bibkey: polak2026finiterobinca authors: Robert Polak year: 2026 title: A Finite Computer-Assisted Verification of Robin’s Inequality via Colossally Abundant Profiles, with Exact Prime-Power Residual Dynamics doi: 10.5281/zenodo.21808589 url: https://doi.org/10.5281/zenodo.21808589 claim: The preprint reports a finite Robin verification and an effective positive full-support core; the core applies at the selected critical clock and, with positive extra-support costs, at arbitrary large GA2 clocks, but does not bound their signed prime-error tails or prove RH. strata_touched: [] license: citation-only triage: anchor

Finite CA-profile verification and the signed residual

The source is Robert Polak, Zenodo preprint, record 21808589, version 1.0.0, published 5 August 2026. The retrieved PDF has 37 pages and SHA-256 9a118f1f0247872458fc9dfbc91dfe067df7bbeb162f46fc1ad11e64f7913b2d. The locators below use printed pages. The full-support identity, Lemma 7, and Proposition 10 with its argument on pp.3–8, and the residual and event interfaces in §8.2–8.8 on pp.18–28 were inspected. This is a primary-source applicability check, not a whole-paper proof audit, independent rerun of its certificates, peer-review claim, or Lean verification.

Finite theorem and residual interface

The preprint claims Robin’s strict inequality for every

$$5041\le n\le 10^{7.1\times 10^{22}}.$$

Its certificate exhausts 3,341,978 colossally abundant exponent profiles, uses an analytic prime-power reduction and a finite-height zero verification, and transfers certified CA endpoints to intervening integers by Robin’s convexity proposition. The stated CA support computation reaches $1.64967\times 10^{23}$.

After the finite theorem, the source derives exact prime-power cell and signed-triangular identities. Its exploratory event scan is explicitly separated from the finite proof: the universal eventwise target needed for an infinite Robin proof remains unproved.

Directly reusable effective core

For a real $x\ge 2$ and a full-support integer $n={\prod }_{p\le x}{p}^{a_p}$ with every $a_p\ge 1$, write

$$\begin{array}{rl}\Delta (n)& =\gamma +\log \log \log n-\log \frac{\sigma (n)}{n},\\ I_{\psi }(x)& ={\int }_x^{\infty }(\psi (t)-t)\frac{1+\log t}{t^2{\log }^{2}t}dt,\\ z(n,x)& =\frac{\log n-x}{x},\\ B_2(n,x)& =\log \left(1+\frac{\log (1+z(n,x))}{\log x}\right)-\frac{z(n,x)}{\log x},\\ R_{\mathrm{core}}(n,x)& =\frac{\log n-\vartheta (x)}{x\log x}-\sum _{p\le x}\log (1-p^{-(a_p+1)}),\\ C_{\mathrm{pp}}(x)& =\frac{\psi (x)-\vartheta (x)}{x\log x}+\sum _{\begin{array}{c}p\le x,m\ge 2\\ p^m>x\end{array}}\frac{1}{m{p}^m}.\end{array}$$

The logarithmic margin is defined when $n>e$; the critical-source application below is entirely above 5040. Proposition 6, p.5, gives

$$\Delta (n)=I_{\psi }(x)+B_2(n,x)+R_{\mathrm{core}}(n,x)-C_{\mathrm{pp}}(x).$$

This is the same $I_{\psi }$ and positive kernel used in the Nicolas comparison. The paper denotes the margin by $\mathcal{G}(n)$; it is not the Gronwall quotient $G(n)=\sigma (n)/(n\log \log n)$ used in the other source notes.

Define the paper’s independent full-support minimum by

$$\begin{array}{rl}{\Phi }_{p,x}(a)& =(a-1)\frac{\log p}{x\log x}-\log (1-p^{-(a+1)}),\\ D^{\ast }(x)& =\sum _{p\le x}\underset{a\in {\mathbb{Z}}_{\ge 1}}{\min }{\Phi }_{p,x}(a)-C_{\mathrm{pp}}(x).\end{array}$$

Lemma 7 and Corollary 8, pp.6–7, give $R_{\mathrm{core}}(n,x)-C_{\mathrm{pp}}(x)\ge D^{\ast }(x)>0$ without a CA, GA1, GA2, or self-tangency premise. The minimization retains full support; its activation rule is not the paper’s CA-family activation rule. Proposition 10, p.8, gives the effective, unbounded continuation

$$\sqrt{x}\log x{D}^{\ast }(x)>D_{\mathrm{lb}}(x)>\frac{1}{2}(x\ge 56048351).$$

The full envelope is explicitly defined on p.7. Put

$$\begin{array}{rlrlrlrl}{L}_x& =\log x,& {\ell }_x& =\frac{L_x+\log 2}{2},& {ϵ}_x& =\frac{3.965}{{\ell }_x^2},& q_x& =1-\sqrt{\frac{2}{x}},\\ a_x& =\frac{L_x}{\sqrt{2}{\ell }_x},& j_x& =\frac{L_x}{\sqrt{2}}\left(\frac{q_x}{{\ell }_x}-\frac{1}{{\ell }_x^2}\right),& & & & \\ s_x& =(1-{ϵ}_x)j_x-2{ϵ}_x{a}_x-\frac{1}{\sqrt{x}},& & & & & & \\ D_{\mathrm{lb}}(x)& =\left(\frac{1}{2}-\frac{1}{3\sqrt{2x}}\right)s_x.& & & & & & \end{array}$$

Here $a_x,j_x,s_x$ are the paper’s $A(L),J(L),S(L)$ at $L=\log x$; $A=\log N$ below is the source clock. Proposition 10 also states that $D_{\mathrm{lb}}$ is increasing on this range. Retaining this published function avoids discarding its effective surplus above $1/2$.

That continuation uses the cited Dusart theta bound and the explicit positive contributions from $\sqrt{2x}<p\le x$. It is separate from the finite all-integer Robin theorem and needs no finite-height RH input in the inspected argument. Its cited theta theorem is an external premise; its original proof was not independently audited here. The finite support sweep and the analytic continuation are reused, not repeated computations or new reserve estimates.

Application at the same critical source and clock

Under a Robin counterexample, select the least global maximizer $N>5040$ by the Caveney–Nicolas–Sondow reduction. Write $A=\log N$, $P=P^{+}(N)=p_k$, and let $P^{+}$ be the next prime. This source is CA with initial prime support and belongs to $U_1$. The existing seven-smooth exclusion rules out $k\le 4$. Hence the Kalyabin endpoint conditions apply and give $P<A<P^{+}$ at this same integer.

There is no prime between $P$ and $A$, so the full-support products and sums above contain exactly the actual primes of $N$ when $x=A$. The same source also attains the full-support core minimum. Its proper GA1 condition gives deletion comparisons in the range $N/p>e$, and GA2 gives every one-prime insertion comparison. The existing archived direct tangent-CA bridge, source §4, thm:direct-bridge, therefore identifies this very $N$ as the regular CA optimizer at

$${\epsilon }_A=\frac{1}{A\log A}.$$

This is precisely the price in Polak’s full-support minimization. Put $Z(m)=\sigma (m)/m$. For each actual $p\le A$, the local price objective and ${\Phi }_{p,A}$ differ by an additive constant and a sign. Equivalently, the source’s minimizer calculation on p.6 gives

$${\Phi }_{p,A}(a+1)-{\Phi }_{p,A}(a)={\epsilon }_A\log p-\log \frac{Z(p^{a+1})}{Z(p^a)}.$$

Thus the actual $a_p=v_p(N)$ attains ${\min }_{a\ge 1}{\Phi }_{p,A}(a)$. Regularity excludes a tied layer. The correspondence uses the clock price ${\epsilon }_A$, rather than identifying it with the support-parametrized CA-family price discussed at the end of Polak’s p.6. Initial support and $P<A<P^{+}$ keep exactly the same prime set in both minimizations. Consequently

$$R_{\mathrm{core}}(N,A)-C_{\mathrm{pp}}(A)=D^{\ast }(A).$$

Moreover $z(N,A)=0$ and $B_2(N,A)=0$ exactly. Proposition 6 now gives the paper-level identity

$$\Delta (N)=I_{\psi }(A)+R_{\mathrm{core}}(N,A)-C_{\mathrm{pp}}(A)=I_{\psi }(A)+D^{\ast }(A).$$

The core optimization slack is zero at this selected source. Its exact strict Robin condition is $I_{\psi }(A)>-D^{\ast }(A)$; equality yields $\Delta (N)=0$ and does not satisfy strict Robin. This application identifies the actual minimizer and pays no new signed estimate.

The numerical threshold also has an existing supplier. Axler’s finite stop, author version 2110.13478v3, Lemma 2.3, p.3, cites strict Robin for $5041\le n\le N_K$, where $K=999999476056$ and $N_K$ is the $K$th primorial. Therefore the counterexample-level source has $N>N_K$, and simply

$$A>\log N_K\ge K\log 2>\frac{K}{2}>56048351.$$

This consumes the cited finite verification; it neither reruns nor extends it, and uses no theta estimate to compare these thresholds. Proposition 10 thus gives at this fixed source

$$\boxed{\Delta (N)>I_{\psi }(A)+\frac{1}{2\sqrt{A}\log A}.}$$

In particular a hypothetical selected counterexample must satisfy $\sqrt{A}\log A{I}_{\psi }(A)<-1/2$. A source-specific lower bound $\sqrt{A}\log A{I}_{\psi }(A)\ge -1/2$ would suffice to exclude it.

The same application retains the complete envelope and gives

$$\sqrt{A}\log A\Delta (N)>\sqrt{A}\log A{I}_{\psi }(A)+D_{\mathrm{lb}}(A).$$

Thus it is enough to prove the weaker source-specific condition

$$\boxed{\sqrt{A}\log A{I}_{\psi }(A)\ge -D_{\mathrm{lb}}(A).}$$

It is weaker than the half-unit condition because $D_{\mathrm{lb}}(A)>1/2$. Conversely, a hypothetical selected counterexample must satisfy the stronger necessary condition $\sqrt{A}\log A{I}_{\psi }(A)<-D_{\mathrm{lb}}(A)$. The strict inequality in Proposition 10 supplies strict Robin even if the sufficient tail condition is attained with equality. The signed condition remains unproved. Its explicit core envelope and cutoff are paid at this same finite source, without an unbounded critical-source sequence or Kalyabin’s existential $K_{\epsilon }$. This does not make that separate asymptotic cutoff effective.

This is reuse of Polak’s core estimate on the endpoint-restricted source, including its full published envelope, not a new reserve theorem, numerical evaluation, or signed Robin estimate. Positivity of $D^{\ast }$ does not assert positivity of $\Delta$; the unknown signed $I_{\psi }(A)$ has been retained. Neither the complete divisor-deletion comparisons nor the all-multiplier GA2 comparisons have yet supplied its needed bound. The application adds no Lean declaration or formal certification of the external analytic or finite-verification premises.

The effective Nicolas–GA2 application retains this same source, minimizer, and clock and supplies

$$\sqrt{A}\log A{D}^{\ast }(A)>\mathcal{E}(\log A)>D_{\mathrm{lb}}(A)+\mathrm{0.01.}$$

Its explicit function $\mathcal{E}$ and uniform comparison use Nicolas’s effective envelope ratio, cancellation of the absent-prime suffix by the source’s GA2 comparisons, and Dusart’s published prime-power bound. All thresholds are paid by the Axler finite stop already used here. Thus the weaker source-specific condition $\sqrt{A}\log A{I}_{\psi }(A)\ge -\mathcal{E}(\log A)$ suffices for strict Robin. That signed condition is unproved. This is a paper-level application, without a new finite verification, originality claim, or Lean result.

Application to arbitrary GA2 sources, including extra prime support

The full-support core can also be used at a GA2 integer that is not known to be GA1, CA, or a global maximizer. Let $N$ be GA2 and put $A=\log N\ge 3$. This application uses its own clock and actual exponents $a_p=v_p(N)$ throughout.

First, GA2 forces every prime $p\le A$ to divide $N$. Indeed, if such a prime were absent, multiplicativity would give

$$\frac{G(Np)}{G(N)}=\left(1+\frac{1}{p}\right)\frac{\log A}{\log (A+\log p)}>1,$$

because

$$\log (A+\log p)<\log A+\frac{\log p}{A}\le \left(1+\frac{1}{p}\right)\log A.$$

The last comparison follows from $p\log p\le A\log A$. This contradicts GA2. This support argument needs only $A>1$; the stronger $A\ge 3$ ensures that the full-support factor

$$n_0=\prod _{p\le A}{p}^{a_p}$$

contains both 2 and 3 and hence is at least $6>e$, as required for its logarithmic margin. No assertion that $N$ has no primes above $A$ is needed.

For an occupied prime $p>A$, define its extra-support cost by

$${\Xi }_{p,A}(a)=a\frac{\log p}{A\log A}-\log Z(p^a),Z(m)=\frac{\sigma (m)}{m},a\ge 1.$$

These costs are strictly positive. The binomial expansion gives $Z(p^a)={\sum }_{j=0}^a{p}^{-j}\le (1+1/p{)}^a$, so

$$\log Z(p^a)\le a\log (1+1/p)<\frac{a}{p}<a\frac{\log p}{A\log A}.$$

The final strict comparison uses $p\log p>A\log A$.

Apply Proposition 6 to $n_0$ at $x=A$. Put $A_0=\log n_0$. The budget transport to the same original $N$ is exactly

$$\Delta (N)-\Delta (n_0)=\log \frac{\log A}{\log A_0}-\sum _{\begin{array}{c}p>A\\ p\mid N\end{array}}\log Z(p^{a_p}),$$

whereas

$$B_2(n_0,A)=\log \frac{\log A_0}{\log A}-\frac{A_0-A}{A\log A}.$$

Their logarithmic terms cancel, and $A-A_0={\sum }_{p>A,p\mid N}{a}_p\log p$. Thus the transported identity is

$$\boxed{\Delta (N)=I_{\psi }(A)+\sum _{p\le A}{\Phi }_{p,A}(a_p)-C_{\mathrm{pp}}(A)+\sum _{\begin{array}{c}p>A\\ p\mid N\end{array}}{\Xi }_{p,A}(a_p).}$$

Taking the existing full-support minimum gives

$$\Delta (N)\ge I_{\psi }(A)+D^{\ast }(A)+\sum _{\begin{array}{c}p>A\\ p\mid N\end{array}}{\Xi }_{p,A}(a_p)\ge I_{\psi }(A)+D^{\ast }(A).$$

This is an inequality at arbitrary GA2 sources. Their actual exponents are not asserted to attain the core minimum. Proper-GA1 deletion conditions and the selected CA source’s equality have not been transported to this larger class. The stronger Nicolas allowance requires the separate actual-core argument linked below.

For $A\ge 56048351$, the same Proposition 10 gives $\sqrt{A}\log A{D}^{\ast }(A)>D_{\mathrm{lb}}(A)>1/2$. Caveney–Nicolas–Sondow’s Fact 2 gives $G(N)\ge e^{\gamma }$, hence $\Delta (N)\le 0$. Consequently every such GA2 source must satisfy

$$\boxed{\sqrt{A}\log A{I}_{\psi }(A)<-D_{\mathrm{lb}}(A)<-\frac{1}{2}.}$$

The Nicolas envelope on actual GA2 cores uses $Z(N)\le \Sigma (N)$ without CA equality and pays the primorial suffix even when its primes already divide $N$. For $N\ge N^{(0)}$ and $\log A\ge 26$, it strengthens the necessary signed condition to $\sqrt{A}\log A{I}_{\psi }(A)<-\mathcal{E}(\log A)$. Its bound is on the whole actual core in the displayed identity, not on $D^{\ast }(A)$.

The published conditional GA2 family has $N_i\to \infty$ under RH failure. Therefore an eventual lower bound $\sqrt{A}\log A{I}_{\psi }(A)\ge -\mathcal{E}(\log A)$ valid at every sufficiently large GA2 clock would contradict that existing family. This route would need no effective starting point for the eventual bound. It does need a proof of the signed bound on these actual clocks; an arbitrary sequence of favorable real cutoffs is insufficient.

The application reuses the published full-support identity, minimum, effective core, and conditional infinitude. It supplies the missing extra-support accounting needed to use that core on the larger source class. It is a paper-level interface assessment, not a new core estimate, RH criterion, signed-tail bound, originality claim, or Lean result. The fixed global source above retains its stronger structural inputs; this unbounded-source route retains the same unpaid signed tail.

Published event dynamics and the unpaid prime-state work

The same manuscript already supplies a continuous-flow and prime-power event program. Reuse this published construction rather than treating an affine recurrence, a cone, or a quadratic energy as a new FIB estimate. The source’s auxiliary buffer is

$$B(t)=\sum _{\rho }\frac{e^{(\rho -1/2)t}}{\rho (1-\rho )},C=B(0)=2+\gamma -\log (4\pi ),$$

with the complete nontrivial-zero multiset. Theorem 15, pp.19–20, makes eventual nonnegativity of $\mathcal{P}(t):=C-B(t)$ equivalent to RH. Here $\mathcal{P}$ is the paper’s $P(t)$, not the support prime $P$ above. It is not $\Delta (N)$ or the normalized $I_{\psi }(A)$.

Equation (8.24), p.22, identifies the regularized buffer exactly with the classical Chebyshev primitive:

$$\hat{B}(\log x)=-\sqrt{x}{\int }_x^{\infty }\frac{\psi (u)-u}{u^2}du-\frac{\log (2\pi )}{\sqrt{x}},x>1.$$

The explicit trivial-zero correction is $B(t)-\hat{B}(t)={\sum }_{j\ge 1}{e}^{-(2j+1/2)t}/(2j(2j+1))$ for $t>0$. This reuses the existing primitive interface; the additional $(1+\log u)/{\log }^{2}u$ weight in $I_{\psi }$ remains distinct.

For $q=p^m$, put ${\tau }_q=\log q$ and $J_q=\Lambda (q)/\sqrt{q}$. Equations (8.26)–(8.30), pp.23–24, define, for $t\ge \log 2$ and $\mathcal{P}(t)>0$,

$$\begin{array}{rlrl}g(t)& =e^{t/2}\left(1-\frac{e^{-3t}}{1-e^{-2t}}\right),& k(t)& =g(t)-C/4,\\ Q(t)& =\sqrt{\mathcal{P}(t{)}^2/4+2k(t)\mathcal{P}(t)},& E_B(t)& ={\mathcal{P}}^{\prime }(t{)}^2-Q(t{)}^2.\end{array}$$

On an open prime-power cell $E_B^{\prime }=-2k^{\prime }\mathcal{P}$; at an event, $\Delta {\mathcal{P}}^{\prime }=-J_q$ and $\Delta E_B=J_q^2-2J_q{\mathcal{P}}^{\prime }({\tau }_q^{-})$. Lemma 18 supplies the initial positive cone at $q=2$. Theorem 19 preserves it, and concludes RH, provided that every next actual event with the inherited cone obeys

$$Q({\tau }_q)+{\mathcal{P}}^{\prime }({\tau }_q^{-})>J_q.$$

This universal event premise remains unpaid. Propositions 20–21, p.25, give its exact signed triangular-window form; they do not bound the signed window. Proposition 22, p.26, gives the central-secant estimate $|\mathcal{P}(t+h)-\mathcal{P}(t-h){|}^2\le 4\mathcal{P}(t)\mathcal{P}(2h)$ assuming RH. It cannot be imported as an unconditional local estimate in a proof of RH. The source’s adaptive window has additive scale $\sqrt{q}\log q$, rather than an independently chosen coarse interval.

The separate energy audit makes the missing joint quantity explicit. For $r\ge 0$, ${ϵ}_r=(1+t{)}^{-r}$ and $E_r=(1-{ϵ}_r/2){\hat{B}}^2+2{ϵ}_r{\hat{B}}^{\prime 2}$, equation (8.52), p.27, gives

$$\Delta E_r=4{ϵ}_r({\tau }_q)J_q{\hat{B}}^{\prime }({\tau }_q^{-})+2{ϵ}_r({\tau }_q)J_q^2.$$

Proposition 23, p.28, supplies ${\sum }_q(1+\log q{)}^{-r}\Lambda (q{)}^2/q<\infty$ exactly when $r>2$. It pays the square-kick budget, not the signed state-prime work in the first term. The source also excludes a fixed positive-definite quadratic form decreasing for every homogeneous cell state, and shows that a positive semidefinite quadratic form invariant under every common translation of its two characteristic coordinates must annihilate $(1,1)$. This restriction concerns universal state-independent energies; it does not exclude an estimate adapted to the actual arithmetic orbit.

These source results identify the required joint estimate and prevent repeating the generic flow, cone, or square-budget work. No bound for the actual signed work, proof of the universal kick premise, or map from the selected CA source or FIB addresses supplying that premise is established here. The selected-source condition $I_{\psi }(A)>-D^{\ast }(A)$ therefore remains unchanged and unproved.

The cited Ford–Soundararajan–Zaharescu sequel and its original fixed smooth-test supplier do identify $-J_q/(2\pi )$ in the ordinate-phase average for each fixed integer $q>1$. Their fixed-scale expansion does not bound the actual weighted buffer or the signed state-prime work above. The golden eigen-scale has no correction density in that theorem; this is a parameter application, not an FIB-to-Robin estimate.

Exact FIB coordinates for the published forced flow

The source’s equation (8.17), p.22, already gives the regularized dynamics

$${\hat{B}}^{\prime \prime }-\frac{1}{4}\hat{B}=-e^{t/2}+\sum _{q=p^m}{J}_q\delta (t-\log q),t>0.$$

Reuse this equation and its finite right-limit initial data. The following is a linear coordinate and clock application, not a new event system or estimate. The existing FIB composition frame, §§3 and 149, has $M=\left(\begin{array}{cc}0& 1\\ 1& 1\end{array}\right)$, $S=M^3$ and $C=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$. Here its coordinates are extended to ${\mathbb{R}}^2$; a real state in these coordinates is not thereby the composition of a finite FIB tree.

Remove the known resonant drift, including its derivative:

$$Y(t)=\hat{B}(t)+t{e}^{t/2},Y^{\prime }(t)={\hat{B}}^{\prime }(t)+(1+t/2)e^{t/2}.$$

Then $Y^{\prime \prime }-Y/4={\sum }_q{J}_q\delta (t-\log q)$. Define the simultaneous state transport by

$$\left(\begin{array}{c}Y\\ Y^{\prime }\end{array}\right)=T\left(\begin{array}{c}a\\ b\end{array}\right),T=\left(\begin{array}{cc}2& 1\\ 0& \sqrt{5}/2\end{array}\right),a=\frac{Y}{2}-\frac{Y^{\prime }}{\sqrt{5}},b=\frac{2Y^{\prime }}{\sqrt{5}}.$$

On each open event cell the homogeneous propagator in this frame is

$$\Phi (s)=\cosh (s/2)I+\frac{2M-I}{\sqrt{5}}\sinh (s/2).$$

Its generator is $(2M-I)/(2\sqrt{5})$ and its eigenvalues are $e^{s/2},e^{-s/2}$. For $h=4\log \phi$,

$$\Phi (h)=M^2,\Phi (3h)=M^6=S^2,C\Phi (s)C^{-1}=\Phi (-s).$$

These are substitutions into the existing hyperbolic flow and FIB matrix identities. $M$ and the single three-position step $S$ have determinant $-1$, whereas this homogeneous propagator has determinant $1$. Thus the even steps above match the continuous flow; an odd FIB step would require an additional orientation reversal. Evaluating $\Phi (h)$ does not assert that an actual event-free cell has length $h$.

The original readout must be transported too:

$$\hat{B}(t)=2a(t)+b(t)-t{e}^{t/2},{\hat{B}}^{\prime }(t)=\frac{\sqrt{5}}{2}b(t)-(1+t/2)e^{t/2}.$$

Neither readout is the original FIB quantity $q(a,b)=2a+3b$. For a passive frame ${\mathbf{z}}_k=C^k\mathbf{z}$, transport the input vector by $C^k$, the propagator by $C^k\Phi C^{-k}$, and each linear readout row $\ell$ by $\ell C^{-k}$. The clock and the explicit drift terms are retained. An active rotation of the state alone changes the forcing direction and is not an invariance of the given prime trajectory.

Retain the actual prime inputs in every step

At ${\tau }_q=\log q$, continuity of $Y$ and the derivative jump $J_q$ give

$$\Delta {\mathbf{z}}_q=\frac{J_q}{\sqrt{5}}\left(\begin{array}{c}-1\\ 2\end{array}\right),\mathbf{z}=(a,b{)}^{\mathsf{T}}.$$

Taking the initial and final states on the right, the actual fixed-time update is therefore

$$\mathbf{z}(t+h)=M^2\mathbf{z}(t)+\sum _{t<{\tau }_q\le t+h}\Phi (t+h-{\tau }_q)\frac{J_q}{\sqrt{5}}\left(\begin{array}{c}-1\\ 2\end{array}\right).$$

For two three-position windows, replace $h$ by $3h$ and $M^2$ by $M^6$; the entire corresponding event sum remains. This is the finite forced propagator formula, with every actual prime power and its timing included. The input term is determined by the prime measure, rather than one of the five fixed integer translations $0,(1,0),(0,1),(2,1),(1,1)$.

Already the event $q=2$ has first component $-\log 2/\sqrt{10}\in (-1,0)$. Thus this kick is outside ${\mathbb{Z}}^2$ in the transported frame. An integral unimodular frame change generated by $M,J,C$ cannot turn it into an integer vector: its inverse preserves ${\mathbb{Z}}^2$. This obstructs identifying these event maps with the original integer-coordinate five-pattern translations. A response computed from a full FIB address can still reconstruct $q$, its event time and $\Lambda (q)$; that is an additional arithmetic response map, whose weights and estimates must be retained.

Read Robin’s actual residual at the same clock

For the tail target, retain the existing prime-power cutoff as a scalar accumulator of this same event history:

$${\mathcal{P}}_{\Pi }(t)={\Pi }_r(e^t)=\sum _{p^m\le e^t}\frac{1}{m{p}^m},\Delta {\mathcal{P}}_{\Pi }({\tau }_q)=\frac{J_q{e}^{-{\tau }_q/2}}{{\tau }_q}.$$

It is constant between events. The existing cross-family signed-tail identity and the source’s characteristic coordinates give, for $t>0$,

$$\begin{array}{rl}\psi (e^t)& =e^{t/2}[-a(t)+(\phi -1)b(t)]-\log (2\pi ),\\ I_{\psi }(e^t)& =\frac{\psi (e^t)-e^t}{e^{t}t}-{\mathcal{P}}_{\Pi }(t)+\log t+\gamma .\end{array}$$

These are transported readouts of already established paper identities, not a new tail formula. The accumulator keeps the distinct $1/\log q$ factor in the event weight; it can also be computed from the complete forced history. It is not replaced by the golden quadratic invariant.

At the selected critical source above, under its cited premises, put $A=\log N$ and $t=\log A$, not $t=\log N$. The exact core application then reads the original Robin margin as

$$\Delta (N)=\frac{\psi (A)-A}{A\log A}-{\Pi }_r(A)+\log \log A+\gamma +D^{\ast }(A).$$

This retains the same integer, cutoff, prime events and complete weights. The fixed $h$ grid is a grid of real cutoffs; it constructs no successor critical integer and supplies no coverage of critical sources by its nodes. The homogeneous matrix correspondence supplies no sign for this centered combination; its required lower bound remains the original signed-tail obligation.

The transported conservation law retains the centering problem

The existing golden form becomes

$$Q_F(a,b)=a^2+ab-b^2=\frac{Y^2}{4}-Y^{\prime 2}.$$

It is constant during homogeneous propagation, while an event changes it by $-2J_q{Y}^{\prime }({\tau }_q^{-})-J_q^2$. Equations (8.19)–(8.21) identify its actual value without new spectral assumptions. With $c=\log (2\pi )$ and ${\psi }_r(x)={\sum }_{q\le x}\Lambda (q)/q$ they give

$$\begin{array}{rl}Y(t)& =e^{t/2}[{\psi }_r(e^t)+\gamma +1]-e^{-t/2}[\psi (e^t)+c],\\ Y^{\prime }(t)& =\frac{1}{2}{e}^{t/2}[{\psi }_r(e^t)+\gamma +1]+\frac{1}{2}{e}^{-t/2}[\psi (e^t)+c],\\ Q_F(\mathbf{z}(t))& =-[{\psi }_r(e^t)+\gamma +1][\psi (e^t)+c].\end{array}$$

The factors are positive and nondecreasing, so this actual product is constant between events and decreases at each event. This is a product law for the counts, not a positive norm bound for the centered buffer. At any fixed time, a fixed negative $Q_F$ allows arbitrarily large $|Y|$ on its real level set; subtracting $t{e}^{t/2}$ does not change that fact. This states the limitation of this scalar invariant on the real carrier, not a no-go for additional constraints on the actual arithmetic orbit. In particular Polak’s retained energy jump still involves the original ${\hat{B}}^{\prime }$ and its signed state-prime work, not just this product.

This coordinate interface exhibits the common expanding and contracting directions and transports the actual forcing and observer. It supplies no estimate for the retained input sum or the centered cancellation, no cone-kick bound, and no proof of $I_{\psi }(A)>-D^{\ast }(A)$ or RH. It is a paper application with no new Lean declaration or originality claim.

The short-interval zero-count supplier refines a specified absolute frequency-block allowance in the actual $I_{\psi }$ formula, with the source’s ordinate thresholds and multiplicities retained. It supplies no sign for the complete response, does not control the remaining zeros, and leaves the same critical-source estimate unpaid.

Earlier support-harmonic route: the envelope is still a target

The author’s Derived Prime-Harmonic Envelope on CA Support, dated 4 June 2026, is a ten-page primary supplement, with PDF SHA-256 6a65507bde32150569791b632866485fe99d86533ffe77104ab1c823408d2a21. Its abstract, §§2–8 and Analytic Target 1 on p.9 were inspected. The author’s scope statement identifies it as an earlier corrected-status paper, rather than a later completion of the finite-verification manuscript. It explicitly derives a required prime-harmonic envelope without proving its infinite-range validity. Its cited estimates and numerical tables are not independently audited or rerun here; no Lean verification is claimed.

To keep the support variable distinct from the current $A=\log N$, write

$${\mathcal{A}}_p(x)=\sum _{p\le x}\frac{1}{p}-\log \log x,$$

and let $B_1$ be the Meissel–Mertens prime constant. On one actual sampled CA-support block $1<Y<x$, with $\nu =\pi (x)-\pi (Y)>0$, put

$$\begin{array}{rl}H& =\log \frac{\log x}{\log Y},\mu =H/\nu ,\\ C_2(Y,x)& =\sum _{Y<p\le x}\frac{1}{p(p-1)},\\ D_{\mathrm{br}}(Y,x)& =\nu [1-(1+\mu )e^{-\mu }].\end{array}$$

The source uses a certified lower divisor-deficit envelope to form the reserve $R^{\Theta }(x)$, and retains the incoming certified upper ledger $U(Y)$. Its equations (11)–(16), pp.6–7, already show that the sufficient first-moment block gate $M_1(Y,x)\le R^{\Theta }(x)-U(Y)$ is equivalent to

$${\mathcal{A}}_p(x)\le {\mathcal{A}}_{\mathrm{req}}^{\Theta }(Y,x):={\mathcal{A}}_p(Y)-C_2(Y,x)+e^{\mu }[R^{\Theta }(x)-U(Y)+D_{\mathrm{br}}(Y,x)].$$

Equations (19)–(22) therefore express the required upper envelope as

$${\mathcal{A}}_p(x)-B_1\le \frac{C_{\mathrm{req}}^{\Theta }(x)}{\sqrt{x}\log x},C_{\mathrm{req}}^{\Theta }(x)=[{\mathcal{A}}_{\mathrm{req}}^{\Theta }(Y,x)-B_1]\sqrt{x}\log x.$$

This is the author’s existing gate and target, not an unconditional prime estimate or a newly derived Robin criterion. The required constant depends on the same block’s incoming ledger, prime count, harmonic remainder and reserve. No fixed lower bound for that constant or infinite-range certification of the gate is supplied. Replacing the actual deficit by its smaller analytic lower envelope reduces the admissible budget, as source equation (20) records. The sampled table does not establish coverage of all critical sources. Closing a non-strict block gate also does not by itself certify the final strict Robin margin; the ledger’s transfer and strictness obligations remain.

At the currently selected integer, the support endpoint is $x=P^{+}(N)$, whereas the retained signed-tail estimate uses $A=\log N$. The already-paid condition $P^{+}(N)<A<P_{\mathrm{next}}^{+}(N)$ does not identify these cutoffs. Applying this earlier route would require a certified incoming ledger and block coverage at that same source, together with the still-missing upper envelope. It does not supply $I_{\psi }(A)>-D^{\ast }(A)$ by renaming the cutoff. The supplement’s $\beta (x)$ is the Euler product ${\prod }_{p\le x}p/(p-1)$, not the FIB atom $\beta =\rho (\alpha )$; equality of the symbol supplies no bridge.

Boundary for FIB

The golden-field Mertens source uses prime-ideal norms, retains a separate $L(s,{\chi }_5)$ response and identifies unit multiples under the ideal observation. Its hypotheses and cutoff accounting do not supply this ordinary same-source tail bound.

The finite certificate is organized by CA exponent profiles and consecutive-CA interpolation, whereas a FIB family is specified by additive Zeckendorf windows or congruence classes. A FIB address alone does not certify the initial prime support or the endpoints used above. Those conditions are supplied here for the selected critical source by the cited arithmetic results. The source supplies no estimate for its remaining signed residual, or for the pointwise signed term in FIB §250. Its finite verification range remains an external boundary check; repeating that computation would not advance RH.

For sufficiently large supports, the native clock is outside the negative critical-damping regime

The repository’s prime-prefix continuation §§441–445 studies a complete normalized first integral, rather than the complete Robin response. Its existing unique-zero and sign results can be applied to the clock supplied above; this application is not a new prime-error estimate or a Lean verification.

Put $z=P$, let $p=P^{+}$ be the next prime, and use the insertion clock $L=\log p$. With the notation of §441,

$$E_P(s)=\prod _{q\le P}(1-q^{-s}),C_P=E_P(1{)}^{-1},F_P(v)=C_P{E}_P(1+v/L),$$

$$H_P(v)=F_P(v)(1-e^{-v})-v,J_P(\sigma )={\int }_0^{\infty }{e}^{-\sigma v}\frac{H_P(v)}{v^2}dv.$$

The zero-displacement first integral at the actual clock $x=A$ uses ${\sigma }_A=\log A/L$: the Laplace change of variable $v=Lu$ sends $e^{-u\log A}$ to $e^{-{\sigma }_{A}v}$. Both prefixes in the insertion comparison retain this same $L$, $A$, and zero displacement. The already supplied $P<A<p$ and Bertrand’s theorem give

$$1-\frac{\log 2}{L}\le \frac{\log P}{\log p}<{\sigma }_A<1.$$

Thus ${\sigma }_A\to 1$ uniformly over $A\in (P,p)$ as $P\to \infty$. In contrast, §§441 and 445 locate the eventual unique damping zero at ${\sigma }_P\sim t_{\ast }{L}^{-C}$, where $C=e^{{\gamma }_E}>1$ and $t_{\ast }>0$. The existing sign on the upper side of this zero therefore yields, for all sufficiently large prime supports and every such $A$,

$$J_P({\sigma }_A)>0,I_p({\sigma }_A)-I_P({\sigma }_A)=\frac{J_P({\sigma }_A)}{p-1}>0.$$

This is an eventual statement over eligible parameters. No effective threshold for the one selected critical integer is supplied here, and no unbounded sequence of critical maximizers is assumed.

The negative correction in §445 is evaluated at $\sigma =t_{\ast }{L}^{-C}$. In the same native-clock substitution this corresponds to $x=\exp (t_{\ast }{L}^{1-C})\to 1$, rather than $x=A\in (P,p)\to \infty$. It therefore cannot be inserted at the actual source by identifying the two damping parameters.

The normalized $J_P(\sigma )$ here is not the integer-row kernel $J_A^{{\eta }_P}(n)$ in the original volume. Insertion changes the Euler normalization; the other density terms, actual rough Möbius weights, all integer rows and the complete complement are still required. Neither insertion sign supplies the sign of $I_{\psi }(A)+D^{\ast }(A)$ or the missing same-source signed-tail bound.

Finite-height residual bounds at the actual critical clock

The effective core application above can pay the complete signed residual on an explicit finite interval. This strengthens a restriction on the selected global maximizer, not Robin’s inequality for every integer in a larger interval. No zero verification or CA-profile enumeration is repeated.

Keep the same selected source $N$, $A=\log N$, $L=\log A$, and $T_A=\sqrt{A}L$. In this paragraph the verified zero height is denoted $H_0=3\cdot 10^{12}$, to distinguish it from $T_A$. Theorem 1 of Platt–Trudgian, arXiv:2004.09765v1, printed p.2, verifies the stronger height $3000175332800$, and hence that all zeros with $0<\Im \rho \le H_0$ have real part $1/2$. The published version is Bulletin of the London Mathematical Society 53 (2021), 792–797, DOI 10.1112/blms.12460. This existing finite-height theorem is an external input; no global RH premise or verification beyond $H_0$ is added.

Reusing the complete residual envelope

Polak’s §5.3–5.5, printed pp.10–12, equations (5.8), (5.11) and (5.14), give at any real $A>1$

$$T_A{I}_{\psi }(A)\ge -C_{\mathrm{low}}(A)-C_{\mathrm{high}}(A),\text{\hspace{1em}(F1)}$$

where, putting $c_0=2+\gamma -\log (4\pi )$,

$$C_{\mathrm{low}}(A)=c_0(1+3/L+4/L^2)+\frac{\log (2\pi )}{\sqrt{A}},$$

$$C_{\mathrm{high}}(A)\le \sqrt{A}(1+A^{-1})\left[(1+1/L)S_2(H_0)+2(1/L+2/L^2)S_3(H_0)\right].\text{\hspace{1em}(F2)}$$

Here $S_j(H_0)={\sum }_{\Im \rho >H_0}(\Im \rho {)}^{-j}$ counts positive ordinates with multiplicity. The source’s (5.13), using the published Hasanalizade–Shen–Wong zero-count bound, supplies the coarser constants

$$S_2(H_0)<1.48\cdot 10^{-12},S_3(H_0)<5\cdot 10^{-25}.\text{\hspace{1em}(F3)}$$

These constants weaken the source’s displayed directed enclosures; the underlying zero-count proof and scalar evaluations are not independently rerun here. Its functional-equation pairing allows off-line quartets and all multiplicities in the unverified high part. The positive trivial-zero contribution is discarded only in the permitted lower-bound direction. Thus (F1) bounds the entire integral over $[A,\infty )$, including its unverified-zero contribution. No tail beyond a finite endpoint is omitted.

The original CA-support certificate also needs its nonlinear bridge loss $C_{B_2}$. At this actual source clock, the already established $B_2(N,A)=0$ exactly, so that term is absent. Equations (F1)–(F3) do not use Büthe’s support-to-size estimate or assume the source’s segmented monotonicity checks outside their certified range.

A uniform finite exclusion

Put

$$X=21\cdot 10^{22}.$$

The Axler finite stop already used above gives $A>K/2>e^{26}$. The elementary constant bounds $0<c_0<0.05$ and $\log (2\pi )<2$ may be used throughout. For example, $\gamma <0.58$ and $\log (4\pi )>2.53$ give the upper bound on $c_0$. The published core application supplies $T_A{D}^{\ast }(A)>\mathcal{E}(L)$, with $\mathcal{E}$ increasing for $L\ge 26$ and $\mathcal{E}(26)>0.65$. It also gives $\mathcal{E}(50)>0.75$. An entirely rational lower comparison is

$$\mathcal{E}(50)>2.828-\frac{2.70\cdot 1.415}{50}+\frac{6.78}{2500}-2.00014-\frac{2.67}{4000}-\frac{101}{6\cdot 10^{10}}>\mathrm{0.75.}\text{\hspace{1em}(F4)}$$

For its exponential bounds, the same positive-series inequalities used above give $e^{25/3}>54^2(25/18)>4000$ and $e^{25}>54^6(8/3)>6\cdot 10^{10}$. Also $e^{25}<(11/4{)}^{25}<10^{11}$.

Apply (F2) uniformly on two overlapping clock ranges. On $e^{26}<A\le e^{50}$, use $L>26$, $\sqrt{A}<10^{11}$ and $1+A^{-1}<1.0001$. On $e^{50}\le A\le X$, use $L\ge 50$, $\sqrt{A}<4.6\cdot 10^{11}$ and the same last factor. The square-root upper bound follows from $(4.6\cdot 10^{11}{)}^2>X$. Equations (F2)–(F3), with $2(1/26+2/26^2)<0.1$ and $2(1/50+2/50^2)<0.05$, then give

For the first low-zero bound, $e^{13}>400000$ bounds its constant term. For the second, $e^{25}>6\cdot 10^{10}$ does so. The high-zero comparisons are the exact rational inequalities

$$10^{11}(1.0001)\left[\frac{27}{26}(1.48\cdot 10^{-12})+0.1(5\cdot 10^{-25})\right]<0.155,$$

$$(4.6\cdot 10^{11})(1.0001)\left[1.02(1.48\cdot 10^{-12})+0.05(5\cdot 10^{-25})\right]<\mathrm{0.695.}$$

The table therefore covers the entire possible interval $K/2<A\le X$, not a selection of endpoints. Combining (F1) with the exact same-source identity $\Delta (N)=I_{\psi }(A)+D^{\ast }(A)$ gives

$$\boxed{\sqrt{A}\log A\Delta (N)>0.001(K/2<A\le X).}\text{\hspace{1em}(F5)}$$

Such a source would satisfy strict Robin and could not be the selected counterexample-level global maximizer. Conditional on the cited source reduction, analytic bounds and finite-verification inputs, a failure of RH must therefore have its selected least global maximizer at

$$\boxed{\log N>21\cdot 10^{22}.}\text{\hspace{1em}(F6)}$$

This bound is on the selected maximizer. It does not bound the least counterexample, certify every integer below $e^X$, or extend Polak’s all-integer finite theorem. The source’s existing all-integer range already forces this maximizer past $7.1\cdot 10^{22}\log 10$; (F6) is a larger source-clock restriction, since $\log 10<5/2$. The derivation of (F5) itself uses the earlier Axler stop, not that stronger all-integer certificate.

At fixed $H_0$, the available high-zero allowance grows proportionally to $\sqrt{A}$; the core allowance tends to a finite constant. Thus this finite application supplies no unbounded signed-tail estimate or RH proof. It adds no Lean result and does not certify the cited papers’ proofs or computations.

Combining finite zero verification with the classical zero-free region

Keep the same selected Robin source, clock $A=\log N$, $L=\log A$, normalization $T_A=\sqrt{A}L$, effective core and verified height $H_0=3\cdot 10^{12}$. This application controls the complete unverified zero contribution with two disjoint height ranges. It uses published inputs and symbolic outward comparisons; no zero or CA-profile computation is repeated, and no originality or Lean result is claimed.

The additional published input

Johnston–Yang, arXiv:2204.01980v2, Lemma 2.7, printed p.5, states that for $|t|\ge 2$ there are no zeta zeros in

$$\beta \ge 1-\frac{1}{R_0\log |t|},R_0=\mathrm{5.5666305.}\text{\hspace{1em}(Z1)}$$

Its footnote attributes the classical region to Mossinghoff–Trudgian and the improved constant to the higher verified height. The versioned PDF is the one identified in the existing supplier note, SHA-256 565993a6def48b237a68a92acba604f2c42f99165e0e71e390f8e21a313b74b2. The additional lemma and footnote were directly inspected; the external zero-free proof is not independently certified here. The existing Platt–Trudgian input supplies the finite verification, rather than a global RH assumption.

Set $U=10^{16}$. For every actual zero with $H_0<\gamma \le U$, (Z1), $R_0<6$, $\log U<40$, and the multiplicity-preserving functional-equation reflection give

$$\frac{1}{240}<\beta <1-\frac{1}{240}.$$

For $A>1$, convexity on this interval therefore gives

$$A^{\beta -1}+A^{-\beta }\le A^{-1/240}+A^{-239/240}=:B(A).\text{\hspace{1em}(Z2)}$$

Apply this to the same reflected pairs in Polak’s (5.12)–(5.14). The middle range keeps the improved factor $B(A)$; the range $\gamma >U$ keeps the original $1+A^{-1}$ factor. Zeros on the critical line, off-line quartets and every multiplicity are included. With the same positive-ordinate tails $S_j$, the complete high-zero allowance satisfies

$$\begin{array}{rl}{C}_{\mathrm{high}}(A)& \le C_{\mathrm{mid}}(A)+C_{>U}(A),\\ C_{\mathrm{mid}}(A)& \le \sqrt{A}B(A)\left[(1+1/L)S_2(H_0)+2(1/L+2/L^2)S_3(H_0)\right],\\ C_{>U}(A)& \le \sqrt{A}(1+A^{-1})\left[(1+1/L)S_2(U)+2(1/L+2/L^2)S_3(U)\right].\end{array}\text{\hspace{1em}(Z3)}$$

The actual ranges in (Z3) are disjoint. The middle-range upper bound uses the larger full tail $S_j(H_0)$ as a positive envelope; this does not omit any zero or identify that envelope with the middle-range sum. Conjugation and reflection carry exactly the same factors as (F2).

Pay the entire range above the second height

Reuse the full parameter formula in Polak’s (5.13), obtained from the Hasanalizade–Shen–Wong zero-count bound. With $(a,b,c)=(0.1038,0.2573,9.3675)$, it gives at $U$

$$S_2(U)\le \frac{\log (U/(2\pi ))+1}{2\pi U}+\frac{2a\log U+a/2+2b\log \log U+b/(2\log U)+2c}{U^2}.\text{\hspace{1em}(Z4)}$$

The elementary bounds $2\pi >6$, $32<\log U<40$, $\log \log U<4$, $a<1/9$, $b<1/3$ and $c<10$ give

$$S_2(U)<\frac{41}{6U}+\frac{32}{U^2}<7\cdot 10^{-16},S_3(U)\le S_2(U)/U<7\cdot 10^{-32}.\text{\hspace{1em}(Z5)}$$

Here $2<\log 10<5/2$ supplies the logarithmic interval, and $e^4>54$ supplies the iterated-logarithm bound. These are scalar outward comparisons of the published formula, not a new zero count or a numerical reconstruction of the source’s directed constants.

The complete signed target on a larger source-clock interval

Consider every selected source with

$$e^{50}\le A\le X_1:=3\cdot 10^{23}.$$

Then $\sqrt{A}<5.5\cdot 10^{11}$, $L\ge 50$ and $1+A^{-1}<1.0001$. The existing positive-series bound $e^{25}>6\cdot 10^{10}$, and $e^t\ge 1+t$ for $t\ge 0$, give

$$B(A)\le \frac{24}{29}+e^{-L/2}<\frac{24}{29}(1.0001).\text{\hspace{1em}(Z6)}$$

Use (F3) at $H_0$, (Z5) at $U$, $1+1/L\le 1.02$ and $2(1/L+2/L^2)<0.05$. The two rational comparisons are

$$(5.5\cdot 10^{11})\frac{24}{29}(1.0001)\left[1.02(1.48\cdot 10^{-12})+0.05(5\cdot 10^{-25})\right]<0.688,$$

$$(5.5\cdot 10^{11})(1.0001)\left[1.02(7\cdot 10^{-16})+0.05(7\cdot 10^{-32})\right]<\mathrm{0.0004.}$$

Consequently $C_{\mathrm{high}}(A)<0.6884$ on the entire stated interval. The existing $C_{\mathrm{low}}(A)<0.054$ and $\mathcal{E}(L)>0.75$ then pay the full signed Robin condition, with the exact same-source identity:

$$\boxed{T_A\Delta (N)>0.75-0.054-0.6884=0.0076>0.007(e^{50}\le A\le X_1).}\text{\hspace{1em}(Z7)}$$

The lower clock interval $K/2<A\le e^{50}$ was already paid in (F5) and is reused. Thus, conditional on the same cited source reduction, core, zero-free region and finite-verification inputs, a failure of RH must have its selected least global maximizer at

$$\boxed{\log N>3\cdot 10^{23}.}\text{\hspace{1em}(Z8)}$$

This improves the existing selected-source restriction $\log N>21\cdot 10^{22}$. It does not constrain the least counterexample, establish Robin for every integer below $e^{X_1}$, or extend Polak’s all-integer finite theorem. The original height $H_0$ is still the only verified height; $U$ is only a partition of the complete unverified-zero contribution.

The unbounded target remains $T_A{I}_{\psi }(A)\ge -\mathcal{E}(\log A)$ at the same selected source. For fixed $H_0,U$, (Z3) still permits a high-zero allowance growing as $A^{1/2-1/240}$, and the part above $U$ retains its $\sqrt{A}$ factor. Hence (Z7) supplies no unbounded signed estimate or RH proof. The separate divisor-order decomposition of the same integral is not added to this spectral allowance; each representation keeps its own complete remainder.

An explicit real-part density allowance for the complete high tail

Keep the conditionally selected least integer $N>5040$ attaining the global maximum of the Robin ratio, its clock $A=\log N$, $L=\log A$, and $T_A=\sqrt{A}L$. This application uses an existing explicit zero-density estimate to sharpen the entire high-zero allowance at that same source. It is a paper application, not a new density theorem, zero verification, all-integer finite Robin theorem or Lean result. The original unbounded signed target remains unchanged.

The existing density input at one tabulated real part

Johnston–Yang, arXiv:2204.01980v2, Lemma 2.6, equation (2.2), printed p.5, and Appendix B, Table 3, printed p.20, give at ${\sigma }_{\ast }=0.98=49/50$

$$N({\sigma }_{\ast },t)\le 16.281t^{4/75}(\log t{)}^{76/25}+2.231(\log t{)}^2.\text{\hspace{1em}(D1)}$$

Here $N(\sigma ,t)$ counts actual zeros with $\sigma <\Re \rho <1$ and $0<\Im \rho <t$, with multiplicity. This is a count in real part, not the total ordinate count. Use only this tabulated row; no interpolation or constants at unlisted real parts are assumed. The versioned PDF and SHA-256 are identified in the existing supplier note.

The paragraph before Lemma 2.6 explicitly replaces the unreliable critical-line bound used in the older Kadiri–Lumley–Ng calculation and recomputes the constants. The older constants are not substituted. Table 3 uses the verified height $H_1=3000175332800$, already cited above. Keep the weaker cut $H=3\cdot 10^{12}$. For $H\le t\le H_1$ the count in (D1) is zero by that same finite-height input; above $H_1$ the density estimate supplies the bound. Its external analytic proof and table calculation are cited, not independently rerun here.

Since $t\ge H>e$, the outward bounds $16.281<17$, $2.231<3$, $4/75<1/10$ and $76/25<4$ give the convenient allowance

$$N({\sigma }_{\ast },t)<20t^{1/10}(\log t{)}^4(t\ge H).\text{\hspace{1em}(D2)}$$

Pay every zero far from the critical line

Partition the actual positive-ordinate high zeros into two disjoint, reflection-invariant sets:

$${\mathcal{Z}}_{\mathrm{bulk}}=\{\rho :\Im \rho >H,1/50\le \Re \rho \le 49/50\},{\mathcal{Z}}_{\mathrm{far}}=\{\rho :\Im \rho >H,\Re \rho <1/50\text{ or }\Re \rho >49/50\}.$$

Reflection $\rho \mapsto 1-\overline{\rho }$ preserves ordinate and multiplicity. Thus the far set has twice the right-hand count in (D1). For $S_j^{\mathrm{far}}={\sum }_{\rho \in {\mathcal{Z}}_{\mathrm{far}}}(\Im \rho {)}^{-j}$, Tonelli applied to the positive counting sum gives

$$S_2^{\mathrm{far}}=4{\int }_H^{\infty }\frac{N({\sigma }_{\ast },t)}{t^3}dt<80{\int }_H^{\infty }{t}^{-29/10}(\log t{)}^{4}dt.\text{\hspace{1em}(D3)}$$

There are no far zeros at or below $H$, so no starting-endpoint term is lost. The integral includes the complete infinite height range. Put $h=\log H$ and $a=19/10>1$. The elementary integral equals

$$H^{-19/10}\left(\frac{h^4}{a}+\frac{4h^3}{a^2}+\frac{12h^2}{a^3}+\frac{24h}{a^4}+\frac{24}{a^5}\right).$$

The existing scalar bounds $e^{25}>6\cdot 10^{10}$ and $e>8/3$ give $e^{30}>H$, hence $h<30$. The bracket is therefore less than $30^4+4\cdot 30^3+12\cdot 30^2+24\cdot 30+24=929544<10^6$. Also $H^{1/10}<20$, since $H<20^{10}$. Consequently

$$S_2^{\mathrm{far}}<\frac{80\cdot 10^6\cdot 20}{H^2}<2\cdot 10^{-16},S_3^{\mathrm{far}}\le S_2^{\mathrm{far}}/H<7\cdot 10^{-29}.\text{\hspace{1em}(D4)}$$

These are symbolic outward comparisons of the published density bound, not a new zero count or numerical certificate.

Pay the remaining bulk at its own real-part range

For every $A>1$, convexity and functional-equation pairing on the bulk give

$$A^{\beta -1}+A^{-\beta }\le A^{-1/50}+A^{-49/50}=:B_{\ast }(A).$$