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.

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<n<2N,\\ 0,&n\ge2N. \end{cases}

Define

\boxed{ W_N = \sum_{n<2N}w_N(n) = \frac{3N^2-N}{2}. }

Define

\boxed{ H_N = \sum_{n<2N} w_N(n)B(n-1), }

and

\boxed{ G_N = \sum_{n<2N} w_N(n)c_n = \sum_{j=N}^{2N-1}B(j). }

The canonical PESC self-correlation is

\boxed{ \mathcal C_N = \sum_{n<2N} w_N(n)c_nB(n-1). }

Define weighted means

\overline B_N = \frac{H_N}{W_N},

\overline c_N = \frac{G_N}{W_N}.

Then:

Theorem 1.1 — Exact Weighted Covariance Decomposition

\boxed{ \mathcal C_N = \sum_{n<2N} w_N(n) [ c_n-\overline c_N ] [ B(n-1)-\overline B_N ] + \frac{ H_NG_N }{ W_N }. }

The first term is the weighted centered covariance.

The second term is the rank-one mean product from Paper 33.

Create:

B-RH-010
PESC_WEIGHTED_COVARIANCE_RANK_ONE_DECOMPOSITION
status:
  CERTIFIED

2. Continuous endpoint kernel

At exponent resolution, scale

n=Nu.

The endpoint weight becomes

w_N(n) = N\omega(u) + O(1),

where

\boxed{ \omega(u) = \begin{cases} 1,&0<u\le1,\\ 2-u,&1<u<2,\\ 0,&u\ge2. \end{cases} }

Define its Mellin moment

\boxed{ I(s) = \int_0^2 \omega(u)u^s\,du. }

For

\Re s>-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:

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:

O-RH-076
CENTERED_COVARIANCE_RETAINS_SMOOTH_DRIFT_EXPONENT
status:
  CERTIFIED

12. Near-linear limit

\beta=1,

the derivative of the drift is constant.

The weighted covariance therefore vanishes exactly:

\boxed{ Q(1)=0. }

\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:

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:

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

status:
  COMPLETE AT FIXED-MODE LEVEL

symbols:
  g(rho), h(rho)

critical-strip zeros:
  none

ZM2 — scalar-product cancellation

status:
  NO UNIFORM FREE CANCELLATION

single mode:
  Omega(N^(2 beta+1)) on subsequences

ZM3 — dyadic phase variation

status:
  PRODUCT OSCILLATES

phase zeros:
  possible at isolated scales

uniform fixed-power escape:
  not available

ZM4 — arithmetic control of first primitive

status:
  CURRENT UNCONDITIONAL CONTROL SUBPOWER

fixed-power control:
  zero-sensitive / unavailable

ZM5 — direct covariance decomposition

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:

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:

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

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:

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

CSM_RH v1.24
  ->
CSM_RH v1.25

with:

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

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.

CSM_RH Paper 34 — Rank-One Mellin Symbols, Weighted Covariance Persistence, and Failure of Low-Frequency Scalar Reduction

CSM_RH Paper 34

Rank-One Mellin Symbols, Weighted Covariance Persistence, and Failure of Low-Frequency Scalar Reduction

0. Trust boundary

1. Weighted PESC decomposition

Theorem 1.1 — Exact Weighted Covariance Decomposition

2. Continuous endpoint kernel

Theorem 2.1 — Endpoint Mellin Symbol

3. Dyadic first-primitive symbol

4. Second-integrated endpoint symbol

5. Relation with the standard integrated explicit formula

6. Rank-one response to one conjugate mode

7. No phase escape for a single mode

Theorem 7.1 — Product-Phase Maximum

8. Real smooth power drift

Total PESC

9. Rank-one coefficient for a smooth drift

Theorem 9.1 — Smooth-Drift Rank-One Coefficient

10. Centered covariance coefficient

11. Strict sign of the covariance

Theorem 11.1 — Smooth-Drift Covariance Persistence

12. Near-linear limit

13. Rank-one is not a complete PESC coordinate

14. Current sieve control of the covariance remains subpower

15. Fixed-power bounds on the linear factors

16. Multiple-zero caveat

17. Campaign 33 track audit

ZM1 — first/second primitive Mellin analysis

ZM2 — scalar-product cancellation

ZM3 — dyadic phase variation

ZM4 — arithmetic control of first primitive

ZM5 — direct covariance decomposition

18. Campaign 33 verdict

19. New certified package

20. Canonical status

21. Campaign 34

22. Campaign 34 tracks

FR1 — polynomial moment projection

FR2 — finite-rank Mellin symbols

FR3 — growing-rank threshold

FR4 — sieve after multi-moment recentering

FR5 — direct comparison with higher-order Selberg

23. Campaign 34 rejection filters

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

25. State transition

26. Final status