bibkey: mantovanelli2026primeworkload authors: Marco Mantovanelli year: 2026 title: Colossally Abundant Numbers, Robin’s Inequality, and an Exact Prime-Layer Workload doi: null url: https://zenodo.org/records/22014299 claim: The archived manuscript supplies the exact regular-return packet identity, a sharp two-moment lower bound, and a cone-envelope reduction for the actual unrestricted CA pressure; none decides the unresolved Robin sign. strata_touched: [] license: citation-only triage: anchor

Actual CA packets and their existing moment bound

The primary used here is paper/colossally_abundant_prime_layer_workload.tex in the author’s Zenodo record 22014299, version 1.0.0, published 19 August 2026. The 88,822-byte archive has MD5 b89a5bb3bf71aec8c4cd5b807aa0097f, matching the record metadata. The manuscript SHA-256 is 8208445c84c6cafc971a6f2a0637b06715a783f680de80d2ed3bb4a3be503d65, matching its archive manifest. These identify the inspected source; the archive’s computational certificates are not premises here and were not rerun.

The record DOI, 10.5281/zenodo.22014299, identifies the computational companion, not a journal publication of the paper. The manuscript is separately licensed CC BY 4.0; this note cites it rather than redistributing the manuscript or code. The ResearchGate page reports an “Improved Version 2” with preprint DOI 10.13140/RG.2.2.24429.76002; correspondence of that version with this archived manuscript is not established. All locators below refer to the inspected LaTeX source, not to assumed PDF pagination, peer review, or Lean verification.

Parameters and the actual packet

Source §2.1 defines $G(n)=[\sigma (n)/n]/\log \log n$ for $n>e$. Definition 3.3 and Appendix A require regular returns to be event-free and to satisfy ${\Phi }_{\mathrm{CA}}(a)=a$. The global prefix includes every tied layer. For the project’s existing pressure, §§98 and 111, the exact parameter map is

$${\eta }_{p,j}={\tau }_{p,j},{\Phi }_{\mathrm{CA}}=A^{+},D_{\mathrm{CA}}(x)=x-A^{+}(x),\hat{\mathcal{F}}(x)=-\mathfrak{D}(x).$$

Here $g(x)=1/(x\log x)$ is the same unrestricted price. On event-free regular endpoints $a<b$, both endpoint conventions agree, the open packet has total mass $L=b-a$, and every simultaneous layer keeps weight $\log p$. Theorem 5.2 (thm:packet-identity) and §7 give the finite probability law

$${\mu }_{a,b}=\frac{1}{L}\sum _{a<{\eta }_{p,j}<b}(\log p){\delta }_{{\eta }_{p,j}},\log \frac{G(C_b)}{G(C_a)}=L\left(\int gd{\mu }_{a,b}-\overline{g}(a,b)\right),$$

where $\overline{g}(a,b)=(\log \log b-\log \log a)/L$. Proposition 7.6 (prop:moment-workload) already supplies the complete moment–workload duality. These identities and their endpoint bookkeeping are reused, not delivered as new project mathematics.

The existing prime-local bridge

Theorem thm:direct-bridge, §4 of the same archived manuscript, starts with an actual integer $N>e$ satisfying

$$N/p>e,G(N)\ge G(N/p)(p\mid N),G(N)\ge G(Np)(p\text{ prime}).$$

It directly gives $N=C_A$, ${\Phi }_{\mathrm{CA}}(A)=A$ and an event-free scale $A=\log N$, with price $1/(A\log A)$. The statement is about every proper prime-local Robin well; neither GA1 alone nor an arbitrary self-tangent return supplies its insertion hypothesis. This bridge is reused without reconstructing its neutral-layer proof.

Lemma lem:external-extremal-input and Theorem thm:persistent-obstruction, §11, select the least global maximizer of $G(n)$ on $n>5040$ under RH failure and place this single proper GA1–GA2 source at a regular return. They state no unbounded regular family with a fixed power-sized excess. The same-source quantified selection uses the prime-local bridge on its selected family; its family and excess quantifiers are separate from that fixed-maximizer reduction. These source statements and their hypothesis correspondence were inspected; no independent full proof audit or Lean certification is claimed.

The sharp two-moment statement is directly reusable

Theorem 7.7 (thm:sharp-two-moment), §7.4, applies to any probability measure on $[a,b]\subset (1,\infty )$ with mean $m<b$ and variance $V$. With $r=m-V/(b-m)$ and $\alpha =(b-m)/(b-r)$ it gives

$$\int gd\mu \ge H_g:=\alpha g(r)+(1-\alpha )g(b).$$

The source proves $a\le r\le m$ using the bounded-support variance inequality and identifies the extremizer $\alpha {\delta }_r+(1-\alpha ){\delta }_b$ by quadratic Hermite interpolation. Its Remark 7.8 states that support, mean and variance alone cannot improve this lower bound. No new generic moment or quadrature theorem is asserted.

On the same actual packet this means

$$\mathfrak{D}(b)-\mathfrak{D}(a)\le L(\overline{g}(a,b)-H_g).$$

Thus $H_g>\overline{g}$ certifies a decrease of the full pressure margin; it does not certify positivity of either endpoint. The relaxed extremizer may have an atom at $b$, while an actual regular packet has none. Validity as a lower bound does not make that measure an arithmetic realization.

Existing completeness and cone reduction

Theorem 7.2 (thm:moment-expansion) already gives a complete positive moment hierarchy: every genuine ascent has a finite-order certificate. Corollary 9.3 (cor:infinite-packet-class) supplies infinitely many such later packets only from a source with $G(C_a)<e^{\gamma }$. It therefore does not address a potentially nonpositive Robin margin. Remark 9.4 explicitly leaves uniform low-moment control on an unbounded family of short packets as the arithmetic problem.

Theorem 10.4 (thm:cone-envelope) identifies, for each regular source, the supremum over all multiples of $C_a$ with the tail supremum of $\hat{\mathcal{F}}$ and then with later regular returns, including the source. This is a same-source extremal reduction, not a sign theorem.

Scope of the project’s additional packet analysis

The project’s §417 uses the already established $A(t)=t+o(t)$ on the actual event packets. It evaluates the existing two-moment expression on fixed and unbounded endpoint ratios and excludes its sufficient certificate uniformly when $b/a\ge 1+d$, for each fixed $d>0$ at sufficiently large $a$. Consequently successful asymptotic certificates require $b/a\to 1$.

That is a repository-derived scale limitation of this particular certificate, not a new general inequality or a certified originality claim. The inspected §§7.4 and 9 state sharpness and the short-packet arithmetic difficulty; they do not supply this fixed-ratio constant comparison or its unbounded-ratio uniform conclusion. This bounded source comparison does not certify an exhaustive literature search. Shrinking width alone still does not force successful actual moments, the full Robin sign, or RH.

The project’s §418 adds a quantitative condition on the same actual packet. For $a\ge e$ and $L=b-a\le a$, let $M={\sup }_{[a,b]}|A^{+}(t)-t|$. A successful two-moment certificate requires $L^3\le 6400a^{2}M$. The uniform-reference Hermite loss and the moment formula are reused from Theorem 7.7 and Proposition 7.6; the additional application controls the change in that expression by the same packet’s workload error. Theorem 8.5 (thm:psi-normal-form), combined with an existing unconditional PNT error supplier, then yields a shrinking necessary relative-width rate. These are restrictions on this sufficient certificate, not a successful actual packet, an endpoint sign theorem, an effective starting threshold, or a solution of §12’s prob:low-moment.

The remaining signed supplier and its sampling conditions

The existing Guth–Maynard note already supplies the uniform short-interval input and its project applications. Within its stated range it applies to actual starting points without an exceptional-center selection step. It still gives an absolute increment allowance, not the signed workload surplus required by the packet certificate. An almost-all estimate has an additional sampling obligation: regular roots are locally finite, so an exceptional set of real measure zero can contain all of them. Their use as noninteger endogenous endpoints also requires the same estimate up to the chosen return, with the actual layer correction retained.

The Caveney–Nicolas–Sondow primary, Theorems 6–7, already gives infinite CA subclasses with and without GA1. Its CA parameters and prime-deletion conclusions do not supply actual regular tangent endpoints or forward signed moments. In particular, Lemma 7’s sufficient price inequality cannot be satisfied by directly substituting the tangent price; another admissible price of the same integer must be checked. The known $720720$ example is already covered by that primary and the project’s §98.5, so it is not a new counterexample.

For a precise arithmetic supplier target, keep the same actual packet and put $E(t)=A^{+}(t)-t$, $L=b-a$, $K=-g^{\prime }$, and $B=g^{\prime \prime }$. Set

$$Q_0={\int }_a^{b}E(t)dt,Q_1={\int }_a^b(t-a)E(t)dt,W=Q_0-\frac{B(a)}{K(a)}{Q}_1.$$

In the near-uniform regime $a\to \infty$, $L=o(a)$, $Q_0=o(L^2)$ and $Q_1=o(L^3)$, the existing moment identities and two-node expression, expanded by ordinary Taylor calculus, give

$$H_g-\overline{g}=\frac{K(a)}{L}\left[W-\left(\frac{1}{36}+o(1)\right)\frac{L^4}{a^2}\right].$$

The first two Taylor terms are exactly $K(a)Q_0/L-B(a)Q_1/L$. The limiting two-node law has weights $3/4,1/4$ at normalized locations $1/3,1$, so its third moment is $5/18$, whereas the uniform third moment is $1/4$. Their difference is $1/36$; using $g^{\prime \prime \prime }(a)/K(a)=-(6+o(1))/a^2$ and the Taylor factor $1/6$ gives the term displayed. The fourth-order remainder is smaller by $O(L/a)$. This is an application of existing moment and calculus interfaces, not a new general moment theorem or an assertion made by the archived paper.

Consequently a fixed positive surplus $W\ge (1/36+{ϵ}_0)L^4/a^2$ on an unbounded actual family satisfying those regime conditions would pay this intermediate two-moment target. It is not established by the cited sources. Even a transferred absolute bound $\sup |E|\le ϵ(a)L$ only gives $|W|\le ϵ(a)L^2(1+O(L/a))$. For $L=a^{\beta }$, $\beta <1$, the short-PNT-type allowance $ϵ(a)L^2$, with $ϵ(a)=\exp [-(\log a{)}^{1/4}]$, remains asymptotically larger than $L^4/a^2$. This compares guaranteed allowances; it neither supplies the actual sign nor proves the packet fails.

An arbitrary unbounded winning family would address the archived intermediate problem. Applying that family to full Robin still requires a selection theorem covering the relevant potentially nonpositive self-matching minima, without assuming those sources are safe.

Source-local multipliers use the existing finite-size and cone results

Section 4’s lem:neutral-point and thm:neutral-mean already give the finite layer’s insertion/deletion thresholds. A successful untied singleton packet has $C_b=p{C}_a$, so its source is improved by one prime insertion. It therefore cannot start at a proper prime-local well, where every such insertion is non-improving. This is a direct application of the cited source, not a new obstruction theorem. Arbitrary winning singleton families do not pay the source-selection obligation at these wells.

The existing thm:cone-envelope gives a less restrictive constructive target: any actual multiple $m$ with $G(m)>G(C_a)$ guarantees an improving later regular return. A joint multiplier need not itself be a CA prefix or end at a prescribed return. Endpoint isolation and a fixed packet length are optional stronger conditions for a particular construction.

The source-local multiplier experiment reuses catalog factorizations and the author’s original rational log enclosure implementation for 109 additional comparisons. At $N=2021649740510400$, all single-prime insertions and deletions decrease $G$. The archive’s existing verification/packet_identity_checks.json already verifies the ascent to return 7, precisely the joint insertion $3\cdot 37$; that positive comparison is directly reused, not recomputed as a new result. At $N=160626866400$, all one- and two-prime insertions decrease it, including repeated primes. Its $2\cdot 29$ descent is also directly reused from the archive’s existing return 4-to-5 check, leaving 65 additional pair comparisons at that source. Universal prime coverage here uses the elementary absent-prime replacement argument stated in the report; the computed intervals themselves cover its finite pool. Neither the integers nor the log-enclosure method are claimed as new discoveries.

These finite applications distinguish actual joint improvement from single-layer improvement, and disallow a universal two-prime shortcut at all proper local wells. They give neither an unbounded family nor a result restricted to potentially persistent Robin-level sources. The full signed source-coverage obligation remains open; these additions have no Lean verification or literature originality certification.

Repetitions can be removed within a bounded multiplier search

The existing layer formula in §2, eq:layer-data, and the finite-size comparison in §4 give a useful paper-level pruning interface. Keep the same integer $N$, put $A=\log N>27/8$, and assume $G(Np)\le G(N)$ for every prime $p$. For every positive integer multiplier $u$ with $W=\log u\le A/2$, let $\mathrm{rad}(u)$ be the product of its distinct prime factors, with $\mathrm{rad}(1)=1$. Then

$$G(N\mathrm{rad}(u))\ge G(Nu),$$

with strict inequality when $u$ has a repeated prime factor. This uses neither a CA hypothesis nor a prior positive Robin margin.

For the calculation, write

$$c_A(w)=\log \log (A+w)-\log \log A={\int }_A^{A+w}g(t)dt,h_{p,j}=\log r_{p,j}.$$

If $S_{p,j}=p+\cdots +p^j$, the classical formula gives $S_{p,j+1}=p(S_{p,j}+1)$. The elementary bounds $\log (1+t)<t$ and $\log (1+t)\ge t/(1+t)$, for $t>0$, hence give

$$h_{p,j+1}<\frac{1}{p(S_{p,j}+1)}\le \frac{h_{p,j}}{p}.$$

Source insertion stability says $h_{p,v_p(N)+1}\le c_A(\log p)$. If a current multiplier still contains at least two copies of $p$, its top inserted layer therefore has reward less than $\frac{1}{2}(\log p)g(A)$. The current integer $m$ has $A\le \log m\le A+W$, and removing that copy saves the denominator cost

$${\int }_{\log m-\log p}^{\log m}g(t)dt\ge (\log p)g(A+W)>\frac{1}{2}(\log p)g(A).$$

The last inequality follows from $W\le A/2$ and $(3A/2)\log (3A/2)<2A\log A$, which is equivalent to $A>27/8$. Thus each such deletion strictly increases $G$. Iterating removes only the repeated copies in $u$, retains $N$ and all required stack prefixes, and reduces the multiplier budget. Intermediate states need not be insertion-stable: the comparison continues to use the original source’s next-layer bound. When $u$ is squarefree the two integers coincide.

Consequently, within this budget, an improving multiplier with at most $k$ inserted layers exists exactly when an improving subset of at most $k$ distinct primes exists. A construction with the explicit bound $W=O(\log A)$ eventually lies in this budget. A fixed layer count alone does not imply that bound. The reduction need not preserve exactly $k$ layers, a prescribed endpoint, or the structure of an actual return packet; it is suitable for the unrestricted-multiple target of thm:cone-envelope. It supplies no improving subset or critical-source coverage.

This is a derived application of the cited layer and finite-size data, not a statement attributed verbatim to the manuscript, a new generic inequality, or a Lean result. No originality certification or full Robin/RH conclusion is supplied.

The complete Euler transform of actual GA2 subset losses

This is a scope check using classical Bernstein complete alternation and finite Euler expansion, not an additional theorem attributed to the archived manuscript or a new Robin estimate. Use the Caveney–Nicolas–Sondow GA2 condition at one actual integer $N>5040$, put $A=\log N$, and assume $P^{+}(N)\le A$. For a finite prime set $\mathcal{B}\subset \{p:p>A\}$, define, for every $\mathcal{S}\subseteq \mathcal{B}$,

$$\begin{array}{rlrlrl}{q}_{\mathcal{S}}& =\prod _{p\in \mathcal{S}}p,& w_{\mathcal{S}}& =\log q_{\mathcal{S}},& b_{\mathcal{S}}& =\sum _{p\in \mathcal{S}}\log (1+1/p),\\ C_A(w)& =\log \log (A+w)-\log \log A,& L_A(\mathcal{S})& =C_A(w_{\mathcal{S}})-b_{\mathcal{S}}.& & \end{array}$$

Multiplicativity and GA2 directly give $L_A(\mathcal{S})=\log [G(N)/G(N{q}_{\mathcal{S}})]\ge 0$. All comparisons retain the same $N$. The empty block is included with $q_{\varnothing }=1$ and $L_A(\varnothing )=0$.

The denominator cost has the classical positive representation

$$C_A(w)={\int }_{(0,\infty )}(1-e^{-wt}){\nu }_A(dt),w\ge 0,$$

with a nonzero positive measure. It can be checked without a general Bernstein theorem: the classical log integral and the Gamma Laplace integral, with $r=s/\log A>0$, give

$$\begin{array}{rl}{C}_A(w)& ={\int }_0^{\infty }\frac{e^{-s}}{s}\left[1-(1+w/A{)}^{-s/\log A}\right]ds\\ & ={\int }_0^{\infty }{\int }_0^{\infty }(1-e^{-wt})\frac{e^{-s}{A}^r{t}^{r-1}{e}^{-At}}{s\Gamma (r)}dtds.\end{array}$$

Tonelli applies to this positive integrand. The density on $s,t>0$ is strictly positive. For every nonempty $\mathcal{U}\subseteq \mathcal{B}$, the interaction integral

$$I_A(\mathcal{U})={\int }_{(0,\infty )}\prod _{p\in \mathcal{U}}(1-p^{-t}){\nu }_A(dt)$$

is finite and strictly positive: selecting $p_0\in \mathcal{U}$, its integrand lies between zero and $1-p_0^{-t}$, whose integral is $C_A(\log p_0)<\infty$.

Set $E_0={\prod }_{p\in \mathcal{B}}(1-1/p)>0$. The actual complete Euler-weighted loss transform satisfies

$$\begin{array}{rl}{\mathsf{T}}_A(\mathcal{B})& :=\sum _{\mathcal{S}\subseteq \mathcal{B}}\frac{\mu (q_{\mathcal{S}})}{q_{\mathcal{S}}}{L}_A(\mathcal{S})\\ & =-E_0\left[\sum _{p\in \mathcal{B}}\frac{L_A(\{p\})}{p-1}+\sum _{\begin{array}{c}\mathcal{U}\subseteq \mathcal{B}\\ |\mathcal{U}|\ge 2\end{array}}\frac{I_A(\mathcal{U})}{{\prod }_{p\in \mathcal{U}}(p-1)}\right].\end{array}$$

For the finite calculation, keep $E_{\mathcal{B}}(t)={\prod }_{p\in \mathcal{B}}(1-p^{-1-t})$ and use

$$\sum _{\mathcal{S}\subseteq \mathcal{B}}\frac{\mu (q_{\mathcal{S}})}{q_{\mathcal{S}}}{C}_A(w_{\mathcal{S}})={\int }_{(0,\infty )}[E_0-E_{\mathcal{B}}(t)]{\nu }_A(dt),$$

$$\sum _{\mathcal{S}\subseteq \mathcal{B}}\frac{\mu (q_{\mathcal{S}})}{q_{\mathcal{S}}}{b}_{\mathcal{S}}=-E_0\sum _{p\in \mathcal{B}}\frac{\log (1+1/p)}{p-1},\frac{E_{\mathcal{B}}(t)}{E_0}=\prod _{p\in \mathcal{B}}\left(1+\frac{1-p^{-t}}{p-1}\right).$$

Expand the last finite product completely. Its singleton terms integrate to $C_A(\log p)$ and combine with the additive benefits to yield the displayed loss transform. Keep $E_0-E_{\mathcal{B}}(t)$ together inside the integral; the individual constant integrals need not converge.

Thus ${\mathsf{T}}_A(\mathcal{B})\le 0$, and it is strictly negative if $|\mathcal{B}|\ge 2$. This conclusion needs only the actual singleton losses to be nonnegative; strict singleton losses are unnecessary. For $\mathcal{B}=\varnothing$ the transform is zero, and for $\mathcal{B}=\{p\}$ it is $-L_A(\{p\})/p$. No subset order is discarded. The particular choice $\mathcal{B}=\{p:A<p\le 2A\}$ is allowed; no limit over an infinite prime tail is asserted.

Consequently neither pointwise GA2 loss positivity nor the additional Bernstein structure makes this particular complete transform nonnegative. Its negative has the displayed positive decomposition, but no sign-preserving comparison with the already recombined actual $I_{\psi }(A)+R(A)$ has been supplied. The FIB volume’s §§434–435 pair the native same-filter kernel window with its complete complement; those are different quantities and are reused directly. This scope check excludes the proposed unflipped loss transform as a positive supplier, not other Abel or cumulative transforms, and gives no Robin-critical margin, source coverage, new Bernstein theorem, or Lean verification.

Divisor-record box geometry does not transfer to Robin records

The inspected primary is Marco Mantovanelli, Prime-Exponent Transition Geometry and Divisor Barriers Between Consecutive Highly Composite Numbers, arXiv:2608.17045v1, 17 August 2026. Its definitions in §2, Proposition 4.1, Theorem 4.2, Corollary 4.3 and layer spectrum in §7.1 were read. This identifies the inspected version; a latest-version comparison and whole-paper proof audit are not supplied. No computation from its companion archive is a premise.

The source uses the closed prime-exponent box

$$\mathcal{B}(m,n)=\left\{\prod _p{p}^{e_p}:\min (v_p(m),v_p(n))\le e_p\le \max (v_p(m),v_p(n))\right\}.$$

Proposition 4.1 supplies the complementary state $z^{♯}=mn/z$ in the same box, with $z{z}^{♯}=mn$ and $d(z)d(z^{♯})\ge d(m)d(n)$. Theorem 4.2 gives $\mathcal{B}(H,H^{\prime })\cap (H,H^{\prime })=\varnothing$ for consecutive strict divisor-count records. Corollary 4.3 consequently puts every interior state of a ceiling-admissible mixed geodesic below $H$. These published results are reused, not reproved. The layer reward in §7.1 is $\log ((j+1)/j)$ for $d(n)$; it is not a layer reward for $Z(n)=\sigma (n)/n$ or $G(n)=Z(n)/\log \log n$.

There is already a published actual pair that prevents transferring the box-gap conclusion to consecutive XA records. Use $H=n_1$ and $H^{\prime }=n_3$ from Nazardonyavi–Yakubovich, Remark 5.5, printed p.26, arXiv:1211.2147v3, as cited in the existing XA source note. Their consecutive XA status is the primary’s reported numerical result, not independently reproduced here. Their published factorizations give

$$H^{\prime }=H\frac{149\cdot 151}{2},v_2(H)=9,v_2(H^{\prime })=8.$$

The other changed coordinates are $v_{149},v_{151}:0\to 1$; all remaining coordinates agree. Thus the actual integers

$$z=149H,z^{♯}=\frac{151H}{2}$$

both belong to $\mathcal{B}(H,H^{\prime })\cap (H,H^{\prime })$, and satisfy $z{z}^{♯}=H{H}^{\prime }$. Membership and the strict size inequalities follow directly from those factorizations; no new enumeration is required. Conditional on the cited record classification, this is a counterexample to the XA version of the proposed box-gap transfer, not to the source’s HCN theorem or to Robin’s inequality.

The normalization explains why the reflected-product argument does not force a contradiction here. Each changed exponent has only its two endpoint choices, so multiplicativity gives $Z(z)Z(z^{♯})=Z(H)Z(H^{\prime })$. Set $t=\log n$ and $b(t)=\log \log t$ for $t>1$. Ordinary calculus gives

$$b^{\prime \prime }(t)=-\frac{1+\log t}{t^2(\log t{)}^2}<0.$$

The two interior logarithms have the same sum as the endpoint logarithms. Strict concavity therefore yields $b(\log z)+b(\log z^{♯})>b(\log H)+b(\log H^{\prime })$, and consequently

$$G(z)G(z^{♯})<G(H)G(H^{\prime }).$$

Thus the size denominator changes the product comparison needed by the HCN proof. This is a direct scope application of the published example and elementary multiplicativity and calculus, not a new general reflection theorem, a signed prime-error estimate, or a Lean result.

There is a separate source-selection obstruction: under a Robin counterexample hypothesis, the project’s selected least global maximizer is the last XA, by the existing source note. It has no later strict $G$-record with which to form the required consecutive pair. Neither replacing it by a divisor-count record nor assuming a later XA retains that source. A usable joint-prime supplier must apply at the same actual selected integer and control the full Robin-normalized comparison; the cited box geometry does not supply it.

Local root spacing controls the sign-exact first-layer sampling error

Use the existing full largest optimizer $C_b$ and let $R(n)=\log G(n)-\gamma$. For $X\ge 4$ and $X\le u<v\le 3X$, put $I=[u,v]$ and $h=v-u$. For any real threshold $t$, let $K_I(t)$ count the complete positive-length plateaus intersecting $I$ in positive length and satisfying $R(C_b)>t$. Define the actual post-event sample count

$$P_I(t)=\sum _{\begin{array}{c}p\text{ prime}\\ u\le {\tau }_{p,1}<v\end{array}}{1}_{\{R(C_{{\tau }_{p,1}})>t\}}.$$

Every simultaneous layer is included in $C_{{\tau }_{p,1}}$. The existing actual-event charging applies with this exact full-state sign, before its prime-prefix upper relaxation. Distinct first-prime samples give distinct plateaus, and only an initial boundary plateau or higher-only event can create an extra counted state. Consequently

$$0\le K_I(t)-P_I(t)\le 1+H_I,H_I=\#\{(p,j):j\ge 2,u\le {\tau }_{p,j}<v\}.\text{\hspace{1em}(LS1)}$$

This reuses the existing bookkeeping rather than giving a second proof of it. The local bound below adds a dependence on the actual interval length; it does not estimate either sample’s Robin sign.

For the usual real layer root $x_j(b)>1$, use

$$F_j(x)=\frac{\log (1+1/S_j(x))}{\log x},S_j(x)=\sum _{k=1}^j{x}^k,F_j(x_j(b))=g(b).$$

The classical decreasing threshold has limits $\infty$ and $0$ at the two ends of $(1,\infty )$, so its inverse root is well defined. For $x\ge 2$ and $j\ge 2$, logarithmic differentiation gives

$$-\frac{F_j^{\prime }(x)}{F_j(x)}=\frac{S_j^{\prime }(x)}{S_j(x)(S_j(x)+1)\log (1+1/S_j(x))}+\frac{1}{x\log x}\ge \frac{j}{4x}.\text{\hspace{1em}(LS2)}$$

Indeed, $\log (1+1/S_j)\le 1/S_j$, $S_j+1\le 2S_j$, $S_j\le 2x^j$ and $S_j^{\prime }\ge j{x}^{j-1}$ give the displayed lower bound. With $-g^{\prime }(b)/g(b)\le 2/b$ for $b\ge e$, implicit differentiation yields $0<x_j^{\prime }(b)\le 8x_j(b)/(jb)$ wherever $x_j(b)\ge 2$.

Put $Q_X=3X\log (3X)/\log 2$ and $d_X=\lfloor \log Q_X/\log 2\rfloor$. The existing activation cutoff, §420.2, proof preceding (420.5), gives $p^j<Q_X$ for every layer active by $3X$, hence only $j\le d_X$ can contribute. On the effective part $x_j\ge 2$ it also gives $x_j<Q_X^{1/j}$. For $j=2$, comparison at $x=\sqrt{2b}$ gives $x_2(b)<\sqrt{2b}$: $x^2\log x=b\log (2b)>b\log b$ and $F_2(x)<1/(x^2\log x)<g(b)$.

Set $r_j(b)=\max (2,x_j(b))$. This increasing root has at most one clipping corner; its derivative bound can be integrated on either side. Integer counting in its image, followed by the derivative estimate, therefore gives the finite bound

$$\begin{array}{rl}{H}_I& \le d_X+\sum _{j=2}^{d_X}[r_j(v)-r_j(u)]\\ & \le d_X+\frac{8h}{X}\left[\frac{\sqrt{6X}}{2}+Q_X^{1/3}\log d_X\right].\end{array}\text{\hspace{1em}(LS3)}$$

The harmonic sum for $j\ge 3$ is at most $\log d_X$. Root endpoints with $u={\tau }_{p,j}$ or $v={\tau }_{p,j}$ are covered by the one-integer allowance per layer; the right price endpoint remains excluded in $H_I$. Together with (LS1), this gives uniformly over these intervals and all real $t$,

$$\boxed{0\le K_I(t)-P_I(t)\ll h/\sqrt{X}+\log X.}\text{\hspace{1em}(LS4)}$$

For the particular fixed exponent of the existing selected-source interval, $h=\lambda A^{1-\eta /2}{e}^{(\log A{)}^{1/4}/2}$, take $X=A$. Its sampling error relative to $h/\log A$ is $O(\log A/\sqrt{A}+(\log A{)}^2/h)=o(1)$. The interval and threshold may depend on actual arithmetic data; the bound is pointwise and assumes no independence.

Unlike a separate prime-product upper envelope, $P_I$ keeps the complete exponent tail and actual size in every sign test. No upper estimate on $P_I(t)$ or its positive-part moments is supplied. This is a paper-level application of the layer geometry and existing charging, not a new prime theorem, originality claim, Lean certification, complete signed Robin estimate or RH proof.

Project supplement: finite rough-shift covariance tail for the actual sample

This supplement is a project derivation attached to the first-prime sampling material above. It uses the complete largest optimizer, including all tied layers, and keeps the optimizer clock $A_p=\log N_p$ distinct from the event price $b_p={\tau }_{(p,1)}$. The existing Mantovanelli manuscript (Marco Mantovanelli, Colossally Abundant Numbers, Robin’s Inequality, and an Exact Prime-Layer Workload, Zenodo record 22014299, version 1.0.0) supplies the packet/workload identity and the actual-clock asymptotic used below. The Caveney–Nicolas–Sondow source, as recorded in the project’s caveney2012sacaga.md, and the project’s CA layer bookkeeping supply the largest-optimizer and first-layer facts. The rough Fourier estimates are named premises from OpenAI/math at commit adc7f1241b42e322a6451854ab7e4b4c146bf78a, specifically the pinned analytic-transfer.tex. The short-interval input is the named theorem of Matomäki–Radziwiłł–Tao, arXiv:1503.05121v3, Definition 1.6, Theorem 1.7 and (1.12). The project locators and their exact text are taken at trureturing commit 8e70a655ae3dc1965637bfc070169d7405526887.

These external results are used as ordinary theorem premises. Their full primary proofs are not reproduced or kernel-certified here. The new content is the finite endpoint calculation, the relative-clock coefficient budget, and its application to the existing complete statistic. No priority claim, Lean claim, official acceptance, Robin theorem, or RH conclusion is made.

Actual sample and finite support

Let $I=[u,v]\subset [X,3X]$, and use the same half-open first-prime sample as in LS1–LS4:

$$P_I=\{p\mathrm{prime}:u\le b_p={\tau }_{(p,1)}<v\},m=|P_I|.$$

The case $m=0$ is settled before taking any minimum or maximum: every sample sum and covariance budget below is then zero. For $m>0$, let $N_p=C_{b_p}$ be the actual largest full optimizer, with every equality layer retained, and put

$$A_p=\log N_p,a=\underset{p\in P_I}{\min }{A}_p,b=\underset{p\in P_I}{\max }{A}_p,T=\lfloor \frac{b}{2}\rfloor .$$

For the large-$X$ range used here, $A_p>1$ is an actual consequence of the unrestricted optimizer. Indeed, with $J_b(n)=\log (\sigma (n)/n)-g(b)\log n$, the elementary estimates $J_b(4)-J_b(2)=\log (7/6)-g(b)\log 2>0$ and $J_b(4)>3/7-1/32>J_b(1)=0$ show that $N_p$ is neither $1$ nor $2$. Thus $a\ge \log 3>1$ before the relative-clock definitions below.

Thus $A_p$ is a clock for the optimizer and $b_p$ is a price/event parameter; no equality $A_p=b_p$ is used. For $A>1$ and $x\ge 1$, write

$$g(t)=\frac{1}{t\log t},W_A(x)=\sum _{k=2}^{\lfloor A\rfloor }(\log k)[g(kx)-g(A){]}_{+}.$$

If $x\ge A/2$, then every $k\ge 2$ has $kx\ge A$, so each positive part vanishes. Hence $W_A(x)=0$ for $x\ge A/2$, and in particular

$$\Gamma (d,e)=\sum _{p\in P_I}(W_{A_p}(d)-\overline{W}(d))(W_{A_p}(e)-\overline{W}(e)),{\Gamma }_z(r)=\Gamma (r,r+z)$$

has ${\Gamma }_z(T+1)=0$ for every $z\ge 1$, where $\overline{W}(x)=m^{-1}{\sum }_p{W}_{A_p}(x)$. The finite support statement is about the actual $A_p$, not about the event values $b_p$.

Named rough Fourier and MRT suppliers

Fix $C_0\ge 1$, $1<\tau <2$, and $\epsilon =1/10000$. For a large parameter $B$, set

$$P_0=\exp (B^{1-\epsilon }),P_0/\tau \le M\le \exp (C_{0}B),D=\lceil M\rceil .$$

An integer is $P_0$-rough when no prime $q<P_0$ divides it. Let $Z$ be any finite set of $P_0$-rough integers in $(M,\tau M]$. Then $D\le z<2D$ for $z\in Z$, and

$$h_Z=\sum _{z\in Z}\frac{1}{z}\le 1.$$

If $Z=\varnothing$, all $Z$-indexed sums, blocks, selectors and corresponding budgets below are zero, including $h_Z,Q,E_r,F_B,{\omega }_r,{\kappa }_L,J,B_Z$ and the tail and sparse allowances. No maximum over $Z$ is taken; definitions for a nonempty rough set apply only when $Z\ne \varnothing$.

For coefficients fixed before the $w$-average, with $|c(z)|\le 1$, put

$$Q(\alpha )=\sum _{z\in Z}\frac{c(z)}{z}e(z\alpha ),e(t)=e^{2\pi it}.$$

The pinned OpenAI/math supplier is the following uniform premise: after one fixed $B_{\mathrm{rough}}=B_{\mathrm{rough}}(C_0,\tau )$, there are fixed constants $C_Q,C_4$ such that

$$\Vert Q{\Vert }_{\infty }\le C_Q{B}^{-1+\epsilon }\log B,{\int }_{\mathbb{T}}|Q(\alpha ){|}^{4}d\alpha \le C_4{M}^{-1}{B}^{-4+4\epsilon }(\log B{)}^6.\text{\hspace{1em}(FQ)}$$

This is the rough-multiplier lemma with arbitrary $z$-only coefficients; it is not a correlation estimate in disguise.

For the second supplier, define the exact MRT distance

$${\mathcal{M}}_{\mu }(r,Q)=\underset{\begin{array}{c}|t|\le r,q\le Q\\ \chi (\mathrm{mod}q)\end{array}}{\inf }\sum _{\begin{array}{c}\ell \mathrm{prime}\\ \ell \le r\end{array}}\frac{1-\Re (\mu (\ell )\overline{\chi (\ell ){\ell }^{it}})}{\ell }.$$

The cutoff in the prime sum and the height bound are both $r$. For a window $H\le r$, use

$$Q_H(r)=\min ((\log r{)}^{1/125},(\log H{)}^5),\ell (H)=\frac{\log \log H}{\log H}.$$

Theorem 1.7 of the MRT primary, with its frequency supremum outside the $y$-integral, gives for every fixed frequency $\alpha$, $r\ge H\ge 10$,

$${\int }_0^r\left|\sum _{y<n\le y+H}\mu (n)e(\alpha n)\right|dy\le C_{\mathrm{MRT}}Hr\left(e^{-{\mathcal{M}}_{\mu }(r,Q_H(r))/20}+\ell (H)+(\log r{)}^{-1/700}\right).\text{\hspace{1em}(MRT)}$$

For every prime $\ell$, $\mu (\ell )=\lambda (\ell )=-1$, so the prime-distance infimum for $\mu$ equals the corresponding Liouville infimum exactly. The quantitative statement (1.12), applied with its fixed small parameter, gives fixed $K_{\mu }$ and $r_0\ge 10$ such that

$${\mathcal{M}}_{\mu }(r,Q)\ge \frac{1}{4}\log \log r-K_{\mu }(r\ge r_0,Q\le (\log r{)}^{1/125}).$$

Thus the exponential term in (MRT) is at most $C_{\mu }(\log r{)}^{-1/80}$ for $r\ge r_0$. Enlarging one fixed constant to ${\tilde{C}}_{\mu }$ pays the finite range $10\le r<r_0$; no numerical value of $r_0$ or of a global $B_0$ is asserted.

Exact finite $D/3D$ reduction

For $r\ge 0$, let

$$S_r(z)=\sum _{1\le w\le r}\mu (w)\mu (w+z),S_0(z)=0.$$

For $r\ge 3D$, define

$$A_y(\alpha )=\sum _{y<n\le y+D}\mu (n)e(\alpha n),B_y(-\alpha )=\sum _{y<m\le y+3D}\mu (m)e(-\alpha m).$$

Since $D\le z<2D$, the first window is contained in the translated second window after imposing $m=n+z$. Integrating over $0\le y\le r$, the exact overlap for a fixed $n$ is

$$L_{D,r}(n)=\mathrm{length}([0,r]\cap [n-D,n))=\max (0,\min (r,n)-\max (0,n-D)).$$

For integer $r\ge D$, this is $D$ only on $D\le n\le r$. The early $n<D$ correction and late triangular tail are retained in the exact identity

$$L_{D,r}(n)=D{1}_{\{1\le n\le r\}}+e_{D,r}(n),e_{D,r}(n)=\{\begin{array}{ll}n-D,& 1\le n<D,\\ r+D-n,& r<n<r+D,\\ 0,& \text{otherwise}.\end{array}$$

The two triangular tails have total absolute mass exactly

$$\sum _{n\ge 1}|e_{D,r}(n)|=D(D-1).$$

Fourier orthogonality therefore gives the finite identity

$$\sum _{z\in Z}\frac{c(z)}{z}{S}_r(z)=\frac{1}{D}{\int }_0^r{\int }_{\mathbb{T}}Q(\alpha )A_y(\alpha )B_y(-\alpha )d\alpha dy-E_r,\text{\hspace{1em}(D3)}$$

where

$$E_r=\frac{1}{D}\sum _{z\in Z}\frac{c(z)}{z}\sum _{n\ge 1}\mu (n)\mu (n+z)e_{D,r}(n),|E_r|\le (D-1)h_Z.$$

The factor $D(D-1)$, rather than twice that number, is the complete endpoint cost. For $1\le r<3D$ we retain the elementary bound

$$\left|\sum _{z\in Z}\frac{c(z)}{z}{S}_r(z)\right|\le r{h}_Z;\text{\hspace{1em}(small-r)}$$

the exact overlap is still available when $D\le r<3D$, but no MRT window is silently invoked there.

For $r\ge 3D$, split at $t_B=B^{-1-\epsilon }$, with $E_B=\{\alpha :|Q(\alpha )|>t_B\}$. From (FQ),

$$|E_B|\le C_4{M}^{-1}{B}^{8\epsilon }(\log B{)}^6.$$

Parseval and Cauchy–Schwarz on $\mathbb{T}\setminus E_B$, where the two window lengths are $D$ and $3D$, give the first contribution $\sqrt{3}r{B}^{-1-\epsilon }$. On $E_B$, use

$$|A_y||B_y|\le \frac{3D|A_y|+D|B_y|}{2},$$

then apply Fubini and (MRT) separately at $H=D$ and $H=3D$. The frequency is fixed before each $y$-average; no supremum over $\alpha$ is moved inside that integral. For $H\in \{D,3D\}$, the prescribed $M$-range gives $\log H\gg B^{1-\epsilon }$ and

$$\ell (H)\le K_{\ell }{B}^{-1+\epsilon }\log B,K_{\ell }=2(1+\log (C_0+2)).$$

Combining the small-frequency term, the fourth-moment measure, the fixed supremum in (FQ), the two $\ell (H)$ terms, the $(\log r{)}^{-1/700}$ terms, and the exact tails in (D3), gives one fixed $C_{20}$ such that

$$\left|\sum _{z\in Z}\frac{c(z)}{z}{S}_r(z)\right|\le C_{20}r\left[B^{-1-\epsilon }+B^{-2+10\epsilon }(\log B{)}^8+B^{-1+9\epsilon }(\log B{)}^7(\log r{)}^{-1/700}\right]+(D-1)h_Z.\text{\hspace{1em}(finite-MRT)}$$

For example, after increasing constants harmlessly one may take

$$C_{20}=\max \{\sqrt{3},6C_{\mathrm{MRT}}{C}_Q{C}_4{K}_{\ell },6C_{\mathrm{MRT}}{C}_Q{C}_4(1+{\tilde{C}}_{\mu })\}.$$

The three bracketed terms respectively record the small-frequency estimate, the fourth-moment times the short-window $\ell (H)$ allowance, and the fourth-moment times the quantitative MRT decay. The final term is the exact triangular endpoint mass.

Choose a fixed $B_0$ large enough to include the rough supplier, $D\ge 10$, and the scalar inequalities

$$B^{-1+11\epsilon }(\log B{)}^8\le 1,B^{-(1/700-10\epsilon )}(\log B{)}^7\le 1.$$

Here $1-11\epsilon >0$ and $1/700-10\epsilon =3/7000>0$. With the corrected rounding

$$R_B=\lceil \max (\exp B,D{B}^2)\rceil ,\text{\hspace{1em}(RB)}$$

every $r\ge R_B$ has $r\ge 3D$ and $\log r\ge B$. Also $(D-1)h_Z\le D\le r{B}^{-2}\le r{B}^{-1-\epsilon }$. Therefore, for a fixed $C_{22}=3C_{20}+1$,

$$\left|\sum _{z\in Z}\frac{c(z)}{z}{S}_r(z)\right|\le C_{22}r{B}^{-1-\epsilon }(r\ge R_B).\text{\hspace{1em}(Corr)}$$

The start $B_0$ exists from the displayed scalar gaps and the named supplier constants; no numerical $B_0$ is claimed. In particular, (RB) is $\lceil \max (\exp B,D{B}^2)\rceil$, never $\lceil M{B}^2\rceil$.

For use at every finite row, define

$$F_B(0)=0,F_B(r)=r{h}_Z(1\le r<3D),F_B(r)=\text{the right side of (finite-MRT)}(r\ge 3D).\text{\hspace{1em}(FB)}$$

Then (FB) bounds the inner sum for every $r\ge 0$, including the rows below the corrected supplier range.

Relative clocks and the actual coefficient mass

Put

$$\ell =\log (b/a),D_a=1+\frac{1}{\log a},K=\ell +\log \frac{\log b}{\log a},C=D_a\left(\ell +\frac{1}{2}\right).$$

When $a=b$, set $K=C=\ell =0$, so every centered covariance below is exactly zero. For $a\le A^{\prime }\le A\le b$, let $U_{A,A^{\prime }}(x)=W_A(x)-W_{A^{\prime }}(x)$. With

$$Q_{A,A^{\prime }}(s)=\sum _{2\le k\le s}\log k,$$

the floor endpoints are harmless because every activation term is zero at its endpoint, and direct integration gives

$$U_{A,A^{\prime }}(x)={\int }_{A^{\prime }}^A(-g^{\prime }(t))Q_{A,A^{\prime }}(t/x)dt.\text{\hspace{1em}(U-int)}$$

Consequently $U\ge 0$, $U$ is decreasing and locally absolutely continuous, and

$$0\le U_{A,A^{\prime }}(x)\le \frac{K}{x}.\text{\hspace{1em}(U-size)}$$

For the bound, use $Q_{A,A^{\prime }}(t/x)\le (t/x)\log t$ and $-g^{\prime }(t)=(\log t+1)/(t^2(\log t{)}^2)$; after multiplying by $x$ the integral is at most ${\int }_{A^{\prime }}^A(1/t+1/(t\log t))dt=K$.

For almost every $x$, activation values cancel at their endpoints and

$$-U_{A,A^{\prime }}^{\prime }(x)=\sum _{\begin{array}{c}k\ge 2\\ A^{\prime }<kx<A\end{array}}(\log k)k[-g^{\prime }(kx)].$$

Since $kx\ge a$, each summand is at most $D_a/(x^{2}k)$. The active shell obeys

$$\sum _{\begin{array}{c}k\ge 2\\ A^{\prime }/x<k<A/x\end{array}}\frac{1}{k}\le \frac{1}{2}+\log {\frac{A}{A}}^{\prime }\le \frac{1}{2}+\ell ,$$

so

$$0\le -U_{A,A^{\prime }}^{\prime }(x)\le \frac{C}{x^2},0\le U_{A,A^{\prime }}(x)-U_{A,A^{\prime }}(y)\le C\left(\frac{1}{x}-\frac{1}{y}\right)(1\le x\le y).\text{\hspace{1em}(U-var)}$$

This is an activation-shell and variation estimate; no convexity of $U$ is claimed. It also covers $A<2$, empty shells, nonintegral clocks and all floor boundaries.

For $F_r(p)=W_{A_p}(r)$ and $H_r(p)=F_r(p)-F_{r+1}(p)$, both are increasing functions of the same clock $A_p$, since (U-var) gives $H_r(A)-H_r(A^{\prime })=U(r)-U(r+1)\ge 0$. The unnormalized covariance identity and Popoviciu’s range bound therefore give

$$0\le {\Gamma }_z(r)\le \frac{m{K}^2}{4r(r+z)},\text{\hspace{1em}(G-size)}$$

and, writing $h_z(r)=1/[r(r+z)]$,

$$0\le \Delta {\Gamma }_z(r):={\Gamma }_z(r)-{\Gamma }_z(r+1)\le \frac{mKC}{4}[h_z(r)-h_z(r+1)].\text{\hspace{1em}(G-var)}$$

Indeed,

$$\Delta {\Gamma }_z(r)={\mathrm{Cov}}_m(H_r,F_{r+z})+{\mathrm{Cov}}_m(F_{r+1},H_{r+z}),$$

where ${\mathrm{Cov}}_m$ is the unnormalized centered sum. Each term is nonnegative by the common-clock ordering, and the range products are $mKC/4$ times

$$\frac{1}{r(r+1)(r+z)}+\frac{1}{(r+1)(r+z)(r+z+1)}=h_z(r)-h_z(r+1).$$

This explicitly supplies the factor $m/4$ and does not use a false monotonicity assertion for a separate $z$-indexed sequence.

Finite Abel identity and coefficient budget

For an integer $L\ge 1$, first settle $T<L$: the block $O_Z^{\ge L}$ and every corresponding sum, selector and budget (including the lower anchor, $J$, and the tail and sparse allowances) are zero. No maximum, selector quotient or logarithmic budget expression is evaluated for an empty block. For the remaining block definitions and formulas, assume $Z\ne \varnothing$ and $1\le L\le T$, and put

$${\omega }_r=\underset{z\in Z}{\max }z\Delta {\Gamma }_z(r),c_r(z)=\{\begin{array}{ll}z\Delta {\Gamma }_z(r)/{\omega }_r,& {\omega }_r>0,\\ 0,& {\omega }_r=0,\end{array}$$

and, for the lower anchor,

$${\kappa }_L=\underset{z\in Z}{\max }z{\Gamma }_z(L),d_L(z)=\{\begin{array}{ll}z{\Gamma }_z(L)/{\kappa }_L,& {\kappa }_L>0,\\ 0,& {\kappa }_L=0.\end{array}$$

The set $Z$, every $c_r$, and $d_L$ are fixed before the inner $w$-sum; they depend on $z$ only, and all nonzero selectors lie in $[0,1]$. For the exact rough-shift block,

$$O_Z^{\ge L}=\sum _{z\in Z}\sum _{w=L}^T\mu (w)\mu (w+z){\Gamma }_z(w),$$

finite summation by parts and ${\Gamma }_z(T+1)=0$ give

$$O_Z^{\ge L}=\sum _{r=L}^T{\omega }_r\sum _{z\in Z}\frac{c_r(z)}{z}{S}_r(z)-\sum _{z\in Z}{\Gamma }_z(L)S_{L-1}(z).\text{\hspace{1em}(Abel)}$$

The lower term is exactly ${\kappa }_L{\sum }_z{d}_L(z)S_{L-1}(z)/z$, with its minus sign retained.

Let $z_{+}=\max Z$ when $Z\ne \varnothing$. Since

$$z[h_z(r)-h_z(r+1)]=\frac{1}{r(r+1)}-\frac{1}{(r+z)(r+z+1)},$$

the maximum direction is increasing in $z$. From (G-size)–(G-var),

$$\begin{array}{rl}J& :=\sum _{r=L}^{T}r{\omega }_r+(L-1){\kappa }_L\\ & \le \frac{m}{4}\left[KC{A}_{L,T}(z_{+})+K^2\frac{(L-1)z_{+}}{L(L+z_{+})}\right],\end{array}\text{\hspace{1em}(J)}$$

where the finite top term is kept in the exact identity

$$A_{L,T}(z)=z\left[L{h}_z(L)+\sum _{r=L+1}^T{h}_z(r)-T{h}_z(T+1)\right]$$

and then bounded by

$$A_{L,T}(z)\le \frac{z}{L+z}+\log \left(1+\frac{z}{L}\right)\le \frac{2z}{L}.$$