bibkey: trureturing2026pricegapconvexity authors: trureturing research synthesis year: 2026 title: Actual Robin price gaps, forced-core tail records and the selected-sign boundary doi: null url: https://github.com/the-omega-institute/trureturing/blob/0d8903a599ca95dc10b460e31a72cc1ebfc3433b/Library/ArithSums/trureturing2026extrawindowpairing.md claim: Under the original source-window hypotheses, the complete selected response retains actual price-gap costs; every inactive-forced-prime core is a strict all-integer right-tail record with original-exponent own-size excess. A synthetic positive-reference model excludes height-and-envelope plus numerical full-Q reasoning alone. Actual favorable slack and RH remain unproved. strata_touched: [] license: Apache-2.0 triage: anchor

Actual price gaps and the same-source signed estimate

Assume the qualifying original window in N1–N2 and PT1–PT11. These sources belong to the supplied RH-failure route. This is a supplier-relative paper application, with no Lean verification, literature priority claim or proof of RH. The existing source construction, finite pressure identity, full-Q estimates and quadrature are reused. The classical convex-order hinge characterization is reused inside the application. The added interface is its correspondence with the actual nonlinear selector and the complete original-source gap response; no new generic convex-order theorem is asserted.

Keep $W(u)=\log (\sigma (u)/u)$, $f(v)=\log \log v$, $\lambda (v)=1/(v\log v)$ and $w=-{\lambda }^{\prime }>0$. Let $K(v)=\log N_v^{\mathrm{str}}$ be the actual strict layer inventory and $Q=K-\psi$, with the complete prime-power function $\psi$. Let $dK$ be the atomic measure assigning every actual activation its full layer mass, including all simultaneous activations. Strict $K$ is left continuous and has $dK([a,b))=K(b)-K(a)$. The weak, right-continuous $\psi$ has $d\psi ((a,b])=\psi (b)-\psi (a)$ and its full von Mangoldt atoms. Define the signed measure $dQ=dK-d\psi$; the mixed function $Q=K-\psi$ is not assigned a common half-open increment convention. All integrations below use these measures with their actual atoms. The original finite core set is

$${\mathcal{C}}_X=\{u\in \mathbb{N}:5040\mid u,\log u\le R,\text{ every prime divisor of }u\text{ is at most }R\}.$$

The nonempty envelope is $V_X(t)={\max }_{u\in {\mathcal{C}}_X}{\Phi }_t(u)$, where ${\Phi }_t(u)=W(u)-f(t+\log u)$. Take all original prime labels in the window, $p_0<\cdots <p_m$, $m\ge 1$, with $t_i=\log p_i$, ${\delta }_i=t_{i+1}-t_i>0$ and $A_i=\log N_{p_i}^{\mathrm{mult}}$. Their canonical least-core selector is ${\mathcal{A}}_X(t)=t+\log u_{\min }(t)$. N1 supplies all endpoint ties and N2 supplies all-choice ordering. The common source start is enlarged so that $\ell >\max \{11,\log R+1\}$, using the existing $\ell /\log R\to \infty$.

Real-parameter active cores retain the actual strict inventory

For every $t\in [t_0,t_m]$ and every active core $u$, put $A=t+\log u$. All-choice ordering and the two actual endpoint all-tie bounds put $A$ in $[\ell ,R]$. Every insertion of a prime $q\le R$ stays in the original finite core domain, since

$$\log (uq)\le R-t+\log R<R.$$

Deleting a layer beyond the mandatory 5040 exponents also stays in that domain. For the actual layer reward per logarithmic mass $r_{q,j}=[W(q^j)-W(q^{j-1})]/\log q$, strict concavity and maximality give

$$r_{q,v_q(u)+1}\le \frac{f(A+\log q)-f(A)}{\log q}<\lambda (A),$$

and, for a removable last layer,

$$r_{q,v_q(u)}\ge \frac{f(A)-f(A-\log q)}{\log q}>\lambda (A).$$

The deletion argument has $A-\log q>1$ by the stated headroom. Mandatory layers are already strictly active by the supplied $A>11$ activation calculation. For $q>R$, the decreasing layer rewards satisfy $r_{q,j}\le r_{q,1}<\lambda (q)<\lambda (A)$. Consequently

$$\boxed{u=N_A^{\mathrm{str}},A\text{ is not an actual activation time}.}\text{\hspace{1em}(PG1)}$$

This extends the finite envelope’s inventory interface to its real parameters. The cores remain literal integers; $e^t$ is not declared a new prime label or a new original arithmetic source.

Full dominance pays a nonnegative prefix cost on every actual gap

For an actual switch $s\in [t_0,t_m)$, let $U,V$ be the smallest and largest active cores and put

$$\alpha =s+\log U,\beta =s+\log V,G_s=(\alpha ,\beta ),Y_s(z)=s+K(z).$$

Ignore only zero-length gaps. PG1 gives $K(\alpha )=\alpha -s$, $K(\beta )=\beta -s$ and no actual activation atom at either endpoint. Every $N_z^{\mathrm{str}}$, $\alpha \le z\le \beta$, satisfies $U\mid N_z^{\mathrm{str}}\mid V$ and belongs to the same ${\mathcal{C}}_X$. Keep its full envelope deficit ${\Delta }_s(z)=V_X(s)-{\Phi }_s(N_z^{\mathrm{str}})\ge 0$.

Reuse the finite pressure identity with the real shift $s$ and keep $C_z(y)=f(z)+\lambda (z)(y-z)-f(y)\ge 0$. It gives

$${\Phi }_s(N_z^{\mathrm{str}})-{\Phi }_s(U)={\int }_{\alpha }^z[Y_s(v)-v]w(v)dv+C_z(Y_s(z)).$$

Thus the continuous prefix cost

$${\mathcal{P}}_s(z)=-{\int }_{\alpha }^z[Y_s(v)-v]w(v)dv={\Delta }_s(z)+C_z(Y_s(z))\ge 0,{\mathcal{P}}_s(\alpha )={\mathcal{P}}_s(\beta )=0\text{\hspace{1em}(PG2)}$$

retains both full-core dominance and the actual concavity cost. Extend it by zero outside $G_s$. Since both $Y_s(z)$ and $z$ lie in $[\alpha ,\beta ]$, the decreasing curvature gives the additional bound

$${\mathcal{P}}_s(z)\ge \frac{w(\beta )}{2}[Y_s(z)-z{]}^2.\text{\hspace{1em}(PG3)}$$

No comparability of $\alpha ,\beta$ is assumed. These are consequences of full dominance, not replacements for dominance over every core. They concern the whole real-parameter switch gap; the previously supplied N4.5 arriving-clock contact and its preceding negative integral are reused.

Two moments and the price-coordinate hinge identity

Use the finite signed event measure $d{\mu }_s=(dK-dv){|}_{G_s}$, with every actual activation mass retained. The own-clock endpoints and the exact switch equality imply

$${\int }_{G_s}d{\mu }_s=0,{\int }_{G_s}\lambda (v)d{\mu }_s(v)=0.\text{\hspace{1em}(PG4)}$$

Indeed $K(\beta )-K(\alpha )=\beta -\alpha$, while the activation reward identity and the tie give $\int_{G_s}\lambda(v)dK(v)=W(V)-W(U) =f(\beta)-f(\alpha)=\int_\alpha^\beta\lambda(v)dv$. Finite integration by parts, or finite Fubini, then gives for every $z>1$

$${\int }_{G_s}(\lambda (v)-\lambda (z){)}_{+}d{\mu }_s(v)=-{\mathcal{P}}_s(z)\le 0.\text{\hspace{1em}(PG5)}$$

For $z$ outside the gap the two moments make this exactly zero. The standard hinge representation, or uniform convex piecewise-linear approximation on the compact price interval, therefore gives for every continuous convex $\phi$

$${\int }_{G_s}\phi (\lambda (v))dK(v)\le {\int }_{G_s}\phi (\lambda (v))dv.\text{\hspace{1em}(PG6)}$$

The coordinate is the decreasing price $\lambda$, not the original clock. All simultaneous activations retain their complete $\log q$ masses.

The original price tent has an exact gap response

For an original canonical clock $A_i$, keep PT1’s $a_i=A_i-A_i^{4/5}$, $b_i=A_i+A_i^{4/5}$, $m_{-,i}=\lambda (a_i)-\lambda (A_i)>0$ and $m_{+,i}=\lambda (A_i)-\lambda (b_i)>0$. For any such clock $A$, the original kernel has the exact expression

$$\begin{array}{rl}{k}_A(v)=& \frac{(\lambda (v)-\lambda (b){)}_{+}}{m_{+}}-\left(\frac{1}{m_{-}}+\frac{1}{m_{+}}\right)(\lambda (v)-\lambda (A){)}_{+}\\ & +\frac{(\lambda (v)-\lambda (a){)}_{+}}{m_{-}}.\end{array}\text{\hspace{1em}(PG7)}$$

Every canonical $A_i$ lies outside each open gap $G_s$. At a switch label the least-core convention selects its left endpoint. Hence ${\mathcal{P}}_s(A_i)=0$, and PG5 yields

$$\boxed{{\int }_{G_s}{k}_{A_i}(v)[dK(v)-dv]=-\frac{{\mathcal{P}}_s(a_i)}{m_{-,i}}-\frac{{\mathcal{P}}_s(b_i)}{m_{+,i}}\le 0.}\text{\hspace{1em}(PG8)}$$

A gap contained in one affine price segment contributes exactly zero. Every remaining contribution is retained through the actual edge costs; no gap is deleted as a sparse exceptional set. PG8 controls complete stock $dK-dv$, not the prime measure alone.

Exact decomposition of the original discrete joint response

Keep the original weights

$$W_X(v)=\sum _{i=0}^{m-1}{\delta }_i{A}_i^{-3/10}{k}_{A_i}(v),L_X=\sum _{i=0}^{m-1}{\delta }_i>0.$$

Let $I_X=[A_0,A_m]$ and $D_X=I_X\setminus {\bigcup }_s{G}_s$. Include switches at $t_0$ and exclude switches at $t_m$. On each fixed-core selector branch, PG1 makes $K$ constant; it also excludes activation atoms at active endpoints. Therefore

$$dK(D_X)=0,|D_X|=L_X.\text{\hspace{1em}(PG9)}$$

Other active cores at a multiple tie can have own clocks inside a gap. Those finitely many points are not removed from ${\mathcal{C}}_X$, have no activation atom by PG1, and do not affect the measure statements. Define the four complete quantities

$$\begin{array}{rl}{\mathsf{E}}_X& ={\int }_{(1,\infty )\setminus I_X}{W}_X(v)[dK(v)-dv],\\ {\mathsf{S}}_X& ={\int }_{D_X}{W}_X(v)dv\ge 0,\\ {\mathsf{G}}_X& =-\sum _s{\int }_{G_s}{W}_X(v)[dK(v)-dv]\ge 0,\\ {\mathsf{Q}}_X& =\int W_X(v)dQ(v).\end{array}\text{\hspace{1em}(PG10)}$$

Writing ${\mathcal{P}}_X={\mathcal{P}}_s$ on each gap and zero elsewhere, PG8 gives the explicit full gap cost

$${\mathsf{G}}_X=\sum _{i=0}^{m-1}{\delta }_i{A}_i^{-3/10}\left[\frac{{\mathcal{P}}_X(a_i)}{m_{-,i}}+\frac{{\mathcal{P}}_X(b_i)}{m_{+,i}}\right].\text{\hspace{1em}(PG11)}$$

Partition the finite support into $I_X^c,D_X,{\bigcup }_s{G}_s$ and use $d\psi -dv=(dK-dv)-dQ$. This gives the exact original response

$$\boxed{\sum _n\Lambda (n)W_X(n)-\int W_X(v)dv={\mathsf{E}}_X-{\mathsf{S}}_X-{\mathsf{G}}_X-{\mathsf{Q}}_X.}\text{\hspace{1em}(PG12)}$$

No new quadrature error, thinning or shortened prime window is introduced. Exterior flux has support only in $[A_0-A_0^{4/5},A_0)\cup (A_m,A_m+A_m^{4/5}]$. Both endpoints, all weights and both costs depend on the same arithmetic. A reference prime-power atom at a switch endpoint remains in $d\psi$. It contributes its negative full mass to $dQ$, even though $dK$ has no activation there; subtraction of ${\mathsf{Q}}_X$ restores its complete $\Lambda$ weight in PG12.

The original two terminal costs are separate from PG2’s switch costs. Keep the actual $Y_{-,i},Y_{+,i}$ from PT2 and define the two costs separately:

$${\mathsf{C}}_X^{-}=\sum _{i=0}^{m-1}\frac{{\delta }_i{A}_i^{-3/10}}{m_{-,i}}{C}_{a_i}(Y_{-,i})\ge 0,{\mathsf{C}}_X^{+}=\sum _{i=0}^{m-1}\frac{{\delta }_i{A}_i^{-3/10}}{m_{+,i}}{C}_{b_i}(Y_{+,i})\ge 0.$$

PT2 still gives

$${\mathsf{E}}_X-{\mathsf{S}}_X-{\mathsf{G}}_X+{\mathsf{C}}_X^{-}+{\mathsf{C}}_X^{+}\le 0.\text{\hspace{1em}(PG13)}$$

The favorable exterior estimate remains the unpaid sign

Reuse the complete pointwise full-Q supplier:

$${\mathsf{Q}}_X=d_0{L}_X+r_X,d_0=(\sqrt{2}-1)/2,|r_X|\le {\eta }_X{L}_X,{\eta }_X=C_Q\underset{\ell \le A\le R}{\sup }{\mathcal{F}}_Q(A)⟶0.\text{\hspace{1em}(PG14)}$$

All seven errors and their common unknown start remain. PT8 requires ${\mathsf{E}}_X-{\mathsf{S}}_X-{\mathsf{G}}_X\ge \epsilon L_X+r_X$. A sufficient still-unproved target is one fixed $\eta >0$ such that, beyond every common analytic/source start, some qualifying original full window satisfies

$$\boxed{{\mathsf{E}}_X\ge {\mathsf{S}}_X+{\mathsf{G}}_X+\eta L_X.}\text{\hspace{1em}(PG15)}$$

Once ${\eta }_X\le \eta /2$, this supplies PT8 with fixed $\epsilon =\eta /2$. It would contradict PG13 on that same window. The cost signs and exact moment cancellations do not establish this lower bound on exterior flux. No favorable prime sign follows from convex order of the complete stock measure or from the old upper comparison. The independent fixed-source signed finite-head obligation and RH also remain unresolved.

For historical reuse, the Erdős–Nicolas benefit is the existing fixed-linear-price loss. At a nonlinear switch the endpoint prices are $\lambda (\alpha )>\lambda (\beta )$; adjacency of these cores in the classical CA sequence has not been proved. Classical neighboring-CA restrictions therefore cannot bound this $V/U$ without that additional correspondence.

The actual inactive-forced-prime core is a strict right-tail record

Keep the original fixed $b,d$, qualifying witness $X$, every original prime label $p$, and every maximizing tie $N=pu$ supplied by N1. Put $t=\log p$, $B=\log u$, $A=B+t$, and $c_p=\log (1+1/p)$. The actual $p$ is absent from $u$ and strictly inactive: $c_p<t\lambda (A)$. Keep $F(n)=W(n)-f(\log n)$ and define

$$D_t(y)=f(y+t)-f(y),a_p=D_t(B)-c_p.$$

Uniformly over all these labels and ties, $B\to \infty$ as $X\to \infty$. Otherwise a bounded sequence of cores has bounded $W(u)$, while $A\ge \ell \to \infty$ and $c_p\le e^{-\ell }$, forcing $F(N)=W(u)+c_p-f(A)\to -\infty$. This contradicts N1’s positive height. Enlarge only the common start until $B>11$.

Strictly decreasing $\lambda =f^{\prime }$ gives $D_t(B)>t\lambda (B+t)>c_p$. Absent-prime multiplicativity therefore gives

$$a_p>0,F(u)=F(N)+a_p,\text{\hspace{1em}(CR1)}$$

and the following actual all-integer comparison holds for every $v>u$:

$$\boxed{F(u)-F(v)\ge \{\begin{array}{ll}{a}_p,& p\mid v,\\ D_t(B)-D_t(\log v),& p\nmid v.\end{array}}\text{\hspace{1em}(CR2)}$$

Both lower bounds are strictly positive. To prove CR2 first suppose $5040\mid v$. If $p\mid v$, full-domain maximality of $N$ gives $F(v)\le F(N)$. Otherwise $pv$ is in that same maximization domain and $F(pv)=F(v)+c_p-D_t(\log v)$. Comparing $pv$ with $pu$ proves the second branch. It is positive because $D_t^{\prime }(y)=\lambda (y+t)-\lambda (y)<0$.

For arbitrary $v>u$, use the actual integer $v^{\ast }=\mathrm{lcm}(5040,v)$. At clocks above $11$ every missing mandatory 5040 layer has reward per logarithmic mass strictly above the current $\lambda$; its finite $f$-increment per mass is strictly below that price. Inserting all missing mandatory layers gives $F(v^{\ast })\ge F(v)$. Since $p\ge 29$, $p\mid v^{\ast }$ iff $p\mid v$. Also $v^{\ast }\ge v>u$ and $D_t(B)-D_t(\log v^{\ast })\ge D_t(B)-D_t(\log v)$. The already proved comparisons at $v^{\ast }$ therefore give CR2.

Consequently the same actual core is the unique maximizer of $F$ on all integers $v\ge u$. This compares every larger integer, not merely multiples of $u$. It is a consequence for every original core; the existing CNS tail-selection mechanism is reused and is not reconstructed as a new source family. CR2 does not make nonlinear switch cores adjacent in the classical CA sequence.

The original power excess transfers without losing its exponent

The same cores retain, with the original fixed $b,d$, the own-size bound

$$\boxed{F(u)-\gamma \ge \frac{d}{64}{B}^{-b},B=\log u.}\text{\hspace{1em}(CR3)}$$

The common eventual start can depend on $b,d$ but not on the label or tie. Indeed CR1 and the original source height give

$$F(u)-\gamma \ge \frac{d}{2}{X}^{-b}+D_t(B)-c_p.\text{\hspace{1em}(CR4)}$$

If $B\ge X$, CR1 immediately gives the stronger $d{B}^{-b}/2$. If $B<X$ and $B\ge t$, then $A\le 2B$, eventually $\log A\le 2\log B$, and the original $\ell =(d/8)X^{1-b}\log X$ gives

$$D_t(B)\ge \frac{t}{A\log A}\ge \frac{\ell }{4B\log B}=\frac{d}{32}{B}^{-b}{\left(\frac{X}{B}\right)}^{1-b}\frac{\log X}{\log B}\ge \frac{d}{32}{B}^{-b}.$$

If $B<X$ and $B<t$, monotonicity in $t$ instead gives

$$D_t(B)\ge f(2B)-f(B)=\log \left(1+\frac{\log 2}{\log B}\right)\ge \frac{\log 2}{2\log B}\ge \frac{d}{32}{B}^{-b}$$

eventually, since the original $b>0$ is fixed. Finally, $c_p{B}^b\le e^{-\ell }{R}^b\to 0$, using $B\le R$ and $\log R=o(\ell )$. Thus $c_p\le d{B}^{-b}/64$ uniformly and CR4 proves CR3. No new zero-abscissa hypothesis or output exponent is substituted. CR3 constrains the actual value at the actual core; it gives no favorable sign for PG15’s compactly supported response.

Maximum-preserving contact recursion is stationary on these tails

For any such core $u$, any nonincreasing functions $q_k$ on $\{n\in \mathbb{N}:n\ge u\}$, any $s_k>0$ and any $a\ge 0$, CR2 directly implies that

$$n⟼e^{s_{k}F(n)}+a{q}_k(n)\text{\hspace{1em}(CR5)}$$

has unique maximum $u$. For every $n>u$ its first term is strictly smaller and its second term is no larger. This elementary implication is an application, not a new generic contact theorem. Restriction to any subset containing $u$ leaves the conclusion unchanged; no CA membership is inferred from the restriction itself.

The existing Musin contact source, Theorem 4.1, concerns a specified classical CA domain and reference, positive decreasing $q_k$, coordinate-transition conditions and a normalization limit. Its reference convention is not transplanted to $u$. CR5 supplies a concrete stationary-tail consequence for the actual cores. A FIB encoding of these same maximum-preserving contacts inherits it; other recursions remain outside this assertion. Cofinal later witnesses can have smaller positive excesses, so cofinality does not force escape from one fixed strict tail maximum.

Height and full-envelope geometry alone do not supply PG15

The following synthetic atomic construction tests only a proposed height-plus-envelope abstraction. It is not actual divisor arithmetic: its rewards are not prime-layer rewards, it has no actual $\psi$ or actual prime-power full-Q supplier, and its model clocks do not replace the original $A_i$. It gives no counterexample to RH or to implication from the full actual RH-failure hypotheses to PG15.

Fix the same $0<b<1/2$, $d>0$, set $c=2bd$, and put

$$h(x)=x+c{x}^{1-b}\log x,z_j=h(j).$$

Choose a fixed sufficiently large integer $j_0\ge 3$. Use states $j\ge j_0-1$, with clocks $j$, and the strict inventory

$$K_{\ast }(v)=j_0-1+\sum _{j\ge j_0}{1}_{\{z_j<v\}}.$$

Realize the initial mass $j_0-1$, if an all-positive-tail measure is needed, by $j_0-1$ fixed unit atoms strictly between $1$ and $h(j_0-1)$. All tested tents eventually have $A-A^{4/5}>h(j_0-1)$; define $\kappa =h^{-1}$ only on that tail. A precise reward normalization is

$${\mathcal{W}}_j=f(j)+\gamma +\sum _{r=j+1}^{\infty }(f(r)-f(r-1)-\lambda (h(r))).$$

It gives ${\mathcal{W}}_j-{\mathcal{W}}_{j-1}=\lambda (z_j)$ and ${\mathcal{W}}_j-f(j)\to \gamma$. The series converges because

$$\lambda (z_j)-[f(j)-f(j-1)]=-c{j}^{-b-1}(1+o(1))$$

is absolutely summable. Its tail gives

$${\mathcal{W}}_j-f(j)-\gamma =(c/b+o(1))j^{-b}=(2d+o(1))j^{-b}.\text{\hspace{1em}(CR6)}$$

These values are eventually strictly decreasing, so all sufficiently large states are strict right-tail records. They also satisfy the numerical H2 upper envelope eventually, because $j^{-b}=o((\log j{)}^{-2})$.

For $t\ge 0$ use the complete state envelope $V_{\ast }(t)={\max }_{j\ge j_0-1}[{\mathcal{W}}_j-f(t+j)]$. The adjacent tie is the unique $s_j$ satisfying

$$f(s_j+j)-f(s_j+j-1)=\lambda (z_j),s_j=z_j-j+\frac{1}{2}+O(j^{-1}).\text{\hspace{1em}(CR7)}$$

The $s_j$ are eventually strictly increasing: their leading difference is $c(1-b)j^{-b}\log j$, which dominates the displayed remainder. The adjacent objective difference is strictly increasing in $t$. Thus for $s_j<t<s_{j+1}$ it is positive through state $j$ and negative afterward, so $j$ dominates every state. The finitely many initial states are irrelevant for sufficiently large $t$. For each fixed $t\ge 0$, the limiting state objective is $\gamma$ and its excess is eventually positive, since $f(j+t)-f(j)=O_t((j\log j{)}^{-1})=o(j^{-b})$. Together with the ordered switches this pays attainment at an actual state. At the active clock $A=t+j$ one has $z_j<A<z_{j+1}$ and $K_{\ast }(A)=j=A-t$. Every switch gap has length one, contains exactly its full unit activation atom, and has zero total and price moment by CR7. Its endpoints do not activate. Full dominance supplies the same nonnegative prefix cost as PG2, with the model rewards.

For every integer $X\to \infty$, retain every ordinary prime label in $[e^{{\ell }_X},e^{2{\ell }_X}]$, with the original numerical ${\ell }_X=(d/8)X^{1-b}\log X$. The candidate state $j=X$ gives uniformly for $t\in [{\ell }_X,2{\ell }_X]$

$$f(X+t)-f(X)\le t\lambda (X)\le \frac{d}{4}{X}^{-b},V_{\ast }(t)-\gamma \ge (7d/4+o(1))X^{-b}\ge \frac{d}{2}{X}^{-b}.\text{\hspace{1em}(CR8)}$$

For $t=\theta {\ell }_X$, $1\le \theta \le 2$, CR7 more precisely gives $j/X=(\theta /(16b){)}^{1/(1-b)}(1+o(1))$ and $A\sim j$, uniformly including ties. Thus $j\asymp X$ and $A\asymp X$ for every active state on these whole windows, with constants depending only on $b,d$. Their original numerical bounds ${\ell }_X\le A\le R_X$ hold eventually. Their selected excesses are $O(X^{-b})$, so the numerical H2 bound holds there as well. Adding the unchanged label reward $c_p$ leaves the model optimizer unchanged and only increases its height. This label use does not turn a state into an actual integer core or its reward into a divisor sum. The H2 scope is only the own-state and selected-clock numerical bounds just proved. For a fixed label $p$ and $j\to \infty$, ${\mathcal{W}}_j+c_p-f(j+\log p)\to \gamma +c_p>\gamma$; the model therefore does not satisfy global arithmetic H2 over all labelled competitors.

Let $\kappa =h^{-1}$ on the sufficiently large real tail. Then $|K_{\ast }(v)-\kappa (v)|\le 1$ and

$${\kappa }^{\prime }(v)-1=-c(1-b)v^{-b}\log v(1+o(1)).$$

Use exactly the original price tent $k_A$ and width $A^{4/5}$. Its existing area asymptotic and $\int |k_A^{\prime }|=2$ are reused. Finite Stieltjes integration by parts bounds the response from $K_{\ast }-\kappa$ by $2$, since the tent vanishes at both support endpoints. Uniformity on that support consequently gives

$$\boxed{A^{-3/10}\int k_A(v)[d{K}_{\ast }(v)-dv]=-c(1-b)A^{1/2-b}\log A(1+o(1))⟶-\infty .}\text{\hspace{1em}(CR9)}$$

Retain all labels, least active states and original positive coefficients in the corresponding model weight $W_X^{\ast }(v)={\sum }_i{\delta }_i(A_i^{\ast }{)}^{-3/10}{k}_{A_i^{\ast }}(v)$. Because $A_i^{\ast }\asymp X$ uniformly, CR9 gives

$$\frac{1}{L_X}\int W_X^{\ast }(v)[d{K}_{\ast }(v)-dv]⟶-\infty ,\frac{{\mathsf{E}}_X^{\ast }-{\mathsf{S}}_X^{\ast }-{\mathsf{G}}_X^{\ast }}{L_X}⟶-\infty .\text{\hspace{1em}(CR10)}$$

Thus fixed cofinal normalized height, strict tail records and complete nonlinear-envelope dominance alone do not force favorable stock sign. This model establishes only the insufficiency of the stated height-and-envelope abstraction; the actual arithmetic interfaces and global labelled-competitor H2 remain outside its scope. CR1–CR10 are supplier-relative paper deductions, with no Lean verification or priority claim; the independent signed finite head, PG15 and RH remain unresolved.

A positive atomic reference can also retain the numerical full-Q estimate

Keep the preceding synthetic model, whose actual-source implications are not used here. It can be equipped with a positive weak reference inventory having the same pointwise full-Q asymptotic form as the reused PT3 supplier; PG14 is its weighted consequence. This remains a synthetic model: none of its reference atoms is identified with an actual prime power, and no actual $\Lambda$, Euler product or zeta explicit formula is supplied.

Set $a_{\ast }=\sqrt{2}-1=2d_0$ and $T(v)=v-a_{\ast }\sqrt{v}$ on a sufficiently large real tail. The map is strictly increasing there. At each model activation $z_j$, let ${\tau }_j$ be the unique later clock satisfying $T({\tau }_j)=z_j$ and define the positive right-continuous reference

$${\Psi }_{\ast }(v)=j_0-1+\sum _{j\ge j_0}{1}_{\{{\tau }_j\le v\}},Q_{\ast }(v)=K_{\ast }(v)-{\Psi }_{\ast }(v),d{Q}_{\ast }=d{K}_{\ast }-d{\Psi }_{\ast }.$$

All reference atoms have positive unit mass and are retained. The baseline constant affects no tested tail measure and can be realized by the same fixed lower positive atoms as the original synthetic stock. No common half-open increment convention is assigned to the mixed function $Q_{\ast }$. For sufficiently large $v$, the existing inverse clock $\kappa =h^{-1}$ gives

$$Q_{\ast }(v)=q(v)+O(1),q(v)=\kappa (v)-\kappa (T(v)),\text{\hspace{1em}(CR11)}$$

with a uniform bounded discrepancy, regardless of strict/weak endpoint contacts. Differentiating the smooth inverse, not an infinite zero series, gives ${\kappa }^{\prime }(v)=1+O(v^{-b}\log v)$ and ${\kappa }^{\prime \prime }(v)=O(v^{-b-1}\log v)$. Therefore

$$q^{\prime }(v)={\kappa }^{\prime }(v)-{\kappa }^{\prime }(T(v))+\frac{a_{\ast }}{2\sqrt{v}}{\kappa }^{\prime }(T(v))=\frac{d_0}{\sqrt{v}}+O(v^{-b-1/2}\log v).\text{\hspace{1em}(CR12)}$$

For the original $h_A=A^{4/5}$, $a=A-h_A$, $b_A=A+h_A$, integrate this identity and retain the bounded discrepancy at both endpoints. Uniformly for $a\le t\le b_A$,

$$Q_{\ast }(t)-Q_{\ast }(a)=\frac{d_0}{\sqrt{A}}(t-a)+O\left(\frac{h_A^2}{A^{3/2}}+h_A{A}^{-b-1/2}\log A+1\right).\text{\hspace{1em}(CR13)}$$

Relative to $h_A/\sqrt{A}$ this error is $O(A^{-1/5}+A^{-b}\log A+A^{-3/10})=o(1)$. For the original fixed $b>0$, $A^{-b}\log A=o(1/\log A)$; thus the error is also bounded by a fixed multiple of the existing seven-term ${\mathcal{F}}_Q(A)$ after a common start. This compares the numerical error form; it does not realize the actual prime-power supplier.

The original tent’s area and variation then give, by finite integration by parts with the same signed event measure,

$$A^{-3/10}\int k_A(v)d{Q}_{\ast }(v)=d_0+o(1).\text{\hspace{1em}(CR14)}$$

Uniformly over the complete model label window, $\int W_X^{\ast }d{Q}_{\ast }=(d_0+o(1))L_X$. CR9–CR10 remain unchanged, and the reference response is exactly

$$\int W_X^{\ast }(v)[d{\Psi }_{\ast }(v)-dv]={\mathsf{E}}_X^{\ast }-{\mathsf{S}}_X^{\ast }-{\mathsf{G}}_X^{\ast }-\int W_X^{\ast }(v)d{Q}_{\ast }(v).\text{\hspace{1em}(CR15)}$$

Its ratio to $L_X$ still tends to $-\infty$. Hence even adding a positive atomic reference and the numerical full-Q response to the retained height and envelope abstraction does not supply PG15. The actual prime/divisor coefficients, integer-source realization and global H2 over all labelled competitors remain absent, with the H2 scope fixed above. Nonnegativity of the reference measure and that asymptotic coefficient alone therefore do not pay the missing comparison. This conclusion does not establish nonimplication from the full actual RH-failure hypotheses, refute RH or prove the actual estimate.

The complete signed-tail consumer remains at the core’s own clock

For each original core $u$ in CR1–CR3 put $B=\log u$, $L_B=\log B$ and $T_B=\sqrt{B}{L}_B$. Its strict all-integer right-tail property gives the existing GA2 condition by taking $v=au$ for each integer $a\ge 2$; $a=1$ is equality. Uniform $B\to \infty$ permits one common source start after which

$$B\ge \max \{\log N^{(0)},e^{26}\},$$

so the supplier thresholds $u\ge N^{(0)}$ and $L_B\ge 26$ both hold. There is no numeric value supplied for that common witness start. Apply of the existing complete GA2 core identity and effective allowance, A3–A5, after one common start. Keep its entire original improper integral

$$I_{\psi }(B)={\int }_B^{\infty }(\psi (v)-v)\frac{1+\log v}{v^2{\log }^{2}v}dv,\gamma -F(u)=I_{\psi }(B)+K_u(B),T_B{K}_u(B)>\mathcal{E}(L_B).$$

Here $K_u(B)$ is the supplier’s extended arithmetic core, distinct from PG1’s strict inventory clock $K(v)$. Every extra support term, prime power, elementary correction and infinite endpoint in that supplier remains. CR3 gives the instantiated all-original-core upper comparison

$$\boxed{T_B{I}_{\psi }(B)<-\mathcal{E}(L_B)-\frac{d}{64}{B}^{1/2-b}\log B.}\text{\hspace{1em}(CR16)}$$

The existing allowance is $\mathcal{E}(L_B)>0.65$ for $L_B\ge 26$, and the original $0<b<1/2$ and $d>0$ are fixed. Thus, for every fixed $M\ge 0$, one common sufficiently large witness start makes $T_B{I}_{\psi }(B)<-M$ for every original label and maximizing tie. This is uniform on the entire original family, with no clock comparison between $B$ and $X$ required.

This uses the existing CNS oscillation/tail consumer with CR3’s newly paid original-core scope and exponent; it does not repeat that consumer as a new analytic theorem or introduce a different kernel. The integral is evaluated at $B=\log u$, not at the forced integer clock $A=\log (pu)$, an auxiliary prime or a freely selected real center. A lower bound at one of those other clocks cannot be transported here without its own estimate. CR16 supplies no lower bound at $B$, no PG15 exterior slack, and no estimate for the independent fixed-source finite head. Those obligations and RH remain unproved. This is a paper application only, with no Lean verification or originality claim.

The full removal gain pays a linear complete-head allowance

Keep the original fixed $b,d$, witness $X$, every label $p$ and every maximizing tie $N=pu$ from CR1–CR4. Write $t=\log p$, $B=\log u$, $A=B+t$, $s_B=\sqrt{B}\log B$, and $v_{p,u}=t/\sqrt{B}$. The same actual core is $u=N_A^{\mathrm{str}}$, so $K(A)=B$.

The existing full-real-clock activation correction is valid for the strict prefix. It gives $K(x)=\psi (x)+O(\sqrt{x}[\log (2x){]}^{5/2})$ for every real $x>1$. Reuse its unconditional PNT input and the original uniform escape of the source clocks. Then, uniformly over all original labels and ties,

$$K(A)=A+o(A),A/B⟶1,t/B⟶0.\text{\hspace{1em}(CR17)}$$

This uses no short-interval estimate, no own-clock optimum at $B$, and no comparison $B\ge X$.

The retained height increment supplies its own coefficient

CR4 retains the full same-source height

$$F(u)-\gamma \ge \frac{d}{2}{X}^{-b}+D_t(B)-c_p,c_p=\log (1+1/p)\le e^{-\ell }.$$

The strictly decreasing price gives $t/(A\log A)\le D_t(B)\le t/(B\log B)$. Define the normalized removal coefficient

$${\chi }_{p,u}=\frac{B\log B}{t}[D_t(B)-c_p].$$

Using the original $t\ge \ell$, $A\le R$ and $\log R=o(\ell )$,

$$\frac{B\log B}{A\log A}-\frac{R\log R}{\ell }{e}^{-\ell }\le {\chi }_{p,u}\le 1.\text{\hspace{1em}(CR18)}$$

The lower correction tends to zero, while CR17 sends the first term to one. Consequently ${\chi }_{p,u}\to 1$ uniformly on the entire original family. Both the original constants and all ties remain; the common analytic/source start is existential, with no numeric value supplied. In particular

$$s_B[D_t(B)-c_p]={\chi }_{p,u}{v}_{p,u}.$$

Reuse the existing unrestricted integer identity OCQ13 at this same core clock. Let $R_{\mathrm{pp}}(B)$ denote its original unrestricted prime-power reserve and $d_B(u)$ its actual own-price defect. They are distinct from the source-window ceiling $R$, strict stock $K(v)$ and extended GA2 core $K_u(B)$ in CR16. The supplier keeps all positive integers, including one, and gives

$$-s_B{I}_{\psi }(B)=s_B(F(u)-\gamma )+s_B{R}_{\mathrm{pp}}(B)+s_B{d}_B(u),R_{\mathrm{pp}}(B)\ge 0,d_B(u)\ge 0.$$

Thus the unabridged height, rather than a second use of CR3’s partly height-derived bound, supplies

$$-s_B{I}_{\psi }(B)\ge \frac{d}{2}\sqrt{B}\log B{X}^{-b}+{\chi }_{p,u}{v}_{p,u}.\text{\hspace{1em}(CR19)}$$

No positive contribution is counted twice.

The residual margin keeps the original witness scale

Fix $0\le \kappa <1$ once. Uniform convergence in CR18 permits one common start with ${\chi }_{p,u}\ge (1+\kappa )/2$. If $B\ge X$, the first term of CR19 is at least $(d/2)X^{1/2-b}\log X$. If $B<X$, then

$$v_{p,u}\ge \frac{\ell }{\sqrt{X}}=\frac{d}{8}{X}^{1/2-b}\log X,$$

and the remaining $({\chi }_{p,u}-\kappa )v_{p,u}$ is at least $d(1-\kappa )X^{1/2-b}\log X/16$. Both branches retain nonnegative other terms. Therefore every original label and maximizing tie satisfies

$$\boxed{-s_B{I}_{\psi }(B)\ge \kappa \frac{\log p}{\sqrt{B}}+\frac{d(1-\kappa )}{16}{X}^{1/2-b}\log X.}\text{\hspace{1em}(CR20)}$$

The last term tends to infinity uniformly because the original $b<1/2$ is fixed. In particular $\kappa =1/2$ leaves the coefficient $d/32$. The boundary $\kappa =1$ is not established by this argument. Neither $v_{p,u}$ alone nor a different source clock is required to have a diverging lower bound.

These are paper applications of the named suppliers, without Lean verification or a claim of literature originality.

The complete original-window head may carry the removal allowance

Keep the entire improper integral in CR16 and all its original zero coefficients, actual real parts, multiplicities, elementary correction and endpoint conventions. For an admissible moving-clock cutoff write a complete finite-head upper enclosure as

$$\begin{array}{rl}U(B,T)=& s_B\left[2\mathrm{Re}\sum _{0<\mathrm{Im}\rho \le T}{m}_{\rho }{F}_B(\rho )+R_{\mathrm{elem}}(B)\right]+E_{\mathrm{full}}(B,T),\\ -s_B{I}_{\psi }(B)& \le U(B,T).\end{array}$$

Here $F_B$ is the supplier’s original coefficient (M1) with its variable $A$ set to this same $B$, and $R_{\mathrm{elem}}(B)$ is that supplier’s pole-and-trivial-zero correction, with its positive normalized sign retained. The existing signed suffix contract (J1)–(J6) applies for $B\ge e^2$ and $T\ge 8$. Its signed prime-power main term keeps its displayed sign. The moving cut $T=B^{1/4}\log B$ pays an asymptotically vanishing suffix allowance; it does not bound the remaining head. The complete $E_{\mathrm{full}}$ must pay every omitted suffix, coefficient approximation and enclosure error exactly once in the supplier’s actual domain. Exact coefficients introduce no coefficient-approximation error. A symbolic remainder or a finite verified zero range does not establish this upper-enclosure contract at arbitrary moving clocks.

CR20 now allows the still-unproved pointwise upper target

$$U(B,T)\le M+\kappa \frac{\log p}{\sqrt{B}}\text{\hspace{1em}(CR21)}$$

with one fixed finite $M\ge 0$ and $0\le \kappa <1$. It is also sufficient to control one complete original window jointly. Keep $p_0<\cdots <p_m$, ${\delta }_i=\log p_{i+1}-\log p_i>0$ for $0\le i<m$, and $L_X={\sum }_{i=0}^{m-1}{\delta }_i$. The final label $p_m$ remains the original window boundary. For each allowed assignment of maximizing ties define

$$V_X^{\mathrm{lag}}=\sum _{i=0}^{m-1}{\delta }_i\frac{\log p_i}{\sqrt{B_i}},B_i=\log u_i.$$

The same positive weights applied to CR20 give

$$\sum _{i=0}^{m-1}{\delta }_i[-s_{B_i}{I}_{\psi }(B_i)]\ge \kappa V_X^{\mathrm{lag}}+\frac{d(1-\kappa )}{16}{X}^{1/2-b}\log X{L}_X.\text{\hspace{1em}(CR22)}$$

A sufficient obligation is: for fixed finite $M\ge 0$ and $0\le \kappa <1$, beyond every common analytic/source start, some original qualifying full window has the following property. Every allowed assignment of maximizing ties admits admissible cutoffs $T_i$, with complete remainder contracts at those same cores, satisfying

$$\boxed{\sum _{i=0}^{m-1}{\delta }_{i}U(B_i,T_i)\le M{L}_X+\kappa V_X^{\mathrm{lag}}.}\text{\hspace{1em}(CR23)}$$

The quantifier order is

$$\begin{array}{c}\exists M\ge 0,\kappa \in [0,1)\forall \text{common finite starts}\exists \text{qualifying original full window}\\ \forall \text{allowed maximizing-tie assignments}\exists \text{admissible cutoffs and complete contracts}.\end{array}$$

The starts do not depend on individual labels or ties. CR22–CR23 would imply

$$\left[\frac{d(1-\kappa )}{16}{X}^{1/2-b}\log X-M\right]L_X\le 0,$$

contradicting the original $X\to \infty$ and $L_X>0$. No thinning, free-center mean or favorable tie selection enters the comparison. CR21 and CR23 are sufficient conditions, not necessary ones; neither has been proved.

Any fixed quadratic lag is already absorbed by this height gain

For the same original sources put $z_{p,u}=t^2/B^{3/2}$. The exact relation

$$z_{p,u}=\frac{t}{B}{v}_{p,u}\text{\hspace{1em}(CR24)}$$

and CR17 show that, for every fixed finite $C\ge 0$ and fixed $0<\kappa <1$, one common eventual start makes $C{z}_{p,u}\le \kappa v_{p,u}$ for every original label and tie. The same comparison holds after applying the original positive window weights. Thus the original full-height gain already permits any fixed quadratic coefficient in the head upper target, including $1/2$ and larger constants. No estimate of the own-price defect is needed for that consumer. This is a relative comparison; it asserts no divergence of either lag allowance.

The linear allowance is a supplier-relative application of the original removal height and actual stock/PNT correspondence. The complete moving finite-head upper condition, cofinal joint upper condition, original PG15 exterior sign, boundary $\kappa =1$ and RH remain unproved. No literature originality, new pressure decomposition or Lean verification is claimed.

A legal integer competitor leaves a strict curvature shortfall

Keep the original sources, labels and maximizing ties from CR1–CR4, with $N=pu$, $A=B+t$, $B=\log u$, $t=\log p$ and $u=N_A^{\mathrm{str}}$. Enlarge the common source start so that $B>11$. Use the existing strict inventory and unrestricted own-price defect, including the integer $1$ in its pressure domain. Put

$$C=N_B^{\mathrm{str}},k=K(B)=\log C,Y=t+k,L_{p,u}=F(N)-F(pC).$$

Strict inventory nesting gives $C\mid u$. The original activation calculation at clocks above $11$ gives $5040\mid C$, and $p\nmid u$ also gives $p\nmid C$. Thus $pC$ is a literal competitor in the original complete domain of multiples of $5040p$, so $L_{p,u}\ge 0$. This uses the original domain and strict-prefix suppliers, N1/N4, and PG1; neither CA adjacency nor GA1 is required.

Reuse PG2’s concavity cost $C_B(y)=f(B)+\lambda (B)(y-B)-f(y)$. The pressure optimizer at $\lambda (B)$ is $C$, so the existing defect definition reads

$$d_B(u)=W(C)-\lambda (B)k-W(u)+\lambda (B)B.$$

Multiplication by the same absent prime adds $c_p=\log (1+1/p)$ to both divisor responses. Substitution into the actual competitor loss therefore gives the exact same-source account

$$t\lambda (B)-D_t(B)-d_B(u)=L_{p,u}+C_B(Y).\text{\hspace{1em}(CR25)}$$

Its strictness has an integer witness. Write $r=u/C\in {\mathbb{N}}_{>0}$. Because $p\nmid u$, $r\ne p$, and

$$Y-B=\log (p/r),|\log (p/r)|\ge \log (1+1/p)=c_p.$$

For $r\ge p+1$ this follows directly; for $r\le p-1$ the stronger bound is $\log (p/(p-1))>c_p$. Both $B$ and $Y$ are greater than $1$ and at most $A$, since $k\le B$. Integrating the decreasing curvature $w=-{\lambda }^{\prime }$ on the interval between $B$ and $Y$ gives $C_B(Y)\ge w(A)(Y-B{)}^2/2$. Consequently every original label and tie satisfies

$$\boxed{d_B(u)\le t\lambda (B)-D_t(B)-\frac{w(A)}{2}{c}_p^2<t\lambda (B)-D_t(B).}\text{\hspace{1em}(CR26)}$$

Even when $pC$ is another maximizing tie and $L_{p,u}=0$, the integer gap remains strictly positive. The same original positive weights give

$$\sum _{i<m}{\delta }_i{s}_{B_i}[d_{B_i}(u_i)-t_i\lambda (B_i)+D_{t_i}(B_i)]=-\sum _{i<m}{\delta }_i{s}_{B_i}[L_{p_i,u_i}+C_{B_i}(Y_i)]<0.$$

Thus the defect alone cannot supply the proposed lower bound $d_B(u)\ge t\lambda (B)-D_t(B)$, even after full-window averaging. This excludes that particular supply condition; it does not exclude a comparison using the original height, reserve or another joint relation. In the complete unrestricted same-core identity, CR25 instead gives

$$\boxed{-s_B{I}_{\psi }(B)=\frac{t}{\sqrt{B}}+s_B\left[F(N)-\gamma +R_{\mathrm{pp}}(B)-L_{p,u}-C_B(Y)-c_p\right].}\text{\hspace{1em}(CR27)}$$

The original height occurs once. A coefficient-one comparison would still have to pay $L_{p,u}+C_B(Y)+c_p$ from that height, the reserve or a proved same-source signed estimate. CR26’s strict gap has no fixed normalized positive lower bound asserted here. The complete coefficient-one condition, CR21/CR23, PG15/J and RH remain unproved. This is an application of the named suppliers, without a new generic pressure theorem, literature originality claim or Lean verification.

Exact linear compensation loses absolute height summability

Fix one original qualifying full window and one allowed assignment of maximizing ties. Keep every $0\le i<m$, the original positive ${\delta }_i=t_{i+1}-t_i$, the last boundary label $p_m$, $A_i=B_i+t_i$ and the exact actual relation $K(A_i)=B_i$. Fix $\kappa >0$. This section examines a way of incorporating the linear allowance into the zero coefficients; it does not redefine the complete upper enclosure $U$ in CR21/CR23.

Use the original complete coefficient (M1), the same elementary correction $r_B=s_B{R}_{\mathrm{elem}}(B)$, and $Q=K-\psi$ with weak, right-continuous $\psi$. For real $x>1$, let ${\Lambda }_{\mathbb{R}}(x)=\log q$ at an actual integer prime power $x=q^j$, and zero otherwise. The classical symmetric-height formula retains the endpoint term:

$$\psi (A)-A=-\underset{T\to \infty }{\lim }\sum _{|\mathrm{Im}\rho |\le T}{m}_{\rho }\frac{A^{\rho }}{\rho }-\log (2\pi )-\frac{1}{2}\log (1-A^{-2})+\frac{1}{2}{\Lambda }_{\mathbb{R}}(A).$$

The actual relation $t_i=A_i-\psi (A_i)-Q(A_i)$ thus forces the compensated coefficient

$$C_{X,\kappa }(\rho )=\sum _{i<m}{\delta }_i\left[s_{B_i}{F}_{B_i}(\rho )-\frac{\kappa A_i^{\rho }}{\sqrt{B_i}\rho }\right].\text{\hspace{1em}(CR28)}$$

It yields the complete signed account

$$\begin{array}{rl}\sum _{i<m}{\delta }_i\left[-s_{B_i}{I}_{\psi }(B_i)-\frac{\kappa t_i}{\sqrt{B_i}}\right]=& \underset{T\to \infty }{\lim }\sum _{|\mathrm{Im}\rho |\le T}{m}_{\rho }{C}_{X,\kappa }(\rho )\\ & +\sum _{i<m}{\delta }_i\left\{r_{B_i}+\frac{\kappa }{\sqrt{B_i}}\left[Q(A_i)-\log (2\pi )-\frac{1}{2}\log (1-A_i^{-2})+\frac{1}{2}{\Lambda }_{\mathbb{R}}(A_i)\right]\right\}.\end{array}\text{\hspace{1em}(CR29)}$$