CSM_RH Paper 30

Fixed-Order Selberg Exponent Invariance and the Growing-Order Complexity Threshold

Project: CSM_RH
Paper: 30
Version: v0.1
Date: 2026-09-07
Parent state: CSM_RH v1.20 / Paper 29
Campaign: 29 — HIGHER_ORDER_SELBERG_AMPLIFIER_ATTACK
Status: generalized-von-Mangoldt amplifier 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 29 showed that the classical fixed-order Selberg feedback is forcing-critical for every fixed Mellin mode $x^\rho$ with $\Re\rho<1$ .

Campaign 29 tests whether generalized von Mangoldt functions

\Lambda_k = \mu*\log^k

can amplify this feedback into a fixed polynomial drift gap.

The result is:

fixed k:
  exponent-neutral

finite fixed-order combinations:
  no certified remainder cancellation

slowly growing k:
  can amplify logarithmic/subpower precision

fixed-power critical order:
  k ~ log X / log log X

at that order:
  pointwise/combinatorial complexity becomes polynomial in X

current fixed-k theorems:
  not uniform enough to cross the critical order

Higher-order Selberg methods are genuine amplifiers, but the audited theory supplies a subpower amplifier rather than a certified fixed-power amplifier.

1. Generalized von Mangoldt functions

For an integer

k\ge1,

define

\boxed{ \Lambda_k(n) = \sum_{d\mid n} \mu(d) \left( \log\frac nd \right)^k. }

Equivalently,

\boxed{ \Lambda_k = \mu*\log^k. }

The Dirichlet series is

\boxed{ \sum_{n\ge1} \frac{\Lambda_k(n)}{n^s} = (-1)^k \frac{\zeta^{(k)}(s)}{\zeta(s)}, \qquad \Re s>1. }

Also,

\boxed{ \Lambda_{k+1} = \Lambda_k\log + \Lambda*\Lambda_k. }

The function is nonnegative and supported on integers with at most $k$ distinct prime factors.

At a prime,

\boxed{ \Lambda_k(p) = (\log p)^k. }

2. Fixed-k summatory polynomial

For each fixed

k\ge1,

there exists a polynomial $R_k$ of degree $k-1$ with leading term

k t^{k-1}

such that

Theorem 2.1 — Fixed-Order Generalized Selberg Summatory Formula

\boxed{ \sum_{n\le x}\Lambda_k(n) = xR_k(\log x) + O_k(x). }

Equivalently, retaining only the leading term,

\boxed{ \sum_{n\le x}\Lambda_k(n) = kx(\log x)^{k-1} + O_k \left( x(\log x)^{k-2} \right). }

For $k=2$ , this contains the classical Selberg symmetry formula.

For $k\ge2$ , the almost-prime observable is much smoother than the prime observable itself.

3. Complete polynomial subtraction

Define

\boxed{ E_k(x) = \sum_{n\le x}\Lambda_k(n) - xR_k(\log x). }

Theorem 2.1 gives

\boxed{ E_k(x) = O_k(x). }

Normalize by the natural leading scale:

\widetilde E_k(x) = \frac{ E_k(x) }{ x(\log x)^{k-1} }.

Then for fixed $k$ ,

\boxed{ \widetilde E_k(x) = O_k \left( (\log x)^{-(k-1)} \right). }

Thus increasing a fixed order can manufacture arbitrarily high fixed logarithmic precision.

It does not yet manufacture a power of $x$ .

4. Fixed-zero response of Lambda-k

Let $\rho$ be a simple nontrivial zero of $\zeta$ .

Near

s=\rho,

write

\zeta(s) = \zeta'(\rho)(s-\rho) + O \left( (s-\rho)^2 \right).

Then

(-1)^k \frac{\zeta^{(k)}(s)}{\zeta(s)}

either:

has a simple pole at $\rho$ , with residue $\boxed{ (-1)^k \frac{ \zeta^{(k)}(\rho) }{ \zeta'(\rho) }, }$ if $\zeta^{(k)}(\rho)\ne0$ ;

has the pole cancelled because the numerator also vanishes there.

Therefore:

Theorem 4.1 — Fixed-Order Zero-Exponent Invariance

Whenever the generalized detector sees the zero $\rho$ , its explicit-formula contribution has exponent

\boxed{ x^\rho. }

Changing fixed $k$ changes the coefficient, not the real exponent.

If the numerator cancels the pole, the detector becomes less sensitive to that particular zero rather than more coercive.

Create:

O-RH-064
FIXED_ORDER_GENERALIZED_SELBERG_ZERO_EXPONENT_INVARIANCE
status:
  CERTIFIED AS STRENGTH AUDIT

5. Fixed-k forcing versus fixed Mellin mode

For a fixed zero or hypothetical Mellin drift with

\Re\rho=\beta<1,

the generalized detector has mode scale

O_{k,\rho}(x^\beta)

whenever the pole survives.

The unconditional complete-model remainder is

O_k(x).

Hence

\boxed{ x^\beta = o(x). }

After normalization by $x(\log x)^{k-1}$ :

forcing

\boxed{ O_k \left( (\log x)^{-(k-1)} \right) }

zero/drift mode

\boxed{ O_{k,\rho} \left( x^{\beta-1} (\log x)^{-(k-1)} \right). }

Their ratio is

\boxed{ x^{\beta-1}. }

It is independent of $k$ at exponent level.

Thus fixed-order amplification sharpens the logarithmic normalization while leaving the fixed Mellin mode buried inside the $O_k(x)$ forcing.

6. Finite fixed-order linear combinations

Let $K$ be fixed and consider

\boxed{ \mathcal L_K(s) = \frac{ \sum_{k=1}^{K} c_k(-1)^k\zeta^{(k)}(s) }{ \zeta(s) }. }

At a simple zero $\rho$ , if

\sum_{k=1}^{K} c_k(-1)^k \zeta^{(k)}(\rho) \ne0,

then $\mathcal L_K$ still has a simple pole at $\rho$ .

Hence the associated zero contribution still has exponent $x^\rho$ .

If the numerator vanishes, the filter misses that zero.

Therefore a fixed-order differential filter cannot move a surviving zero mode to a better exponent.

It can only alter or remove its residue.

7. Main-term cancellation does not cancel unknown remainders

For fixed $K$ , known polynomial main terms

xR_k(\log x)

can be combined exactly.

But the current unconditional statements provide separate errors

E_k(x)=O_k(x).

A finite linear combination gives only

\boxed{ \sum_{k\le K}c_kE_k(x) = O_K \left( x\sum_{k\le K}|c_k| \right) }

without additional joint information.

One cannot use cancellation among independent big- $O$ errors as theorem authority.

Create:

O-RH-065
FIXED_ORDER_REMAINDER_NONCOHERENCE
status:
  CERTIFIED AS CURRENT-THEOREM AUDIT

This does not say that the true remainders never cancel.

It says that current fixed-order summatory formulae do not certify such cancellation.

8. Historical calibration: higher weight really does amplify

Higher-weight Selberg/Bombieri/Wirsing methods are not useless.

Historically, elementary methods were strengthened from the original PNT to error terms of the form

\boxed{ \psi(x)-x = O_A \left( x(\log x)^{-A} \right) }

for every fixed $A>0$ .

Later Diamond–Steinig-type methods obtained stretched-exponential subpower remainders of the form

\boxed{ x \exp \left( -c(\log x)^\theta \right) }

for fixed $\theta>0$ in the known elementary range.

Thus higher weighting and recursion can increase cumulative amplification beyond every fixed logarithmic power.

But these estimates remain

\boxed{ x^{1-o(1)}, }

not $x^{1-\delta}$ .

This is consistent with the CSM_RH subpower / fixed-power distinction.

9. Critical growing order

Suppose one tries to turn a formal logarithmic factor

(\log X)^{-k}

into a fixed power

X^{-\delta}.

The balance equation is

k\log\log X \sim \delta\log X.

Therefore the critical order is

Theorem 9.1 — Fixed-Power Order Scale

\boxed{ k_{\mathrm{crit}} \asymp \delta \frac{ \log X }{ \log\log X }. }

Any order satisfying

k = o \left( \frac{\log X}{\log\log X} \right)

can generate at most a subpower factor from powers of $\log X$ .

10. Pointwise weight complexity at the critical order

At a prime,

\Lambda_k(p) = (\log p)^k.

k = c \frac{\log X}{\log\log X},

we have

\boxed{ (\log X)^k = X^{c+o(1)}. }

Thus the generalized detector itself acquires polynomial-size spikes exactly when the formal logarithmic amplifier enters fixed-power territory.

The pointwise bound

0\le\Lambda_k(n)\le(\log n)^k

therefore ceases to be exponent-neutral.

11. Factorial proof-complexity scale

The elementary fixed- $k$ summatory argument expands powers of logarithms and integrates expressions of the form

\log^k y.

The natural coefficient scale includes factorial-size quantities.

Stirling gives

\boxed{ \log(k!) = k\log k-k+O(\log k). }

At the critical order

k = c \frac{\log X}{\log\log X},

one gets

\boxed{ k! = X^{c+o(1)}. }

Therefore any factorial-type dependence hidden inside the fixed- $k$ constant $O_k(1)$ becomes polynomial in $X$ at exactly the order required for fixed-power amplification.

The existing theorem

E_k(x)=O_k(x)

is stated for fixed $k$ and provides no uniform control capable of excluding this exponent cost.

Create:

O-RH-066
GROWING_ORDER_FACTORIAL_COMPLEXITY_THRESHOLD
status:
  CERTIFIED AS CURRENT-METHOD UNIFORMITY BARRIER

The statement is not that every possible proof must lose $k!$ .

It is that the current fixed- $k$ theorem contains no uniformity strong enough to cross the critical order, and factorial-scale dependence is already exponent-critical there.

12. Support complexity at large order

The generalized function $\Lambda_k$ is supported on integers with at most $k$ distinct prime factors.

For fixed $k$ , this is a strong smoothing from primes to bounded-order almost primes.

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

the allowed number of distinct prime factors is far larger than the typical order $\log\log X$ .

Thus the support restriction becomes weak for typical integers.

This is a qualitative warning:

small fixed k:
  strong almost-prime smoothing

critical growing k:
  broad support, large weights

The very mechanism that smooths the prime problem at fixed order becomes less selective when the order approaches fixed-power scale.

13. Growing-order verdict

There are three regimes.

Regime I — fixed k

k=O(1).

Result:

arbitrary fixed log powers possible;
fixed zero exponent unchanged;
fixed-power coercivity absent.

Regime II — subcritical growing k

k = o \left( \frac{\log X}{\log\log X} \right).

Formal log amplification remains

X^{-o(1)}.

This is compatible with stretched-exponential/subpower error improvements.

Regime III — critical order

k \asymp \frac{\log X}{\log\log X}.

Formal amplification can become a fixed power.

But:

pointwise weights:
  polynomial in X

factorial-type constants:
  polynomial in X

fixed-k summatory theorem:
  no authorized uniformity

almost-prime support:
  substantially less selective

No certified net fixed power is obtained.

14. Campaign 29 track audit

H1 — fixed-k generalized Selberg response

status:
  CLOSED

zero exponent:
  invariant

forcing:
  O_k(x)

H2 — finite linear-combination forcing cancellation

status:
  NO CERTIFIED CANCELLATION OF O_k(x) REMAINDERS

fixed-zero exponent:
  unchanged if pole survives

H3 — growing-k amplification

status:
  SUBCRITICAL -> SUBPOWER

critical order:
  complexity/uniformity barrier

H4 — generalized almost-prime to PESC transfer

status:
  NO LOWER-STRENGTH FIXED-POWER TRANSFER FOUND

H5 — spectral polynomial in Selberg operator

status:
  FIXED ORDER CHANGES RESIDUES, NOT ZERO EXPONENTS

no coercive filter found

15. Campaign 29 verdict

No fixed-power theorem is proved.

The higher-order Selberg route is classified as:

real amplifier:
  YES

arbitrary fixed log-power:
  YES

subpower / stretched-log potential:
  YES

certified fixed-power:
  NO

fixed-order route:
  CLOSED AS EXPONENT-NEUTRAL

growing-order route:
  OPEN IN PRINCIPLE BUT CURRENTLY UNAUTHORIZED AT CRITICAL ORDER

The root target remains PESC.

16. New certified obstruction package

Create:

O-RH-064
FIXED_ORDER_GENERALIZED_SELBERG_ZERO_EXPONENT_INVARIANCE
CERTIFIED

O-RH-065
FIXED_ORDER_REMAINDER_NONCOHERENCE
CERTIFIED AS CURRENT-THEOREM AUDIT

O-RH-066
GROWING_ORDER_FACTORIAL_COMPLEXITY_THRESHOLD
CERTIFIED AS CURRENT-METHOD UNIFORMITY BARRIER

No new frontier is created.

17. Return to the arithmetic root

The generalized Selberg shell has now been audited at:

order 2;
fixed higher order;
finite fixed-order combinations;
subcritical growing order;
critical growing-order exponent budget.

No lower-strength fixed-power bridge is found.

Therefore CSM_RH returns again to:

F-RH-010
PESC

with no additional surrogate.

18. Campaign 30

The next campaign is:

CSM_RH Campaign 30
PARITY_SENSITIVE_PESC_ATTACK

The goal is not to introduce another almost-prime smoothing layer.

It is to identify arithmetic information that distinguishes the true prime residual from the family of generalized almost-prime models.

19. Campaign 30 tracks

Q1 — PESC parity decomposition

Decompose the residual prime detector according to Liouville/Möbius parity information while preserving the signed endogenous primitive.

The target must remain bilinear.

Q2 — asymptotic-sieve scalar defect

Use the Bombieri asymptotic-sieve perspective: higher generalized von Mangoldt information leaves a parity-sensitive scalar undetermined.

Identify the analogue of that undetermined scalar inside PESC.

Q3 — parity-breaking bilinear input

Revisit Friedlander–Iwaniec-type parity-breaking hypotheses, but require a direct bilinear bridge to PESC rather than EMBF/PACPSA surrogate proliferation.

Q4 — prime-versus-almost-prime drift comparator

Construct an observable whose value on $\Lambda$ differs at fixed exponent from every bounded-order generalized almost-prime model.

Q5 — direct residual sign structure

Search for a prime-specific signed identity involving $f=\Lambda-\Lambda^\sharp$ that is not shared by $\Lambda_k$ for $k\ge2$ .

20. Campaign 30 rejection filters

Reject a candidate if:

R1. It only introduces another $\Lambda_k$ or almost-prime average.

R2. It reproduces EMBF/EMDQO without a direct PESC bridge.

R3. It uses parity-breaking as a slogan without an exponent ledger.

R4. It obtains only logarithmic/subpower precision.

R5. It assumes a fixed zero strip.

R6. It creates another positive higher-moment target.

21. External calibration

Current/classical calibration:

For fixed $k$ , $$ \sum_{n\le x}\Lambda_k(n)

xR_k(\log x)+O_k(x), $$ with $R_k$ of degree $k-1$ and leading term $k(\log x)^{k-1}$ .
The generalized von Mangoldt functions satisfy $$ \Lambda_{k+1}

\Lambda_k\log
\Lambda*\Lambda_k. $$
Higher-weight elementary methods historically improve the PNT remainder through arbitrary logarithmic powers and stretched-exponential subpower scales.
Bombieri's asymptotic-sieve perspective shows that very rich higher-almost-prime information can still leave the prime/parity component undetermined.

These facts motivate the next parity-sensitive return to PESC.

22. State transition

CSM_RH v1.20
  ->
CSM_RH v1.21

with:

Campaign 29
  CLOSED_AS_HIGHER_ORDER_SELBERG_AMPLIFIER_AUDIT

O-RH-064
  CREATED / CERTIFIED

O-RH-065
  CREATED / CERTIFIED AS CURRENT-THEOREM AUDIT

O-RH-066
  CREATED / CERTIFIED AS CURRENT-METHOD UNIFORMITY BARRIER

F-RH-010
  PESC
  REMAINS OPEN / ROOT TARGET

F-RH-016
  MLEPG
  REMAINS OPEN

Campaign 30
  PARITY_SENSITIVE_PESC_ATTACK
  READY

23. Final status

RH = OPEN

PESC = OPEN

MLEPG = OPEN

FIXED-ORDER HIGHER SELBERG = EXPONENT-NEUTRAL

FINITE FIXED-ORDER FILTER = NO CERTIFIED REMAINDER CANCELLATION

SUBCRITICAL GROWING ORDER = SUBPOWER

FIXED-POWER CRITICAL ORDER = log X / log log X

CURRENT UNIFORMITY AT CRITICAL ORDER = INSUFFICIENT

HIGHER SELBERG = REAL SUBPOWER AMPLIFIER, NOT CERTIFIED FIXED-POWER AMPLIFIER

NEXT CAMPAIGN = 30

The central scale law is

\boxed{ (\log X)^{-k} = X^{-\delta} \quad\Longleftrightarrow\quad k \sim \delta \frac{\log X}{\log\log X}. }

At precisely this order,

\boxed{ (\log X)^k = X^{\delta+o(1)} }

and factorial-type $k$ -dependence also enters polynomial exponent scale.

The current fixed- $k$ generalized Selberg theory does not provide a certified net fixed power across this threshold.

CSM_RH Paper 30 — Fixed-Order Selberg Exponent Invariance and the Growing-Order Complexity Threshold

CSM_RH Paper 30

Fixed-Order Selberg Exponent Invariance and the Growing-Order Complexity Threshold

0. Trust boundary

1. Generalized von Mangoldt functions

2. Fixed-k summatory polynomial

Theorem 2.1 — Fixed-Order Generalized Selberg Summatory Formula

3. Complete polynomial subtraction

4. Fixed-zero response of Lambda-k

Theorem 4.1 — Fixed-Order Zero-Exponent Invariance

5. Fixed-k forcing versus fixed Mellin mode

forcing

zero/drift mode

6. Finite fixed-order linear combinations

7. Main-term cancellation does not cancel unknown remainders

8. Historical calibration: higher weight really does amplify

9. Critical growing order

Theorem 9.1 — Fixed-Power Order Scale

10. Pointwise weight complexity at the critical order

11. Factorial proof-complexity scale

12. Support complexity at large order

13. Growing-order verdict

Regime I — fixed k

Regime II — subcritical growing k

Regime III — critical order

14. Campaign 29 track audit

H1 — fixed-k generalized Selberg response

H2 — finite linear-combination forcing cancellation

H3 — growing-k amplification

H4 — generalized almost-prime to PESC transfer

H5 — spectral polynomial in Selberg operator

15. Campaign 29 verdict

16. New certified obstruction package

17. Return to the arithmetic root

18. Campaign 30

19. Campaign 30 tracks

Q1 — PESC parity decomposition

Q2 — asymptotic-sieve scalar defect

Q3 — parity-breaking bilinear input

Q4 — prime-versus-almost-prime drift comparator

Q5 — direct residual sign structure

20. Campaign 30 rejection filters

R1. It only introduces another Λk\Lambda_kΛk​ or almost-prime average.

R2. It reproduces EMBF/EMDQO without a direct PESC bridge.

R3. It uses parity-breaking as a slogan without an exponent ledger.

R4. It obtains only logarithmic/subpower precision.

R5. It assumes a fixed zero strip.

R6. It creates another positive higher-moment target.

21. External calibration

For fixed kkk, $$ \sum_{n\le x}\Lambda_k(n)

The generalized von Mangoldt functions satisfy $$ \Lambda_{k+1}

22. State transition

23. Final status

R1. It only introduces another $\Lambda_k$ or almost-prime average.

For fixed $k$ , $$ \sum_{n\le x}\Lambda_k(n)