CSM_RH Paper 35

Finite-Rank Low-Frequency Projection Invariance and the Polynomial-Rank Fixed-Power Threshold

Project: CSM_RH
Paper: 35
Version: v0.1
Date: 2026-09-07
Parent state: CSM_RH v1.25 / Paper 34
Campaign: 34 — FINITE_RANK_LOW_FREQUENCY_PROJECTION_AUDIT
Status: finite-rank projection closure / growing-rank complexity 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 34 showed that rank-one mean subtraction changes the coefficient structure of a smooth low-frequency drift but not its fixed-power exponent.

Campaign 34 asks whether projecting out several scale-local low-frequency moments can do better.

The answer is:

fixed rank:
  coefficient rearrangement only

fixed-rank correction:
  same PESC exponent on a Mellin drift

polynomial basis:
  endpoint singularity gives only algebraic approximation in rank

fixed-power residual suppression by ordinary polynomials:
  requires polynomial rank in N

finite-rank projection:
  not a lower-strength PESC mechanism

No new frontier is created.

1. Discrete weighted Hilbert space

For each dyadic scale $N$ , let

\mathcal H_N = \mathbb C^{\{1,\ldots,2N-1\}}

with weighted inner product

\boxed{ \langle u,v\rangle_N = \sum_{n<2N} w_N(n) u_n\overline{v_n}. }

Let

c_n

be the centered prime detector and let

b_n = B(n-1)

be its cumulative-error coordinate.

Then PESC is

\boxed{ \mathcal C_N = \langle c,b\rangle_N. }

2. Polynomial low-frequency subspace

For a fixed integer

r\ge0,

define

v_j(n) = \left( \frac nN \right)^j, \qquad 0\le j\le r.

Let

V_{N,r} = \operatorname{span} \{ v_0,\ldots,v_r \}.

Let

P_{N,r}

be the orthogonal projection onto $V_{N,r}$ in $\mathcal H_N$ .

3. Exact finite-rank PESC decomposition

Orthogonality gives

Theorem 3.1 — Finite-Rank PESC Projection Identity

\boxed{ \mathcal C_N = \langle P_{N,r}c, P_{N,r}b \rangle_N + \langle (I-P_{N,r})c, (I-P_{N,r})b \rangle_N. }

The two cross terms vanish exactly.

Create:

B-RH-011
FINITE_RANK_PESC_PROJECTION_IDENTITY
status:
  CERTIFIED

For $r=0$ , this reduces to the rank-one weighted-mean decomposition of Paper 34.

4. Moment-matrix form of the finite-rank correction

Define the Gram matrix

\boxed{ G^{(r)}_{ij} = \langle v_i,v_j\rangle_N, \qquad 0\le i,j\le r. }

Define moment vectors

\boxed{ m_c^{(r)} = ( \langle c,v_0\rangle_N, \ldots, \langle c,v_r\rangle_N )^T, }

and

\boxed{ m_b^{(r)} = ( \langle b,v_0\rangle_N, \ldots, \langle b,v_r\rangle_N )^T. }

Then

Theorem 4.1 — Finite-Rank Correction Matrix

\boxed{ \langle P_{N,r}c, P_{N,r}b \rangle_N = (m_c^{(r)})^\ast (G^{(r)})^{-1} m_b^{(r)}. }

Thus removing several low-frequency moments does not eliminate their contribution.

It exports them into an explicit finite correction matrix.

5. Continuum endpoint measure

Scale

n=Nu.

The endpoint weight converges to

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

Define

\boxed{ \langle f,g\rangle_\omega = \int_0^2 \omega(u) f(u)\overline{g(u)} \,du. }

Let $P_r$ denote the corresponding orthogonal projection onto

\Pi_r = \operatorname{span} \{ 1,u,\ldots,u^r \}.

For fixed $r$ , the discrete projection converges to this scale-local projection at exponent resolution.

6. Smooth Mellin drift shapes

Let

B(x)=x^\beta, \qquad 0<\beta<1.

At scale $n=Nu$ ,

B(n) = N^\beta \phi_\beta(u),

where

\boxed{ \phi_\beta(u)=u^\beta. }

Its discrete derivative has leading shape

c_n = N^{\beta-1} \psi_\beta(u) + O(N^{\beta-2}),

where

\boxed{ \psi_\beta(u) = \beta u^{\beta-1}. }

7. Fixed-rank exponent invariance

The full PESC coefficient is

\langle \psi_\beta, \phi_\beta \rangle_\omega.

By orthogonal projection,

Theorem 7.1 — Fixed-Rank Mellin-Mode Split

\boxed{ \langle \psi_\beta, \phi_\beta \rangle_\omega = \langle P_r\psi_\beta, P_r\phi_\beta \rangle_\omega + \langle (I-P_r)\psi_\beta, (I-P_r)\phi_\beta \rangle_\omega. }

Returning to the dyadic arithmetic scale gives

\boxed{ \mathcal C_N = N^{2\beta+1} [ K_r^{\mathrm{low}}(\beta) + K_r^{\mathrm{high}}(\beta) ] + o(N^{2\beta+1}), }

for fixed $r$ .

Both coefficients depend on $r$ and $\beta$ but not on $N$ .

Therefore fixed-rank projection changes only coefficients.

It cannot change the exponent

\boxed{ 2\beta+1. }

Create:

O-RH-078
FIXED_RANK_LOW_FREQUENCY_PROJECTION_EXPONENT_INVARIANCE
status:
  CERTIFIED

8. Moment-vector exponent ledger

The same conclusion follows directly from the correction matrix.

For a smooth drift:

\boxed{ m_c^{(r)} = N^{\beta+1} [ \mathbf c_r(\beta)+o(1) ], }

\boxed{ m_b^{(r)} = N^{\beta+2} [ \mathbf b_r(\beta)+o(1) ], }

while

\boxed{ G^{(r)} = N^2 [ \mathbf G_r+o(1) ]. }

Therefore

\boxed{ (m_c^{(r)})^\ast (G^{(r)})^{-1} m_b^{(r)} = N^{2\beta+1} [ R_r(\beta)+o(1) ]. }

Every fixed number of low-frequency moment corrections remains PESC-scale.

9. No fixed finite basis can annihilate the Mellin continuum

The functions

u^\beta

with distinct real exponents are linearly independent on every interval contained in $(0,\infty)$ .

Therefore a fixed finite-dimensional polynomial or Müntz space cannot contain

u^\beta

for every

\beta

in an interval.

A projection could be chosen to annihilate one known exponent exactly by inserting that exponent into the basis.

But doing so would require prior knowledge of the drift / zero parameter and is disallowed as an RH proof mechanism.

Thus fixed rank cannot uniformly remove the continuum of possible off-axis Mellin modes.

10. Polynomial approximation route

Suppose the proof strategy attempts to make the projected residual uniformly small:

\boxed{ \| u^\beta-p_r(u) \|_{L^\infty([0,1])} \le \varepsilon_r }

for a polynomial $p_r$ of degree at most $r$ .

Classical Bernstein approximation theory gives, for noninteger fixed $\beta>0$ ,

\boxed{ E_r ( u^\beta;[0,1] ) \asymp_\beta r^{-2\beta}. }

More precisely, the normalized quantity

(2r)^{2\beta} E_r ( u^\beta;[0,1] )

has a finite positive limit.

Therefore ordinary algebraic polynomials approximate the endpoint branch singularity only at an algebraic rate in rank.

11. Fixed-power polynomial-rank threshold

Suppose one needs

\boxed{ \varepsilon_r \le N^{-\delta} }

for a fixed

\delta>0.

The Bernstein rate implies

Theorem 11.1 — Polynomial Projection Fixed-Power Rank Scale

\boxed{ r \ge N^{\delta/(2\beta)+o(1)}. }

Thus a uniform-norm polynomial recentering route reaches fixed-power residual suppression only at polynomial rank.

Create:

O-RH-079
POLYNOMIAL_PROJECTION_FIXED_POWER_REQUIRES_POLYNOMIAL_RANK
status:
  CERTIFIED FOR THE UNIFORM-APPROXIMATION ROUTE

This statement does not claim that every possible growing-rank basis has the same rate.

It closes the ordinary polynomial-moment projection route.

12. Orthogonal-projection calibration

Polynomial spectral projections of endpoint algebraic singularities are also known to converge only algebraically.

This is consistent with the exact finite-rank scaling analysis above.

The endpoint branch point at $u=0$ prevents ordinary fixed-degree polynomial spaces from giving exponential-in-rank approximation.

Hence switching from best uniform projection to Legendre/Jacobi-style orthogonal projection does not create a hidden fixed-rank exponential mechanism.

13. Comparison with higher-order Selberg

Paper 30 found that a formal logarithmic amplifier reaches fixed-power scale at

k \asymp \frac{\log N}{\log\log N},

but current generalized-Selberg constants lose uniformity there.

The polynomial projection route is even more expensive.

For a smooth fixed drift, the ordinary polynomial approximation mechanism needs

\boxed{ r=N^{\Omega(\delta)} }

to create an $N^{-\delta}$ coefficient.

Thus:

higher-order Selberg formal critical order:
  log N / log log N

ordinary polynomial low-frequency projection:
  polynomial in N

Finite-rank polynomial recentering is not a cheaper replacement for the higher-order Selberg amplifier.

14. Multi-moment sieve recentering

Suppose one removes the first $r+1$ weighted polynomial moments of the PESC perturbation before applying a signed sieve theorem.

The exact finite-rank identity shows that the removed component reappears as

\boxed{ (m_c^{(r)})^\ast (G^{(r)})^{-1} m_b^{(r)}. }

For a fixed Mellin drift, this finite correction matrix has size

N^{2\beta+1}.

Thus multi-moment recentering generalizes the Paper-33 rank-one phenomenon:

one removed moment:
  rank-one PESC-scale correction

r+1 removed moments:
  finite-rank PESC-scale correction matrix

Create:

O-RH-080
MULTI_MOMENT_RECENTERING_EXPORTS_PESC_SCALE_CORRECTION_MATRIX
status:
  CERTIFIED

15. Finite-rank Mellin windows

The same fixed-rank scaling argument is not special to monomials.

Let

\phi_1,\ldots,\phi_r

be any fixed scale-local basis independent of $N$ and of the unknown zero set.

For a Mellin drift $N^\beta u^\beta$ , every resulting moment is a fixed Mellin transform times a power of $N$ .

Finite-dimensional projection therefore changes only the transfer coefficient.

It does not change the $N$ exponent.

A basis containing the exact unknown $u^\beta$ would annihilate that mode, but that is zero-parameter-dependent filtering and is not an admissible proof input.

16. Campaign 34 track audit

FR1 — polynomial moment projection

status:
  EXACT FINITE-RANK DECOMPOSITION

fixed rank:
  exponent invariant

FR2 — finite-rank Mellin symbols

status:
  COEFFICIENT FILTER ONLY

unknown zero-dependent basis:
  disallowed

FR3 — growing-rank threshold

polynomial basis:
  fixed-power suppression requires polynomial rank

no lower-complexity theorem obtained

FR4 — sieve after multi-moment recentering

status:
  LOW MOMENTS REMOVED FROM INPUT

hard component:
  exported to finite correction matrix at PESC exponent

FR5 — comparison with higher-order Selberg

status:
  FIXED ORDER BEHAVES THE SAME AT EXPONENT LEVEL

growing polynomial rank:
  even more expensive than formal Selberg critical order

17. Campaign 34 verdict

No lower-strength fixed-power route is obtained.

The finite-rank low-frequency family is classified as:

rank one:
  diagnostic

fixed finite rank:
  coefficient rearrangement

growing ordinary polynomial rank:
  algebraic approximation only

fixed-power polynomial suppression:
  polynomial-rank complexity

PESC exponent:
  unchanged

No new frontier is created.

18. New certified package

Create:

B-RH-011
FINITE_RANK_PESC_PROJECTION_IDENTITY
CERTIFIED

O-RH-078
FIXED_RANK_LOW_FREQUENCY_PROJECTION_EXPONENT_INVARIANCE
CERTIFIED

O-RH-079
POLYNOMIAL_PROJECTION_FIXED_POWER_REQUIRES_POLYNOMIAL_RANK
CERTIFIED FOR UNIFORM-APPROXIMATION ROUTE

O-RH-080
MULTI_MOMENT_RECENTERING_EXPORTS_PESC_SCALE_CORRECTION_MATRIX
CERTIFIED

19. Canonical status

F-RH-010
PESC
OPEN / ROOT TARGET

F-RH-016
MLEPG
OPEN / DIRECT THEOREM CANDIDATE

finite-rank projections:
  diagnostic / closed as lower-strength route

20. Campaign 35

The polynomial finite-rank route is closed.

One mathematically distinct approximation family remains worth auditing because its approximation complexity is genuinely different:

CSM_RH Campaign 35
RATIONAL_MUNTZ_LOW_FREQUENCY_FILTER_AUDIT

This is a mechanism audit, not a new frontier.

21. Campaign 35 tracks

RA1 — rational approximation complexity

Best rational approximation of $u^\beta$ has root-exponential rather than algebraic convergence.

Determine the rank required for an $N^{-\delta}$ approximation coefficient.

RA2 — Stieltjes/resolvent representation

Represent fractional powers using resolvent kernels.

Translate those kernels into explicit weighted prime-error observables.

RA3 — arithmetic control of rational moments

Determine whether the resulting resolvent/Stieltjes prime observables are known with fixed-power accuracy or are already zero-sensitive.

RA4 — Müntz basis without zero knowledge

Use a predetermined family of noninteger powers

u^{\lambda_j}

and test whether finite/growing rank can approximate the whole critical-strip Mellin family without encoding the unknown exponent.

RA5 — complexity versus zero sensitivity

If rational/Müntz approximation reaches fixed-power coefficient suppression at polylogarithmic rank, audit whether the arithmetic estimates needed for each basis observable already have fixed-strip strength.

22. Campaign 35 rejection filters

Reject a candidate if:

R1. Its poles/exponents are chosen from the unknown zeta zero set.

R2. It approximates the drift efficiently but requires fixed-power integrated PNT estimates for every basis observable.

R3. It produces only a new representation with no prime-side estimate.

R4. The coefficient condition number consumes the approximation gain.

R5. It reduces to the already-audited higher-order Selberg filter.

R6. It assumes rational approximation error as an arithmetic theorem.

23. External calibration

Classical approximation theory gives a sharp contrast.

For noninteger fixed $\beta>0$ :

Polynomial approximation

E_r^{\mathrm{poly}} ( u^\beta;[0,1] ) \asymp r^{-2\beta}.

Rational approximation

Best diagonal rational approximation has root-exponential behavior of the form

\boxed{ E_r^{\mathrm{rat}} ( u^\beta;[0,1] ) = \exp ( -\Theta_\beta(\sqrt r) ). }

Thus rational filters could formally reach an $N^{-\delta}$ coefficient with rank of order

(\log N)^2.

Whether the corresponding prime observables are arithmetically controllable at lower strength is completely separate and remains unaudited.

This is the reason for Campaign 35.

24. State transition

CSM_RH v1.25
  ->
CSM_RH v1.26

with:

Campaign 34
  CLOSED_AS_FINITE_RANK_LOW_FREQUENCY_PROJECTION_AUDIT

B-RH-011
  FINITE_RANK_PESC_PROJECTION_IDENTITY
  CREATED / CERTIFIED

O-RH-078
  FIXED_RANK_LOW_FREQUENCY_PROJECTION_EXPONENT_INVARIANCE
  CREATED / CERTIFIED

O-RH-079
  POLYNOMIAL_PROJECTION_FIXED_POWER_REQUIRES_POLYNOMIAL_RANK
  CREATED / CERTIFIED FOR UNIFORM-APPROXIMATION ROUTE

O-RH-080
  MULTI_MOMENT_RECENTERING_EXPORTS_PESC_SCALE_CORRECTION_MATRIX
  CREATED / CERTIFIED

F-RH-010
  PESC
  REMAINS OPEN / ROOT TARGET

Campaign 35
  RATIONAL_MUNTZ_LOW_FREQUENCY_FILTER_AUDIT
  READY

25. Final status

RH = OPEN

PESC = OPEN

MLEPG = OPEN

RANK-ONE PROJECTION = DIAGNOSTIC

FIXED FINITE-RANK PROJECTION = EXPONENT-NEUTRAL

MULTI-MOMENT RECENTERING = PESC-SCALE CORRECTION MATRIX

ORDINARY POLYNOMIAL GROWING RANK = ALGEBRAIC APPROXIMATION

FIXED-POWER POLYNOMIAL RESIDUAL SUPPRESSION = POLYNOMIAL RANK

FINITE-RANK LOW-FREQUENCY ROUTE = CLOSED

NEXT CAMPAIGN = 35

The decisive exact decomposition is

\boxed{ \mathcal C_N = \langle P_{N,r}c, P_{N,r}b \rangle_N + \langle (I-P_{N,r})c, (I-P_{N,r})b \rangle_N. }

For a fixed Mellin drift, both terms inherit the same outer factor

N^{2\beta+1}.

Finite-rank projection changes coordinates.

It does not change the RH-strength exponent.

CSM_RH Paper 35 — Finite-Rank Low-Frequency Projection Invariance and the Polynomial-Rank Fixed-Power Threshold

CSM_RH Paper 35

Finite-Rank Low-Frequency Projection Invariance and the Polynomial-Rank Fixed-Power Threshold

0. Trust boundary

1. Discrete weighted Hilbert space

2. Polynomial low-frequency subspace

3. Exact finite-rank PESC decomposition

Theorem 3.1 — Finite-Rank PESC Projection Identity

4. Moment-matrix form of the finite-rank correction

Theorem 4.1 — Finite-Rank Correction Matrix

5. Continuum endpoint measure

6. Smooth Mellin drift shapes

7. Fixed-rank exponent invariance

Theorem 7.1 — Fixed-Rank Mellin-Mode Split

8. Moment-vector exponent ledger

9. No fixed finite basis can annihilate the Mellin continuum

10. Polynomial approximation route

11. Fixed-power polynomial-rank threshold

Theorem 11.1 — Polynomial Projection Fixed-Power Rank Scale

12. Orthogonal-projection calibration

13. Comparison with higher-order Selberg

14. Multi-moment sieve recentering

15. Finite-rank Mellin windows

16. Campaign 34 track audit

FR1 — polynomial moment projection

FR2 — finite-rank Mellin symbols

FR3 — growing-rank threshold

FR4 — sieve after multi-moment recentering

FR5 — comparison with higher-order Selberg

17. Campaign 34 verdict

18. New certified package

19. Canonical status

20. Campaign 35

21. Campaign 35 tracks

RA1 — rational approximation complexity

RA2 — Stieltjes/resolvent representation

RA3 — arithmetic control of rational moments

RA4 — Müntz basis without zero knowledge

RA5 — complexity versus zero sensitivity

22. Campaign 35 rejection filters

R1. Its poles/exponents are chosen from the unknown zeta zero set.

R2. It approximates the drift efficiently but requires fixed-power integrated PNT estimates for every basis observable.

R3. It produces only a new representation with no prime-side estimate.

R4. The coefficient condition number consumes the approximation gain.

R5. It reduces to the already-audited higher-order Selberg filter.

R6. It assumes rational approximation error as an arithmetic theorem.

23. External calibration

Polynomial approximation

Rational approximation

24. State transition

25. Final status