# CSM_RH Paper 58 ## Euler-Product Positivity Walls, Higher-Derivative Sign Loss, and the Heath-Brown Joint-Covariance Requirement **Project:** CSM_RH **Paper:** 58 **Version:** v0.1 **Date:** 2026-09-08 **Campaign:** 46 — `SEEDED_ARITHMETIC_STRIP_GAP_GENERATION` **Track:** SG4 — `NEW_ARITHMETIC_CONTRACTION_THEOREM` **Status:** STANDARD POSITIVITY / LINEAR-FACTORIZATION ROUTES CLOSED AS NON-AMPLIFYING; DIRECT ARITHMETIC FRONTIERS REMAIN OPEN **Canonical entry state:** v1.48 / Paper 57 v0.1 **RH_PROVED:** FALSE **RH_DISPROVED:** FALSE **GLOBAL_RH_CERTIFICATE:** FALSE --- # Abstract Campaign 46 seeks a genuine arithmetic theorem which upgrades a seed exponent $$ \operatorname{PESC}(\kappa) $$ to $$ \operatorname{PESC}(\kappa+\eta) $$ for some fixed $$ \eta>0. $$ Papers 55–57 showed that this is exactly a fixed zero-strip improvement problem and that principal-arc projection, standard moment-Markov methods, and classical Turan-Gallagher single-zero detectors do not generate the missing exponent. The present paper audits the remaining standard candidates based on Euler-product positivity, higher logarithmic derivatives, prime support positivity, and Heath-Brown factorization. First, positivity in the absolute-convergence half-plane has an intrinsic logarithmic-distance wall. Let $$ F(s) = -\frac{\zeta'(s)}{\zeta(s)}. $$ For $$ \sigma>1, $$ the Dirichlet coefficients of $F$ are nonnegative: $$ F(s) = \sum_{n\ge1} \frac{\Lambda(n)}{n^s}. $$ A nonnegative trigonometric polynomial with nonnegative cosine coefficients therefore produces the classical positivity inequality $$ \sum_{j=0}^{J} a_j \Re F(\sigma+ijt) \ge0. $$ If $$ \rho=\beta+i\gamma $$ is the target zero and $t=\gamma$, the $j=1$ term contains the negative zero pole $$ -\frac{a_1}{\sigma-\beta}. $$ However, the same explicit formula contains an archimedean contribution of order $$ \frac12 \left( \sum_{j\ge1}a_j \right) \log|\gamma|. $$ In the standard one-target zero-dropping positivity architecture, a contradiction is therefore possible only if $$ \boxed{ 1-\beta \ll \frac1{\log|\gamma|}. } $$ A zero at a fixed seeded distance $$ 1-\beta = \frac{\kappa}{2} $$ is asymptotically invisible to this scale comparison. This explains why modern positivity-based improvements continue to sharpen constants in logarithmic zero-free regions rather than create fixed strips. In 2026, Bellotti, Trudgian and Yang improved the classical-shaped region to $$ \sigma \ge 1-\frac1{4.896\log t}, $$ but the shape remains logarithmic. Second, one cannot remove the logarithmic wall simply by taking higher derivatives of the logarithmic derivative. Define $$ F_m(s) = (-1)^m \frac{d^m}{ds^m} F(s) = \sum_{n\ge1} \frac{ \Lambda(n)(\log n)^m }{ n^s }, $$ which again has nonnegative Dirichlet coefficients for $\Re s>1$. A target zero pole is amplified to order $$ m! (s-\rho)^{-m-1}. $$ But for every $$ m\ge1, $$ the zero-side kernel $$ \Re (s-\rho)^{-m-1} $$ changes sign as the ordinate separation varies. Thus the crucial one-sign Poisson-kernel property of $m=0$ is lost. Other zeros can no longer be discarded in a positivity upper bound. Higher derivatives amplify the target pole and simultaneously destroy the zero-side positivity that made the classical argument possible. Third, raw positivity of the prime measure is far below the seeded boundary scale. A boundary zero mode contributes a short-interval error of relative size $$ N^{-\kappa/2}. $$ Since $$ \psi(x+H)-\psi(x) = H+U_H(x) $$ and $$ N^{-\kappa/2}=o(1), $$ the positivity condition $$ \psi(x+H)-\psi(x)\ge0 $$ is automatically compatible with such a mode. Prime support positivity therefore supplies no fixed-power boundary suppression. Fourth, the exact Heath-Brown factorization route is already algebraically calibrated by Paper 47. The complete alternating cross- $j$ recombination returns the original von Mangoldt root object. If the lag vectors of the finitely many factorization components are each controlled only at seed scale $$ NH^2N^{-\kappa+o(1)}, $$ componentwise triangle or Cauchy estimates can recover at best the same seed scale. A strict amplifier requires a genuinely joint covariance deficit of fixed-power size, for example $$ \boxed{ \left\| \sum_j c_jX_j \right\|_2^2 \le N^{-\eta} \sum_j|c_j|^2\|X_j\|_2^2. } $$ Such a theorem is not supplied by the identity itself. Paper 47's exact recoupling and Papers 43–46's component barriers show that alternating signs, Ramaré extraction, structured-weight smallness, and componentwise Type-II estimates do not create this fixed-power joint deficit for free. Finally, the paper calibrates positivity inside the critical strip. Recent work on the real part of the logarithmic derivative of the Riemann xi-function shows that positivity near the critical line may remain compatible with hypothetical off-critical zeros outside local neighborhoods of those zeros. Thus positivity of the xi log-derivative is not, by itself, a global off-critical exclusion principle. The conclusion is a closure theorem for the standard SG4 candidates: ```text Euler-product coefficient positivity: non-amplifying at fixed-strip scale. Higher log-derivative positivity: zero-side sign structure fails. Raw prime-measure positivity: boundary error is too small relative to the positive main term. Existing Heath-Brown factorization: requires a new fixed-power joint cross-j covariance theorem. ``` No new equivalent criterion is needed. The preferred direct arithmetic frontier remains Paper 56's F-RH-017: $$ \boxed{ \#\left\{ x: |U_H(x)|>HN^{-\nu} \right\} \ll N^{1-c}, \qquad \nu>\frac{\kappa}{2}, \qquad c>2(1-\alpha). } $$ Campaign 46 has therefore exhausted its standard structural mechanisms. The next progress must be a genuinely new prime-side excess estimate. No RH theorem is claimed. --- # 1. Entry state Assume $$ \operatorname{PESC}(\kappa) $$ for fixed $$ 0<\kappa<1. $$ The inherited facts are: $$ \boxed{ \beta_* \le 1-\frac{\kappa}{2} } $$ and $$ \boxed{ |\psi(x)-x| \ll x^{1-\kappa/2+o(1)}. } $$ Define $$ \boxed{ d_{\rm seed} = \frac{\kappa}{2}. } $$ A hypothetical zero on the seed boundary has $$ \beta = 1-d_{\rm seed}. $$ Campaign 46 asks for an arithmetic mechanism proving a fixed additional distance from $\Re s=1$. --- # 2. Absolute-convergence positivity For $$ \Re s>1, $$ define $$ \boxed{ F(s) = -\frac{\zeta'(s)}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\Lambda(n)}{n^s}. } $$ The coefficients are nonnegative. Let $$ P(\theta) = a_0 + \sum_{j=1}^{J} a_j\cos(j\theta) $$ satisfy $$ P(\theta)\ge0 $$ for all real $\theta$, with $$ a_j\ge0. $$ Then for every $$ \sigma>1 $$ and real $t$, $$ \boxed{ \sum_{j=0}^{J} a_j \Re F(\sigma+ijt) = \sum_{n\ge1} \frac{ \Lambda(n) }{ n^\sigma } P(t\log n) \ge0. } $$ This is the basic Euler-product positivity mechanism used in the classical zero-free-region method. --- # 3. One-target positivity ledger Let $$ \rho=\beta+i\gamma $$ be a hypothetical target zero, with $$ |\gamma| $$ large. Set $$ t=\gamma $$ and $$ \sigma=1+\eta, \qquad \eta>0. $$ Write $$ d=1-\beta. $$ Then $$ \sigma-\beta = \eta+d. $$ The completed-zeta explicit formula gives, uniformly in the relevant right half-plane, $$ \Re F(\sigma+i\tau) = \frac12\log(|\tau|+3) - \sum_\rho \frac{ \sigma-\Re\rho }{ (\sigma-\Re\rho)^2+(\tau-\Im\rho)^2 } + O_\sigma(1) $$ for nonzero $\tau$, with the pole at $1$ treated separately at $\tau=0$. In the $j=1$ term, the target zero contributes $$ \boxed{ -\frac1{\eta+d}. } $$ Dropping all other negative zero contributions yields the standard one-target upper bound. At $j=0$, $$ F(1+\eta) = \frac1{\eta} + O(1). $$ For $j\ge1$, $$ \Re F(1+\eta+ij\gamma) \le \frac12 \log(|j\gamma|+3) + O_\eta(1), $$ except that the target negative pole is retained for $j=1$. Hence $$ \begin{aligned} 0 &\le \sum_{j=0}^{J} a_j \Re F(1+\eta+ij\gamma) \\ &\le \frac{a_0}{\eta} + \frac12 \left( \sum_{j\ge1}a_j \right) \log|\gamma| - \frac{a_1}{\eta+d} + O_{P,\eta} \left( 1+\log(J+2) \right). \end{aligned} $$ This is the classical positivity budget in the form relevant to the seed audit. --- # 4. The logarithmic-distance wall Set $$ A_+ = \sum_{j\ge1}a_j. $$ Since $$ a_1\le A_+, $$ the target zero term satisfies $$ \frac{a_1}{\eta+d} \le \frac{A_+}{d}. $$ If $$ d \gg \frac1{\log|\gamma|}, $$ then $$ \frac{A_+}{d} \ll A_+\log|\gamma|. $$ Thus the target pole is no larger than the unavoidable archimedean scale retained in the standard zero-dropping positivity ledger. Therefore: ## Theorem 4.1 — Positive-coefficient Euler-product logarithmic wall Within the standard one-target isolation architecture based on: 1. the nonnegative Dirichlet coefficients of $-\zeta'/\zeta$ ; 2. a nonnegative trigonometric polynomial with nonnegative cosine coefficients; and 3. discarding all non-target zero contributions by sign, the method can exclude zeros only at horizontal distance $$ \boxed{ 1-\beta = O\left( \frac1{\log|\gamma|} \right). } $$ It cannot by itself create a fixed zero-free strip. Create: ```text O-RH-139 POSITIVE_COEFFICIENT_EULER_PRODUCT_ZERO_DROPPING_HAS_INTRINSIC_LOGARITHMIC_DISTANCE_WALL CERTIFIED_AS_METHOD_BARRIER ``` This is a statement about the standard positivity architecture, not about every possible use of the Euler product. --- # 5. Seeded boundary is far outside the positivity detection scale Under PESC $(\kappa)$, the active boundary distance is $$ d_{\rm seed} = \frac{\kappa}{2}, $$ which is fixed as $$ |\gamma|\to\infty. $$ Therefore $$ d_{\rm seed} \log|\gamma| \to\infty. $$ The boundary zero contribution in the absolute-line positivity ledger is only $$ O_\kappa(1), $$ while the natural archimedean budget is $$ \asymp \log|\gamma|. $$ Hence the classical Euler-product positivity mechanism is asymptotically less sensitive than the seed. This is consistent with the modern state of the art. Bellotti, Trudgian and Yang proved in 2026 the explicit region $$ \boxed{ \zeta(\sigma+it)\ne0 \quad \text{for} \quad \sigma \ge 1- \frac1{4.896\log t}, \qquad t\ge3. } $$ The constant is substantially improved, but the asymptotic distance scale remains $$ 1/\log t. $$ Thus modern refinements of this positivity family improve constants inside the same logarithmic geometry rather than supplying a fixed-strip bootstrap. --- # 6. Why higher logarithmic derivatives do not repair the wall Define $$ \boxed{ F_m(s) = (-1)^m \frac{d^m}{ds^m} F(s). } $$ For $$ \Re s>1, $$ $$ \boxed{ F_m(s) = \sum_{n\ge1} \frac{ \Lambda(n)(\log n)^m }{ n^s }, } $$ so the Dirichlet coefficients remain nonnegative. Near a zero $\rho$, $$ F_m(s) $$ contains the amplified pole $$ \boxed{ -\frac{ m! }{ (s-\rho)^{m+1} }. } $$ This looks promising because the target zero pole becomes high order. However, the zero-side sign structure changes. For $$ s-\rho = a+ib, \qquad a>0, $$ $$ \boxed{ \Re \frac1{(a+ib)^{m+1}} = \frac{ \cos \left( (m+1)\arctan(b/a) \right) }{ (a^2+b^2)^{(m+1)/2} }. } $$ When $$ m=0, $$ the cosine is positive for every real $b$. When $$ m\ge1, $$ the cosine changes sign as $b/a$ varies. For example, take $$ \arctan(b/a) = \frac{\pi}{m+2}. $$ Then $$ \cos \left( \frac{(m+1)\pi}{m+2} \right) <0. $$ Thus: ## Theorem 6.1 — Higher-derivative zero-side sign-loss theorem For every $$ m\ge1, $$ the real zero kernel associated with $F_m$ is not one-signed in the half-plane to the right of the zeros. Therefore the classical step ```text keep the target zero drop all other zero contributions by sign ``` is no longer valid. Create: ```text O-RH-140 HIGHER_LOG_DERIVATIVE_POLE_AMPLIFICATION_DESTROYS_ZERO_SIDE_ONE_SIGN_STRUCTURE CERTIFIED ``` Higher derivatives amplify the target pole but simultaneously remove the positivity mechanism needed to isolate it. --- # 7. Relation to Turan detectors One can attempt to recover target isolation from higher derivatives using Turan power-sum selection. That is exactly the architecture audited in Paper 57. The result was: ```text target isolation is recovered, but at a fixed distance-exponent loss C, and the classical detector cannot amplify the seed when C>2. ``` Thus the two routes fit together: ```text m = 0: one-sign zero kernel, but logarithmic-distance wall. m >= 1: strong pole amplification, but zero-side sign is lost. Turan recovery: restores target sensitivity, but pays a non-amplifying detector constant. ``` There is no missing free derivative trick between Papers 57 and 58. --- # 8. Raw prime-measure positivity is boundary-blind Let $$ H=N^\alpha $$ and $$ U_H(x) = \psi(x+H)-\psi(x)-H. $$ Since $\Lambda(n)\ge0$, $$ \boxed{ \psi(x+H)-\psi(x) \ge0. } $$ Equivalently, $$ U_H(x)\ge-H. $$ A boundary zero under PESC $(\kappa)$ contributes at scale $$ \boxed{ |U_H(x)| \asymp H N^{-\kappa/2} } $$ in the critical model. Since $$ N^{-\kappa/2} =o(1), $$ $$ HN^{-\kappa/2} \ll H. $$ Therefore the positivity lower bound $$ U_H(x)\ge-H $$ is compatible with the entire boundary-mode oscillation. Create: ```text O-RH-141 RAW_PRIME_MEASURE_POSITIVITY_IS_BLIND_TO_SEEDED_BOUNDARY_MODE CERTIFIED_AS_SCALE_BARRIER ``` Prime positivity controls order-one relative negativity. The required strip amplifier lives at a vanishing relative scale. --- # 9. Positivity of the xi logarithmic derivative does not give global exclusion The completed zeta function satisfies $$ \boxed{ \frac{\xi'}{\xi}(s) = \sum_\rho \frac1{s-\rho} } $$ in the standard paired sense. If $$ \Re s>\beta_*, $$ every individual Poisson kernel has positive real part, so $$ \Re \frac{\xi'}{\xi}(s) >0. $$ This is simply positivity to the right of the complete zero set. It contains no mechanism for moving $\beta_*$. Recent work by Grigutis and Turcinskas studies positivity of $$ \Re \xi'/\xi $$ near the critical line. Their 2026 analysis explicitly considers hypothetical off-critical zeros and finds that positivity may persist away from relatively small neighborhoods of those zeros. This is external evidence for the internal conclusion: ```text xi-log-derivative positivity is compatible with off-critical zeros and is not itself a global strip-gap theorem. ``` --- # 10. Heath-Brown exact recoupling is unchanged by seeding Paper 47 certified the exact cross- $j$ recombination of the Heath-Brown identity. After all $j$ -levels are retained before squaring and the inherited output range is respected, $$ \boxed{ \sum_j c_j H_j = \Lambda } $$ on the relevant root range. The translated lag operator is linear. Therefore, if $$ X_j $$ denotes the lag vector of component $H_j$, $$ \boxed{ X_\Lambda = \sum_j c_jX_j. } $$ The seed PESC exponent does not change this identity. It only changes the scale against which the component and joint estimates are judged. --- # 11. Componentwise estimates cannot create a joint fixed power for free Assume there are only $$ N^{o(1)} $$ effective $j$ -levels / dyadic component cells after the fixed Heath-Brown order and dyadic partition are accounted for. Suppose every component is controlled only at the seed lag scale: $$ \boxed{ \|X_j\|_2^2 \ll NH^2N^{-\kappa+o(1)}. } $$ Then triangle inequality gives $$ \begin{aligned} \|X_\Lambda\|_2 &\le \sum_j |c_j| \|X_j\|_2 \\ &\ll N^{o(1)} \left( NH^2N^{-\kappa} \right)^{1/2}. \end{aligned} $$ Hence $$ \boxed{ \|X_\Lambda\|_2^2 \ll NH^2N^{-\kappa+o(1)}. } $$ No fixed improvement occurs. Therefore: ## Theorem 11.1 — Seed-scale componentwise Heath-Brown no-amplification theorem A finite or $N^{o(1)}$ Heath-Brown decomposition whose components are controlled only through separate seed-scale norm bounds cannot yield a strict MLEPG exponent $$ \delta>\kappa. $$ Create: ```text O-RH-142 COMPONENTWISE_HEATH_BROWN_SEED_SCALE_BOUNDS_CANNOT_CREATE_FIXED_POWER_AMPLIFICATION CERTIFIED ``` This is independent of the detailed form of the component bounds. --- # 12. The exact missing Heath-Brown theorem is joint Expand $$ \boxed{ \|X_\Lambda\|_2^2 = \sum_{j,k} c_j \overline{c_k} \langle X_j,X_k \rangle. } $$ A strict improvement at the component natural scale would follow from a genuinely joint inequality such as $$ \boxed{ \left\| \sum_j c_jX_j \right\|_2^2 \le N^{-\eta} \sum_j |c_j|^2 \|X_j\|_2^2 } $$ for some fixed $$ \eta>0, $$ or from another estimate of equivalent fixed-power strength. This is a covariance theorem. It is not contained in the Heath-Brown identity. Paper 47 already proved that the exact alternating signs restore the original $\Lambda$ object rather than automatically creating a power saving. Papers 43–46 further certified that: - isolated Liouville components; - Ramaré extraction; - prime-harmonic leverage; - structured-weight $L^2$ smallness; and - factor-by-factor determinant geometry do not supply the missing fixed power. Thus SG4B is not "apply Heath-Brown again." It is: ```text prove a new seeded joint cross-j covariance theorem. ``` No such theorem is certified. --- # 13. Boundary-mode interpretation of the joint covariance problem Suppose the root error contains a seed-boundary mode at exponent $$ \beta = 1-\frac{\kappa}{2}. $$ Paper 55 showed that its lag energy is $$ NH^2N^{-\kappa} $$ and is concentrated on the principal Fejer arc. Since the complete Heath-Brown sum reconstructs $\Lambda$, the component covariance matrix must reconstruct that boundary mode as well. Therefore a fixed-power joint covariance deficit would imply that the complete reconstructed boundary mode is smaller than the seed scale. By Papers 54–55, that is already a zero-strip improvement. Hence: ## Corollary 13.1 — Joint covariance improvement is genuine strip-gap arithmetic Any seeded Heath-Brown joint inequality strong enough to yield $$ \delta>\kappa $$ is itself a new fixed zero-strip theorem. It cannot be justified solely by the formal alternating signs of the identity. This is a calibration, not a circularity objection. --- # 14. SG4A verdict: prime support and positivity The original SG4A candidate was: ```text PRIME_SUPPORT_AND_POSITIVITY_VERSUS_SMOOTH_BOUNDARY_MODE ``` Verdict: ```text STANDARD POSITIVITY ROUTE CLOSED AS NON-AMPLIFYING. ``` Reasons: 1. raw prime-measure positivity only controls the relative scale $1$ ; 2. the boundary signal has relative scale $N^{-\kappa/2}$ ; 3. Euler-product positivity in $\Re s>1$ has only logarithmic zero-distance resolution; 4. near-critical xi positivity is compatible with hypothetical off-critical zeros away from local neighborhoods. A new positivity theorem would need quantitative information beyond coefficient nonnegativity. --- # 15. SG4B verdict: seeded Heath-Brown bilinear route The original SG4B candidate was: ```text SEEDED_HEATH_BROWN_BILINEAR_MAJOR_ARC_INCOMPATIBILITY ``` Verdict: ```text EXISTING COMPONENTWISE ROUTE CLOSED. ``` The missing theorem is explicitly joint: $$ \boxed{ \text{fixed-power cross-j covariance deficit}. } $$ The seed does not manufacture it. Record: ```text SG4B-JOINT OPEN ``` as a possible but high-cost arithmetic subfrontier. --- # 16. SG4C verdict: nonlinear Euler-product boundary detector The original SG4C candidate was: ```text NONLINEAR_EULER_PRODUCT_BOUNDARY_MODE_DETECTOR ``` The standard candidates are exhausted: ```text first log derivative: logarithmic-distance wall higher log derivatives: zero-side sign loss Turan derivative selection: Paper 57 detector saturation xi log-derivative positivity: compatible with hypothetical off-line zeros ``` Therefore close the standard SG4C architecture as: ```text CLOSED_AS_NO_STANDARD_EULER_POSITIVITY_OR_DERIVATIVE_ROUTE_CROSSES_THE_SEEDED_FIXED_BOUNDARY ``` A genuinely nonlinear Euler-product theorem remains logically possible but would require new information not contained in ordinary coefficient positivity. --- # 17. SG4D status SG4D was: ```text MULTISCALE_PRIME_FACTOR_COHERENCE_BREAKING ``` No certified theorem currently separates the von Mangoldt sequence from the critical boundary mode at a fixed-power level. This remains the only genuinely open structural-arithmetic direction inside SG4. However, it is not yet as concrete as F-RH-017. Therefore Campaign 46 should not create a new canonical frontier merely to rename the gap. The preferred direct target remains: ```text F-RH-017 SEEDED_SUPERCRITICAL_SHRINKING_THRESHOLD_EXCEPTIONAL_SET ``` because it has explicit parameters and a certified amplification map. The secondary detector target remains: ```text F-RH-018 SEEDED_TURAN_GALLAGHER_DETECTOR_EXCESS_POWER ``` but is quantitatively more expensive. --- # 18. Current external calibration ## 18.1. 2026 Heath-Brown-inspired zero-free region Chiara Bellotti, Tim Trudgian and Andrew Yang, *Zero-free regions inspired by work of Heath-Brown*, arXiv:2603.21490. They prove $$ \zeta(\sigma+it)\ne0 $$ for $$ t\ge3, \qquad \sigma \ge 1-\frac1{4.896\log t}. $$ The result substantially improves the constant in the classical logarithmic region but does not alter its $1/\log t$ shape. URL: https://arxiv.org/abs/2603.21490 ## 18.2. Positivity of the xi log derivative Andrius Grigutis and Lukas Turcinskas, *Note on the positivity of the real part of the log-derivative of the Riemann xi-function near the critical line*, Lithuanian Mathematical Journal, 2026. The paper studies positivity near the critical line and explicitly considers scenarios with hypothetical off-critical zeros. Positivity can persist away from local neighborhoods of such zeros. URL: https://arxiv.org/abs/2509.18963 These results are used only as calibration of method scope. --- # 19. Campaign 46 structural status After Papers 56–58: ```text SG1 principal-arc standard structure: CLOSED arithmetic suppression: OPEN SG2 seeded shrinking-threshold bridge: CERTIFIED F-RH-017: OPEN SG3 standard Turan-Gallagher / Mellin detector: CLOSED AS NON-AMPLIFYING F-RH-018: OPEN SECONDARY SG4A standard positivity: CLOSED AS NON-AMPLIFYING SG4B componentwise Heath-Brown: CLOSED joint covariance theorem: OPEN HIGH-COST SG4C standard Euler-product derivative positivity: CLOSED SG4D new nonlinear prime coherence: OPEN / UNSPECIFIED ``` The campaign has reached the point where further formal architecture is unlikely to reduce the gap. --- # 20. State transition Advance the candidate state from $$ v1.48 $$ to $$ v1.49. $$ Add: ```text O-RH-139 POSITIVE_COEFFICIENT_EULER_PRODUCT_ZERO_DROPPING_HAS_INTRINSIC_LOGARITHMIC_DISTANCE_WALL CERTIFIED_AS_METHOD_BARRIER ``` Add: ```text O-RH-140 HIGHER_LOG_DERIVATIVE_POLE_AMPLIFICATION_DESTROYS_ZERO_SIDE_ONE_SIGN_STRUCTURE CERTIFIED ``` Add: ```text O-RH-141 RAW_PRIME_MEASURE_POSITIVITY_IS_BLIND_TO_SEEDED_BOUNDARY_MODE CERTIFIED_AS_SCALE_BARRIER ``` Add: ```text O-RH-142 COMPONENTWISE_HEATH_BROWN_SEED_SCALE_BOUNDS_CANNOT_CREATE_FIXED_POWER_AMPLIFICATION CERTIFIED ``` No new root theorem and no RH certificate is created. --- # 21. Recommended next action The most economical remaining theorem is still F-RH-017. Choose $$ H=N^{1-\tau} $$ with fixed small $\tau>0$. Prove $$ \boxed{ \#\left\{ x\in[N,2N]: |\psi(x+H)-\psi(x)-H| > HN^{-\nu} \right\} \ll N^{1-c} } $$ with $$ \boxed{ \nu>\frac{\kappa}{2}, \qquad c>2\tau. } $$ Paper 56 then converts this directly into $$ \operatorname{PESC}(\kappa+\eta) $$ for an explicit fixed $\eta>0$. This target is weaker and more concrete than: - full natural Selberg variance; - a large-constant Turan detector excess; - a new joint Heath-Brown covariance theorem; or - an unspecified nonlinear Euler-product principle. The next mathematical campaign should therefore attack F-RH-017 directly rather than generating another representation. --- # 22. Conclusion The standard positivity idea has now been separated into its real strengths and its real limits. Positive von Mangoldt coefficients are powerful enough to prove nonvanishing near $\Re s=1$. They are not fine enough to see a seeded boundary at fixed distance. Higher derivatives increase pole sensitivity but lose one-sign zero geometry. Raw positivity of prime counts is blind to a vanishing relative boundary oscillation. Heath-Brown factorization exposes arithmetic structure but, after exact cross- $j$ recoupling, does not create a fixed covariance deficit by formal algebra alone. Thus Campaign 46 has exhausted its standard structural mechanisms. The missing theorem is no longer hidden. It is a genuinely new prime-side excess estimate beyond the critical seed scale. Among the currently certified formulations, F-RH-017 remains the cheapest direct target.