# CSM_RH Paper 34 ## Rank-One Mellin Symbols, Weighted Covariance Persistence, and Failure of Low-Frequency Scalar Reduction **Project:** `CSM_RH` **Paper:** `34` **Version:** `v0.1` **Date:** `2026-09-07` **Parent state:** `CSM_RH v1.24 / Paper 33` **Campaign:** `33 — PESC_LOW_FREQUENCY_RANK_ONE_ATTACK` **Status:** low-frequency scalar / covariance decomposition audit; not a proof or disproof of RH --- # 0. Trust boundary This paper does not prove or disprove the Riemann Hypothesis. ```text RH_PROVED = FALSE RH_DISPROVED = FALSE GLOBAL_RH_CERTIFICATE = FALSE CSM_RH_ROOT_STATUS = OPEN ``` Paper 33 exposed the rank-one scalar $$ \mathcal R_N^{(1)} = \frac{H_NG_N}{W_N} $$ after mean-zero recentering of the PESC perturbation. Campaign 33 asks whether this scalar is a lower-strength low-frequency target which can be controlled more easily than full PESC. The answer is negative. The rank-one term is genuinely sensitive to fixed Mellin drift modes, but the centered covariance remainder retains the same polynomial exponent for every fixed smooth drift $x^\beta$ with $0<\beta<1$. Thus the decomposition changes coefficients, not the fixed-power exponent class. No new frontier is created. --- # 1. Weighted PESC decomposition Let $$ B(j) = \sum_{n\le j}c_n $$ and define the canonical endpoint weight $$ w_N(n) = \begin{cases} N,&n\le N,\\ 2N-n,&N-1, $$ a direct calculation gives: ## Theorem 2.1 — Endpoint Mellin Symbol $$ \boxed{ I(s) = \frac{ 2^{s+2}-1 }{ (s+1)(s+2) }. } $$ In particular, $$ \boxed{ I(0)=\frac32. } $$ --- # 3. Dyadic first-primitive symbol Test a Mellin mode $$ \boxed{ B(x)=c x^\rho, } $$ with $$ 0<\beta=\Re\rho<1. $$ The first primitive observable is $$ G_N = \sum_{j=N}^{2N-1}B(j). $$ At exponent level, $$ \boxed{ G_N = c g(\rho) N^{\rho+1} + O_\rho(N^\beta), } $$ where $$ \boxed{ g(\rho) = \int_1^2u^\rho\,du = \frac{ 2^{\rho+1}-1 }{ \rho+1 }. } $$ For $0<\Re\rho<1$, $$ \boxed{ g(\rho)\ne0. } $$ Indeed, $$ |2^{\rho+1}| = 2^{\beta+1} > 1. $$ Thus the numerator cannot vanish. --- # 4. Second-integrated endpoint symbol The rank-one factor $H_N$ has exponent-level form $$ \boxed{ H_N = c h(\rho) N^{\rho+2} + O_\rho(N^{\beta+1}), } $$ where $$ \boxed{ h(\rho) = I(\rho) = \frac{ 2^{\rho+2}-1 }{ (\rho+1)(\rho+2) }. } $$ Again, $$ \boxed{ h(\rho)\ne0 } $$ throughout the critical strip because $$ |2^{\rho+2}| = 2^{\beta+2} > 1. $$ Therefore neither linear factor of the rank-one scalar has a Mellin blind spot in the critical strip. --- # 5. Relation with the standard integrated explicit formula For the von-Mangoldt error $$ E(x)=\psi(x)-x, $$ define $$ \psi_1(x) = \int_0^x\psi(t)\,dt. $$ The classical explicit formula is $$ \boxed{ \psi_1(x) = \frac{x^2}{2} - \sum_\rho \frac{ x^{\rho+1} }{ \rho(\rho+1) } + \text{elementary lower-order terms}. } $$ The zero series converges absolutely because $$ \sum_\rho|\rho|^{-2}<\infty. $$ Thus integration genuinely smooths the zero packet while preserving every zero exponent. A standard isolated-off-line-zero calibration gives $$ \boxed{ \psi_1(x) = \frac{x^2}{2} + C_\rho x^{1+\beta} \cos ( \gamma\log x+\phi_\rho ) + O(x^{3/2}) } $$ when one zero pair $\beta\pm i\gamma$, $\beta>1/2$, is the only pair to the right of the critical line. This is classical calibration, not a hypothesis used to prove RH. --- # 6. Rank-one response to one conjugate mode Let the real drift associated with one conjugate pair be $$ B_\rho(x) = c x^\rho + \overline c x^{\overline\rho}. $$ Then $$ G_N = 2 |c g(\rho)| N^{\beta+1} \cos ( \gamma\log N+\phi_g ) + o(N^{\beta+1}), $$ and $$ H_N = 2 |c h(\rho)| N^{\beta+2} \cos ( \gamma\log N+\phi_h ) + o(N^{\beta+2}). $$ Since $$ W_N = \frac32N^2 + O(N), $$ the rank-one scalar is $$ \boxed{ \mathcal R_N^{(1)} = \frac83 |c|^2 |g(\rho)h(\rho)| N^{2\beta+1} \cos(\theta_N+\phi_g) \cos(\theta_N+\phi_h) + o(N^{2\beta+1}), } $$ where $$ \theta_N=\gamma\log N. $$ --- # 7. No phase escape for a single mode For any real phase difference $\Delta$, $$ \cos\theta\cos(\theta+\Delta) = \frac12 [ \cos(2\theta+\Delta) + \cos\Delta ]. $$ Therefore ## Theorem 7.1 — Product-Phase Maximum $$ \boxed{ \max_\theta | \cos\theta\cos(\theta+\Delta) | = \frac{ 1+|\cos\Delta| }{2} \ge \frac12. } $$ Hence the rank-one response of one nonzero conjugate Mellin mode is not identically small. For continuous scale $N$ there are arbitrarily large scales with $$ \boxed{ |\mathcal R_N^{(1)}| \gg_\rho N^{2\beta+1}. } $$ Integer-scale approximation preserves the same exponent. Thus a uniform fixed-power rank-one upper bound already has isolated-zero fixed-strip strength. Create: ```text O-RH-075 RANK_ONE_ISOLATED_MELLIN_MODE_FIXED_STRIP_LOCK status: CERTIFIED AS FIXED-MODE STRENGTH AUDIT ``` This is not a full multiple-zero coefficient-isolation theorem. --- # 8. Real smooth power drift Now take the nonoscillatory strength model $$ \boxed{ B(x)=x^\beta, \qquad 0<\beta<1. } $$ Then $$ c_n = B(n)-B(n-1) = \beta n^{\beta-1} + O_\beta(n^{\beta-2}). $$ The endpoint Mellin symbol gives: ## Total PESC $$ \boxed{ \mathcal C_N = K(\beta) N^{2\beta+1} + o(N^{2\beta+1}), } $$ with $$ \boxed{ K(\beta) = \beta I(2\beta-1) = \frac{ 2^{2\beta+1}-1 }{ 2(2\beta+1) }. } $$ --- # 9. Rank-one coefficient for a smooth drift We have $$ H_N = I(\beta) N^{\beta+2} + o(N^{\beta+2}), $$ $$ G_N = \beta I(\beta-1) N^{\beta+1} + o(N^{\beta+1}), $$ and $$ W_N = I(0)N^2 + o(N^2). $$ Therefore ## Theorem 9.1 — Smooth-Drift Rank-One Coefficient $$ \boxed{ \mathcal R_N^{(1)} = R(\beta) N^{2\beta+1} + o(N^{2\beta+1}), } $$ where $$ \boxed{ R(\beta) = \frac{ \beta I(\beta)I(\beta-1) }{ I(0) }. } $$ Thus the rank-one term has exactly the same polynomial exponent as full PESC. --- # 10. Centered covariance coefficient Define $$ \boxed{ Q(\beta) = K(\beta)-R(\beta). } $$ Then the centered covariance term satisfies $$ \boxed{ \operatorname{Cov}_{w_N}(c,B) = Q(\beta) N^{2\beta+1} + o(N^{2\beta+1}). } $$ The two pieces therefore share the same exponent. --- # 11. Strict sign of the covariance Under the positive measure $$ d\mu(u) = \omega(u)\,du, $$ the two functions $$ u^\beta $$ and $$ \beta u^{\beta-1} $$ have opposite monotonicity for $$ 0<\beta<1. $$ The first is strictly increasing. The second is strictly decreasing. The weighted covariance inequality therefore gives: ## Theorem 11.1 — Smooth-Drift Covariance Persistence For every fixed $$ 0<\beta<1, $$ $$ \boxed{ Q(\beta)<0. } $$ In particular, $$ \boxed{ Q(\beta)\ne0. } $$ Thus mean-zero recentering does not eliminate the fixed-power smooth mode from the covariance sector. Create: ```text O-RH-076 CENTERED_COVARIANCE_RETAINS_SMOOTH_DRIFT_EXPONENT status: CERTIFIED ``` --- # 12. Near-linear limit At $$ \beta=1, $$ the derivative of the drift is constant. The weighted covariance therefore vanishes exactly: $$ \boxed{ Q(1)=0. } $$ As $$ \beta\uparrow1, $$ $$ \boxed{ Q(\beta)\to0. } $$ Meanwhile $$ R(\beta)\to K(\beta). $$ Thus the rank-one term captures an increasingly large fraction of a near-linear drift. This explains why the Section-6 zeroth-mode leakage found in Paper 33 is a natural low-frequency obstruction. But for every fixed $\beta<1$, the covariance coefficient remains nonzero and the exponent remains $2\beta+1$. --- # 13. Rank-one is not a complete PESC coordinate The decomposition $$ \mathcal C_N = \operatorname{Cov}_{w_N}(c,B) + \mathcal R_N^{(1)} $$ does not reduce PESC to the rank-one scalar. For a fixed smooth mode: ```text rank-one term: N^(2 beta+1) centered covariance: N^(2 beta+1) full PESC: N^(2 beta+1) ``` Only the coefficients differ. Therefore controlling the rank-one scalar alone does not supply a fixed-power PESC theorem. Create: ```text O-RH-077 RANK_ONE_SCALAR_NOT_A_COMPLETE_PESC_COORDINATE status: CERTIFIED BY SMOOTH-DRIFT COUNTERMODEL ``` --- # 14. Current sieve control of the covariance remains subpower The centered perturbation has zero weighted mean. This removes the classical zeroth-mode leakage from the coupled-sieve input. However the covariance term remains an endogenous signed prime correlation. Current sieve / higher-uniformity technology gives logarithmic or subpower precision for the relevant prime residual inputs. Since the covariance smooth-mode contribution has the same $N^{2\beta+1}$ exponent, arbitrary logarithmic control does not produce a fixed PESC exponent. Thus mean-zero sieve stability does not close the covariance sector. --- # 15. Fixed-power bounds on the linear factors The symbols $$ g(\rho) $$ and $$ h(\rho) $$ have no zeros in the critical strip. Therefore fixed-power control of either linear observable over all scales would be strongly zero-sensitive. The classical explicit formula for the first integrated prime error confirms this directly for isolated off-line zeros. Thus the factors $$ G_N, \qquad H_N $$ are not cheap low-frequency quantities whose fixed-power bounds are known independently of the PESC problem. --- # 16. Multiple-zero caveat For a general zero packet, $G_N$ and $H_N$ are log-scale exponential sums. Their product may have substantial phase cancellation at individual scales. The present campaign does not prove a universal coefficient-isolation theorem for the rank-one product. This is another reason not to promote $$ \mathcal R_N^{(1)} $$ to a canonical frontier. The fixed-mode audit is sufficient to show that the scalar is not obviously lower-strength. --- # 17. Campaign 33 track audit ## ZM1 — first/second primitive Mellin analysis ```text status: COMPLETE AT FIXED-MODE LEVEL symbols: g(rho), h(rho) critical-strip zeros: none ``` ## ZM2 — scalar-product cancellation ```text status: NO UNIFORM FREE CANCELLATION single mode: Omega(N^(2 beta+1)) on subsequences ``` ## ZM3 — dyadic phase variation ```text status: PRODUCT OSCILLATES phase zeros: possible at isolated scales uniform fixed-power escape: not available ``` ## ZM4 — arithmetic control of first primitive ```text status: CURRENT UNCONDITIONAL CONTROL SUBPOWER fixed-power control: zero-sensitive / unavailable ``` ## ZM5 — direct covariance decomposition ```text status: EXACT centered covariance: retains same smooth-drift exponent ``` --- # 18. Campaign 33 verdict The rank-one scalar is useful diagnostically but is not a lower-strength replacement for PESC. The low-frequency decomposition produces: ```text rank-one mean product: captures most near-linear drift mean-zero covariance: smaller coefficient near beta=1 but same fixed polynomial exponent fixed-power theorem: still absent ``` No new frontier is created. The canonical root remains PESC. --- # 19. New certified package Create: ```text B-RH-010 PESC_WEIGHTED_COVARIANCE_RANK_ONE_DECOMPOSITION CERTIFIED O-RH-075 RANK_ONE_ISOLATED_MELLIN_MODE_FIXED_STRIP_LOCK CERTIFIED AS FIXED-MODE STRENGTH AUDIT O-RH-076 CENTERED_COVARIANCE_RETAINS_SMOOTH_DRIFT_EXPONENT CERTIFIED O-RH-077 RANK_ONE_SCALAR_NOT_A_COMPLETE_PESC_COORDINATE CERTIFIED ``` --- # 20. Canonical status ```text F-RH-010 PESC OPEN / ROOT TARGET F-RH-016 MLEPG OPEN / DIRECT THEOREM CANDIDATE rank-one scalar: diagnostic only mean-zero covariance: diagnostic subcomponent ``` --- # 21. Campaign 34 The next campaign is: ```text CSM_RH Campaign 34 FINITE_RANK_LOW_FREQUENCY_PROJECTION_AUDIT ``` This campaign tests one final natural extension of the rank-one idea: > can projecting out several low-frequency polynomial/Mellin moments make the remaining signed covariance genuinely easier at fixed exponent? No new frontier may be created unless the projection produces a proved fixed-power bridge weaker than PESC. --- # 22. Campaign 34 tracks ## FR1 — polynomial moment projection Project the cumulative error against $$ 1,u,u^2,\ldots,u^r $$ under the endpoint weight. Compute the response of a drift $u^\beta$. ## FR2 — finite-rank Mellin symbols Use a finite family of scale-local Mellin windows. Determine whether any fixed rank can change the exponent $2\beta+1$, rather than only its coefficient. ## FR3 — growing-rank threshold If fixed rank only creates coefficient zeros at $\beta=1$, determine the rank growth needed to turn coefficient suppression into an $X$ -power. Track condition number and arithmetic complexity. ## FR4 — sieve after multi-moment recentering Test whether eliminating several signed low moments removes all classical asymptotic-sieve leakage terms or merely exports them into a finite-rank correction matrix. ## FR5 — direct comparison with higher-order Selberg Determine whether finite-rank recentering is mathematically distinct from the already-audited higher-order generalized Selberg amplifier. --- # 23. Campaign 34 rejection filters Reject a candidate if: ## R1. Fixed rank changes only coefficients, not exponents. ## R2. Growing rank reaches the same log X/log log X complexity threshold as Paper 30 without new uniformity. ## R3. The projection uses the unknown zero set. ## R4. The finite-rank correction matrix contains PESC-scale Mellin modes. ## R5. The result is merely another representation of the same low-frequency drift. ## R6. It assumes fixed-power integrated PNT estimates. --- # 24. External calibration The fixed-mode calibration is consistent with standard explicit-formula theory. The first integrated Chebyshev function has an absolutely convergent zero expansion: $$ \psi_1(x) = \frac{x^2}{2} - \sum_\rho \frac{x^{\rho+1}}{\rho(\rho+1)} + \text{lower-order terms}. $$ A single hypothetical zero pair with real part $\beta>1/2$ produces a term of size $$ x^{1+\beta}\cos(\gamma\log x). $$ Thus integrated low-frequency observables remain directly sensitive to off-critical zeros; integration damps high ordinates but does not improve the real exponent. --- # 25. State transition ```text CSM_RH v1.24 -> CSM_RH v1.25 ``` with: ```text Campaign 33 CLOSED_AS_RANK_ONE_AND_COVARIANCE_STRENGTH_AUDIT B-RH-010 PESC_WEIGHTED_COVARIANCE_RANK_ONE_DECOMPOSITION CREATED / CERTIFIED O-RH-075 RANK_ONE_ISOLATED_MELLIN_MODE_FIXED_STRIP_LOCK CREATED / CERTIFIED AS FIXED-MODE STRENGTH AUDIT O-RH-076 CENTERED_COVARIANCE_RETAINS_SMOOTH_DRIFT_EXPONENT CREATED / CERTIFIED O-RH-077 RANK_ONE_SCALAR_NOT_A_COMPLETE_PESC_COORDINATE CREATED / CERTIFIED F-RH-010 PESC REMAINS OPEN / ROOT TARGET Campaign 34 FINITE_RANK_LOW_FREQUENCY_PROJECTION_AUDIT READY ``` --- # 26. Final status ```text RH = OPEN PESC = OPEN MLEPG = OPEN RANK-ONE LOW-FREQUENCY SCALAR = DIAGNOSTIC RANK-ONE FIXED-MODE RESPONSE = FIXED-STRIP SCALE MEAN-ZERO COVARIANCE = SAME POWER EXPONENT ON SMOOTH DRIFT LOW-FREQUENCY SPLIT = COEFFICIENT REARRANGEMENT, NOT EXPONENT REDUCTION NEXT CAMPAIGN = 34 ``` The central coefficient law is $$ \boxed{ \mathcal C_N = [ Q(\beta)+R(\beta) ] N^{2\beta+1} + o(N^{2\beta+1}), } $$ with $$ \boxed{ Q(\beta)<0, \qquad R(\beta)>0, \qquad 0<\beta<1. } $$ Both pieces preserve the same fixed-power exponent. The rank-one projection therefore isolates a low-frequency component, but it does not reduce the RH-strength exponent class.