bibkey: triangularprimework2026scope authors: trureturing research synthesis year: 2026 title: A conditional all-center obstruction for the actual shrinking triangular prime work doi: null url: https://doi.org/10.2298/PIM0476081K claim: A paper application of the classical explicit formula and positive Gaussian testing, conditional on RH, rules out an eventual finite lower bound for the actual triangular work normalized by h squared over square root A at all real centers; it does not decide the selected-source lower estimate. strata_touched: [] license: citation-only triage: anchor

Conditional scope of a uniform triangular-work lower estimate

The actual forward consumer in the Nicolas note uses

$$w(t)=\frac{1+\log t}{t^2{\log }^{2}t},T_{\psi }(A,h)={\int }_0^h[\psi (A+u)-\psi (A)-u]\frac{w(A+u)}{w(A)}du.$$

$\psi$ is the actual Chebyshev prime-power sum, with every $\Lambda (n)$ weight retained. A lower reference atom at $A$ is excluded; an upper atom at $A+h$ has zero integrated weight. The existing sufficient condition asks for a favorable lower estimate on particular original maximizing sources beyond every required common finite start. It does not ask for that lower estimate at all real centers.

The following is a conditional paper argument, with no Lean verification or general literature-novelty claim. It reuses the classical RH estimate $\psi (x)-x=O(\sqrt{x}{\log }^{2}x)$, the distributional explicit formula, the Riemann–von Mangoldt zero count, and simultaneous recurrence of finitely many real frequencies. The Gaussian explicit-formula supplier is already recorded in the Nicolas note’s real-parameter Gaussian interface: Kamiya–Suzuki, An asymptotic formula for a sum involving zeros of the Riemann zeta-function, Lemma 2.1, DOI 10.2298/PIM0476081K. That formula itself is unconditional. RH is an explicit additional premise throughout the application below, not a derived conclusion.

Actual shrinking-window normalization

Fix $0<\alpha <1$ and set $h=A^{\alpha }$. The paper claim is

$$\boxed{\text{assuming RH},\underset{A\to \infty }{\mathrm{liminf}}\frac{\sqrt{A}}{A^{2\alpha }}{T}_{\psi }(A,A^{\alpha })=-\infty .}\text{\hspace{1em}(TW1)}$$

Thus, even under RH, no finite constant $K$ can give $T_{\psi }(A,A^{\alpha })\ge -K{A}^{2\alpha -1/2}$ at every sufficiently large real $A$. At $\alpha =4/5$, the scale is exactly $A^{11/10}$, the scale in the selected-source consumer. This compares quantifiers; it does not refute that consumer’s existence statement on its actual source family.

A positive Gaussian sees the density derivative

Write $y=\log A$ and

$$E(y)=e^{-y/2}[\psi (e^y)-e^y],S=E^{\prime }+\frac{1}{2}E,$$

where the derivative is distributional. Under RH, $E(y)$ has at most quadratic growth as $y\to +\infty$, and $E(y)=-e^{y/2}$ for $y<\log 2$. Hence $S$ is a tempered distribution. Its actual arithmetic form is

$$S(y)=\sum _{n\ge 2}\frac{\Lambda (n)}{\sqrt{n}}{\delta }_{\log n}(y)-e^{y/2}.\text{\hspace{1em}(TW2)}$$

Let $G_\sigma(v)=(\sqrt{2\pi}\sigma)^{-1} e^{-v^2/(2\sigma^2)}$,withfixed$\sigma>0$. The classical distributional explicit formula, tested with this Gaussian, gives

$$⟨S,G_{\sigma }(\cdot -t)⟩=-\sum _{\rho }{e}^{i{\gamma }_{\rho }t}{e}^{-{\sigma }^2{\gamma }_{\rho }^2/2}+o(1),t\to \infty ,\text{\hspace{1em}(TW3)}$$

under RH. Every actual zero occurs with its multiplicity and both conjugate signs. The derivative $E^{\prime }+E/2$ cancels the $1/\rho$ coefficient of the usual $\psi$ explicit formula. The constant $-\log (2\pi )$ has derivative zero; the trivial-zero derivative decays on every fixed positive half-line. A smooth cutoff below a fixed positive $y$ contributes $o(1)$ against the translated Gaussian. Riemann–von Mangoldt makes the displayed zero sum absolutely convergent at each fixed $\sigma$.

This is an application of the existing explicit formula, not a new trace formula. In the Kamiya–Suzuki notation $H(u,v)={\sum }_{\rho }{e}^{u{\rho }^2-v\rho }$, the real-parameter correspondence is $u={\sigma }^2/2$, $v=u-t$:

$$e^{-t/2+u/4}H(u,u-t)=\sum _{\rho }{e}^{-u{\gamma }_{\rho }^2}{e}^{i{\gamma }_{\rho }t}\text{under RH}.$$

The Gaussian width is fixed while $t$ tends to infinity. No uniform small-width asymptotic from the external paper is assumed.

Transport the complete triangular work before taking a limit

For large $y$ put

$$F_{\alpha }(y)=e^{(1/2-2\alpha )y}{T}_{\psi }(e^y,e^{\alpha y}),{ϵ}_y=e^{-(1-\alpha )y}.$$

Multiply by a fixed smooth cutoff equal to one for $y\ge 2$ and zero for $y\le 1$, and extend by zero to the left. This changes no eventual lower-bound statement. Finite interval integration followed by the change $s=e^y+e^{\alpha y}r$, $0\le r\le 1$, gives the exact distributional representation, before this cutoff,

$$F_{\alpha }(y)={\int }_0^1{a}_y(r)S(y+\log (1+{ϵ}_{y}r))dr,\text{\hspace{1em}(TW4)}$$

$$a_y(r)=(1+{ϵ}_{y}r{)}^{-1/2}{\int }_r^1\frac{w(e^y(1+{ϵ}_{y}s))}{w(e^y)}ds.$$

The equality means pullback and averaging of the distribution in (TW2); the resulting locally integrable function agrees with the original work except at irrelevant endpoint values. It retains all prime-power atoms. To verify the factor, first swap the finite triangular integral with $d[\psi (s)-s]$ on $(A,A+h]$. Its inner weight is $h{\int }_r^{1}w(A+hv)/w(A)dv$, and $d[\psi (e^z)-e^z]=e^{z/2}S(z)dz$. The factors $\sqrt{A}/h^2$, $h$, $\sqrt{A}\sqrt{1+ϵr}$ and $ϵ/(1+ϵr)$ combine to $(1+ϵr{)}^{-1/2}$.

For each $r$, the map $g_r(y)=y+\log (1+r{ϵ}_y)$ is strictly increasing, since

$$g_r^{\prime }(y)=\frac{1+\alpha r{ϵ}_y}{1+r{ϵ}_y}>0.$$

For $y\ge 2$, uniformly in $0\le r\le 1$, the coefficient $a_y(r)$ and its first two $y$ derivatives differ from $1-r$, $0$, $0$ by $O_{\alpha }({ϵ}_y)$. The shift $g_r(y)-y$ and its first two derivatives are also $O_{\alpha }({ϵ}_y)$, and $g_r^{\prime }(y)$ is bounded below by $\alpha$. These bounds follow from the displayed formulas; their constants are independent of the large translation $t$.

Changing variables separately in every pullback gives an adjoint Gaussian test $H_t$ for (TW4). The preceding bounds give

$${\int }_{\mathbb{R}}(1+|z|{)}^2\left(|H_t(z)-\frac{1}{2}{G}_{\sigma }(z-t)|+|H_t^{\prime }(z)-\frac{1}{2}{G}_{\sigma }^{\prime }(z-t)|\right)dz=o(1).\text{\hspace{1em}(TW5)}$$

Indeed the part with $z\ge 2$ is bounded by a constant times $(1+t{)}^2{e}^{-(1-\alpha )t}$ after integrating the fixed-width Gaussian and its derivatives; the cutoff part has a polynomial times Gaussian decay in $t$. The factor $1/2$ is exactly ${\int }_0^1(1-r)dr$, not an imposed normalization of the atoms. Since $S=E^{\prime }+E/2$ and $|E(z)|\ll (1+|z|{)}^2$, (TW5) proves

$${\int }_{\mathbb{R}}{G}_{\sigma }(v)F_{\alpha }(t+v)dv=\frac{1}{2}⟨S,G_{\sigma }(\cdot -t)⟩+o(1).\text{\hspace{1em}(TW6)}$$

Only a weak, Gaussian-tested limit is used. There is no claim that the sharply sampled density distribution has a pointwise function limit, or that $S$ itself is bounded.

Simultaneous recurrence forces arbitrarily negative averages

Combine (TW3) and (TW6), pairing conjugate ordinates:

$${\int }_{\mathbb{R}}{G}_{\sigma }(v)F_{\alpha }(t+v)dv=-\sum _{\gamma >0}{m}_{\gamma }{e}^{-{\sigma }^2{\gamma }^2/2}\cos (\gamma t)+o(1).\text{\hspace{1em}(TW7)}$$

The coefficients are positive and absolutely summable. Simultaneous recurrence of every finite list of ordinates, followed by its absolutely bounded tail, gives

$$\underset{t\to \infty }{\mathrm{liminf}}{\int }_{\mathbb{R}}{G}_{\sigma }(v)F_{\alpha }(t+v)dv=-\sum _{\gamma >0}{m}_{\gamma }{e}^{-{\sigma }^2{\gamma }^2/2}.\text{\hspace{1em}(TW8)}$$

No linear-independence hypothesis is needed: simultaneous recurrence approximates phase zero, and makes no prescription of independent nonzero phases. The sum on the right tends to infinity as $\sigma \downarrow 0$, because there are infinitely many actual positive zero ordinates, with multiplicities retained.

If $F_{\alpha }(y)\ge -K$ eventually for a finite $K$, positivity and unit mass of every fixed $G_{\sigma }$ would make the left liminf at least $-K$. The bounded initial interval contributes $o(1)$ to the translated average. Choose a fixed sufficiently small $\sigma$ with the positive sum in (TW8) greater than $K$. This contradiction proves the conditional paper claim (TW1). The two limits are taken in this order: fix $\sigma$, then let $t\to \infty$, and only then choose a smaller fixed width.

What this excludes and what it leaves open

For $\alpha =4/5$, an all-center replacement of the actual $-\kappa A^{11/10}$ lower estimate is excluded by this argument even under RH. A Gaussian or other positive average cannot be used to infer that all individual centers have the requested sign. This does not exclude a favorable cofinal subset, and provides no information about whether the required original maximizing sources meet that subset.

The original joint selected-source/prime-error relation remains unproved. The argument certifies no all-integer Robin range, no unconditional counterexample to Robin, no RH proof and no Lean-verified mathematical result. It supplies a concrete conditional benchmark for the strength of a proposed auxiliary, using the actual triangular kernel rather than a virtual sign model.