bibkey: caveney2012sacaga authors: Geoffrey Caveney, Jean-Louis Nicolas, and Jonathan Sondow year: 2012 title: On SA, CA, and GA numbers doi: null url: https://arxiv.org/abs/1112.6010v2 claim: The paper supplies a critical-source reduction, an unbounded GA2 family conditional on RH failure, and necessary prime-layer restrictions for proper GA1 integers; these do not supply a signed forward packet bound. strata_touched: [] license: citation-only triage: anchor

Published GA1 source constraints and the actual tangent parameter

The inspected primary is arXiv:1112.6010v2, whose first page identifies the version as 17 July 2012. It has 29 pages; the retrieved PDF SHA-256 is 7bf39ed12be2f708dc6c4dca747d7d9316470c11d8f09f071cdbc79e6f1148ae. Locators below use its printed pages. The relevant statements and their proofs in §§5.1–5.2 and §6, together with the recalled reduction and facts on printed p.5 and Theorem 5(iii) with its proof on pp.15–16, were read; no independent audit of the whole paper, journal-version correspondence, or Lean verification is claimed.

The source defines $G(n)=\sigma (n)/(n\log \log n)$ for $n>1$ and calls a composite $N$ GA1 when $G(N)\ge G(N/q)$ for every prime divisor $q$. The packet applications here use the positive-logarithm range $n>e$; Lemma 7 itself ensures $N/q\ge 6$.

Which potentially critical Robin source is covered

Theorem 4(ii), printed p.5, recalls the published same-source reduction: if any counterexample to Robin’s strict inequality exists above 5040, then ${\max }_{n>5040}G(n)$ exists, and the least integer $N>5040$ attaining it is extraordinary. Here extraordinary means composite and both GA1 and GA2; GA2 requires $G(N)\ge G(aN)$ for every positive integer $a$. This selects a global extremal source under the counterexample hypothesis; it does not say that every counterexample, regular return, or tangent minimum is GA1. Theorem 4(i) already states the equivalent RH criterion that 4 is the only extraordinary number. Both reductions are reused.

The source’s definition following Fact 1 on the same page calls a GA1 integer proper exactly when $\Omega (N)\ge 3$, counting prime factors with multiplicity. Fact 1 classifies the improper GA1 integers as 4 and $2p$ for primes $p\ge 7$. Fact 2 gives $G(N)\ge e^{\gamma }$ for every GA2 integer. The selected source above 5040 is therefore proper: 4 is excluded by size, while $\sigma (2p)/(2p)=\frac{3}{2}(1+1/p)<2$ and $\log \log (2p)>2$ when $2p>5040$, so $G(2p)<1<e^{\gamma }$ excludes GA2. This is hypothesis bookkeeping for the existing reduction, not a new critical-source theorem or a new enumeration.

The published unbounded GA2 supplier under RH failure

Theorem 5(iii), printed p.13, states that if RH is false, infinitely many GA2 integers exist. It also states that $\mu ={\max }_{n>5040}G(n)>e^{\gamma }$ and that every integer attaining this global maximum is both GA2 and CA. The infinitude and the CA classification have different scopes.

Its proof on pp.15–16 uses Robin’s published positive oscillation on CA integers to obtain ${\max }_{n\ge t}G(n)>e^{\gamma }$ for every $t$. Together with Gronwall’s $\mathrm{limsup}G(n)=e^{\gamma }$, this permits a sequence of tail maximizers $N_i$, with successive tails beginning after the largest previous maximizer. Thus

$$N_i⟶\infty ,G(N_i)>e^{\gamma },n>N_i⟹G(n)\le G(N_i).$$

Every $N_i$ is therefore GA2. This is the source’s construction, with its integer $A_i$ renamed $N_i$ to distinguish it from the logarithmic clock $A=\log N$. The proof classifies global maximizers as CA; it does not classify all these later tail maximizers as GA1, CA, or maximizers on the original domain $n>5040$.

The all-GA2 core application uses this existing unbounded family without requiring those stronger classifications. An eventual signed estimate covering every sufficiently large GA2 clock can therefore be tested against these actual sources. This supplier does not give that estimate. It is distinct from the fixed least global maximizer and from the finite XA family under RH failure; no new source-selection theorem, infinitude proof, originality claim, or Lean verification is asserted.

Ceiling replacements constrain the same right-tail source

The following applications use the all-integer comparisons of the published tail sources, together with the classical Euler-layer rearrangement. They retain one actual final integer for every comparison; GA2 comparisons with multiples alone do not supply this argument.

Let $N>e$ satisfy $G(m)\le G(N)$ for every integer $m\ge N$, and put $A=\log N$, $L=\log A$, $T=\sqrt{A}L$ and $W(n)=\log Z(n)$. For any actual integer $1\le d\le N$, reuse the ceiling replacement and retain its exact clock:

$$c=\lceil \frac{N}{d}\rceil ,m=cd,H=\log (N/d),s=\log (m/N),C_A(s)=\log \frac{\log (A+s)}{L}.$$

Thus $N\le m<N+d$ and $0\le s<\log (1+e^{-H})$. With $b_p=v_p(d)$ and $r_p=v_p(c)$, the actual added Euler reward is

$$B_d(c)=\log \frac{Z(cd)}{Z(d)}=\sum _p\log \frac{1-p^{-(b_p+r_p+1)}}{1-p^{-(b_p+1)}}\ge 0.$$

Shared factors of $c$ and $d$ are included. The exact forward comparison and the existing concavity tangent give

$$\boxed{W(d)-W(N)+B_d(c)\le C_A(s)\le \frac{s}{AL}.}\text{\hspace{1em}(R1)}$$

If the left side exceeds $C_A(s)$, the same actual $cd>N$ improves $G(N)$. No comparison at an intermediate deletion integer is required.

Rearrange all occupied layers in one integer

For the same $N$, let $S_k=\{p:v_p(N)\ge k\}$ and $r_k=|S_k|$, and let $q_j$ be the $j$th ordinary prime. Set

$$d_N=\prod _{k\ge 1}\prod _{j\le r_k}{q}_j,{\beta }_k(p)=\log \frac{Z(p^k)}{Z(p^{k-1})}=\log \left(1+\frac{1}{p+\cdots +p^k}\right).$$

The prefixes are nested because $r_{k+1}\le r_k$, so $d_N$ is one realizable exponent inventory. Monotonicity of $\log p$ and ${\beta }_k(p)$ gives the simultaneous savings

$$\begin{array}{rl}{H}_N& =\sum _k\left(\sum _{p\in S_k}\log p-\sum _{j\le r_k}\log q_j\right)=\log (N/d_N)\ge 0,\\ {\Gamma }_N& =\sum _k\left(\sum _{j\le r_k}{\beta }_k(q_j)-\sum _{p\in S_k}{\beta }_k(p)\right)=W(d_N)-W(N)\ge 0.\end{array}$$

All sums are finite. Apply (R1) to this complete $d_N$, with $c_N=\lceil N/d_N\rceil$ and $s_N=\log (c_N{d}_N/N)$:

$$\boxed{{\Gamma }_N+B_{d_N}(c_N)\le C_A(s_N)<\frac{e^{-H_N}}{AL},0\le T{\Gamma }_N<\frac{e^{-H_N}}{\sqrt{A}}.}\text{\hspace{1em}(R2)}$$

The inequality counts the total rearrangement and its actual padding, not separately attainable optimum layers. It gives no positive lower funding from a failure of prime-prefix order at the critical scale. If $d_N=N$, then $c_N=1$ and both rewards are zero.

A padding prime outside the replacement gives a strict improvement

Suppose additionally that $Z(d)\ge Z(N)$ and $AL\ge 2$. Then

$$\boxed{\mathrm{supp}(\lceil N/d\rceil )\subseteq \mathrm{supp}(d).}\text{\hspace{1em}(R3)}$$

For $d=N$ the multiplier is one. Otherwise $d$ is not a divisor of $N$: a proper divisor has strictly smaller $Z$. Thus $c-1<N/d<c$, $c\ge 2$, $cd>N$, and $s<\log (c/(c-1))$. If a prime $p\mid c$ is absent from $d$, then $B_d(c)\ge \log (1+1/p)\ge \log (1+1/c)$, so

$$\log \frac{G(cd)}{G(N)}>\log \frac{c+1}{c}-\frac{1}{AL}\log \frac{c}{c-1}>0.$$

The last inequality uses $\log (c/(c-1))<2\log ((c+1)/c)$ for $c\ge 2$, equivalently $c^2-c-1>0$. This contradicts the actual forward comparison. Even when (R3) holds, (R1) keeps all added layers in the same budget: $B_d(c)<d/(NAL)$. Both restrictions apply to $d_N$ and to an actual integer attaining $\Sigma (N)$.

The argument does not force a violation of (R1)–(R3), identify $N$ as CA or GA1, or control the original signed $I_{\psi }(A)$. A FIB address of either comparator must retain this actual prime inventory and clock; the address alone supplies no strict improvement. These are applications of existing source comparisons and Euler factors, without a priority claim, new rearrangement theorem, or Lean verification.

Inherited oscillation relaxes the eventual signed target

The displayed positive oscillation on printed p.15 of the same CNS v2 primary supplies more than the existence of arbitrarily large violations. Under RH failure, set $\Theta ={\sup }_{\zeta (\rho )=0}\Re \rho >1/2$ and fix $1-\Theta <\eta <1/2$. The source applies Robin’s oscillation result to CA integers $C$ and obtains, for some fixed $c_{\eta }>0$, arbitrarily large actual witnesses satisfying

$$G(C)>e^{\gamma }\left(1+c_{\eta }(\log C{)}^{-\eta }\right).\text{\hspace{1em}(O1)}$$

This is the oscillation used in the published infinitude proof, with an explicit positive constant; its underlying Robin proof is cited through that primary and is not independently reproved here.

Choose such a $C_j>\max \{e,N_{j-1}\}$ and let $N_j$ be the rightmost maximizer of $G$ on the integers $n\ge C_j$. The maximum is attained with finitely many ties: $G(C_j)>e^{\gamma }=\mathrm{limsup}G(n)$ places all sufficiently large integers strictly below $G(C_j)$. This is the same tail-selection mechanism as on printed p.16, with the tail starting at the chosen oscillation witness. Thus

$$N_j\ge C_j,N_j⟶\infty ,G(N_j)\ge G(C_j),m>N_j⟹G(m)\le G(N_j).$$

This particular selection can also retain CA status by direct application of the same primary’s Lemma 2, printed p.10, and Lemma 6, p.13. If $N_j=C_j$, it already has that status. If $N_j>C_j$, set ${ϵ}_j=1/(\log N_j\log \log N_j)$. Lemma 6 makes

$$g_{{ϵ}_j}(t)={ϵ}_j\log t-\log \log \log t$$

minimal at $t=N_j$ on $t>e$. Hence, for every integer $n\ge C_j$,

$$Z(n)n^{-{ϵ}_j}=G(n)e^{-g_{{ϵ}_j}(n)}\le G(N_j)e^{-g_{{ϵ}_j}(N_j)}=Z(N_j)N_j^{-{ϵ}_j}.$$

Lemma 2 applies with $N_0=C_j$ and $N=N_j$ and supplies global CA optimization at ${ϵ}_j$. Its comparison with a CA parameter for $C_j$ is a conclusion of the lemma, not an additional assumption. The equality branch $N_j=C_j$ does not identify its optimizing price with ${ϵ}_j$. This selection does not classify every member of the original integer-tail family, or make these $N_j$ GA1 or global maximizers on the original $n>5040$ domain.

At their actual clocks $A_j=\log N_j$, $L_j=\log A_j$, $T_j=\sqrt{A_j}{L}_j$, and with $\Delta (N)=\gamma -\log G(N)$, positivity of $\eta$ gives

$$\Delta (N_j)<-\log (1+c_{\eta }{A}_j^{-\eta }).\text{\hspace{1em}(O2)}$$

There is no bound needed on $N_j/C_j$: increasing the chosen integer only decreases $A_j^{-\eta }$ from $(\log C_j{)}^{-\eta }$. Reuse the exact same-source identity $\Delta (N)=I_{\psi }(A)+K_N(A)$ and the actual-core allowance. Its thresholds hold eventually along this family. Equations (O1)–(O2) then give

$$\begin{array}{rl}{T}_j{I}_{\psi }(A_j)& <-\mathcal{E}(L_j)-T_j\log (1+c_{\eta }{A}_j^{-\eta })\\ & \le -\mathcal{E}(L_j)-\frac{c_{\eta }}{2}{A}_j^{1/2-\eta }{L}_j⟶-\infty .\end{array}\text{\hspace{1em}(O3)}$$

The second line uses $c_{\eta }{A}_j^{-\eta }\le 1$ eventually. The complete original improper integral $I_{\psi }$ and its clock remain unchanged; no zero sum, upper endpoint, or elementary correction is discarded. In particular, under RH failure, these actual CA right-tail sources carry an unbounded negative normalized signed response. Their classical prime-prefix support and ordered exponents give $d_{N_j}=N_j$, so the sorting reward ${\Gamma }_{N_j}$ and its padding reward in (R2) are zero. Their existing strict SA property also gives $\Sigma (N_j)=Z(N_j)$ and $D_{N_j}=0$. Neither defect supplies a uniform strictly positive funding term on this selectable critical family. The source constraints do not establish a favorable-cutoff principle.

A weaker sufficient target for an eventual contradiction is therefore an independently established finite lower bound: find fixed $M\ge 0$ and $A_0$ such that every actual CA all-integer right-tail maximizer with $A=\log N\ge A_0$ satisfies

$$\boxed{\sqrt{A}\log A{I}_{\psi }(A)\ge -M.}\text{\hspace{1em}(O4)}$$

Any such fixed $M$ would contradict (O3) under RH failure. This target does not require $M$ to equal the effective core allowance. The lower bound (O4) is not proved; the application only relaxes the outstanding same-source signed obligation using the already published oscillation. It is not a new source theorem, priority claim, Lean result, or proof of RH.

The backward record classification is already published

Nazardonyavi and Yakubovich, Superabundant numbers, their subsequences and the Riemann hypothesis, arXiv:1211.2147v3, 26 February 2013, already supply the relevant strict-record language. The inspected PDF has 32 pages and SHA-256 c184db590a50ea81daa88660e5885d3c64980b9324f6e545d27bbae4e1eba155. Definition 2.1, Theorems 2.3–2.4 and 4.32, and the numerical scope and example in §5 were read against that primary. This is a citation and applicability assessment, without a whole-paper proof audit, independent reproduction of its numerical claims, or Lean verification.

Definition 2.1, printed p.4, calls $10080$ extremely abundant (XA), and calls $n>10080$ XA exactly when

$$G(m)<G(n)(10080\le m<n).$$

Proposition 2.2 gives $\mathrm{XA}\subseteq \mathrm{SA}$. Theorem 2.3, printed p.5, states that the least Robin counterexample is XA; its proof reports the finite check on $5040<n\le 10080$. That check is cited, not rerun. Theorem 2.4, on the same page, already proves that RH is equivalent to infinitely many XA numbers, and its proof discusses attainment of the global supremum under failure of RH. Neither criterion nor the record classifier needs to be reconstructed here.

Apply these existing results to the same least global maximizer $N$ selected above under the counterexample hypothesis. The cited finite check places it above $10080$. Minimality among maximizers gives

$$G(m)<G(N)(5040<m<N),$$

so it is XA, while global maximality prevents any larger integer from being XA. Thus it is the last XA under these hypotheses. This is direct application of the published record definition and supremum argument, not a new source-selection result. The least global maximizer need not be the least counterexample. In particular, the available strict backward comparisons include every divisor deletion with $1<d\mid N$ and $N/d>5040$, and also nondivisor comparators in that range; they do not authorize a comparison with $N/d\le 5040$.

Theorem 4.32, printed p.24, supplies $P^{+}(n)<\log n$ on XA sources. It supplies no effective signed prime-error estimate. Under the counterexample hypothesis there are only finitely many XA by Theorem 2.4, so neither an unbounded XA critical family nor an infinite continuation from this selected maximizer is available.

The recursive route must also retain exponent changes. Remark 5.5, printed p.26, reports consecutive XA numbers

$$n_1=(139\#)(13\#)(5\#)(3\#{)}^2{2}^4,n_3=(151\#)(13\#)(5\#)(3\#{)}^2{2}^3,$$

where $p\#$ denotes the primorial. Their $2$-exponents decrease from $9$ to $8$, so XA records do not form a divisibility chain. This cited example is not a new numerical search and does not refute nesting in a separately chosen CA parameter chain. Properties 5.1–5.4 are explicitly finite observations from §5, not uniform theorems; in particular their consecutive exponent-change and largest-prime assertions cannot be used as unbounded recursion laws.

These suppliers pay backward record classification and its source scope. They leave the same-source quantitative coupling of deletion and insertion losses, a strict Robin sign and full finite-source coverage unpaid. Reusing them does not yield a new Robin estimate.

Directly reusable prime and stack envelopes

Keep the same proper GA1 integer $N$ and put $A=\log N$. The following published restrictions are available without any CA or regular-return hypothesis.

Theorem 9, §6.1, printed pp.20–21, gives, for every prime $p\mid N$ and integer $1\le r\le v_p(N)$,

$$p\le (rA{)}^{1/r}\le A.$$

Its displayed equation (34) also gives $p^r\log p\le A\log A$. Thus the same source’s $r$th occupied prime layer has an explicit size restriction; its prime support and its exponents are not separate freely selectable data.

Theorem 10, §6.2, printed pp.21–22, proves that for each fixed integer $k\ge 3$ there are only finitely many GA1 integers with $\Omega (N)=k$. Its proof already supplies the quantitative bound

$$\Omega (N)\ge \frac{A}{\log A}.$$

Theorem 11, §6.3, printed pp.22–23, defines $R=h(A)$ using the inverse of $t\mapsto 2^t/t$ on $[2,\infty )$, so $2^R/R=A$, and gives the divisibility envelope

$$N\mid M(N),M(N)=\prod _{r=1}^{\lfloor R\rfloor }\prod _{\begin{array}{c}p\mathrm{prime}\\ p\le (rA{)}^{1/r}\end{array}}p.$$

Lemma 9 on printed p.22 gives $2\le R\le 3.08\log A$. These are necessary restrictions and a finite divisor envelope at a given size; dividing $M(N)$ does not certify GA1, a tangent realization, or a favorable packet. The envelope is retained as a cited supplier, not reconstructed as another candidate enumerator.

Theorem 12, §6.3, printed p.23, gives $v_{P(N)}(N)=1$, where $P(N)$ is the largest prime factor. Its proof’s check of the 84 divisors of 43200 belongs to the published proof and is not rerun here. Theorem 13, §6.4, printed p.24, gives

$$P(N)\sim \log N\text{as }N\to \infty \text{ through proper GA1 integers}.$$

It supplies neither an effective numerical threshold nor a forward signed workload estimate. The above restrictions, including the growing $\Omega (N)$ bound, are published results rather than project discoveries.

Host structure and the FIB atom cutoff are different inputs

For the actual packet interface, these suppliers constrain the factorization of the Robin host $N$ at $A=\log N$. They can be used together on the same proper GA1 source, including the global maximizer selected by Theorem 4(ii).

The project’s §423 instead sorts the raw FIB correction by odd squarefree atom kernels with a fixed number $k$ of distinct prime factors at an atom-index cutoff $X$. Its $k$, $X$, and kernel are not $\Omega (N)$, $N$, and the host’s prime factorization. Theorem 10 does not by itself transfer to a growing- kernel cancellation estimate, and Theorem 11 does not couple those signed layers to a packet from that host.

A use of the host envelope in the packet problem still needs the same integer’s CA/tangent correspondence where that interface is required, and an estimate of its actual joint-layer debt or an improving multiple. Theorem 4’s selected source is GA2, so an improving multiple of that source would contradict its defining extremality. Constructing such an improvement under the critical-source hypotheses is the remaining task; neither the divisor envelope nor $P(N)\sim A$ establishes it. A uniformly positive full Robin margin on all proper GA1 integers above 5040 would also suffice by the recalled reduction, but is not supplied here.

Directly reusable infinite subclasses

Lemma 7, §5.1, printed p.17, states that a CA number $N$ for parameter $ϵ>0$, with largest prime factor $p=P(N)\ge 5$, is GA1 if

$$ϵ>\frac{1}{\log (N/p)\log \log (N/p)}.$$

Its proof gives the strict deletion inequalities $G(N/q)<G(N)$ for all prime factors $q$. Theorem 6, printed pp.17–18, already proves infinitely many CA numbers are GA1 by choosing the largest CA integer at the layer price $ϵ=F(p,1)$ and using the existing positive Chebyshev oscillation. This infinite subclass is not a new project target.

Lemma 8, §5.2, printed p.18, takes the largest CA integer $N$ for $ϵ=F(p,1)$, $p\ge 3$, where

$$F(p,1)=\frac{\log (1+1/p)}{\log p}.$$

If $ϵ<1/(\log N\log \log N)$, it gives $G(N/p)>G(N)$, so $N$ is not GA1. Theorem 7, printed pp.18–19, already proves infinitely many CA numbers are not GA1 using the negative Chebyshev oscillation. These are actual CA integers, not measures chosen independently of the primes. Neither theorem estimates a forward regular packet’s moments.

Keep the CA price and the tangent price distinct

For the actual packet interface, write $a=\log N$ and $q(t)=1/(t\log t)$ on $t>1$. This $q$ is the project’s price; it is not the auxiliary function called $g$ in equations (20)–(21) of the present primary. A regular tangent state uses the same integer $N=C_a$ at price $ϵ=q(a)$.

At that price, Lemma 7’s sufficient condition would be $q(a)>q(a-\log p)$, whereas $q$ is strictly decreasing and $a-\log p=\log (N/p)>1$. Thus that particular substitution cannot satisfy the condition. This does not exclude a tangent state from GA1: the same CA integer may have another admissible price that does satisfy Lemma 7. To use the sufficient condition, retain the same integer and check its admissible price interval, not an independently selected CA state.

Lemma 8 has a different obstruction. For its largest CA integer, the first $p$-layer is selected at $F(p,1)=ϵ<q(a)$. That layer is absent at the tangent price $q(a)$. Hence that integer is not the actual prefix $C_a$; the negative-oscillation construction in Theorem 7 does not supply regular tangent states. This follows by applying the existing activation rule and does not invalidate the theorem about CA numbers.

Theorem 6’s GA1 conclusion alone likewise supplies neither an event-free root $A(a)=a$ nor a later regular endpoint $b$, a short width $b-a=o(a)$, or the required signed packet moment. Those are joint conditions on the same actual source. They are not consequences of having two separately infinite classes.

A known finite example needs no new enumeration

The source’s §5.2, printed p.18, explicitly states that CA integers with $P(N)=13$ are not GA1. The project’s existing §98.5 already records $720720$ as a strict self-matching integer, with the existing CA-prefix correspondence. Since $720720=2^4\cdot 3^2\cdot 5\cdot 7\cdot 11\cdot 13$, these existing results already distinguish regular self-matching from GA1. No new counterexample search, numerical verification, generic CA theorem, or Lean wrapper is needed to make that distinction.

The unresolved packet task concerns actual joint selection: a suitable unbounded tangent family and a forward signed surplus. For full Robin, any proposed selection also has to cover the relevant potentially nonpositive minima; an arbitrary infinite GA1 or safe-source family does not supply that coverage.

A proposed oscillation-geometry supplier loses its shrinking margin

Thomas Schwabhäuser, Preventing Exceptions to Robins InEquality, arXiv:1308.3678v3, is a proposed CA-multiplier argument using the Alaoglu–Erdős conjecture. The inspected PDF has 23 pages and SHA-256 050c4444d706c1d6dd89ed3d227f57a196fececaade4d3b0648c7e8de8e062db. The scope here is §4.3: equation (4.1), Fact 4.21, Lemma 4.22, Proposition 4.25 and its use in Corollaries 4.27 and 4.31, printed pp.13–16. This is a paper-level assessment of these statements, without a complete proof audit or Lean verification. The elementary exponential limit used below is reused, not claimed as new oscillation theory.

For fixed $b>0$ and $0<\delta <1$, retain the source’s function

$$g(\mu ,\nu )=\frac{\mu }{\nu }\frac{1+\delta e^{-b\mu }}{1-\delta e^{-b\nu }},{ϵ}_{\mu ,\nu }=\log \frac{1+\delta e^{-b\mu }}{1-\delta e^{-b\nu }}.$$

For a fixed angle $0<\varphi <\pi /2$, both coordinates of $(\mu ,\nu )=(r\cos \varphi ,r\sin \varphi )$ tend to infinity. Thus

$${ϵ}_{r\cos \varphi ,r\sin \varphi }⟶0,g(r\cos \varphi ,r\sin \varphi )⟶\cot \varphi .$$

Fact 4.21 and Lemma 4.22 instead use the positive limiting constant ${ϵ}_{\infty }=\log ((1+\delta )/(1-\delta ))$ and the resulting $e^{{ϵ}_{\infty }}\cot \varphi$. That constant is not the limit of the printed function. In particular, there is no positive limiting angular margin separating its contour from the diagonal.

An increasing arithmetic progression tests the actual auxiliary claim

The source defines $\mathcal{M}=\{(\mu ,\nu ):\nu >\mu ,g(\mu ,\nu )>1\}$. Its Proposition 4.25 claims that an increasing real sequence with $\arctan (a_{n+1}/a_n)\to \pi /4$ has an adjacent pair in $\mathcal{M}$. For $a>0$ and $h>0$, the denominator above is positive and its exact crossing condition is

$$g(a,a+h)>1⟺h<\delta (a{e}^{-ba}+(a+h)e^{-b(a+h)}).$$

Choose $C\ge 1/b$ large enough that $2\delta t{e}^{-bt}<1$ for every $t\ge C$, and put $a_n=C+n$. Such a $C$ exists because $t{e}^{-bt}\to 0$. The function $t{e}^{-bt}$ is decreasing on $t\ge 1/b$. Consequently every adjacent pair has $h=1$ and

$$\delta (a_n{e}^{-b{a}_n}+(a_n+1)e^{-b(a_n+1)})\le 2\delta a_n{e}^{-b{a}_n}<1.$$

Therefore $g(a_n,a_{n+1})<1$ for every $n$, while the sequence is strictly increasing and $\arctan (a_{n+1}/a_n)\to \pi /4$. This directly contradicts the stated Proposition 4.25. Relative spacing converging to one does not by itself beat the exponentially shrinking margin. The same crossing condition excludes distinct integer pairs with $\mu \ge C$, since then $h\ge 1$; hence Corollary 4.27’s claimed infinitely many consecutive-prime pairs in $\mathcal{M}$ also fails in this parameter range.

This assessment does not exclude close pairs of actual CA logarithmic clocks, whose additive gaps may tend to zero. It shows that Proposition 4.25 does not supply their required quantitative comparison. Corollary 4.31 uses that proposition in its proposed source-selection step; its printed derivation cannot therefore be imported as the needed same-source gain. The Alaoglu–Erdős conjecture, alternative proofs, and the actual selected Robin source are not settled by this auxiliary counterexample. The original full signed-tail target and RH remain unproved; no new general criterion or certified theorem is claimed.

Relaxed forward records and the unfilled Robin step

Nazardonyavi and Yakubovich’s later note, Delicacy of the Riemann hypothesis and certain subsequences of superabundant numbers, arXiv:1306.3434v2, is distinct from the earlier 1211.2147v3 source used for XA above. Definition 1.3 and Lemma 1.4, printed pp.2–3, use the same actual pair of integers $m<n$:

$$\frac{Z(n)}{Z(m)}>1+\frac{\log (n/m)}{\log n\log \log m},Z(j)=\frac{\sigma (j)}{j}.$$

Lemma 1.4 gives such a successor for a member $m$ of the source’s $X^{\prime }$; Theorem 1.5 states that $X^{\prime }$ is infinite without an RH premise. Definition 1.6, printed p.4, uses the stronger relaxed threshold

$$\frac{Z(n)}{Z(m)}>1+\frac{2\log (n/m)}{(\log n+\log m)\log \log m}$$

to define $X^{\prime \prime }$. The displayed Theorem 1.5 concerns $X^{\prime }$, not $X^{\prime \prime }$; the latter’s reported finite counts are not adopted as an infinitude or a uniform jump estimate. The printed definitions say to find the next integer and do not explicitly specify least successors. No canonical-selection or arbitrary-branch SA assertion is used here.

The exact normalization shows what these rules guarantee. Put

$$\ell =\log \log m>0,r=\frac{\log n}{\log m}>1,G(j)=\frac{Z(j)}{\log \log j}.$$

An actual Robin-ratio improvement needs $Z(n)/Z(m)>1+\log r/\ell$. The two published relaxed rules instead use

$${\varphi }_1(r)=1-r^{-1},{\varphi }_2(r)=\frac{2(r-1)}{r+1},{\varphi }_1(r)<{\varphi }_2(r)<\log r.$$

For either rule, on the same pair $m,n$, the exact readout is

$$\frac{Z(n)}{Z(m)}>1+\frac{{\varphi }_j(r)}{\ell }⟺\frac{G(n)}{G(m)}>1-\frac{\log r-{\varphi }_j(r)}{\ell +\log r}.$$

The allowed loss is positive. With $h=\log r\downarrow 0$, its two numerators are respectively

$$h-1+e^{-h}=\frac{h^2}{2}+O(h^3),h-2\tanh (h/2)=\frac{h^3}{12}+O(h^5).$$

These are normalizations of the published rules and standard elementary expansions, not a new record theorem or analytic prime-error estimate. They distinguish an available forward increase of $Z$ from an increase of the normalized Robin quotient.

In particular, retain the same conditional least global maximizer $N$ from the earlier reduction. If an actual forward pair starts at $m=N$ and meets either rule, its lower bound for $G(n)/G(N)$ is below one and is compatible with the known global upper bound $G(n)/G(N)\le 1$. This supplies no improving multiple or contradiction to GA2. No new membership or successor-selection theorem at $N$ is asserted.

A usable recursive estimate still requires control of the actual logarithmic jumps and their accumulated loss along one realized path, or another jointly signed bound. The statements inspected here supply no such uniform jump control and no bound for the complete original $I_{\psi }(\log N)$. The earlier XA criterion and the later relaxed infinitude statement are reused without reconstructing their proofs. The same selected integer, actual zero real parts and multiplicities, all heights, explicit-formula terms and strict Robin core remain unchanged. RH remains unproved; this source application is not Lean verified and claims no mathematical originality.

Positive psi points supply a strict actual CA increase

The Pintz source application locates positive $\psi$-error points in $[Y^r,Y]$ under its stated fixed-exponent conditions. The following correspondence consumes that input and the present primary’s actual activation rules, equations (7)–(12), pp.7–9. It does not repeat Theorem 6’s infinite GA1 supplier. Use the source’s full layer price

$$F(x,k)=\frac{\log (1+1/(x+\cdots +x^k))}{\log x},x>1.$$

Let $t\to \infty$ satisfy $\psi (t)>t$, and let $p$ be the largest prime at most $t$. The prime number theorem gives $p\sim t$, while the full prime-power accounting gives

$$\psi (t)-\vartheta (t)=\vartheta (\sqrt{t})+\sum _{k=3}^{\lfloor \log t/\log 2\rfloor }\vartheta (t^{1/k})=(1+o(1))\sqrt{t}.$$

The remaining sum is $O(t^{1/3}\log t)$. Since $\vartheta (p)=\vartheta (t)$, positivity implies $\vartheta (p)-p>-(1+o(1))\sqrt{p}$.

At the actual price ${ϵ}_p=F(p,1)$, let $x_k$ be the root of $F(x_k,k)={ϵ}_p$, so $x_1=p$, and retain

$$H=\prod _{k:x_k\ge 2}\prod _{q\le x_k}q,D=H/p.$$

Here $q$ ranges over ordinary primes. The published activation rule makes $H$ the largest CA integer at this price, including every tied layer. The first $p$-layer is tied and no higher $p$-layer is occupied, so $D$ is another actual CA integer at the same price.

For each fixed $c>0$, $F(c\sqrt{p},2)/F(p,1)\to 2/c^2$; monotonicity therefore gives $x_2\sim \sqrt{2p}$. Lemma 1, p.9, and equation (15), p.11, give $x_3<(3p{)}^{1/3}$ and $O(\log p)$ occupied layers. Thus all layers yield

$$\log H=\vartheta (p)+(\sqrt{2}+o(1))\sqrt{p}+O(p^{1/3}\log p),\log D\sim \log H\sim t.$$

In particular $\log D>p+(\sqrt{2}-1+o(1))\sqrt{p}-\log p>p+1$ eventually. The sign of the gain is then exact:

$$\begin{array}{rl}\log \frac{G(H)}{G(D)}& ={ϵ}_p\log p-{\int }_{\log D}^{\log H}\frac{du}{u\log u}>0,\\ {ϵ}_p& >\frac{1}{(p+1)\log (p+1)}.\end{array}$$

Indeed $Z(H)/Z(D)=1+1/p$, $\log H-\log D=\log p$, and $u\mapsto 1/(u\log u)$ decreases throughout that interval. The positive $\sqrt{2}-1$ term permits a positive $\psi$-point even if $\vartheta (p)<p$. The full layer accounting is retained, and possible activation ties remain allowed. This correspondence supplies no GA1 or event-free assertion.

Quantitative selection into the self-clock-price class

Define the actual source class

$${\mathcal{S}}_{\ast }=\left\{N>e:N\text{ is globally CA at }{ϵ}_N=\frac{1}{(\log N)\log \log N},G(m)\le G(N)\text{for every integer }m\ge N\right\}.$$

Under RH failure, the preceding correspondence permits an unbounded family entirely in this class while retaining a fixed power-sized excess. This is a paper-level synthesis of the cited inputs, not a priority claim, Lean certification, or bound for the signed tail.

Retain the zero abscissa $1/2<\Theta \le 1$. Choose fixed parameters

$$1-\Theta <\eta <\frac{1}{4\Theta },2\eta <r<\frac{1}{2\sigma },$$

where $\Theta <\sigma <\min \{1,1/(4\eta )\}$ if $\Theta <1$, and $\sigma =1$ if $\Theta =1$. These choices exist because $4\Theta (1-\Theta )<1$. They satisfy both $r\sigma <1/2$ for the Pintz application and $0<\eta /r<1/2$ for the final excess. This selects an allowed fixed exponent in (O1); it is not a construction for every previously chosen $\eta <1/2$.

Take the published CA witnesses $C$ from (O1), with $a=\log C\to \infty$, and put $Y=a/8$. Select a positive $\psi$-point $t\in [Y^r,Y]$, then construct the actual $D,H$ above. Eventually

$$\log D\ge \frac{1}{2}(a/8{)}^r,\log H\le 2t\le a/4,D<H<C,G(H)>G(D).\text{\hspace{1em}(S1)}$$

The selection of $t$ can be specified without a free continuum choice: take the least member of the finite set $\{Y^r\}\cup \{q^k:Y^r<q^k\le Y,q\text{ prime},k\ge 1\}$ where $\psi (t)>t$. Such a member exists since $\psi (u)-u$ decreases between its prime-power jumps.

Let $N$ be the rightmost maximizer of $G$ over the integers $n\ge D$. The tail contains $C$ with $G(C)>e^{\gamma }=\mathrm{limsup}G(n)$, so this uses the already recalled attained-tail-maximum mechanism and has finitely many ties. The strict gain in (S1) gives $N>D$. Lemmas 6 and 2 therefore apply with CA base $N_0=D$, selected maximum $N>D$, and price ${ϵ}_N$; they supply global CA optimization at that exact price. Hence $N\in {\mathcal{S}}_{\ast }$, with all activation ties still allowed.

Put $A=\log N$, $L=\log A$, $T=\sqrt{A}L$ and $k_0=1/(28^r)$. On the same final integer, (S1) gives $A\ge k_0{a}^r$, and consequently

$$a^{-\eta }\ge k_0^{\eta /r}{A}^{-\eta /r},G(N)\ge G(C)>e^{\gamma }(1+c_{\ast }{A}^{-{\eta }^{\prime }}),{\eta }^{\prime }=\eta /r\in (0,1/2),c_{\ast }=c_{\eta }{k}_0^{{\eta }^{\prime }}>0.\text{\hspace{1em}(S2)}$$

Also $A\to \infty$. The possible positions $D<N<C$, $N=C$, and $N>C$ are all covered: the lifting comparison is $N>D$. The earlier procedure whose tail starts at $C$ keeps its stated equality-branch price limitation. Here the price identification follows from $N>D$ even if $N=C$.

Apply (O3) with the fixed $({\eta }^{\prime },c_{\ast })$ supplied by (S2). Under RH failure it yields $T{I}_{\psi }(A)\to -\infty$ on an unbounded family inside ${\mathcal{S}}_{\ast }$. The same-source identity, complete original integral and actual core $K_N(A)$ are evaluated at $A=\log N$, not at any auxiliary clock $t,p,\log D$ or $\log C$.

Accordingly a sufficient remaining target can be restricted to

$$\exists M\ge 0,A_0\forall N\in {\mathcal{S}}_{\ast },A=\log N\ge A_0:\sqrt{A}\log A{\int }_A^{\infty }(\psi (u)-u)\frac{1+\log u}{u^2{\log }^{2}u}du\ge -M.\text{\hspace{1em}(S3)}$$

No bound in (S3) is established. The increment is the quantified source-selection interface justifying this narrower class, with the original signed obligation preserved. It supplies neither GA1 nor regular or event-free states, a favorable cutoff, eliminated ties, positive sorting-defect funding, or RH. Those restrictions cannot be added to (S3) without another same-source selection argument.

The same power-excess family also supplies proper GA1 sources

Keep the actual $D,H,N$ and all parameters of (S1)–(S2). The following application of equations (10)–(12) and Lemma 6, printed pp.8–9 and p.13, provides the additional same-source selection required above. It does not reconstruct the paper’s known infinite CA–GA1 supplier or claim priority for the application.

First the selected maximum satisfies $N\ge H$, not only $N>D$. The two endpoints optimize at the common price ${ϵ}_p$, with $Q=Z(D)D^{-{ϵ}_p}=Z(H)H^{-{ϵ}_p}$. For $D<m<H$, global CA optimization gives

$$G(m)\le Q{e}^{g_{{ϵ}_p}(m)},g_{ϵ}(t)=ϵ\log t-\log \log \log t.$$

Lemma 6’s strict decrease followed by strict increase implies $g_{{ϵ}_p}(m)<\max \{g_{{ϵ}_p}(D),g_{{ϵ}_p}(H)\}$. Since $G(D)<G(H)$, the larger endpoint is $H$. Together with the separate strict comparison at $m=D$, every $D\le m<H$ has $G(m)<G(H)$; tail maximality therefore forces $N\ge H$.

Put $q_0(u)=1/(u\log u)$. The same integer’s own price now satisfies

$${ϵ}_N=q_0(A)\le q_0(\log H)<q_0(p+1)<{ϵ}_p.$$

Every occupied $H$ layer has price at least ${ϵ}_p$ and hence is mandatory at ${ϵ}_N$, by the cited activation rule. Thus $H\mid N$. For any ordinary prime divisor $q$ of $N$, if $q\le p$ then $N/q\ge H/q\ge H/p=D$. If $q>p$, the largest prime factor of $H$ is $p$, so $H\mid N/q$ and again $N/q\ge H>D$. These comparisons cover every multiplicity. The defining maximum over integers at least $D$ therefore supplies $G(N/q)\le G(N)$ for every such $q$. Eventually $p\ge 5$, so $\Omega (N)\ge \Omega (H)\ge 3$; this is proper GA1 in the source’s exact sense. The power excess (S2) holds on this same $N$.

Apply the published prime-local regularity bridge

Define the narrower class

$${\mathcal{S}}_{\mathrm{reg}}={\mathcal{S}}_{\ast }\cap \{N:N\text{ is proper GA1}\}.$$

Every member meets the hypotheses of the published prime-local bridge: properness gives $N/q\ge 4>e$ for every prime divisor $q$, GA1 gives the deletion comparisons, and right-tail maximality gives $G(Nq)\le G(N)$ for every ordinary prime $q$. The cited theorem therefore directly supplies the event-free identity $N=C_A$ and ${\Phi }_{\mathrm{CA}}(A)=A$ at $A=\log N$. No general regularity or activation-tie proof is repeated here. The same manuscript’s fixed-maximizer reduction in §11 does not supply the unbounded family and fixed power excess in (S2).

Restrict the unpaid signed estimate to this joint source class

The selected family lies in ${\mathcal{S}}_{\mathrm{reg}}$ and retains (S2). Therefore (O3) still forces $T{I}_{\psi }(A)\to -\infty$ on that same unbounded family if RH fails. It suffices to establish

$$\exists M\ge 0,A_0\forall N\in {\mathcal{S}}_{\mathrm{reg}},A=\log N\ge A_0:\sqrt{A}\log A{I}_{\psi }(A)\ge -M.\text{\hspace{1em}(S4)}$$

This is the complete original signed integral, with every prime-power layer and explicit-formula term retained. The existing regular-return endpoint application now applies to this same family: it already gives $A-\vartheta (A)=\sqrt{2A}(1+o(1))$ and a negligible squared endpoint penalty at the actual primorial cutoff. Its proof and the archived packet and cone results are reused rather than repeated.

No bound in (S4) follows from those endpoint identities. The added interface is joint selection of proper GA1 regular sources with the same quantified excess, not a proof that every member of ${\mathcal{S}}_{\ast }$ is GA1. The full signed estimate and RH remain unproved; this application is not Lean certified and claims no mathematical originality.

The existing Johnston weighted-sign theorem also gives a negative primitive with this kernel, anchored at 2. Writing that primitive as $J$ leaves the complete target $I_{\psi }(A)=J_{\infty }-J(A)$. Its negative bias does not supply the required comparison with $J_{\infty }$ at these same actual integers; the uniform floor in (S4) on the already justified source class remains unpaid.

Transport one selected source into an actual CA price interval

The Euler optimization and tangent minimum in equations (7)–(12) and Lemma 6 of the inspected primary give the following application. It quantifies the sampling requirement for a future arithmetic mean or exceptional-set estimate; it is not a new optimization theorem, a signed-tail bound, an originality claim, or Lean certification. The existing quadratic tail interpolation already supplies the $A^{3/4}$ bounded-loss scale for a different comparison. That interpolation and the tangent minimum are reused. The tangent gap $R(A,b)$ below is also the existing $c_A(b)$ in the Nicolas pressure application, equation (PH1). Its global Robin maximum hypothesis is not imported here: the comparison uses the original source’s right-tail maximality only at $C_b\ge N$ and retains actual Gronwall values, rather than identifying them with pressure away from a self-clock point.

Fix the same $N\in {\mathcal{S}}_{\mathrm{reg}}$, put $A=\log N>1$, $L=\log A$, $T=\sqrt{A}L$, and retain

$$g(b)=\frac{1}{b\log b},k(b)=-g^{\prime }(b)=\frac{1+\log b}{b^2{\log }^{2}b}.$$

For $b\ge A$, let $C_b$ be the largest full CA optimizer at price $g(b)$, including all activation ties and repeated prime layers. Write $B(b)=\log C_b$ and

$$X_N=\log G(N)-\gamma ,X(b)=\log G(C_b)-\gamma .$$

The already cited prime-local bridge gives $N=C_A$. The full layer rule gives $N\mid C_b$, hence $B(b)\ge A>1$. In general $B(b)\ne b$: no self-clock identity, GA1 property, or tail maximality is imposed on the later $C_b$.

Global optimization at $g(b)$ supplies $\log Z(N)-g(b)A\le \log Z(C_b)-g(b)B(b)$, where $Z(n)=\sigma (n)/n$. Lemma 6, in the logarithmic variable $t$, says that $F_b(t)=g(b)t-\log \log t$ has its minimum at $t=b$. Thus $X_N-X(b)\le F_b(A)-F_b(B(b))\le F_b(A)-F_b(b)$. The original $N$’s all-integer right-tail maximality separately gives $X(b)\le X_N$. Together these existing comparisons yield

$$\begin{array}{rl}0\le X_N-X(b)& \le R(A,b),\\ R(A,b)& =\log \frac{\log b}{\log A}-\frac{b-A}{b\log b}={\int }_A^b(u-A)k(u)du\\ & \le \frac{k(A)}{2}(b-A{)}^2.\end{array}\text{\hspace{1em}(P1)}$$

The integral follows by differentiating the explicit tangent gap; the last inequality uses the existing decreasing kernel. No ordinary prime error has been assigned a favorable sign in this comparison.

For $h>0$, define the actual-price mean ${\mathcal{M}}_A(h)=h^{-1}{\int }_A^{A+h}X(b)db$. There are finitely many CA activation events on each such interval; the mean is over the actual stepwise arithmetic values, with endpoint ties of zero integration weight. Integrating (P1) gives

$$0\le X_N-{\mathcal{M}}_A(h)\le \frac{1}{h}{\int }_A^{A+h}(A+h-u)(u-A)k(u)du\le \frac{k(A)h^2}{6}.\text{\hspace{1em}(P2)}$$

At $h=A^{3/4}$ this is $0\le T[X_N-{\mathcal{M}}_A(h)]\le (1+1/L)/6$. It supplies a bounded transport cost, not an upper bound for $T{\mathcal{M}}_A(h)$. In the existing identity $I_{\psi }(A)+K_N(A)=-X_N$, a bound on that positive mean would still need an independent arithmetic supplier before it could fund (S4).

A power excess occupies a quantified price interval

Use the particular fixed ${\eta }^{\prime }\in (0,1/2)$ and $c_{\ast }>0$ supplied by (S2), rather than choosing a different exponent. Eventually $X_N\ge c{A}^{-{\eta }^{\prime }}$ for some fixed $c>0$ on that selected family. Choose a fixed $\lambda >0$ with ${\lambda }^2\le c/2$, and set

$$h_{{\eta }^{\prime }}(A)=\lambda A^{1-{\eta }^{\prime }/2}\sqrt{\log A}=o(A).$$

For $L\ge 1$, (P1) gives $R(A,A+h_{{\eta }^{\prime }}(A))\le {\lambda }^2{A}^{-{\eta }^{\prime }}$. Consequently every actual optimizer on this entire interval satisfies

$$X(b)\ge \frac{c}{2}{A}^{-{\eta }^{\prime }}(A\le b\le A+h_{{\eta }^{\prime }}(A)).\text{\hspace{1em}(P3)}$$

This provides a quantitative interface for the existing exceptional-center sampling obstruction. For example, put $\delta =c/(42^{{\eta }^{\prime }})$ and ${\mathcal{E}}_X=\{b\in [X,3X]:X(b)>\delta X^{-{\eta }^{\prime }}\}$. For any selected source with $A\in [X,2X]$, (P3)’s interval lies in $[X,3X]$ eventually and is contained in ${\mathcal{E}}_X$. Its length is at least $\lambda X^{1-{\eta }^{\prime }/2}\sqrt{\log X}$. An unconditional estimate

$$|{\mathcal{E}}_X|=o\left(X^{1-{\eta }^{\prime }/2}\sqrt{\log X}\right)\text{\hspace{1em}(P4)}$$

would therefore exclude such selected sources at all sufficiently large scales, contradicting the existing RH-failure supplier. Here $|\cdot |$ is Lebesgue measure in the actual CA price variable $b$. No estimate (P4), or uniform upper bound for the mean in (P2), is established. Qualitative density $o(X)$ does not reach this required rate; an average over integers, prime cutoffs, moduli, or characters cannot be substituted without a proved transport of the averaging measure. The interval comparison adds no new signed ordinary-prime control and leaves the complete original Robin tail and RH unresolved.

Use the existing short-interval PNT on the actual CA price path

The uniform short-interval theorem in Guth–Maynard, Corollary 1.3 can be applied to the same actual path in (P1). The centered prime-count conversion and wider interpolation scale are already in the FIB theory volume, §260, equations (260.5) and (260.9). They are reused here. The additional application is the transport from one selected $N\in {\mathcal{S}}_{\mathrm{reg}}$ to all the actual later $C_b$, including their higher layers and activation ties. It is a paper-level application of existing results, without an originality or Lean certification claim.

Retain $A,L,T,g,k,B,X_N,X$ from (P1), in particular $B(A)=A$. Put

$$u=L^{1/4},{\epsilon }_A=L{e}^{-u},t_0=A^{2/3}.$$

For all sufficiently large $A$, uniformly for $t_0\le t\le A$, the existing arithmetic inputs give

$$|B(A+t)-B(A)-t|\le C_0{\epsilon }_{A}t.\text{\hspace{1em}(P5)}$$

The existing uniform extension (GS4) gives the relative error $\ll (\log Y)e^{-(\log Y{)}^{1/4}}+Y^{-1/10}+Y^{-1/6}$ on all intervals in $[Y/2,3Y/2]$ of length at least $Y^{2/3}$. For $2^{2/3}{A}^{2/3}\le t\le A$, apply it with $Y=2A$ to $[A,A+t]$. For $A^{2/3}\le t<2^{2/3}{A}^{2/3}$, use directly the existing centered conversion (260.9), whose range contains this whole segment for large $A$, and its accompanying ordinary prime-power correction. Both give

$$|\psi (A+t)-\psi (A)-t|\ll {\epsilon }_{A}t(t_0\le t\le A),$$

including the already paid ordinary prime-power corrections. The existing full activation correction gives $|B(x)-\psi (x)|\ll \sqrt{x}[\log (2x){]}^{5/2}$ for every large real price coordinate $x$. Its two endpoint corrections are $O(\sqrt{A}{L}^{5/2})=o({\epsilon }_A{t}_0)$, proving (P5). No regularity or GA1 hypothesis is required of any future $C_b$.

Use the existing pressure from the Nicolas application, with its almost-everywhere derivative:

$$\begin{array}{rl}P(x)& =\gamma +\log \log x-g(x)x-\underset{m\ge 1}{\max }\{\log Z(m)-g(x)\log m\},\\ P^{\prime }(x)& =k(x)[x-B(x)].\end{array}$$

The concavity tangent gap is

$$H(b,B)=\log \log b+g(b)(B-b)-\log \log B\ge 0.$$

Since $B(A)=A$, the definitions give the exact identity $X_N-X(b)=P(b)-P(A)-H(b,B(b))$. The original $N$’s right-tail maximality supplies its nonnegative left side. For $0\le t<t_0$, monotonicity gives $A+t-B(A+t)\le t$; for $t\ge t_0$, (P5) and $B(A)=A$ give $A+t-B(A+t)\le C_0{\epsilon }_{A}t$. Integrating the existing derivative against decreasing $k$ yields, for one fixed $C_1>0$ and all $A\le b\le 2A$,

$$0\le X_N-X(b)\le C_{1}k(A)\left[A^{4/3}+{\epsilon }_A(b-A{)}^2\right].\text{\hspace{1em}(P6)}$$

The short initial segment is paid by $A^{4/3}$, rather than assigning a favorable sign to any ordinary-prime error. The pressure comparison retains $B(b)$ in the tangent gap; it does not identify $B(b)$ with $b$.

For example, at the already documented wider scale $h=A^{3/4}{e}^{u/2}/L$, (P6) implies

$$\underset{A\le b\le A+h}{\sup }T[X_N-X(b)]\ll A^{-1/6}+\frac{1}{L}⟶0.\text{\hspace{1em}(P7)}$$

This is the selected-source application of that existing scale, not a new interpolation theorem or an upper bound for $X(b)$.

A wider actual-price interval for the same supplied power excess

Keep the particular fixed ${\eta }^{\prime }\in (0,1/2)$ and $c>0$ from (S2) and (P3). Choose a sufficiently small fixed $\lambda >0$, depending only on $c,C_1$, and put

$${\tilde{h}}_{{\eta }^{\prime }}(A)=\lambda A^{1-{\eta }^{\prime }/2}\exp \left(\frac{(\log A{)}^{1/4}}{2}\right).$$

For every fixed ${\eta }^{\prime }>0$, this is $o(A)$; no lower bound on ${\eta }^{\prime }$ beyond positivity is imposed; the sufficiently large threshold may depend on that fixed exponent. The reused extension (GS4) in (P5) therefore also covers small fixed exponents for which ${\tilde{h}}_{{\eta }^{\prime }}(A)>A^{0.99}$ eventually. Since

$$\begin{array}{rl}k(A){\epsilon }_A{\tilde{h}}_{{\eta }^{\prime }}(A{)}^2& ={\lambda }^2\left(1+\frac{1}{L}\right)A^{-{\eta }^{\prime }},\\ k(A)A^{4/3}& =o(A^{-{\eta }^{\prime }}),\end{array}$$

(P6), with $\lambda$ small enough, gives eventually

$$X(b)\ge \frac{c}{2}{A}^{-{\eta }^{\prime }}\left(A\le b\le A+{\tilde{h}}_{{\eta }^{\prime }}(A)\right).\text{\hspace{1em}(P8)}$$

The actual-price interval is wider than (P3)’s by a factor $e^{(\log A{)}^{1/4}/2}/\sqrt{\log A}\to \infty$, up to fixed constants. The improvement uses the published uniform arithmetic estimate, with the same actual integer and all activation layers; it does not assert the sign of the remaining Robin tail.

For the same $\delta =c/(42^{{\eta }^{\prime }})$ and ${\mathcal{E}}_X$ from (P4), any selected $A\in [X,2X]$ places this entire interval in ${\mathcal{E}}_X\subseteq [X,3X]$ eventually. Hence a sufficient still-missing exceptional-set estimate can be weakened to

$$|{\mathcal{E}}_X|=o\left(X^{1-{\eta }^{\prime }/2}\exp \left(\frac{(\log X{)}^{1/4}}{2}\right)\right).\text{\hspace{1em}(P9)}$$

This is a required rate in Lebesgue measure of the actual price variable. No bound (P9), directed mean estimate, effective cutoff, complete signed floor (S4), or proof of RH is obtained. Qualitative density $o(X)$ and averages in other variables still require the quantitative same-source measure transport described after (P4).

A partial zero-free envelope leaves the supplied excess unpaid

Keep the actual family (S2), proper GA1 and the event-free bridge above. Write its original oscillation exponent as $u$ and its output exponent as ${\eta }_{\mathrm{out}}=u/r\in (0,1/2)$, so that

$$1-\Theta <u,\Theta <\sigma ,2u<r<\frac{1}{2\sigma },X_N\ge c{A}^{-{\eta }_{\mathrm{out}}},A=\log N.$$

These are the existing source parameters, not a new choice of a smaller output exponent. Assume conditionally that $\Theta \le \nu <1$. The already recorded complete explicit formula is directly reused, including all prime powers, actual zero multiplicities, the pole and trivial-zero terms, and the infinite endpoint. Its original coefficient is

$$F_A(s)=\frac{1}{s}{\int }_A^{\infty }{t}^{s-2}\frac{\log t+1}{{\log }^{2}t}dt.$$

For $A\ge 2$, its logarithmic Laplace representation gives

$$F_A(\rho )=\frac{A^{\rho -1}}{\rho }{\int }_0^{\infty }\frac{(1+v)A^{-v}}{1+v-\rho }dv,|F_A(\rho )|\le \frac{A^{\Re \rho -1}}{|\rho (1-\rho )|}\left(\frac{1}{\log A}+\frac{1}{{\log }^{2}A}\right).$$

Here $|1+v-\rho |\ge |1-\rho |$ on the actual critical strip. The finite classical weight $C_{\zeta }={\sum }_{\rho }{m}_{\rho }/|\rho (1-\rho )|$ is already used in that owning note; no new zero computation or numerical value is needed. The complete pole/trivial-zero contribution has magnitude at most $\log (2\pi )/(A\log A)$. Thus the conditional supplied envelope is

$$|I_{\psi }(A)|\le {\mathcal{E}}_{\nu }(A):=C_{\zeta }{A}^{\nu -1}\left(\frac{1}{\log A}+\frac{1}{{\log }^{2}A}\right)+\frac{\log (2\pi )}{A\log A}.$$

This applies the existing integrated formula, rather than introducing a new Chebyshev or zero-free estimate. Its upper allowance is eventually smaller than $c{A}^{-{\eta }_{\mathrm{out}}}$ when ${\eta }_{\mathrm{out}}\le 1-\nu$, including equality because of the logarithmic factor. The actual supplied parameters instead satisfy

$${\eta }_{\mathrm{out}}>2\Theta (1-\Theta )>1-\Theta \ge 1-\nu .$$

In particular, the cited seven-eighths input would give ${\eta }_{\mathrm{out}}>7/32$, whereas that envelope funds only ${\eta }_{\mathrm{out}}\le 1/8$. The external zero-free result remains a conditional source input without this repository’s accepted compiled dependency/axiom closure. Failure of the allowance to fund this family is not a lower bound on the actual tail or a proof that the joint arithmetic conditions cannot yield a stronger estimate.

For the same fixed exponent, the unpaid signed part can be isolated without truncating the original tail. Define, over the actual distinct nontrivial zeros, with their original multiplicities $m_{\rho }$,

$${\mathcal{Z}}_{{\eta }_{\mathrm{out}}}(A)=\Re \sum _{1-{\eta }_{\mathrm{out}}<\Re \rho \le \nu }{m}_{\rho }{F}_A(\rho ).$$

The same formula and absolute weight give the exact decomposition

$$I_{\psi }(A)=-{\mathcal{Z}}_{{\eta }_{\mathrm{out}}}(A)+e(A),|e(A)|\le {\mathcal{E}}_{1-{\eta }_{\mathrm{out}}}(A)=O\left(\frac{A^{-{\eta }_{\mathrm{out}}}}{\log A}\right).$$