϵ _{N}}{L ^{2}} (\frac{h ^{2} D ^{2}}{N} + h^{3} D), 2 \leq D \leq N . (I97)$ For the second row, the interior sum uses the unchanged interval-sum and variation bounds in (I92)–(I93). Its endpoint uses the first row of (I97) and the actual $e_{N, H} ≪ h^{3} ϵ_{N}$ from (I75). Thus the endpoint’s factor $lo g (2 D)$ in the earlier bound is removed by retaining cancellation between primitive numerators of each $q$ .

With $κ_{⋆}$ from (I94), fix any $0 \leq η < 1$ and set $D_{N} = ⌊ \frac{N}{h} e^{κ_{⋆} Ω} L^{η} ⌋, T_{N, > D_{N}} = R_{N} - d_{N} - T_{N, \leq D_{N}} .$ For sufficiently large $N$ , $2 \leq D_{N} \leq N$ and $D_{N} ≍ N^{3/4} e^{(κ_{⋆} - λ) Ω} L^{η}$ . Substituting this cut in (I97) and using linearity yields $A_{N, H} = J_{N} (d_{N} + T_{N, > D_{N}}) + O (e^{- (c_{⋆} - 2 κ_{⋆}) Ω} L^{2 η - 2} + e^{- (Δ_{⋆} - κ_{⋆}) Ω} L^{η - 2}) . (I98)$ Both exponential factors are bounded by one, and both logarithmic powers tend to zero for each fixed $η < 1$ . No uniform convergence as $η ↑ 1$ is asserted. At $η = 0$ the same cut as (I95) has a smaller endpoint allowance; every fixed $η > 0$ also pays a larger complete frequency sector.

The retained aliases have $q > D_{N}$ and $j < N / D_{N}$ . Their actual signs, coprimality, endpoint and constant channel remain coupled. The still required estimate is $J_{N} (d_{N} + T_{N, > D_{N}}) = o (1)$ . This is a refinement of the project application of existing arithmetic inputs, with no mathematical-priority claim or Lean certification. It supplies no complete same-integer signed Robin bound or RH proof.

The uniform estimate (I97) also permits using its two allowances directly, without restricting the logarithmic gain to a fixed power. Define $D_{int} = \frac{N}{h} ϵ_{N}^{- 1/2} L, D_{end} = h^{- 3} ϵ_{N}^{- 1} L^{2}, D_{N}^{\circ} = ⌊ \frac{min ( D _{int} , D _{end} )}{lo g L} ⌋ . (I99)$ The original parameter family has $h ≍ N^{- 1/4} e^{λ Ω}$ , hence both allowances are $N^{3/4} e^{O (Ω)}$ times a fixed logarithmic power. Since $Ω = o (L)$ , eventually $2 \leq D_{N}^{\circ} \leq N$ . Equation (I97) then reads $∣ J_{N} (T_{N, \leq D_{N}^{\circ}}) ∣ A_{N, H} ≪ (\frac{D _{N}^{\circ}}{D _{int}})^{2} + \frac{D _{N}^{\circ}}{D _{end}} ≪ \frac{1}{( lo g L ) ^{2}} + \frac{1}{lo g L} = o (1), = J_{N} (d_{N} + T_{N, > D_{N}^{\circ}}) + O (1/ lo g L), T_{N, > D_{N}^{\circ}} = R_{N} - d_{N} - T_{N, \leq D_{N}^{\circ}} . (I100)$ In the fixed regime $Δ_{⋆} \geq c_{⋆} /2$ , the interior allowance is eventually smaller and $D_{N}^{\circ} ≍ (N / h) e^{c_{⋆} Ω/2} L / lo g L$ . In the fixed regime $Δ_{⋆} < c_{⋆} /2$ , the endpoint allowance is eventually smaller and $D_{N}^{\circ} ≍ (N / h) e^{Δ_{⋆} Ω} L^{2} / lo g L$ . These comparisons follow from $\mathfrak D_{\mathrm{end}}/\mathfrak D_{\mathrm{int}} \asymp e^{(\Delta_\star-c_\star/2)\Omega}L$; the equality case belongs to the first regime. Thus $D_{N}^{\circ} / D_{N} \to \infty$ for each fixed $0 \leq η < 1$ in (I98). This deduction uses the uniform finite- $D$ estimate (I97), rather than substituting a varying $η$ into its fixed-parameter consequence (I98).

This is an optimization of the two existing error allowances, with no additional cancellation theorem or mathematical-priority claim. The original aliases, endpoint and constant are retained. The still missing estimate is $J_{N} (d_{N} + T_{N, > D_{N}^{\circ}}) = o (1)$ ; the larger paid sector does not settle that estimate, the signed Robin bound or RH.

Pay a complete top band of reduced denominators

Reuse the all-start classical short-interval Mobius estimate in the existing source contract. Use its same fixed $θ_{0} = 7/12 + η$ from (I48), with $c_{⋆} \leq c_{θ_{0}}$ as allowed by the original common-constant choice in (I54)–(I59). Keep that $c_{⋆}$ , $λ$ and $H$ fixed. No new Mobius distribution theorem is assumed or proved here.

For any fixed $C > 0$ , let the integer $W$ satisfy $N^{3/4} e^{- C Ω} \leq W \leq N^{3/4} e^{C Ω}$ , and put $T_{top, W} = \sum_{N - W < q \leq N} b_{N, q} Π_{q}$ . This band eventually has $q > N /2$ , so its complete coefficients (I19) have exactly the alias $j = 1$ . Apply the already owned inversion (I96) and write $q = t d$ . Squarefreeness gives the exact finite identity $T_{top, W} (m) = \frac{1}{L} t \leq N \sum \frac{μ ( t ) ^{2}}{t} (N - W) / t < d \leq N / t (d, t) = 1 \sum μ (d) lo g \frac{N}{t d} S_{d} (m) . (I101)$ Indeed, on squarefree $q$ , $μ (q) μ (q / d) = μ (d)$ ; $μ (t)^{2}$ , $μ (d)$ and $(d, t) = 1$ retain precisely those terms. This is not a replacement of the original signs by absolute values.

Set $K = ⌈ ϵ_{N}^{- 2} ⌉$ and $U = ⌈ ϵ_{N}^{- 4} ⌉$ . For $t \leq K$ put $x = (N - W) / t$ , $y = W / t$ , and let $S (t)$ be the positive integers whose prime factors all divide $t$ . The standard finite Euler-factor convolution is $μ (n) 1_{(n, t) = 1} = s ∣ n s \in S (t) \sum μ (n / s) .$ For $s \leq U$ , the interval start $x / s = N exp (O_{C} (Ω))$ and its full length $y / s = N^{3/4} exp (O_{C} (Ω))$ . Moreover $Ω (x / s) = Ω (N) + O_{C} (Ω^{2} / L) = Ω (N) + o (1)$ . Apply the same $θ_{0}$ supplier to prefixes of admissible length; shorter prefixes use the trivial bound. The fixed power surplus $3/4 - θ_{0} > 0$ absorbs the latter in $y ϵ_{N} / s$ . Thus each such prefix is $O_{C} (y ϵ_{N} / s)$ , with the unchanged exponential constant.

The tail $s > U$ must also be retained. Set $P_{t} = \prod_{p ∣ t} (1 - p^{- 1/2})^{- 1}$ . Ordering the distinct prime factors and using $p_{i} \geq i + 1$ gives $P_{t} \leq exp (O (lo g (2 t))) = exp (O (Ω))$ for $t \leq K$ . The convergent positive Euler product implies $s > U s \in S (t) \sum s^{- 1} \leq U^{- 1/2} P_{t}, # {s \leq N : s \in S (t)} \leq N P_{t} .$ An interval contains at most $y / s + 1$ integers. Its total tail is therefore at most $y U^{- 1/2} P_{t} + N P_{t} = O_{C} (y ϵ_{N})$ . Summing the small- $s$ bounds with $\sum_{s \in S (t)} s^{- 1} = t / φ (t)$ yields $0 \leq u \leq y sup x < d \leq x + u (d, t) = 1 \sum μ (d) ≪_{C} \frac{t}{φ ( t )} \frac{W}{t} ϵ_{N}, t \leq K . (I102)$ This uses the mother theorem on all those starts; the three scalar bounds for $A, B, δ_{N, H}$ alone would not imply (I102).

For integer $N \leq m \leq 2 N$ , the actual weight on $[x, x + y]$ is $w_{t, m} (z) = lo g \frac{N}{t z} (\frac{m + 1/2}{z} - ⌊ m / z ⌋ - \frac{1}{2}) .$ At integer $z = d$ this equals the weight in (I101). Since $t W / N = o (1)$ for $t \leq K$ , there are only $O (1)$ floor jumps, each of size $O (W / N)$ . On smooth pieces $∣ w_{t, m}^{'} ∣ ≪ t / N + W t^{2} / N^{2}$ . Consequently $∥ w_{t, m} ∥_{\infty} + Var_{[x, x + y]} w_{t, m} ≪_{C} W / N . (I103)$ The values at the actual integer thresholds are included. Abel summation with (I102) controls the $t \leq K$ terms in (I101) by $O_{C} (W^{2} ϵ_{N} / (N L))$ , because the already owned first moment $\sum_{t \leq X} t / φ (t) ≪ X$ gives $\sum_{t} (t / φ (t)) / t^{2} < \infty$ by dyadic summation. For $t > K$ , use $∣ S_{d} (m) ∣ \leq 1/2$ on the integer grid: $\frac{W}{N L} K < t \leq N \sum \frac{1}{t} (\frac{W}{t} + 1) ≪ \frac{W ^{2}}{N L K} + \frac{W}{N} ≪_{C} \frac{W ^{2} ϵ _{N}}{N L} .$ Here $K^{- 1} \leq ϵ_{N}^{2}$ and $W ϵ_{N} ≫ L$ . Thus the complete primitive band, including every coprime numerator, satisfies $N \leq m \leq 2 N max ∣ T_{top, W} (m) ∣ ≪_{C} \frac{W ^{2} ϵ _{N}}{N L}, ∣ J_{N} (T_{top, W}) ∣ ≪_{C} \frac{h ^{2} W ^{2} ϵ _{N}^{2}}{N L ^{2}} . (I104)$ For the second bound use the actual coefficients in (I91): $\sum_{m = N}^{2 N} ∣ k_{N, m} ∣ + ∣ e_{N, H} ∣ ≪ h^{2} ϵ_{N}$ . The term $e_{N, H} T_{top, W} (2 N)$ is included.

Take $W_{N}^{\circ} = ⌊ \frac{N}{h ϵ _{N}} \frac{L}{lo g L} ⌋, R_{N}^{\circ} = d_{N} + D_{N}^{\circ} < q \leq N - W_{N}^{\circ} \sum b_{N, q} Π_{q} . (I105)$ This width remains in the stated $N^{3/4} exp (O (Ω))$ family. Both $D_{N}^{\circ}$ and $W_{N}^{\circ}$ are $o (N)$ , so the two deleted bands are disjoint eventually. Equations (I100) and (I104) give $∣ J_{N} (T_{top, W_{N}^{\circ}}) ∣ ≪ (lo g L)^{- 2}, A_{N, H} = J_{N} (R_{N}^{\circ}) + O (1/ lo g L) . (I106)$ The width satisfies $W_{N}^{\circ} / H ≍ e^{Δ_{⋆} Ω} L / lo g L \to \infty$ . A direct absolute-value estimate for the top band loses the extra $ϵ_{N}$ in its pointwise bound, giving at this width an allowance $O (ϵ_{N}^{- 1} / (lo g L)^{2})$ that does not vanish. The coprime-prefix estimate is the quantitative gain used here.

This is an application of existing short-interval and finite-moment inputs to a reduced-denominator band, distinct from the original-index band (I49). It makes no new-supplier, priority or Lean-certification claim. Every remaining coefficient still uses the original cutoff $N$ and its full alias list; $R_{N}^{\circ}$ is not a residual reweighted at $N - W_{N}^{\circ}$ . The constant and endpoint are retained. The estimate $J_{N} (R_{N}^{\circ}) = o (1)$ remains missing, as do the complete same-integer signed Robin bound and RH proof.

The Bettin–Chandee applicability note records the direct alias-split boundary and excludes the withdrawn stronger Kloosterman-fraction claim from the available suppliers.

A narrower numerator list for the interior linear functional

The numerator cut (I28)–(I30) controls the weighted residual norm. Here use instead the bounded variation of the actual linear weight in (I91)–(I93). This is a direct application of the same geometric sum, sine and finite-moment inputs; it is not a new exponential-sum estimate. Keep $D_{N}^{\circ}, W_{N}^{\circ}, R_{N}^{\circ}$ from (I99)–(I106), with every original coefficient and alias.

Define the finite transform of the interior weight by $F_{N} (θ) = m = N \sum 2 N k_{N, m} e (m θ) .$ The geometric sum and the already supplied bound on $∥ k_{N} ∥_{\infty} + Var k_{N}$ give, by discrete Abel summation, $∣ F_{N} (θ) ∣ ≪ \frac{h ^{2} ϵ _{N}}{N ∥ θ ∥}, θ \in / Z . (I107)$ This uses the actual closed integer interval and both its boundary values. It does not include the separate $e_{N, H} f (2 N)$ term.

Write $r^{*} = min (r, q - r)$ . For any integer $s \geq 1$ , consider only $D_{N}^{\circ} < q \leq N - W_{N}^{\circ}$ and $r^{*} > s$ . Using (I107), $∣1 - e (- r / q) ∣ \geq 4 r^{*} / q$ and the already owned first positive moment, their complete contribution to the interior part of $J_{N}$ is bounded by $\frac{1}{L} q \sum 1 \leq r < q (r, q) = 1 r^{*} > s \sum \frac{b _{N, q} F _{N} ( r / q )}{q ( 1 - e ( - r / q ))} ≪ \frac{h ^{2} ϵ _{N}}{N L} q \leq N \sum q ∣ b_{N, q} ∣ r^{*} > s \sum \frac{1}{( r ^{*} ) ^{2}} ≪ \frac{N h ^{2} ϵ _{N}}{L ^{2} s} . (I108)$ At most two numerators have any given $r^{*}$ , so the inner tail is $O (1/ s)$ . The other factor follows from $\sum_{q \leq N} q ∣ b_{N, q} ∣ ≪ N^{2} / L$ , exactly the finite-moment input of (I92). Coprimality is relaxed only in the positive allowance; the retained sum still has its actual coprime numerators.

Choose the integer cutoff $s_{N}^{\circ} = ⌈ \frac{N h ^{2} ϵ _{N} lo g L}{L ^{2}} ⌉ . (I109)$ Its argument tends to infinity, and $s_{N}^{\circ} ≍ N^{1/2} e^{- Δ_{⋆} Ω} \frac{lo g L}{L ^{2}}, s_{N}^{\circ} = o (D_{N}^{\circ}) .$ The allowance in (I108) is $O (1/ lo g L)$ , matching the paid error in (I106).

The remaining finite scalar is $C_{N}^{\circ} = A_{N, H} = d_{N} F_{N} (0) - D_{N}^{\circ} < q \leq N - W_{N}^{\circ} \sum 1 \leq r < q (r, q) = 1 r^{*} \leq s_{N}^{\circ} \sum \frac{b _{N, q} F _{N} ( r / q )}{q ( 1 - e ( - r / q ))} + e_{N, H} R_{N}^{\circ} (2 N), \frac{C _{N}^{\circ}}{L} + O (1/ lo g L) . (I110)$ The last line follows from (I106) and (I108). The interior sum has at most $2 N s_{N}^{\circ}$ frequency entries, of order at most $N^{3/2} e^{- Δ_{⋆} Ω} lo g L / L^{2}$ . This is a bound on that list, not on the cost of computing its coefficients or on the complete endpoint. Conjugate pairing makes $C_{N}^{\circ}$ real.

The endpoint remains the full $e_{N, H} R_{N}^{\circ} (2 N)$ , including its constant and every retained numerator. No high-numerator endpoint deletion is paid by (I107)–(I108). In particular this is a reduction of a linear functional, not replacement of the response by one frequency-truncated function at every evaluation port.

The still missing estimate is $C_{N}^{\circ} = o (L)$ for every sufficiently large original integer $N$ in the fixed family. The paid error makes this equivalent to $A_{N, H} = o (1)$ within this reduction; neither estimate is established here. The comparison shrinks the internal list relative to (I30) at $m ≍ N$ , where that norm cut retains numerators up to $N / L^{b}$ , $0 < b < 2$ . It supplies no new cancellation theorem, complete signed Robin bound, RH proof, mathematical-priority claim or Lean certification.

An angular cut that also pays the actual endpoint

Retain the same original $N, H, A, B, e_{N, H}$ and coefficients. The following application strengthens the frequency reduction using integer sample locations, the primitive inversion (I96), geometric summation and the classical divisor bound. It also controls the actual endpoint; it does not estimate the remaining signed sum.

For $V \geq 2$ , let $Π_{q, V}^{far}$ denote the original primitive Fourier sum $Π_{q}$ restricted to $∥ r / q ∥ \geq 1/ V$ , and put $Π_{q, V}^{near} = Π_{q} - Π_{q, V}^{far}$ . Define its nonprimitive counterpart $S_{d, V} (m) = - \frac{1}{d} 1 \leq r < d ∥ r / d ∥ \geq 1/ V \sum \frac{e ( m r / d )}{1 - e ( - r / d )}, S_{1, V} = 0.$ The cutoff depends only on the frequency value and is unchanged by reduction. Therefore the same finite grouping and inversion as in (I96) give, for any $I \subseteq {2, \dots, N}$ , $Π_{q, V}^{far} (m) T_{I, V} (m) := q \in I \sum b_{N, q} Π_{q, V}^{far} (m) γ_{I, N} (d) = d ∣ q \sum \frac{μ ( q / d )}{q / d} S_{d, V} (m), = d \leq N \sum γ_{I, N} (d) S_{d, V} (m), = t \leq N / d d t \in I \sum \frac{μ ( t )}{t} b_{N, d t} . (I111)$ Every $b_{N, d t}$ here keeps all of its original aliases. By $d t / φ (d t) \leq [d / φ (d)] [t / φ (t)]$ , the supplied coefficient bound and first moment imply, uniformly in $I$ , $∣ γ_{I, N} (d) ∣ ≪ \frac{d}{φ ( d )} \frac{1 + lo g ( N / d )}{L} ≪ \frac{d}{φ ( d )} \leq τ (d) . (I112)$ The harmonic weighted first moment follows by partial summation; $τ$ is the ordinary divisor-count function. These are positive bounds on the exact regrouping, not replacement coefficients.

A discrete primitive for the truncated nonprimitive response is $P_{d, V} (m) P_{I, V} (m) P_{I, V} (m + 1) - P_{I, V} (m) = \frac{1}{4 d} 1 \leq r < d ∥ r / d ∥ \geq 1/ V \sum \frac{e ( m r / d )}{sin ^{2} ( π r / d )}, P_{1, V} = 0, = d \leq N \sum γ_{I, N} (d) P_{d, V} (m), = T_{I, V} (m) . (I113)$ Indeed $(e (x) - 1) / (4 sin^{2} (π x)) = - 1/ (1 - e (- x))$ term by term.

For integer $m$ and $d ∤ m$ , put $j_{d} (m) = min_{k \in Z} ∣ m - k d ∣$ . It is a positive integer, at most $d /2$ . On the central numerator interval the supremum plus total variation of the two coefficients in $S_{d, V}$ and $P_{d, V}$ is respectively $O (V / d)$ and $O (V^{2} / d)$ . Abel summation in $r$ , using the geometric-sum bound $O (d / j_{d} (m))$ , gives $∣ S_{d, V} (m) ∣ ∣ S_{d, V} (m) ∣ ≪ V / j_{d} (m), \leq 1/2, ∣ P_{d, V} (m) ∣ ∣ P_{d, V} (m) ∣ ≪ V^{2} / j_{d} (m) ≪ V (d ∤ m), (d ∣ m) . (I114)$ For the second row the cutoff is symmetric under $r \mapsto d - r$ , and $[1 - e (- x)]^{- 1} + [1 - e (x)]^{- 1} = 1$ , which gives the first bound also for a self-paired numerator. The sine bound gives the other: $d \sum_{r \geq m a x (1, ⌈ d / V ⌉)} r^{- 2} ≪ V$ . Thus exact resonances are included without division by a zero distance.

Write $τ_{3} (n) = \sum_{d ∣ n} τ (d) \leq τ (n)^{2}$ . Directly reuse the arbitrary-integer divisor bound in Nicolas’s existing source note, printed p.117, equations (6)–(8): for each fixed $0 < ν < 1$ , its fixed maximizing reference gives $τ (n) ≪_{ν} n^{ν}$ . No new divisor theorem is needed. If $j_{d} (m) = j > 0$ , then $d$ divides $m - j$ or $m + j$ . Hence (I112)–(I114) give $∣ T_{I, V} (m) ∣ ∣ P_{I, V} (m) ∣ N \leq m \leq 2 N + 1 sup ∣ T_{I, V} (m) ∣ N \leq m \leq 2 N + 1 sup ∣ P_{I, V} (m) ∣ ≪ τ_{3} (m) + V 1 \leq j \leq N /2 \sum \frac{τ _{3} ( m - j ) + τ _{3} ( m + j )}{j}, ≪ V τ_{3} (m) + V^{2} 1 \leq j \leq N /2 \sum \frac{τ _{3} ( m - j ) + τ _{3} ( m + j )}{j}, ≪_{η} N^{η} V, ≪_{η} N^{η} V^{2} (η > 0 fixed) . (I115)$ These suprema are over integers. All divisor arguments are positive and at most $3 N$ . To obtain the final two lines, choose $ν$ sufficiently small in the cited bound and absorb the harmonic logarithm into the remaining fixed $N^{η}$ allowance. The constants are independent of $V$ and $I$ .

Apply the exact discrete integration-by-parts identity $m = N \sum 2 N k_{N, m} T_{I, V} (m) = k_{N, 2 N} P_{I, V} (2 N + 1) - k_{N, N} P_{I, V} (N) - m = N + 1 \sum 2 N (k_{N, m} - k_{N, m - 1}) P_{I, V} (m) .$ The original weight variation and $∣ e_{N, H} ∣ ≪ h^{3} ϵ_{N}$ therefore pay both terms of the unchanged functional: $∣ J_{N} (T_{I, V}) ∣ ≪_{η} \frac{N ^{η} ϵ _{N}}{L} (\frac{h ^{2} V ^{2}}{N} + h^{3} V) . (I116)$ In particular the second allowance explicitly bounds $∣ e_{N, H} T_{I, V} (2 N) ∣/ L$ . The exact $2 N + 1$ primitive sample in integration by parts was included in (I115).

For any fixed $0 < κ < 1/4$ , set $V_{N} R_{N, κ}^{♯} (m) = ⌊ N^{3/4 - κ} ⌋, = d_{N} + D_{N}^{\circ} < q \leq N - W_{N}^{\circ} \sum b_{N, q} Π_{q, V_{N}}^{near} (m) .$ Take $η = κ /4$ in (I116). Its two allowances are $N^{η - 2 κ} e^{- Δ_{⋆} Ω} / L$ and $N^{η - κ} e^{(3 λ - c_{⋆}) Ω} / L$ . Since $Ω = o (lo g N)$ , (I106) and (I116) imply $∣ J_{N} (T_{I, V_{N}}) ∣ A_{N, H} # {retained middle frequency entries} ≪_{κ} N^{- κ /2}, = J_{N} (R_{N, κ}^{♯}) + O (1/ lo g L) + O_{κ} (N^{- κ /2}), ≪ N^{2} / V_{N} + N ≪ N^{5/4 + κ} . (I117)$ In the first line use the retained denominator set for $I$ . No condition $3 λ < c_{⋆}$ is added: a positive subpower exponential in the second allowance is paid by the fixed $N^{- κ}$ margin. Every retained numerator satisfies $r^{*} < q / V_{N} ≪ N^{1/4 + κ}$ .

Unlike the interior-only cut (I110), this reduction uses the same truncated response at every original evaluation, including the actual endpoint. The endpoint change is paid in (I116), rather than assumed harmless. The count concerns frequency entries only, not coefficient computation or overall arithmetic cost. The constant $d_{N}$ and all original aliases remain. In particular $q > N /2$ still has its actual $j = 1$ alias, and $r = 1, q - 1$ remain in that sector. There is no newly created second long averaging variable.

This is a repository application of existing finite Fourier, integer-divisor and weight-variation inputs, not a new Möbius cancellation theorem or a claim of mathematical priority. The argument needs fixed $κ > 0$ and supplies no uniform $κ = 0$ or varying- $κ$ endpoint. It has not been Lean compiled. The remaining same-integer estimate $J_{N} (R_{N, κ}^{♯}) = o (1)$ , the complete signed Robin inequality and RH remain unresolved.

A fixed Mertens input sufficient for the original functional

Keep the original family (I59), its actual band, all coefficients and the complete functional (I91). Suppose, as an additional input, that a fixed $1/2 \leq b < 5/8$ satisfies $ζ (s) \neq = 0$ for $ℜ s > b$ . Choose $b < β < 5/8$ . Reuse (I60) for both errors $M (x) ≪_{β} x^{β}$ and $ψ (x) - x ≪_{β} x^{β}$ ; no new zero-free or Mertens theorem is used.

On $Q \leq t \leq N$ , the actual prefix $M (t) - M (Q)$ is $O_{β} (N^{β})$ . The supremum plus variation of the weights $lo g (N / t)$ , $lo g (N / t) / t$ and $lo g (N / t) (1/ t - 1/ N)$ is respectively $O (h)$ , $O (h / N)$ and $O (h^{2} / N)$ . Here $H / N ≍ h = (H - 1/2) / N$ , so the original half-integer convention and the strict lower cutoff do not change. Abel summation, with those same prefixes, gives $∣ A ∣ + ∣ B ∣/ N ∣ δ_{N, H} ∣ + ∣ e_{N, H} ∣ ≪_{β} h N^{β - 1}, ≪_{β} h^{2} N^{β - 1} . (I118)$ For the second line use the exact endpoint identity (I75), including $B / [2 N (2 N + 1)]$ , and $N h ≫ 1$ . This is a conditional estimate for the actual endpoint, not its deletion.

The residual estimate already derived after (I60) is uniform on the whole original integer grid, including $m = 2 N$ : $sup ∣ R_{N} (m) ∣ ≪_{β} N^{β} / L$ . Moreover the original weights satisfy $\sum_{m = N}^{2 N} ∣ k_{N, m} ∣ ≪ ∣ A ∣ + ∣ B ∣/ N$ . Apply these facts and (I118) directly to (I91), retaining its endpoint: $∣ A_{N, H} ∣ = ∣ J_{N} (R_{N}) ∣ ≪_{β} \frac{h N ^{2 β - 1}}{L ^{2}} ≪_{β} \frac{N ^{2 β - 5/4} e ^{λ Ω}}{L ^{2}} = o (1) . (I119)$ The last step uses fixed $β < 5/8$ and $Ω = o (lo g N)$ . It applies to every sufficiently large original $N$ under the added input, with the same actual $A, B$ and endpoint.

A global bound $M (x) ≪ x^{ϑ}$ for one fixed $1/2 \leq ϑ < 5/8$ is also sufficient, with exponent slack. Indeed the classical partial-summation formula for $1/ ζ$ is holomorphic on $ℜ s > ϑ$ under that bound and agrees with $1/ ζ$ on $ℜ s > 1$ . Its continuation excludes zeros on the former half-plane. Choose $ϑ < β < 5/8$ and reuse (I60). The residual bound at the identical exponent $ϑ$ is not being deduced from bandwise Abel summation alone: (I33) also contains the prime error. The slack supplies that error through the existing analytic interface.

For example, the existing local fourth-moment source application identifies a sufficient input $M_{4, c} (N) ≪ N^{ν}$ for fixed $c > 0$ and $0 \leq ν < 1/32$ . That deterministic estimate is not established here. The recorded boundaries near $7/8$ in the OpenAI and Liu sources do not meet the $5/8$ hypothesis. This is a conditional parameter application of existing theorems, not a new RH criterion, an unconditional bound for (I117)’s retained near sector, a mathematical-priority claim or Lean certification. The complete signed Robin objective and RH remain unresolved.

A sharper fixed fourth-moment input for the same functional

The classical polynomial-evaluation application gives a stronger interface than the first-derivative moment estimate used above. Keep the original family (I59), the actual functional (I91), and one fixed $c > 0$ . Assume, as an additional deterministic input, that $M_{4, c} (N) ≪ N^{ν}$ for every sufficiently large integer $N$ , with one fixed $0 \leq ν < 1/8$ . The source’s same local arc and Möbius coefficients are retained.

The cited application supplies $∣ M (N) ∣ ≪_{c} lo g (2 N) (N M_{4, c} (N) + 1) .$ Consequently, with $θ = 1/2 + ν$ , for every fixed $δ > 0$ , $M (x) ≪_{c, ν, δ} x^{θ + δ} . (I120)$ The implied constants also retain the assumed moment-bound constant. To use the required prime error, take any $ℜ s > θ$ , choose $0 < δ < ℜ s - θ$ , and apply the already used partial-summation continuation of $1/ ζ$ with (I120). This excludes zeros throughout $ℜ s > θ$ . Choose a fixed $θ < β < 5/8$ and reuse (I60) for both $M (x)$ and $ψ (x) - x$ at exponent $β$ ; there is no same-exponent prime-error assertion obtained from bandwise Abel summation alone.

Apply (I118)–(I119) with this $β$ . For every fixed $η > 0$ , choose $β - θ$ sufficiently small and absorb the original $e^{λ Ω}$ and logarithmic factors into the remaining $N^{η}$ allowance. The result is $∣ A_{N, H} ∣ = ∣ J_{N} (R_{N}) ∣ ≪_{c, ν, η} N^{2 ν - 1/4 + η} . (I121)$ All original fixed parameters may also enter the constant. Taking $0 < η < 1/4 - 2 ν$ gives $o (1)$ on every sufficiently large original integer $N$ . This reuses the complete functional, including $e_{N, H} = B / (2 N + 1) - A /2$ and $m = 2 N$ , with the same strict lower band cutoff and half-integer convention. It supplies no uniform $ν = 1/8$ endpoint or estimate for a moment hypothesis given only on a subsequence.

Using the exact correlation identity in the same source application, another sufficient input is $C_{4, c} (N) ≪ N^{5/2 - ξ}$ for a fixed $0 < ξ \leq 1/2$ . It gives $ν = 1/8 - ξ /4$ and hence $∣ J_{N} (R_{N}) ∣ ≪_{η} N^{- ξ /2 + η}$ , with decay when $0 < η < ξ /2$ . This weakens the previously recorded $N^{17/8 - ξ}$ sufficient correlation input; it does not prove either bound or replace the signed fourfold sum by a diagonal.

The new parameter range is a repository application of classical polynomial evaluation, Taylor approximation and the already owned analytic interfaces, without a mathematical-priority claim or Lean certification. The deterministic moment or correlation bound remains unpaid. The complete signed Robin estimate and RH remain unresolved.

A single actual endpoint already obstructs a uniform response supplier

Keep (I59), the complete mean $d_{N}$ from (I17), and the cutoffs of (I105) and (I117). The following is a repository-derived analytic application, without a mathematical-priority claim or Lean certification. Its finite-height input is the already cited Platt–Trudgian theorem; no zero computation is repeated.

For $q > N /2$ , the original alias sum has exactly $j = 1$ , so $b_{N, q} = μ (q) lo g (N / q) / L$ . Define only the first pair of its actual harmonics, together with the unchanged original mean: $P_{q}^{(1)} (m) R_{N}^{(1)} (m) = - \frac{1}{q} [\frac{e ( m / q )}{1 - e ( - 1/ q )} + \frac{e ( - m / q )}{1 - e ( 1/ q )}], = d_{N} + N /2 < q \leq N - W_{N}^{\circ} \sum \frac{μ ( q ) lo g ( N / q )}{L} P_{q}^{(1)} (m) . (I122)$ Eventually $D_{N}^{\circ} < N /2 < N - W_{N}^{\circ}$ and $V_{N} < N /2$ . Both numerators $1, q - 1$ therefore remain in (I117)’s near sector. This selects actual terms of that response; it does not replace the Möbius coefficients or assert that separately bounding this selection is necessary to bound their signed total.

For any fixed $1/2 \leq α < 1$ , the additional hypothesis $L ∣ R_{N}^{(1)} (2 N) ∣ ≪_{η} N^{α + η} for every fixed η > 0 and every sufficiently large original integer N (I123)$ implies $ζ (s) \neq = 0$ on $ℜ s > α$ . In particular its $α = 1/2$ case implies RH. The estimate (I123) remains unproved.

Transferring the original endpoint without deleting its top cutoff

Pairing the two Fourier terms gives exactly $P_{q}^{(1)} (m) = - \frac{1}{q} [cos (2 πm / q) + cot (π / q) sin (2 πm / q)] .$ Since $q^{- 1} cot (π / q) = π^{- 1} + O (q^{- 2})$ , at $m = 2 N$ this is $- π^{- 1} sin (4 π N / q) + O (N^{- 1})$ for $q > N /2$ . The accumulated error after multiplying $R_{N}^{(1)}$ by $L$ is $O (1)$ . For real $y > 1$ , put $F (y) = lo g y + \frac{1}{2} n \leq y \sum μ (n) lo g (y / n) (1 - 1/ n) - \frac{1}{π} y /2 < n \leq y \sum μ (n) lo g (y / n) sin (4 π y / n) . (I124)$ Its first two terms at $y = N$ are exactly $L d_{N}$ . If $W = W_{N}^{\circ}$ and $w = N - q < W$ , then $lo g (N / q) \leq 2 w / N$ and $∣ sin (4 π N / q) ∣ = ∣ sin (4 π w / q) ∣ \leq 8 π w / N$ eventually. Both vanish at the actual top endpoint. Hence $L R_{N}^{(1)} (2 N) = F (N) + O (1 + W^{3} / N^{2}), W^{3} / N^{2} = N^{1/4 + o (1)} . (I125)$ This pays the original top cutoff using absolute values and requires no extra condition $3 λ < c_{⋆}$ . For $N \leq y < N + 1$ , the common terms in the first sum change by $O (N^{- 1})$ each. The derivative of each common summand in the second sum is $\frac{1}{y} sin (4 π y / n) + \frac{4 π}{n} lo g (y / n) cos (4 π y / n) = O (N^{- 1}) (n ≍ N) .$ Only a bounded number of bounded terms enter or leave at its moving limits. Thus $F (y) - F (N) = O (1)$ . Together with (I123)–(I125), this gives $F (y) ≪_{η} y^{α + η}$ for every fixed $η > 0$ .

A Mellin kernel with no high-height cancellation

Consequently $F (s) = \int_{1}^{\infty} F (y) y^{- s - 1} d y$ is holomorphic on $ℜ s > α$ . Initially for $ℜ s > 1$ , absolute convergence and the classical Möbius Dirichlet series give $F (s) K (s) = \frac{1}{s ^{2}} - \frac{1}{2 s ^{2} ζ ( s + 1 )} + \frac{K ( s )}{ζ ( s )}, = \frac{1}{2 s ^{2}} - \frac{1}{π} \int_{1}^{2} lo g v sin (4 π v) v^{- s - 1} d v . (I126)$ The term $1/ (2 s^{2})$ in $K$ comes from the original mean and must remain. Since $1/ ζ (s + 1)$ is holomorphic on this larger half-plane, the identity theorem continues the holomorphic identity $(s - 1) ζ (s) [F (s) - \frac{1}{s ^{2}} + \frac{1}{2 s ^{2} ζ ( s + 1 )}] = (s - 1) K (s) (I127)$ to $ℜ s > α$ . A zero $ρ$ there, of any multiplicity $m_{ρ}$ , would force $ord_{ρ} K \geq m_{ρ}$ . Neither zero simplicity nor a reciprocal-derivative moment is used.

Here the necessary cancellation can be ruled out at every high height: $∣ K (s) ∣ \geq ∣ s ∣^{- 2} (1/2 \leq ℜ s \leq 1, ∣ℑ s ∣ \geq 2^{16}) . (I128)$ To prove this quantitative bound, set $a = lo g 2$ and $f (u) = u sin (4 π e^{u})$ . Then $f (0) = f (a) = f^{'} (0) = 0$ and $f^{'} (a) = 8 πa$ . Three integrations by parts in (I126) give $K (s) B (s) = \frac{1/2 + 8 a 2 ^{- s}}{s ^{2}} - \frac{B ( s )}{π s ^{3}}, = f^{''} (0) - f^{''} (a) 2^{- s} + \int_{0}^{a} f^{'''} (u) e^{- s u} d u .$ For $z = 4 π e^{u}$ the derivatives are $f^{''} (u) f^{'''} (u) = (u + 2) z cos z - u z^{2} sin z, = (u + 3) z cos z - (3 u + 3) z^{2} sin z - u z^{3} cos z .$ Using $0 \leq u \leq a < 1$ , $z < 32$ and $3 < π < 4$ yields $∣ f^{''} (0) ∣ < 32$ , $∣ f^{''} (a) ∣ < 96$ and $\int_{0}^{a} ∣ f^{'''} (u) ∣ d u < 4 \cdot 32 + 6 \cdot 3 2^{2} + 3 2^{3}$ . Thus $∣ B (s) ∣ < 39168 < 2^{16}$ for $ℜ s \geq 0$ . On $1/2 \leq ℜ s \leq 1$ , $∣1/2 + 8 a 2^{- s} ∣ \geq 4 lo g 2 - 1/2 > 3/2$ . For $∣ s ∣ \geq 2^{16}$ the remainder in $s^{2} K (s)$ is less than $1/ π < 1/3$ , proving (I128).

For the remaining finite heights, directly reuse Platt–Trudgian, Theorem 1, printed p.2, as already recorded in the finite-height source application. Its verified height $3000175332800$ exceeds $2^{16}$ . Conjugation covers negative heights, and the classical absence of real zeros in $(0, 1)$ covers height zero. Thus there are no low-height zeros with real part greater than $α \geq 1/2$ ; this does not assert low-height nonvanishing of $K$ . Every higher possible zero contradicts (I127)–(I128). The standard zero-free region $ℜ s \geq 1$ completes the stated half-plane conclusion. At $α = 1/2$ , functional-equation symmetry gives RH.

The method boundary concerns the response supplier, not signed cancellation

The original coefficient budget in (I91) gives only $∣ J_{N} (f) ∣ ≪ \frac{h ^{2} ϵ _{N}}{L} ∥ f ∥_{\infty}, h^{2} ϵ_{N} ≍ N^{- 1/2} e^{- Δ_{⋆} Ω},$ where the supremum is over the original integer grid $N \leq m \leq 2 N$ . A sufficient response bound through this budget alone would be $∥ R_{N}^{(1)} ∥_{\infty} = o (L N e^{Δ_{⋆} Ω}) . (I129)$ Since $L^{2} e^{Δ_{⋆} Ω} = N^{o (1)}$ , (I129) already implies (I123) at $α = 1/2$ , hence RH. The obstruction is specific to one actual endpoint and a specified actual harmonic pair plus mean; unlike the compact kernel discussion around (I68), its high-height nonvanishing is quantitative.

This does not show that $J_{N} (R_{N}^{(1)}) = o (1)$ or $J_{N} (R_{N, κ}^{♯}) = o (1)$ implies RH. Their actual signed weights can use joint relations or inter-sector cancellation without supplying (I129). In particular the total endpoint coefficient is $(k_{N, 2 N} + e_{N, H}) / L = O (h^{3} ϵ_{N} / L)$ , which is smaller than the budget used above; paying that contribution does not require a square-root bound on the response itself. Nor does this invalidate (I118)–(I119), which also improve the band coefficients under their additional hypothesis. No new bound for the signed functional, complete Robin inequality or RH follows from this method obstruction.

An absolute contour estimate cannot relax the fixed half-plane threshold

Keep the original family (I59), its exact coupled kernel (I65), and the additional fixed zero-free-half-plane premise of (I60): $1/2 \leq b < 1$ and no zero of $ζ$ in $ℜ s > b$ . Choose any fixed $b < σ, τ < 1$ . There is a fixed compact real interval $I$ of positive length such that, for every sufficiently large original integer $N$ , the finite absolute majorant satisfies $B_{N} (σ, τ; I) := \frac{1}{L ^{2}} \int_{1/ (4 h)}^{1/ (2 h)} \int_{I} \frac{N ^{σ + τ - 1 + i (t + y)} K _{N} ( σ + i t , τ + i y )}{ζ ( σ + i t ) ζ ( τ + i y )} d y d t ≫_{b, σ, τ} \frac{h N ^{σ + τ - 1}}{L ^{2}} . (I130)$ The interval and lower-bound constant depend only on fixed parameters. This measures the cost of taking absolute values inside the two contour integrals. It gives no lower bound on the signed functional. The application below reuses (I60), (I64), (I70) and the already owned expansion of $H_{q}$ at zero; no new classical mean-value theorem, mathematical-priority claim or Lean certification is asserted.

A nonzero part of the exact kernel survives at height comparable to 1/h

Use $Φ$ from (I67). The integral defining $H_{q}$ is holomorphic on $ℜ v > 0$ . The expansion after (I88), uniform for $1 \leq q \leq 2$ by (I70), gives $Φ (v) = \frac{lo g 2 - 3/4}{2 v ^{2}} + O (1) (v ⟶ 0) . (I131)$ Here $lo g 2 < 3/4$ , for example by the strict trapezoidal upper bound for the convex function $1/ x$ on $[1, 2]$ . Thus $Φ$ is not identically zero. The identity theorem implies that it cannot vanish on the entire line $τ + i R$ . Choose $I$ about a nonzero point so that $∣Φ (τ + i y) ∣$ is bounded below on $I$ . The actual $∣1/ ζ (τ + i y) ∣$ also has a positive minimum there: the segment is compact, zero-free, and avoids the pole at one.

Put $G (z) = \int_{0}^{1} r e^{- i zr} d r$ . For $u = σ + i t$ with $1/4 \leq h t \leq 1/2$ , substitute $x = 1 - h r$ into the original $W_{a}$ . Uniformly on this growing-height rectangle, $- lo g (1 - h r) = h r + O (h^{2})$ , $(1 - h r)^{σ - 1} = 1 + O_{σ} (h)$ and $e^{i t l o g (1 - h r)} = e^{- ih t r} (1 + O (h))$ . Consequently $W_{a} (u) W_{a} (u - 1) - W_{a} (u) = h^{2} G (h t) + O_{σ} (h^{3}), = \int_{a}^{1} x^{u - 2} (1 - x) lo g (1/ x) d x = O_{σ} (h^{3}) .$ The second estimate holds for every real $t$ . Reuse the fixed-compact relations from (I64)–(I67), $p_{N} (v) = O (1)$ and $p_{N} (v) - N q_{N} (v) = Φ (v) + O (1/ N)$ , to obtain $K_{N} (u, v) = W_{a} (u) [p_{N} (v) - N q_{N} (v)] + [W_{a} (u - 1) - W_{a} (u)] p_{N} (v) = h^{2} G (h t) Φ (v) + O (h^{3} + h^{2} / N) . (I132)$ This calculation is uniform for $v = τ + i y$ , $y \in I$ . It does not extend the compact- $u$ error term of (I67) to this height. Since $ℜ G (z) \geq cos (1/2) /2 > 0$ on $1/4 \leq z \leq 1/2$ , (I132) gives $∣ K_{N} (u, v) ∣ ≫ h^{2}$ throughout the rectangle for all sufficiently large $N$ . Both original endpoint contributions remain in $p_{N}, q_{N}$ .

Actual reciprocal-zeta mass supplies the full width of the rectangle

The reciprocal growth bound in (I60), with exponent allowance $1/2$ , is uniform for $σ \leq ℜ s \leq 2$ at large heights. The reciprocal is holomorphic on $ℜ s > b$ , with a removable zero at the pole of $ζ$ . Cauchy’s theorem between the lines $σ$ and $2$ gives $\int_{T}^{2 T} \frac{d t}{ζ ( σ + i t )} = \int_{T}^{2 T} \frac{d t}{ζ ( 2 + i t )} + O_{b, σ} (T^{1/2}) = T + O_{b, σ} (T^{1/2}) . (I133)$ For the final equality directly apply the absolutely convergent Möbius Dirichlet series underlying (I62). Its $n = 1$ term contributes $T$ ; the remaining integral is at most $2 \sum_{n \geq 2} n^{- 2} / lo g n < \infty$ in modulus. Thus $\int_{T}^{2 T} ∣1/ ζ (σ + i t) ∣ d t \geq T /2$ eventually. Take $T = 1/ (4 h)$ . Combine this actual mass, the fixed- $v$ reciprocal minimum and (I132). Since the modulus of the outer $N$ -power is $N^{σ + τ - 1}$ , this proves (I130) for each sufficiently large original integer, without an average over $N$ or zero-residue assumptions.

The paid two-scale smoothing preserves the obstruction

The same lower bound holds with the complete smoothed kernel $K_{N}$ of (I89) in place of $K_{N}$ : $B_{N} (σ, τ; I) ≫_{b, σ, τ} \frac{h N ^{σ + τ - 1}}{L ^{2}} . (I134)$ Use exactly the already paid scales $η = H^{- 1}$ and $ν = N^{- 1/2}$ . Their bump multipliers satisfy $∣ F_{ℓ} (s) - 1∣ \leq e^{ℓ ∣ s ∣} - 1$ by positivity and unit mass of $ϕ$ . On this rectangle $η ∣ u ∣ = O (η / h) \to 0$ , $η ∣ u - 1∣ \to 0$ , and $ν ∣ v ∣ \to 0$ , since $η / h = 1/ (H h) ≍ N / H^{2} \to 0$ . Hence all three multipliers in (I89) tend uniformly to one here. The fixed-compact bounds also give $N q_{N} (v) = O (1)$ , so its exact formula and (I132) imply $K_{N} (u, v) - K_{N} (u, v) = o (h^{2})$ uniformly. The same positive lower bound therefore survives. This uses the original smoothing scales and retains its actual scalar and endpoint accounting; it introduces no new contour-shift claim.

The limitation concerns taking absolute values on fixed lines

At the original bandwidth, the lower-bound scale is $\frac{h N ^{σ + τ - 1}}{L ^{2}} ≍ \frac{N ^{σ + τ - 5/4} e ^{λ Ω}}{L ^{2}} . (I135)$ For $σ + τ \geq 5/4$ this tends to infinity, including equality, because $λ > 0$ and $Ω/ lo g L \to \infty$ . If $b \geq 5/8$ , every pair of fixed lines $b < σ, τ < 1$ has this problem; unequal fixed lines do not remove it. On equal lines the displayed power is the same as the allowance in (I119).

Thus improving only an absolute bound for the growing-height coupled kernel cannot relax that fixed-line threshold, even after the already paid smoothing. An improved bound needs information beyond this absolute majorant, such as signed oscillation in $N^{i (t + y)}$ or cancellation with other parts of the exact functional. No necessity of $b < 5/8$ for the signed functional is proved, and no conclusion about estimates on $N$ -dependent lines is asserted. The original same-source joint upper bound, complete signed Robin estimate and RH remain unresolved.

One same-source product remains under a fixed half-plane below 3/4

Retain the original family (I59), the exact functional (I91), and all its integer-grid observations. Suppose, as an additional premise, that $1/2 \leq b < 3/4$ and $ζ$ has no zeros in $ℜ s > b$ . Fix $b < β < 3/4$ . The Abel argument in (I118) uses $β < 1$ for its coefficient estimates; the restriction $β < 5/8$ enters only at the final vanishing of (I119). Apply that same argument and the residual estimate after (I60), without a new Mertens or prime-error theorem: $∣ B ∣ ≪_{β} h N^{β}, ∣ δ_{N, H} ∣ ≪_{β} h^{2} N^{β - 1}, N \leq m \leq 2 N max ∣ L R_{N} (m) ∣ ≪_{β} N^{β} .$

Write $S_{N} (m) = L R_{N} (m)$ and, for $N \leq m < 2 N$ , put $u_{N, m} = \frac{m + \frac{1}{2} - \frac{3}{2} N}{m ( m + 1 )}, p_{N, m} = \frac{m + \frac{1}{2}}{m ( m + 1 )}, T_{N} (f) = m = N \sum 2 N - 1 u_{N, m} f (m) .$ The identity $A = B / N + δ_{N, H}$ in the original coefficients gives exactly $L^{2} J_{N} (R_{N}) = \frac{B}{N} T_{N} (S_{N}) + δ_{N, H} m = N \sum 2 N - 1 p_{N, m} S_{N} (m) + (k_{N, 2 N} + e_{N, H}) S_{N} (2 N) . (I136)$ This is the full response, including its actual mean and every Fourier alias. The endpoint is retained in its combined original coefficient. Since $\sum_{m = N}^{2 N - 1} p_{N, m} ≪ 1$ , and the exact coefficients of (I91) and (I75) give $∣ k_{N, 2 N} + e_{N, H} ∣ ≪ ∣ δ_{N, H} ∣ + ∣ B ∣/ N^{2}$ , the last two terms divided by $L^{2}$ have total bound $J_{N} (R_{N}) - \frac{B}{N L ^{2}} T_{N} (S_{N}) ≪_{β} \frac{h ^{2} N ^{2 β - 1} + h N ^{2 β - 2}}{L ^{2}} ≪_{β} \frac{N ^{2 β - 3/2} e ^{2 λ Ω} + N ^{2 β - 9/4} e ^{λ Ω}}{L ^{2}} = o (1) . (I137)$ Both fixed negative powers absorb their exponential-log factors when $β < 3/4$ .

Consequently, under this additional fixed-half-plane premise, the original all-integer target $J_{N} (R_{N}) = o (1)$ is equivalent to $B T_{N} (S_{N}) = o (N L^{2})$ for those same integers and the specified bandwidth. The coefficients and response in this product cannot be chosen or optimized independently. This pays the secondary and endpoint contributions over a wider range than (I119), but proves no new bound for the remaining signed product and does not improve its $5/8$ sufficient threshold. It is an application of the owned analytic estimates, not a new cancellation theorem or a Lean-certified statement. The complete signed Robin estimate and RH remain unresolved.

A zero-density contour for the actual band coefficient

Keep the original integers, bandwidth and endpoint conventions of (I59), (I62), (I83) and (I136). Suppose additionally that $1/2 \leq b < β < 3/4$ and $ζ$ has no zeros in $ℜ s > b$ . This section reuses the local disk argument already transported to ordinary $ζ$ in (I60), Inoue’s owned good heights, and the published Ingham estimate recorded in the finite Robin source note. The additional result is their application to the actual coefficient $B$ and the complete same-integer functional, not a new disk theorem, zero-density estimate, zero-free half-plane or mathematical-priority claim. No Lean certification is asserted.

Write $Z (σ, U) = # {ρ : ζ (ρ) = 0, ℜ ρ \geq σ, 0 < ℑ ρ \leq U}, d (σ) = \frac{3 ( 1 - σ )}{2 - σ},$ counting multiplicities. Chourasiya–Simonič, arXiv:2507.15184v2, Corollary 1 and Table 1, cover $1/2 \leq σ \leq 1$ by closed subintervals and imply, uniformly on the fixed substrip used here, $Z (σ, U) ≪ U^{d (σ)} lo g^{3} (2 U)$ . Their stated lower height is $3 \cdot 1 0^{12}$ ; the bounded lower heights are included by increasing the implicit constant. This is the published input, reused without reproof.

For every fixed $ξ > 0$ , the band coefficient satisfies $∣ B ∣ ≪_{b, β, ξ} N^{ξ} (h N^{1/2} + h^{2 - d (β)} N^{β}) + 1. (I138)$ The proof keeps the complete high-frequency decay of the actual weight and charges the rightward contour excursions to zeros.

The local reciprocal input and the original smoothing

Use exactly $η = H^{- 1}$ , $a = 1 - h$ and the multiplier of (I83), (I86). Those formulas already give $B = B + O (1)$ and $\widetilde B=(2\pi i)^{-1}\int_{(2)} N^sW_a(s)F_\eta(s)/\zeta(s),ds$. On a fixed real strip, $∣ W_{a} (σ + i t) ∣ ≪ h min {h, (1 + ∣ t ∣)^{- 1}} .$ Indeed, with $ℓ = lo g (1/ a) ≍ h$ , its exact expression is $\int_{0}^{ℓ} x e^{- σ x} e^{- i t x} d x$ . The direct integral bounds it by $O (h^{2})$ ; one integration by parts bounds it by $O (h /∣ t ∣)$ because both the terminal value and total variation of $x e^{- σ x}$ are $O (h)$ .

Take the already supplied good height $T \in [N^{4}, N^{4} + N^{4/3}]$ and write $Y = lo g T$ . Choose a fixed sufficiently small $d > 0$ with $b + 3 d < β$ , $1/2 + 3 d < β$ and $d < 1 0^{- 3}$ . The local form of the disk argument used in (I60) supplies $∣1/ ζ (σ + i t) ∣ ≪_{d, ϵ} (3 + ∣ t ∣)^{ϵ}$ whenever $σ \geq 1/2 + 2 d$ , $σ \leq β$ and the rectangle $ℜ z > σ - d$ , $∣ℑ z - t ∣ \leq Y$ is zero free, for $Y > 2$ . Here is the parameter correspondence needed for that reuse. In its cited lem:logarithmic-control disk proof, put $a_{0} = σ - d$ and $e_{0} = d /8$ . The outer radius $2 - a_{0} - 2 e_{0} = 2 - σ + 3 d /4$ about $2 + i t$ lies inside this rectangle; the control radius $2 - a_{0} - 6 e_{0} = 2 - σ + d /4$ contains the point $σ + i t$ . Use the ordinary regularized function $(s - 1) ζ (s) / (s + 1)$ , its classical polynomial strip growth and bounded Euler logarithm, exactly as in (I60). The printed Hecke/Dedekind statement is not being quoted as literally an ordinary- $ζ$ theorem. Only its already documented ordinary disk argument is used, and that argument needs nonvanishing on the disk, not a global zero-free half-plane with boundary $σ - d$ .

Count every excursion, including its horizontal joins

Put $g_{d} (x) = 1/2 + d ⌈(x - 1/2) / d ⌉$ . At $4 Y \leq ∣ t ∣ \leq T$ , define $c_{T} (t) = max {\frac{1}{2} + 2 d, g_{d} (ℜ ρ) + 2 d : ℜ ρ > \frac{1}{2}, ∣ℑ ρ - t ∣ \leq 2 Y} . (I139)$ Ignore an empty zero set in the maximum. The fixed-half-plane hypothesis implies $1/2 + 2 d \leq c_{T} (t) \leq b + 3 d < β$ . Use the vertical pieces at $ℜ s = c_{T} (t)$ , joining their jumps horizontally. For $∣ t ∣ \leq 4 Y$ use $ℜ s = β$ , with horizontal connectors at $t = \pm 4 Y$ ; call the resulting contour $Γ$ . Every nontrivial zero inside the truncated rectangle lies strictly to its left, so the shift from $ℜ s = 2$ crosses no poles of $1/ ζ$ . Its zero at one causes no residue.

The required local rectangle is zero free at every high-height vertical piece. At a jump, zeros within distance $Y$ of that height are present on both sides of the jump; the entering or leaving zeros are at distance $2 Y$ . Thus the rectangle is zero free even at the smaller real endpoint, and hence along the entire horizontal join. The same reasoning applies to each connector from $c_{T} (\pm 4 Y)$ to $β$ . Their reciprocal bound is local; these connectors need not lie in $ℜ s > b$ . The low-height $β$ line instead uses the global reciprocal bound already established in (I60). Consequently $∣1/ ζ (s) ∣ \leq N^{o_{d} (1)}$ on $Γ$ . The low-height line and its two connectors contribute $≪ h^{2} N^{β} N^{o_{d} (1)}$ .

Let $Γ^{hi}$ contain only the high vertical pieces and their jump joins, excluding the low-height line and both transition connectors just paid. In a dyadic block $U \leq ∣ t ∣ \leq 2 U$ , $U \geq 4 Y$ , separate the baseline pieces, whose real parts are at most $1/2 + 4 d$ . Their length is $O_{d} (U lo g T)$ , including horizontal joins: the vertical length is $O (U)$ and there are $O (U lo g U)$ possible zero-window endpoints. For any remaining real strip $σ \leq ℜ s < σ + d$ , its total contour length satisfies $length (Γ_{σ, U}^{hi}) ≪_{d} Y Z (σ - 3 d, 4 U) . (I140)$ For vertical pieces, a zero forcing the maximum into that strip has $ℜ ρ \geq σ - 3 d$ , and each such zero influences at most $4 Y$ of vertical length. Each horizontal jump occurs at an endpoint of a zero’s influence interval. A zero responsible for raising or lowering the maximum across the strip also has $ℜ ρ \geq σ - 3 d$ ; each zero supplies at most two such endpoints and at most length $d$ in the strip per endpoint. Positive and negative zeros are handled together by conjugation. Since $U \geq 4 Y$ , all responsible heights have absolute value below $4 U$ . These charges prove (I140) for the complete contour pieces.

Use $∣ F_{η} ∣ ≪ 1$ on this strip. The density bound, (I140), and $d (σ - 3 d) = d (σ) + O (d)$ give the following block bound, for a finite real mesh $G_{d}$ : $\int_{Γ^{hi} \cap {U \leq ∣ℑ s ∣ \leq 2 U}} \frac{N ^{s} W _{a} ( s ) F _{η} ( s )}{ζ ( s )} ∣ d s ∣ ≪_{d} N^{O (d) + o_{d} (1)} h min {h, U^{- 1}} \cdot (N^{1/2} U + σ \in G_{d} \sum N^{σ} U^{d (σ)}) . (I141)$ Here the nonbaseline mesh has $σ > 1/2 + 3 d$ and $σ \leq β$ ; the real-width cost $N^{d}$ and logarithms are included in the displayed loss. With $U \leq T ≍ N^{4}$ , the density-exponent displacement also costs only $N^{O (d)}$ .

For the baseline, dyadic summation gives $\sum_{U \leq T} h min (h, U^{- 1}) U ≪ h lo g T$ . For each fixed $σ > 1/2$ , $0 < d (σ) < 1$ , and splitting at $U = h^{- 1}$ gives $\sum_{U\le T}h\min(h,U^{-1})U^{\mathfrak d(\sigma)} \ll_\sigma h^{2-\mathfrak d(\sigma)}$. The original bandwidth has $lo g (1/ h) / lo g N = 1/4 + o (1)$ . Moreover, $\frac{d}{d σ} lo g (N^{σ} h^{- d (σ)}) = lo g N - \frac{3 lo g ( 1/ h )}{( 2 - σ ) ^{2}} > 0 (1/2 \leq σ \leq β < 3/4)$ for all sufficiently large $N$ . Thus every nonbaseline term is bounded by the one at $β$ . The low-height allowance $h^{2} N^{β} N^{o_{d} (1)}$ is absorbed too.

The top and bottom edges use the owned good-height reciprocal bound and the original multiplier’s decay. For fixed $K = 20$ they cost $≪_{ϵ} N^{2} (h / T) T^{ϵ} (T / H)^{- 20}$ . The tails on the original line cost $≪ N^{2} h (T / H)^{- 20}$ , since $1/ ζ (2 + i t)$ is bounded. Both are $O (N^{- 10})$ at the specified $T$ and $H$ . Choose $d$ small enough in terms of $b, β, ξ$ , then take $N$ large enough to absorb all remaining logarithms and $N^{o_{d} (1)}$ into $N^{ξ}$ . Finally (I83) restores the original integer-band coefficient at cost $O (1)$ . This proves (I138) for every sufficiently large original integer, without an average over cutoffs.

Apply the coefficient gain to the complete original functional

The existing (I136)–(I137) decomposition and $∣ T_{N} (S_{N}) ∣ ≪_{β} N^{β}$ give $∣ J_{N} (R_{N}) ∣ ≪_{b, β, ξ} \frac{N ^{ξ}}{L ^{2}} (h N^{β - 1/2} + h^{2 - d (β)} N^{2 β - 1}) + \frac{h ^{2} N ^{2 β - 1} + h N ^{2 β - 2} + N ^{β - 1}}{L ^{2}} . (I142)$ The final $N^{β - 1} / L^{2}$ is the cost of restoring the unsmoothed $B$ , acting on the actual response. The secondary coefficient and the combined original endpoint retain their paid (I137) allowance. The response, its mean and every alias remain unchanged.

Compared with (I119), the controlling term has gained $h^{1 - d (β)} = N^{- γ (β) + o (1)}$ , where $γ (β) = (2 β - 1) / (4 (2 - β))$ . The strict condition $γ (β) > 2 β - 5/4$ is equivalent, in the present range, to $β < β_{c} := \frac{19 - 73}{16} = 0.653499765 \dots . (I143)$ Thus the additional zero-free-half-plane premise with $b < β_{c}$ is sufficient for the complete same-integer target $J_{N} (R_{N}) = o (1)$ , improving this application’s previous $5/8$ sufficient boundary. A convenient fixed value is $β = 13/20$ : $d (β) = 7/9$ , and the controlling scale is $\frac{h ^{11/9} N ^{3/10}}{L ^{2}} ≍ \frac{N ^{- 1/180} e ^{11 λ Ω/9}}{L ^{2}} . (I144)$ Since $Ω = o (lo g N)$ , under $b < 13/20$ this gives $∣ J_{N} (R_{N}) ∣ ≪_{b, ξ} N^{- 1/180 + ξ}$ for every fixed $ξ > 0$ . The other powers at this checkpoint are $- 1/10$ , $- 1/5$ , $- 19/20$ and $- 7/20$ , all strictly smaller.

The gain comes from the length of the rightward excursions on a variable contour, not from a smaller fixed-line double absolute majorant. It therefore does not contradict (I130)–(I135). It does not provide the zero-free-half-plane premise required here, a new unconditional Robin bound, or a proof of RH. For larger $b$ the complete original signed estimate remains unresolved.

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