bibkey: platttrudgian2021rh
authors: Dave Platt and Tim Trudgian
year: 2021
title: The Riemann hypothesis is true up to 3·10^12
doi: 10.1112/blms.12460
url: https://arxiv.org/abs/2004.09765v1
claim: Published verification supplies the finite-height RH premise. Exact heat-loss factorization and the complete arithmetic block bound give a whole-source negative allowance of order (log T)^(-4) in the original sixth moment; fixed height still leaves a negative allowance. A support-independent Gamma-only lower bound for the prime Hankel form fails on all compact half-line tests. An even actual-source partition controls the joint metric defect and bounds supporting multipliers of negative cap-four minima from below, up to the finite-height signed zero tail. Its null supporting vector bootstraps every fixed logarithmic order below five and recovers individual branch moments with an explicit inverse-multiplier cost; separation of the finite reconstruction intervals replaces its N-fold Fourier-tail payment by a harmonic factor, giving every fixed joint-metric exponent below five on actual negative minima and a fifth-power rate with a log-log factor on nonpositive-energy sources under RH(T), without improving the general exponent-four negative allowance. For hypothetical unit negative cap-four minimizers with a positive supporting multiplier, retaining the errors' own moments also gives every finite logarithmic order and arbitrary fixed logarithmic-power branch rates. A bounded analytic Sobolev regularizer gives positive-order regularity by commuting with the complete energy and paying both original metric defects; its constants deteriorate exponentially as the multiplier tends to zero and do not improve the uniform negative allowance. The same regularity controls finite physical dilation in the complete energy and both original metrics, including the source error itself, with explicit multiplier costs; the resulting sublinear squared-norm rate gives no dilation derivative or uniform allowance improvement. Directly retaining the regularizer's original moment coordinate then improves the squared-error range to every rho < 2s, with a larger inverse-multiplier cost and still no first-order or uniform allowance conclusion. A fixed actual source with every subcritical Sobolev order has a signed critical metric defect of order (log N)^2 while its regularized original moment is O(log N), excluding a universal endpoint absorption when lambda differs from mu even for unit original sources in any prescribed moment cap K>1; no negative/null-minimizer obstruction is inferred. The same global null equation gives weighted Gamma control at nonnegative multiplier and additional original physical moments at positive multiplier, retaining every theta dilation and the complete mixed debit. Direct physical truncation of the same negative minimizer then preserves its normalized cap and approximates its complete energy at any fixed rate exp(-2 sigma L), sigma<1/4, with constants independent of the supporting multiplier on the bounded branch; a negative deficit delta has a completed physical core of radius O_p(delta^(-p)) for every p>2. No positivity, Gaussian atom-count or uniform allowance improvement follows.
strata_touched: []
license: citation-only
triage: anchor

Verified zero height and the complete heat loss

Published suppliers and the additional application

Platt–Trudgian, The Riemann hypothesis is true up to $3\cdot 10^{12}$, Bulletin of the London Mathematical Society 53 (2021), no. 3, 792–797, DOI 10.1112/blms.12460. Theorem 1 of the arXiv v1 primary, printed p.2, states verification through height $3000175332800$. This published computation is reused, without repeating or independently certifying it.

For $T\ge 3$, write $\mathsf{RH}(T)$ for the assertion that every nontrivial zeta zero with $|\Im \rho |\le T$, including the height boundary, lies on $\Re \rho =1/2$. The source supplies this premise at $T_0=3\cdot 10^{12}$; it does not supply unbounded verified heights.

The existing actual-source account, W359 and W365–W371, supplies the exact full-form identification, weighted profile bounds and entire signed zero tail. Its allowance is $O(M(F)(\log T{)}^{-1/3})$. These results are reused unchanged. The general two-pole arithmetic interface is already owned by the non-even localization account. The strip transport below reuses Tao, Proposition 3(i), with its integrable strip-decay hypothesis verified on a smooth compact core. Trudgian, Theorem 1 supplies the all-prime-power Chebyshev bound. None of these suppliers is reproved as new content.

The additional, repo-derived application factors the ENTIRE heat loss into the same arithmetic form evaluated on an odd auxiliary test. The positive logarithmic Gamma part then pays its complete quadratic arithmetic debit, with no mixed-error estimate. The ingredients are classical contour transport, weighted block estimates and scalar minimization. No new general Fourier or explicit-formula theorem, literature-priority claim or Lean/kernel certification is made.

The original full-source allowance

Retain W317’s actual domain $\mathcal{V}$, COMPLETE energy $\mathfrak{E}(F)$ and original sixth moment $M(F)$, and W338’s even profile $g_F$. There is a finite absolute $C$, independent of $T$, source, scalar field, coefficients, count, width and support, such that

$$\mathsf{RH}(T)⟹\mathfrak{E}(F)\ge -\frac{CM(F)}{(\log T{)}^4}(T\ge 3,F\in \mathcal{V}).\text{\hspace{1em}(VHR1)}$$

It suffices to consider $\mathfrak{E}(F)\le 0$: positive-energy sources already satisfy this lower bound. If $M(F)=0$, the source is zero. Otherwise put $M=M(F)>0$, $g=g_F$, $b=5/2$, and write

$$\Vert v{\Vert }_a^2={\int }_{\mathbb{R}}{e}^{2a|x|}|v(x){|}^{2}dx,\mathcal{L}(v)={\int }_{\mathbb{R}}\log (2+|\tau |)|\hat{v}(\tau ){|}^{2}d\tau .$$

Use the unitary Fourier convention with kernel $e^{-i\tau x}$. W366, ONLY on this nonpositive-energy class, gives $\Vert g{\Vert }_b^2+\mathcal{L}(g)\le CM$. W359 gives exactly $\mathfrak{E}(F)=\alpha {\mathcal{Q}}_W(g)$, $\alpha =2/\sqrt{\pi }>0$. No Gamma bound from the moment alone is assumed.

Factor the complete filtering loss with an analytic signed root

Define

$$j(z)=\frac{1-e^{-2z^2}}{z^2},j(0)=2.$$

Its nonzero zeros satisfy $z^2=\pi ik$, $k\in \mathbb{Z}\setminus \{0\}$. Thus it has no zero on the simply connected strip $|\Im z|<\sqrt{\pi /2}$. Let $s$ be the holomorphic square root there with $s(0)=\sqrt{2}$, and set $m(z)=zs(z)$. Uniqueness makes $s$ even and Schwarz-real, so $m$ is odd and Schwarz-real, and

$$m(z{)}^2=1-e^{-2z^2},|m(z){|}^2\le 1+e^{2(\Im z{)}^2},|m(\tau )|\le 1(\tau \in \mathbb{R}).\text{\hspace{1em}(VHR2)}$$

This is the SIGNED analytic root on the real line, rather than its nonnegative absolute value, which is not analytic at zero. Fix $0<\eta <\sqrt{\pi /2}$ and restrict to sufficiently small $ϵ>0$ that $\sqrt{ϵ}b<\eta$. Define

$$\hat{h}(\tau )=e^{-ϵ{\tau }^2}\hat{g}(\tau ),\hat{u}(\tau )=m(\sqrt{ϵ}\tau )\hat{g}(\tau ).\text{\hspace{1em}(VHR3)}$$

The multiplier in the second formula is holomorphic on a neighborhood of the closed strip $|\Im \tau |\le b$ and bounded there by a constant depending only on $\eta$, uniformly in small $ϵ$. For smooth compactly supported $g$, the product with $\hat{g}$ has rapid decay on every smaller closed strip. Apply the cited strip-translation identity to this product, with angular-frequency normalization, and then Plancherel on the two lines $\Im \tau =\pm b$:

$$\begin{array}{rl}\Vert e^{\pm bx}u{\Vert }_2& =\Vert m(\sqrt{ϵ}(\tau \pm ib))\hat{g}(\tau \pm ib){\Vert }_2\\ & \le C_{\eta }\Vert e^{\pm bx}g{\Vert }_2.\end{array}$$

The margin beyond $b$ verifies the source theorem’s open-strip hypothesis. Since $e^{2b|x|}$ and $e^{2bx}+e^{-2bx}$ are comparable, smooth compact approximation in the weighted $L^2$ norm extends this estimate to every present $g$. Real-axis contraction also gives $\mathcal{L}(u)\le \mathcal{L}(g)$. The existing weighted heat bound supplies the same domain control for $h$. Consequently

$$\Vert u{\Vert }_b^2+\Vert h{\Vert }_b^2+\mathcal{L}(u)+\mathcal{L}(h)\le CM.\text{\hspace{1em}(VHR4)}$$

The auxiliary $u$ is ODD, also for complex even $g$. Neither $u$ nor $h$ is asserted to be another actual source profile.

Keep both poles in the exact loss identity

For arbitrary tests in this weighted logarithmic form domain, use the existing full arithmetic expression

$$\begin{array}{rl}{\mathcal{Q}}_{\mathrm{full}}(v)& ={\Gamma }_q(v)-2\sum _{n\ge 2}\frac{\Lambda (n)}{\sqrt{n}}{R}_v(\log n)+P_{\mathrm{full}}(v),\\ {\Gamma }_q(v)& =\int q(\tau )|\hat{v}(\tau ){|}^{2}d\tau ,q(\tau )=\Re \mathrm{digamma}(1/4+i\tau /2)-\log \pi ,\\ R_v(t)& =\Re \int v(x)\overline{v(x-t)}dx,\\ A_{\pm }(v)& =\int e^{\pm x/2}v(x)dx,P_{\mathrm{full}}(v)=2\Re (A_{+}(v)\overline{A_{-}(v)}).\end{array}$$

Here $\Lambda$ includes EVERY prime power. This agrees with ${\mathcal{Q}}_W$ on even tests. Absolute arithmetic convergence follows from the block estimate below, and the Gamma integrals are finite by VHR4, so all three displayed values are defined.

Plancherel and VHR2–VHR3 give $R_g(t)-R_h(t)=R_u(t)$ for every shift $t$, and the Gamma difference is exactly ${\Gamma }_q(u)$. At the poles, Fourier–Laplace transport on the interior lines $\Im \tau =\pm 1/2$ gives

$$A_{\pm }(h)=e^{ϵ/4}{A}_{\pm }(g),A_{\pm }(u)=m(\pm i\sqrt{ϵ}/2)A_{\pm }(g).$$

Schwarz symmetry and VHR2 imply

$$A_{+}(u)\overline{A_{-}(u)}=(1-e^{ϵ/2})A_{+}(g)\overline{A_{-}(g)}.$$

Thus the pole difference also matches, and

$$\boxed{{\mathcal{Q}}_W(g)-{\mathcal{Q}}_W(h)={\mathcal{Q}}_{\mathrm{full}}(u).}\text{\hspace{1em}(VHR5)}$$

In particular, the odd $u$ has a nonpositive pole term; an even-only positive pole square would give a false identity. The formula uses the full ARITHMETIC form on $u$, without asserting a new non-even zero-side explicit formula or an equality of operator domains.

Bound the entire arithmetic debit at its block endpoint

For $I_j=[j,j+1)$ and $0<c<b$, put

$$a_j(v)=\Vert v{\Vert }_{L^2(I_j)},{\mathcal{A}}_c(v)=\sum _{j\in \mathbb{Z}}{e}^{c|j|}{a}_j(v).$$

The classical two-norm split gives

$${\mathcal{A}}_c(v)\le C_{b,c}\Vert v{\Vert }_0^{1-c/b}\Vert v{\Vert }_b^{c/b}.\text{\hspace{1em}(VHR6)}$$

For completeness, split at an integer $N\ge 0$. Cauchy–Schwarz and geometric sums bound the central and tail sums by $C_{b,c}(e^{cN}\Vert v{\Vert }_0+e^{-(b-c)N}\Vert v{\Vert }_b)$; the weighted block squared norm is comparable with $\Vert v{\Vert }_b^2$. Choose $N$ within one of $b^{-1}\log (\Vert v{\Vert }_b/\Vert v{\Vert }_0)$, with the zero case separate. This proves the displayed application without a limiting $c\downarrow 1/2$ constant or an extra logarithmic factor.

The cited all-prime-power theorem gives $\Psi (X)\le CX$ for $X\ge 2$; its explicit error factor is bounded and the remaining bounded range is absorbed in $C$. It follows that, for integers $d$,

$$\sum _{\begin{array}{c}n\ge 2\\ \log n\in [d-1,d+1]\end{array}}\frac{\Lambda (n)}{\sqrt{n}}\le C{e}^{|d|/2}.$$

For a fixed block pair $(j,k)$, the correlation of $v{1}_{I_j}$ and $w{1}_{I_k}$ is supported in $t\in [j-k-1,j-k+1]$ and bounded by $a_j(v)a_k(w)$. Tonelli and $|j-k|\le |j|+|k|$ therefore give

$$\sum _{n\ge 2}\frac{\Lambda (n)}{\sqrt{n}}\left|\int v(x)\overline{w(x-\log n)}dx\right|\le C{\mathcal{A}}_{1/2}(v){\mathcal{A}}_{1/2}(w).$$

The negative shifts have the same bound. Both exponential pole functionals have absolute value at most $C{\mathcal{A}}_{1/2}(v)$. Writing ${\mathcal{B}}_{\mathrm{full}}$ for the complete prime-and-pole part, VHR6 with $c=1/2$, $b=5/2$ consequently gives

$$|{\mathcal{B}}_{\mathrm{full}}(v,v)|\le C\Vert v{\Vert }_0^{8/5}\Vert v{\Vert }_b^{2/5}.\text{\hspace{1em}(VHR7)}$$

This is an absolute bound on the entire prime-and-pole part, with all prime powers and both poles. It assumes no favorable arithmetic sign.

Pay this quadratic loss from the same Gamma contribution

Set $H=\log (1/ϵ)$ and split frequency at $R={ϵ}^{-1/4}$. Since $|m(\sqrt{ϵ}\tau ){|}^2=1-e^{-2ϵ{\tau }^2}$ on the real line,

$${\int }_{|\tau |<R}|\hat{u}(\tau ){|}^{2}d\tau \le 2\sqrt{ϵ}\Vert g{\Vert }_0^2\le C\sqrt{ϵ}M.$$

The owned digamma bounds give $q\ge q_0=q(0)$ everywhere and $q(\tau )\ge cH$ for $|\tau |\ge R$ when $H$ is sufficiently large. Keeping the high-frequency contribution and paying the low part yields

$${\Gamma }_q(u)\ge cH\Vert u{\Vert }_0^2-CH\sqrt{ϵ}M.$$

Put $r=\Vert u{\Vert }_0^2/M$. VHR4–VHR7, all on the SAME actual $g$ and its ONE loss test $u$, give

$${\mathcal{Q}}_W(g)\ge {\mathcal{Q}}_W(h)+M[cHr-C{r}^{4/5}-CH\sqrt{ϵ}].\text{\hspace{1em}(VHR8)}$$

Direct minimization over the one real variable $r\ge 0$ gives $cHr-C{r}^{4/5}\ge -C^{\prime }{H}^{-4}$; the minimizer is proportional to $H^{-5}$. Also $H{e}^{-H/2}=O(H^{-4})$. Hence

$${\mathcal{Q}}_W(g)\ge {\mathcal{Q}}_W(h)-CM{H}^{-4}.\text{\hspace{1em}(VHR9)}$$

The auxiliary quadratic loss, rather than a mixed comparison with $g-h$, is what permits this absorption. No desired full-form sign has been assumed.

Reuse the signed zero tail and return to original sources

Under $\mathsf{RH}(T)$, W369–W370 already give, for $ϵ=1/T$,

$${\mathcal{Q}}_W(h)\ge -CM\sqrt{T}\log (2+T)e^{-T}.\text{\hspace{1em}(VHR10)}$$

Only the inclusively verified zero head is replaced by nonnegative squares. The entire remaining signed conjugate pairing, multiplicities and domain passage are reused unchanged. VHR9–VHR10 and W359 prove VHR1 for large $T$. W317’s uniform $\mathfrak{E}(F)\ge -CM(F)$ pays the bounded range $3\le T\le T_{\ast }$ by enlarging one absolute constant.

The original-source normalization W257 and the completed minima in W371 consequently give, under the SAME premise,

$$\begin{array}{c}{e}_K\ge -\frac{CK}{(\log T{)}^4}(K>1),-\frac{4C}{(\log T{)}^4}\le e_4\le 0,\\ 4{\pi }^{3}A{Q}_{{\mathcal{K}}_A^g}(\sigma )\ge -\frac{C}{(\log T{)}^4}{Q}_{6,A}(\sigma ).\end{array}\text{\hspace{1em}(VHR11)}$$

These are the original source, full form and sixth moment, with arbitrary admitted count, width and real or complex coefficients. No auxiliary source is normalized or substituted for the actual one.

The constants have not been numerically evaluated, so this is an asymptotic rate improvement, without a claimed numerical improvement at $T_0$. No optimality of exponent $4$ is asserted. A fixed verified height still leaves a negative allowance. Removing it requires unbounded verified heights or an independent signed estimate. VHR1–VHR11 establish neither $e_4=0$, all-source positivity, RH, the same-integer signed Robin tail bound nor strict Robin for every integer $n>5040$. The application remains a paper derivation, without Lean/kernel certification.

A Gamma-only bound cannot pay the prime Hankel term on all compact tests

The odd loss in VHR3 motivates separating the one-sided translation and reflection channels. The corresponding even split is already owned by W338–W340, and W284/W313 already retain and bound the complete prime Hankel correction. A new copy of those interfaces is unnecessary. There is, however, an obstruction to replacing their weighted allowance by a uniform Gamma-plus-$L^2$ lower bound on ALL compact half-line tests.

This is a repo-derived application of the classical Dirichlet series for $-{\zeta }^{\prime }/\zeta$, the nonnegative Laplace-transform abscissa theorem and the existence of a nonzero critical-line zero. The Laplace supplier is the same Landau–Widder theorem already consumed by the fixed-shape detector: Widder, The Laplace Transform, Chapter II, Theorem 5b, also explicitly used in Suzuki, arXiv:2209.12832v1, section 3, printed pp.3–4. These classical results are reused, not reproved or claimed as new Hankel theory. The following argument has not been Lean/kernel certified and carries no literature-priority claim.

For a real $v\in C_c^{\infty }((0,\infty ))$, let $v^{\mathrm{odd}}$ be its odd full-line extension and retain EVERY prime power in

$$H_p(v)=\sum _{n\ge 2}\frac{\Lambda (n)}{\sqrt{n}}{\int }_0^{\log n}v(x)v(\log n-x)dx.$$

This sum is finite for each such $v$. Set

$${\Gamma }_{+}(w)={\int }_{\mathbb{R}}[q(\tau )-q(0)]|\hat{w}(\tau ){|}^{2}d\tau \ge 0.$$

There is NO finite constant $K\ge 0$ such that

$$H_p(v)\ge -K(\Vert v{\Vert }_2^2+{\Gamma }_{+}(v^{\mathrm{odd}}))\text{for every real }v\in C_c^{\infty }((0,\infty )).\text{\hspace{1em}(VHO1)}$$

In particular, no fixed finite coefficients on Gamma and $L^2$ can provide the proposed bound without a further source-dependent term or a restriction of the source domain.

Translate a fixed test and retain its exact Laplace transform

Suppose the lower bound in VHO1 held for a fixed $K$. Fix real $f\in C_c^{\infty }((0,L))$, put $v_R(x)=f(x-R)$ for $R\ge 0$, and define

$$F(z)={\int }_{\mathbb{R}}{e}^{zx}f(x)dx,\kappa =f\ast f,h_f(R)=H_p(v_R).$$

Choose $F(1/2)=0$. Reflection invariance, the nonnegative even Gamma multiplier and the Fourier triangle inequality give

$${\Gamma }_{+}(v_R^{\mathrm{odd}})\le 4{\Gamma }_{+}(f),h_f(R)\ge -K_f,K_f=K(\Vert f{\Vert }_2^2+4{\Gamma }_{+}(f)).\text{\hspace{1em}(VHO2)}$$

The bound is uniform in $R$. Chebyshev gives $h_f(R)=O_f(e^R)$. For $\Re s>1$, absolute convergence and the substitution $t=\log n-2R$ give exactly

$$\begin{array}{rl}{D}_f(s)& :={\int }_0^{\infty }{e}^{-sR}{h}_f(R)dR\\ & =\frac{1}{2}F(s/2{)}^2\left(-\frac{{\zeta }^{\prime }}{\zeta }\right)((s+1)/2)+E_f(s),\\ E_f(s)& =-\frac{1}{2}\sum _{\log n<2L}\Lambda (n)n^{-(s+1)/2}{\int }_{\log n}^{2L}{e}^{st/2}\kappa (t)dt.\end{array}\text{\hspace{1em}(VHO3)}$$

The endpoint correction is entire because its sum is finite. The factor $F(1/2{)}^2$ cancels the pole at $s=1$. The right side is analytic near every positive real $s$ and near $s=0$: zeta has no real zero in $[1/2,1)$ or $(1,\infty )$. Since $h_f+K_f\ge 0$, the cited nonnegative Laplace theorem forces its convergence abscissa to be at most zero. Thus its actual integral converges and is holomorphic on $\Re s>0$, where it agrees with $D_f(s)+K_f/s$.

Already this consequence would exclude any zero with $\Re \rho >1/2$: it would give a pole at $s=2\rho -1$ unless $F(\rho -1/2)=0$. For any designated such zero, choosing $f=(d/dx+1/2)w$ with a sufficiently short nonnegative smooth bump $w$ gives $F(z)=(1/2-z)\int e^{zx}w(x)dx$ and avoids that cancellation. The following boundary argument gives a contradiction even using only an already known critical-line zero.

A known boundary pole contradicts a support-independent constant

Fix a nonzero critical-line zero ${\rho }_0=1/2+i{\gamma }_0$ of multiplicity $m_0\ge 1$. VHO3 has residue $-m_{0}F(i{\gamma }_0{)}^2$ at $s=2i{\gamma }_0$. Positivity of the same Laplace integral gives, for $\eta >0$,

$$\eta \left|D_f(\eta +2i{\gamma }_0)+\frac{K_f}{\eta +2i{\gamma }_0}\right|\le \eta D_f(\eta )+K_f.$$

Letting $\eta \downarrow 0$, analyticity at zero and that exact residue therefore imply

$$m_0|F(i{\gamma }_0){|}^2\le K_f.\text{\hspace{1em}(VHO4)}$$

Now choose real $\chi \in C_c^{\infty }((1,2))$ with $\int \chi \ne 0$ and, for $a\ge 1$, put

$$w_a(x)=\chi (x/a)\cos ({\gamma }_{0}x),f_a=(d/dx+1/2)w_a.$$

Integration by parts gives $F_a(1/2)=0$, while Riemann–Lebesgue and the fixed-frequency derivative bounds give

$$\begin{array}{rl}|F_a(i{\gamma }_0){|}^2& \sim \frac{(1/4+{\gamma }_0^2)(\int \chi {)}^2}{4}{a}^2,\\ \Vert f_a{\Vert }_2^2+{\Gamma }_{+}(f_a)& =O_{{\gamma }_0,\chi }(a).\end{array}\text{\hspace{1em}(VHO5)}$$

For the second estimate it suffices to use the owned digamma bound $0\le q(\tau )-q(0)\le C(1+{\tau }^2)$ and $\Vert f_a{\Vert }_{H^1}^2=O_{{\gamma }_0,\chi }(a)$. For each fixed $a$, use $L=2a$ in VHO3 and take $\eta \downarrow 0$ first; only afterwards let $a\to \infty$. No bound on $E_{f_a}$ uniform in $a$ is needed. VHO4 would bound a positive multiple of $a^2$ by $O_K(a)$, a contradiction. No selection of simultaneous phases of all zeros, new zero computation or full non-even zero-side form identity is used.

What this obstruction leaves open

VHO1 concerns ALL compact tests and has no original-moment remainder. It does not refute a bound restricted to the actual VHR3 loss family with $\mathfrak{E}(F)\le 0$, or one retaining $M(F)\eta (H)$. Long supports and translations can have exponentially large weighted moments even when their Gamma and unweighted norms have the estimates above. Such a remainder cannot be discarded in this argument. No negative value of the complete Weil form, RH counterexample or improvement of VHR1 follows. A usable improvement still needs a joint estimate on the SAME actual loss test, including its original moment and a sufficiently small quantified remainder.

Keep the heat partition inside the actual source range

The VHR loss is odd, so the actual even-source variational inequality cannot be applied to it. A fixed positive shift in the heat symbol instead gives two EVEN actual profiles. The useful estimate is a JOINT defect of the ORIGINAL moment and norm, on the bounded Gamma/moment class. A moment-only uniform defect would be false.

Reuse VHR2–VHR5’s full-form splitting calculation, W324’s actual inverse, W338’s forward profile comparison, W359’s complete-form identity, and W331–W332’s small-interval and arithmetic-boundary bounds. The ordinary binomial series, Gaussian convolution and supporting lines of finite convex functions are classical inputs. The additional application below keeps both reconstructed sources and both original metrics in one variational comparison. All statements in this section are paper derivations, without Lean/kernel certification or a literature-priority claim.

An even partition retains both pole terms exactly

Fix $c>5/2$ and, for sufficiently small $ϵ>0$, put

$$a_{ϵ}(z)=e^{-ϵ(z^2+c^2)},b_{ϵ}(z)=\sqrt{1-e^{-2ϵ(z^2+c^2)}}.$$

Choose the square root positive on the real axis. Its first zeros are $\pm ic$; the others solve $z^2=-c^2+\pi ik/ϵ$, $k\ne 0$, and have imaginary part of absolute value greater than $c$. Thus it is an EVEN Schwarz-real holomorphic function on $|\Im z|<c$. Both multipliers are uniformly bounded on each smaller closed strip. For $g=g_F$ define

$$\hat{h}=a_{ϵ}\hat{g},\hat{v}=b_{ϵ}\hat{g}.$$

Both profiles are even, including for complex $F$. Their weighted and logarithmic norms are finite. Recover their actual features $F_h,F_v$ using the EXISTING inverse: for $j=h,v$, set

$${\beta }_j(t)=j(t/2),u_j(t)={\beta }_j(t)+\frac{3}{2}{\int }_0^{\infty }{e}^{s/4}{\beta }_j(t+s)ds,U_j(x)=x^{-9/4}{u}_j(\log (x/\pi )),F_j=\mathcal{S}{U}_j.$$

The positive exponential weight makes the integral bounded and W324 supplies the actual inverse. Domain membership is checked BEFORE using W359. For each fixed $ϵ$, the heat multiplier and $1-b_{ϵ}$ decay rapidly at high real frequency on smaller closed strips. Their convolutions with $g$ have weighted $H^1$ regularity. The inverse Volterra integral preserves that regularity: if $Y(t)={\int }_0^{\infty }{e}^{s/4}{\beta }_j(t+s)ds$, then $Y^{\prime }=-{\beta }_j-Y/4$, with $Y$ bounded in the positive exponential weight. Hence the reconstructed $u_h$ and $u_{g-v}$ are $H^1$ on the closed positive half-line. Their zero extensions have at most one jump. Subtracting the trace times $e^{-t}{1}_{+}$ leaves a full-line $H^1$ function; the jump term’s Fourier transform is a constant times $(1+i\xi {)}^{-1}$. Both have finite logarithmic Gamma form, so $F_h,F_{g-v}\in \mathcal{V}$. Set $F_v=F-F_{g-v}$. The forward identity $B{B}^{-1}{\beta }_j={\beta }_j$ now proves $g_{F_h}=h$ and $g_{F_v}=v$. This is an explicit range check, not a converse assumed from W338 or a bare unweighted Volterra inverse.

On the real axis $a_{ϵ}^2+b_{ϵ}^2=1$. At BOTH pole frequencies $\pm i/2$ the two multipliers are real and satisfy the same identity. Consequently VHR5’s Gamma and every prime-power correlation split, and the pole squares split with them:

$$\mathfrak{E}(F)=\mathfrak{E}(F_h)+\mathfrak{E}(F_v).\text{\hspace{1em}(VHP1)}$$

No odd loss is identified with an actual even profile. The fixed shift costs a scalar $e^{-ϵc^2}$ in the heat branch and removes the previous root’s zero at the origin.

Pay the two metric defects on the same bounded-energy source

Write $\mathcal{H}(F)=\Vert F{\Vert }_H^2$ and $L_{ϵ}=\log (1/ϵ)$. There is a fixed $C$, depending on the chosen $c$, such that for every $F\in \mathcal{V}$ with $M(F)+\mathcal{P}(F)\le D$,

$$\begin{array}{rl}|M(F_h)+M(F_v)-M(F)|+|\mathcal{H}(F_h)+\mathcal{H}(F_v)-\mathcal{H}(F)|& \le CD{L}_{ϵ}^{-2/5},\\ \Vert F_h-F{\Vert }_H& \le C\sqrt{D}{L}_{ϵ}^{-1/5},\\ M(F_h)+M(F_v)& \le CM(F).\end{array}\text{\hspace{1em}(VHP2)}$$

Here the first line controls a joint defect; it does not assert that $M(F_v)$ tends to zero uniformly over that class. The last line is a moment-only bound on all actual sources, without the Gamma hypothesis. It follows from the full-line reconstruction and boundedness of both exponentially weighted filters below.

Use $q=5/4$, the even full-line function $\beta (t)=g(t/2)$, and $w(t)=e^{qt}\beta (t)$. On the full line let

$$S=\sum _{n\ge 1}\mu (n)n^{-3}{T}_{2\log n},(T_{a}f)(t)=f(t+a),K=I+\frac{3}{2}{J}_1,(J_{1}f)(t)={\int }_0^{\infty }{e}^{-s}f(t+s)ds,X=KSw.$$

All these are bounded on full-line $L^2$, and $S,K$ commute with full-line Fourier multipliers. Their restrictions for $t\ge 0$ are exactly the owned source reconstruction, since all shifts are forward:

$$M(F)={\pi }^{-7/2}\Vert 1_{+}X{\Vert }_2^2,\mathcal{H}(F)={\pi }^{-7/2}{\int }_0^{\infty }{e}^{-6t}|X(t){|}^{2}dt.\text{\hspace{1em}(VHP3)}$$

Also $\Vert w{\Vert }_2+\Vert X{\Vert }_2\le C\sqrt{M(F)}$. Apply the Fourier partition with symbols $a_{ϵ}(2\xi ),b_{ϵ}(2\xi )$ to this full-line representation. Conjugating either filter by $e^{qt}$ changes it in operator norm by at most $C\sqrt{ϵ}\log (2+1/ϵ)$. Thus its reconstructed $X_j$ differs from the corresponding filter applied to $X$ by at most this factor times $\sqrt{M(F)}$.

For completeness, the kernel estimates needed here follow directly from the ordinary binomial expansion

$$b_{ϵ}(2D)=I-k_{ϵ}\ast ,k_{ϵ}=\sum _{m\ge 1}{d}_m{e}^{-2mϵc^2}{K}_{8mϵ},d_m=\frac{\left(\genfrac{}{}{0ex}{}{2m}{m}\right)}{4^m(2m-1)}>0,\sum _m{d}_m=1,$$

where $K_t$ is the normalized heat kernel. Since $d_m=O(m^{-3/2})$ and $c>2q$, Gaussian integration gives

$$\int (e^{q|s|}-1)k_{ϵ}(s)ds\le C\sqrt{ϵ}\log (2+1/ϵ),{\int }_{|s|>r}{k}_{ϵ}(s)ds\le C\sqrt{ϵ}/r(r\ge \sqrt{ϵ}).\text{\hspace{1em}(VHP4)}$$

The heat branch satisfies the same bounds. For the second estimate, sum $d_m\min (1,mϵ/r^2)$; for the first, split the Gaussian exponential moment sum at $m=1/ϵ$. These are estimates for the actual filters, not an assumed multiplier-domain separation.

Let $P_{+}$ be multiplication by $1_{+}$. For either filter $A$, its commutator on $X$ obeys

$$\Vert [P_{+},A]X{\Vert }_2\le C\Vert 1_{[-r,r]}X{\Vert }_2+C\sqrt{ϵ}{r}^{-1}\Vert X{\Vert }_2.\text{\hspace{1em}(VHP5)}$$

Split the input into that interval and its complement; on the latter, any kernel crossing the cut has displacement greater than $r$. For multiplication by $1_{+}{e}^{-6t}$ there is additionally a $C\sqrt{ϵ}\log (2+1/ϵ)\Vert X{\Vert }_2$ term, since $e^{-6\max (t,0)}$ is bounded and Lipschitz.

The remaining local mass is the SAME arithmetic-boundary estimate already paid in W332. W338 and W332’s frequency split give $\Vert 1_I\beta {\Vert }_2^2\le CD/\log (1/r)$ on every interval of length $2r$. For $n\le N$ near $t=0$ the factor $n^{-3}{e}^{q(2\log n)}$ is $n^{-1/2}$, so their triangle sum is at most $C\sqrt{DN/\log (1/r)}$. The rest of $S$ has norm at most $C{N}^{-2}$. Finally $J_1:L^2\to L^{\infty }$ is bounded. Hence

$$\Vert 1_{[-r,r]}X{\Vert }_2\le C\sqrt{D}\left[\sqrt{N/\log (1/r)}+N^{-2}+\sqrt{r}\right].\text{\hspace{1em}(VHP6)}$$

Choose $r={ϵ}^{1/4}$ and $N$ comparable to $L_{ϵ}^{1/5}$. Equations VHP4–VHP6 give $C\sqrt{D}{L}_{ϵ}^{-2/5}$ for the commutators. The identity $A^2+B^2=I$ for the real-axis filters turns each metric defect into $⟨X,(A[P_{+},A]+B[P_{+},B])X⟩$, with the indicated bounded weight for $\mathcal{H}$. The exponential-conjugation error has the smaller order already stated. This proves the first line of VHP2 while retaining the sharp arithmetic boundary.

For its second line, the same frequency split as W331 gives $\Vert h-g{\Vert }_0^2\le CD/L_{ϵ}$, while their weight-$5/2$ norms remain bounded by $C\sqrt{D}$. Interpolate to weight $3/2$ in $r$, or $3/4$ in $t$. At that weight the displayed Volterra inverse is bounded, and W331’s EXISTING $H_5$ Möbius inverse has norm at most $\zeta (2)$. This gives the asserted original-$H$ error.

The Gamma hypothesis cannot be dropped from the first line of VHP2. For a nonzero real $\varphi \in C_c^{\infty }(-2,-1)$, concentrate ${\beta }_s(t)=s^{-1/2}\varphi ((t-2\log 2)/s)$ on the positive half-line, evenly extend it, and take $s=2\sqrt{ϵ}$. Recover its actual source by the same inverse. The original packet lies below the first nontrivial Möbius shift. After filtering, the $n=2$ channel crosses $t=0$. Its limit gives

$$\frac{{\Delta }_M}{M(F_s)}⟶\frac{2^{-6}}{\Vert \varphi {\Vert }_2^2}{\int }_0^{\infty }\left[|e^{-D^2}\varphi {|}^2+|\sqrt{1-e^{-2D^2}}\varphi {|}^2\right]dt>0.\text{\hspace{1em}(VHP6a)}$$

In the weighted coordinates above, $J_1$ applied to these concentrated packets tends to zero in $L^2$. The $n=1,2$ outputs concentrate at different locations; each fixed $n\ge 3$ output vanishes on the positive half-line and the remaining tail is bounded by $C{N}^{-2}$. These facts justify the displayed limit. Here $\mathcal{P}(F_s)/M(F_s)$ grows as $\log (1/s)$; this family is not a counterexample on the nonpositive complete-energy class used below.

Consume the joint defect at an actual negative minimum

Suppose $e_4=-\delta <0$, and choose its actual unit minimizer $F$. The usual supporting multiplier can be taken GLOBAL, not merely as a weak stationary equation:

$$Q_{\lambda }(G):=\mathfrak{E}(G)+\lambda M(G)-\mu \mathcal{H}(G)\ge 0(G\in \mathcal{V}),Q_{\lambda }(F)=0,\mu =-\delta +4\lambda ,0\le \lambda \le \frac{\delta }{4-{\kappa }_{\circ }}.\text{\hspace{1em}(VHP7)}$$

Indeed the existing real extraction and imaginary mixing used in W345/W351 show that the finite function $K\mapsto e_K$ is convex and nonincreasing for $K>1$: mix two real minimizers with coefficients $\sqrt{\theta },i\sqrt{1-\theta }$, then extract at the SAME moment. A classical supporting line at $K=4$ supplies $-\lambda$. Its bound at the actual minimizer gives complementary slackness, and testing on W350’s unit null vector gives the upper bound in VHP7. No unconfined self-adjoint operator or negative penalized eigenvalue is assumed.

W317 gives $M(F)+\mathcal{P}(F)\le C$ on these minimizers. With ${\Delta }_M=M(F_h)+M(F_v)-M(F)$ and the corresponding ${\Delta }_H$, VHP1, VHP2 and VHP7 give

$$0\le Q_{\lambda }(F_h),Q_{\lambda }(F_v),Q_{\lambda }(F_h)+Q_{\lambda }(F_v)=\lambda {\Delta }_M-\mu {\Delta }_H\le C\delta L_{ϵ}^{-2/5}.\text{\hspace{1em}(VHP8)}$$

The crucial factor $\delta$ multiplies the metric error; no ${\lambda }^{-1}$ or lower-multiplier premise was used.

Now assume $\mathsf{RH}(T)$, take $ϵ=1/T$, and put ${\eta }_T=\sqrt{T}\log (2+T)e^{-T}$. W369–W370, already used in VHR10, apply to the scalar-shifted heat branch and give $\mathfrak{E}(F_h)\ge -C{\eta }_T$. For all sufficiently large $T$, VHP2 also gives $\mathcal{H}(F_h)\ge 1/2$, while always $M(F_h)\ge \mathcal{H}(F_h)$. Therefore

$$\boxed{\lambda \ge \left[\frac{1}{3}-C(\log T{)}^{-2/5}\right]\delta -C{\eta }_T.}\text{\hspace{1em}(VHP9)}$$

To check the direction, VHP8 and the height bound imply $(\delta-3\lambda)\mathcal H(F_h) \le C\delta(\log T)^{-2/5}+C\eta_T$. Dividing by its positive norm proves VHP9. In particular, at a sufficiently large starting height depending only on the fixed $c$,

$$\lambda \le \delta /6⟹\delta \le C\sqrt{T}\log (2+T)e^{-T}.\text{\hspace{1em}(VHP10)}$$

This includes the previously allowed zero multiplier. Outside that exponentially small allowance the actual supporting multiplier has a positive lower bound proportional to $\delta$, as well as the existing upper bound. The constants and starting height have not been numerically evaluated.

VHP9–VHP10 do not improve VHR1’s allowance for the remaining positive-multiplier case. They supply the lower-multiplier premise needed by such a case’s future estimates, up to the displayed height tail. They neither establish higher-log regularity nor remove the negative allowance in the original complete form. Full RH and strict Robin for every integer $n>5040$ remain unproved.

Bootstrap the actual minimizing source before paying its moment loss

The global supporting form in VHP7 supplies more than a lower multiplier: it gives a source-specific logarithmic bootstrap. The following repo-derived application reuses VHP’s actual even branches, W331’s weighted inverse and W338’s full Gamma comparison. No generic nonlocal eigenfunction regularity theorem is substituted for these interfaces. All estimates concern the SAME hypothetical negative cap-four minimizer. They remain paper derivations without Lean/kernel certification or a priority claim.

Square the local mass in the joint metric defect

For $k\ge 1$ put

$${\mathcal{L}}_k(g)={\int }_{\mathbb{R}}[\log (2+|\tau |){]}^k|\hat{g}(\tau ){|}^{2}d\tau ,L=\log (1/ϵ).$$

On the actual class $M(F)+{\mathcal{L}}_k(g_F)\le D$, the same branches as VHP satisfy the stronger joint estimate

$$|{\Delta }_M|+|{\Delta }_H|\le C_{k}D{L}^{-4k/5}.\text{\hspace{1em}(VHB1)}$$

The range check in VHP is unchanged; ${\mathcal{L}}_k$ with $k\ge 1$ pays its logarithmic domain. To improve the estimate, use the self-adjoint defect of the REAL-axis filters $A=a_{ϵ}(2D)$ and $B=b_{ϵ}(2D)$ in VHP3:

$$D_P=APA+BPB-P=A[P,A]+B[P,B],P=1_{+}.$$

For $Z_r=1_{|t|\ge r}$, VHP4’s cross-cut kernel bound gives $\Vert D_P{Z}_r\Vert \le C\sqrt{ϵ}/r$. Since $D_P=D_P^{\ast }$ and $\Vert D_P\Vert \le C$, splitting $X=(1-Z_r)X+Z_{r}X$ gives

$$|⟨X,D_{P}X⟩|\le C\Vert 1_{[-r,r]}X{\Vert }_2^2+C\sqrt{ϵ}{r}^{-1}\Vert X{\Vert }_2^2.\text{\hspace{1em}(VHB2)}$$

Both mixed terms use the SAME $D_P{Z}_r$ estimate. For $P{e}^{-6\max (t,0)}$ the bounded Lipschitz factor adds only $C\sqrt{ϵ}\log (2+1/ϵ)\Vert X{\Vert }_2^2$ by VHP4. The exponential-conjugation errors have this same smaller order.

A Fourier split on $\beta (t)=g(t/2)$ now gives local squared mass $C_{k}D/[\log (1/r){]}^k$. Reuse VHP6’s finite Möbius reconstruction, without replacing its arithmetic coefficients:

$$\Vert 1_{[-r,r]}X{\Vert }_2^2\le C_{k}D\left[\frac{N}{[\log (1/r){]}^k}+N^{-4}+r\right].$$

Take $r={ϵ}^{1/4}$ and $N$ comparable to $L^{k/5}$. This proves VHB1. It is the self-adjoint quadratic defect, rather than the norm of its first commutator alone, that retains the squared local mass.

Keep the reconstruction just above its critical weight

Fix $0<\eta <2$, and set

$${\theta }_{\eta }=\frac{4-2\eta }{5}>0,d=F-F_h.$$

On the same class, the actual original-source errors satisfy

$$\mathcal{N}(d{)}^2+\mathcal{N}(F_v{)}^2\le C_{k,\eta }D{L}^{-{\theta }_{\eta }k}.\text{\hspace{1em}(VHB3)}$$

Indeed both profile differences $(1-a_{ϵ})\hat{g}$ and $b_{ϵ}\hat{g}$ have unweighted squared norm at most $C_{k}D{L}^{-k}$ by splitting at $|\tau |={ϵ}^{-1/4}$. Their weight-$5/2$ norms remain bounded by $C\sqrt{M(F)}$, as in VHP. In the $t$ coordinate interpolate to the positive exponential weight $a_{\eta }=1/4+\eta /2$, below $q=5/4$. The resulting squared norm has exponent $k(1-a_{\eta }/q)={\theta }_{\eta }k$.

Here the ACTUAL Volterra inverse is bounded because $a_{\eta }>1/4$. This is precisely the source space $H_{4+\eta }$, with squared norm ${\int }_{\pi }^{\infty }(x/\pi {)}^{4+\eta }|F(x){|}^{2}dx$. The actual Möbius inverse has norm at most $\zeta (1+\eta )$ there, since its $n$th dilation has norm $n^{-1-\eta }$. Finally $\mathcal{N}(G{)}^2\le C_{\eta }\Vert G{\Vert }_{H_{4+\eta }}^2$. This proves VHB3 without an endpoint $H_4$ inverse. The constants may diverge as $\eta \downarrow 0$; no endpoint assertion is made.

Use the null supporting vector to gain logarithmic regularity

Let $F$ be an actual unit minimizer with $e_4=-\delta <0$ and use VHP7’s global $Q_{\lambda }$. The existing complete-energy floor gives $\delta \le C$ and $M(F)+\mathcal{P}(F)\le C$. Thus W338 supplies ${\mathcal{L}}_1(g_F)\le C$. Suppose at some stage that ${\mathcal{L}}_k(g_F)\le C_k$, uniformly on these minimizing vectors. VHP1 and VHB1 give

$$0\le Q_{\lambda }(F_h),Q_{\lambda }(F_v)\le C_k\delta L^{-4k/5}.$$

Since $Q_{\lambda }$ is nonnegative and $Q_{\lambda }(F)=0$, its polarized pairing with $F$ vanishes. In particular $Q_{\lambda }(d)=Q_{\lambda }(F_h)$. Applying W317’s COMPLETE debit bound to $d$ and $F_v$, using VHB3 and $|\mu |\le C\delta \le C$, proves

$$\begin{array}{rl}& \mathcal{P}(d)+\mathcal{P}(F_v)+\lambda [M(d)+M(F_v)]\\ & \le C_{k,\eta }{L}^{-{\theta }_{\eta }k}.\end{array}\text{\hspace{1em}(VHB4)}$$

No division by $\lambda$ is used in the Gamma estimate. W338 applied to the ACTUAL difference $d$ bounds its full logarithmic profile norm by $C[\mathcal{P}(d)+\mathcal{N}(d{)}^2]$. On $|\tau |\ge {ϵ}^{-1/2}$ the factor $1-a_{ϵ}(\tau )$ is bounded below by a fixed positive constant. Therefore

$${\int }_{|\tau |\ge R}\log (2+|\tau |)|{\hat{g}}_F(\tau ){|}^{2}d\tau \le C_{k,\eta }[\log (2+R){]}^{-{\theta }_{\eta }k}.$$

Layer integration gives ${\mathcal{L}}_{k^{\prime }}(g_F)\le C_{k^{\prime }}$ for every $k^{\prime }<1+{\theta }_{\eta }k$. Starting at $k=1$, choose $\eta >0$ so that $1/(1-{\theta }_{\eta })$ exceeds a given target $k_{\ast }<5$, and iterate a finite number of STRICT gains. Consequently

$$\boxed{{\mathcal{L}}_k(g_F)\le C_k(1\le k<5).}\text{\hspace{1em}(VHB5)}$$

The constants depend on the chosen exponent and fixed $c$, but not on $\delta$, $\lambda$ or the minimizing vector. No $k=5$ or $k>5$ claim follows by taking a limit of these constants.

Retain the actual sixth moment and expose its remaining cost

For each fixed $0<s<4$, choose $k<5$ and $\eta >0$ with ${\theta }_{\eta }k>s$. VHB1 and VHB3–VHB5 give

$$\begin{array}{c}|{\Delta }_M|+|{\Delta }_H|\le C_s{L}^{-s},Q_{\lambda }(F_h),Q_{\lambda }(F_v)\le C_s\delta L^{-s},\\ \mathcal{P}(d)+\mathcal{P}(F_v)+\lambda [M(d)+M(F_v)]\le C_s{L}^{-s},\\ |\mathcal{H}(F_h)-1|\le C_s{L}^{-s},\\ \lambda >0⟹M(F_v)+|M(F_h)-M(F)|\le C_s{\lambda }^{-1}{L}^{-s}.\end{array}\text{\hspace{1em}(VHB6)}$$

The fourth line uses the joint $H$ defect together with $\mathcal{H}(F_v)\le C\mathcal{N}(F_v{)}^2$, keeping all terms on the same source. The last line uses the joint $M$ defect and $\lambda M(F_v)\le C_s{L}^{-s}$; the uniform upper bound on $\lambda$ absorbs its harmless factor $1+{\lambda }^{-1}$. For $\lambda >0$, complementary slackness gives $M(F)=4$. Under the same finite-height premise as VHP9, at sufficiently large $T$ depending on the fixed $s,c$, this also improves that multiplier comparison to

$$\lambda \ge [1/3-C_s(\log T{)}^{-s}]\delta -C{\eta }_T(0<s<4).\text{\hspace{1em}(VHB7)}$$

This is a genuine single-minimizer moment recovery, unlike a sum-of-moments defect alone. Under $\mathsf{RH}(T)$, for $\delta \ge C_0{\eta }_T$, with $C_0$ sufficiently large and $T$ sufficiently large, VHP9 permits ${\lambda }^{-1}\le C{\delta }^{-1}$, so the final moment error still costs $C_s{\delta }^{-1}{L}^{-s}$. Inserting it into the heat-branch supporting equation only yields a negative allowance of order $L^{-s}$ for $s<4$. It does NOT improve the already owned VHR1 exponent $4$. An estimate beyond the critical logarithmic tail, or a new signed joint estimate, is still required. Full RH and strict Robin for every integer $n>5040$ remain unproved.

Use the verified height to reach a conditional boundary endpoint

The same finite-height premise supplies an endpoint estimate for the actual boundary defect. This is not a limit of VHB5’s constants. Let $F\in \mathcal{V}$, $\mathfrak{E}(F)\le 0$, and assume $\mathsf{RH}(T)$, with $T$ sufficiently large depending on fixed $c$. Write $L=\log T$, retain ${\eta }_T$ from VHP9, and use the SAME actual branches at any $T^{-1}\le \sigma \le {\sigma }_0$, where ${\sigma }_0<1$ is fixed and sufficiently small. Put $s_{\sigma }=\Vert g_{F_{v,\sigma }}{\Vert }_0^2$ and $L_{\sigma }=\log (1/\sigma )$. Then

$$s_{\sigma }\le CM(F)\left[L_{\sigma }^{-5}+\sqrt{\sigma }+{\eta }_T/L_{\sigma }\right].\text{\hspace{1em}(VHB8)}$$

Indeed the finite-height heat-tail argument of VHP9 is uniform over these COARSER filters. For a zero frequency $z=u+iv$, $|v|<1/2$,

$$|a_{\sigma }(z{)}^2|=e^{-2\sigma (u^2+c^2-v^2)}\le e^{-2u^2/T}.$$

Thus the existing signed zero-tail majorant gives $\mathfrak{E}(F_{h,\sigma })\ge -CM(F){\eta }_T$. The exact energy partition and the original nonpositive energy imply $\mathfrak{E}(F_{v,\sigma })\le CM(F){\eta }_T$; the high-pass branch is NOT assumed to have nonpositive energy. Its weighted profile norm is bounded by $C\sqrt{M(F)}$ by VHP’s full-$M$ branch bound. On $|\tau |\le {\sigma }^{-1/4}$ its squared Fourier mass is at most $CM(F)\sqrt{\sigma }$. Retain the Gamma contribution on the complement and apply the EXISTING complete prime-and-pole bound VHR7 to obtain

$$a{L}_{\sigma }{s}_{\sigma }\le BM(F{)}^{1/5}{s}_{\sigma }^{4/5}+CM(F)L_{\sigma }\sqrt{\sigma }+CM(F){\eta }_T,a>0.$$

Young’s inequality absorbs half the left side at cost $CM(F)L_{\sigma }^{-4}$. Division by $L_{\sigma }$ proves VHB8. This uses the full arithmetic debit and never applies W366’s nonpositive-energy hypothesis to the high-pass branch.

At the TARGET scale $ϵ=1/T$, the actual joint metric defect satisfies the endpoint

$$|{\Delta }_M|+|{\Delta }_H|\le CM(F)L^{-4}.\text{\hspace{1em}(VHB9)}$$

To pay the arithmetic boundary, take the coarser $\sigma =T^{-1/4}$, set $r=T^{-1/4}$ and $R=T^{1/8}$, and retain $\beta (t)=g_F(t/2)$. The symbol $b_{\sigma }(2\xi {)}^2$ is bounded below on $|\xi |\ge R$. VHB8 therefore gives Fourier-tail squared mass $CM(F)L^{-5}$ there. The low-frequency projection has local squared mass at most $CrRM(F)$ on every interval of length $2r$. Consequently the finite actual reconstruction used in VHP6 gives

$$\Vert 1_{[-r,r]}X{\Vert }_2^2\le CM(F)\left[N{L}^{-5}+N^{-4}+N{r}^{1/2}+r\right].$$

Choose $N$ comparable to $L$. The SQUARED local mass in VHB2, together with the smaller kernel and exponential-conjugation errors, proves VHB9. This is a conditional estimate on the original nonpositive-energy class, not a vanishing defect on the full $M$ domain.

For the SAME hypothetical unit negative cap-four minimizer, VHP7 and VHB9 give, under $\mathsf{RH}(T)$ at sufficiently large $T$,

$$\begin{array}{c}0\le Q_{\lambda }(F_h),Q_{\lambda }(F_v)\le C\delta L^{-4},\\ \lambda \ge [1/3-C{L}^{-4}]\delta -C{\eta }_T.\end{array}\text{\hspace{1em}(VHB10)}$$

The last line uses the existing $\mathcal{H}(F_h)\ge 1/2$ and $M(F_h)\ge \mathcal{H}(F_h)$ in VHP9’s supporting-form comparison. No new multiplier argument or source normalization is needed.

Finite $T$ in VHB8 does not give a strong logarithmic fifth moment, or a fifth-order Fourier-tail rate continuing uniformly as the frequency tends to infinity. VHB9–VHB10 supply a conditional boundary endpoint; they do not remove the inverse-multiplier cost in original-moment recovery or improve VHR1’s general fourth-power negative allowance. A new signed arithmetic estimate is still required. These applications remain paper derivations without Lean/kernel certification.

Pay one Fourier tail across the separated reconstruction intervals

The finite actual reconstruction in VHP6 and VHB9 admits a stronger local estimate than their termwise triangle bound. Reuse exactly VHP3’s $S,K,w,X$, VHB2’s self-adjoint metric defect, VHB5’s minimizer regularity and VHB8’s finite-height tail. The ingredients are classical Cauchy–Schwarz and disjoint input intervals; W268 in the actual-source account already uses disjoint dilation intervals and harmonic weights for a different source contract. The additional application here concerns the joint metrics of the actual even heat branches. It introduces no new general localization theorem or literature-priority claim and remains a paper derivation without Lean/kernel certification.

Separate only the finite reconstruction before cutting frequencies

For $0<r\le 1$, $R\ge 1$ and an integer $N\ge 1$ with $2rN<1$, put

$$I=[-r,r],{\mathcal{T}}_{\beta }(R)={\int }_{|\xi |\ge R}|\hat{\beta }(\xi ){|}^{2}d\xi ,S_N=\sum _{n\le N}\mu (n)n^{-3}{T}_{2\log n}.$$

The same source satisfies

$$\boxed{\Vert 1_{I}X{\Vert }_2^2\le C\left[(1+\log N){\mathcal{T}}_{\beta }(R)+M(F)(rRN+N^{-4}+r)\right].}\text{\hspace{1em}(VHB11)}$$