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CSM_RH Paper 36 — Rational_Stieltjes Filter Closure, Resolvent Drift Persistence, and the Representation–Arithmetic Separation Principle

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CSM_RH Paper 36

Rational/Stieltjes Filter Closure, Resolvent Drift Persistence, and the Representation–Arithmetic Separation Principle

Project: CSM_RH
Paper: 36
Version: v0.1
Date: 2026-09-07
Parent state: CSM_RH v1.26 / Paper 35
Campaign: 35 — RATIONAL_MUNTZ_LOW_FREQUENCY_FILTER_AUDIT
Status: rational/Müntz approximation 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 35 closed ordinary finite-rank polynomial projection as a lower-strength route.

Campaign 35 asks whether rational or Müntz filters change that conclusion because they approximate endpoint branch singularities much more efficiently.

The answer is:

rational approximation complexity:
  dramatically better

prime-side arithmetic strength:
  not automatically better

fixed finite resolvent rank:
  exponent-neutral

root-exponential rank:
  polylogarithmic rank can represent N^(-delta) function error

derivative/correlation control:
  not implied by uniform function approximation

clustered rational poles:
  enter polynomial conditioning scale at fixed-power accuracy

Müntz coordinates:
  are Mellin coordinates and retain hard arithmetic moments

Thus approximation efficiency and arithmetic proof strength separate.

No new frontier is created.

1. Stieltjes representation of a fractional power

For

0<β<1,0<\beta<1,

the power function is a complete Bernstein function and has the Stieltjes representation

uβ=sin⁡(πβ)π∫0∞tβ−1uu+t dt,u>0.\boxed{ u^\beta = \frac{\sin(\pi\beta)}{\pi} \int_0^\infty t^{\beta-1} \frac{u}{u+t} \,dt, \qquad u>0. }

Define the resolvent kernel

rt(u)=uu+t,t>0.\boxed{ r_t(u) = \frac{u}{u+t}, \qquad t>0. }

The fractional Mellin drift is therefore a positive continuum superposition of rational resolvent coordinates.

2. Rational approximation complexity

Let

Er,r(uβ;[0,1])E_{r,r}(u^\beta;[0,1])

be the best uniform type- (r,r)(r,r) rational approximation error.

Stahl's theorem gives

Er,r(uβ;[0,1])=Cβexp⁡(−2πβr)[1+o(1)],\boxed{ E_{r,r}(u^\beta;[0,1]) = C_\beta \exp \left( -2\pi\sqrt{\beta r} \right) [ 1+o(1) ], }

where

Cβ=41+β∣sin⁡(πβ)∣.C_\beta = 4^{1+\beta} |\sin(\pi\beta)|.

Therefore:

Theorem 2.1 — Rational Fixed-Power Rank Scale

To reach a purely approximation-theoretic target

Er,r≤N−δ,E_{r,r} \le N^{-\delta},

it is sufficient at leading order to take

r≍δ24π2β(log⁡N)2.\boxed{ r \asymp \frac{ \delta^2 }{ 4\pi^2\beta } (\log N)^2. }

This is exponentially cheaper in rank than the ordinary polynomial route of Paper 35.

3. Pole geometry

For

0<β<1,0<\beta<1,

the poles and zeros of the best rational approximants lie on the negative real axis.

Near-best rational and lightning constructions achieve root-exponential convergence by exponential or tapered-exponential clustering of poles toward the branch point u=0u=0.

A representative clustering law is

tmin⁡=exp⁡[−Θβ(r)].\boxed{ t_{\min} = \exp [ -\Theta_\beta(\sqrt r) ]. }

At the fixed-power approximation scale

r=Θ((log⁡N)2),r=\Theta((\log N)^2),

this becomes

tmin⁡=N−Θβ(1).\boxed{ t_{\min} = N^{-\Theta_\beta(1)}. }

This is used only as route calibration for clustered-pole rational schemes.

4. Resolvent arithmetic observables

Return to the PESC endpoint Hilbert space.

For fixed t>0t>0, define the scale-local rational coordinate

vt(n)=rt(n/N)=n/Nn/N+t.\boxed{ v_t(n) = r_t(n/N) = \frac{n/N}{n/N+t}. }

Define the cumulative-error moment

MB(t;N)=∑n<2NwN(n)B(n−1)vt(n).\boxed{ M_B(t;N) = \sum_{n<2N} w_N(n) B(n-1) v_t(n). }

Define the detector moment

Mc(t;N)=∑n<2NwN(n)cnvt(n).\boxed{ M_c(t;N) = \sum_{n<2N} w_N(n) c_n v_t(n). }

These are the natural correction coordinates in a rational low-frequency projection.

5. Fixed resolvent response to a smooth drift

Take

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

Then

cn=βnβ−1+Oβ(nβ−2).c_n = \beta n^{\beta-1} + O_\beta(n^{\beta-2}).

At scale n=Nun=Nu,

MB(t;N)=Nβ+2JB(β,t)+o(Nβ+2),M_B(t;N) = N^{\beta+2} J_B(\beta,t) + o(N^{\beta+2}),

where

JB(β,t)=∫02ω(u)uβuu+t du.\boxed{ J_B(\beta,t) = \int_0^2 \omega(u) u^\beta \frac{u}{u+t} \,du. }

Likewise,

Mc(t;N)=Nβ+1Jc(β,t)+o(Nβ+1),M_c(t;N) = N^{\beta+1} J_c(\beta,t) + o(N^{\beta+1}),

where

Jc(β,t)=β∫02ω(u)uβ−1uu+t du.\boxed{ J_c(\beta,t) = \beta \int_0^2 \omega(u) u^{\beta-1} \frac{u}{u+t} \,du. }

For real

0<β<1,t>0,0<\beta<1, \qquad t>0,

both integrands are positive.

Therefore:

Theorem 5.1 — Fixed Resolvent Drift Persistence

JB(β,t)>0,Jc(β,t)>0.\boxed{ J_B(\beta,t)>0, \qquad J_c(\beta,t)>0. }

Every fixed positive resolvent coordinate retains the full smooth-drift exponent.

6. Finite rational projection remains exponent-neutral

Let

Vr=span⁡{rt1,…,rtr}V_r = \operatorname{span} \{ r_{t_1},\ldots,r_{t_r} \}

for fixed positive nodes tjt_j independent of NN and of the unknown zero set.

Paper 35's finite-rank projection theorem applies verbatim.

For

B(x)=xβ,B(x)=x^\beta,

the low-rank correction has scale

N2β+1\boxed{ N^{2\beta+1} }

and the residual covariance has the same outer exponent whenever its coefficient is nonzero.

Thus:

Theorem 6.1 — Fixed Rational Rank Exponent Invariance

A fixed finite family of resolvent coordinates can change the Mellin transfer coefficient but not the PESC smooth-drift exponent.

Create:

O-RH-081
FIXED_RATIONAL_RESOLVENT_PROJECTION_EXPONENT_INVARIANCE
status:
  CERTIFIED

7. Representation error is not arithmetic error

Suppose a rational function satisfies

∥uβ−rr(u)∥∞≤N−δ.\boxed{ \|u^\beta-r_r(u)\|_\infty \le N^{-\delta}. }

This is a statement about representing the cumulative drift profile.

PESC, however, couples the cumulative profile to its arithmetic derivative.

Uniform approximation of a function does not imply comparable approximation of its derivative.

Indeed, in general one may have functions eNe_N with

∥eN∥∞≤N−δ\|e_N\|_\infty \le N^{-\delta}

but

∥eN′∥∞≍1\|e_N'\|_\infty \asymp1

or larger.

Therefore a fixed-power rational approximation coefficient cannot be inserted directly into the PESC derivative/correlation ledger.

Create:

O-RH-082
UNIFORM_FUNCTION_APPROXIMATION_DOES_NOT_CONTROL_PESC_DERIVATIVE
status:
  CERTIFIED

8. Resolvent derivative conditioning

The resolvent derivative is

rt′(u)=t(u+t)2.\boxed{ r_t'(u) = \frac{t}{(u+t)^2}. }

Hence

∥rt′∥L∞([0,2])=1t.\boxed{ \|r_t'\|_{L^\infty([0,2])} = \frac1t. }

For clustered-pole schemes with

tmin⁡=exp⁡[−Θ(r)],t_{\min} = \exp[-\Theta(\sqrt r)],

the most singular basis derivative has size

exp⁡[Θ(r)].\boxed{ \exp[\Theta(\sqrt r)]. }

At

r=Θ((log⁡N)2),r=\Theta((\log N)^2),

this becomes

NΘ(1).\boxed{ N^{\Theta(1)}. }

Thus the same pole clustering which gives fixed-power function approximation introduces polynomial derivative conditioning at the PESC fixed-power rank scale.

Create:

O-RH-083
CLUSTERED_RESOLVENT_DERIVATIVE_CONDITIONING_ENTERS_FIXED_POWER_SCALE
status:
  CERTIFIED FOR EXPONENTIALLY CLUSTERED RESOLVENT ROUTES

This is not a universal theorem about every rational basis.

9. Stieltjes quadrature interpretation

The Stieltjes formula expresses the drift as a positive continuum of resolvents:

uβ=Cβ∫0∞tβ−1rt(u) dt.u^\beta = C_\beta \int_0^\infty t^{\beta-1} r_t(u) \,dt.

A rational quadrature discretizes this continuum into a finite family

∑j=1rαjrtj(u).\boxed{ \sum_{j=1}^{r} \alpha_jr_{t_j}(u). }

The approximation-theoretic gain is obtained by selecting nodes and coefficients so that the continuum profile is reconstructed efficiently.

But the corresponding arithmetic projection requires the moments

MB(tj;N),Mc(tj;N),M_B(t_j;N), \qquad M_c(t_j;N),

or equivalent linear combinations.

Theorem 5.1 shows that these moments themselves retain the full smooth-drift exponent.

Thus the approximation does not supply their arithmetic smallness.

10. Rational correction matrix

For a finite rational basis, define the Gram matrix

Gij=⟨rti,rtj⟩N.G_{ij} = \langle r_{t_i}, r_{t_j} \rangle_N.

Define moment vectors

mB=(MB(t1;N),…,MB(tr;N))T,m_B = ( M_B(t_1;N),\ldots,M_B(t_r;N) )^T, mc=(Mc(t1;N),…,Mc(tr;N))T.m_c = ( M_c(t_1;N),\ldots,M_c(t_r;N) )^T.

The exact low-rank PESC correction is

mc∗G−1mB.\boxed{ m_c^\ast G^{-1} m_B. }

For fixed rank, this has the same smooth-drift exponent as PESC.

For growing rank, proving that this correction is fixed-power small requires arithmetic control of a growing family of resolvent moments and of the conditioning of G−1G^{-1}.

Approximation theory alone supplies neither.

11. Representation–arithmetic separation principle

Create:

O-RH-084
REPRESENTATION_EFFICIENCY_DOES_NOT_IMPLY_ARITHMETIC_FIXED_POWER
status:
  CERTIFIED AS CSM_RH MECHANISM PRINCIPLE

Statement:

Efficient approximation of the shape of a hypothetical low-frequency prime-error drift does not imply a fixed-power estimate for the arithmetic coordinates required to subtract that drift from PESC.

The rational route dramatically improves the first problem.

It does not solve the second.

12. Müntz coordinates

A Müntz family has the form

1,uλ1,uλ2,…\boxed{ 1, u^{\lambda_1}, u^{\lambda_2}, \ldots }

with positive exponents.

Müntz's theorem states that, under the standard hypotheses, the span is dense in C[0,1]C[0,1] precisely when

∑j1λj=∞.\boxed{ \sum_j \frac1{\lambda_j} = \infty. }

Thus a zero-independent infinite exponent family can be representationally complete.

But every basis vector is itself a Mellin coordinate.

13. Müntz moment scaling

For a basis exponent λ\lambda, define the cumulative-error moment

MB(λ;N)=∑n<2NwN(n)B(n−1)(nN)λ.M_B(\lambda;N) = \sum_{n<2N} w_N(n) B(n-1) \left( \frac nN \right)^\lambda.

For

B(x)=xβ,B(x)=x^\beta, MB(λ;N)=Nβ+2I(β+λ)+o(Nβ+2),\boxed{ M_B(\lambda;N) = N^{\beta+2} I(\beta+\lambda) + o(N^{\beta+2}), }

where

I(s)=2s+2−1(s+1)(s+2).I(s) = \frac{ 2^{s+2}-1 }{ (s+1)(s+2) }.

Likewise,

Mc(λ;N)=βNβ+1I(β−1+λ)+o(Nβ+1).\boxed{ M_c(\lambda;N) = \beta N^{\beta+1} I(\beta-1+\lambda) + o(N^{\beta+1}). }

For real nonnegative λ\lambda, these coefficients are nonzero.

Thus a fixed Müntz basis is exponent-neutral.

14. Unknown-exponent issue

If one inserts the exact unknown exponent

λ=β\lambda=\beta

into a basis, representation of the drift becomes trivial.

But such a choice uses the unknown drift / zero parameter as a proof input.

This is disallowed.

A predetermined dense exponent family avoids that circularity.

However the arithmetic correction moments remain Mellin-weighted integrated prime-error observables.

No current fixed-power estimates for those growing families are supplied by the approximation theorem.

15. Müntz density is not a quantitative arithmetic theorem

The divergence condition

∑1/λj=∞\sum1/\lambda_j=\infty

is a completeness statement.

It does not give:

a fixed-power approximation rate for the entire critical-strip drift family;
uniform conditioning of the resulting arithmetic correction matrix;
fixed-power estimates for the Mellin prime-error moments.

Therefore Müntz completeness does not create a lower-strength PESC route.

Create:

O-RH-085
MUNTZ_COMPLETENESS_WITHOUT_ARITHMETIC_MOMENT_CONTROL
status:
  CERTIFIED AS MECHANISM AUDIT

16. Campaign 35 track audit

RA1 — rational approximation complexity

status:
  ROOT-EXPONENTIAL

formal N^(-delta) representation rank:
  O((log N)^2)

RA2 — Stieltjes/resolvent representation

status:
  EXACT

basis:
  u/(u+t)

RA3 — arithmetic control of rational moments

status:
  FIXED RESOLVENTS RETAIN FULL DRIFT EXPONENT

fixed-power moment theorem:
  not available

RA4 — Müntz basis without zero knowledge

status:
  REPRESENTATIONALLY COMPLETE IN PRINCIPLE

arithmetic moments:
  Mellin-hard

RA5 — complexity versus zero sensitivity

status:
  APPROXIMATION COMPLEXITY IMPROVED

arithmetic fixed-power complexity:
  not improved by current theorem set

derivative conditioning:
  exponent-critical in clustered-pole route

17. Campaign 35 verdict

The rational route is the first audited low-frequency filter whose approximation rank is genuinely attractive:

r=O((log⁡N)2)r = O((\log N)^2)

can represent a fixed-power branch-profile error.

But this does not produce a prime theorem.

The correction coordinates retain the same Mellin drift exponent, and derivative-sensitive PESC control is not inherited from uniform approximation.

Müntz completeness has the same representation/arithmetic separation.

Therefore:

rational/Müntz filtering:
  valuable representation technology

lower-strength RH route:
  not obtained

No new frontier is created.

18. New certified package

Create:

O-RH-081
FIXED_RATIONAL_RESOLVENT_PROJECTION_EXPONENT_INVARIANCE
CERTIFIED

O-RH-082
UNIFORM_FUNCTION_APPROXIMATION_DOES_NOT_CONTROL_PESC_DERIVATIVE
CERTIFIED

O-RH-083
CLUSTERED_RESOLVENT_DERIVATIVE_CONDITIONING_ENTERS_FIXED_POWER_SCALE
CERTIFIED FOR CLUSTERED-POLE ROUTES

O-RH-084
REPRESENTATION_EFFICIENCY_DOES_NOT_IMPLY_ARITHMETIC_FIXED_POWER
CERTIFIED

O-RH-085
MUNTZ_COMPLETENESS_WITHOUT_ARITHMETIC_MOMENT_CONTROL
CERTIFIED

19. Canonical root status

F-RH-010
PESC
OPEN / ROOT TARGET

F-RH-016
MLEPG
OPEN / DIRECT THEOREM CANDIDATE

low-frequency projection families:
  exhausted as representation-only mechanisms

20. Campaign 36

After Papers 17–36, surrogate representation generation is stopped again.

The next campaign is:

CSM_RH Campaign 36
MINIMAL_FIXED_POWER_BREAKTHROUGH_GATE

Its purpose is to consolidate the audited closure graph and admit only genuinely new prime-side fixed-power lemmas.

21. Campaign 36 admission rules

A candidate is admitted only if it satisfies all of:

G1 — new arithmetic estimate

It contains a theorem estimate not already equivalent by deterministic identities to PESC, MLEPG, or one of the closed auxiliary gates.

G2 — fixed exponent source

The proof contains an explicit mechanism generating a fixed positive exponent.

G3 — no zero-strip input

No fixed zero-free half-plane is assumed.

G4 — no representation-only gain

Improved approximation, projection, decomposition, filtering, or basis efficiency does not count unless accompanied by a new prime estimate.

G5 — direct bridge

A proved chain returns the candidate to PESC / fixed PNT mean-square.

22. Campaign 36 preferred theorem families

B1 — direct signed PESC contraction

A genuinely new arithmetic inequality for the endogenous prime self-correlation.

B2 — direct MLEPG power theorem

A fixed-power lag-energy estimate proved without importing a fixed strip.

B3 — shrinking-threshold short-interval theorem

A theorem with polynomially shrinking relative error and power-sized exceptional set.

B4 — fixed-power principal Fejer deconcentration

A prime exponential-sum estimate on the principal arc with an actual fixed exponent.

B5 — new arithmetic recursion

A scale recurrence whose cumulative contraction mass is provably linear in log⁡N\log N from prime-specific information.

23. Campaign 36 hard rejects

Reject:

new equivalent criteria;
new positive p-moment surrogates;
new zero-density-only routes;
new fixed-order Selberg filters;
new finite-rank low-frequency projections;
new rational/Müntz representations without arithmetic estimates;
logarithmic or stretched-log savings labelled as fixed power;
fixed zero-strip assumptions.

24. State transition

CSM_RH v1.26
  ->
CSM_RH v1.27

with:

Campaign 35
  CLOSED_AS_RATIONAL_MUNTZ_REPRESENTATION_ARITHMETIC_AUDIT

O-RH-081
  CREATED / CERTIFIED

O-RH-082
  CREATED / CERTIFIED

O-RH-083
  CREATED / CERTIFIED FOR CLUSTERED-POLE ROUTES

O-RH-084
  CREATED / CERTIFIED

O-RH-085
  CREATED / CERTIFIED

F-RH-010
  PESC
  REMAINS OPEN / ROOT TARGET

F-RH-016
  MLEPG
  REMAINS OPEN / DIRECT THEOREM CANDIDATE

Campaign 36
  MINIMAL_FIXED_POWER_BREAKTHROUGH_GATE
  READY

25. Final status

RH = OPEN

PESC = OPEN

MLEPG = OPEN

POLYNOMIAL PROJECTION = CLOSED

RATIONAL REPRESENTATION = ROOT-EXPONENTIALLY EFFICIENT

RATIONAL ARITHMETIC CONTROL = OPEN / NO POWER GAIN

MUNTZ COMPLETENESS = REPRESENTATIONAL ONLY

FIXED-POWER DERIVATIVE CONTROL = NOT PROVIDED BY FUNCTION APPROXIMATION

LOW-FREQUENCY REPRESENTATION SHELL = CLOSED

NEXT CAMPAIGN = 36

The central lesson is:

approximating the shape of a hard drift is not the same as proving that primes cannot contain that drift.\boxed{ \text{approximating the shape of a hard drift is not the same as proving that primes cannot contain that drift.} }

Rational approximation solves the former with impressive efficiency.

CSM_RH still needs the latter.