bibkey: nicolas2025comparison authors: Jean-Louis Nicolas year: 2025 title: Comparison of large values of n/φ(n) and σ(n)/n doi: null url: https://hal.science/hal-05389053v1 claim: The manuscript gives unconditional asymptotic and effective comparisons between the two separate extremal envelopes Φ(X) and Σ(X); it does not identify common maximizers, arbitrary near-extremal prime profiles, or source-weighted concentration in a prescribed residue class. strata_touched: [] license: citation-only triage: anchor

Comparison of large values of n/φ(n) and σ(n)/n

The primary manuscript is Jean-Louis Nicolas’s single-author HAL hal-05389053v1, with versioned full text. Its title page is dated 15 July 2025; the HAL cover and submission record say 29 November 2025. The PDF has 38 numbered manuscript pages plus the HAL cover. These dates are distinct from later changes to the record’s metadata. No journal publication or DOI is asserted here.

This source is not Axler–Nicolas, Large values of n/φ(n) and σ(n)/n, Acta Arithmetica 209 (2023), 357–383, DOI 10.4064/aa220705-12-10. The latter is reference [2] in the 2025 manuscript and supplies an earlier effective comparison.

This card records the definitions, main theorem statements and relevant proof interfaces checked in the primary manuscript. It is not a full verification of all analytic proofs, external estimates or Maple computations, and supplies no Lean verification.

The two envelopes and the actual quantifiers

For real $X \geq 1$ , equation (1.7), printed p.3, defines

$Φ (X) = 1 \leq n \leq X max \frac{n}{φ ( n )}, Σ (X) = 1 \leq n \leq X max \frac{σ ( n )}{n} .$

The maxima range over positive integers and are taken separately. The strict record holders for $n / φ (n)$ are the primorials; those for $σ (n) / n$ are the superabundant integers. Remark 1.1 reports that the only integers that are both primorial and superabundant are 2 and 6. This observation concerns strict records: because $n / φ (n)$ can tie, it does not imply that the two sets of maximizers are always disjoint. No common maximizing integer is supplied for a general $X$ .

Theorem 1.2, printed p.4, states that for every fixed integer $J \geq 1$ , as real $X \to \infty$ ,

$\frac{Φ ( X )}{Σ ( X )} = 1 + j = 1 \sum J \frac{c _{j} 2}{lo g X ( lo g lo g X ) ^{j}} + O_{J} (\frac{1}{lo g X ( lo g lo g X ) ^{J + 1}}),$

where, with $a = lo g 2$ ,

$c_{1} = 2, c_{2} = - (2 + a), c_{3} = 8 + 4 a + \frac{3 a ^{2}}{4} .$

The statement is unconditional. The error constant may depend on $J$ ; there is no uniform assertion for $J$ growing with $X$ , and this is not a moment asymptotic with exponent $s$ .

Equation (1.10) defines $τ (X)$ by the exact equality, for $X > e$ ,

$\frac{Φ ( X )}{Σ ( X )} = 1 + \frac{2 2}{lo g X lo g lo g X} - \frac{( 2 + lo g 2 ) 2}{lo g X ( lo g lo g X ) ^{2}} + \frac{τ ( X )}{lo g X ( lo g lo g X ) ^{3}} .$

Theorem 1.3, on the same page, supplies unconditional effective bounds:

$6.78 \leq τ (X) \leq 94.73$ for every real $X \geq N^{(0)}$ ;
$0.153 \leq τ (X) \leq 129.08$ for every real $4 \leq X \leq N^{(0)}$ .

Here $N^{(0)}$ is the specific CA integer defined by its prime factorization in (3.13), with $lo g N^{(0)} = 1000014552.11 \dots$ in (3.14). The displayed decimal is a description of the threshold, not its definition. The finite part of the proof uses bounds from Axler–Nicolas 2023; this source inspection has not independently rerun that computation.

What prime-profile information the argument uses

Section 3 parametrizes CA reference integers by the usual decreasing prime-power thresholds $F (ξ_{k}, k) = ϵ$ . Lemmas 3.2–3.4 estimate these thresholds, particularly the square layer $ξ_{2}$ . Lemmas 3.5–3.6 compare $ξ$ with $lo g X$ after choosing adjacent parameter-defined CA integers $N^{'} \leq X < N$ .

Section 4, printed pp.23–24, then uses

$\frac{σ ( N ^{'} )}{N ^{'}} \leq Σ (X) < \frac{σ ( N )}{N}$

and the primorial attaining $Φ (X)$ to compare the envelopes. Its “excess” is

$E (X) = lo g N - ϑ (ξ) = k \geq 2 \sum ϑ (ξ_{k}),$

which measures the higher-prime-power content of that CA reference. It is not the deficit $J_{s} (d)$ of an arbitrary actual divisor from the FIB source.

These are forward estimates of extremizers and their envelopes. The checked theorem and lemma statements do not give an inverse assertion that every integer whose abundancy is close to the envelope shares a specified common core. They also do not preserve the original integer or its residue class when replacing it by a CA or primorial extremizer.

Relation to the current actual-divisor estimates

Current object or goal	What this source supplies	Additional obligation
Replacing $Z (n) = σ (n) / n$ by $n / φ (n)$	Quantifies the discrepancy of the two maxima up to the same $X$ , including its leading $22 / (lo g X lo g lo g X)$ term.	It is not a lower bound for $[n / φ (n)] / Z (n)$ at the same integer. Separate maximizers cannot be treated as jointly realized.
A bound for an actual $n \leq X$	Any positive lower bound $L (X)$ for $Φ (X) /Σ (X)$ gives the valid global bound $Z (n) \leq Σ (X) \leq Φ (X) / L (X)$ .	Proving this bound below the actual Robin budget requires an additional comparison with $e^{γ} lo g lo g n$ ; the envelope theorem does not already make that comparison.
The low-loss divisor set $J_{s} (d) \leq O (R)$ for $w_{s} (d) = b_{s} (d) / d$	CA prime-layer estimates provide related extremal arithmetic background.	No stated comparison identifies its CA reference objective with $b_{s} = μ * Z^{s}$ , its normalized weights, or $J_{s} = lo g (max w_{s} / w_{s})$ .
Uniform editing/common-core bounds for every low-loss divisor	No directly matching inverse theorem was located in the inspected statements.	The source-specific uniform bound remains to be justified; the unweighted §232 count already has the separate classical benefit/PNT simplification.
$s = lo g A lo g lo g A$ , $R = lo g A / (lo g lo g A)^{2}$ , and the full complementary moment	The paper studies fixed-order expansions of extremal envelopes.	It supplies no $s$ -uniform source power sum, $J_{s}$ -tail estimate, or actual-divisor incidence count.
Every prescribed growing reduced residue class	No progression or residue-class hypothesis is present in its main results.	A map to an extremizer is not a map preserving $n \equiv c (mod m)$ , divisor incidence, or the candidate’s complete joint weight.

The unconditional status of Theorems 1.2–1.3 should be kept separate from RH-conditional bounds in Nicolas’s other work. In particular, the frequently quoted bound with a $0.095/ lo g n$ subtraction is not Theorem 1.2 or Theorem 1.3 of this manuscript; no such corollary is stated here. This card does not reclassify any result from the distinct 2022 paper.

The useful addition is an explicit comparison between the global Euler and divisor-sum envelopes. It does not by itself remove the possible singleton in §232, establish the missing FIB-to-CA transfer, or certify originality of the actual-source weighted concentration argument. The remaining common-implementation and uniformity requirements must not be erased by the shared terminology “large values.”

Explicit comparison at the same candidate

The following is an application of the cited envelope statements, not an additional theorem attributed to Nicolas. For an actual integer $n > 5040$ , put $v = lo g n$ , $L = lo g v$ and $t_{n} = e^{γ} L$ . Let $a (n) = 0.153$ for $n < N^{(0)}$ and $a (n) = 6.78$ for $n \geq N^{(0)}$ , with the exact threshold as above. Theorem 1.3 supplies the positive lower bound

$R_{-} (n) = max {1, 1 + \frac{2 2}{v L} - \frac{( 2 + lo g 2 ) 2}{v L ^{2}} + \frac{a ( n )}{v L ^{3}}} \leq \frac{Φ ( n )}{Σ ( n )} .$

Here $Φ (n) > Σ (n)$ for $n \geq 2$ : at any integer attaining $Σ (n)$ , which is larger than one, $Z (m) < m / φ (m) \leq Φ (n)$ . Thus the maximum with one preserves a lower bound without assuming that a truncated expression is positive at every parameter.

Write $D_{n} = lo g (Σ (n) / Z (n)) \geq 0$ . If $d_{n} \leq D_{n}$ is independently established, then

$lo g \frac{Φ ( n )}{t _{n}} < lo g R_{-} (n) + d_{n}$

suffices for strict Robin at this same $n$ . Taking $d_{n} = 0$ uses only the cited envelope comparison. Defining $D_{n}$ does not establish a positive lower bound for it.

Let $p$ be the last prime for which $P_{p} = \prod_{q \leq p} q \leq n$ , so that $P_{p} \leq n < P_{p^{+}}$ . The Euler envelope is exactly $Φ (n) = \prod_{q \leq p} (1 - 1/ q)^{- 1}$ . Consequently the left side above equals

$E_{M} (p) + lo g \frac{lo g p}{L}, E_{M} (p) = lo g \frac{\prod _{q \leq p} ( 1 - 1/ q ) ^{- 1}}{e ^{γ} lo g p} .$

Both terms come from the same prime sequence; the second cannot be deleted or assigned a favorable sign. Writing $v = ϑ (p) + h$ gives $0 \leq h < lo g p^{+}$ , and variation of $h$ contributes only $O (1/ v)$ as $n \to \infty$ .

The Dusart estimates already used in the project give a coarse $O (L^{- 2})$ bound for this joint expression. Its guaranteed error scale is larger than the envelope saving $lo g R_{-} (n) \sim 22 / (v L)$ by a factor of order $v / L$ . This compares the available bounds, not the actual signed error. Even a fixed inverse-logarithmic error order does not reach the required scale; increasing the fixed order in Nicolas’s ratio expansion alone does not repair it.

For fixed $C_{0} > 1$ , the growing moment $s = y lo g y$ and $n \in [e^{y}, C_{0} e^{y}]$ , the logarithmic envelope saving is $s lo g R_{-} (n) \sim 22 y$ . This is $o (y / (lo g y)^{2})$ , the scale of the existing global moment excess. It therefore cannot by itself pay that global relaxation cost. The individual candidate need not pay that global cost, so this observation does not refute a bound using additional information about the actual candidate.

Finally, the fixed-price benefit is not $D_{n}$ . If $C$ maximizes $Z (m) m^{- 1/ s}$ and $Q_{s} = Z (C)^{s} / C$ , then

$D_{n} = Ben_{C, 1/ s} (n) - K_{s} (n), K_{s} (n) = \frac{lo g Q _{s} + lo g n}{s} - lo g Σ (n) \geq 0.$

A benefit lower bound must therefore be accompanied by a bound for this support-line gap before it supplies $d_{n}$ . The FIB theory volume, §234 gives the separate finite comparison between the actual increment loss and this classical price objective. Neither comparison establishes the missing fixed-residue candidate exclusion.

Signed prime error at the actual primorial cutoff

The following application combines the envelope comparison above with classical partial summation. It introduces no new prime-distribution estimate. Its sufficient condition remains unproved at the FIB candidates; it is not a proof of Robin’s inequality or a claim of originality.

For the same actual integer $N$ , put $Y = lo g N$ and let $x$ be the prime satisfying $P_{x} \leq N < P_{x^{+}}$ , where $x^{+}$ is the next prime. Write

$P (x) = p \leq x \prod (1 - 1/ p)^{- 1} = Φ (N), f (x) = \frac{e ^{γ} lo g ϑ ( x )}{P ( x )}, b_{N} = lo g \frac{lo g Y}{lo g ϑ ( x )} \geq 0.$

Here $ϑ (u) = \sum_{p \leq u} lo g p$ and $ψ (u) = \sum_{p^{j} \leq u} lo g p$ . For any proved positive $L (N) \leq Φ (N) /Σ (N)$ , the exact normalization gives

$lo g \frac{Z ( N )}{e ^{γ} lo g Y} \leq - lo g f (x) - lo g L (N) - b_{N} .$

This retains the correlation between the Euler product and the primorial logarithm. Also $0 \leq Y - ϑ (x) < lo g x^{+}$ , so the prime number theorem gives $x \sim Y$ and $b_{N} = O (1/ x) = o (1/ (x lo g x))$ as $N \to \infty$ .

The unconditional Nicolas input and its endpoint term

Jean-Louis Nicolas, Small values of the Euler function and the Riemann hypothesis, arXiv:1202.0729v2, Lemma 2.1, printed p.4, states for $x \geq 121$ that

$I_{ϑ} (x) - \frac{( ϑ ( x ) - x ) ^{2}}{x ^{2} lo g x} \leq lo g f (x) \leq I_{ϑ} (x) + \frac{1}{2 ( x - 1 )},$

where

$k (u) = \frac{1 + lo g u}{u ^{2} lo g ^{2} u}, I_{ϑ} (x) = \int_{x}^{\infty} (ϑ (u) - u) k (u) d u, I_{ψ} (x) = \int_{x}^{\infty} (ψ (u) - u) k (u) d u .$

The paper calls these integrals $K (x)$ and $J (x)$ in (1.15)–(1.16), and attributes Lemma 2.1 to Proposition 1 of its 1983 reference [6]. This lemma has no RH hypothesis. The later RH-dependent bounds in Lemma 2.4 and Proposition 2.1 are not used here.

Consequently, for $x \geq 121$ , the independently defined margin

$m_{N} = I_{ϑ} (x) - \frac{( ϑ ( x ) - x ) ^{2}}{x ^{2} lo g x} + lo g L (N) + b_{N}$

gives $Z (N) \leq e^{γ} lo g lo g N e^{- m_{N}}$ . Positivity of this margin is sufficient, but has not been established for all actual candidates.

Partial summation also gives an exact version of the endpoint accounting. Put $a = (ϑ (x) - x) / x$ , $v = lo g x$ and

$Q (x) = \frac{a}{v} - lo g (1 + \frac{lo g ( 1 + a )}{v}), T_{2} (x) = p > x \sum j \geq 2 \sum \frac{1}{j p ^{j}} > 0.$

For $x \geq 121$ , all logarithms are defined, and

$lo g f (x) = I_{ϑ} (x) - Q (x) + T_{2} (x) .$

Indeed, writing $B_{1}$ for the Meissel–Mertens constant, partial summation yields $\sum_{p \leq x} 1/ p = lo g lo g x + B_{1} + a / v - I_{ϑ} (x)$ . Add the higher-power terms in $lo g P (x)$ and use $B_{1} + \sum_{p} \sum_{j \geq 2} 1/ (j p^{j}) = γ$ . This proves the identity and the positive sign of $T_{2}$ . Thus $I_{ϑ} - Q + lo g L + b_{N}$ is another valid lower margin. It comes from the same classical identity, not a new analytic estimate.

What remains after the prime-square contribution

For this paragraph choose the specific $L (N) = R_{-} (N)$ above, which has Nicolas’s leading correction $22 / (Y lo g Y)$ . An arbitrary positive envelope-ratio lower bound need not have that correction. The unconditional prime number theorem gives

$ψ (u) - ϑ (u) = ϑ (u) + O (u^{1/3} lo g u) = (1 + o (1)) u .$

Integrating against the positive kernel gives

$I_{ψ} (x) - I_{ϑ} (x) = \frac{2 + o ( 1 )}{x lo g x} .$

It follows that

$x lo g x m_{N} = 22 - 2 + x lo g x I_{ψ} (x) - \frac{( ϑ ( x ) - x ) ^{2}}{x ^{3/2}} + o (1) .$

Thus a sufficient candidate-specific input is: there are fixed $η > 0$ and $N_{1}$ such that at the associated prime cutoff of every actual candidate $N \geq N_{1}$ ,

$\frac{( ϑ ( x ) - x ) ^{2}}{x ^{3/2}} - x lo g x I_{ψ} (x) \leq 22 - 2 - η .$

This would give strict Robin with a positive margin of order $1/ (lo g N lo g lo g N)$ for sufficiently large candidates. It permits compensation between the signed tail and endpoint error. The endpoint penalty cannot be discarded from the prime number theorem alone: its being negligible at this scale requires $ϑ (x) - x = o (x^{3/4})$ . The exact $Q$ version has

$x lo g x Q (x) = \frac{( ϑ ( x ) - x ) ^{2}}{2 x ^{3/2}} (1 + \frac{1}{lo g x} + O (∣ a ∣)) .$

For example, bounds $I_{ψ} (x) \geq - κ / (x lo g x)$ and $∣ ϑ (x) - x ∣ \leq K x^{3/4}$ with fixed nonnegative constants satisfying $κ + K^{2} /2 < 22 - 2$ would suffice. These are additional prime-distribution hypotheses, not consequences of a large divisor core or of a FIB address. No equivalence between this restricted sufficient condition and RH is asserted.

The endpoint condition at actual self-tangent sources

There is a more specific source condition under which the squared endpoint term above is already negligible. Take an actual regular self-tangent CA integer $N = C_{A}$ with $A = lo g N$ as in the archived workload interface. Keep $x$ as the primorial cutoff of this same $N$ : $ϑ (x) \leq A < ϑ (x^{+})$ , with $x^{+}$ the next prime. It is not the largest prime factor of $C_{A}$ or the clock coordinate $A$ .

The archived manuscript’s §8, thm:theta-normal-form, equation eq:self-tangent-theta, already gives

$A - ϑ (A) = 2 A (1 + o (1))$

along regular returns tending to infinity. The source uses a strict prime cutoff; changing to the weak $ϑ$ used here costs at most $O (lo g A)$ and preserves this assertion. This return identity is reused, not rederived. The manuscript’s §11, lem:external-extremal-input and thm:persistent-obstruction, also supplies the existing reduction to a proper GA1 regular source at or above the Robin level if RH fails. The estimates here apply to that selected source class; GA1, CA status, or a five-window address alone is not the self-tangency hypothesis.

The existing uniform short-interval supplier and inverse- $ϑ$ argument can now be applied at this actual $A$ . With $H = A^{2/3}$ , it gives $ϑ (A + H) - ϑ (A) \sim H$ . Since the return deficit is only $O (A)$ , eventually $ϑ (A + H) > A$ . There is a prime in $(A, A + H]$ ; at the first such prime the added mass is $O (lo g A)$ , smaller than the return deficit, so its $ϑ$ value is still below $A$ . Monotonicity therefore locates the same primorial cutoff and its residual as

$0 < x - A \leq A^{2/3}, 0 \leq h_{N} := A - ϑ (x) < lo g x^{+} = O (lo g A) .$

Consequently $ϑ (x) - x = - (x - A) - h_{N}$ and

$\frac{( ϑ ( x ) - x ) ^{2}}{x ^{3/2}} = O (A^{- 1/6}) = o (1), x lo g x Q (x) = O (A^{- 1/6}) .$

The second estimate uses the already displayed expansion of the exact endpoint $Q$ . The exponent $2/3$ is a convenient choice within the cited uniform short-interval range, not a new prime-gap estimate. This application does not require an effective starting threshold or assume RH, and it does not claim the endpoint estimate at arbitrary FIB candidates.

There is also a controlled transfer of the unnormalized tail. The same manuscript’s §8, thm:psi-normal-form, gives $ψ (A) - A = - (2 - 1) A + o (A)$ at a regular return. At the primorial cutoff, $ϑ (x) = A - h_{N}$ and the classical prime-power decomposition gives $ψ (x) = A + O (A)$ . For $A \leq u \leq x$ , monotonicity of $ψ$ hence bounds $∣ ψ (u) - u ∣ = O (A^{2/3})$ . Integrating over the same interval against $k (u) = O (A^{- 2} / lo g A)$ yields

$I_{ψ} (x) = I_{ψ} (A) + O (\frac{A ^{- 2/3}}{lo g A}) .$

With $Z_{ψ} (t) = t lo g t I_{ψ} (t)$ this is precisely

$Z_{ψ} (x) = c_{N} Z_{ψ} (A) + O (A^{- 1/6}), c_{N} = \frac{x lo g x}{A lo g A} = 1 + O (A^{- 1/3}) .$

The factor $c_{N}$ must remain unless a suitable bound on $Z_{ψ} (A)$ is available; $x / A \to 1$ alone does not justify an additive $o (1)$ comparison of the normalized tails. A fixed one-sided lower bound at $A$ can be transported with this factor. No such lower bound follows from the return identity, which fixes a point value of $ψ$ rather than its entire tail.

For $L (N) = R_{-} (N)$ , the existing comparison margin therefore simplifies on this source class to

$x lo g x m_{N} = 22 - 2 + Z_{ψ} (x) + o (1) .$

This pays the endpoint requirement of that comparison, not its signed-tail requirement. The source-clock pressure identity and its reserve constant $2 (2 - 1)$ are already recorded in the FIB volume, §§87 and 93; rewriting the remaining tail condition is not a new RH criterion or a stronger Robin estimate. This is a paper application connecting two existing observation cutoffs, with no Lean or originality certification.

The 2026 signed formula and the available absolute-error scale

Broadbent–Fiori–Kadiri–Ng–Wilk, Bounds for Mertens sums, arXiv:2608.01498v1, Proposition 13(i), equation (55), gives an unconditional explicit formula for $I_{ψ} (x)$ for $x \geq 2$ . Its proof, equations (82)–(83), also gives the form

$I_{ψ} (x) = \frac{lo g x + 1}{lo g ^{2} x} ρ \sum \frac{x ^{ρ - 1}}{ρ ( ρ - 1 )} - ρ \sum \frac{1}{ρ ( ρ - 1 )} \int_{x}^{\infty} t^{ρ - 2} (\frac{1}{lo g ^{2} t} + \frac{2}{lo g ^{3} t}) d t - \frac{lo g ( 2 π )}{x lo g x} - \frac{1}{2} κ_{0} (x),$

where the sums run over the nontrivial zeta zeros with multiplicity and $κ_{0} (x) = \int_{x}^{\infty} lo g (1 - t^{- 2}) k (t) d t$ . The proof uses absolute convergence after integration by parts to pass to the infinite endpoint. This is a signed formula; Proposition 13(ii) is a separate absolute-value estimate. Merely expressing the integral through zeros gives no needed one-sided bound at FIB-selected cutoffs. Equation (54) is also the partial-summation identity used above. The relevant statements and these proof steps were inspected; no complete proof audit or Lean verification of the preprint is claimed.

Fiori–Jaskari, Explicit bounds for the prime number theorem, arXiv:2609.23222v1, Theorem 1.1 and Table 1, state for $u \geq e^{3}$ that

$\frac{∣ ψ ( u ) - u ∣}{u} < 0.2390 exp [- D (1 + \frac{lo g lo g lo g u}{15 lo g lo g u}) \frac{( lo g u ) ^{3/5}}{( lo g lo g u ) ^{1/5}}], D = \frac{5}{2} (\frac{5}{3})^{1/5} (\frac{2000}{161967})^{3/5} .$

This is a September 2026 unconditional preprint statement. Its theorem and table were read in the primary text; its full analytic proof and computations have not been independently audited. Even this error shape, with any fixed constant $c > 0$ in its leading exponential, is asymptotically larger than the required scale:

$\frac{exp [ - c ( lo g x ) ^{3/5} / ( lo g lo g x ) ^{1/5} ]}{1/ ( x lo g x )} ⟶ \infty.$

Integration introduces no improvement to a power of $x$ : with a slightly smaller fixed $c$ , the same exponential shape bounds $∣ I_{ψ} (x) ∣$ . This compares the guaranteed upper bounds, not the unknown actual signed error, and does not prove that the candidate condition fails.

There is a stronger obstruction to a different proposed shortcut. Diamond–Pintz, Oscillation of Mertens’ product formula, JTNB 21 (2009), Theorem 1.1, printed p.524, prove that $x (P (x) - e^{γ} lo g x)$ has arbitrarily large positive and negative values. Hence no fixed $K$ gives $P (x) \leq e^{γ} lo g x + K / x$ for all sufficiently large $x$ . This refutes that raw-product certificate; it does not refute Robin or the comparison normalized by $lo g ϑ (x)$ .

Return to the actual source budget

Use the notation of the FIB volume’s §233.5: $t = e^{γ} lo g lo g A$ , $s = lo g A lo g lo g A$ , and $C_{*} + H_{*} = (Z (N) / t)^{s}$ at the possible actual candidate. Put $B_{N} = (lo g lo g N / lo g lo g A)^{s}$ . Either valid margin above gives

$C_{*} + H_{*} \leq B_{N} e^{- s m_{N}} .$

Therefore $B_{N} e^{- s m_{N}} + ε_{A} < B_{N}$ suffices for the already stated target $C_{*} + ε_{A} < B_{N}$ . For $0 < ε_{A} < B_{N}$ , the needed margin for this certificate is exactly

$m_{N} > - \frac{1}{s} lo g (1 - ε_{A} / B_{N}) .$

The fixed-slack prime condition above is stronger. In a fixed-ratio window $N \in [A, C_{0} A]$ , with $C_{0} > 1$ fixed and $y = lo g A$ , it supplies $s m_{N} ≫ y$ and $B_{N} = (N / A) (1 + o (1))$ . Conditional on the previously stated high-loss estimate $ε_{A} = exp [- (π^{2} /9) y / (lo g y)^{2} + o (y / (lo g y)^{2})]$ , the budget then holds eventually. This application stays at the same integer throughout. It does not supply the missing signed-error hypothesis, certify an effective starting threshold, or extend a candidate-only result to the full RH criterion.

At a common primorial cutoff, the signed integral and endpoint terms coincide. FIB information must independently restrict the attained cutoffs or supply an actual-candidate lower bound for $D_{N} = lo g (Σ (N) / Z (N))$ ; an address change alone supplies neither.

On the full CA test set this self-cutoff deficit is identically zero, including after restriction to each integer’s actual residue; see the classical CA scope note. A positive-deficit estimate is therefore only an option for other candidates. The signed comparison remains applicable to CA integers with $d_{C} = 0$ , without providing the missing prime-error estimate.

The Guth–Maynard short-interval application makes one limitation precise. On the cutoffs at the lower endpoints of all prime-index FIB windows, a uniform eventual square-root-scale lower bound for $I_{ψ}$ already implies RH: monotonicity controls interpolation between samples, and the cited short-interval theorem bounds their gaps. The same implication has not been established for the cutoffs of actually existing low-loss candidates, whose gaps are unknown. These two sets of cutoffs must remain distinct when using the sufficient condition above.

The exponent-excess bound at an actual tied CA candidate

The following connects the existing CA estimate to the actual exception budget in the Pollack application. It is an application of the cited estimates, not a new analytic theorem or a Lean-verified result.

Fix a threshold parameter $ϵ_{i}$ from (3.3), and let $N = N_{ϵ_{i}} > N^{(0)}$ be the representative defined by (3.8), including every prime-power layer whose activation price equals $ϵ_{i}$ . Let $n$ be any actual CA maximizer at this same parameter. Remark 3.1, printed p.11, describes both ordinary and extraordinary ties: every such $n$ divides $N$ . This comparison transports no canonical composition or discriminant from $N$ to $n$ .

The exponent excess is monotone under divisibility, so

$B_{exp} (n) = lo g \frac{n}{rad ( n )} \leq lo g \frac{N}{rad ( N )} = E = k \geq 2 \sum ϑ (ξ_{k}) . (C1)$

Here $rad (N) = \prod_{p \leq ξ} p$ follows from (3.8); the final equality is (4.3)–(4.4), printed p.23. To apply Lemma 4.1, printed p.24, take $X = N^{'} = N_{ϵ_{i - 1}}$ . Since $N > N^{(0)}$ , the adjacent representative satisfies $X = N^{'} \geq N^{(0)}$ and $X < N$ . Thus, with $t = lo g ξ$ ,

$E \leq U (ξ) := 2 ξ (1 - \frac{lo g 2}{2 t} + \frac{19.512}{t ^{2}}) . (C2)$

The boundary $N = N^{(0)}$ is not covered by this particular instantiation of the lemma. For the same actual $n$ with canonical unit bit one and nonsquare signed discriminant, reuse the Pollack application’s $H_{1}, T_{1}, R_{1}, m_{1}$ . Combining its exponent inequality with (C1)–(C2) gives the explicit comparison

$lo g H_{1} (n) \leq lo g 5 + U (ξ) . (C3)$

This includes intermediate tied maximizers. It changes neither their actual $c mod H_{1}$ , their primitive conductor $q$ , nor the signed Euler budget.

For clarity, the upper bound already recorded in the Pollack application is $m_{1} \leq U_{n} := ∣ D ∣ R_{1} / T_{1}^{2}$ . Its numerical value itself satisfies

$U_{n} \geq \frac{∣ D ∣}{H _{1}^{2}} \geq \frac{∣ D ∣}{25} exp [- 2 U (ξ)], (C4)$

because $T_{1} ∣ H_{1}$ and $R_{1} \geq 1$ . Under the additional same-candidate assumptions $∣ D ∣ ≍ n$ and $ξ \sim lo g n$ , this lower bound on $U_{n}$ is $n^{1 - o (1)}$ . Consequently this upper-bound certificate cannot establish $m_{1}^{b} \leq lo g n$ for fixed $b > 0$ in that regime, even after maximizing the allowed supported square-factor removal. This is not a lower bound on $m_{1}$ . Independent conductor information or square factors outside this supported exception budget could still give a smaller modulus. The assumption $∣ D ∣ ≍ n$ has not been established for the actual remaining extremal candidates; (C4) is conditional in that comparison.

An effective core bound at the selected GA2 source

This application improves the effective positive-core allowance, while leaving the complete signed Robin tail unbounded. It uses the same critical integer and clock throughout. The envelope theorem, GA2 comparisons, prime-power estimates, and core identity are existing results; no originality or Lean verification is claimed for their combination.

Under a Robin counterexample, use the selected source in the Polak application: $N$ is CA and GA2, $A = lo g N$ , $P = P^{+} (N)$ , and $P < A < P^{+}$ . Its actual exponents attain the full-support minimum at price $1/ (A lo g A)$ , so

$Δ (N) = I_{ψ} (A) + D^{*} (A), D^{*} (A) = R_{core} (N, A) - C_{pp} (A) . (G1)$

Set $L = lo g A$ and $T = A L$ . The classical CA record property in Alaoglu–Erdős gives $Σ (N) = Z (N)$ , where $Z (n) = σ (n) / n$ . Let $x$ be the primorial cutoff $ϑ (x) \leq A < ϑ (x^{+})$ , so $Φ (N) = P (x)$ , with $P (u) = \prod_{q \leq u} (1 - 1/ q)^{- 1}$ . Since $ϑ (P) \leq A$ , we have $x \geq P$ . Every prime in $(P, x]$ is absent from $N$ and exceeds $A$ .

Cancelling the suffix at the actual clock

For each such prime $q$ , the GA2 comparison with $qN$ gives

$lo g (1 + 1/ q) \leq lo g \frac{lo g ( A + lo g q )}{lo g A} \leq \frac{lo g q}{A lo g A} . (G2)$

The final inequality is the concavity tangent for $u \mapsto lo g lo g (A + u)$ at $u = 0$ . Equivalently, the already established CA optimality at price $1/ (A lo g A)$ supplies this local upper bound. It is not a new gain from jointly optimizing different integers. Using $lo g (q / (q - 1)) = lo g (1 + 1/ q) - lo g (1 - q^{- 2})$ yields

$lo g \frac{P ( x )}{P ( P )} \leq \frac{ϑ ( x ) - ϑ ( P )}{A lo g A} + q > A \sum - lo g (1 - q^{- 2}) . (G3)$

Put $m = ⌊ A ⌋$ . The prime sum is at most the integer sum, which telescopes:

$q > A \sum - lo g (1 - q^{- 2}) \leq n = m + 1 \sum \infty - lo g (1 - n^{- 2}) = lo g \frac{m + 1}{m} < \frac{1}{A - 1} . (G4)$

Consequently Nicolas’s effective $R_{-} (N)$ above and $Σ (N) = Z (N)$ give

$R_{core} (N, A) = \frac{A - ϑ ( P )}{A lo g A} + lo g \frac{P ( P )}{Z ( N )} > lo g R_{-} (N) + \frac{A - ϑ ( x )}{A lo g A} - \frac{1}{A - 1} \geq lo g R_{-} (N) - \frac{1}{A - 1} . (G5)$

This uses the primorial cutoff only to cancel its missing-prime suffix. It neither transports $I_{ψ} (A)$ to $x$ nor requires an effective prime-gap bound or an endpoint estimate for $ϑ (x) - x$ .

An effective allowance stronger than the existing core envelope

The Dusart prime-power input, Proposition 3.2, gives $T C_{pp} (A) < 2.00014 + 2.67 A^{- 1/6}$ . For $N \geq N^{(0)}$ and $L \geq 26$ , put

$b (L) = 22 - \frac{( 2 + lo g 2 ) 2}{L} + \frac{6.78}{L ^{2}} .$

Then $0 < b (L) < 3$ , and $R_{-} (N) = 1 + b (L) / T$ . The elementary inequality $lo g (1 + u) \geq u - u^{2} /2$ , $u \geq 0$ , gives $T lo g R_{-} (N) > b (L) - 5/ T$ . Also $T / (A - 1) < 2 L / A$ . Combining these inequalities with (G1) and (G5) gives the explicit bound

$A lo g A D^{*} (A) > E (L), (G6)$

where

$E (L) = 22 - \frac{( 2 + lo g 2 ) 2}{L} + \frac{6.78}{L ^{2}} - 2.00014 - 2.67 e^{- L /6} - (2 L + 5/ L) e^{- L /2} . (G7)$

This is strictly stronger than the published $D_{lb} (A)$ on the whole range $L \geq 26$ , not just its half-unit simplification. To check the comparison, set $a = lo g 2$ and

$U (L) = \frac{1}{2} (\frac{L}{L + a} - \frac{2 L}{( L + a ) ^{2}}) .$

The displayed definition of $D_{lb}$ in the Polak note, with $j_{A} > 0$ and $s_{A} > 0$ , gives $D_{lb} (A) < j_{A} /2 < U (L)$ . For $L \geq 26$ ,

$U^{'} (L) = \frac{1}{2} (\frac{a}{( L + a ) ^{2}} + \frac{2 ( L - a )}{( L + a ) ^{3}}) < \frac{a + 2}{2 L ^{2}} .$

Both exponential error terms in (G7) decrease, hence

$(E - U)^{'} (L) > \frac{1}{L ^{2}} (\frac{a + 2}{2} - \frac{13.56}{L}) > 0.$

Elementary rational bounds $1.414 < 2 < 1.415$ , $0.69 < lo g 2 < 0.70$ , $e^{13/3} > 75$ , and $e^{13} > 400000$ give $E (26) > 0.65$ and $U (26) < 0.64$ . For the exponential bounds one can use $\sum_{j = 0}^{9} 4^{j} / j! > 54$ , $e^{1/3} > 25/18$ , and $e > 8/3$ . The comparison therefore has the uniform strict surplus

$E (L) > D_{lb} (A) + 0.01 (L \geq 26) . (G8)$

Moreover $E$ is increasing on this range and $lim_{L \to \infty} E (L) = 22 - 2.00014$ . This is the limit of the lower-bound function, not a new assertion about the attained core’s asymptotic. The core asymptotic $2 (2 - 1)$ is already recorded in the FIB volume, §93.

Paying the thresholds and retaining the signed obligation

The existing Axler finite stop, Lemma 2.3, puts this selected source above the $K$ th primorial, $K = 999999476056$ . Hence $A > K lo g 2 > K /2$ and $L > 26$ . For the latter comparison, $e < 11/4$ and $(11/4)^{26} < 4 \cdot 1 0^{11} < K /2$ suffice.

The exact $N^{(0)}$ in Nicolas’s (3.13), printed p.12, has largest prime $1000000007$ and maximum exponent $33$ . It divides the $33$ rd power of that primorial. Dusart’s Theorem 5.2, $k = 0$ , therefore gives

$lo g N^{(0)} \leq 33 ϑ (1000000007) < 66 (1000000007) < K /2 < A .$

Thus the $6.78$ branch of Theorem 1.3 is paid using the same finite supplier as the previous core application. The decimal approximation to $lo g N^{(0)}$ is not used to define or certify its threshold. No new finite computation, unbounded source sequence, or additional large finite Robin theorem is needed for (G6)–(G8).

At this fixed source the resulting sufficient condition is

$A lo g A I_{ψ} (A) \geq - E (lo g A) . (G9)$

The strict core inequality then supplies strict Robin. For a zero-response representation, retain the elementary terms from the full signed formula above. With $L = lo g A$ , write

$R_{elem} (A) r_{A} = \frac{lo g ( 2 π )}{A L} + \frac{1}{2} \int_{A}^{\infty} lo g (1 - x^{- 2}) k (x) d x, = A L R_{elem} (A) .$

Thus $R_{elem} = lo g (2 π) / (A L) - T_{triv}$ in the notation of (V4)–(V6), and it is $R_{Ak} (1; A)$ in the Akatsuka formula below. For $A \geq e^{2}$ , the already used inequality $- lo g (1 - u) \leq u / (1 - u)$ and $\int_{A}^{\infty} k (x) d x = 1/ (A L)$ give

$0 < \frac{lo g ( 2 π ) - 1/ [ 2 ( A ^{2} - 1 )]}{A} \leq r_{A} \leq \frac{lo g ( 2 π )}{A} .$

This is the elementary correction to the nontrivial-zero response, not another zero term. In particular, the exact source identity and (G9) read

$- A L I_{ψ} (A) = A L Z_{orig} (A) + r_{A}, A L Z_{orig} (A) + r_{A} \leq E (L) .$

The second inequality is equivalent to (G9). Although $r_{A} \to 0$ , its positive sign must be retained in exact budgets. Every subsequent approximation to $Z_{orig}$ still requires this correction when used to bound the full integral. These are applications of the existing explicit formula, not a new signed estimate or Lean verification.

A selected counterexample would instead have to satisfy $A lo g A I_{ψ} (A) < - E (lo g A)$ . The new allowance is weaker as a tail requirement than the existing $- D_{lb} (A)$ requirement; neither signed tail bound has been proved. The full integral over $[A, \infty)$ remains in (G9). This paper-level improvement does not prove RH, and finite source checks and source inspection do not certify its external premises.

The same effective allowance on actual GA2 cores

The allowance $E$ in (G7) also bounds the actual extended core of an arbitrary large GA2 integer. This uses the envelope theorem at the original integer, without identifying it with an envelope maximizer or a CA optimizer. It does not strengthen the independent $D^{*} (A)$ .

Let $N$ be GA2, $A = lo g N$ , $L = lo g A$ , and $T = A L$ , with $N \geq N^{(0)}$ and $L \geq 26$ . The GA2 support argument gives every $p \leq A$ as a divisor of $N$ . In particular $ϑ (A) \leq A$ . Let $x$ be the last prime for which $ϑ (x) \leq A < ϑ (x^{+})$ ; its prime set includes all primes at most $A$ . Numeric $x \geq A$ is unnecessary. As above, $Φ (N) = P (x)$ and $Z (N) \leq Σ (N)$ , so

$\frac{P ( x )}{Z ( N )} \geq \frac{Φ ( N )}{Σ ( N )} \geq R_{-} (N) . (A1)$

Every prime $q$ in the suffix $(A, x]$ satisfies $lo g (1 + 1/ q) < 1/ q < lo g q / (A lo g A)$ , whether or not $q$ divides $N$ . The Euler-factor splitting used in (G3), followed by the same integer-tail payment (G4), therefore gives

$lo g \frac{P ( x )}{P ( A )} \leq \frac{ϑ ( x ) - ϑ ( A )}{A lo g A} + q > A \sum - lo g (1 - q^{- 2}) < \frac{ϑ ( x ) - ϑ ( A )}{A lo g A} + \frac{1}{A - 1} . (A2)$

The non-strict first comparison and strict positive tail allowance also cover an empty suffix. No absent-prime premise is used here.

The actual core in the linked extra-support identity is

$K_{N} (A) Δ (N) = \frac{A - ϑ ( A )}{A lo g A} + lo g \frac{P ( A )}{Z ( N )} - C_{pp} (A) = p \leq A \sum Φ_{p, A} (v_{p} (N)) - C_{pp} (A) + p > A p ∣ N \sum Ξ_{p, A} (v_{p} (N)), = I_{ψ} (A) + K_{N} (A) . (A3)$

Combining (A1)–(A2) and $ϑ (x) \leq A$ gives

$K_{N} (A) > lo g R_{-} (N) - \frac{1}{A - 1} - C_{pp} (A) . (A4)$

The existing estimates used in (G6)–(G7) apply to this right-hand side unchanged. They give

$T K_{N} (A) > E (L) > D_{lb} (A) + 0.01. (A5)$

The $D_{lb}$ comparison is (G8), not a new computation. Since GA2 gives $Δ (N) \leq 0$ , every such actual source must satisfy

$T I_{ψ} (A) < - E (L) . (A6)$

Thus the conditional unbounded GA2 family can be used with this stronger allowance. An eventual lower bound $T I_{ψ} (A) \geq - E (L)$ on all sufficiently large GA2 clocks would contradict that family under RH failure. This signed lower bound remains unproved.

The result estimates $K_{N} (A)$ , not $D^{*} (A)$ : the inequality $K_{N} (A) \geq D^{*} (A)$ does not transport a lower bound backwards to the minimum. The selected CA source has equality by (G1); arbitrary GA2 sources have no such equality here. This is a source-scope application of the existing envelope and payment estimates, without a new core theorem, originality claim, finite verification, or Lean result.

All-integer forward comparisons bound the envelope deficit

The CNS tail maximizers also retain more information than GA2 alone provides for this argument. Let $N > e$ satisfy $G (m) \leq G (N)$ for every integer $m \geq N$ , and keep the existing $D_{N} = lo g (Σ (N) / Z (N)) \geq 0$ . Choose an actual $d \leq N$ attaining $Σ (N)$ and put

$m = d ⌈ \frac{N}{d} ⌉, N \leq m < N + d \leq 2 N .$

Divisibility monotonicity gives $Z (d) \leq Z (m)$ . Applying the same forward record condition at this actual $m$ gives

$\frac{Σ ( N )}{Z ( N )} \leq \frac{Z ( m )}{Z ( N )} \leq \frac{lo g lo g m}{lo g lo g N} < \frac{lo g lo g ( 2 N )}{lo g lo g N} .$

Hence, with the same $A = lo g N$ and $L = lo g A$ ,

$0 \leq D_{N} < lo g \frac{lo g ( A + lo g 2 )}{lo g A} < \frac{lo g 2}{A lo g A} .$

The last comparison is the strict concavity tangent used in (G2). The case $d = N$ or $m = N$ has $D_{N} = 0$ and satisfies these strict positive allowances. The constructed $m$ need not be a multiple of $N$ , so this proof uses the all-integer forward comparisons of the published tail maximizers, rather than just their GA2 comparisons.

Along the CNS family under RH failure,

$0 \leq A lo g A D_{N} < \frac{lo g 2}{A} ⟶ 0.$

Thus replacing $Z (N)$ by $Σ (N)$ loses only a vanishing amount at the original critical scale on these actual sources. Conversely, an upper bound of this size cannot supply a fixed positive lower funding term from $D_{N}$ . It neither identifies an exponent profile nor makes $N$ CA or GA1. This is a short application of standard divisibility monotonicity and the existing source comparisons, without a priority claim, new Lean wrapper, or signed-tail estimate.

A fixed positive scale mixture cannot remove the functional-equation weight

The signed formula above and the actual-source condition (G9) are retained here. Durkan–Hughes–Pearce-Crump, Generalisations of the Landau–Gonek theorem and applications to mean values of zeta, arXiv:2601.18025v1, Theorem 5, instead estimates the dyadic sum

$D_{T} (X) = T < Im ρ \leq 2 T \sum χ (ρ) X^{ρ}, ζ (s) = χ (s) ζ (1 - s) .$

The sum uses the actual zeros with multiplicity. Its complex, zero-dependent weight is part of the theorem; the RH-dependent errors cannot be used unconditionally. This application addresses only one proposed weight-removal interface. It does not reassess that theorem, claim an original transform obstruction, or provide Lean verification.

Fix the same cutoff $A > 1$ throughout. The coefficient contributed by the original tail to a zero $s$ is

$F_{A} (s) = \frac{1}{s} \int_{A}^{\infty} u^{s - 2} \frac{1 + lo g u}{lo g ^{2} u} d u, 0 < Re s < 1. (M1)$

In the signed explicit formula its contribution is $- F_{A} (ρ)$ . The pole term and the trivial-zero integral remain those displayed above. Integration by parts identifies (M1) with the two corresponding terms of that existing formula; no zero tail is truncated here.

The original coefficient already has a positive Mellin representation

Put $k (u) = (1 + lo g u) / (u^{2} lo g^{2} u)$ . Since

$\int_{v}^{\infty} k (u) d u = \frac{1}{v lo g v} (v > 1),$

integrating $x^{s - 1}$ over $0 < x < u$ and changing the order gives

$F_{A} (s) = \int_{0}^{\infty} \frac{x ^{s - 1}}{max ( A , x ) lo g max ( A , x )} d x . (M2)$

For real $0 < s < 1$ this follows by Tonelli. For complex $s$ in the same strip, the absolute integral is the finite integral with exponent $Re s$ , so Fubini gives the identical formula. Thus $F_{A} (s) = \int x^{s} d ν_{A} (x)$ for the positive scale measure

$d ν_{A} (x) = \frac{d x}{x max ( A , x ) lo g max ( A , x )} .$

This representation by itself supplies no one-sided bound for the signed zero sum. Applying it directly to $D_{T}$ would retain the unwanted $χ (ρ)$ factor.

The reciprocal weight destroys fixed positive scale representability

The classical functional equation, in DLMF 25.4.2, has

$χ (s) = 2^{s} π^{s - 1} sin (π s /2) Γ (1 - s) .$

Consequently, for real $σ ↑ 1$ , $χ (σ) > 0$ and $χ (σ)^{- 1} \sim (1 - σ) /2$ . For $0 < σ < 1$ , $F_{A} (σ) / χ (σ)$ is strictly positive, but

$σ ↑ 1 lim \frac{F _{A} ( σ )}{χ ( σ )} = 0. (M3)$

To verify the endpoint, set $ϵ = 1 - σ$ and $w (u) = (1 + lo g u) / lo g^{2} u$ , which tends to zero. For every $η > 0$ , choose $B \geq A$ with $w (u) \leq η$ for $u \geq B$ . The finite-prefix contribution to $ϵ \int_{A}^{\infty} u^{- 1 - ϵ} w (u) d u$ tends to zero, while its tail is at most $\eta\epsilon\int_B^\infty u^{-1-\epsilon}du =\eta B^{-\epsilon}\le\eta$. This proves (M3), including the complete infinite tail and the fixed original cutoff.

Suppose a nonnegative Borel measure $μ_{A}$ on $(0, \infty)$ had finite real moments throughout $0 < σ < 1$ and satisfied the exact all-strip transport identity

$F_{A} (s) = χ (s) \int_{0}^{\infty} X^{s} μ_{A} (d X), 0 < Re s < 1. (M4)$

It may depend on $A$ , but is fixed as $s$ varies. Fatou along any real sequence $σ ↑ 1$ and (M3) imply

$0 \leq \int_{0}^{\infty} X μ_{A} (d X) \leq σ ↑ 1 lim inf \int_{0}^{\infty} X^{σ} μ_{A} (d X) = 0.$

Since $X > 0$ everywhere on the scale domain, $μ_{A} = 0$ . This contradicts the strict positivity of $F_{A} (σ) / χ (σ)$ . Hence no such nonnegative measure exists, even without requiring finite total mass or a finite first moment in advance. Matching the real interval alone already produces the contradiction.

If (M4) held, each finite dyadic actual-zero multiset would obey $\sum F_{A} (ρ) = \int D_{T} (X) μ_{A} (d X)$ , with its multiplicities unchanged. The result rules out this fixed positive, all-strip kernel transport before any summation over heights. It does not rule out signed or complex measures, interpolation only on the actual zero set, height-dependent transforms, or approximation with a separately bounded remainder. Those are different interfaces requiring their own integrability, uniform error and sign estimates. In particular, an all-strip identity is a sufficient universal matching requirement, not a necessary condition for every use of the weighted theorem.

No estimate for the original signed $I_{ψ} (A)$ follows. Condition (G9), at the same selected integer and with the full tail, remains unproved; RH remains unproved.

Exact signed scale transport and the cost of its large-scale tail

The preceding positive-measure exclusion leaves signed and complex transports open. For the same fixed $A > 1$ and original coefficient $F_{A}$ in (M1), a classical cosine–Mellin calculation constructs a real signed transport. Its absolute moments also explain why the displayed error in Durkan–Hughes–Pearce-Crump’s Theorem 5 cannot simply be integrated over all scales. This is an application of classical transforms and the already assessed theorem, with no originality or Lean-verification claim.

A reciprocal-coordinate cosine transform

Let $h_{A} (u) = 1/ [max (A, u) lo g max (A, u)]$ and put

$v_{A} (x) = \frac{h _{A} ( 1/ x )}{x} = {1/ lo g (1/ x), 1/ (A lo g A x), 0 < x \leq 1/ A, x \geq 1/ A . (S1)$

The two branches agree at $1/ A$ . Set $v_{A} (0) = 0$ . The function is locally absolutely continuous, tends to zero at infinity, and has the integrable derivative

$v_{A}^{'} (x) = {1/ [x lo g^{2} (1/ x)], - 1/ (A lo g A x^{2}), 0 < x < 1/ A, x > 1/ A, \int_{0}^{\infty} ∣ v_{A}^{'} (x) ∣ d x = \frac{2}{lo g A} .$

Define the improper cosine transform for $X > 0$ by

$g_{A} (X) = 2 \int_{0}^{\infty} v_{A} (x) cos (2 π X x) d x = - \frac{1}{π X} \int_{0}^{\infty} v_{A}^{'} (x) sin (2 π X x) d x . (S2)$

The first integral converges by Dirichlet on its $1/ x$ tail; the second is absolutely convergent. Integration by parts gives the equality without a boundary term. In particular,

$∣ g_{A} (X) ∣ \leq \frac{2}{π X lo g A} . (S3)$

At the other endpoint, splitting the tail at $1/ X$ gives

$g_{A} (X) = \frac{2}{A lo g A} lo g (1/ X) + O_{A} (1) (X ↓ 0) . (S4)$

Indeed, the compact part is bounded. In the tail integral, subtracting one from the cosine between $1/ A$ and $1/ X$ has a bounded integral after the substitution $t = X x$ , and the integral from $1/ X$ to infinity is bounded by Dirichlet. Thus $\int_{0}^{\infty} X^{σ - 1} ∣ g_{A} (X) ∣ d X < \infty$ for every $0 < σ < 1$ .

The standard cosine moment in DLMF 5.9.6, followed by integration by parts, yields

$\int_{0}^{\infty} t^{s - 2} sin t d t = \frac{Γ ( s ) cos ( π s /2 )}{1 - s} (0 < Re s < 1) .$

Applying this to the second integral in (S2) is a justified Fubini step: its absolute double integral is a finite constant depending on $σ = Re s$ times $\int_{0}^{\infty} ∣ v_{A}^{'} (x) ∣ x^{1 - σ} d x < \infty$ . Integration by parts in $x$ then gives

$\int_{0}^{\infty} X^{s - 1} g_{A} (X) d X = 2 (2 π)^{- s} Γ (s) cos (π s /2) \int_{0}^{\infty} v_{A} (x) x^{- s} d x = \frac{F _{A} ( s )}{χ ( s )} . (S5)$

Here $x = 1/ u$ identifies the last integral with (M2), and the classical functional equation identifies its prefactor with $χ (1 - s) = 1/ χ (s)$ . Thus $d μ_{A} (X) = g_{A} (X) d X / X$ is a locally finite real signed Borel scale measure with finite absolute moments throughout the open strip; finite total variation is not required. The preceding positive-measure result and the strictly positive real moments imply that this density has both signs; no large- $X$ pointwise asymptotic or sign location is assumed.

For each finite actual dyadic zero multiset, (S5) gives the exact coefficient identity

$T < Im ρ \leq 2 T \sum F_{A} (ρ) = \int_{0}^{\infty} D_{T} (X) μ_{A} (d X) . (S6)$

Absolute strip moments justify the finite sum interchange and preserve all multiplicities. Summing over infinitely many height blocks is a separate interchange or remainder obligation; (S6) alone does not pay it.

Every absolutely integrable exact transport has a missing higher moment

There is a sharper endpoint than (M3). With $L = lo g A$ and $ϵ = 1 - σ ↓ 0$ , substituting $u = e^{y}$ in (M1) gives

$F_{A} (1 - ϵ) = \frac{1}{1 - ϵ} \int_{L}^{\infty} e^{- ϵy} (\frac{1}{y} + \frac{1}{y ^{2}}) d y = lo g (1/ ϵ) + O_{A} (1) .$

For the $1/ y$ integral, split at $1/ ϵ$ : replacing the exponential by one below that point changes the answer by at most one, while the rescaled tail is a fixed convergent integral. The $1/ y^{2}$ integral is at most $1/ L$ . Since $χ (1 - ϵ)^{- 1} = ϵ /2 + O (ϵ^{2})$ ,

$\frac{F _{A} ( 1 - ϵ )}{χ ( 1 - ϵ )} = \frac{ϵ}{2} lo g (1/ ϵ) + O_{A} (ϵ) . (S7)$

Suppose any locally finite signed or complex Borel scale measure $μ$ , allowing infinite total variation, had finite absolute moments at every $0 < σ < 1$ , realized the exact all-strip identity (M4), and also had a finite absolute moment at $1 + δ$ for some $δ > 0$ . For $σ$ in a neighborhood of one, $X^{σ} lo g X$ is dominated against $∣ μ ∣$ by a fixed subunit moment on $0 < X < 1$ and the $1 + δ$ moment on $X \geq 1$ . Dominated convergence therefore makes $\int X^{σ} μ (d X)$ differentiable at one, with a finite derivative. Its value there is zero by (M3). But (S7) forces its left difference quotient to tend to $- \infty$ , a contradiction. Hence every such exact transport has

$\int_{0}^{\infty} X^{1 + δ} ∣ μ ∣ (d X) = \infty for every δ > 0. (S8)$

This includes the constructed real density. Its divergent higher moment comes from $X \geq 1$ , because its absolute subunit moments already bound the contribution of $0 < X < 1$ . No explicit asymptotic for $g_{A}$ is needed for this moment obstruction.

Consequence for the existing error supplier

Theorem 5, printed p.4, equation (2.1), is uniform for $X \geq 1$ and $T > 1$ . One term in its displayed absolute-error majorant is

$\frac{( lo g T ) ^{2}}{T} X^{1 + 1/ l o g T} .$

For every fixed $T > 1$ , (S8) shows that integrating this term against the total variation of any exact all-strip transport is infinite. Thus the displayed error majorant cannot, by direct absolute integration over $[1, \infty)$ , control the complete signed transport in (S6). This is a limitation of that guaranteed majorant, not a claim that the actual error integral diverges. The source theorem also leaves the scale interval $(0, 1)$ to a separate argument.

Using a scale truncation requires an independent bound for its complement. A cancellation-sensitive integrated remainder, a different large-scale error estimate, or matching only on the actual zero set could change the conclusion. No such supplier is established here. The original infinite $I_{ψ} (A)$ tail, same selected integer, and sufficient condition (G9) remain unchanged and unproved; RH remains unproved.

A finite first absolute moment controls a scale complement

The higher-moment obstruction (S8) does not assert that the first absolute moment is infinite. For the constructed density, that endpoint is finite and has an explicit tail allowance. The following estimates use (S1)–(S6), without repeating the cosine–Mellin transport proof or the Durkan–Hughes–Pearce-Crump theorem. They are paper-level applications of the elementary Dirichlet estimate for an oscillatory integral; no originality, numerical-experiment or Lean-verification claim is made.

An explicit bound at the large-scale endpoint

Let $A \geq e^{2}$ , $L = lo g A$ and $k = 2 π X \geq A$ . On $0 < x \leq 1/ A$ the function

$u (x) = \frac{1}{x lo g ^{2} ( 1/ x )}$

is decreasing: its derivative is $- (lo g (1/ x) - 2) / [x^{2} lo g^{3} (1/ x)] \leq 0$ . Split the first branch of $v_{A}^{'}$ in (S2) at $1/ k$ . On its initial part,

$\int_{0}^{1/ k} u (x) sin (k x) d x \leq k \int_{0}^{1/ k} \frac{d x}{lo g ^{2} ( 1/ x )} \leq \frac{1}{lo g ^{2} k} .$

For a nonnegative decreasing function $w$ on $[a, b]$ , integrating against the primitive $\int_{a}^{x} sin (k t) d t$ , whose modulus is at most $2/ k$ , gives $∣ \int_{a}^{b} w (x) sin (k x) d x ∣ \leq 2 w (a) / k$ . Apply this once to $u$ on $[1/ k, 1/ A]$ , and once to $1/ (A L x^{2})$ on $[1/ A, \infty)$ . These contributions are bounded by $2/ lo g^{2} k$ and $2 A / (k L)$ respectively. Thus

$∣ g_{A} (X) ∣ \leq \frac{3}{π X lo g ^{2} ( 2 π X )} + \frac{A}{π ^{2} X ^{2} L} (2 π X \geq A \geq e^{2}) . (C1)$

This is a bound rather than an asserted asymptotic or an eventual sign for the density. Integrating it gives, for $R \geq A$ ,

$J_{A} (R) := \int_{R}^{\infty} ∣ g_{A} (X) ∣ d X \leq \frac{3}{π lo g ( 2 π R )} + \frac{A}{π ^{2} R L} < \frac{2}{lo g R} . (C2)$

For the last inequality, put $v = R / A \geq 1$ and use $1 + lo g v / L \leq v$ , so $A / (R L) \leq 1/ lo g R$ ; also $3/ π + 1/ π^{2} < 2$ . The existing small- $X$ estimate (S4) and local continuity of (S2) then give

$\int_{0}^{\infty} X ∣ μ_{A} ∣ (d X) = \int_{0}^{\infty} ∣ g_{A} (X) ∣ d X < \infty. (C3)$

Dominated convergence at the endpoint $σ ↑ 1$ , using (S4), (C3) and the exact strip moments, also transports (M3) to the signed identity

$\int_{0}^{\infty} X μ_{A} (d X) = \int_{0}^{\infty} g_{A} (X) d X = 0. (C4)$

This signed cancellation coexists with a finite, positive first absolute moment and with the divergent higher moments in (S8).

Bound the actual block remainder on its scale complement

Keep the actual positive-ordinate zero multiset $Z_{T} = {ρ : T < Im ρ \leq 2 T}$ , with every multiplicity, and $D_{T}$ from the preceding note. Let $M_{T} (X)$ be the three-branch main expression in the already assessed Theorem 5, and define its actual remainder by

$E_{T} (X) = D_{T} (X) - M_{T} (X), X \geq 1.$

For $X \geq R \geq max (A, T)$ its third branch applies. With $y = π X / T > 1$ it is

$M_{T} (X) = - X y \leq n < 2 y \sum \frac{Λ ( n )}{n} e^{2 πi X / n} .$

The already inspected Dusart inputs $ϑ (t) < 2 t$ and $ψ (t) - ϑ (t) < 1.00007 t + 1.78 t^{1/3}$ give $ψ (t) < 5 t$ for $t \geq 1$ . Hence

$∣ M_{T} (X) ∣ \leq \frac{X}{y} ψ (2 y) < 10 X .$

Define the finite block factor

$K_{T} = ρ \in Z_{T} \sum ∣ χ (ρ) ∣ T^{Re ρ - 1} .$

Because $0 < Re ρ < 1$ and $X \geq T$ , $∣ D_{T} (X) ∣ \leq X K_{T}$ . Thus the actual remainder, independently of its printed superlinear majorant, obeys

$\int_{R}^{\infty} ∣ E_{T} (X) ∣ ∣ μ_{A} ∣ (d X) \leq (K_{T} + 10) J_{A} (R) < \frac{2 ( K _{T} + 10 )}{lo g R} (R \geq max (A, T), A \geq e^{2}) . (C5)$

The factor $K_{T}$ refers to the same actual finite multiset; it is not an RH assumption, a numerical zero certificate or a selected favorable configuration. In particular (C5) does not change the real parts to $1/2$ . It shows that this constructed transport’s actual large-scale remainder is absolutely integrable for each fixed block, although direct integration of the published majorant remains infinite. The two statements concern different integrands and do not contradict one another.

The omitted scale complements can receive a full height budget

Let $T_{j} = 2^{j - 1} T_{0}$ , $j \geq 1$ , for any fixed $T_{0} > 1$ . For any chosen $ε > 0$ , select finite cutoffs

$R_{j} \geq max (A, T_{j}), lo g R_{j} \geq \frac{2 ^{j + 1} ( K _{T_{j}} + 10 )}{ε} . (C6)$

Then the entire, unbounded sequence of scale-complement remainders has the absolute allowance

$j \geq 1 \sum \int_{R_{j}}^{\infty} E_{T_{j}} (X) μ_{A} (d X) < ε . (C7)$

This follows directly from (C5) and $\sum_{j \geq 1} 2^{- j} = 1$ . All ordinates above $T_{0}$ remain assigned to their original dyadic blocks. No finite-height RH verification, critical-line substitution or discarded infinite-height suffix is involved. The cutoffs depend on $A$ , the block factor and the requested allowance; a fixed moderate cutoff is not asserted to suffice.

For each block, its exact coefficient sum in (S6) can consequently be kept in the grouped form

$ρ \in Z_{T_{j}} \sum F_{A} (ρ) = \int_{0}^{1} D_{T_{j}} (X) μ_{A} (d X) + \int_{1}^{\infty} M_{T_{j}} (X) μ_{A} (d X) + \int_{1}^{R_{j}} E_{T_{j}} (X) μ_{A} (d X) + r_{j}, j \geq 1 \sum ∣ r_{j} ∣ < ε . (C8)$

The small-scale term is retained, and the main-expression integral exists by (C3) and its finite-scale branches. The original integrated explicit formula already supplies absolute convergence of the $F_{A} (ρ)$ zero series. Together with (C7), this permits summing (C8) with its first three terms grouped per block; it does not permit separating those three infinite series without further estimates. In the real signed explicit formula the conjugate blocks give $- 2 Re \sum F_{A} (ρ)$ , so the complementary error allowance is $2 ε$ before the original $A lo g A$ normalization. Pole and trivial-zero contributions are unchanged.

This pays one complement obligation without proving the original one-sided Robin bound. The remaining grouped small-scale, main and retained-error terms still require a signed estimate at the same selected cutoff. The printed theorem’s error can be integrated on each finite retained scale interval, but (C6) supplies no bound making that increasing cost fit the Robin reserve. No such uniform balance, effective zero computation or proof of RH is asserted here.

Centering the large-scale arithmetic main expression

The first-moment cancellation (C4) also removes the continuous prime-density contribution of the third branch of $M_{T}$ . The previously assessed Fiori–Jaskari application supplies an independent bound for the remainder. This concerns the arithmetic main expression, with the same kernel and height blocks; it does not estimate the actual zero remainder $E_{T}$ or reproduce the prime-number-theorem proof. These deductions are paper-level applications with no originality or Lean-verification claim.

Keep the moving frequency in the prime-error estimate

Reuse the explicit positive constant

$d = \frac{5}{2} (\frac{5}{3})^{1/5} (\frac{2000}{161967})^{3/5}$

from that existing application. Its unconditional conclusion is

$∣ ψ (u) - u ∣ \leq 0.239 u e^{- d l o g u} (u \geq e^{3}) . (P1)$

No new inspection or certification of its source computation is claimed. In particular the prime powers are part of $ψ$ .

Put $ψ_{-} (u) = \sum_{n < u} Λ (n)$ . The endpoint difference $ψ (u) - ψ_{-} (u)$ is at most $lo g u$ . Also $d < 1$ : use $5/3 < 2$ and $2000/161967 < 1/64$ to get $d < 5/16$ . For $v \geq 3$ , $v - lo g v \geq v$ , hence $lo g u \leq u e^{- d l o g u}$ . Consequently

$∣ ψ_{-} (u) - u ∣ < 2 u e^{- d l o g u} (u \geq e^{3}) . (P2)$

This includes the endpoint convention of the actual interval $y \leq n < 2 y$ , including when $y$ or $2 y$ is a prime power.

For $T > 1$ , $X$ with $y = π X / T \geq e^{3}$ , write

$S_{T} (X) = y \leq n < 2 y \sum \frac{Λ ( n )}{n} e^{2 πi X / n}, b_{T} = \int_{T}^{2 T} \frac{e ^{i t}}{t} d t .$

Substituting $t = 2 π X / u$ gives $\int_{y}^{2 y} u^{- 1} e^{2 πi X / u} d u = b_{T}$ . For $w_{X} (u) = u^{- 1} e^{2 πi X / u}$ , Stieltjes integration with the retained endpoints gives exactly

$S_{T} (X) - b_{T} = w_{X} (2 y) [ψ_{-} (2 y) - 2 y] - w_{X} (y) [ψ_{-} (y) - y] - \int_{y}^{2 y} [ψ (u) - u] w_{X}^{'} (u) d u . (P3)$

The integral may use either endpoint version of $ψ$ , since their difference is supported at integers. As $∣ w_{X}^{'} (u) ∣ \leq u^{- 2} + 2 π X u^{- 3}$ , (P1)–(P2), with $δ_{y} = e^{- d l o g y}$ , give

$∣ S_{T} (X) - b_{T} ∣ \leq (4 + 2 lo g 2 + 2 T) δ_{y} < 8 T δ_{y} . (P4)$

The factor $T$ records the moving phase. It has not been replaced by a fixed-frequency constant or omitted after changing variables. Ordinary integration by parts also gives $∣ b_{T} ∣ \leq 2/ T$ .

A signed cancellation and a controlled centered complement

Since $M_{T} (X) = - X S_{T} (X)$ on this large-scale branch, define the centered expression on all $X \geq 1$ by

$M_{T} (X) = M_{T} (X) + X b_{T} .$

Here the other two branches of $M_{T}$ remain their original formulas; (P4) is used only where its threshold holds. Combining (P4) with (C2) gives the independent complement estimate

$\int_{R}^{\infty} ∣ M_{T} (X) ∣ ∣ μ_{A} ∣ (d X) < \frac{16 T}{lo g R} e^{- d l o g (π R / T)} (R \geq max (A, e^{3} T / π), A \geq e^{2}) . (P5)$

The linear part is eliminated by the full signed moment, not by discarding a part of the scale interval:

$\int_{1}^{\infty} M_{T} (X) μ_{A} (d X) = \int_{1}^{\infty} M_{T} (X) μ_{A} (d X) + b_{T} \int_{0}^{1} g_{A} (X) d X . (P6)$

Thus the small-scale term in (C8) changes to $\int_{0}^{1} [D_{T} (X) + X b_{T}] μ_{A} (d X)$ when its main term is replaced by the centered one. The actual zero remainder $E_{T} = D_{T} - M_{T}$ is unchanged. This retains all endpoint, small-scale and conjugate contributions at the same cutoff.

Cutoffs for the centered arithmetic contribution

With the same $T_{j} = 2^{j - 1} T_{0}$ , any requested $ε > 0$ has explicit sufficient main-term cutoffs

$V_{j} Q_{j} = max (3, d^{- 2} [max (0, lo g \frac{16 T _{j} 2 ^{j}}{ε})]^{2}), = max (A, \frac{T _{j}}{π} e^{V_{j}}) . (P7)$

Since $lo g Q_{j} \geq 2$ , (P5) gives

$j \geq 1 \sum \int_{Q_{j}}^{\infty} ∣ M_{T_{j}} (X) ∣ ∣ μ_{A} ∣ (d X) < ε . (P8)$

For fixed $A, T_{0}, ε$ , this choice has $lo g Q_{j} = O (j^{2})$ as $j \to \infty$ . It is a sufficient allowance for the centered arithmetic main complement, not a necessary cutoff or a replacement for (C6)’s zero-remainder cutoffs. It uses the same cumulative error law for each prime interval and does not combine independently favorable phases.

These estimates control another scale complement and make its continuous cancellation explicit. They do not bound the signed combination left in (C8): the small-scale zero sum, retained centered main and actual zero remainder must still be compared together. The growing-scale error cost and the original full Robin inequality remain unpaid; no bound for $A lo g A I_{ψ} (A)$ at the selected integer or proof of RH follows.

A finite scale band already costs a growing absolute-error allowance

The infinite-moment obstruction (S8) and complement estimates (C5)–(C8) leave a different question: can a retained scale cutoff make the printed Durkan–Hughes–Pearce-Crump error majorant small enough? For the constructed transport, a finite band gives a quantitative obstruction even before its large-scale complement is considered. This is a paper-level application of (S2) and the already assessed Theorem 5, not a new source audit, originality claim or Lean-verified result.

Locate a positive band of the same signed kernel

Let $A \geq e^{8}$ , $L = lo g A$ , and write $X = A ξ$ and $k = 2 π ξ$ . Rescale the two branches of $v_{A}^{'}$ in (S2), without changing the original cutoff. Then

$A L g_{A} (A ξ) = \frac{2}{k} [B (k) - L I_{L} (k)], B (k) = \int_{1}^{\infty} \frac{sin ( k t )}{t ^{2}} d t, I_{L} (k) = \int_{0}^{1} \frac{sin ( k t )}{t ( L - lo g t ) ^{2}} d t . (R1)$

For $0 < k \leq 1/2$ , the sine is nonnegative on $[1, π / k]$ . Its chord bound on $[1, π / (2 k)]$ , followed by an absolute bound on the remaining tail, gives

$B (k) \geq \frac{2 k}{π} lo g \frac{π}{2 k} - \frac{k}{π} > \frac{k}{4} . (R2)$

Here $lo g (π / (2 k)) \geq lo g π > 1$ and $π < 4$ . Also $sin (k t) \leq k t$ on $0 < t < 1$ , so $L I_{L} (k) \leq k / L \leq k /8$ . Thus (R1) gives the uniform, explicit positive band

$g_{A} (X) > \frac{1}{4 A lo g A} (\frac{A}{8 π} \leq X \leq \frac{A}{4 π}, A \geq e^{8}) . (R3)$

This does not claim positivity on all scales; (C4)’s complete signed first moment is still zero. The band is contained in $[1, R]$ for every $R \geq A$ . From $d ∣ μ_{A} ∣ (X) = ∣ g_{A} (X) ∣ d X / X$ , (R3), $lo g 2 > 1/2$ and $π < 4$ imply

$\int_{1}^{R} ∣ μ_{A} ∣ (d X) > \frac{1}{8 A lo g A}, \int_{1}^{R} X^{1 + δ} ∣ μ_{A} ∣ (d X) > \frac{( A /32 ) ^{δ}}{128 lo g A} (δ > 0) . (R4)$

Both lower bounds concern the transport’s absolute weights, not an actual prime or zero error. No critical-line assumption, zero count or finite zero computation is used.

Balance the two printed height costs at every height

Theorem 5, printed p.4, (2.1), includes the two error shapes

$T (lo g T)^{2}, \frac{( lo g T ) ^{2}}{T} X^{1 + 1/ l o g T} (T > 1, X \geq 1) .$

Their uniform implied constants are not numerically certified here. Define the unit-coefficient majorant allowance, after the original Robin normalization, by

$B_{A} (T, R) = A lo g A (lo g T)^{2} \int_{1}^{R} (T + \frac{X ^{1 + 1/ l o g T}}{T}) ∣ μ_{A} ∣ (d X) . (R5)$

For $h = lo g T > 0$ and $R \geq A \geq e^{8}$ , (R4) gives

$B_{A} (T, R) > h^{2} [\frac{1}{8} T / A + \frac{1}{128} A / T (A /32)^{1/ h}] \geq \frac{h ^{2}}{16} exp (\frac{lo g ( A /32 )}{2 h}) . (R6)$

For any $c > 0$ , the minimum of $h^{2} e^{c / (2 h)}$ over $h > 0$ is $e^{2} c^{2} /16$ , attained at $h = c /4$ . Consequently

$B_{A} (T, R) > \frac{e ^{2}}{256} [lo g (A /32)]^{2} for every T > 1, R \geq A \geq e^{8} . (R7)$

This is a lower bound on the allowance produced by integrating these two positive majorant shapes. It is not a lower bound on $∣ E_{T} ∣$ , on its actual integral, or on a Robin violation. The other two printed error shapes can only increase this particular absolute allowance. If the two shapes receive any fixed positive coefficients $c_{1}, c_{2}$ , the same argument has (R7)’s right side multiplied by $c_{1} c_{2}$ . No numerical value for those coefficients or effective failure threshold is inferred from big- $O$ notation.

The core reserve $E (lo g A)$ in (G7) remains bounded as $A \to \infty$ . In contrast, (R7) grows quadratically in $lo g A$ for every choice of height $T$ , including a height depending on $A$ . Thus this direct absolute-majorant transfer on retained intervals $[1, R]$ with $R \geq A$ cannot provide a uniform bounded Robin allowance uniformly as the clock tends to infinity. No unbounded sequence of selected sources is asserted. Increasing those cutoffs, or improving only their omitted complements, does not remove the finite band. An actual-remainder bound that improves on these shapes, integration that controls the remainder’s sign, or a different scale decomposition could change this conclusion; none is excluded by (R7).

The result specifies a finite retained-scale obstruction to one estimate method. It does not supply the required signed estimate at the same selected integer, drop any zero multiplicity or infinite height block, or settle RH. The centered arithmetic-main estimates (P5)–(P8) remain valid independently; they do not alter the actual zero remainder used in (R5).

Fixed Gaussian probes leave a coefficient remainder at unbounded height

Moriya, A Gaussian-Perron Prime-Side Defect and Local Profiles Near Critical-Line Zeros of the Riemann Zeta Function, arXiv:2607.04316v2, Theorem 3.3, printed p.7, gives an exact smoothed prime-defect formula with crossed zero, pole, trivial-zero and shifted-contour terms. Its zero coefficient, at a fixed observation point $z$ , is

$G_{X, α, z} (s) = \frac{X ^{s - z} exp ( α ^{2} lo g X ( s - z ) ^{2} )}{s - z}, X > 1, α > 0.$

Only zeros in the source’s crossed strip contribute to that zero term. The conditional localization in Theorem 7.4 requires its stated damping, pole and contour hypotheses and concerns a fixed simple critical-line zero; Theorem 7.6 additionally assumes RH. Those local conclusions are not used below. The following paper-level application compares the coefficient in Theorem 3.3 with the unchanged original $F_{A}$ in (M1), without repeating the source’s explicit-formula proof or asserting originality or Lean verification.

The original coefficient has a uniform algebraic lower bound

Fix $A \geq e^{2}$ , put $L = lo g A$ , and define

$w (u) = \frac{1 + lo g u}{lo g ^{2} u}, h (u) = u w^{'} (u) = - \frac{lo g u + 2}{lo g ^{3} u} .$

For $s = σ + i t$ with $0 < σ < 1$ , two integrations by parts in (M1) give

$F_{A} (s) = \frac{w ( A ) A ^{s - 1}}{s ( 1 - s )} + \frac{h ( A ) A ^{s - 1} + \int _{A}^{\infty} u ^{s - 1} h ^{'} ( u ) d u}{s ( 1 - s ) ^{2}} . (H1)$

The boundary terms at infinity vanish in this strip. Moreover $h^{'} (u) = 2 (lo g u + 3) / (u lo g^{4} u) > 0$ and $h (u) \to 0$ , so the numerator of the remainder has modulus at most $2∣ h (A) ∣ A^{σ - 1}$ . Relative to the leading term its bound is

$\frac{2∣ h ( A ) ∣}{w ( A ) ∣1 - s ∣} = \frac{2 ( L + 2 )}{L ( L + 1 ) ∣1 - s ∣} \leq \frac{4}{3∣ t ∣} .$

Thus, uniformly throughout the actual critical strip,

$∣ F_{A} (σ + i t) ∣ \geq \frac{2 w ( A ) A ^{σ - 1}}{3∣ s ∣∣1 - s ∣} \geq \frac{2 w ( A )}{3 A ( 1 + t ^{2} )} (A \geq e^{2}, ∣ t ∣ \geq 4) . (H2)$

No real part has been replaced by $1/2$ . This is a lower bound on an individual complex coefficient’s modulus, not on the signed zero sum or on the Robin margin.

Compare any finite family of fixed probes

For this same $A$ , choose any finite nonempty family $(X_{j}, α_{j}, z_{j}, b_{j})$ , with $X_{j} > 1$ , $α_{j} > 0$ and arbitrary complex coefficients $b_{j}$ . The family may depend on $A$ , but its parameters are fixed as the zero height varies. Put

$a_{j} = α_{j}^{2} lo g X_{j}, a_{0} = j min a_{j} > 0, H = j max ∣ Im z_{j} ∣,$

$C_{j} = 0 \leq σ \leq 1 max X_{j}^{σ - Re z_{j}} e^{a_{j} (σ - Re z_{j})^{2}}, C = j \sum ∣ b_{j} ∣ C_{j} .$

Let $η_{j} (s)$ be the source’s crossed-strip indicator; the same argument permits any $∣ η_{j} (s) ∣ \leq 1$ . Define the zero coefficient of the proposed finite reconstruction by

$Q (s) = j \sum b_{j} η_{j} (s) G_{X_{j}, α_{j}, z_{j}} (s) .$

For $0 < σ < 1$ and $∣ t ∣ \geq max (4, 2 H)$ , each denominator has modulus at least $∣ t ∣/2$ , and each Gaussian factor has its height part at most $e^{- a_{0} t^{2} /4}$ . Consequently

$∣ Q (σ + i t) ∣ \leq \frac{2 C}{∣ t ∣} e^{- a_{0} t^{2} /4} . (H3)$

The discrepancy even has an explicit height threshold. Since $e^{x} \geq x^{2} /2$ for $x > 0$ , (H3) is at most $64 C / (a_{0}^{2} ∣ t ∣^{5})$ . Set

$Γ = max (4, 2 H, [\frac{384 A C}{a _{0}^{2} w ( A )}]^{1/3}) .$

For $∣ t ∣ \geq Γ$ , this upper bound is at most $w (A) / (3 A (1 + t^{2}))$ . The triangle inequality and (H2) therefore give

$∣ F_{A} (σ + i t) - Q (σ + i t) ∣ \geq \frac{w ( A )}{3 A ( 1 + t ^{2} )} (0 < σ < 1, ∣ t ∣ \geq Γ) . (H4)$

The classical zero-counting theorem supplies actual nontrivial zeros at unbounded positive ordinates. Applying (H4) to those zeros shows that this finite family cannot match $F_{A} (ρ)$ at every actual zero, even with complex coefficients. Multiplicities remain unchanged: multiplying a coefficient by its positive multiplicity cannot remove the mismatch. Matching only on the actual zero set already fails in this class; an all-strip identity is not required for the conclusion.

This restriction concerns exact coefficient reconstruction by finitely many probes with positive fixed Gaussian parameters. It does not exclude an identity for the aggregate signed sum, an inequality using these probes, or an approximation with an independently paid remainder. No conclusion is asserted here for infinite or height-dependent reconstructions; unbounded center heights or $in f_{j} a_{j} = 0$ are not covered by (H3). Pole, trivial-zero and shifted-contour terms from the source still need their own treatment; they have not been discarded or declared to satisfy the original Robin budget.

The calculation identifies a remainder that any such finite exact replacement would otherwise omit. Controlling its signed sum, or constructing a different reconstruction with a uniform remainder at the same selected integer, remains necessary. Condition (G9), its full infinite tail and the strict core remain unchanged and unproved; RH remains unproved.

The discrepancy also survives real conjugate pairing

The modulus restriction (H4) alone does not control a real part. The existing Ford–Zaharescu fixed-phase input, Corollary 2, supplies a way to select actual zero phases where the real coefficient remains large. This is an application of that published theorem and (H1)–(H3), not a new phase-distribution theorem, source proof audit or Lean-verified result.

Keep the same fixed $A \geq e^{2}$ , $L = lo g A$ and finite fixed family $Q$ above. For $s = σ + i t$ with $0 < σ < 1$ , put

$B_{A} (σ, t) = \frac{w ( A ) A ^{σ - 1}}{t ^{2}} > 0.$

Since $s (1 - s) = t^{2} + σ (1 - σ) + i t (1 - 2 σ)$ , with $0 < σ (1 - σ) \leq 1/4$ and $∣1 - 2 σ ∣ \leq 1$ , comparison of its reciprocal with $t^{- 2}$ and the remainder estimate in (H1) give

$F_{A} (σ + i t) - B_{A} (σ, t) e^{i t L} \leq B_{A} (σ, t) (\frac{7}{3∣ t ∣} + \frac{1}{4 t ^{2}}) \leq \frac{1}{3} B_{A} (σ, t) (∣ t ∣ \geq 8) . (H5)$

In particular $Re F_{A} (s) \geq B_{A} /6$ when $cos (t L) \geq 1/2$ , and $Re F_{A} (s) \leq - B_{A} /6$ when $cos (t L) \leq - 1/2$ . The actual real part $σ$ is still present in $B_{A}$ .

For an actual zero $ρ = β + iγ$ with $γ > 0$ , define the real discrepancy of its conjugate pair by

$Δ_{A} (ρ) = 2 Re F_{A} (ρ) - Re [Q (ρ) + Q (\overline{ρ})] .$

This definition allows arbitrary complex probe weights; $Q$ need not commute with conjugation. The same estimate (H3) applies at both ordinates. For

$Γ_{R} = max (8, 2 H, [\frac{768 A C}{a _{0}^{2} w ( A )}]^{1/3}),$

the elementary exponential bound used above gives

$∣ Q (ρ) + Q (\overline{ρ}) ∣ \leq \frac{128 C}{a _{0}^{2} γ ^{5}} \leq \frac{w ( A )}{6 A γ ^{2}} \leq \frac{1}{6} B_{A} (β, γ) (γ \geq Γ_{R}) .$

Consequently, at those actual heights,

$cos (γ L) \geq \frac{1}{2} cos (γ L) \leq - \frac{1}{2} ⟹ Δ_{A} (ρ) \geq \frac{1}{6} B_{A} (β, γ) \geq \frac{w ( A )}{6 A γ ^{2}}, ⟹ Δ_{A} (ρ) \leq - \frac{1}{6} B_{A} (β, γ) \leq - \frac{w ( A )}{6 A γ ^{2}} . (H6)$

Both signs occur on a positive proportion of actual zero pairs

Use the published theorem only with the fixed frequency $η = L / (2 π) > 0$ . Take the fixed $C^{2} (T)$ minorants

$f_{+} (u) = 8 (cos (2 π u) - \frac{1}{2})_{+}^{3}, f_{-} (u) = f_{+} (u - \frac{1}{2}) .$

Here $v_{+} = max (v, 0)$ . They lie in $[0, 1]$ and have the same mean $c_{0}$ . On $∣ u ∣ \leq 1/12$ modulo one, $cos (2 π u) \geq 3 /2 > 3/4$ , so $f_{+} (u) \geq 1/8$ and $c_{0} \geq 1/48$ . Let $N (T)$ count actual nontrivial zeros with $0 < γ \leq T$ , including every multiplicity. The fixed smooth-test sum below ranges over distinct zeros, with $m_{ρ}$ supplying their multiplicities. The expansion already supplied by Ford–Zaharescu, together with the classical $N (T) \sim T lo g T / (2 π)$ , gives

$\frac{1}{N ( T )} 0 < γ_{ρ} \leq T \sum m_{ρ} f_{\pm} (η γ_{ρ}) ⟶ c_{0} .$

The source’s correction is $O_{A} (T)$ and its error is $o_{A} (T)$ , so neither changes this normalized limit. No interval indicator is substituted into its $C^{2}$ theorem.

Let $C_{+} (T)$ count, with multiplicity, the zeros satisfying $Γ_{R} \leq γ \leq T$ and $Δ_{A} (ρ) \geq B_{A} (β, γ) /6$ ; define $C_{-} (T)$ using $Δ_{A} (ρ) \leq - B_{A} (β, γ) /6$ . The support of $f_{\pm}$ lies in the corresponding phase region of (H6). Removing the finite head below $Γ_{R}$ does not change the normalized limit, hence

$T \to \infty lim inf \frac{C _{\pm} ( T )}{N ( T )} \geq c_{0} \geq \frac{1}{48} . (H7)$

Conjugate zeros have the same positive multiplicity, so this also prevents the fixed finite family from reproducing every real conjugate-pair coefficient. The conclusion is asymptotic for each fixed $A$ and probe family; it supplies no effective first qualifying height and no uniform transition when $A$ or the family varies with $T$ .

The original zero contribution has the negative of these real pair coefficients. Both signs of the discrepancy therefore remain in the original bookkeeping. Counts of phases cannot replace its weights $A^{β - 1} / γ^{2}$ or control their joint signed sum. Neither (H6) nor (H7) pays the full Robin remainder, rules out an aggregate identity or a paid approximation, or changes the sufficient target (G9) at the original selected integer. That bound and RH remain unproved.

A signed infinite-height allowance from the existing Landau formula

The existing Gonek uniform Landau input, Theorem 1, printed pp.92–93, can be applied with the arithmetic variable fixed at the original $A$ . Integrating its actual-zero sum in height retains a prime-power main term. This is different from the continuum scale pairing in that note, where the point-supported main term has zero ordinary integral. The following applies the already inspected unconditional theorem and (H5); it is not a new Landau formula, source proof audit, numerical zero computation, originality claim or Lean certification.

The existing real-part density allowance, (W1)–(W11), already pays a complete infinite-height tail on a finite source-clock range and restricts the selected source to $A > 1 0^{34}$ . Those results are reused. The application here instead estimates the signed height suffix uniformly in both $A$ and a variable cut $T$ . Its implied constants are not numerically certified, and its moving finite head remains unpaid.

Preserve the arithmetic variable and the whole original coefficient

Let $A \geq e^{2}$ , $L = lo g A$ , $T \geq 8$ , and $J (A) = lo g (2 A) lo g lo g (3 A)$ . Sums below range over distinct actual nontrivial zeros, with $m_{ρ}$ supplying their multiplicities exactly once. Define

$S_{A} (U) = 0 < γ_{ρ} \leq U \sum m_{ρ} A^{ρ}, ρ = β + i γ_{ρ} .$

Gonek’s uniform formula, with $x = A$ and height $U$ , gives

$S_{A} (U) = - \frac{U}{2 π} Λ (A) + E_{A} (U), ∣ E_{A} (U) ∣ ≪ A [J (A) + lo g (2 U)] + LU (U \geq 8) . (J1)$

The implied constant is absolute on the stated domain. To obtain this weaker uniform remainder from the displayed source errors, use only

$L min (U, \frac{A}{⟨ A ⟩}) \leq LU, min (\frac{lo g U}{L}, U lo g U) \leq \frac{lo g U}{L} \leq A lo g (2 U) .$

No separation from prime powers is assumed. $Λ (A)$ remains the source’s nonnegative point-supported real-variable function, equal to $lo g p$ at $A = p^{k}$ and zero otherwise. It is not replaced by a prime measure, and no assertion that $A$ avoids prime powers is needed.

Keep the complete original coefficient $F_{A}$ from (M1), and set

$T_{A} (T) = γ_{ρ} > T \sum m_{ρ} F_{A} (ρ) .$

The more precise first inequality in (H5) retains the actual real part and gives

$F_{A} (ρ) - \frac{w ( A )}{A} \frac{A ^{ρ}}{γ _{ρ}^{2}} ≪ \frac{w ( A ) A ^{β - 1}}{γ _{ρ}^{3}} \leq \frac{w ( A )}{γ _{ρ}^{3}} .$

The classical zero count gives $\sum_{γ > T} m_{ρ} γ^{- 3} ≪ lo g (2 T) / T^{2}$ . Thus, with uniform implied constants,

$T_{A} (T) = \frac{w ( A )}{A} γ_{ρ} > T \sum \frac{m _{ρ} A ^{ρ}}{γ _{ρ}^{2}} + O (\frac{w ( A ) lo g ( 2 T )}{T ^{2}}) . (J2)$

Both sums converge absolutely for each fixed $A$ : use $∣ A^{ρ} ∣ \leq A$ and the reciprocal-square zero count, together with the preceding remainder. This step pays the coefficient approximation; it does not replace $F_{A}$ by its leading term without a remainder.

Integrate in zero height with the endpoint convention fixed

Stieltjes partial summation, with the inclusive head and exclusive suffix specified above, gives

$γ_{ρ} > T \sum \frac{m _{ρ} A ^{ρ}}{γ _{ρ}^{2}} = - \frac{S _{A} ( T )}{T ^{2}} + 2 \int_{T}^{\infty} \frac{S _{A} ( u )}{u ^{3}} d u . (J3)$

The boundary term at infinity vanishes. If $T$ is a zero ordinate, its full multiplicity stays in $S_{A} (T)$ and outside the suffix; (J3) retains that convention. The linear main term in (J1) contributes exactly $- Λ (A) / (2 π T)$ .

For the remainder use

$\int_{T}^{\infty} \frac{d u}{u ^{3}} = \frac{1}{2 T ^{2}}, \int_{T}^{\infty} \frac{lo g ( 2 u )}{u ^{3}} d u = \frac{lo g ( 2 T )}{2 T ^{2}} + \frac{1}{4 T ^{2}}, \int_{T}^{\infty} \frac{d u}{u ^{2}} = \frac{1}{T} .$

Combining (J1)–(J3) therefore bounds the entire suffix:

$T_{A} (T) = - \frac{w ( A ) Λ ( A )}{2 π A T} + O (w (A) [\frac{J ( A ) + lo g ( 2 T )}{T ^{2}} + \frac{L}{A T}]), A \geq e^{2}, T \geq 8. (J4)$

This is an unconditional uniform application for actual complex zeros, not a critical-line formula. No infinite height block, real part or multiplicity has been omitted. Keeping $A$ fixed during the height integration is compatible with the uniform estimate holding for all $A, T$ in the stated domain.

A moving cut makes this signed suffix allowance tend to zero

In the original explicit formula the high-zero contribution is $- 2 Re T_{A} (T)$ . Its normalized prime-power main term is favorable. Equation (J4) gives an absolute constant $C_{*} > 0$ such that

$A L [- 2 Re T_{A} (T)] \geq \frac{( 1 + 1/ L ) Λ ( A )}{π A T} - C_{*} [\frac{A [ J ( A ) + lo g ( 2 T )]}{T ^{2}} + \frac{L + 1}{A T}] . (J5)$

The nonnegative main term can be kept or discarded for this lower bound; it must not be assigned an adverse sign. No numerical value of $C_{*}$ or resulting effective source-clock threshold is asserted.

For the admissible moving cut $T = A^{1/4} L$ , the adverse allowance in (J5) is

$O (\frac{J ( A ) + lo g ( 2 A ^{1/4} L )}{L ^{2}} + \frac{L + 1}{A ^{3/4} L}) = O (\frac{lo g L}{L}) ⟶ 0. (J6)$

Thus the complete infinite-height suffix has an asymptotically vanishing one-sided allowance at this moving cut. This gains height cancellation over a direct reciprocal-square absolute sum, whose normalized generic allowance is of order $A lo g (2 T) / T$ . It leaves the actual finite signed head $0 < γ \leq A^{1/4} lo g A$ in the original formula, including all its real parts and multiplicities. That head eventually exceeds the existing verified height and has not been bounded uniformly here. The pole and trivial-zero terms are still those in the original explicit formula.

For any prescribed positive allowance, (J6) gives an existential large- $A$ threshold for this suffix alone. It supplies no certified numerical threshold at the conditional source $A > 1 0^{34}$ , no new finite-zero verification, and no assertion of an unbounded sequence of selected sources. The original same-source condition (G9), with its complete signed integral and strict core, remains unproved. RH remains unproved. The remaining obstacle includes the signed moving finite head; the infinite suffix cannot simply be dropped.

A continuous Gaussian approximation of the complete original zero response

The fixed-family comparison (H3)–(H7) leaves continuous scale mixtures and paid approximations available. The following application uses the same original coefficient (M1), the already assessed Gaussian coefficient of Moriya’s Theorem 3.3, and the classical Riemann–von Mangoldt count. It supplies a uniform coefficient-error estimate, rather than repeating any source proof or claiming an original analytic theorem or Lean verification. No nontrivial zero is moved to the critical line.

Fix $A \geq e^{2}$ , $L = lo g A$ and $0 < a \leq 1/16$ . Retain $w (X) = (1 + lo g X) / lo g^{2} X$ . At observation $z = 0$ choose, separately for each $X \geq A$ ,

$α_{X} = a / lo g X, G_{X, α_{X}, 0} (s) = \frac{X ^{s} e ^{a s^{2}}}{s} .$

The positive scale measure $w (X) X^{- 2} d X$ gives the exact identity

$Q_{A, a} (s) := \int_{A}^{\infty} G_{X, α_{X}, 0} (s) \frac{w ( X )}{X ^{2}} d X = e^{a s^{2}} F_{A} (s), 0 < Re s < 1. (U1)$

For every such $s$ , the scale integral is absolutely convergent. The width parameter varies with $X$ , while the Gaussian coefficient $a$ is fixed throughout the integral. This realizes the damped coefficient exactly; it does not claim exact reconstruction of $F_{A}$ by a finite family of fixed probes.

A bound valid before taking the infinite scale endpoint

For $R \geq A$ put

$F_{A, R} (s) = \frac{1}{s} \int_{A}^{R} X^{s - 2} w (X) d X .$

Since $w$ is positive and decreasing, one integration by parts gives, for $σ = Re s < 1$ and $s \neq = 0$ ,

$∣ F_{A, R} (s) ∣ \leq \frac{2 w ( A ) A ^{σ - 1}}{∣ s ∣∣1 - s ∣} . (U2)$

Indeed the two endpoint terms and the derivative integral are bounded by $A^{σ - 1} [w (A) + w (R) + \int_{A}^{R} ∣ w^{'} (X) ∣ d X]$ , which is $2 A^{σ - 1} w (A)$ . The bound is uniform in $R$ . Passing to the infinite endpoint gives the same bound for $F_{A}$ . In particular, for an actual zero $ρ = β + iγ$ with $γ \geq 4$ ,

$∣ F_{A, R} (ρ) ∣, ∣ F_{A} (ρ) ∣ \leq \frac{2 w ( A )}{γ ^{2}} .$

This is the same reciprocal-square coefficient control used in the existing Fiori density application, now also applied to finite scale intervals. It will justify the scale-endpoint passage below; it is not a new zero-density input.

Pay the discrepancy over all actual zeros

With multiplicities $m_{ρ}$ retained, uniformly in $A$ and $a$ ,

$Im ρ > 0 \sum m_{ρ} ∣ F_{A} (ρ) - Q_{A, a} (ρ) ∣ ≪ w (A) a lo g (2/ a) . (U3)$

To prove this, set $U = a^{- 1/2} \geq 4$ . For $4 \leq γ \leq U$ ,

$∣ e^{a ρ^{2}} - 1∣ = a ρ^{2} \int_{0}^{1} e^{t a ρ^{2}} d t \leq a e^{a} ∣ ρ ∣^{2} ≪ a γ^{2},$

because $0 < β < 1$ . The preceding coefficient bound and $N (U) ≪ U lo g (2 U)$ give a contribution $O (w (A) a U lo g (2 U))$ . For $γ > U$ use $∣ e^{a ρ^{2}} - 1∣ \leq 1 + e^{a}$ and the classical consequence $\sum_{γ > U} m_{ρ} / γ^{2} ≪ lo g (2 U) / U$ of the same zero count. Their sum has the size in (U3).

The finite multiset $0 < γ < 4$ is not omitted or assumed empty. Directly from the defining integral,

$∣ F_{A} (ρ) ∣ \leq \frac{w ( A )}{∣ ρ ∣ ( 1 - β )},$

so its discrepancy is at most $a e^{a} w (A) \sum_{0 < γ < 4} m_{ρ} ∣ ρ ∣/ (1 - β) = O (a w (A))$ . This fixed zeta-dependent constant is finite and independent of $A, a$ ; it is absorbed into (U3). No finite-height verification is needed for this step. All implied constants here are uniform but not numerically certified.

Consequently the original conjugate-paired response has allowance

$A L 2 Re γ > 0 \sum m_{ρ} (F_{A} (ρ) - Q_{A, a} (ρ)) ≪ A a lo g (2/ a) . (U4)$

The admissible choice

$a_{A} = \frac{1}{A ( lo g A ) ^{4}}$

makes (U4) $O (1/ lo g A)$ as $A \to \infty$ . This estimate covers the complete positive-ordinate multiset and infinite height tail, with actual real parts and multiplicities unchanged. It pays unsmoothing of the zero response, while giving no signed lower bound for that response. Positivity of the scale measure in (U1) is not positivity of a sum of complex zero coefficients.

Transport to a centered Gaussian prime integral with the contour paid

The coefficient approximation (U1)–(U4) can be realized on the prime side without discarding the pole, the shifted contour, or the original trivial-zero correction. Use only Moriya, arXiv:2607.04316v2, Proposition 2.2, equations (17)–(19), Theorem 3.3, equations (35)–(36), and Lemma 3.1. This is an application of those assessed interfaces; none of their explicit-formula proofs or RH-dependent localization claims is reproved or used.

Center the pole before the infinite scale integration

For $A \geq e^{2}$ and $0 < a \leq 1/16$ define

$Ψ_{a} (X) = n \geq 1 \sum Λ (n) \frac{1}{2} erfc (\frac{lo g n - lo g X}{2 a}) . (V1)$

In the cited formula take observation $s = 0$ , right line $c = 2$ , left line $- d = - 1/2$ , and $α_{X}^{2} lo g X = a$ . The observation is not a zero or the pole of $ζ$ . All nontrivial zeros and the pole at $1$ are crossed; no trivial zero is crossed. The left line avoids all zeros and the pole. The source’s prime term is $- Ψ_{a} (X)$ , and $ζ^{'} / ζ (0) = lo g (2 π)$ , so its exact formula reads

$Ψ_{a} (X) - e^{a} X = - ρ \sum m_{ρ} \frac{X ^{ρ} e ^{a ρ^{2}}}{ρ} - lo g (2 π) - R_{a} (X), (V2)$

where

$R_{a} (X) = \frac{1}{2 πi} \int_{Re z = - 1/2} \frac{ζ ^{'}}{ζ} (z) \frac{X ^{z} e ^{a z^{2}}}{z} d z .$

For each fixed $a > 0$ this contour integral and the zero sum in (V2) converge absolutely. Define the centered improper integral

$J_{a} (A) = R \to \infty lim \int_{A}^{R} [Ψ_{a} (X) - e^{a} X] \frac{w ( X )}{X ^{2}} d X . (V3)$

The prime and pole terms must be centered before taking this limit. Their separate integrals diverge, since $w (X) / X \sim 1/ (X lo g X)$ and $Ψ_{a} (X) \sim e^{a} X$ . The latter asymptotic also follows from (V2): its Gaussian zero series divided by $X$ tends to zero by absolute domination, and the fixed- $a$ contour is $O_{a} (X^{- 1/2})$ .

For finite $R$ , Gaussian decay allows exchange of the scale integral and the zero sum. Its zero coefficients are $e^{a ρ^{2}} F_{A, R} (ρ)$ . The bound (U2) and the finite low-zero bound dominate them independently of $R$ by a summable multiset. Thus dominated convergence passes $R \to \infty$ . This uses finite scale intervals first; no absolute exchange over the infinite scale range is assumed. The constant integrates exactly to $lo g (2 π) / (A lo g A)$ .

For the contour, the scale exchange over $[A, \infty)$ is absolutely convergent for each fixed $a > 0$ , using $X^{- 5/2} w (X)$ on the scale side and Gaussian decay on the vertical line. Therefore (V3) exists and

$J_{a} (A) = - 2 Re γ > 0 \sum m_{ρ} Q_{A, a} (ρ) - \frac{lo g ( 2 π )}{A lo g A} - C_{a} (A), (V4)$

$C_{a} (A) = \frac{1}{2 πi} \int_{Re z = - 1/2} \frac{ζ ^{'}}{ζ} (z) e^{a z^{2}} F_{A} (z) d z .$

Here $F_{A}$ is defined by the same integral (M1) at $Re z < 1$ , $z \neq = 0$ . No new analytic continuation assumption is needed.

Bound the contour after integrating the scale

The infinite-endpoint version of (U2) on $z = - 1/2 + i t$ gives

$∣ F_{A} (- 1/2 + i t) ∣ ≪ \frac{w ( A ) A ^{- 3/2}}{1 + t ^{2}} .$

This line has a fixed positive distance from every zero and the pole, so the source’s Lemma 3.1 gives $∣ ζ^{'} / ζ (- 1/2 + i t) ∣ ≪ lo g^{2} (3 + ∣ t ∣)$ . Moreover $∣ e^{a z^{2}} ∣ \leq e^{a /4}$ . Taking absolute values after the scale integration, rather than before it, therefore gives

$∣ C_{a} (A) ∣ ≪ w (A) A^{- 3/2} (A \geq e^{2}, 0 < a \leq 1/16), (V5)$

since $lo g^{2} (3 + ∣ t ∣) / (1 + t^{2})$ is integrable. The implied constant is independent of the shrinking width $a$ , but is not numerically certified.

The original signed formula already recorded above is

$I_{ψ} (A) = - 2 Re γ > 0 \sum m_{ρ} F_{A} (ρ) - \frac{lo g ( 2 π )}{A lo g A} + T_{triv} (A), T_{triv} (A) = - \frac{1}{2} \int_{A}^{\infty} lo g (1 - X^{- 2}) \frac{w ( X )}{X ^{2}} d X .$

In particular $0 \leq T_{triv} (A) ≪ w (A) A^{- 3}$ . Thus the pole constant matches exactly, the Gaussian shifted contour is paid by (V5), and the original trivial term is retained. Combining (U4) and (V5) yields the full prime-side approximation

$A lo g A ∣ I_{ψ} (A) - J_{a} (A) ∣ ≪ A a lo g (2/ a) + A^{- 1} . (V6)$

For $a_{A} = 1/ [A (lo g A)^{4}]$ this is $O (1/ lo g A + 1/ A) \to 0$ . The bound alone does not assert convergence at fixed $A$ as $a \to 0$ , because its $A^{- 1}$ allowance remains.

The estimate still needed at the selected Robin source

At the same conditional least integer $N > 5040$ attaining the global Robin-ratio maximum, retain $A = lo g N$ and condition (G9). The existing density restriction requires $A > 1 0^{36}$ if RH fails and its cited inputs hold. An eventual bound of the form

$A lo g A J_{a_{A}} (A) \geq - E (lo g A) + η (A)$

would imply (G9) wherever $η (A)$ pays the discrepancy in (V6). This is a sufficient condition, not an estimate established here; its validity at this selected source remains unproved. Without a numerical constant in (V6), no explicit margin or new source threshold is certified. Neither positive prime weights nor positive scale weights give a lower bound after subtracting the pole main term.

The application closes the full-coefficient approximation and its prime-side contour transport, not the signed Robin estimate. It does not discard an infinite zero tail, use RH-dependent localization, replace real parts by $1/2$ , or turn the selected global maximizer into the least counterexample. RH remains unproved; all conclusions are paper-level applications without Lean certification.

A signed heat-weight application permits a broader Gaussian window

The absolute all-zero discrepancy (U3) pays a narrow Gaussian window. The uniform actual-zero phase sum (J1) also controls a signed heat-weight discrepancy. The following application combines those already assessed inputs, (H5)’s paid coefficient approximation and the classical zero count. It does not repeat Landau’s formula or claim an original phase theorem or Lean verification.

Keep $A \geq e^{2}$ , $L = lo g A$ , $0 < a \leq 1/16$ , and $J (A) = lo g (2 A) lo g lo g (3 A)$ . All sums run over distinct actual zeros $ρ = β + iγ$ with $γ > 0$ , with each multiplicity $m_{ρ}$ included once. Define

$b_{a} (u) = \frac{1 - e ^{- a u^{2}}}{u ^{2}} = \int_{0}^{a} e^{- v u^{2}} d v (u > 0), b_{a} (0) = a .$

This is a nonnegative decreasing real function. Its total variation and first weighted variation are

$\int_{0}^{\infty} (- b_{a}^{'} (u)) d u = a, \int_{0}^{\infty} u (- b_{a}^{'} (u)) d u = \int_{0}^{\infty} b_{a} (u) d u = πa . (W1)$

For the final equality, integrate $(1 - e^{- a u^{2}}) / u^{2}$ by parts and use the classical Gaussian integral. Boundary terms vanish both at zero and at infinity. With $U = a^{- 1/2} \geq 4$ , the bounds $b_{a} (u) \leq min (a, u^{- 2})$ also give

$\int_{0}^{\infty} lo g (2 + u) (- b_{a}^{'} (u)) d u = a lo g 2 + \int_{0}^{\infty} \frac{b _{a} ( u )}{2 + u} d u \leq a lo g (2 + U) + a /2 ≪ a lo g (2/ a) . (W2)$

Apply the existing Landau sum to this variation measure

Use exactly the sum $S_{A} (u) = \sum_{0 < γ \leq u} m_{ρ} A^{ρ}$ from (J1). Its remainder $E_{A} (u) = S_{A} (u) + u Λ (A) / (2 π)$ has the bound

$∣ E_{A} (u) ∣ ≪ A [J (A) + lo g (2 + u)] + Lu (u \geq 0) .$

For $u \geq 8$ this is (J1). For $0 \leq u < 8$ the same weaker envelope follows from the fixed finite zero count, $∣ A^{ρ} ∣ \leq A$ and $0 \leq Λ (A) \leq L$ ; no low zero is assumed absent or verified. The implied constant remains independent of $A, u$ .

Stieltjes partial summation over the entire positive-ordinate multiset now yields

$γ > 0 \sum m_{ρ} A^{ρ} b_{a} (γ) = - \int_{0}^{\infty} S_{A} (u) b_{a}^{'} (u) d u = - \frac{Λ ( A ) a}{2 π} + O (A a [J (A) + lo g (2/ a)] + L a) . (W3)$

At zero $S_{A} (0) = 0$ ; at infinity the boundary term vanishes by (J1) and $b_{a} (u) \leq u^{- 2}$ . The series is absolutely convergent for fixed $A$ , by $∣ A^{ρ} ∣ \leq A$ and the reciprocal-square zero count. Equations (W1)–(W2) pay the remainder integral. This applies the existing uniform formula to a new weight; it is not a reproof of that formula. $Λ (A)$ retains its point-supported real-variable meaning, and no nonzero main term is assumed at a selected source.

Pay the coefficient and the actual-real-part phase changes

The first inequality in (H5) gives, uniformly for $γ \geq 8$ ,

$F_{A} (ρ) = \frac{w ( A )}{A} \frac{A ^{ρ}}{γ ^{2}} + O (\frac{w ( A ) A ^{β - 1}}{γ ^{3}}) .$

After multiplication by $1 - e^{- a γ^{2}}$ , the total error is $O (w (A) a lo g^{2} (2/ a))$ . Indeed, the classical count gives

$γ \geq 8 \sum m_{ρ} \frac{min ( a γ ^{2} , 1 )}{γ ^{3}} ≪ a lo g^{2} (2/ a),$

by splitting at $U$ and using the existing reciprocal-cube tail estimate. The finite multiset $0 < γ < 8$ contributes $O (a w (A))$ to this comparison, using the pointwise finite-zero bound in (U3). Its fixed constants are independent of $A, a$ ; it is retained even when $U < 8$ .

The Gaussian in (U1) contains the actual square $ρ^{2}$ , not just $- γ^{2}$ . The additional phase change is explicitly bounded by

$∣ e^{a ρ^{2}} - e^{- a γ^{2}} ∣ \leq a e^{a} (1 + 2 γ) e^{- a γ^{2}},$

since $0 < β < 1$ . Applying (U2) and the same zero count gives

$γ > 0 \sum m_{ρ} ∣ F_{A} (ρ) ∣∣ e^{a ρ^{2}} - e^{- a γ^{2}} ∣ ≪ w (A) a lo g^{2} (2/ a) . (W4)$

For large ordinates split at $U$ : below it $\sum m_{ρ} / γ ≪ lo g^{2} (2 U)$ ; above it Gaussian decay and the same count give $O (lo g (2 U))$ for the weighted reciprocal-first sum. The finite low multiset again costs $O (a w (A))$ . These are absolute error estimates for the changes, not critical-line replacements of the zeros themselves.

Consequently the full signed coefficient discrepancy obeys

$D_{A, a} := γ > 0 \sum m_{ρ} F_{A} (ρ) (1 - e^{a ρ^{2}}) = - \frac{w ( A ) Λ ( A ) a}{2 π A} + O (w (A) [a (J (A) + lo g^{2} (2/ a)) + \frac{L a}{A}]) . (W5)$

This is a uniform estimate for a complex signed sum. The actual real parts, multiplicities, finite low zeros and infinite height tail have not been discarded. Its cancellation comes from (J1), while (W4) pays the mismatch between real-ordinate damping and the actual Gaussian coefficient. No numerical value for the implied constants is asserted.

The broader window retains a vanishing full Robin transport allowance

Use the exact centered prime integral $J_{a} (A)$ in (V3), the matching pole constant in (V4), the uniform contour bound (V5), and the original trivial-zero correction already retained there. Subtracting their exact formulas gives

$I_{ψ} (A) - J_{a} (A) = - 2 Re D_{A, a} + T_{triv} (A) + C_{a} (A) .$

Therefore (W5) supplies the complete normalized transport estimate

$A L [I_{ψ} (A) - J_{a} (A)] = \frac{( 1 + 1/ L ) Λ ( A ) a}{π A} + O (A a [J (A) + lo g^{2} (2/ a)] + \frac{L a}{A} + \frac{1}{A}) . (Y1)$

The nonnegative displayed main term is favorable for a lower bound on $I_{ψ}$ in terms of $J_{a}$ . No positive main term is presumed at the selected integer, and no effective value for the error constant is certified. The pole, contour and trivial-zero contributions are those of the full exact formulas, with no further truncation.

A larger admissible Gaussian width

For $A \geq e^{2}$ choose

$a_{A}^{♯} = \frac{1}{A ( lo g A ) ^{3}} \leq \frac{1}{16} .$

Then $lo g (2/ a_{A}^{♯}) = L /2 + 3 lo g L + lo g 2 = O (L)$ and $J (A) = O (L lo g L)$ , so the adverse allowance in (Y1) is

$O (\frac{1}{L} + \frac{lo g L}{L ^{2}} + \frac{1}{A ^{3/4} L} + \frac{1}{A}) = O (1/ L) . (Y2)$

The displayed main term is also $O (A^{- 3/4} / L)$ , because $Λ (A) \leq L$ . Thus the full normalized absolute discrepancy is $O (1/ lo g A)$ at this larger width as well.

Compared with $a_{A} = 1/ [A (lo g A)^{4}]$ in (V6), $a_{A}^{♯} / a_{A} = A lo g A \to \infty$ . The reciprocal Gaussian height scale changes from $A (lo g A)^{2}$ to $A^{1/4} (lo g A)^{3/2}$ . This specifies damping of the full response, not a cutoff that permits any zero to be omitted.

At the larger width, the earlier generic absolute allowance $A a lo g (2/ a)$ grows like $A^{1/4} / lo g A$ . Equation (Y1) instead uses the existing signed Landau cancellation and the paid actual-real-part phase error to obtain (Y2). This compares the two guaranteed allowances; it does not claim the earlier bound is attained by the actual discrepancy.

The unchanged selected-source obligation

At the same conditional least integer $N > 5040$ attaining the global Robin-ratio maximum, set $A = lo g N$ and retain the existing condition (G9), its complete infinite integral and strict core. The existing restriction $A > 1 0^{36}$ under RH failure and the cited inputs remains unchanged. A lower bound

$A lo g A J_{a_{A}^{♯}} (A) \geq - E (lo g A) + η (A)$

would suffice wherever $η (A)$ pays the adverse allowance in (Y1). Such a signed lower bound is not proved here. The new estimate allows a broader Gaussian window with a vanishing transport cost; it does not certify that cost numerically at $A > 1 0^{36}$ , control the remaining signed response, verify a new zero height, or turn the selected global maximizer into the least counterexample. All inputs are reused paper-level interfaces without Lean certification; RH remains unproved.

A smaller moving cut from joint density and zero-free support

The complete coefficient (M1) can use a smaller moving height than (J6), by combining two existing inputs on the same actual zero multiset. This controls the infinite suffix and reduces the height of the still-uncontrolled signed head. It does not establish that head’s sign or the original selected-source Robin condition.

Corollary 1 and Table 1, printed p.2, of Chourasiya–Simonič, arXiv:2507.15184v2 also extend the existing count (W1) in the Polak application to $5/8 \leq σ < 1$ , with the same larger constants $47, 10, 168$ . The additional rows have $B_{1} \leq 13.66$ , $B_{2} \leq 8.290$ , $B_{3} \leq 147.0$ . Lemma 2.10, printed p.6, of Johnston–Yang, arXiv:2204.01980v2 supplies Ford’s zero-free region

$β < 1 - η (γ), η (t) = \frac{1}{R ( lo g t ) ^{2/3} ( lo g lo g t ) ^{1/3}}, R = 57.54, γ \geq 3.$

Their analytic proofs and finite verified-height input remain external premises. The constant is sufficient for this interface; no optimal-constant claim or source proof rerun is needed.

Put $L = lo g A$ , $Ω = (L / lo g L)^{1/3}$ and

$κ = \frac{1}{28 R}, T = A^{1/4} e^{- κ Ω}, U = A^{2/7} .$

Work for sufficiently large $A$ so $T \geq H = 3 \cdot 1 0^{12}$ . Then $U > T$ , $T \to \infty$ , and $T / (A^{1/4} L) \to 0$ . No numerical starting clock is certified here. Define

$B_{2} (A; T) = γ > T \sum \frac{m _{ρ} ( A ^{β - 1/2} + A ^{1/2 - β} )}{γ ^{2}} .$

The full coefficient estimate (H5), including its remainder, and multiplicity-preserving reflection give

$A L 2 γ > T \sum m_{ρ} ∣ F_{A} (ρ) ∣ \leq \frac{4}{3} (1 + 1/ L) B_{2} (A; T) .$

Thus an absolute bound on $B_{2}$ pays the original signed suffix. The original head $γ \leq T$ , pole and trivial-zero terms remain.

Use the existing layer calculation with $σ_{0} = 5/8$ and $Q_{A} (σ) = A^{σ - 1/2} + A^{1/2 - σ}$ . For the inclusive count and exclusive suffix, the reflected count has the endpoint term

$F_{2} (σ; T) = - \frac{2 N ( σ , T )}{T ^{2}} + 4 \int_{T}^{\infty} \frac{N ( σ , t )}{t ^{3}} d t \leq 4 \int_{T}^{\infty} \frac{N ( σ , t )}{t ^{3}} d t .$

Consequently,

$B_{2} (A; T) \leq Q_{A} (σ_{0}) S_{2} (T) + 4 \int_{σ_{0}}^{1} Q_{A}^{'} (σ) \int_{T}^{\infty} N (σ, t) t^{- 3} d t d σ, S_{2} (T) = γ > T \sum \frac{m _{ρ}}{γ ^{2}} .$

Tonelli preserves all heights, real parts and multiplicities. Write $ϵ = 1 - σ$ . The existing density exponent satisfies $d = 3 ϵ / (1 + ϵ) \leq 3 ϵ$ , $d \leq 9/11$ and $q = 2 + d < 3$ . Uniformly on this range, $N (σ, t) ≪ t^{d} (lo g t)^{3}$ and $Q_{A}^{'} (σ) \leq L A e^{- ϵ L}$ .

The classical full zero count gives $S_{2} (T) ≪ lo g T / T$ . The base layer therefore costs

$Q_{A} (σ_{0}) S_{2} (T) ≪ L A^{- 1/8} e^{κ Ω} .$

For $T < t \leq U$ , finite verification and monotonicity of the zero-free region jointly imply $N (σ, t) = 0$ when $σ \geq 1 - η (U)$ . On this same band,

$A^{- ϵ} t^{d} \leq e^{- ϵ L /7}, η (U) \geq \frac{1}{R L ^{2/3} ( lo g L ) ^{1/3}}, \int_{η (U)}^{3/8} e^{- ϵ L /7} d ϵ \leq \frac{7}{L} e^{- η (U) L /7} .$

If $η (U) > 3/8$ the band is empty. Otherwise these bounds, $lo g t \leq L$ and $\int_{T}^{U} t^{- 3} d t \leq 1/ (2 T^{2})$ give the whole finite-band allowance $O (L^{3} e^{- Ω/ (14 R)})$ .

For $t > U$ , use the density bound without imposing the fixed- $U$ zero-free restriction. Uniformly for $0 \leq d \leq 9/11$ , $\int_{U}^{\infty} t^{d - 3} (lo g t)^{3} d t ≪ U^{d - 2} (lo g U)^{3}$ . Also $A^{- ϵ} U^{d} \leq e^{- ϵ L /7}$ , whose layer integral cancels the outer factor $L$ . The complete infinite band therefore costs $O (L^{3} A / U^{2}) = O (L^{3} A^{- 1/14})$ .

Combining the three allowances yields

$A L 2 Re γ > T \sum m_{ρ} F_{A} (ρ) ≪ L A^{- 1/8} e^{κ Ω} + L^{3} e^{- Ω/ (14 R)} + L^{3} A^{- 1/14} ⟶ 0.$

The absolute-sum bound above satisfies the same estimate. Every implied constant is independent of $A$ ; no ordinate tail or coefficient correction is dropped. The limit follows from $Ω/ lo g L \to \infty$ and $Ω/ L \to 0$ . Using the old $3/4$ base clip would instead leave $O (L e^{κ Ω})$ , so it cannot support this vanishing claim.

This is an asymptotic application of established density and zero-free results. The same selected integer, its original clock, strict core and complete signed target are retained. The actual finite head still grows without bound and remains uncontrolled; neither a new finite source-clock exclusion nor RH is proved. There is no Lean verification of this application.

Pay coefficient error below the whole-response cut

The full-response suffix estimate above and the existing coefficient comparisons (H1), (H5) can be used together at two different heights. Keep the same selected source $A = lo g N > 1 0^{36}$ , every actual zero real part and multiplicity, and the original complete coefficient (M1). This is a paper-level application of the already cited density and zero-free inputs, without a new source theorem, numerical starting clock, originality claim or Lean verification.

Put $L = lo g A$ , $Ω = (L / lo g L)^{1/3}$ , $R = 57.54$ and $κ = 1/ (28 R)$ as above. Define

$S = A^{1/6} e^{- κ Ω}, T = A^{1/4} e^{- κ Ω}, U = A^{1/5} .$

Work for sufficiently large $A$ with $S \geq H = 3 \cdot 1 0^{12}$ . Then $S < U < T$ eventually. The cut $S$ concerns coefficient error; $T$ remains the previously paid cut for the full original response. Both cuts tend to infinity, so fixed unverified zero heights remain in the exact lower head.

For an actual zero $ρ = β + iγ$ with $γ > 0$ , set

$G_{A} (ρ) = \frac{w ( A ) A ^{ρ - 1}}{γ ^{2}}, B_{3} (A, S) = γ > S \sum m_{ρ} \frac{A ^{β - 1/2} + A ^{1/2 - β}}{γ ^{3}}, w (A) = \frac{1 + L}{L ^{2}} .$

Reflection of this same multiplicity-weighted zero multiset gives $B_{3} (A, S) = 2 \sum_{γ > S} m_{ρ} A^{β - 1/2} / γ^{3}$ . The first inequality of (H5) therefore pays the full approximation error by

$A L 2 γ > S \sum m_{ρ} ∣ F_{A} (ρ) - G_{A} (ρ) ∣ \leq (1 + 1/ L) (\frac{7}{3} + \frac{1}{4 S}) B_{3} (A, S) . (RC1)$

The actual weight $A^{β - 1/2}$ and phase $e^{iγ L}$ remain in $G_{A}$ . If a transport instead uses $P_{A} (ρ) = w (A) A^{ρ - 1} / [ρ (1 - ρ)]$ , (H1) gives the supplementary bound

$A L 2 γ > S \sum m_{ρ} ∣ F_{A} (ρ) - P_{A} (ρ) ∣ \leq \frac{2 ( L + 2 )}{L ^{2}} B_{3} (A, S) . (RC2)$

This is an alternative coefficient comparison; it is not an additional error to add to (RC1).

The same layer estimate with a reciprocal cube

Reuse the real-part layer identity with $σ_{0} = 5/8$ and $Q_{A} (σ) = A^{σ - 1/2} + A^{1/2 - σ}$ . Replacing the reciprocal square by a reciprocal cube changes the reflected partial-summation formula to

$F_{3} (σ; S) = - \frac{2 N ( σ , S )}{S ^{3}} + 6 \int_{S}^{\infty} N (σ, t) t^{- 4} d t .$

The endpoint term is nonpositive. Dropping it only for an upper bound, with the same inclusive head and exclusive suffix convention, gives

$B_{3} (A, S) \leq Q_{A} (σ_{0}) S_{3} (S) + 6 \int_{σ_{0}}^{1} Q_{A}^{'} (σ) \int_{S}^{\infty} N (σ, t) t^{- 4} d t d σ, S_{3} (S) = γ > S \sum \frac{m _{ρ}}{γ ^{3}} ≪ \frac{lo g S}{S ^{2}} . (RC3)$

The already retained density bound on $5/8 \leq σ < 1$ is $N (σ, t) \leq C_{0} t^{d (σ)} (lo g t)^{3}$ for $t \geq H$ , where, with $ε = 1 - σ$ , $d = 3 ε / (1 + ε) \leq 3 ε$ and $d \leq 9/11$ . All constants below are independent of $A$ and $σ$ .

The base layer in (RC3) is $O (L A^{- 5/24} e^{2 κ Ω})$ . On $S < t \leq U$ , use the same actual-zero support $N (σ, t) = 0$ when $σ \geq 1 - η (U)$ . Here

$A^{- ε} U^{d} \leq e^{- 2 ε L /5}, η (U) \geq \frac{Ω}{R L}, \int_{η (U)}^{3/8} e^{- 2 ε L /5} d ε \leq \frac{5}{2 L} e^{- 2 η (U) L /5},$

when $η (U) \leq 3/8$ ; otherwise this count band is empty. Using $lo g t \leq L$ and $\int_{S}^{U} t^{- 4} d t \leq 1/ (3 S^{3})$ , its allowance is $O (L^{3} e^{- 41Ω/ (140 R)})$ .

Above $U$ , no fixed- $U$ zero-free support is imposed. Since $3 - d \geq 24/11$ , the complete height integral is bounded uniformly by $C_{1} U^{d - 3} (lo g U)^{3}$ . Integrating $e^{- 2 ε L /5}$ over $ε \geq 0$ cancels the outer $L$ ; this entire infinite band costs $O (L^{3} A^{- 1/10})$ . Thus

$B_{3} (A, S) ≪ L A^{- 5/24} e^{2 κ Ω} + L^{3} e^{- 41Ω/ (140 R)} + L^{3} A^{- 1/10} ⟶ 0. (RC4)$

All three terms vanish because $Ω/ lo g L \to \infty$ . Equations (RC1) and (RC2) pay complete absolute coefficient errors above $S$ , rather than just at the larger whole-response cut $T$ .

Retain the exact lower head and the oscillatory middle

Let $Z_{\rm orig}(A)=2\operatorname{Re}\sum_{\gamma>0} m_\rho F_A(\rho)$ denote the original paired zero sum, and set

$Z_{split} (A) = 2 Re 0 < γ \leq S \sum m_{ρ} F_{A} (ρ) + S < γ \leq T \sum m_{ρ} G_{A} (ρ) .$

Its contribution to $I_{ψ}$ is $- Z_{orig}$ ; the approximation there uses $- Z_{split}$ with the same sign. All elementary terms remain unchanged. Combining (RC1) with the already retained complete absolute suffix estimate gives

$A L ∣ Z_{orig} (A) - Z_{split} (A) ∣ \leq (1 + 1/ L) (\frac{7}{3} + \frac{1}{4 S}) B_{3} (A, S) + A L 2 γ > T \sum m_{ρ} ∣ F_{A} (ρ) ∣ ⟶ 0. (RC5)$

This restores every coefficient error and every infinite height. It reduces the range requiring exact-coefficient treatment while preserving the actual signed oscillatory middle. Its sign, the exact lower head, the original full sufficient Robin bound and RH remain unproved. This asymptotic reduction does not certify a finite numerical clock exclusion or a uniform safe margin.

Localized complex Mellin transport of the moving middle

Reuse (RC2), (RC4) and the complete original suffix above $T$ ; their density and zero-free-region proofs are not repeated. The following application of Mellin inversion keeps the actual real parts and multiplicities. It supplies a finite-frequency interface to the unconditional Theorem 2 of Garunkštis–Sourmelidis–Steuding, version 1, without repeating that theorem’s proof or using its RH-conditional corollary. This is a paper-level transport, with no originality or Lean-verification claim.

Keep $A, L, S, T, B_{3}$ as in (RC1)–(RC5), and assume $A$ is sufficiently large that $S \geq H$ , $S \geq L \geq 1$ and $L / lo g S \leq 7$ . These are eventual conditions, not a certified numerical starting threshold. Partition $(S, T]$ into disjoint bands $(V, W]$ , where $V = 2^{j} S$ and $W = min (2 V, T)$ . A zero at a shared endpoint belongs to exactly one band. Every sum below retains $m_{ρ}$ .

Localize the kernel and pay the compensating weight

For $ρ = β + iγ$ put

$r = i (1/2 - ρ) = γ + i δ, δ = 1/2 - β, f_{L} (r) = \frac{e ^{i L r}}{r ^{2} + 1/4} .$

The rational coefficient from (RC2) satisfies the exact identity $A L P_{A} (ρ) = (1 + 1/ L) f_{L} (r)$ . For each band set $c_{loc} = 3/2$ , $a_{V} = L / V^{2}$ and

$f_{V} (r) = e^{- a_{V} (r - c_{loc} V)^{2}} f_{L} (r), h_{V} (v) = \frac{e ^{a_{V} (v - c_{loc} V)^{2}}}{v}, J_{V} (τ) = V < γ \leq W \sum m_{ρ} h_{V} (γ) r^{- i τ} . (LM1)$

All complex powers use the principal logarithm in $Re r > 0$ . The kernel reconstructed with this height weight is

$h_{V} (γ) r f_{V} (r) = \frac{r}{γ} e^{E_{ρ}} f_{L} (r), E_{ρ} = a_{V} δ^{2} - 2 i a_{V} δ (γ - c_{loc} V) .$

Since $∣ δ ∣ \leq 1/2$ and $∣ γ - c_{loc} V ∣ \leq V /2$ , $∣ E_{ρ} ∣ \leq L / (4 V^{2}) + L / (2 V) \leq 3 L / (4 V) \leq 1$ . Using $∣ e^{z} - 1∣ \leq e ∣ z ∣$ on $∣ z ∣ \leq 1$ and $e < 3$ gives

$\frac{r}{γ} e^{E_{ρ}} - 1 \leq \frac{4 L}{V} .$

Also $∣ r^{2} + 1/4∣ = ∣ ρ (1 - ρ) ∣ \geq γ^{2}$ and $∣ e^{i L r} ∣ = A^{β - 1/2}$ . Thus $γ \leq 2 V$ and reflection in $β = 1/2$ give the complete paired approximation allowance

$2 (1 + 1/ L) bands \sum V < γ \leq W \sum m_{ρ} ∣ h_{V} (γ) r f_{V} (r) - f_{L} (r) ∣ \leq 8 (1 + 1/ L) L B_{3} (A, S) ⟶ 0. (LM2)$

The final limit uses the explicit terms of (RC4), which remain vanishing after multiplication by $L$ . This error pays both the local Gaussian compensation and the outer $γ$ versus $r$ weight. No replacement $β = 1/2$ is made.

Absolute inversion on the actual complex zeros

Define the real-axis Mellin transform

$M_{V} (τ) = \int_{0}^{\infty} e^{- a_{V} (u - c_{loc} V)^{2}} \frac{e ^{i Lu}}{u ^{2} + 1/4} u^{i τ} d u .$

In the logarithmic coordinate the profile is $g_{V} (z) = e^{z} f_{V} (e^{z})$ . Shift its Fourier contour to $Im z = \pm α_{V}$ , where $α_{V} = 1/ V$ . There are no poles in this strip and $cos (2/ V) \geq 1/2$ . On either boundary line the real part of the exponential exponent is at most

$- a_{V} cos (2/ V) u^{2} + b u - a_{V} c_{loc}^{2} V^{2}, b \leq 2 a_{V} c_{loc} V + L / V = 4 L / V .$

Completing the square bounds it by $8 L - 9 L /4 = 23 L /4 < 6 L$ . The denominator satisfies $∣ u^{2} e^{\pm 2 i / V} + 1/4∣ \geq \frac{1}{2} (u^{2} + 1/4)$ ; the integral of $(u^{2} + 1/4)^{- 1}$ on $(0, \infty)$ is $π$ . Consequently

$∣ M_{V} (τ) ∣ \leq 2 π e^{6 L} e^{- ∣ τ ∣/ V} . (LM3)$

The vertical contour sides vanish: near zero the logarithmic profile is $O (e^{Re z})$ , and at infinity the positive $cos (2/ V)$ gives quadratic damping. Fourier inversion and analytic continuation therefore give

$f_{V} (r) = \frac{1}{2 π r} \int_{R} M_{V} (τ) r^{- i τ} d τ (∣ ar g r ∣ < 1/ V) .$

For every actual zero in the band, $∣ ar g r ∣ \leq 1/ (2 γ) \leq 1/ (2 V)$ , so this inverse converges absolutely. Define

$Q_{V} = \frac{1 + 1/ L}{π} Re \int_{R} M_{V} (τ) J_{V} (τ) d τ .$

Finite summation with multiplicity commutes with the absolute integral, and exactly $Q_V=2(1+1/L)\operatorname{Re}\sum_{V<\gamma\le W} m_\rho h_V(\gamma)r f_V(r)$.

Pay the entire frequency tail within a fixed source range

Let $n_{V} = \sum_{V < γ \leq W} m_{ρ}$ and let $Q_{V, K}$ use only $∣ τ ∣ \leq K$ . Since

$h_{V} (γ) \leq e^{L /4} / V, ∣ J_{V} (τ) ∣ \leq \frac{n _{V} e ^{L /4}}{V} e^{∣ τ ∣/ (2 V)},$

(LM3) pays the complete two-sided omitted frequency integral:

$∣ Q_{V} - Q_{V, K} ∣ \leq 8 (1 + 1/ L) n_{V} e^{25 L /4} e^{- K / (2 V)} .$

Choose $K_{V} = 20 V L$ . Then this allowance is $8 (1 + 1/ L) n_{V} A^{- 15/4}$ . The bands are disjoint, so $\sum n_{V} \leq N_{0} (T) ≪ T lo g T$ . Using the retained $T = A^{1/4} e^{- κ Ω}$ gives

$bands \sum ∣ Q_{V} - Q_{V, K_{V}} ∣ ≪ L A^{- 7/2} ⟶ 0. (LM4)$

Moreover $K_{V} / (V lo g V) = 20 L / lo g V \leq 140$ under the stated eventual conditions. The positive frequencies lie within the fixed $O (V lo g V)$ range of GSS Theorem 2; its implied constants must be those for this prescribed fixed range. Reflection preserves $h_{V} (γ)$ and sends $r$ to $\overline{r}$ , hence $J_{V} (- τ) = \overline{J_{V} (τ)}$ . This accesses negative frequencies without assuming RH. The frequency-range statement alone supplies no signed bound.

Height weights, endpoint atoms and the remaining source error

For $V \leq v \leq W$ let $I_{V} (v, τ) = \sum_{V < γ \leq v} m_{ρ} r^{- i τ}$ . Ordinary finite partial summation yields

$J_{V} (τ) = h_{V} (W) I_{V} (W, τ) - \int_{V}^{W} h_{V}^{'} (v) I_{V} (v, τ) d v, ∣ h_{V}^{'} (v) ∣ \leq \frac{e ^{L /4} ( L + 1 )}{V ^{2}} .$

The source uses $[V, v)$ rather than $(V, v]$ . Define the exact atom $E_{z} (τ) = \sum_{γ = z} m_{ρ} r^{- i τ}$ . Then

$I_{V} (v, τ) = I_{GSS} (V, v, τ) + E_{v} (τ) - E_{V} (τ) .$

Keep these atoms in the endpoint term and in the partial-summation integral. In that integral $E_{v}$ is supported on finitely many ordinates and contributes zero as a Lebesgue integral; the lower $E_{V}$ term and the upper $E_{W}$ term still require their exact contributions. Equivalently, the exact weighted correction to the source partial sum is $h_{V} (W) E_{W} (τ) - h_{V} (V) E_{V} (τ)$ . At $v = V$ , use the empty sum. Every other pair $(V, v)$ is comparable; the whole interval $(S, T]$ is never substituted as a comparable source interval.

Fix the independent GSS splitting parameter $c_{GSS} = 1/2$ ; its low/high-frequency transition is $τ = V lo g 2/2$ . GSS’s high-frequency remainder includes

$e^{τ / (2 V)} ((lo g V)^{2} + (τ / V)^{2} lo g V) + \frac{τ ^{3/2}}{V} e^{τ / (2 V) - τ / v} .$

Its low-frequency remainder is $O ((lo g V)^{2})$ . Retain both, as well as the main prime sum and the exact endpoint atoms. The elementary absolute envelope (LM3) combined with the displayed height-weight bounds gives, for the integrated source remainder on one band, only the coarse allowance

$O (A^{25/4} (L^{3} + V^{1/2} L)) .$

For example the first term uses $\int_0^\infty e^{-\tau/(2V)} [(\log V)^2+(\tau/V)^2\log V]d\tau=O(VL^2)$; the second uses $v \leq 2 V$ and $\int_{0}^{\infty} τ^{3/2} e^{- τ / V} d τ = O (V^{5/2})$ . The partial-summation weight has total bound $e^{L /4} (L + 2) / V$ . There are $O (L)$ bands and $\sum V = O (T)$ , so the corresponding allowance for the entire middle is $O (A^{25/4} (L^{4} + T L))$ . This growing upper allowance does not control the desired signed response. It is neither a lower bound for the actual remainder nor an impossibility result for cancellation or a sharper transport.

Restore the complete original response

Put

$Y (A) = A L 2 Re 0 < γ \leq S \sum m_{ρ} F_{A} (ρ) + bands \sum Q_{V, K_{V}} .$

(RC2), (LM2), (LM4) and the retained complete original suffix give

$∣ A L Z_{orig} (A) - Y (A) ∣ \leq (\frac{2 ( L + 2 )}{L ^{2}} + 8 (1 + 1/ L) L) B_{3} (A, S) + A L 2 γ > T \sum m_{ρ} ∣ F_{A} (ρ) ∣ + O (L A^{- 7/2}) ⟶ 0. (LM5)$

Every actual real part, multiplicity, coefficient and height is retained or covered by an explicit error. The contribution to $I_{ψ}$ is $- Z_{orig}$ , so its normalized approximation is $- Y (A)$ ; all elementary terms and the strict core remain unchanged. The exact low head, the signed transformed middle and its integrated GSS prime and error contributions are still unpaid. This interface does not prove the full signed Robin estimate, a numerical threshold, a uniform strict margin or RH.

A norm floor for the local GSS absolute endpoint allowance

The local inversion permits a sharper method diagnosis than the growing sufficient allowance displayed above. Retain exactly the kernel, height weight and cutoff in (LM1)–(LM5), on the same bands $S \leq V < W \leq min (2 V, T)$ , with the same eventual conditions. Put $τ_{0} = V lo g 2/2$ , the fixed GSS splitting point. As in (R5)–(R7), distinguish a positive majorant allowance from the actual signed source error. The following is a paper-level application of the existing inverse, with no originality or Lean-verification claim.

The actual transform has an in-range weighted norm floor

Use the analytic test point $r_{*} = c_{loc} V - i /2$ , with $c_{loc} = 3/2$ . It is not an asserted zeta zero. Since $∣ ar g r_{*} ∣ \leq 1/ (3 V)$ , the absolute inverse from (LM3) gives

$\int_{R} ∣ M_{V} (τ) ∣ e^{∣ τ ∣/ (2 V)} d τ \geq 2 π ∣ r_{*} f_{V} (r_{*}) ∣ \geq \frac{2 π A}{3 V} . (LM6)$

For the last inequality write $x = c_{loc} V \geq 1$ . The Gaussian factor has modulus $e^{a_{V} /4} \geq 1$ and $∣ e^{i L r_{*}} ∣ = A$ . Moreover

$\frac{∣ r _{*} ∣}{∣ r _{*}^{2} + 1/4∣} = \frac{x ^{2} + 1/4}{x x ^{2} + 1} \geq \frac{1}{2 x} = \frac{1}{3 V} .$

This uses the chosen analytic kernel at a test point, without replacing any actual zero’s real part.

The floor cannot be assigned only to frequencies outside GSS’s retained range. In the defining real-axis integral for $M_{V}$ , split at $V /2$ . On the lower part the Gaussian is at most $A^{- 1}$ ; on the upper part $(u^{2} + 1/4)^{- 1} \leq 4/ V^{2}$ and the complete real Gaussian integral is $π V / L$ . Thus, for every real $τ$ ,

$∣ M_{V} (τ) ∣ \leq \frac{π}{A} + \frac{4 π}{V L} .$

Consequently the weighted norm on $∣ τ ∣ \leq τ_{0}$ is at most $2 πV / A + 8 π / L$ . From (LM3), the entire weighted norm beyond $K_{V} = 20 V L$ is at most

$\int_{∣ τ ∣ > K_{V}} ∣ M_{V} (τ) ∣ e^{∣ τ ∣/ (2 V)} d τ \leq 8 πV e^{6 L - K_{V} / (2 V)} = 8 πV A^{- 4} .$

Subtract both parts from (LM6) to obtain

$\int_{τ_{0} < ∣ τ ∣ \leq K_{V}} ∣ M_{V} (τ) ∣ e^{∣ τ ∣/ (2 V)} d τ \geq \geq \frac{2 π A}{3 V} - \frac{2 πV}{A} - \frac{8 π}{L} - 8 πV A^{- 4} \frac{π A}{3 V} eventually, uniformly for S \leq V \leq T . (LM7)$

Indeed $A / V \geq A^{1/4} e^{κ Ω}$ grows, whereas each subtracted bound vanishes uniformly. The source range and negative-frequency reflection are precisely those already retained in (LM4); no new source theorem or range calculation is required.

The defined endpoint budget cannot tend to zero

GSS’s high-frequency remainder includes the positive majorant shape $e^{τ / (2 V)} (lo g V)^{2}$ . Define the unit-coefficient allowance for its term-by-term absolute propagation through the weighted height endpoint by

$B_{V}^{GSS} (A) = \frac{1 + 1/ L}{π} h_{V} (W) (lo g V)^{2} \int_{τ_{0} < ∣ τ ∣ \leq K_{V}} ∣ M_{V} (τ) ∣ e^{∣ τ ∣/ (2 V)} d τ .$

Since $h_{V} (W) \geq 1/ W \geq 1/ (2 V)$ , (LM7) gives

$B_{V}^{GSS} (A) \geq \frac{A ( lo g V ) ^{2}}{6 V ^{2}} \geq \frac{L ^{2} e ^{2 κ Ω}}{294} ⟶ \infty. (LM8)$

Here $lo g V \geq lo g S \geq L /7$ and $V \leq T = A^{1/4} e^{- κ Ω}$ . Multiplying the printed majorant shape by any fixed positive source coefficient multiplies this floor by that coefficient. No numerical value for it or effective starting threshold follows from big- $O$ notation. No unbounded sequence of selected integer sources is asserted, nor a failure at a particular selected $A$ .

This is a lower bound for the explicitly defined allowance, not for the actual remainder, its integral, the total propagated error or a Robin violation. The second printed source-error shape, derivative terms and endpoint atoms retain their roles in (LM5). Because (LM8) uses the actual transform, merely sharpening its crude upper bound $e^{6 L}$ cannot make this same absolute endpoint budget tend to zero. Joint cancellation, stronger source information, another kernel or a favorable bound for the combined main and exact low head could change the budget comparison; none is excluded or supplied here. The original full signed Robin condition and RH remain unproved, with all coefficients, real parts, multiplicities and height ranges unchanged.

A shrinking strip improvement alone leaves the first-band allowance large

Retain the kernel and complete frequency range of (LM1)–(LM8). Grant, only for this comparison, a strengthened source endpoint majorant in which $e^{∣ τ ∣/ (2 V)}$ is replaced by $e^{(1/2 - δ (V)) ∣ τ ∣/ V}$ , where $δ (V) \geq 0$ and $δ (V) \to 0$ . No such strengthened GSS estimate is inferred merely from a zero-free region. Define $B_{V, δ}$ by this replacement in the same unit-coefficient absolute allowance $B_{V}^{GSS}$ .

Since the retained range has $∣ τ ∣ \leq K_{V} = 20 V L$ , direct comparison of the existing positive integrals gives

$B_{V, δ} (A) \geq e^{- 20 δ (V) L} B_{V}^{GSS} (A) \geq e^{- 20 δ (V) L} \frac{A ( lo g V ) ^{2}}{6 V ^{2}} .$

On the first band, $V = S = A^{1/6} e^{- κ Ω}$ and eventually $W = 2 S \leq T$ , the already retained $lo g S \geq L /7$ therefore yields

$B_{S, δ} (A) \geq \frac{L ^{2}}{294} exp (\frac{L}{6} + 2 κ Ω - 20 δ (S) L) ⟶ \infty.$

Here $S \to \infty$ , $δ (S) \to 0$ and $Ω = o (L)$ . For example, the existing gap $δ (V) = η (2 V)$ from the Vinogradov–Korobov input above has these properties. Thus even granting this exponential improvement, while retaining the other endpoint prefactors, cannot make the same independently absolute allowance uniformly bounded on all middle bands. Any fixed positive source coefficient preserves the conclusion.

This is a direct application of (LM8), not a new norm, density or zero-free theorem. The lower bound concerns only the defined allowance; it is not a lower bound for the actual remainder, a Robin violation or a failure at a certified numerical clock. Additional prefactor savings, joint signed cancellation, other kernels or a favorable combined head/main bound retain their roles. The original complete signed estimate and RH remain unproved.

Actual zero jumps obstruct a uniform inverse-height prefactor

The classical jump principle gives a further application boundary for the retained source interface. Reuse GSS Theorem 2, printed p.3 of arXiv:2505.14228v1, and its Riemann–von Mangoldt inputs (3)–(4), printed p.2. No source proof is repeated. This evaluates a proposed strengthening of that particular interface; it is not a mathematical originality claim or an improved Robin estimate.

With the source’s half-open convention, write

$I (V, v, τ) = V \leq γ < v \sum m_{ρ} r_{ρ}^{- i τ}, r_{ρ} = γ + i (1/2 - β),$

and retain its complete high-frequency main term

$P (V, v, τ) = \frac{e ^{i (τ + π /4)} τ ^{1/2 - i τ}}{2 π} e^{τ / v} \leq n \leq e^{τ / V} \sum \frac{Λ ( n ) ( lo g n ) ^{i τ}}{n ^{1/2} lo g n} .$

Here sums over $ρ$ list distinct zeros and attach their actual multiplicities $m_{ρ}$ . Suppose one attempted to strengthen only the first printed error shape by a factor $V^{- 1}$ , leaving the main term and second shape unchanged:

$∣ I (V, v, τ) - P (V, v, τ) ∣ \leq C [V^{- 1} e^{τ / (2 V)} ((lo g V)^{2} + (τ / V)^{2} lo g V) + \frac{τ ^{3/2}}{V} e^{τ / (2 V) - τ / v}] . (JP1)$

The proposed $C$ is fixed and uniform in $v, τ$ on the same comparable-height source range, including $21 V /20 < v < 6 V /5$ and the frequencies used below. Such a uniform strengthening cannot hold, unconditionally and even under RH.

For every sufficiently large $V$ , the cited count gives an actual zero ordinate $Γ \in (21 V /20, 23 V /20)$ . Choose

$\frac{7}{4} V lo g V < τ < \frac{7}{4} V lo g V + V$

outside the finitely many values $Γ lo g n$ in this interval. Then $P (V, v, τ)$ is locally constant as $v$ crosses $Γ$ : its lower cutoff crosses no integer there, and its upper cutoff and coefficients are independent of $v$ . The frequency lies in the printed fixed $O (V lo g V)$ regime. For $S \leq V \leq T$ it also lies below the retained $K_{V} = 20 V L$ eventually, using the existing scale relations, without changing that frequency range.

By contrast, the two one-sided values of $I$ differ by the exact atom

$J_{Γ} (τ) = ℑ ρ = Γ \sum m_{ρ} (Γ + i (1/2 - β))^{- i τ} .$

The actual functional-equation reflection $ρ \mapsto 1 - \overline{ρ}$ preserves this ordinate and multiplicity and sends $r_{ρ}$ to $\overline{r}_{ρ}$ . For a reflected noncritical pair, with the usual logarithm in the right half-plane, its contribution is

$m_{ρ} (r^{- i τ} + \overline{r}^{- i τ}) = 2 m_{ρ} cosh (τ ar g r) e^{- i τ l o g ∣ r ∣} .$

A critical-line zero is counted once, with its original multiplicity and phase $e^{- i τ l o g Γ}$ . Since $∣1/2 - β ∣ < 1/2$ , every reflected pair has phase displacement from this central phase at most

$0 \leq τ lo g (∣ r ∣/Γ) \leq \frac{τ}{8 Γ ^{2}} = O (lo g V / V) .$

Choose $k \in {0, 1, 2, 3}$ so that the central phase after multiplication by $i^{k}$ is within $π /4$ of the positive real axis. Eventually every pair and critical-line contribution is within $π /3$ , and $2 cosh (τ ar g r) \geq 2$ . Therefore

$Re (i^{k} J_{Γ} (τ)) \geq \frac{1}{2} ℑ ρ = Γ \sum m_{ρ} \geq \frac{1}{2} . (JP2)$

The four phases here are output observation directions. This projection does not assert a new intertwining between the FIB operators $M, J, C = M J$ and the actual zeta spectrum.

For $v$ on either side sufficiently close to $Γ$ , take $v \leq 6 V /5$ . The two proposed majorants in (JP1) satisfy

$V^{- 1} e^{τ / (2 V)} ((lo g V)^{2} + (τ / V)^{2} lo g V) = O (V^{- 1/8} (lo g V)^{3}),$

$\frac{τ ^{3/2}}{V} e^{τ / (2 V) - τ / v} = O (V^{- 1/12} (lo g V)^{3/2}) .$

Both tend to zero, uniformly in this neighborhood. A locally constant main and two vanishing one-sided residuals cannot produce the nonvanishing jump (JP2). More explicitly, one of the actual one-sided residual magnitudes is at least $∣ J_{Γ} ∣/2 \geq 1/4$ . This contradicts (JP1) for any fixed $C$ . The conclusion also rules out a still smaller first shape obtained by combining this factor $V^{- 1}$ with a nonnegative exponential reduction $δ (V)$ .

The preceding allowance comparison separately shows why this restricted prefactor question matters. If its first shape alone were multiplied by $V^{- p}$ , with fixed $p \geq 0$ and $δ (V) \to 0$ , its defined first-band allowance would have the lower bound

$S^{- p} B_{S, δ} (A) \geq \frac{L ^{2}}{294} exp (\frac{1 - p}{6} L + (2 + p) κ Ω - 20 δ (S) L) . (JP3)$

For every fixed $p < 1$ this diverges. Thus a uniform pure inverse-height-power repair of the first shape cannot supply the required bounded independent allowance: powers below one retain this allowance floor, while powers at least one encounter the actual-jump obstruction. This is not a sufficiency equivalence; the unchanged second shape and all other terms still matter.

The lower bound $1/4$ concerns the actual pointwise residual near the chosen zero jumps, not its signed Mellin integral or its height-weighted integral. Restricted parameter ranges, smoothing, explicit atomic main terms, joint signed cancellation and other kernels can escape this particular model. Keep the exact endpoint atoms in (LM5), the actual real parts and multiplicities, the original rational coefficient, the low head and all heights. The complete signed Robin estimate and RH remain unproved. These application calculations are paper-level and have no Lean verification or certified numerical starting clock.

Scope of the Gaussian single-zero-sum formula

Kamiya–Suzuki, An asymptotic formula for a sum involving zeros of the Riemann zeta-function, Publications de l’Institut Mathématique 76(90) (2004), 81–88, DOI 10.2298/PIM0476081K, primary text, studies

$H (u, v) = ρ \sum e^{u ρ^{2} - v ρ} .$

Theorem 1.1 is unconditional and retains actual zero real parts and multiplicities. Its $u ↓ 0$ resonance estimates at $v = \pm lo g m$ have constants depending on the fixed integer $m > 2$ ; its off-resonance estimates are uniform on fixed closed intervals contained in the positive or negative half-line and avoiding the corresponding prime-power logarithms. These intervals exclude zero. Neither statement supplies uniform control when the frequency parameter grows with $A$ . The subsequent RH specialization is an illustration, not a hypothesis of the theorem.

Lemma 2.1 gives the exact Gaussian explicit formula for every $u > 0$ and real $v$ . It is a special case of Weil’s formula, already available as an interface. Lemma 3.1 bounds its real convolution residual between zero and one; this does not bound the complete prime and archimedean contributions.

For the localized kernel above, put $a = L / V^{2}$ , $c = 3/2$ and $r = i (1/2 - ρ)$ . Its numerator has the exact parameter correspondence

$e^{- a (r - c V)^{2} + i L r} = e^{a /4 - L /2 + ia c V - a c^{2} V^{2}} e^{a ρ^{2} - v_{V} ρ}, v_{V} = a - L + 2 ia c V .$

Here $v_{V}$ is complex, while the printed lemma takes real $v$ . Analytic continuation of an identity alone supplies no uniform signed estimate in this moving parameter range. The rational denominator $r^{2} + 1/4 = ρ (1 - ρ)$ , the height restrictions and the compensating weight $h_{V} (γ)$ also remain to be transported. Thus the printed Gaussian results do not pay the signed middle, its combined prime and error contribution, or the exact low head in (LM5). They are reused within their stated scope, without excluding a future weighted application or establishing the original Robin bound or RH.

A real-parameter heat integral transports the rational high response

The exact real-parameter formula of Kamiya–Suzuki, Lemma 2.1, can also be used with a different kernel. Keep the original selected integer, $A, L, S$ and complete coefficient $F_{A}$ . Reuse (RC2)–(RC4), the reciprocal-square zero count and the printed Gaussian formula; their source proofs are not repeated. This application pays a transport error, not the sign of the resulting prime expression. It makes no mathematical originality or Lean-verification claim.

Put

$u_{*} = \frac{L}{S ^{2}}, H (u, v) = ρ \sum m_{ρ} e^{u ρ^{2} - v ρ}, H_{\leq S} (u, v) = ∣ Im ρ ∣ \leq S \sum m_{ρ} e^{u ρ^{2} - v ρ} .$

These sums list distinct nontrivial zeros, with their actual multiplicities. Both conjugate signs of the ordinate are included. The head includes every zero at the cut. Work eventually with $L \geq 2$ , $S \geq H$ and $u_{*} \leq 1/16$ ; no numerical starting clock is certified. The head below $S$ includes unverified heights above $H$ and is retained exactly.

For an actual $ρ = β + iγ$ , $Re [ρ (1 - ρ)] = γ^{2} + β (1 - β) > 0$ . Consequently the elementary scalar resolvent identity gives

$\frac{A ^{ρ - 1/2}}{ρ ( 1 - ρ )} = A^{- 1/2} \int_{0}^{\infty} e^{u ρ^{2} - (u - L) ρ} d u . (HR1)$

The Gaussian parameter $v = u - L$ is real throughout. No continuation of the printed lemma to complex $v$ is needed. Its fixed-parameter asymptotic theorem is not used in this moving range.

For $∣ γ ∣ > S$ , the modulus of the normalized integrand in (HR1) is at most $A e^{- u γ^{2}}$ . Thus summation and integration are absolutely interchangeable, since $\sum_{γ > S} m_{ρ} / γ^{2} ≪ lo g S / S$ . The entire discarded heat-time suffix, still summing every zero height, has allowance

$A^{- 1/2} \int_{u_{*}}^{\infty} ∣ γ ∣ > S \sum m_{ρ} ∣ e^{u ρ^{2} - (u - L) ρ} ∣ d u \leq 2 A e^{- u_{*} S^{2}} γ > S \sum \frac{m _{ρ}}{γ ^{2}} ≪ A^{- 1/2} \frac{lo g S}{S} ⟶ 0. (HR2)$

Here $e^{- u_{*} S^{2}} = A^{- 1}$ . This truncates heat time, not the zero height range. In particular every zero above the previously paid whole-response cut $T$ is included before this bound is applied.

Write

$G_{u} (t) = \frac{e ^{- t^{2} / (4 u)}}{4 π u}, D_{A, u_{*}} = A^{- 1/2} \int_{0}^{u_{*}} [A - n \geq 2 \sum Λ (n) G_{u} (lo g n - L + u)] d u . (HR3)$

The pole is centered at the same arithmetic clock as the primes. Indeed the continuous background is exactly

$\int_{0}^{\infty} G_{u} (lo g x - L + u) d x = A .$

Retain this difference as a combined signed expression; its definition gives no estimate for its sign or size. The integrated prime terms are not replaced by an average over other clocks.

At $v = u - L$ , the two pole terms in the literal source formula are $A + 1$ . Its first prime sum is precisely the one in (HR3). Its remaining terms contribute a vanishing allowance after multiplying by $A^{- 1/2}$ and integrating over $0 < u < u_{*}$ . To see this using only the printed interfaces, its logarithmic archimedean term becomes

$\frac{e ^{- u /4}}{2 π} \int_{R} lo g ∣1/4 + i t /2∣ e^{- u t^{2} + i t L} d t .$

Since $∣ lo g ∣1/4 + i t /2∣∣ ≪ lo g (2 + ∣ t ∣)$ , its complete absolute integral over heat time is

$O (\int_{0}^{u_{*}} u^{- 1/2} lo g (2/ u) d u) = O (u_{*} lo g (2/ u_{*})) ⟶ 0.$

The source’s Lemma 3.1 gives $0 \leq (E * G_{u}) (u - L) \leq 1$ . Its convolution term and the constant pole $1$ together cost at most $2 A^{- 1/2} u_{*}$ . The $lo g π$ term and the second prime sum have allowance

$O (A^{- 1/2} u_{*} e^{- L^{2} / (16 u_{*})}) .$

For the second sum this follows from $L - u \geq L /2$ and $(L - u) / (2 u) \geq 1$ :

$n \geq 2 \sum \frac{Λ ( n )}{n} e^{- (u - L - l o g n)^{2} / (4 u)} \leq e^{- L^{2} / (16 u)} n \geq 2 \sum \frac{lo g n}{n ^{2}},$

where the last series is finite. All these terms retain the signs and coefficients of Lemma 2.1; only their absolute propagation is bounded. Because $S = A^{1/6} e^{- κ Ω}$ , $u_{*} lo g (2/ u_{*}) = O (L^{3/2} / S) \to 0$ .

No endpoint value at $u = 0$ is substituted. For each fixed $A$ , the Gaussian zero sum is integrable on this finite heat interval: the reciprocal-square count controls its high part, and its head is finite. All other source terms just bounded are integrable there. The nonnegative first prime sum is consequently integrable by the exact source identity; Tonelli then permits its prime summation and heat integral to be interchanged. This includes the case where $A$ is exactly a prime power, without importing fixed-resonance asymptotics. Integrating the two pole terms separately to infinite heat time would diverge and is not used.

Define the exact combined head

$J_{head} (A) = A L 2 Re 0 < γ \leq S \sum m_{ρ} F_{A} (ρ) - (1 + 1/ L) A^{- 1/2} \int_{0}^{u_{*}} H_{\leq S} (u, u - L) d u . (HR4)$

The finite subtraction belongs to the same zero multiset and clock; it is not an independently optimized head allowance. Combining (HR1)–(HR4) with the single coefficient comparison (RC2) gives the complete original response

$A L Z_{orig} (A) = J_{head} (A) + (1 + 1/ L) D_{A, u_{*}} + o (1) . (HR5)$

More precisely, the absolute discrepancy is at most

$\frac{2 ( L + 2 )}{L ^{2}} B_{3} (A, S) + O (A^{- 1/2} \frac{lo g S}{S} + u_{*} lo g (2/ u_{*}) + A^{- 1/2} u_{*} + A^{- 1/2} u_{*} e^{- L^{2} / (16 u_{*})}), (HR6)$

which tends to zero by the already retained (RC4). All actual real parts, conjugate and reflected multiplicities, original coefficient errors and infinite zero heights are covered. No complex-v extension, GSS pointwise remainder or independent GSS endpoint allowance is used in this alternative transport.

The nontrivial-zero contribution to the original $I_{ψ}$ is still $- Z_{orig}$ . Paying (G9) through (HR5) requires an upper bound for its combined right side, plus the positive elementary correction $r_{A}$ defined after (G9) and an allowance for the signed discrepancy (HR6). The strict core at the same integer remains the one already proved. Neither $D_{A, u_{*}}$ nor $J_{head}$ is bounded here by the required margin. In particular the head subtraction does not establish positivity or an estimate for the unverified head. The full original Robin bound and RH remain unproved.

The heat prime response has a paid finite arithmetic window

The positive kernel in (HR3) permits an unconditional localization of its prime sum. This uses only $Λ (n) \leq lo g n$ , the Gaussian bound below and the same $A, L, S, u_{*}$ . It is an application estimate for this profile, not a new prime-distribution theorem, an originality claim or a Lean-verified result.

For $0 < U \leq 1/16$ and $d \in R$ , put

$K_{U} (d) = \int_{0}^{U} G_{u} (d + u) d u .$

Expanding the square inside the Gaussian gives

$0 \leq K_{U} (d) = e^{- d /2} \int_{0}^{U} \frac{e ^{- d^{2} / (4 u) - u /4}}{4 π u} d u \leq U / π e^{- d /2} e^{- d^{2} / (4 U)} . (HR7)$

For $A \geq e^{2}$ , $L = lo g A$ and any $D > 0$ , split the last Gaussian exponent into two equal parts. On $∣ d ∣ > D$ one part is at most $e^{- D^{2} / (8 U)}$ . Consequently

$∣ l o g (n / A) ∣ > D \sum Λ (n) K_{U} (lo g (n / A)) ≪ U e^{- D^{2} / (8 U)} n \geq 2 \sum f_{A, U} (n),$

where

$f_{A, U} (x) = lo g x (A / x)^{1/2} exp [- \frac{( lo g x - L ) ^{2}}{8 U}], x \geq 1.$

This nonnegative function vanishes at $1$ and at infinity and has just one maximum. Indeed, for $t = lo g x > 0$ , its logarithmic derivative with respect to $t$ is $1/ t - 1/2 - (t - L) / (4 U)$ , which is strictly decreasing from positive infinity to negative infinity. The elementary unimodal sum–integral comparison therefore gives

$n \geq 2 \sum f_{A, U} (n) \leq \int_{1}^{\infty} f_{A, U} (x) d x + 2 x \geq 1 sup f_{A, U} (x) .$

Write $d = lo g x - L$ . Completing the square first with $e^{- d /2}$ and then with $e^{d /2}$ gives, uniformly in the stated parameters,

$x \geq 1 sup f_{A, U} (x) \leq e^{U /2} (L + 2 U + 2 U) ≪ L + 1,$

$\int_{1}^{\infty} f_{A, U} (x) d x \leq A e^{U /2} \int_{R} (L + ∣ d ∣) e^{- (d - 2 U)^{2} / (8 U)} dd ≪ A U (L + 1) .$

These comparisons account for the integer grid as well as the continuous background. Combining them with (HR7) proves the complete positive arithmetic tail bound

$∣ l o g (n / A) ∣ > D \sum Λ (n) K_{U} (lo g (n / A)) ≪ (L + 1) (A U + U) e^{- D^{2} / (8 U)} . (HR8)$

No prime powers outside the displayed window are dropped before this bound is applied, and no prime-counting asymptotic or average over $A$ is assumed.

Now use the same $U = u_{*} = L / S^{2}$ as (HR1)–(HR6), and take

$D_{*} = 4 u_{*} L = \frac{4 L}{S}, W_{A} = {n \geq 2 : A e^{- D_{*}} \leq n \leq A e^{D_{*}}} .$

Keep both endpoints in this finite window. Define

$D_{A, u_{*}}^{loc} = A^{- 1/2} [A u_{*} - n \in W_{A} \sum Λ (n) K_{u_{*}} (lo g (n / A))] .$

Since $e^{- D_{*}^{2} / (8 u_{*})} = A^{- 2}$ , (HR8) yields

$0 \leq D_{A, u_{*}}^{loc} - D_{A, u_{*}} ≪ (L + 1) (A u_{*} + \frac{u _{*}}{A}) A^{- 2} ⟶ 0. (HR9)$

The direction is relevant to Robin: removing the outside positive prime terms increases $D$ , hence gives an upper estimate for its contribution to $Z_{orig}$ and a lower estimate for the corresponding contribution $- Z_{orig}$ to $I_{ψ}$ . The separate bound in (HR9) also pays the absolute discrepancy. Together with (HR5)–(HR6), it gives

$A L Z_{orig} (A) = J_{head} (A) + (1 + 1/ L) D_{A, u_{*}}^{loc} + o (1), (HR10)$

with no discarded zero heights or unpaid arithmetic tail.

The relative log window $D_{*} = 4 L / S$ tends to zero. Its continuous arithmetic width is asymptotic to $8 A L / S$ ; this is a window at $A = lo g N$ , not a window at the original integer $N$ . The weighted prime-power sum within it is still not estimated by the required signed margin. Nor is the exact unknown head in (HR4) paid by localization. The original selected-source joint estimate, strict core, all actual real parts and multiplicities remain unchanged, and RH remains unproved. All limits are eventual, with no certified numerical starting clock.

Whole multiplicative sums do not control prescribed prime blocks

Granville–Lamzouri, Large values of exponential sums with multiplicative coefficients, arXiv:2604.02306v1, Corollary 1.1 and Example 1.2, printed pp.5–6, give a useful one-way interface. Write $e (t) = exp (2 πi t)$ and

$S_{f} (x, α) = n \leq x \sum f (n) e (α n), f : N \to {z : ∣ z ∣ \leq 1} multiplicative .$

For fixed $ϵ \in (0, 1/10)$ , Corollary 1.1 assumes $∣ α - a / q ∣ \leq 1/ (q x)$ , $(a, q) = 1$ and $(lo g x)^{2 + ϵ} \leq q \leq x / (lo g x)^{3 + ϵ}$ . An inequality $∣ S_{f} (x, α) ∣ \geq c x / lo g x$ , with fixed $c \in (0, 1)$ , forces a dyadic prime sum with large modulus for some positive integer harmonic $h ≪ c^{- 2} lo g (1/ c)$ and a block location $c^{2} x / lo g (1/ c) ≪ z \leq x / (2 h)$ . This is an existential conclusion, not a bound for a prescribed prime block.

The source’s Example 1.2 supplies a completely multiplicative $f$ with $∣ f ∣ \leq 1$ , a cutoff $y \sim 3 x /4$ , $z = y /2$ and $δ \in {0, 1}$ such that

$S_{f} (x, α) = e (α) + δ, z < p \leq 2 z \sum f (p) e (α p) \sim \frac{x}{4 lo g x} .$

Thus multiplicativity and cancellation of the whole integer sum alone do not imply cancellation of each prime block. This printed counterexample is reused directly; it concerns the source’s selectable coefficients, not a counterexample for the actual $Λ (n) n^{- i t}$ Gaussian response.

The actual von Mangoldt weight is not multiplicative: $Λ (6) = 0$ while $Λ (2) Λ (3) > 0$ ; multiplication by a fixed nonzero scalar preserves that obstruction. Putting $n^{- i t}$ into an admissible multiplicative coefficient instead leaves the prime support, $Λ$ weight, Gaussian cutoff and signed transport to be justified. Corollary 1.1 supplies no such bridge or original-kernel estimate. The same selected integer, complete rational coefficient, actual zero real parts and multiplicities, exact low head and all remaining height ranges therefore retain their roles in (LM5) and (G9). The required joint signed estimate and RH remain unproved.

The same-source price minimum gives a nonnegative heat cost

The existing unrestricted price pressure and selected-source contact can be transported into the real heat kernel (HR7). This application identifies the direction and scale of the resulting constraint. It reuses the pressure, derivative and limit in the FIB theory volume, §98, the actual tangent optimum (G1), and (HR1)–(HR10). It does not reprove those results or the packet and cone theorems. No mathematical originality or Lean verification is claimed.

Keep the same conditional least global Robin-ratio maximizer $N > 5040$ , $A = lo g N$ , $L = lo g A$ and $U = L / S^{2}$ . Write

$g (x) = \frac{1}{x lo g x}, q (x) = - g^{'} (x) = \frac{lo g x + 1}{x ^{2} ( lo g x ) ^{2}},$

$B (ϵ) = m \geq 1 max {lo g Z (m) - ϵ lo g m}, P (x) = γ + lo g lo g x - g (x) x - B (g (x)) .$

Here $P$ is the existing $D$ , and $q$ is its positive derivative weight, not the price. Let $a_{C} (x) = lo g C_{x}$ for an unrestricted optimum at price $g (x)$ ; ties may use either consistent prefix convention. Local finiteness of activation events and the existing pressure derivative give

$P^{'} (x) = q (x) [x - a_{C} (x)] almost everywhere, P (x) ⟶ 0.$

The selected $N$ attains the optimum at $g (A)$ , so $P (A) = γ + lo g lo g A - lo g Z (N)$ . Choose a fixed $x_{0} > 1$ such that every optimum for $x \geq x_{0}$ exceeds $5040$ ; it suffices that the first thirteen $2$ -layers are strictly active. The actual global Robin maximum and the concavity tangent for $b (t) = γ + lo g lo g t$ imply

$0 \leq P (x) - P (A) \leq c_{A} (x) (x \geq x_{0}), c_{A} (x) = lo g lo g x - lo g lo g A + g (x) (A - x) . (PH1)$

For the lower inequality use the Robin comparison at this same $C_{x}$ and $b (a_{C}) \leq b (x) + g (x) (a_{C} - x)$ . For the upper inequality insert the same $N$ as a competitor in $B (g (x))$ . Concavity gives $c_{A} (x) \geq 0$ for all $x > 1$ . The global comparison does not cover optima at most $5040$ ; their fixed interval is paid below, not included in (PH1).

A positive kernel row with the correct atom

Put $ℓ = lo g x$ , $d = lo g (x / A)$ and $a (ℓ) = ℓ^{2} / (ℓ + 1)$ . The integrated Gaussian obeys, in the distributional sense on the $d$ -line,

$K_{U}^{''} (d) + K_{U}^{'} (d) = G_{U} (d + U) - δ_{0} (d) .$

Indeed, this is the heat equation integrated over $0 < u < U$ ; the initial Gaussian is a unit atom. Its one-sided derivatives are

$K_{U}^{'} (d) = {\frac{1}{2} e^{- d} erfc ((- d + U) / (2 U)), - \frac{1}{2} e^{- d} erfc ((d - U) / (2 U)), d < 0, d > 0.$

Thus the derivative jumps by $- 1$ at $0$ . Define

$h (x) = x a (ℓ) K_{U}^{'} (d), r (x) = a (ℓ) G_{U} (d + U) + a^{'} (ℓ) K_{U}^{'} (d) .$

Using one-sided values at $A$ gives the exact row identity

$d h = r (x) d x - A a (L) δ_{A}, π_{A, U} (d x) = \frac{r ( x )}{A a ( L )} d x . (PH2)$

The atom has coefficient $A a (L)$ , including the Jacobian $δ_{0} (lo g (x / A)) = A δ_{A} (x)$ . For $d < 0$ every term in $r$ is nonnegative. For $d > 0$ put $t = (d - U) / (2 U) \geq - 1/8$ , using $U \leq 1/16$ . Then

$\frac{- K _{U}^{'} ( d )}{G _{U} ( d + U )} = π U e^{t^{2}} erfc (t) \leq 5 U .$

For $t \geq 0$ use $e^{t^{2}} erfc (t) \leq 1$ ; for $- 1/8 \leq t < 0$ use $erfc (t) \leq 2$ and $e^{t^{2}} \leq e^{1/64}$ . Since $x > A$ , $L \geq 2$ and

$\frac{a ( ℓ )}{a ^{'} ( ℓ )} = \frac{ℓ ( ℓ + 1 )}{ℓ + 2} \geq \frac{3}{2} > 5 U,$

we again have $r \geq 0$ . Moreover $h (1 +) = h (\infty) = 0$ ; the Gaussian decay controls the latter endpoint. Integrating (PH2) therefore gives $\int_{1}^{\infty} r (x) d x = A a (L)$ . Consequently $π_{A, U}$ is an actual nonnegative probability measure on $(1, \infty)$ , formed from the same $A$ and kernel. It is not an average over independently selected extremal integers.

The arithmetic correction vanishes at the paid heat scale

Write $Δ (x) = ψ (x) - a_{C} (x)$ , so

$ψ (x) - x = - \frac{P ^{'} ( x )}{q ( x )} + Δ (x) .$

Only a crude uniform activation estimate is needed here. The first active prime layer differs from the cutoff $p \leq x$ only for $x - 1 < p \leq x$ : use $1/ (p + 1) < lo g (1 + 1/ p) < 1/ p$ in the activation rule. Every active higher layer satisfies $p^{j} lo g p < x lo g x$ , so there are $O (lo g x)$ layers and all their primes lie below $x lo g x / lo g 2$ . With the elementary $ϑ (t) \leq t lo g t$ , the higher layers of $a_{C}$ and $ψ$ give the uniform bound

$∣Δ (x) ∣ ≪ x [lo g (2 x)]^{5/2} (x > 1) .$

This estimate follows from activation, not from imposing GA1 on every intervening CA prefix. In particular it uses neither a new source-selector theorem nor a square-root PNT error. For $d \neq = 0$ , differentiating the integral defining $K_{U}$ and taking absolute values gives the almost-everywhere estimate

$e^{d /2} ∣ K_{U}^{'} (d) ∣ \leq \int_{0}^{U} (\frac{∣ d ∣}{2 u} + \frac{1}{2}) G_{u} (d) e^{- u /4} d u .$

Gaussian moments and $0 < U \leq 1/16$ therefore imply

$\int_{R} e^{d /2} (1 + ∣ d ∣)^{5/2} ∣ K_{U}^{'} (d) ∣ dd ≪ U .$

At $d = 0$ use the one-sided jump in (PH2); differentiation under the integral there does not supply either one-sided derivative.

After $x = A e^{d}$ , the complete arithmetic correction satisfies

$A^{- 1/2} \int_{1}^{\infty} Δ (x) \frac{K _{U}^{'} ( lo g ( x / A ))}{x} d x ≪ (L + 1)^{5/2} U = O (L^{3} / S) ⟶ 0. (PH3)$

All actual prime powers are present in this comparison. No fixed-beta or zero-height simplification is made.

Exact direction and the remaining upper requirement

Stieltjes integration by parts in (HR3), including the interval $0 < x \leq 1$ , yields

$D_{A, U} = A^{- 1/2} [\int_{0}^{1} K_{U} (lo g (x / A)) d x - K_{U} (- L) + \int_{1}^{\infty} (ψ (x) - x) \frac{K _{U}^{'} ( lo g ( x / A ))}{x} d x] .$

The first two terms have allowance $O (U e^{- L^{2} / (4 U)})$ by (HR7). For the pressure term, $1/ (x q (x)) = x a (ℓ)$ , so integration by parts with (PH2) gives

$- A^{- 1/2} \int_{1}^{\infty} P^{'} (x) h (x) d x = A a (L) [\int P (x) π_{A, U} (d x) - P (A)] .$

There are no pressure atoms at activation events: the existing pressure is continuous and locally absolutely continuous. Near $1$ , $B (g (x)) = 0$ and $P (x) = γ + lo g ℓ - 1/ ℓ$ , while $a (ℓ) = O (ℓ^{2})$ and $a^{'} (ℓ) = O (ℓ)$ . These facts justify the lower boundary and integrability. At infinity reuse $P (x) \to 0$ and Gaussian decay.

Set the independently nonnegative cost

$V_{A, U} = \int_{x_{0}}^{\infty} [P (x) - P (A)] π_{A, U} (d x) \geq 0.$

The omitted fixed interval $1 < x < x_{0}$ costs at most $O_{x_{0}} (U^{- 1/2} e^{- (L - l o g x_{0})^{2} / (4 U)}) = o (1)$ after multiplying by $A a (L)$ . To obtain this bound, use the preceding integrability at $1$ , boundedness of $P (A)$ from its existing limit, and the negative- $d$ derivative formula. No small-prefix Robin comparison is used. Together with (PH3), this proves the specific transport

$D_{A, U} = A a (L) V_{A, U} + o (1) . (PH4)$

Combining with (HR5)–(HR10) retains the exact same-source head:

$A L Z_{orig} = J_{head} + (1 + 1/ L) A a (L) V_{A, U} + o (1) . (PH5)$

All actual zero real parts, multiplicities and heights remain as specified in those equations. In particular the unknown head above the verified height is not paid by (PH4).

The existing support gap in (PH1) does give an upper allowance, but not one of Robin strength. Since $c_{A} (A) = 0$ and $c_{A}^{'} (x) = q (x) (x - A)$ , (PH2) and integration by parts give

$0 \leq A a (L) V_{A, U} \leq A a (L) \int_{1}^{\infty} c_{A} (x) π_{A, U} (d x) = A^{- 1/2} [\int_{1}^{\infty} K_{U} (lo g (x / A)) d x - (A - 1) K_{U} (- L)] \leq A U = L A^{1/6} e^{2 κ Ω} . (PH6)$

The final normalization uses the already fixed $S = A^{1/6} e^{- κ Ω}$ . The displayed allowance diverges. This is not a lower bound on the actual cost and does not rule out a sharper joint estimate. It is the same ceiling obtained by merely dropping positive prime terms in (HR3); the price constraint has not improved it.

Thus the price minimum supplies $D_{A, U} \geq - o (1)$ . To apply (G9), retain the elementary correction $r_{A} > 0$ defined there. Let $ε_{PH5} (A)$ be the signed discrepancy in (PH5), so $ε_{PH5} (A) = o (1)$ by (HR6), (PH3) and the fixed-prefix allowance. The exact complete response is

$- A L I_{ψ} (A) = J_{head} + (1 + 1/ L) A a (L) V_{A, U} + r_{A} + ε_{PH5} (A) .$

Consequently the remaining joint target, equivalent to (G9), is

$J_{head} + (1 + 1/ L) A a (L) V_{A, U} + r_{A} + ε_{PH5} (A) \leq E (L) .$

For any proved allowance $δ (A) \geq ∣ ε_{PH5} (A) ∣$ , a sufficient upper estimate is

$J_{head} + (1 + 1/ L) A a (L) V_{A, U} \leq E (L) - r_{A} - δ (A) .$

The absolute allowance is sufficient, not necessary. Its constants have not been numerically certified at the selected finite clock. A vanishing discrepancy alone does not establish (G9) without enough signed slack; the strict core remains the one already proved. No such bound, numerical starting clock, unbounded sequence of selected sources, or proof of RH is established here. The nonnegative quantity is independently constructed; naming it does not prove that its size fits the available Robin budget.

An unconditional primary source already contains the full response coefficient

Akatsuka, Maximal order for divisor functions and zeros of the Riemann zeta-function, arXiv:2411.19259v1, Proposition 4.4, printed p.15, supplies an unconditional explicit formula at divisor-weight exponent $1$ . Write $ν$ for the paper’s $κ$ ; this exponent is distinct from the moving-cut constant used above. This application reuses that proposition and the project’s existing finite-tail identity; it does not reprove the explicit formula or claim new mathematical content. The selected source statements, exponential integral convention and parameter correspondence were inspected, not the complete source proofs or a Lean implementation.

Let $P (X) = \sum_{2 \leq n \leq X} Λ (n) / (n lo g n)$ , using the prime-power cutoff rather than the full Euler-product logarithm. Equations (4.9) and (4.11) give

$P (X) - \frac{ψ ( X ) - X}{X lo g X} = lo g lo g X + γ + Z_{Ak} (1; X) + R_{Ak} (1; X),$

where

$Z_{Ak} (1; X) = - ρ \sum [Ei ((ρ - 1) lo g X) - \frac{X ^{ρ - 1}}{ρ lo g X}],$

$R_{Ak} (1; X) = \frac{ζ ^{'} ( 0 )}{ζ ( 0 ) X lo g X} - k \geq 1 \sum [\frac{X ^{- 2 k - 1}}{2 k lo g X} + Ei (- (1 + 2 k) lo g X)] .$

The zeros retain their actual complex values and multiplicities. The exponential integral defined immediately before Proposition 4.4 uses the cut $[0, \infty)$ and the horizontal integral from left infinity. That convention matters: adding a lateral $iπ$ constant would change the individual coefficient and its absolute-convergence statement.

For the existing $A = lo g N$ , $L = lo g A$ , write (M1) in the source’s convention:

$F_{A} (s) = \frac{A ^{s - 1}}{s L} - Ei ((s - 1) L), 0 < ℜ s < 1.$

This is the defining integral in the source’s convention: the endpoint term at infinity vanishes in this strip. Thus $Z_{Ak} (1; A) = Z_{orig}$ with the complete $F_{A}$ , without replacing it by a leading asymptotic term or moving zeros onto the critical line. Substitution into the existing finite-tail identity retains the other explicit terms as well:

$I_{ψ} (A) = - Z_{Ak} (1; A) - R_{Ak} (1; A) .$

The source’s maximal-order Theorem 1 concerns fixed $ν \in [1/2, 1)$ ; its boundedness condition is equivalent to the stated zero-free half-plane, not an unconditional upper estimate. Definition 2.1 optimizes $σ_{ν} (n) / n^{ν (1 + ε)}$ , so its half-power extremum is not identified with the selected ordinary Robin extremum. Lemma 4.5 gives $O_{ν}$ bounds only for $0 < ν < 1$ ; it supplies neither a uniform limit as $ν \to 1$ nor the missing signed bound at exponent $1$ . The formula above is an available primary representation of the original response. The required same-source upper bound on $Z_{Ak} (1; A) + R_{Ak} (1; A)$ , equivalently the original lower bound on $I_{ψ} (A)$ , remains unproved, as do the strict Robin margin and RH.

Quantitative Tauberian hypotheses for the complete response

Pierce, Turnage-Butterbaugh and Zaman, A guide to Tauberian theorems for arithmetic applications, arXiv:2504.16233v4, §2.3, Hypothesis B and Theorem B, give a quantitative result for a general Dirichlet series $D (s) = \sum_{j} a_{j} λ_{j}^{- s}$ with nonnegative coefficients. Write their parameters as $α_{T} > 0$ , $0 < δ_{T} < α_{T}$ , $k_{T} > 0$ and $m \geq 1$ for the quoted positive-growth version, to distinguish them from the FIB atoms and the moving-cut parameter. Hypothesis B permits $k_{T} = 0$ as well. Its assumptions require analytic continuation throughout $ℜ s \geq α_{T} - δ_{T}$ except for the pole of order $m$ at the single real point $α_{T}$ , the strip bound

$∣ (s - α_{T})^{m} D (s) ∣ \leq M_{1} exp (∣ s ∣^{M_{2}}), α_{T} - δ_{T} \leq ℜ s \leq α_{T},$

and, on the left boundary, a uniform bound

$∣ D (s) ∣ \leq C (1 + ∣ℑ s ∣)^{k_{T}} (lo g (3 + ∣ℑ s ∣))^{m - 1} .$

Under these hypotheses, for $X \geq 2$ ,

$λ_{j} \leq X \sum a_{j} = s = α_{T} Res \frac{D ( s ) X ^{s}}{s} + O (X^{α_{T} - δ_{T} / (k_{T} + 1)} (lo g X)^{m - 1}) .$

The implied constant depends on $α_{T}, δ_{T}, k_{T}, m, C, D (α_{T} + δ_{T})$ , and is independent of $M_{1}, M_{2}$ . Remark 9.3.1 gives the $k_{T} = 0$ version with remainder $O (X^{α_{T} - δ_{T}} (lo g X)^{m})$ . The parameters are fixed; applying the result to a changing family requires controlling this dependence. In particular, the stated saving is $δ_{T} / (k_{T} + 1)$ , rather than a saving of $δ_{T}$ from analytic continuation alone.

Theorems B.4–B.5 supply limiting examples for $k_{T} > 1/2$ : general Dirichlet series satisfying Hypothesis B can have a remainder of size $Ω (X^{α_{T} - δ_{T} / (k_{T} + 1/2)} (lo g X)^{m - 1})$ , including examples with $0 \leq a_{j} \leq 1$ . These examples do not identify the exact optimum between the two displayed exponents. The B.4 construction can use integer frequencies when $(k_{T} + 1/2) / δ_{T}$ is a positive integer. The B.5 bounded-coefficient construction uses general real frequencies without giving this integer-frequency guarantee. Neither result asserts a counterexample for the Riemann zeta function or the selected Robin integer.

For the direct prime-counting series $D (s) = - ζ^{'} (s) / ζ (s)$ , $λ_{j} = j$ , $a_{j} = Λ (j)$ and $α_{T} = m = 1$ . Each actual zeta zero in the proposed half-plane is an additional pole, so the single-pole hypothesis must be verified for all heights. A finite zero subtraction supplies no such verification for the remaining zeros. Applying Theorem B to a resulting remainder also requires a proved nonnegative-coefficient representation and the stipulated uniform growth bounds; none is supplied by this source application.

The same conditional least integer $N > 5040$ attaining the global Robin-ratio maximum remains fixed, with $A = lo g N > 1 0^{36}$ and $L = lo g A$ . The original complete $A L I_{ψ} (A)$ lower allowance, or equivalently the full zero-plus-remainder upper allowance above, has not been obtained from these hypotheses. The actual zero real parts and multiplicities, every height, all remaining explicit terms and the strict core remain present. The published theorem and examples are reused without reconstructing their proofs; this applicability check supplies neither a new prime-error estimate nor a Robin/RH proof.

A bounded-shape Gamma mixture retains an original coefficient remainder

Balanzario, Cárdenas Romero and Chacón Serna, A smooth version of Landau’s explicit formula, arXiv:2311.04347v1, Theorem 1, supplies the Gamma density

$w_{α, λ} (x) = \frac{x ^{α - 1} e ^{- x / λ}}{λ ^{α} Γ ( α )}, G_{α, λ} (s) = λ^{s - 1} \frac{Γ ( α + s - 1 )}{Γ ( α )} .$

For $α, λ > 0$ , $α \in / N$ and $μ = α λ$ , the stated formula is

$n \geq 1 \sum Λ (n) w_{α, λ} (n) = 1 - ρ \sum m_{ρ} G_{α, λ} (ρ) - R (μ, α) .$

This statement has no RH premise. The source defines $R$ by two separate series involving falling factorials and $ζ^{'} / ζ$ ; those terms remain present, with no assumed sign or uniform budget. Its Theorem 2, which gives a finite-ordinate formula near a natural-number center $μ$ , assumes RH and fixes its window parameters. That truncated formula is not an unconditional supplier at $A = lo g N$ .

The existing algebraic lower bound (H2) also specifies a restriction on exact replacement by these Gamma coefficients. Fix $A \geq e^{2}$ and a finite $B > 0$ . Let $ν_{A}$ be a locally finite signed or complex measure on

${(α, λ) : 0 < α \leq B, α \in / N, λ > 0}$

which is fixed as the zero height varies, and assume only the critical-line absolute moment

$M_{A} = \int λ^{- 1/2} d ∣ ν_{A} ∣ (α, λ) < \infty.$

This does not require finite total mass or moments at the endpoints of the critical strip. Set

$Q_{A} (1/2 + i t) = \int G_{α, λ} (1/2 + i t) d ν_{A} (α, λ), t \neq = 0.$

The standard vertical-line Stirling formula, DLMF 5.11.9, is uniform for bounded real arguments. Here $α - 1/2 \in [- 1/2, B - 1/2]$ and $1/Γ (α)$ is bounded for $0 < α \leq B$ . Consequently, for a constant $C_{B}$ and sufficiently large $∣ t ∣$ ,

$∣ Q_{A} (1/2 + i t) ∣ \leq C_{B} M_{A} ∣ t ∣^{B - 1} e^{- π ∣ t ∣/2} .$

In contrast, (H2) on this same line gives, for $∣ t ∣ \geq 4$ ,

$∣ F_{A} (1/2 + i t) ∣ \geq \frac{2 w ( A )}{3 A ( t ^{2} + 1/4 )}, w (A) = \frac{1 + lo g A}{( lo g A ) ^{2}} .$

Exponential decay is eventually smaller than this algebraic bound. Hence there is a threshold depending on $A, B, M_{A}$ such that

$∣ F_{A} (1/2 + i t) - Q_{A} (1/2 + i t) ∣ \geq \frac{w ( A )}{3 A ( t ^{2} + 1/4 )}$

above that threshold. There are unconditionally infinitely many actual critical-line zeros, as recalled in DLMF §25.10(i). Thus the mismatch occurs at actual zeros of arbitrarily large height; it is not merely an off-spectrum difference on the whole strip. Their positive multiplicities do not remove the coefficient difference.

This applies even to infinitely many scales and to complex mixture weights under the stated absolute moment. It reuses (H2), uniform Stirling and the classical critical-line zero theorem; no new version of those results or mathematical originality is claimed. Shapes and weights may depend on $A$ , but their bound is fixed at each such $A$ . The conclusion concerns exact coefficient matching at every actual zero. It does not give a sign for a paired residual or exclude equality of an aggregate signed sum, unbounded shape support, nonabsolute constructions, or approximation on a finite height range with a paid remainder. In particular the original suffix above $T$ is already paid independently; this restriction does not invalidate that transport or prohibit a useful approximation of its lower head. The complete same-selected-source signed Robin estimate and RH remain unproved. This application is not Lean verified.

Uniform control of the combined Gamma remainder

The two remainder series in Balanzario, Cárdenas Romero and Chacón Serna, arXiv:2311.04347v1, Theorem 1, need not be integrated separately. Their combined remainder has a uniform bound under the existing canonical scale measure. The following is a repo-derived paper-level application of that theorem, the classical explicit formula displayed as equation (1) in the same source, the existing coefficient estimate (H5), and unconditional PNT bounds. It does not reconstruct those source proofs or claim mathematical originality or Lean verification.

Let $A \geq e^{2}$ , $L = lo g A$ , and retain (M2):

$h_{A} (μ) = \frac{1}{max ( A , μ ) lo g max ( A , μ )}, k (u) = \frac{1 + lo g u}{u ^{2} lo g ^{2} u} .$

For noninteger $α \geq 4$ , write the source’s prime response as

$P_{α} (μ) = n \geq 1 \sum Λ (n) \frac{( α / μ ) ^{α} n ^{α - 1} e ^{- α n / μ}}{Γ ( α )} = 1 - ρ \sum m_{ρ} G_{α, μ / α} (ρ) - R (μ, α) .$

The sum indexes distinct zero locations, and $m_{ρ}$ carries their actual multiplicities. No zero is moved to the critical line. Define

$\overline{R}_{α} (A) = \int_{0}^{\infty} h_{A} (μ) R (μ, α) d μ, M_{3} (α) = \frac{α ^{3}}{( α - 1 ) ( α - 2 ) ( α - 3 )} .$

This integral exists absolutely for each such $α$ . There is an absolute constant $K$ , independent of both $A$ and $α$ , for which

$\overline{R}_{α} (A) - \frac{lo g ( 2 π )}{A lo g A} \frac{α}{α - 1} \leq \frac{K M _{3} ( α )}{A ^{3} lo g A} . (GR1)$

Since $M_{3} (α) \leq 32/3$ and $α / (α - 1) \leq 4/3$ , this is uniform even when the noninteger shape grows with $A$ or approaches an integer. In particular,

$A lo g A \overline{R}_{α} (A) = \frac{lo g ( 2 π )}{A} \frac{α}{α - 1} + O (A^{- 5/2}) ⟶ 0 (GR2)$

uniformly over the stated shapes. The constant is not numerically certified; no finite starting clock follows.

Convergence and the dilation representation

Put $D (x) = ψ (x) - x$ . Reuse an unconditional PNT bound in the global form

$∣ D (x) ∣ \leq C_{ψ} \frac{x}{lo g ^{2} ( 2 + x )} (x > 0) . (GR3)$

An absolute $C_{ψ}$ exists by PNT with a fixed logarithmic saving, with the bounded initial interval absorbed into the constant. This is not a square-root prime-error estimate. Integration by parts against the fixed-shape Gamma density, splitting its argument at $μ$ , gives $P_{α} (μ) - 1 = O_{α} ((lo g μ)^{- 2})$ at infinity. At zero, $P_{α} (μ) = O_{α} (μ^{- α} e^{- α / μ})$ . Consequently $h_{A} (1 - P_{α})$ is absolutely integrable. These convergence constants may depend on the fixed shape; they are not used as the uniform constants in (GR1).

For fixed $α$ , vertical-line Stirling gives exponential decay of the Gamma coefficient. Moreover (M2) implies, for $0 < β < 1$ ,

$F_{A} (β) \leq \frac{A ^{β - 1}}{L} (\frac{1}{β} + \frac{1}{1 - β}) .$

The already cited zero-free region and its reflection bound the two reciprocal endpoint distances by powers of logarithms of the ordinate. Together with the classical zero count this makes

$ρ \sum m_{ρ} \int_{0}^{\infty} h_{A} (μ) ∣ G_{α, μ / α} (ρ) ∣ d μ < \infty.$

The source identity therefore proves absolute convergence of the combined $\overline{R}_{α} (A)$ , without integrating either printed remainder series term by term. It also justifies

$\overline{R}_{α} (A) = J_{α} (A) - ρ \sum m_{ρ} H_{α} (ρ) F_{A} (ρ), J_{α} (A) = \int_{0}^{\infty} h_{A} (μ) (1 - P_{α} (μ)) d μ, (GR4)$

where the existing coefficient identity is $H_{α} (s) = Γ (α + s - 1) / [Γ (α) α^{s - 1}]$ . The constant pole and the prime response have been kept together.

Let $C > 0$ have Gamma shape $α$ and rate $α$ , so its mean is one. The source primitive and its negative moments are

$U_{α} (μ) = E \frac{ψ ( C μ )}{C}, U_{α}^{'} (μ) = P_{α} (μ), E C^{- 1} = \frac{α}{α - 1}, E C^{- 3} = M_{3} (α) .$

The derivative follows by differentiating the Gamma integral for each prime-power jump. Thus integration by parts gives

$J_{α} (A) = E J_{C} (A), J_{c} (A) = - \frac{1}{c} \int_{A}^{\infty} D (c u) k (u) d u . (GR5)$

Here the exchange is an absolute one. For $u \geq A$ , (GR3) and the split $C \geq u^{- 1/2}$ or $C < u^{- 1/2}$ give

$E \frac{∣ D ( C u ) ∣}{C} \leq C_{ψ} u [\frac{4}{lo g ^{2} u} + \frac{M _{3} ( α ) u ^{- 3/2}}{( lo g 2 ) ^{2}}] .$

Multiplication by $k (u)$ is integrable. The same bound pays the integration-by-parts boundary at infinity; the boundary at zero vanishes by $∣ D (x) ∣ ≪ x$ . No separate infinite-mass integral of the pole term has been taken.

A bound valid on both sides of the unit cutoff

For each $c > 0$ , define the absolutely convergent zero sum and its arithmetic remainder

$Z_{c} (A) = ρ \sum m_{ρ} c^{ρ - 1} F_{A} (ρ), r_{c} (A) = J_{c} (A) - Z_{c} (A) .$

The existing (H5) and $\sum_{ρ} m_{ρ} / γ^{2} < \infty$ give

$∣ Z_{c} (A) ∣ \leq ρ \sum m_{ρ} ∣ c^{ρ - 1} F_{A} (ρ) ∣ \leq \frac{C _{0}}{L} max {1, (c A)^{- 1}} . (GR6)$

If $c A \geq 1$ , applying the classical explicit formula to (GR5) gives

$r_{c} (A) = \frac{lo g ( 2 π )}{c A L} + \frac{1}{2} \int_{c A}^{\infty} lo g (1 - x^{- 2}) \frac{1 + lo g ( x / c )}{x ^{2} lo g ^{2} ( x / c )} d x . (GR7)$

Take finite scale endpoints first and use the existing finite-endpoint bound (U2), then dominated convergence for the zero coefficients. The prime-power half-value convention changes no integral. At $c A = 1$ the logarithmic singularity is integrable. For $1 \leq c A \leq 2$ , use

$\int_{1}^{\infty} \frac{- lo g ( 1 - x ^{- 2} )}{x ^{2}} d x = 2 - 2 lo g 2 < 1.$

For $c A \geq 2$ , use $- lo g (1 - x^{- 2}) \leq 4/ (3 x^{2})$ . Since $lo g (x / c) \geq L$ , both ranges give

$r_{c} (A) - \frac{lo g ( 2 π )}{c A L} \leq \frac{K _{1}}{( c A ) ^{3} L} (c A \geq 1) . (GR8)$

For $c A < 1$ , do not extend the explicit formula below one. There $D (x) = - x$ . Substitution $x = c u$ in (GR5) and direct integration over $[c A, 1]$ give, with $d = - lo g (c A) > 0$ ,

$\int_{c A}^{1} \frac{1 + lo g ( x / c )}{x lo g ^{2} ( x / c )} d x = lo g (1 + d / L) + \frac{1}{L} - \frac{1}{L + d} \leq \frac{d + 1}{L} \leq \frac{2}{c A L} .$

On $x \geq 1$ , (GR3) bounds the remaining absolute integral by

$\frac{C _{1}}{L} \int_{1}^{\infty} \frac{d x}{x lo g ^{2} ( 2 + x )} \leq \frac{C _{2}}{L} .$

Together with (GR6), this yields $∣ r_{c} (A) ∣ \leq C_{3} / (c A L)$ . Because $(c A)^{- 1} \leq (c A)^{- 3}$ in this range, (GR8), with a larger absolute constant, holds for every $c > 0$ . This pays the entire small-scale contribution, including $c A < 1$ .

Finally, $E C^{ρ - 1} = H_{α} (ρ)$ . Equation (GR6) and $E C^{- 1} < \infty$ justify averaging the complete zero series. Equations (GR4)–(GR5) therefore give $\overline{R}_{α} (A) = E r_{C} (A)$ . Average (GR8) and use the exact inverse-third moment to obtain (GR1).

The result bounds the combined source remainder, not its individual series. It adds no Gamma coefficient approximation or replacement for the existing complete Gaussian transport. At the same conditional least global Robin maximizer $N > 5040$ , $A = lo g N > 1 0^{36}$ , the full centered prime response and any signed comparison of $H_{α} (ρ) F_{A} (ρ)$ with the original $F_{A} (ρ)$ still require control. All actual zeros, multiplicities, heights, original pole and trivial terms, and the strict core remain present. The complete signed Robin estimate and RH remain unproved.

A directed prime-weight comparison for the complete Gamma response

The combined-remainder estimate (GR1) can be paired with a one-sided arithmetic comparison. This repo-derived application uses the same canonical measure (M2), nonnegative von Mangoldt weights, the standard unconditional Chebyshev bound, and the Gamma law already used in (GR4)–(GR5). It does not replace the original coefficient by a pointwise approximation or claim mathematical originality or Lean verification.

Keep $A \geq e^{2}$ , $L = lo g A$ and noninteger $α \geq 4$ . Let $C$ have Gamma shape $α$ and rate $α$ , and put

$h_{A} (t) = \frac{1}{max ( A , t ) lo g max ( A , t )}, w_{c} (t) = \frac{1}{c} h_{A} (t / c), Δ_{α} (t) = E w_{C} (t) - h_{A} (t) .$

The resulting complete signed comparison is

$- I_{ψ} (A) \leq J_{α} (A) + \frac{ψ ( A )}{( α - 1 ) A lo g A} . (GP1)$

Here $J_{α}$ is the centered prime response in (GR4), not the uncentered integral of its pole or prime terms. No source extremality assumption is needed for (GP1); it therefore applies at the same selected Robin integer without changing that integer.

A global tangent majorant on the infinite prime tail

Fix $t \geq A$ and write $s = t / A \geq 1$ , $v = lo g t$ . As a function of the dilation $c > 0$ , the transported prime weight is

$f_{t} (c) = w_{c} (t) = {[t (v - lo g c)]^{- 1}, (A L c)^{- 1}, 0 < c \leq s, c \geq s .$

On $0 < c < s$ ,

$f_{t}^{''} (c) = \frac{2 - ( v - lo g c )}{t c ^{2} ( v - lo g c ) ^{3}} \leq 0,$

because $v - lo g c \geq L \geq 2$ . Thus its tangent at $c = 1$ majorizes the first branch:

$f_{t} (c) \leq \frac{1}{t v} + \frac{c - 1}{t v ^{2}} (0 < c \leq s) .$

The same affine function is increasing and majorizes $f_{t} (s)$ . The second branch decreases for $c \geq s$ , so this is a global majorant, even though $f_{t}$ is not globally concave. At $s = 1$ use the left tangent and the same argument. Since $E C = 1$ , averaging gives

$Δ_{α} (t) \leq 0 (t \geq A) . (GP2)$

For $0 < t < A$ , $h_{A} (t) = 1/ (A L)$ and $h_{A} (t / c) \leq 1/ (A L)$ . The existing inverse-first Gamma moment therefore gives

$Δ_{α} (t) \leq \frac{E C ^{- 1} - 1}{A L} = \frac{1}{( α - 1 ) A L} . (GP3)$

Centering and absolute convergence of the full prime difference

The pole and prime integrals are not subtracted as separate infinite quantities. First fix $c > 0$ and set $T = A max (1, c)$ . Reuse a fixed unconditional bound $ψ (x) \leq B x$ . Below $T$ , the absolute prime-weight difference is bounded by

$n \leq T \sum Λ (n) ∣ w_{c} (n) - h_{A} (n) ∣ \leq \frac{B}{L} (1 + c^{- 1}) max (1, c) .$

For $n > T$ both weights use their logarithmic branch, and

$∣ w_{c} (n) - h_{A} (n) ∣ = \frac{∣ lo g c ∣}{n lo g n lo g ( n / c )} \leq \frac{∣ lo g c ∣ ( 1 + ∣ lo g c ∣/ L )}{n lo g ^{2} n} .$

Partial summation with the same Chebyshev bound gives $\sum_{n > T} Λ (n) / (n lo g^{2} n) \leq 2 B / L$ . Since $∣ lo g c ∣ + (lo g c)^{2} ≪ c + c^{- 1}$ , these estimates yield an absolute constant $B_{1}$ such that

$n \geq 2 \sum Λ (n) ∣ w_{c} (n) - h_{A} (n) ∣ \leq \frac{B _{1}}{L} (c + c^{- 1}) . (GP4)$

The analogous Lebesgue integral is also absolutely convergent. Changing variables in finite integrals shows

$\int_{0}^{\infty} (w_{c} (t) - h_{A} (t)) d t = 0 :$

for sufficiently large $R$ , the difference of integrals up to $R$ is $lo g lo g (R / c) - lo g lo g R$ , which tends to zero. Integration by parts against $D (t) = ψ (t) - t$ , using (GR3) at infinity and $D (t) = - t$ below one, therefore gives

$J_{1} (A) - J_{c} (A) = n \geq 2 \sum Λ (n) [w_{c} (n) - h_{A} (n)], J_{1} (A) = - I_{ψ} (A) .$

The $J_{c}$ here is the deterministic dilation in (GR5); $J_{1}$ is its value at dilation one, not a Gamma response with shape one. All boundary terms vanish. Equation (GP4), $E C = 1$ and $E C^{- 1} < \infty$ justify averaging and termwise summation. Reusing (GR5) yields the absolutely convergent identity

$- I_{ψ} (A) - J_{α} (A) = n \geq 2 \sum Λ (n) Δ_{α} (n) . (GP5)$

Apply (GP2) to the entire infinite tail, (GP3) to the finite lower part, and $Λ (n) \geq 0$ to prove (GP1). An atom at $n = A$ has the nonpositive sign in (GP2); using $ψ (A)$ in the upper allowance remains valid.

Pay the remainder and retain the original signed target

Define the complete real zero response

$Z_{α} (A) = ρ \sum m_{ρ} H_{α} (ρ) F_{A} (ρ), H_{α} (s) = \frac{Γ ( α + s - 1 )}{Γ ( α ) α ^{s - 1}} .$

Its absolute convergence and reality under conjugate pairing follow from the existing fixed-shape argument in (GR4). No zero real part, multiplicity or height is changed. Since $J_{α} = Z_{α} + \overline{R}_{α}$ , (GR1) and (GP1) give

$A L [- I_{ψ} (A)] \leq A L Z_{α} (A) + \frac{ψ ( A )}{( α - 1 ) A} + \frac{lo g ( 2 π )}{A} \frac{α}{α - 1} + \frac{K M _{3} ( α )}{A ^{5/2}} . (GP6)$

This is a directed bound for the original full response, with a paid finite prime budget and the complete combined remainder. It is not an absolute coefficient approximation, a bound on the sum of coefficient moduli, or a deletion of the original explicit-formula terms.

For shapes with $α (A) / A \to \infty$ , the added allowance in (GP6) tends to zero by the existing PNT bound. This gives a sufficient transfer from a full signed Gamma-response upper estimate to (G9), provided the estimate has enough slack to pay the displayed allowance. The signed upper estimate on $Z_{α} (A)$ is not established here. At the same conditional least global Robin maximizer $N > 5040$ , $A = lo g N > 1 0^{36}$ , the original strict core, complete infinite response and target remain unchanged. Neither (G9) nor RH is proved.

An explicit one-sided bound for the combined Gamma remainder

For finite source clocks, the absolute constant in (GR1) need not be assigned an unproved numerical value. The following upper bound instead uses the existing finite-height verification and zero-count constants in (F3) of the Polak application, together with (U2), (GR5) and (GR7). These inputs are reused without a new zero computation or reconstruction of their source proofs.

Put $H = 3 \cdot 1 0^{12}$ , and let every zero sum retain the full actual nontrivial-zero multiset, with multiplicities. Write

$c_{0} = 2 + γ - lo g (4 π) < 0.05, S_{2} (H) = ℑ ρ > H \sum \frac{m _{ρ}}{( ℑ ρ ) ^{2}} < 1.48 \cdot 1 0^{- 12} .$

The same sources supply criticality up to $H$ and the classical identity $\sum_{ρ} m_{ρ} / [ρ (1 - ρ)] = c_{0}$ . For $ρ = β + iγ$ in the critical strip,

$ℜ \frac{1}{ρ ( 1 - ρ )} = \frac{γ ^{2} + β ( 1 - β )}{∣ ρ ( 1 - ρ ) ∣ ^{2}} > 0.$

On the verified head the denominator is the positive real number $γ^{2} + 1/4$ . On the remaining zeros, $∣ ρ ∣∣1 - ρ ∣ \geq γ^{2}$ . Consequently the complete absolute weight obeys

$C_{ζ} := ρ \sum \frac{m _{ρ}}{∣ ρ ∣∣1 - ρ ∣} \leq c_{0} + 2 S_{2} (H) < 0.051. (GE1)$

The factor two accounts for the two ordinate signs, not an additional independently chosen zero population.

Keep $A \geq e^{2}$ and $L = lo g A$ . For the same deterministic dilation $r_{c} (A) = J_{c} (A) - Z_{c} (A)$ in (GR5)–(GR7), one has the uniform upper bound

$r_{c} (A) < \frac{5}{c A L} (c > 0) . (GE2)$

If $c A \geq 1$ , (GR7) immediately gives $r_{c} (A) \leq lo g (2 π) / (c A L) < 2/ (c A L)$ , since its logarithmic integral is nonpositive. This includes the integrable endpoint $c A = 1$ .

If $c A < 1$ , put $B = 1/ c > A$ and $d = lo g (B / A) > 0$ . Since $ψ (c u) - c u = - c u$ on $A < u < B$ , (GR5) gives exactly

$r_{c} (A) = lo g (1 + d / L) + \frac{1}{L} - \frac{1}{L + d} + r_{c} (B) - ρ \sum m_{ρ} c^{ρ - 1} F_{A, B} (ρ) . (GE3)$

The initial elementary term is at most $2/ (c A L)$ , as already computed in the proof of (GR8). At $c B = 1$ , (GR7) gives $r_{c} (B) < 2/ lo g B \leq 2/ (c A L)$ . The finite-endpoint bound (U2), (GE1) and $(c A)^{β - 1} \leq (c A)^{- 1}$ give

$ρ \sum m_{ρ} c^{ρ - 1} F_{A, B} (ρ) \leq \frac{2 ( 1 + 1/ L ) C _{ζ}}{c A L} \leq \frac{3 C _{ζ}}{c A L} < \frac{0.153}{c A L} .$

Thus (GE2) also holds below the unit cutoff. In particular, no classical explicit formula has been extended to $c u < 1$ and no small Gamma scales have been discarded.

For the same noninteger shapes $α \geq 4$ as in (GR1), averaging (GE2) by (GR5) and the existing inverse-first Gamma moment yields

$\overline{R}_{α} (A) \leq \frac{5 M _{1} ( α )}{A L}, M_{1} (α) = \frac{α}{α - 1} . (GE4)$

This is an explicit upper bound, not an absolute bound or a replacement for the sharper asymptotic assertion (GR1). With the original full prime-integral comparison (GP1), it gives the finite inequality

$A L [- I_{ψ} (A)] \leq A L Z_{α} (A) + \frac{ψ ( A )}{( α - 1 ) A} + \frac{5 M _{1} ( α )}{A} . (GE5)$

The full elementary and trivial contributions are paid through $\overline{R}_{α}$ ; the left side is the original complete integral. No signed main estimate or unbounded source-clock conclusion follows from (GE5) alone.

A larger finite clock from the verified-height Gamma response

The explicit directed inequality (GE5) can be paired with a uniform Gamma modulus estimate. It bounds the original signed integral on a larger finite interval, while keeping the same conditional least integer $N > 5040$ attaining the global Robin-ratio maximum. This is not a reduction to the least counterexample or an all-integer finite Robin verification. No arithmetic profiles or zero ordinates are newly enumerated.

Uniform damping with the actual real parts retained

Use the classical product NIST DLMF 5.8.3, for $x > 0$ :

$\frac{Γ ( x + i t )}{Γ ( x )} = k = 0 \prod \infty (1 + \frac{t ^{2}}{( x + k ) ^{2}})^{- 1/2} .$

The summand $lo g (1 + t^{2} / (x + u)^{2})$ is positive and decreasing in $u$ . Using $lo g (1 + v) \geq v / (1 + v)$ in its integral gives

$k = 0 \sum \infty lo g (1 + \frac{t ^{2}}{( x + k ) ^{2}}) \geq ∣ t ∣ arctan (∣ t ∣/ x) \geq \frac{t ^{2}}{x + ∣ t ∣} .$

The last inequality uses $arctan v \geq v / (1 + v)$ for $v \geq 0$ . For $0 < β < 1$ , the already used Gamma law has $E C^{β - 1} \leq (E C^{- 1})^{1 - β} \leq M_{1} (α)$ by Hölder. Set $x = α + β - 1 \leq α$ in the product. It follows that

$∣ H_{α} (β + i t) ∣ \leq M_{1} (α) exp (- \frac{t ^{2}}{2 ( α + ∣ t ∣ )}) . (GH1)$

This is uniform over the whole actual critical strip. Criticality is used only on the already verified head below $H$ .

Passing to the infinite endpoint in the existing (U2) gives

$A L ∣ F_{A} (ρ) ∣ \leq 2 (1 + 1/ L) \frac{A ^{β - 1/2}}{∣ ρ ∣∣1 - ρ ∣} . (GH2)$

The function $t^{2} / [2 (α + t)]$ increases for $t > 0$ . Split the complete $Z_{α}$ at the verified height $H$ . Use the positive $c_{0}$ identity on the critical head, and (GH1), (GH2), $β < 1$ , and $2 S_{2} (H)$ on every remaining zero. This gives

$A L ∣ Z_{α} (A) ∣ \leq 2 M_{1} (α) (1 + 1/ L) [c_{0} + 2 A S_{2} (H) exp (- \frac{H ^{2}}{2 ( α + H )})] . (GH3)$

Every height beyond $H$ is included to infinity. Replacing the actual $A^{β - 1/2}$ by $A$ in this upper bound does not assume RH outside the verified head.

Pay the constants at the same selected source

Take the exact noninteger shape and clock range

$α = 1 0^{23} + \frac{1}{2}, 1 0^{36} < A \leq 1 0^{45}, L = lo g A .$

Then $L > 78$ , $A < 3.2 \cdot 1 0^{22}$ , $M_{1} (α) < 1.0001$ , and

$\frac{H ^{2}}{2 ( α + H )} > 40, e^{40} > 1 0^{17} .$

The latter follows from $e > 8/3$ and $8^{40} > 1 0^{17} 3^{40}$ . The two contributions in (GH3) are bounded by the rational inequalities

$2 (1.0001) (1.02) (0.05) < 0.103,$

$4 (1.0001) (1.02) (3.2 \cdot 1 0^{22}) (1.48 \cdot 1 0^{- 12}) 1 0^{- 17} < 0.000003. (GH4)$

For the prime-comparison allowance, reuse Dusart’s Theorem 5.2, $k = 0$ , and Proposition 3.2: $ϑ (A) < 2 A$ and $ψ (A) - ϑ (A) < 1.00007 A + 1.78 A^{1/3}$ . On this range they give $ψ (A) < 2.01 A$ . Therefore

$\frac{ψ ( A )}{( α - 1 ) A} < \frac{2.01 ( 3.2 \cdot 1 0 ^{22} )}{0.9999 \cdot 1 0 ^{23}} < 0.644, \frac{5 M _{1} ( α )}{A} < 6 \cdot 1 0^{- 18} . (GH5)$

Combining (GE5) and (GH3)–(GH5) pays the full original signed integral:

$- A L I_{ψ} (A) < 0.103 + 0.000003 + 0.644 + 6 \cdot 1 0^{- 18} < \frac{3}{4} (1 0^{36} < A \leq 1 0^{45}) . (GH6)$

The same selected source still satisfies $Δ (N) = I_{ψ} (A) + D^{*} (A)$ and the strict core (G6). The already proved bound (X5) in the Polak application gives $E (L) > 0.778$ for every $L \geq 78$ . Thus at this actual source and clock,

$A L Δ (N) > 0.778 - 0.75 = 0.028 (1 0^{36} < A \leq 1 0^{45}) . (GH7)$

The existing restriction $A > 1 0^{36}$ and (GH7) therefore force $lo g N > 1 0^{45}$ for the same hypothetical least global maximizer. This conclusion is specific to that selected source; it does not assert strict Robin for every integer below $exp (1 0^{45})$ .

This application reuses the published finite-height verification, zero-count and prime estimates, the classical Gamma product, and the existing core and directed-transport results. It makes no mathematical priority claim and has no Lean certification. The fixed shape and verified height give no uniform bound as $A \to \infty$ : the comparison and high-zero allowances in (GE5), (GH3) still grow with $A$ . The full unbounded signed Robin target and RH remain unproved.

A one-sided entire envelope retaining the original cutoff

The original prime weight has a derivative jump at its cutoff. A signed mixture of published power-function extremals gives entire lower and upper envelopes with an exact error while retaining that cutoff. Use the power-function corollary in Carneiro, Littmann and Vaaler, arXiv:1008.4969v2, Part III, “Extremal Functions for $∣ x ∣^{σ}$ ”, can instead be applied before mixing the actual second-derivative measure. This is a paper-level application of that published corollary and elementary signed-measure integration, with no mathematical priority or Lean-verification claim.

Retain $A \geq e^{2}$ , $L = lo g A$ , and the original weight and derivative

$H_{A} (x) = \frac{1}{max ( A , ∣ x ∣ ) lo g max ( A , ∣ x ∣ )}, k (t) = \frac{1 + lo g t}{t ^{2} lo g ^{2} t}, κ_{A} = k (A) .$

The distributional second derivative is the signed measure

$μ_{A} = μ_{A}^{+} - μ_{A}^{-}, μ_{A}^{+} (d u) = - k^{'} (∣ u ∣) 1_{{∣ u ∣ > A}} d u, μ_{A}^{-} = κ_{A} (δ_{- A} + δ_{A}) . (CE1)$

Here $k^{'} < 0$ . Both parts have mass $2 κ_{A}$ , and $\int (1 + ∣ u ∣) d ∣ μ_{A} ∣ (u) < \infty$ . The negative cutoff atoms have not been dropped. Direct integration gives the absolutely convergent potential representation

$H_{A} (x) = \frac{1}{2} \int_{R} ∣ x - u ∣ d μ_{A} (u) . (CE2)$

For example, differentiation off $\pm A$ gives the original derivative; the value at zero is $- A κ_{A} + \int_{A}^{\infty} u [- k^{'} (u)] d u = 1/ (A L)$ . These facts and continuity identify the potential with $H_{A}$ .

Use the source corollary at $σ = 1$ , where its normalization is $γ (- 1) = - 2 π$ and $γ (2) = 1/ π$ . Reversing the inequalities when dividing by $- 2 π$ , and then rescaling, provides even real entire functions $P_{δ}, Q_{δ}$ of exponential type at most $2 π δ$ for every $δ > 0$ , satisfying

$P_{δ} (x) \leq ∣ x ∣ \leq Q_{δ} (x), \int_{R} (∣ x ∣ - P_{δ} (x)) d x = \frac{1}{6 δ ^{2}}, \int_{R} (Q_{δ} (x) - ∣ x ∣) d x = \frac{1}{12 δ ^{2}} . (CE3)$

This reuses the source result rather than reconstructing its proof. The two nonnegative errors sum to the integrable entire function $Q_{δ} - P_{δ}$ . The standard real-line bound for an integrable entire function of finite exponential type makes that difference bounded; hence both power extremals are $O_{δ} (1 + ∣ x ∣)$ on the real line. The usual polynomial-growth version of the Paley–Wiener bound then gives $O_{K, δ} (1 + ∣ u ∣)$ for their translates $z - u$ , uniformly for $z$ in a fixed compact subset $K$ of the complex plane. This bound and the first moment in (CE1) justify locally uniform integration below, including differentiation on compact subsets.

Define

$ℓ_{A, δ} (z) u_{A, δ} (z) = \frac{1}{2} [\int P_{δ} (z - u) d μ_{A}^{+} (u) - \int Q_{δ} (z - u) d μ_{A}^{-} (u)], = \frac{1}{2} [\int Q_{δ} (z - u) d μ_{A}^{+} (u) - \int P_{δ} (z - u) d μ_{A}^{-} (u)] . (CE4)$

They are even real entire functions of exponential type at most $2 π δ$ , and the signs in (CE4) give

$ℓ_{A, δ} (x) \leq H_{A} (x) \leq u_{A, δ} (x) (x \in R) . (CE5)$

Tonelli applies to the nonnegative differences from (CE2). Using the actual two masses in (CE1), rather than treating the signed measure as positive, gives the exact errors

$\int_{R} (H_{A} - ℓ_{A, δ}) d x = \int_{R} (u_{A, δ} - H_{A}) d x = \frac{κ _{A}}{4 δ ^{2}} . (CE6)$

Evenness therefore yields

$\int_{0}^{\infty} (H_{A} - ℓ_{A, δ}) d t = \frac{κ _{A}}{8 δ ^{2}} . (CE7)$

These are explicit envelopes for the unchanged cutoff weight. They do not assert extremality for $H_{A}$ , arithmetic sampling convergence, a compact zero-height cutoff, or a sign estimate on the original complete zero response. Fourier support on the additive real variable is not spectral truncation at the actual complex zeta zeros.

A finite centered prime comparison

The lower envelope (CE5) gives a finite arithmetic consumer without assuming that an unweighted $L^{1}$ error controls an infinite von-Mangoldt weighted sum. Keep $R \geq A \geq e^{2}$ , $V = lo g R$ , $D (t) = ψ (t) - t$ , and the existing unconditional constant $C_{ψ}$ in (GR3). Define the finite centered response

$J_{A, δ} (R) = \int_{0}^{R} ℓ_{A, δ} (t) d t - 2 \leq n \leq R \sum Λ (n) ℓ_{A, δ} (n) . (CP1)$

Both terms are finite. No separately divergent pole or prime integral has been taken. Set $e = H_{A} - ℓ_{A, δ} \geq 0$ . For this same $R$ , nonnegative von Mangoldt weights and (CE7) give

$\int_{0}^{R} H_{A} (t) d t - 2 \leq n \leq R \sum Λ (n) H_{A} (n) = J_{A, δ} (R) + \int_{0}^{R} e (t) d t - 2 \leq n \leq R \sum Λ (n) e (n) \leq J_{A, δ} (R) + \frac{κ _{A}}{8 δ ^{2}} . (CP2)$

An atom at $n = A$ satisfies the same pointwise order. No zero real part, multiplicity, or height is altered in the original response.

Partial summation, with the right endpoint included in $ψ (R)$ , keeps the full original integral:

$- I_{ψ} (A) = [\int_{0}^{R} H_{A} (t) d t - 2 \leq n \leq R \sum Λ (n) H_{A} (n)] + D (R) H_{A} (R) - \int_{R}^{\infty} D (t) k (t) d t . (CP3)$

The existing (GR3) implies $∣ D (t) ∣ \leq C_{ψ} t / lo g^{2} t$ for $t > 1$ . Its boundary allowance is $C_{ψ} / V^{3}$ , and direct integration pays all of the remaining tail by

$\int_{R}^{\infty} ∣ D (t) ∣ k (t) d t \leq C_{ψ} (\frac{1}{2 V ^{2}} + \frac{1}{3 V ^{3}}) .$

Consequently the finite directed upper bound is

$- A L I_{ψ} (A) \leq A L J_{A, δ} (R) + \frac{1 + L}{8 A ^{3/2} L δ ^{2}} + C_{ψ} A L (\frac{1}{2 V ^{2}} + \frac{4}{3 V ^{3}}) . (CP4)$

This pays the approximation and the entire omitted arithmetic tail for any displayed parameters. It neither requires nor proves convergence of an infinite prime sample of the envelope error. For instance, $δ = 1$ and $R = exp (A^{1/4} lo g A)$ make both displayed added allowances tend to zero as $A \to \infty$ ; this is asymptotic, since $C_{ψ}$ has not been numerically certified. It gives no starting clock and no claim that evaluating this finite prime sum is inexpensive.

At the unchanged conditional least global Robin-ratio maximizing integer, a sufficiently strong upper estimate on the actual $J_{A, δ} (R)$ would still be needed to pay the strict core after (CP4). No such main estimate is supplied. The original $F_{A}$ , actual zero multiset, complete elementary contributions and unbounded Robin/RH goal remain unchanged; no compact spectral truncation or RH inference follows from additive exponential type. This is a paper-level comparison, not a Lean-verified result.

Joint short-interval cancellation in the Gamma comparison

The allowance in (GP6) discards the cancellation between the two sides of the original cutoff. The uniform short-interval input already recorded in the Guth–Maynard note, Corollary 1.3 of arXiv:2405.20552v2, supplies a two-sided comparison with a smaller admissible Gamma shape. The global PNT input is the Fiori–Jaskari theorem quoted above. These source theorems, the absolute centering in (GP4)–(GP5), and the full combined remainder (GR1) are reused. The following is a paper-level application, without mathematical-priority or Lean-verification claims.

Retain $A \to \infty$ , $L = lo g A$ , $D (t) = ψ (t) - t$ , $h_{A}, w_{c}, J_{α}$ and the Gamma law from (GP1)–(GP5). Define

$η_{0} (A) η_{si} (A) = t \geq A sup \frac{∣ D ( t ) ∣}{t}, = A /2 \leq u < v \leq 3 A /2 v - u \geq A^{2/3} sup \frac{∣ D ( v ) - D ( u ) ∣}{v - u} . (GS1)$

There are absolute constants $K, c > 0$ such that, for all sufficiently large $A$ and every noninteger $α \geq 4$ ,

$A L ∣ J_{1} (A) - J_{α} (A) ∣ \leq K {\frac{A}{α} [η_{0} (A) + η_{si} (A) + A^{- 1/30}] + A α A^{- 9/10} + A e^{- c α}} . (GS2)$

The starting value and constants are not numerically certified. This is an asymptotic comparison, not an additional finite Robin clock.

The local prime input and its uniform range

The retained global PNT theorem gives, for some fixed $c_{0} > 0$ ,

$η_{0} (A) ≪ exp [- c_{0} \frac{L ^{3/5}}{( lo g L ) ^{1/5}}] . (GS3)$

Use Guth–Maynard Corollary 1.3 with the fixed parameter $ϵ = 1/20$ . For $u \in [A /2, 3 A /2]$ and $A^{2/3} \leq h \leq A^{9/10}$ , its required range $u^{37/60} \leq h \leq u^{0.99}$ holds uniformly for sufficiently large $A$ . Multiplying the prime count by $lo g u$ introduces an error $O (h^{2} / (A L))$ from the variation of $lo g p$ . Thus

$ϑ (u + h) - ϑ (u) = h + O (h [L e^{- L^{1/4}} + A^{- 1/10}]) .$

Here $(lo g u)^{1/4} = L^{1/4} + O (L^{- 3/4})$ uniformly. If $A^{9/10} < h \leq A$ , with $u + h \leq 3 A /2$ , partition into equal pieces of lengths between $A^{9/10} /2$ and $A^{9/10}$ and add the same estimates. The classical Chebyshev bound and prime-power identity give $ψ (t) - ϑ (t) = O (t)$ ; paying this at the two endpoints, with $h \geq A^{2/3}$ , proves

$η_{si} (A) ≪ L e^{- L^{1/4}} + A^{- 1/10} + A^{- 1/6} . (GS4)$

This uses the uniform corollary, not the separate almost-all theorem. It neither assumes a short-interval relative error from global PNT nor substitutes $π$ or $ϑ$ for $ψ$ without paying prime powers.

Remove only the exactly centered linear dilation

Choose the following endpoint convention and retain it in the prime sum:

$a_{A} (t) = {- 1/ (A L), 1/ (t lo g^{2} t), 0 < t \leq A, t > A . B_{c} (t) = w_{c} (t) - h_{A} (t) - (c - 1) a_{A} (t) . (GS5)$

Both $a_{A}$ and $w_{c} - h_{A}$ are absolutely integrable against $d t$ and $d ψ (t)$ ; for the latter use (GP4). Moreover $\int_{0}^{\infty} a_{A} (t) d t = - 1/ L + 1/ L = 0$ and $\int_{0}^{\infty} (w_{c} - h_{A}) d t = 0$ . The absolute $d ψ + d t$ norm of $a_{A}$ is $O (1/ L)$ by Chebyshev and partial summation. Consequently (GP5), Fubini, and $E (C - 1) = 0$ give the unchanged arithmetic difference

$J_{1} - J_{α} = E \int_{0}^{\infty} B_{C} (t) d D (t) . (GS6)$

No divergent pole and prime quantities are separated in this identity.

For $1/2 \leq c \leq 3/2$ , put $e = c - 1$ and split $B_{c} = b_{c} + v_{c}$ , where

$b_{c} (t) = ⎩ ⎨ ⎧ e^{2} / (c A L), \frac{1}{t ( lo g t - lo g c )} - \frac{1}{t lo g t} - \frac{e}{t lo g ^{2} t}, 0 < t \leq A, t > A . (GS7)$

This regular part agrees with $B_{c}$ outside the interval between $A$ and $c A$ . Taylor’s theorem in $c$ , uniformly on $[1/2, 3/2]$ , gives

$∣ b_{c} (t) ∣ ≪ \frac{e ^{2}}{t lo g ^{2} t}, ∣ b_{c}^{'} (t) ∣ ≪ \frac{e ^{2}}{t ^{2} lo g ^{2} t} (t > A) .$

Partial summation on $(A, \infty)$ , together with the constant part on $(0, A]$ , therefore yields

$\int_{0}^{\infty} b_{c} (t) d D (t) ≪ η_{0} (A) e^{2} / L . (GS8)$

In particular, the boundary contribution is paid using $D (A)$ ; the change of formula at $A$ has not been ignored.

The local residual $v_{c}$ is zero outside the closed interval with endpoints $A, c A$ . Inside that interval its nonzero formula is

$⎩ ⎨ ⎧ \frac{1}{c A L} - \frac{1}{t lo g ( t / c )}, \frac{1}{t lo g ( t / c )} - \frac{1}{c A L}, c > 1, A < t < c A, c < 1, c A < t < A .$

The actual endpoint values are those of $B_{c} - b_{c}$ in (GS5)–(GS7). The first derivative of $1/ [t lo g (t / c)]$ is $O (1/ (A^{2} L))$ in this interval. Including the possible jump at $A$ gives

$∥ v_{c} ∥_{\infty} + Var (v_{c}) ≪ ∣ e ∣/ (A L) . (GS9)$

Pay the shrinking cutoff interval, including its atoms

Set $H_{0} = A^{2/3}$ , $H_{1} = A^{7/10}$ and $h = A ∣ e ∣$ . For $h \geq H_{1}$ , (GS1) bounds $∣ D (t) - D (A) ∣$ by $η_{si} (A) h$ whenever $∣ t - A ∣ \geq H_{0}$ . For smaller distances, monotonicity of $ψ$ and (GS4) applied to $[A - H_{0}, A + H_{0}]$ give the bound $O (H_{0})$ . Since $H_{0} / H_{1} = A^{- 1/30}$ , it follows uniformly on the support of $v_{c}$ that

$∣ D (t) - D (A) ∣ ≪ [η_{si} (A) + A^{- 1/30}] h .$

The same bound holds for one-sided limits after paying a possible endpoint prime-power atom of size at most $lo g (2 A) = O (lo g A)$ , absorbed in $H_{0}$ . Integration of a bounded-variation function against these increments, using (GS9), gives

$\int v_{c} (t) d D (t) ≪ [η_{si} (A) + A^{- 1/30}] e^{2} / L (H_{1} \leq h \leq A /2) . (GS10)$

For $h < H_{1}$ , the support lies in $[A - H_{1}, A + H_{1}]$ . Its total $d ψ + d t$ mass is $O (H_{1})$ by (GS4), with the endpoint atom included. Therefore $∣ \int v_{c} d D ∣ ≪ ∣ e ∣ H_{1} / (A L)$ . The maximum of the Gamma density is $O (α)$ uniformly for $α \geq 4$ , by the standard Stirling bound at its mode. Hence

$E [∣ C - 1∣ 1_{{∣ C - 1∣ < H_{1} / A}}] A L E [\int v_{C} d D 1_{{∣ C - 1∣ < H_{1} / A}}] ≪ α (H_{1} / A)^{2}, ≪ A α A^{- 9/10} . (GS11)$

For the remaining event $C \in / [1/2, 3/2]$ , (GP4) and the absolute norm of $a_{A}$ give $∣ \int B_{c} d D ∣ ≪ (1 + c + c^{- 1}) / L$ for every $c > 0$ . The Gamma moment-generating function and Chernoff’s inequality give $Pr (C \in / [1/2, 3/2]) \leq 2 e^{- c_{1} α}$ for an absolute $c_{1} > 0$ . The second moment of $1 + C + C^{- 1}$ is bounded uniformly for $α \geq 4$ , using $E C^{2} = 1 + 1/ α$ and $E C^{- 2} = α^{2} / [(α - 1) (α - 2)]$ . Cauchy–Schwarz thus pays the entire event by $O (A e^{- c_{1} α /2})$ after normalization. In particular no scales with $c A < 1$ have been discarded.

Finally average (GS8) and (GS10), use $E (C - 1)^{2} = 1/ α$ , and add (GS11) and this Gamma tail. This proves (GS2) uniformly over its stated shapes.

A smaller shape family with vanishing full comparison error

Fix $0 < b_{-} \leq b_{+} < \infty$ , and take any noninteger shape satisfying

$b_{-} a_{*} (A) \leq α \leq b_{+} a_{*} (A), a_{*} (A) = A e^{- L^{1/4} /2} . (GS12)$

For $b_{-} = 1/2$ and $b_{+} = 2$ , for example, $α = ⌊ a_{*} (A)⌋ + 1/2$ is lawful eventually. The first term of (GS2), using (GS3)–(GS4), is $O_{b_{-}, b_{+}} (L e^{- L^{1/4} /2})$ . The small-interval term is $O_{b_{-}, b_{+}} (A^{- 3/20} e^{- L^{1/4} /4})$ , and the Gamma tail is smaller. Reusing the complete elementary remainder in (GR1) gives

$- A L I_{ψ} (A) = A L Z_{α} (A) + O_{b_{-}, b_{+}} (L e^{- L^{1/4} /2}) . (GS13)$

Every zero in $Z_{α}$ still has its actual real part, multiplicity and height, and the coefficient remains $H_{α} (ρ) F_{A} (ρ)$ . No main spectral response has been bounded. In (GS12) the old allowance $A / α$ diverges, whereas the joint arithmetic comparison tends to zero. This is the gained transport regime; it supplies neither a sign nor a finite numerical threshold for the remaining main response.

At the same conditional least integer $N > 5040$ attaining the global Robin-ratio maximum, this estimate uses $A = lo g N$ without changing the selected source, the strict core, or the existing conclusion $A > 1 0^{45}$ . The asymptotic starting point in (GS2) has not been numerically compared with this finite exclusion. A full signed upper bound for $Z_{α} (A)$ with enough strict slack remains required. Robin’s criterion and RH remain unproved by these applications.

Reusing the coefficient remainder on a larger finite clock

Keep the same conditional least integer $N > 5040$ attaining the global Robin-ratio maximum under the Robin-violation hypothesis. The preceding finite-clock application already gives $A = lo g N > 1 0^{45}$ ; put $L = lo g A$ . The following application uses the existing coefficient remainder (H1), directed comparison (GE5), Gamma damping (GH1), verified height and complete zero-count bound. It does not construct a new smoothing transform or repeat their proofs.

Use the full coefficient remainder separately on the two height ranges

The existing (H1) and its numerator estimate give, throughout the actual critical strip $0 < β < 1$ ,

$A L ∣ F_{A} (β + i t) ∣ \leq (1 + 1/ L) \frac{A ^{β - 1/2}}{∣ β + i t ∣∣1 - β - i t ∣} (1 + \frac{2 ( L + 2 )}{L ( L + 1 ) ∣1 - β - i t ∣}) . (GJ1)$

This retains the complete original coefficient, including its remainder; it sharpens the coarser factor two in (GH2) by direct use of the already available estimate. No zero coefficient is replaced by only its leading term. For $L > 78$ , the verified critical head has $∣1 - ρ ∣ \geq 1/2$ , so its parenthesized factor is less than $1.054$ :

$\frac{4 ( L + 2 )}{L ( L + 1 )} < \frac{4 \cdot 80}{78 \cdot 79} < 0.054.$

For the actual unverified tail $∣ t ∣ > H = 3 \cdot 1 0^{12}$ , $∣1 - ρ ∣ \geq ∣ t ∣ > H$ instead bounds that factor by $1.0001$ . Criticality is used only on the verified head. The original real parts and the complete infinite multiset of tail zeros are retained.

Using the same positive $c_{0}$ identity, both ordinate signs and (GH1), (GJ1) gives the finite bound

$A L ∣ Z_{α} (A) ∣ < M_{1} (α) (1 + 1/ L) [1.054 c_{0} + 2 (1.0001) A S_{2} (H) exp (- \frac{H ^{2}}{2 ( α + H )})] . (GJ2)$

Here $M_{1} (α) = α / (α - 1)$ , $c_{0} < 0.05$ and $S_{2} (H) < 1.48 \cdot 1 0^{- 12}$ are the same previously paid quantities. In particular, this argument does not need a new verified height or a new zero enumeration.

Pay the complete response on the added interval

Take the exact noninteger shape and additional interval

$α = \frac{29}{20} 1 0^{23} + \frac{1}{2}, 1 0^{45} < A \leq 1 0^{46} .$

Then $L > 78$ , $A \leq 1 0^{23}$ , $M_{1} (α) < 1.0001$ , and

$\frac{H ^{2}}{2 ( α + H )} > 31, e^{31} > 1 0^{13} .$

For the exponential inequality, $e > 8/3$ and $8^{31} > 1 0^{13} 3^{31}$ suffice. The head and complete tail in (GJ2) therefore obey the exact rational budgets

$(1.0001) (1.02) (1.054) (0.05) < 0.054,$

$2 (1.0001)^{2} (1.02) (1 0^{23}) (1.48 \cdot 1 0^{- 12}) 1 0^{- 13} < 0.031. (GJ3)$

For the arithmetic comparison term, reuse Dusart’s Theorem 5.2 and Proposition 3.2. The original v1 theorem’s $k = 2$ , $η_{2} = 3.965$ , $x_{2} = 2$ row and the complete prime-power correction give

$\frac{ψ ( A )}{A} < 1 + \frac{3.965}{L ^{2}} + 1.00007 A^{- 1/2} + 1.78 A^{- 2/3} < 1.001.$

The final inequality follows from $3.965/7 8^{2} < 0.0007$ and $1.00007 \cdot 1 0^{- 22} + 1.78 \cdot 1 0^{- 30} < 0.0003$ . The source row applies throughout the added interval; no asymptotic starting threshold is substituted for its explicit $x_{2} = 2$ . Since $α - 1 > 1.4499 \cdot 1 0^{23}$ , the two remaining terms of (GE5) satisfy

$\frac{ψ ( A )}{( α - 1 ) A} < \frac{1.001}{1.4499} < 0.691, \frac{5 M _{1} ( α )}{A} < 6 \cdot 1 0^{- 22} . (GJ4)$

Combining (GE5) with (GJ2)–(GJ4) pays the original full signed response:

$- A L I_{ψ} (A) < 0.054 + 0.031 + 0.691 + 6 \cdot 1 0^{- 22} < 0.777, 1 0^{45} < A \leq 1 0^{46} . (GJ5)$

All elementary and trivial contributions, including Gamma scales below the unit cutoff, are retained through the already proved (GE5). The strict core for this same selected source still gives $A L D^{*} (A) > E (L) > 0.778$ . Hence

$A L Δ (N) > 0.001, 1 0^{45} < A \leq 1 0^{46} . (GJ6)$

Together with the preceding $A > 1 0^{45}$ restriction, this forces $lo g N > 1 0^{46}$ for the same hypothetical least global maximizer. It is a paper-level enlargement of the excluded selected-source window, with no new classical theorem, priority claim, Lean certification or all-integer finite Robin verification. The fixed verified height and shape still provide no uniform main bound as $A \to \infty$ ; the complete unbounded signed Robin target and RH remain unproved.

An effective directed Gamma comparison retaining the cutoff sign

Retain the original $h_{A}, w_{c}, J_{1}, J_{α}, D = ψ - id$ and the endpoint convention in (GS5). Reuse the absolutely convergent centering (GP4)–(GP5), (GS6), and the decomposition (GS7)–(GS9). The following explicit upper comparison preserves the sign of the local residual against the actual nonnegative prime-power measure. No new short-interval theorem or smoothing transform is required.

For $A \geq e^{78}$ , $L = lo g A$ , and every noninteger $α \geq 4$ , put $η_{0} (A) = sup_{t \geq A} ∣ ψ (t) - t ∣/ t$ as in (GS1). Then

$A L [J_{1} (A) - J_{α} (A)] \leq [0.52 + 2 η_{0} (A)] \frac{A}{α} + 400 A e^{- α /60000} . (GK1)$

This is a directed inequality. The absolute asymptotic comparison (GS2) retains its stated scope; (GK1) supplies explicit constants without assigning a numerical starting point to that asymptotic result.

Pay the regular part using only the global error

Fix $99/100 \leq c \leq 101/100$ and write $e = c - 1$ . For $t > A$ , put $v = lo g t$ and $u (c, t) = 1/ [t (v - lo g c)]$ . Its derivatives are

$\partial_{c}^{2} u = \frac{2 - ( v - lo g c )}{t c ^{2} ( v - lo g c ) ^{3}}, \partial_{t} \partial_{c}^{2} u = \frac{( v - lo g c ) ^{2} - 6}{t ^{2} c ^{2} ( v - lo g c ) ^{4}} .$

Throughout the segment joining $1$ to $c$ , one has $v - lo g c \geq v - 1/100 \geq (999/1000) v$ and $c \geq 99/100$ . Taylor’s formula and $[(99/100)^{2} (999/1000)^{2}]^{- 1} < 1.024$ therefore give

$∣ b_{c} (t) ∣ \leq \frac{0.512 e ^{2}}{t lo g ^{2} t}, ∣ b_{c}^{'} (t) ∣ \leq \frac{0.512 e ^{2}}{t ^{2} lo g ^{2} t} (t > A) . (GK2)$

On $(0, A]$ , the same existing regular part equals $e^{2} / (c A L)$ . Partial summation against $D$ , including its actual value at $A$ and the change of formula there, gives

$\int b_{c} d D = [\frac{e ^{2}}{c A L} - b_{c} (A +)] D (A) - \int_{A}^{\infty} D (t) b_{c}^{'} (t) d t .$

The upper boundary vanishes by the global PNT input already used in (GS3). Applying (GK2), $∣ D (t) ∣ \leq η_{0} (A) t$ for all $t \geq A$ , and $\int_{A}^{\infty} d t / (t lo g^{2} t) = 1/ L$ yields

$\int b_{c} d D \leq \frac{η _{0} ( A ) e ^{2}}{L} [\frac{100}{99} + 0.512 + \frac{0.512}{L}] \leq \frac{2 η _{0} ( A ) e ^{2}}{L} . (GK3)$

This uses a bound on the regular part against the global error; no short-interval bound has been inferred from global PNT.

Retain the negative local residual, including the endpoint atoms

The actual local residual $v_{c} = B_{c} - b_{c}$ from (GS5)–(GS7) is nonpositive. For $c > 1$ its formula on $A < t < c A$ is $1/ (c A L) - 1/ [t lo g (t / c)]$ , and $v_{c} (A) = v_{c} (c A) = 0$ . For $c < 1$ its formula on $c A < t \leq A$ is $1/ [t lo g (t / c)] - 1/ (c A L)$ , with $v_{c} (c A) = 0$ . It is zero elsewhere. Since $1/ [t lo g (t / c)]$ decreases with $t$ on these intervals, both displayed formulas are nonpositive. These values also fix the sign at a possible prime-power atom at $A$ or $c A$ .

Consequently the actual measure $d ψ \geq 0$ gives

$\int v_{c} d D = \int v_{c} d ψ - \int v_{c} d t \leq - \int v_{c} d t .$

On the entire interval with endpoints $A, c A$ , $t \geq (99/100) A$ and $lo g (t / c) \geq L - 1/100$ . Thus

$\frac{d}{d t} \frac{1}{t lo g ( t / c )} \leq \frac{1.034}{A ^{2} L} .$

Indeed, the ratio to $1/ (A^{2} L)$ is at most $[(99/100)^{2}]^{- 1} [L / (L - 1/100)] [1 + 1/ (L - 1/100)] < 1.034$ for $L \geq 78$ . The local residual vanishes at $c A$ , so integrating the resulting triangular upper bound gives

$\int v_{c} d D \leq \frac{1.034 ( A ∣ e ∣ ) ^{2}}{2 A ^{2} L} \leq \frac{0.52 e ^{2}}{L} . (GK4)$

An atom contributes to the nonpositive prime term; it is not replaced by a Lebesgue density or discarded with an unknown sign.

Pay all Gamma scales outside the narrow interval

An explicit global bound needed only for this tail follows directly from the existing Dusart Theorem 5.2 $k = 0$ row and Proposition 3.2: $ψ (t) \leq 4 t$ for $t > 0$ . For $t < 2$ , $ψ (t) = 0$ ; for $t \geq 2$ use $ϑ (t) < 2 t$ and $1.00007 t^{- 1/2} + 1.78 t^{- 2/3} < 2$ . Applying the already established (GP4) estimates with this constant gives, for every $c > 0$ ,

$\int ∣ w_{c} - h_{A} ∣ d ψ \leq \frac{16 ( c + c ^{- 1} )}{L}, \int ∣ w_{c} - h_{A} ∣ d t \leq \frac{4 ( c + c ^{- 1} )}{L} .$

For clarity, below $A max (1, c)$ the corresponding bound with $ψ (t) \leq Bt$ is $2 B (c + c^{- 1}) / L$ . Above that point it is also $2 B (c + c^{- 1}) / L$ : use the logarithmic difference and $∣ lo g c ∣ + \frac{1}{2} (lo g c)^{2} \leq c + c^{- 1}$ , since $L \geq 2$ . This is a numerical use of the existing all-scale convergence argument. The affine derivative in (GS5) satisfies $\int ∣ a_{A} ∣ (d ψ + d t) \leq 14/ L$ by the same partial summation. Since $∣ c - 1∣ \leq c + c^{- 1}$ , these estimates pay the complete centered kernel, with room in the constant:

$\int B_{c} d D \leq \frac{100 ( c + c ^{- 1} )}{L} (c > 0) . (GK5)$

This includes every scale with $c A < 1$ . Only absolutely convergent differences are integrated, as in (GP4)–(GP5).

Let $C$ have the same mean-one Gamma law with shape and rate $α$ . Use its existing moment-generating function and Chernoff bound with $δ = 1/100$ . The two exponents satisfy $δ - lo g (1 + δ) \geq δ^{2} / [2 (1 + δ)] > 1/30000$ and $- δ - lo g (1 - δ) \geq δ^{2} /2 > 1/30000$ . Therefore

$Pr (C \in / [99/100, 101/100]) \leq 2 e^{- α /30000}, E (C + C^{- 1})^{2} \leq \frac{5}{4} + 2 + \frac{8}{3} < 6.$

Cauchy–Schwarz and (GK5) bound the normalized contribution from that whole event by $400 A e^{- α /60000}$ . On its complement, (GK3)–(GK4) apply. Averaging with $E (C - 1)^{2} = 1/ α$ proves (GK1) using the globally centered identity (GS6). No conditional mean-one assertion has been used.

Preserve the complete original zero response

Reusing $J_{1} = - I_{ψ}$ , $J_{α} = Z_{α} + \overline{R}_{α}$ and the full elementary upper bound (GE4) gives

$- A L I_{ψ} (A) \leq A L Z_{α} (A) + [0.52 + 2 η_{0} (A)] \frac{A}{α} + 400 A e^{- α /60000} + \frac{5 M _{1} ( α )}{A} . (GK6)$

Here $Z_{α}$ retains the original $H_{α} (ρ) F_{A} (ρ)$ , every actual real part, multiplicity and height. The improvement is an effective directed arithmetic payment obtained from the retained local sign, not a new bound on the main response. The unbounded signed Robin estimate and RH remain unproved. This is a paper-level application of the existing prime bounds and kernel identities, without a mathematical priority or Lean-certification claim.

A further finite clock from the effective cutoff-sign payment

Keep the same conditional least integer $N > 5040$ attaining the global Robin-ratio maximum under the Robin-violation hypothesis. The preceding application already excludes $A = lo g N \leq 1 0^{46}$ ; put $L = lo g A$ . Apply (GK6) on the additional interval

$1 0^{46} < A \leq \frac{7}{2} 1 0^{46}, α = \frac{7}{5} 1 0^{23} + \frac{1}{2} .$

This uses the effective directed comparison, the full original coefficient bound (GJ1), Gamma damping (GH1), and the existing complete elementary remainder. No primes, arithmetic profiles or zero ordinates are newly enumerated.

Pay the actual global arithmetic error

Dusart’s original Theorem 5.2 $k = 2$ , $η_{2} = 3.965$ , $x_{2} = 2$ row and Proposition 3.2, already used in (GJ4), give for every $t \geq A$

$\frac{∣ ψ ( t ) - t ∣}{t} \leq \frac{3.965}{lo g ^{2} t} + 1.00007 t^{- 1/2} + 1.78 t^{- 2/3} < 0.001.$

Hence $η_{0} (A) < 0.001$ . Moreover $L > 78$ , $A < 1.871 \cdot 1 0^{23}$ and $α \geq 1.4 \cdot 1 0^{23}$ . The normalized central comparison term in (GK6) is therefore

$[0.52 + 2 η_{0} (A)] \frac{A}{α} < \frac{0.522 ( 1.871 )}{1.4} < 0.698. (GL1)$

The Gamma tail is explicit: $α /60000 > 1 0^{18}$ and $e^{x} > x^{3} /6$ for $x > 0$ imply $e^{α /60000} > 1 0^{53}$ . Thus $400 A e^{- α /60000} < 8 \cdot 1 0^{- 28}$ . The entire elementary contribution is less than $6 \cdot 1 0^{- 23}$ by (GE4). This includes Gamma scales below the unit cutoff.

Pay every original zero above the verified height

The existing (GJ2) gives, with $H = 3 \cdot 1 0^{12}$ ,

$A L ∣ Z_{α} (A) ∣ < M_{1} (α) (1 + 1/ L) [1.054 c_{0} + 2 (1.0001) A S_{2} (H) e^{- H^{2} / [2 (α + H)]}] .$

Here $c_{0} < 0.05$ , $S_{2} (H) < 1.48 \cdot 1 0^{- 12}$ and $M_{1} (α) < 1.0001$ are unchanged. For the selected shape, $H^{2} / [2 (α + H)] > 32$ and $e^{32} > 4 \cdot 1 0^{13}$ ; the latter follows from $e > 8/3$ and $8^{32} > 4 \cdot 1 0^{13} 3^{32}$ . Both ordinate signs, the actual tail real parts and all heights to infinity remain included. The head and tail satisfy the exact rational budgets

$(1.0001) (1.02) (1.054) (0.05) < 0.054,$

$2 (1.0001)^{2} (1.02) (1.871 \cdot 1 0^{23}) (1.48 \cdot 1 0^{- 12}) (4 \cdot 1 0^{13})^{- 1} < 0.015. (GL2)$

Combining (GK6), (GL1) and (GL2) pays the original full signed response:

$- A L I_{ψ} (A) < 0.054 + 0.015 + 0.698 + 8 \cdot 1 0^{- 28} + 6 \cdot 1 0^{- 23} < 0.768. (GL3)$

The same selected source obeys the strict core $A L D^{*} (A) > E (L) > 0.778$ and $Δ (N) = I_{ψ} (A) + D^{*} (A)$ . Therefore

$A L Δ (N) > 0.01, 1 0^{46} < A \leq \frac{7}{2} 1 0^{46} . (GL4)$

Together with the preceding exclusion, this forces $lo g N > (7/2) 1 0^{46}$ for the same hypothetical least global maximizer. It is a selected-source finite exclusion with a normalized margin, not an all-integer finite Robin verification or an unbounded main estimate. The full RH objective remains unproved. All classical prime, zero-count and verified-height inputs are reused; this paper-level application carries no mathematical-priority or Lean-certification claim.

Absolute full-weight cost at a fixed negative heat time

Keep the same conditional least global Robin-ratio maximizer $N$ , the arithmetic clock $A = lo g N > (7/2) 1 0^{46}$ , and $L = lo g A$ . The result below also holds for every fixed $A > 1$ . It concerns a particular absolute transport through negatively deformed zeros; it supplies no upper bound for the original signed response $- A L I_{ψ} (A)$ .

The inputs are Alexander Dobner, A proof of Newman’s conjecture for the extended Selberg class, Acta Arithmetica 201 (2021), 29–62, inspected in arXiv:2005.05142v2, Theorems 4–5, Lemma 3 and §3.1, and the NIST DLMF principal exponential integral and sector asymptotic expansion. These published results are reused as literature-attested inputs. The combined full-weight convergence restriction is a repo-derived paper-level source application; no mathematical-priority or Lean-certification claim is made.

Continue the complete coefficient, keeping its cancellation

For $0 < ℜ s < 1$ , put $v = lo g u$ in the unchanged coefficient (M1). The DLMF principal $E_{1}$ definition and one integration by parts give

$F_{A} (s) = \frac{1}{s} \int_{L}^{\infty} e^{(s - 1) v} (\frac{1}{v} + \frac{1}{v ^{2}}) d v = \frac{A ^{s - 1}}{s L} + E_{1} ((1 - s) L) . (NH1)$

Thus the right side defines the natural principal continuation $F_{A}$ on $C ∖ ({0} \cup [1, \infty))$ . Continuation defines these individual terms beyond the integral’s half-plane of convergence; it does not itself establish a spectral transport identity.

Let $s = x + i y$ , $y \to + \infty$ , with $∣ x ∣ \leq B lo g y$ for a fixed $B$ . Then $z = (1 - s) L$ lies, eventually, in $∣ ar g z ∣ \leq 3 π /4$ . DLMF (8.20.2), with its parameter $p = 1$ , supplies uniformly there

$E_{1} (z) = e^{- z} (z^{- 1} - z^{- 2} + O (∣ z ∣^{- 3})) .$

Substitution into the full (NH1) yields

$F_{A} (s) = A^{s - 1} (\frac{1}{L s ( 1 - s )} - \frac{1}{L ^{2} ( 1 - s ) ^{2}} + O_{A, B} (y^{- 3})) = \frac{1 + L}{L ^{2}} \frac{A ^{s - 1}}{y ^{2}} (1 + O_{A, B} (\frac{lo g y}{y})) . (NH2)$

The terms $1/ (s L)$ and $1/ ((1 - s) L)$ cancel at order $y^{- 1}$ . The $- z^{- 2}$ term supplies the additional $1/ L^{2}$ in the nonzero leading coefficient. Neither part of the original weight has been discarded. All asymptotics here fix $A$ before taking the height limit.

Obtain enough distinct actual zeros from the source’s shifts

Fix one $ϵ > 0$ throughout and use Dobner’s heat parameter $t = - ϵ$ . In the Riemann specialization his equations (11)–(12) give

$D_{ϵ} (s) = n \geq 1 \sum e^{- ϵ l o g^{2} n /4} n^{- s}, J_{ϵ} (s) = s + \frac{ϵ}{4} Log \frac{s}{2 π} . (NH3)$

The completion has the normalization $ξ_{t} ((1 + i z) /2) = 8 H_{t} (z)$ , with the same heat time $t$ . These are deformed $ξ_{- ϵ}$ zeros, not original zeta zeros.

Dobner’s Lemma 3 supplies a zero of $D_{ϵ}$ . His §3.1 proof, using Theorems 4–5, supplies actual zeros of $ξ_{- ϵ} (J_{ϵ} (s))$ in sufficiently high translates of one fixed zero-isolating circle of radius $r$ . The same shift sequence satisfies

$m \to \infty lim inf (τ_{m + 1} - τ_{m}) > 0, m \to \infty lim sup \frac{τ _{m}}{m} < \infty.$

In particular $τ_{m} ≍_{ϵ} m$ . Choose $d > 0$ such that the consecutive gaps are eventually at least $d$ , and retain every $q$ -th shift, where $q d > 2 r + ϵ π /4$ . On the upper half-plane, $J_{ϵ}$ adds an imaginary offset in $(0, ϵ π /4)$ . Consequently the images of the retained circles have disjoint height intervals. Choosing one actual zero in each produces distinct zeros $w_{m}$ of the same $ξ_{- ϵ}$ with

$ℑ w_{m} ≍_{ϵ} m, ℜ w_{m} = \frac{ϵ}{4} lo g (ℑ w_{m}) + O_{ϵ} (1) . (NH4)$

This uses the source’s shift-density information, not merely the existence of infinitely many zeros. The selected preimages stay in one fixed bounded real strip, and $ℑ J_{ϵ} (s) = ℑ s + O_{ϵ} (1)$ , which proves the real-part statement in (NH4).

A necessary absolute-budget condition, including equality

Write $κ = ϵ L /4$ . Equations (NH2) and (NH4) imply, for all sufficiently large $m$ ,

$∣ F_{A} (w_{m}) ∣ ≍_{A, ϵ} m^{κ - 2} . (NH5)$

The constants are positive because $(1 + L) / L^{2} > 0$ and $A^{ℜ w_{m} - 1} ≍_{A, ϵ} (ℑ w_{m})^{κ}$ . Hence, counting the actual deformed zeros with multiplicities,

$ϵ lo g A \geq 4 ⟹ ξ_{- ϵ} (w) = 0 ℑ w > 0 \sum ∣ F_{A} (w) ∣ = \infty. (NH6)$

At equality the selected subseries dominates the harmonic series; for larger $ϵ L$ it dominates a divergent power series. Thus $ϵ lo g A < 4$ is a necessary condition for this principal-continued, complete-weight, absolute full-spectrum transport. For $ϵ lo g A < 4$ , (NH5) makes this particular selected subseries converge, but establishes no convergence or uniform bound for the rest of the deformed spectrum.

The conclusion is about absolute cost at one fixed negative heat time. It proves neither signed divergence nor a failure of Robin or RH, and does not exclude transports that retain cancellation or use additional comparison terms. Such a route still needs an all-height identity and a paid signed error returning to the original zeros, their real parts and multiplicities, the original elementary correction and the strict core at the same $N$ . No changing heat time per zero, finite-spectrum truncation or unproved positive-time certificate is used to discharge that obligation.

The subcritical full-spectrum absolute boundary

The necessary condition (NH6) has a converse for every fixed $A > 1$ and fixed $ϵ > 0$ . The additional step is a uniform right-half-plane comparison, not a use of Dobner’s Theorem 4 outside its stated $∣ℜ s ∣ \leq C (ℑ s)^{1/4}$ range. Reuse his exact negative-heat convolution (9), the Riemann completion, and the already retained (NH1)–(NH6). The DLMF digamma expansion (5.11.2) is used in its fixed sector $∣ ar g z ∣ \leq π /2$ . These are literature-attested inputs; the uniform comparison and combined convergence boundary below are repo-derived paper-level applications, with no mathematical-priority or Lean-certification claim.

A uniform comparison on the entire right half-plane

Write

$γ (s) = \frac{1}{2} s (s - 1) π^{- s /2} Γ (s /2), c = ϵ /4, J_{ϵ} (s) = s + c Log (s / (2 π)),$

$γ_{- ϵ} (s) = γ (s) exp ((s - J_{ϵ} (s))^{2} / ϵ) .$

For fixed $ϵ > 0$ , the comparison is

$\frac{ξ _{- ϵ} ( J _{ϵ} ( s ))}{γ _{- ϵ} ( s )} = D_{ϵ} (s) + O_{ϵ} (∣ s ∣^{- 1}), ℜ s \geq 2, ∣ s ∣ ⟶ \infty. (NH7)$

Uniformity includes arbitrarily large $ℜ s$ , both ordinate signs, and the real axis. It is not uniform in $ϵ$ as $ϵ \to 0$ .

Here is the required interface proof. The DLMF expansion gives uniformly in this half-plane

$\frac{γ ^{'}}{γ} (s) = \frac{1}{s} + \frac{1}{s - 1} - \frac{1}{2} lo g π + \frac{1}{2} \frac{Γ ^{'}}{Γ} (s /2) = \frac{1}{2} Log (s / (2 π)) + O (∣ s ∣^{- 1}) .$

Its imaginary part is bounded on the whole half-plane: the principal argument is bounded, and the remaining bounded part is covered by continuity on a compact set. Consequently, for an absolute $M$ and all real $v$ ,

$\frac{γ ( s + i v )}{γ ( s )} \leq e^{M ∣ v ∣} .$

Set

$R_{s} (v) = \frac{γ ( s + i v )}{γ ( s )} exp (- \frac{i v}{2} Log (s / (2 π))), r = ∣ s ∣.$

Then $∣ R_{s} (v) ∣ \leq e^{(M + π /4) ∣ v ∣}$ for all $v$ . Integrating the logarithmic derivative along the vertical segment, for $∣ v ∣ \leq r^{1/4}$ and large $r$ , gives

$∣ R_{s} (v) - 1∣ ≪ \frac{∣ v ∣ + v ^{2}}{r} .$

Indeed, $∣ s + i u ∣ \geq r /2$ on that segment and $∣ Log (s + i u) - Log s ∣ \leq 2∣ u ∣/ r$ . The integrated exponent is $O ((∣ v ∣ + v^{2}) / r) = o (1)$ uniformly on this range. No bounded-real-part Stirling estimate is used at an unbounded real part.

In Dobner’s (9), shift the vertical contour to $ℜ z = ℜ s$ and write $z = s + i v$ . For each fixed $s$ this is a finite contour shift; the Gaussian and the usual vertical decay kill the horizontal ends. After division by $γ_{- ϵ} (s)$ the exact identity is

$\frac{ξ _{- ϵ} ( J _{ϵ} ( s ))}{γ _{- ϵ} ( s )} = \frac{1}{π ϵ} \int_{R} e^{- v^{2} / ϵ} R_{s} (v) ζ (s + i v) d v . (NH8)$

Replacing $R_{s}$ by $1$ gives exactly $D_{ϵ} (s)$ by the Gaussian Fourier integral and the absolutely convergent Dirichlet series. On $ℜ s \geq 2$ , $∣ ζ (s + i v) ∣ \leq ζ (2)$ . The local difference therefore has integral $O_{ϵ} (r^{- 1})$ . On $∣ v ∣ > r^{1/4}$ the uniform exponential bound for $R_{s}$ gives $O_{ϵ} (exp (- r^{1/2} / (2 ϵ))) = o_{ϵ} (r^{- 1})$ . This proves (NH7), without estimating every Dirichlet term separately.

Every sufficiently high zero lies in a logarithmic region

On $ℜ s \geq 2$ , $∣ D_{ϵ} (s) - 1∣ \leq ζ (2) - 1 < 1$ . The completion $γ_{- ϵ}$ is nonzero there, so (NH7) shows that $ξ_{- ϵ} (J_{ϵ} (s)) \neq = 0$ for all sufficiently large $∣ s ∣$ in this entire half-plane.

To cover the rightmost region, let $w$ have sufficiently large modulus and satisfy $ℜ w \geq c lo g (∣ w ∣/ (2 π)) + 3$ . Put $s_{0} = w - c Log (w / (2 π))$ . On $∣ s - s_{0} ∣ = 1/2$ one has $ℜ s \geq 5/2$ and

$J_{ϵ} (s) - w = (s - s_{0}) + c [Log (s / (2 π)) - Log (w / (2 π))] .$

The second term is $O_{ϵ} (lo g ∣ w ∣/∣ w ∣) < 1/2$ . The logarithms and the intervening segments stay in the right half-plane. Rouché’s theorem gives a preimage $s$ inside the circle; it satisfies $ℜ s \geq 2$ and $∣ s ∣ ≍ ∣ w ∣$ . Thus there is no such zero $w$ . Apply the same argument to $1 - w$ using $ξ_{- ϵ} (1 - w) = ξ_{- ϵ} (w)$ . Together these bounds imply, for every zero of sufficiently large modulus, $∣ℜ w ∣ \leq c lo g ∣ w ∣ + O_{ϵ} (1)$ . This forces $∣ℑ w ∣ \to \infty$ with $∣ w ∣$ and then gives

$1 - c lo g y - O_{ϵ} (1) \leq ℜ w \leq c lo g y + O_{ϵ} (1), y = ℑ w ⟶ + \infty. (NH9)$

All actual zeros are covered; this is not the lower-density subsequence in (NH4). To replace $∣ w ∣$ by $y$ , note that the bound on $∣ℜ w ∣$ precludes $∣ℜ w ∣ \geq ∣ℑ w ∣$ at arbitrarily large modulus, and hence $∣ w ∣ \leq 2 ∣ℑ w ∣$ eventually.

Count the whole spectrum and use the complete original coefficient

The original Fourier kernel $Φ$ in Dobner’s (2) is positive for $u \geq 0$ : each summand has the positive factor $2 π n^{2} e^{4 u} - 3$ . Evenness gives positivity on the whole real line. For $∣ s - 1/2∣ \leq R$ , its exact Fourier representation implies

$∣ ξ_{- ϵ} (s) ∣ \leq \int_{R} Φ (u) e^{2 R ∣ u ∣} d u \leq 2 ξ_{0} (1/2 + R), ξ_{- ϵ} (1/2) > 0.$

Classical real Stirling growth gives a logarithm $O (R lo g (R + 2))$ for the upper bound. Jensen’s zero-count theorem, applied at $1/2$ with radii $R$ and $2 R$ , therefore gives $O_{ϵ} (R lo g (R + 2))$ zeros in the disk, counted with multiplicities. Together with (NH9), this gives the complete height count

$N_{ϵ} (T) = # {w : ξ_{- ϵ} (w) = 0, 0 < ℑ w \leq T} ≪_{ϵ} T lo g (T + 2) . (NH10)$

The bounded-height exceptions are finite by (NH9) and analyticity. This step reuses the standard Jensen theorem, not an original-zeta zero count applied to a different function.

Fix $A > 1$ as well, write $L = lo g A$ and $κ = ϵ L /4$ . The retained complete-weight asymptotic (NH2), with (NH9), now gives for every sufficiently high actual zero

$∣ F_{A} (w) ∣ ≪_{A, ϵ} (ℑ w)^{κ - 2} .$

For $κ < 1$ , the mass in a dyadic height band is at most $O_{A, ϵ} (j 2^{j (κ - 1)})$ by (NH10), and these bounds sum. Thus the converse to (NH6) is

$ξ_{- ϵ} (w) = 0 ℑ w > 0 \sum ∣ F_{A} (w) ∣ < \infty ⟺ ϵ lo g A < 4, A > 1, ϵ > 0. (NH11)$

Every zero has its actual real part and multiplicity, and all positive heights are retained. (NH6) supplies divergence at and above the boundary; (NH7)–(NH10) supply full-spectrum convergence below it. The tail below the boundary is $O_{A, ϵ} (T^{κ - 1} lo g (T + 2))$ as $T \to \infty$ , with fixed parameters. No uniform allowance as $ϵ \to 0$ or $ϵ lo g A \to 4$ is asserted.

This determines the absolute convergence domain of the particular continued complete-weight deformation. It provides no numerical Robin budget, signed transport identity, or comparison error returning to the original zero multiset. The same selected integer $N$ , $A = lo g N > (7/2) 1 0^{46}$ , the original $F_{A}$ , positive elementary correction and strict core remain in the unproved target (G9). In particular, absolute convergence of the deformed trace does not prove its signed upper bound or RH.

A joint small-negative-time comparison of the complete signed trace

The fixed-parameter convergence in (NH11) can be supplemented with a quantitative comparison to the original trace. Reuse (NH8), its global vertical ratio bound, the principal continuation (NH1), and the all-spectrum count and convergence (NH9)–(NH11). The new interface is uniform in the arithmetic clock and in small negative heat time; no individual zero trajectories or zero-gap bounds are required.

Write $H_{ϵ} = ξ_{- ϵ}^{'} / ξ_{- ϵ}$ , with $H_{0}$ for the original completion. Define the full real trace

$Z_{ϵ} (A) = ξ_{- ϵ} (w) = 0 \sum F_{A} (w) .$

All zeros here are nonreal: positivity of the original Fourier kernel makes $ξ_{- ϵ} (x) > 0$ for every real $x$ . The trace includes both ordinate signs and every multiplicity. At $ϵ = 0$ it is precisely $Z_{orig} (A)$ from (G9). The branch continuation is used for deformed zeros with real part beyond $1$ ; their coefficients are not replaced by convergent original integrals there.

There are absolute constants $ϵ_{0} > 0$ and $C < \infty$ such that

$∣ Z_{ϵ} (A) - Z_{orig} (A) ∣ \leq C ϵ A^{2}, A \geq e^{2}, 0 < ϵ \leq ϵ_{0}, ϵ lo g A \leq 1. (NH12)$

This is a repo-derived paper-level application of the retained convolution and spectral interfaces, together with the standard Cauchy, Hadamard and argument-principle theorems. It does not assert mathematical priority, Lean certification, a numerical value of $C$ or $ϵ_{0}$ , or a signed Robin upper bound.

Center the small-time error before taking a logarithmic derivative

Keep $J_{ϵ}, γ_{- ϵ}, R_{s}$ from (NH7)–(NH8), and set

$U_{ϵ} (s) = \frac{ξ _{- ϵ} ( J _{ϵ} ( s ))}{γ _{- ϵ} ( s )} .$

For $ℜ s \geq 2$ , (NH8) gives the exact Gaussian average of $R_{s} (v) ζ (s + i v)$ . The retained log-Gamma estimates imply that the first two $v$ derivatives of this product are bounded uniformly for $∣ v ∣ \leq 1$ and $ℜ s \geq 2$ . To see the needed uniformity, use

$\partial_{v} lo g R_{s} (v) = i [\frac{γ ^{'}}{γ} (s + i v) - \frac{1}{2} Log (s / (2 π))] .$

The bracket and its $v$ derivative are bounded there. The DLMF sector estimate supplies the large- $∣ s ∣$ bound, Cauchy’s derivative estimate supplies its derivative, and the remaining compact region has no Gamma pole or zero. The series for $ζ, ζ^{'}, ζ^{''}$ are uniformly absolutely convergent in $ℜ s \geq 2$ .

Taylor-expand the product at $v = 0$ and subtract its linear term. The centered Gaussian integrates that linear term to zero, while its second moment is $ϵ /2$ . On $∣ v ∣ > 1$ , the retained $e^{(M + π /4) ∣ v ∣}$ envelope pays the whole tail by $O (ϵ)$ for $0 < ϵ \leq 1/2$ , including the subtracted affine term. Hence

$U_{ϵ} (s) = ζ (s) + O (ϵ) (ℜ s \geq 2), (NH13)$

with an absolute constant. This improves the parameter dependence needed here; (NH7)’s fixed- $ϵ$ height estimate is not assigned a uniform constant without proof.

Since $∣ ζ (s) ∣ \geq 2 - ζ (2) > 0$ , choose an absolute $ϵ_{0} \leq 1/2$ small enough that $U_{ϵ}$ is bounded away from zero on the entire half-plane. On $ℜ s = 3$ , Cauchy’s estimate on radius- $1/2$ disks also gives $U_{ϵ}^{'} - ζ^{'} = O (ϵ)$ . The denominators are now paid, so $U_{ϵ}^{'} / U_{ϵ} - ζ^{'} / ζ = O (ϵ)$ .

The exact normalization obeys

$\frac{γ _{- ϵ}^{'}}{γ _{- ϵ}} (s) = \frac{γ ^{'}}{γ} (s) + \frac{ϵ}{8} \frac{Log ( s / ( 2 π ))}{s} .$

Consequently, on the whole line $ℜ s = 3$ ,

$K_{ϵ} (s) := H_{ϵ} (J_{ϵ} (s)) J_{ϵ}^{'} (s) = H_{0} (s) + O (ϵ), ∣ H_{0} (s) ∣ ≪ lo g (∣ s ∣ + 2) . (NH14)$

Both bounds have absolute constants. No function-value estimate is substituted for a logarithmic-derivative estimate without this step.

Transport the complete coefficient on these curves

Differentiating the exact (NH1), including its exponential-integral part, gives

$F_{A}^{'} (z) = A^{z - 1} [\frac{1}{z ( 1 - z )} - \frac{1}{L z ^{2}}], L = lo g A . (NH15)$

This derivative is single-valued away from $0, 1$ ; the original weight itself still has its branch cut. Put $s = 3 + i y$ , $r = ∣ s ∣$ , $κ = ϵ L /4 \leq 1/4$ , and

$Q_{ϵ} (s) = F_{A} (J_{ϵ} (s)) + F_{A} (1 - J_{ϵ} (s)) .$

On the segments from $s$ to $J_{ϵ} (s)$ and their reflections, $∣ z ∣, ∣1 - z ∣ ≍ r$ . They remain respectively in the right and left half-planes, and the first segment has the same ordinate sign at both ends. Its real part is at most $3 + (ϵ /4) max (lo g (r / (2 π)), 0)$ . For bounded $r$ , any analogous reflected increase is bounded by $ϵ L \leq 1$ . Thus (NH15) gives

$∣ Q_{ϵ} (s) - Q_{0} (s) ∣ ≪ ϵ A^{2} lo g (r + 2) r^{κ - 2} .$

One also has $∣ Q_{0} (s) ∣ ≪ A^{2} r^{- 2}$ uniformly for $A \geq e^{2}$ . This full-coefficient bound can be obtained from (NH15) without losing its cancellation: integrate from $3 + i y$ to the corresponding imaginary infinity, where (NH2) gives zero, and integrate $e^{i L t}$ once by parts. The rational bracket is $O ((1 + t^{2})^{- 1})$ and its derivative is $O ((1 + ∣ t ∣^{3})^{- 1})$ ; $L \geq 2$ pays the integration factor. Use the same calculation on $- 2 - i y$ , whose exponential factor is $A^{- 3}$ . Each ordinate sign is treated on its own branch; at $y = 0$ take the one-sided limits. It follows that

$∣ Q_{0} (s) ∣ ∣ Q_{ϵ} (s) ∣ ∣ Q_{ϵ} (s) - Q_{0} (s) ∣ ≪ A^{2} r^{- 2}, ≪ A^{2} lo g (r + 2) r^{κ - 2}, ≪ ϵ A^{2} lo g (r + 2) r^{κ - 2} . (NH16)$

No leading-term replacement of $F_{A}$ is used.

The whole trace has an exact contour, cut and pole balance

Let $Γ_{ϵ}$ be $J_{ϵ} (3 + i R)$ , oriented upward, and put $b_{ϵ} = J_{ϵ} (3)$ . The curve and its reflection enclose all zeros. For completeness, $J_{ϵ}$ is injective on $ℜ s > 2$ : the real part of

$\frac{J _{ϵ} ( s _{1} ) - J _{ϵ} ( s _{2} )}{s _{1} - s _{2}} = 1 + \frac{ϵ}{4} \int_{0}^{1} \frac{d t}{s _{2} + t ( s _{1} - s _{2} )}$

is positive. Its boundary line maps to a graph, since $ℑ J_{ϵ} (b + i y) = y + (ϵ /4) arctan (y / b)$ is strictly increasing and onto. Properness at infinity then identifies the half-plane image with the region to the right of that graph. By (NH13) there are no zeros in this image. The reflected exterior is zero-free by the functional equation. In particular all zeros lie between $Γ_{ϵ}$ and $1 - Γ_{ϵ}$ .

Use the argument principle in this intervening region, slit along $[1, b_{ϵ}]$ and with the pole at $0$ accounted for. The already cited DLMF (6.2.4), $E_{1} (z) = Ein (z) - Log z - γ$ , shows

$F_{A} (x + i 0) - F_{A} (x - i 0) = 2 πi (x > 1), Res_{z = 0} F_{A} (z) = \frac{1}{A L} .$

The upper side of the slit is traversed from $1$ to $b_{ϵ}$ ; its contribution is therefore $lo g (ξ_{- ϵ} (b_{ϵ}) / ξ_{- ϵ} (1))$ . The logarithms here are real logarithms of positive values. The circle at the logarithmic branch point $1$ tends to zero. Reflection changes $H_{ϵ} (1 - z)$ to $- H_{ϵ} (z)$ , so the two exterior curves combine into the exact balance

$Z_{ϵ} (A) = \frac{1}{2 πi} \int_{3 - i \infty}^{3 + i \infty} Q_{ϵ} (s) K_{ϵ} (s) d s + lo g \frac{ξ _{- ϵ} ( b _{ϵ} )}{ξ _{- ϵ} ( 1 )} - \frac{H _{ϵ} ( 0 )}{A L} . (NH17)$

The two integral halves use the corresponding principal boundary values at the real crossing; the integral is absolutely convergent by (NH14)–(NH16). Both ordinate signs and all multiplicities remain. At $ϵ = 0$ this is the identical contour balance for $Z_{orig}$ , not a different kernel or another maximizing integer.

The infinite-contour limit also needs its horizontal ends paid. The retained Fourier growth and a positive lower bound at $1/2$ give the (NH10) disk/height counts uniformly for $0 \leq ϵ \leq ϵ_{0}$ ; the lower bound may use $ξ_{- ϵ_{0}} (1/2) > 0$ . In each large dyadic interval choose heights whose unit neighborhoods contain $O (lo g T)$ zeros and whose distance from every zero ordinate is at least $c / lo g T$ . These heights exist by averaging the complete count and removing intervals of that radius with sufficiently small $c$ .

The standard genus-one Hadamard logarithmic derivative is

$H_{ϵ} (z) = B_{ϵ} + w \sum (\frac{1}{z - w} + \frac{1}{w}) .$

Its finite part at these heights is $O (T lo g T)$ : only $O (lo g T)$ zeros have ordinate distance below $1$ , their reciprocals cost $O (lo g^{2} T)$ , and the other $O (T lo g T)$ terms have distance at least $1$ . The $1/ w$ terms cost $O (lo g^{2} T)$ , and the $∣ w ∣ > 4 T$ tail costs $O (lo g T)$ by the complete count. Here $∣ z ∣ ≍ T$ on the horizontal ends, and the endpoints’ real width is $O (lo g T)$ . The complete-coefficient estimate (NH2) therefore pays these ends by $O_{A, ϵ} (T^{κ - 1} lo g^{2} T) \to 0$ . The Hadamard theorem is used for the deformed entire function itself; no original-zeta count or simple-zero assumption is imported. This justifies (NH17) over the full spectrum, including collisions.

Pay the signed trace difference on the same arithmetic clock

Subtract (NH17) at zero heat time. Equations (NH14)–(NH16) give

$\frac{1}{2 πi} \int [Q_{ϵ} K_{ϵ} - Q_{0} H_{0}], d s ≪ ϵ A^{2} \int_{R} ∣3 + i y ∣^{κ - 2} lo g^{2} (∣3 + i y ∣ + 2) d y ≪ ϵ A^{2},$

uniformly for $κ \leq 1/4$ . The exact slit and pole differences are $O (ϵ)$ as well. Indeed, $b_{ϵ} - 3 = O (ϵ)$ , and the Fourier representation has uniformly bounded $x$ and $ϵ$ derivatives on the fixed real compact set $0 \leq x \leq 3$ , $0 \leq ϵ \leq ϵ_{0}$ . Positivity bounds its denominators away from zero there. The required moments are finite by the original super-exponential kernel decay. Hence the real logarithm difference and $H_{ϵ} (0) - H_{0} (0)$ are $O (ϵ)$ ; $1/ (A L) \leq 1$ . This proves (NH12), with every contour correction retained.

For one common heat time at each $A$ , choose

$ϵ (A) = min (ϵ_{0} /2, A^{- 3}) .$

It satisfies all the preceding hypotheses for $A \geq e^{2}$ . Using the unchanged positive elementary term $r_{A}$ from (G9), (NH12) gives

$- A L I_{ψ} (A) = A L Z_{ϵ (A)} (A) + r_{A} + O (L A^{- 1/2}) . (NH18)$

The implicit constant is independent of $A$ . The same selected $N$ and $A = lo g N > (7/2) 1 0^{46}$ remain the source, and the original zero real parts, multiplicities and infinite height range are paid through the complete comparison. The time depends on the one arithmetic clock, never on individual zeros. No uniform fixed negative time is used as $A \to \infty$ .

The comparison provides no upper bound for the signed main trace $A L Z_{ϵ (A)} (A)$ . In particular its absolute convergence and its zero-free exterior do not establish the Robin budget $E (L)$ . The constants have not been numerically certified against the selected finite clock, so an $o (1)$ comparison alone is not an exact finite Robin certificate. The original strict core and positive $r_{A}$ still require enough signed slack. RH remains unproved.

Project supplement: a uniform selected-source interval through 4.03e46

This is a project derivation from the named premises below, extending (GL4) on one additional interval. The interval choice and rational enclosures are project calculations, not results attributed to Nicolas’s manuscript or the other cited papers. This is a conditional ordinary proof and application, without a literature-priority, Lean-certification, or all-integer finite-verification claim.

Keep the selected integer, exact clock and source premises

Suppose Robin’s strict inequality fails at some integer above 5040. By the Caveney–Nicolas–Sondow reduction, Theorems 4(ii) and 5(iii), select the same least global maximizer $N > 5040$ of $G (n) = σ (n) / (n lo g lo g n)$ . Put

$A = lo g N, L = lo g A, T = A L, W (n) = lo g \frac{σ ( n )}{n}, λ = \frac{1}{A L} .$

The selected level is at least $e^{γ}$ , so $Δ (N) = γ + lo g L - W (N) \leq 0$ . This $N$ is not the least counterexample. Reuse the finite ladder ending in (GL4), with its Axler finite-stop premise and its cited zero-free and density inputs, to obtain $A > (7/2) 1 0^{46}$ .

For comparison, at an arbitrary integer $n > 5040$ use its own $A = lo g n$ , $L = lo g A$ and $λ = 1/ (A L)$ , and define $Δ (n) = γ + lo g L - W (n)$ . Here primes and positive powers are used in $P_{pp}$ , and the pressure maximum is over integers:

$M (λ) P_{pp} (A) I_{ψ} (A) = m \in Z_{\geq 1} max {W (m) - λ lo g m}, = p^{j} \leq A \sum \frac{1}{j p ^{j}}, = \int_{A}^{\infty} (ψ (t) - t) \frac{1 + lo g t}{t ^{2} lo g ^{2} t} d t . d_{n} R (A) = M (λ) - W (n) + λ A \geq 0, = P_{pp} (A) - λ ψ (A) - M (λ),$

The existing pressure and tail identities give $Δ (n) = I_{ψ} (A) + R (A) + d_{n}$ ; $d_{n}$ remains in this all-integer identity. At the selected $N$ alone, the same-source transport gives $d_{N} = 0$ and $R (A) = D^{*} (A)$ , hence (G1). Its premises are explicit: CNS Fact 1 and GA2 make $N$ proper, so $N / p \geq 4 > e$ for deletion; Mantovanelli’s archived direct bridge at the exact price $λ$ gives the event-free CA optimizer; the existing unrestricted-exponent SevenSmooth exclusion rules out support index $k \leq 4$ ; and Kalyabin’s Theorem 2 endpoint conditions give $P = P^{+} (N) < A < P^{+}$ and exponent one at $P$ . Thus primes $p \leq A$ are exactly the actual support, every actual exponent attains Polak’s full-support minimum over $a \geq 1$ , and his Proposition 6 has $z (N, A) = B_{2} (N, A) = 0$ . There is no transfer of this zero defect to arbitrary integers.

Reuse (G6)–(G7), including Nicolas’s effective Theorem 1.3 branch $6.78$ , whose exact $N^{(0)}$ threshold is paid above. The primorial cutoff $x$ , support endpoint $P$ and clock $A$ stay distinct: (G2)–(G5) pay the absent-prime suffix by GA2 and use $Σ (N) = Z (N)$ from CA; they do not assume common maximizers of $Φ$ and $Σ$ .

The analytic inputs are the full classical explicit formula and its endpoint-inclusive Stieltjes identities, Dusart v1 Theorem 5.2 and Proposition 3.2, Platt–Trudgian’s verification through height $3000175332800$ , and the Hasanalizade–Shen–Wong full zero count used in (F3) and (Z4) of the Polak note. Keep $H = 3 \cdot 1 0^{12}$ , $c_{0} = 2 + γ - lo g (4 π) < 0.05$ and $S_{2} (H) = \sum_{ℑ ρ > H} m_{ρ} / (ℑ ρ)^{2} < 1.48 \cdot 1 0^{- 12}$ . The preceding project derivations (GE1)–(GE4), (GH1), (GJ1)–(GJ2) and (GK1)–(GK6) are used with exactly these premises. The cited finite Robin and zero-verification computations, analytic proofs and density-table computations are not rerun here. The Mantovanelli, Kalyabin, Polak and Nicolas inputs retain their stated preprint/archive status; no complete independent primary-proof audit or fresh SevenSmooth compilation is asserted. Polak’s separate huge finite sweep is not a premise of the added interval.

Uniform arithmetic and Gamma enclosures

Assume, for contradiction, that the same selected clock satisfies

$\frac{7}{2} 1 0^{46} < A \leq \frac{403}{100} 1 0^{46}, α = \frac{73}{50} 1 0^{23} + \frac{1}{2} = \frac{292000000000000000000001}{2} . (GM1)$

Use the whole law $C\sim\operatorname{Gamma}(\text{shape }\alpha, \text{rate }\alpha) $, n o t sc a l e$ \alpha$. Its mean is one, $E (C - 1)^{2} = 1/ α$ and $M_{1} = E C^{- 1} = α / (α - 1)$ . The shape is noninteger and at least $4$ , as required in (GK6). With the complete coefficient $F_{A} (s) = s^{- 1} \int_{A}^{\infty} u^{s - 2} (1 + lo g u) / lo g^{2} u d u$ , put

$H_{α} (s) = \frac{Γ ( α + s - 1 )}{Γ ( α ) α ^{s - 1}}, Z_{α} (A) = ρ \sum m_{ρ} H_{α} (ρ) F_{A} (ρ) .$

Every actual nontrivial zero, both ordinate signs and all multiplicities are included. The preceding absolute-convergence arguments justify this sum; conjugate pairing makes it real. Criticality is used only for $∣ℑ ρ ∣ \leq H$ ; the tail retains every actual $0 < ℜ ρ < 1$ .

On the whole interval (GM1),

$L > 107, 1 0^{23} < A < r := \frac{251}{125} 1 0^{23}, M_{1} < \frac{10001}{10000}, \frac{H ^{2}}{2 ( α + H )} > \frac{1541}{50} . (GM2)$

Indeed, $lo g 10 > 2.3$ and $lo g (7/2) > 1.25$ give $L > 107$ ; $r^{2} = 4.032064 \cdot 1 0^{46} > 4.03 \cdot 1 0^{46}$ . The moment and damping bounds are direct rational comparisons at the fixed $α$ , with no sampling of $A$ .

Dusart’s stronger Theorem 5.2 row is exactly $k = 2$ , $η_{k} = 0.01$ , $x_{k} = 7713133853$ . It applies to every $t \geq A$ in (GM1). Proposition 3.2 includes all higher prime powers, so the decreasing right side gives

$η_{0} (A) := t \geq A sup \frac{∣ ψ ( t ) - t ∣}{t} \leq \frac{0.01}{L ^{2}} + 1.00007 A^{- 1/2} + 1.78 A^{- 2/3} < \frac{0.01}{10 7 ^{2}} + \frac{1.00007}{1 0 ^{23}} + \frac{1.78}{1 0 ^{30}} < 1 0^{- 6} . (GM3)$

This stronger constant is used with its own effective threshold; it is not assigned to the earlier $3.965$ row.

The directed (GK6) and the absolute (GJ2) now give

$- T I_{ψ} (A) \leq T ∣ Z_{α} (A) ∣ < T Z_{α} (A) + (0.52 + 2 η_{0} (A)) \frac{A}{α} + 400 A e^{- α /60000} + \frac{5 M _{1}}{A}, M_{1} (1 + 1/ L) [1.054 c_{0} + 2 (1.0001) A S_{2} (H) e^{- H^{2} / [2 (α + H)]}] .$

Centering in (GK6) occurs before splitting the whole Gamma law at $[0.99, 1.01]$ . Its regular part uses (GM3) for all $t \geq A$ ; its local residual is nonpositive against the positive prime-power measure, including endpoint atoms. The $c < 1$ local part below $A$ uses this sign, not an unprovided prime-error bound below $A$ . All outside scales are paid by (GK5). In particular, $c A < 1$ and the finite endpoint response in (GE3) are retained; (GE4) is used only as a directed upper bound, not an absolute bound.

For $x > 0$ , $e^{x} > \sum_{j = 0}^{149} x^{j} / j!$ . Exact rational evaluation at $x = 1541/50$ gives $e^{1541/50} > (121/50) 1 0^{13}$ . Also $α /60000 > 1 0^{18}$ and $e^{x} > x^{3} /6$ give $e^{α /60000} > 1 0^{53}$ . Substituting these and (GM2)–(GM3) gives the following complete rational allowances:

Contribution	Rational upper allowance	Strict outward bound
Verified head with full coefficient remainder	$(10001/10000) (108/107) (1.054) (0.05)$	$0.0532$
All heights above $H$ , both signs	$2 (10001/10000)^{2} (108/107) r (1.48 \cdot 1 0^{- 12}) / [(121/50) 1 0^{13}]$	$0.0248$
Centered regular and signed local payment	$(0.520002) r / α$	$0.7152$
Every outside Gamma scale	$400 r /1 0^{53}$	$9 \cdot 1 0^{- 28}$
Complete elementary/trivial response, including $c A < 1$	$5 (10001/10000) /1 0^{23}$	$6 \cdot 1 0^{- 23}$

For clarity, the first three rational allowances are respectively $142304229/2675000000$ , $25079915230749/1011484375000000$ , and $208832803200000000000000/292000000000000000000001$ . Summing the rational allowances, or their strict outward bounds, yields

$- T I_{ψ} (A) < 0.79321 throughout (GM1) . (GM4)$

The damping exponent $t^{2} / [2 (α + t)]$ increases for $t > 0$ ; the full $S_{2} (H)$ therefore pays every height above $H$ through infinity. No tail real part is moved to $1/2$ , and no sign or coefficient remainder is omitted in passing from $T Z_{α}$ to its absolute upper allowance.

Transcendental enclosures and the strict core

All decimal constants here denote exact terminating rationals. The load-bearing logarithms can be enclosed by the convergent positive series, with $z = (y - 1) / (y + 1)$ and $1 \leq y \leq 2$ ,

$q_{m} (y) = 2 j = 0 \sum m - 1 \frac{z ^{2 j + 1}}{2 j + 1}, 0 \leq lo g y - q_{m} (y) \leq \frac{2 z ^{2 m + 1}}{( 2 m + 1 ) ( 1 - z ^{2} )} .$

This follows by integrating the geometric series for $1/ (1 - z^{2})$ ; each omitted denominator is at least $2 m + 1$ . Write any rational $y \geq 1$ as $2^{k} v$ , $1 \leq v < 2$ , and add the brackets for $v$ and $k lo g 2$ . With $m = 64$ the resulting rational comparisons give $lo g 2 < 0.6931472$ , $lo g 10 > 2.3$ and $lo g (7/2) > 1.25$ , as used above. Squaring positive rational endpoints gives $1.4142135 < 2 < 1.4142136$ . Positive Taylor prefixes through degree 149 give $e^{17} > 2 \cdot 1 0^{7}$ and $e^{52} > 1 0^{22}$ , as well as the damping comparison used in (GM4). These are analytic enclosures with explicit remainders or one-sided positive prefixes, not floating-point values.

For the existing core function $E$ in (G7), differentiation gives

$E^{'} (L) = \frac{( 2 + lo g 2 ) 2}{L ^{2}} - \frac{13.56}{L ^{3}} + \frac{2.67}{6} e^{- L /6} + (L - 2 + \frac{5}{2 L} + \frac{5}{L ^{2}}) e^{- L /2} > 0 (L \geq 26) .$

The first two terms have sum greater than $(2 L - 13.56) / L^{3} > 0$ ; the other terms are positive. Thus $L > 107$ gives $E (L) > E (107)$ uniformly. Since $107/6 > 17$ and $107/2 > 52$ , the outward enclosures just proved yield

$E (107) > 2 (1.4142135) - \frac{( 2 + 0.6931472 ) ( 1.4142136 )}{107} + \frac{6.78}{10 7 ^{2}} - 2.00014 - \frac{2.67}{2 \cdot 1 0 ^{7}} - \frac{214 + 5/107}{1 0 ^{22}} > 0.79328. (GM5)$

The last comparison is exact rational arithmetic. Together with (G6), this is a whole-interval strict lower bound for the core at the same $N$ .

Contradiction and remaining boundary

By (G1), (GM4) and (GM5),

$T Δ (N) = T I_{ψ} (A) + T D^{*} (A) > 0.79328 - 0.79321 = \frac{7}{100000} > 0 throughout (GM1) . (GM6)$

This contradicts the selected source’s $Δ (N) \leq 0$ . Together with (GL4), it proves the conditional restriction $lo g N > (403/100) 1 0^{46}$ for the same least global maximizer. The open lower endpoint is supplied by (GL4), and the upper endpoint is included by the uniform enclosures; there is no uncovered clock.

This excludes a selected-source interval, not every integer below $exp ((403/100) 1 0^{46})$ , and gives no least-counterexample bound. For arbitrary $n$ the exact $d_{n}$ remains. At surviving selected clocks, the full signed requirement (G9), with its positive elementary correction, remains unpaid. A substantive formal frontier is the directed centered-Gamma transport (GK6), including the endpoint atoms, whole mean-one law and $c A < 1$ branch; scalar wrappers of the rational budget do not close it. No such formal theorem is supplied here. The RH/Robin research target remains active and unproved.

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