CSM_RH Paper 24

Prime-Like Spike–Drift Countermodels and Fourth-Moment $H$ -Exponent Amplification

Project: CSM_RH
Paper: 24
Version: v0.1
Date: 2026-09-06
Parent state: CSM_RH v1.14 / Paper 23
Campaign: 23 — PRIME_SIDE_FIXED_POWER_GENERATION_II
Status: prime-side candidate generation / one-point obstruction 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

Campaign 23 asks for prime-specific information which can stop a smooth low-frequency drift from coexisting with large microscopic prime-like jump energy.

Two results are obtained.

One-point prime-like constraints are still insufficient: an explicit sparse nonnegative spike sequence can have prime-scale jump energy and a persistent smooth cumulative drift.
A centered fourth-moment exponent gap in the interval length $H$ is a genuine fixed-power generator. The hypothesis contains no power of $X$ ; a power of $X$ appears only after setting $H=X^\alpha$ and applying Hölder plus the residue-chain bridge.

No fixed-power fourth-moment theorem is proved.

1. Prime-like one-point data

On a dyadic block around $X$ , the prime-only or von-Mangoldt increment has several elementary one-point features:

nonnegative uncentered detector;
typical support density about 1/log X;
spike height about log X;
centered increment bounded below by -1;
one-step centered energy about X log X.

Paper 23 showed that large one-step energy alone cannot stop a smooth low-frequency drift.

The next question is whether adding the one-sided / sparse spike structure changes this conclusion.

It does not.

2. Spike–drift construction

Fix

\frac12<\beta<1.

Let

L=L_X

be an integer with

L\asymp\log X.

Define

\boxed{ M_\beta(n) = \left\lfloor \frac{ n+n^\beta }{ L } \right\rfloor. }

For sufficiently large $n$ ,

0< \frac{ (n+1)+(n+1)^\beta - n-n^\beta }{ L } <1.

Hence

M_\beta(n)-M_\beta(n-1) \in \{0,1\}.

Define the nonnegative sparse detector

\boxed{ q_n = L [ M_\beta(n)-M_\beta(n-1) ]. }

Thus

\boxed{ q_n\in\{0,L\}. }

Define the centered increment

\boxed{ c_n=q_n-1. }

Then

\boxed{ c_n\ge-1. }

3. Exact cumulative drift

Let

B(n)=\sum_{k\le n}c_k.

By telescoping,

B(n) = L M_\beta(n)-n + O(L)

depending only on the harmless lower endpoint convention.

Therefore:

Theorem 3.1 — Prime-Like Smooth Drift

\boxed{ B(n) = n^\beta + O(L). }

Thus a sparse nonnegative spike sequence with prime-like one-point scale can carry a persistent sublinear power drift.

4. Correct spike density

On a dyadic block

X<n\le2X,

the number of spikes is

\begin{aligned} M_\beta(2X)-M_\beta(X) &= \frac{ X+(2^\beta-1)X^\beta }{ L } + O(1) \\ &\sim \frac XL. \end{aligned}

Therefore the support density is

\boxed{ \frac1L \asymp \frac1{\log X}. }

The uncentered total mass is

\sum_{X<n\le2X}q_n = X + (2^\beta-1)X^\beta + O(L),

which is a main term $X$ plus a sublinear drift.

5. One-step energy

At a spike,

c_n=L-1.

Away from a spike,

c_n=-1.

Let $K_X\sim X/L$ be the number of spikes in the dyadic block.

Then

\begin{aligned} \sum_{X<n\le2X}c_n^2 &= K_X(L-1)^2 + (X-K_X) \\ &= XL + O(X). \end{aligned}

Hence:

Theorem 5.1 — Prime-Scale Jump Energy

\boxed{ \sum_{X<n\le2X}c_n^2 \asymp X\log X. }

The model therefore reproduces the prime jump-energy exponent.

6. Polynomial-scale lag drift

Define

U_H(x) = B(x+H)-B(x).

For

x\asymp X, \qquad H=o(X),

Theorem 3.1 and Taylor expansion give

\boxed{ U_H(x) = \beta H X^{\beta-1} + O_\beta \left( H^2X^{\beta-2} + L \right) }

uniformly at exponent scale.

\boxed{ H X^{\beta-1} \gg L, }

then the smooth drift dominates the floor/spike discrepancy.

For

H=X^\alpha,

this holds whenever

\boxed{ \alpha>1-\beta. }

7. Long-scale energy and defect

In the drift-dominated regime,

\boxed{ \mathcal S_{\rm drift}(X,H) \asymp H^2X^{2\beta-1}. }

The adjacent-block smooth defect has size

U_H(x+H)-U_H(x) = O_\beta \left( H^2X^{\beta-2} + L \right).

Consequently,

\boxed{ \frac{ \mathcal D(X,H) }{ \mathcal S(X,H) } \ll_\beta \left( \frac HX + \frac{ L }{ HX^{\beta-1} } \right)^2. }

For every fixed

\alpha>1-\beta, \qquad H=X^\alpha,

the right side tends to zero.

Thus the model has:

prime-like sparse nonnegative spikes;
centered lower bound -1;
jump energy X log X;
smooth cumulative power drift;
polynomial-scale critical locking.

8. One-point arithmetic uncertainty is insufficient

Create:

O-RH-051
PRIME_LIKE_ONE_POINT_CONSTRAINTS_INSUFFICIENT
status:
  CERTIFIED BY EXPLICIT COUNTERMODEL

Statement:

Nonnegativity of the uncentered detector, support density $1/\log X$ , spike height $\log X$ , centered lower bound $-1$ , correct total mass at leading order, and one-step energy $X\log X$ do not force polynomial-scale spectral deconcentration or PDSD.

Therefore any successful low-frequency arithmetic theorem must use genuinely higher-order information about prime locations.

9. Nonlinear moment route

The countermodel motivates a higher-correlation candidate.

Return to the actual centered von Mangoldt sequence

a_n=\Lambda(n)-1

on a finite prime block and define

U_H(x) = \sum_{r=1}^{H}a_{x+r}.

Define the centered fourth lag moment

\boxed{ \mathcal M_{4,X}(H) = \sum_x |U_H(x)|^4. }

The completely trivial pointwise estimate gives the scale

\mathcal M_{4,X}(H) \ll XH^4 (\log X)^{O(1)}.

The new candidate asks only for any fixed improvement in the exponent of $H$ .

10. Fourth-Moment $H$ -Exponent Gap

Fix

0<\alpha<1, \qquad H=X^\alpha.

Fix

\eta>0.

Define:

C4HEG $(\alpha,\eta)$

\boxed{ \mathcal M_{4,X}(H) \ll XH^{4-\eta} (\log X)^{O(1)}. }

The acronym is:

C4HEG
CENTERED FOURTH-MOMENT H-EXPONENT GAP

No factor $X^{-\delta}$ appears in the hypothesis.

The only saving is a fixed power of the interval length $H$ .

11. Hölder amplification

There are only

O(X)

nonzero zero-extended interval positions when $H<X$ .

Therefore Hölder / Cauchy gives

\begin{aligned} \mathcal S_X(H) &= \sum_x|U_H(x)|^2 \\ &\le O(X)^{1/2} \mathcal M_{4,X}(H)^{1/2}. \end{aligned}

Under C4HEG,

\boxed{ \mathcal S_X(H) \ll XH^{2-\eta/2} (\log X)^{O(1)}. }

Since

H=X^\alpha,

this becomes

\boxed{ \mathcal S_X(H) \ll XH^2 X^{-\alpha\eta/2} (\log X)^{O(1)}. }

Thus a fixed $H$ -exponent gap becomes a fixed $X$ -power.

12. Global mean-square consequence

Apply Paper 17's corrected residue-chain bridge:

\sum_{n\le2X}|A(n)|^2 \ll \left( \frac XH \right)^2 \mathcal S_X(H) + XH^2,

where

A(n)=\psi(n)-n.

Under C4HEG,

\boxed{ \sum_{n\le2X}|A(n)|^2 \ll X^3H^{-\eta/2} (\log X)^{O(1)} + XH^2. }

For

H=X^\alpha,

the exponents are

3-\frac{\alpha\eta}{2}

and

1+2\alpha.

Therefore:

Theorem 12.1 — Fourth-Moment Exponent Amplification

C4HEG $(\alpha,\eta)$ implies

\boxed{ \sum_{n\le2X} |\psi(n)-n|^2 \ll X^{3-\kappa+o(1)} }

for every sufficiently small fixed

\boxed{ 0<\kappa < \min \left\{ \frac{\alpha\eta}{2}, 2-2\alpha \right\}. }

Paper 20's Mellin pole recovery then produces a fixed zero-free strip.

Thus C4HEG is a genuine prime-side fixed-power generator.

13. General $p$ -moment amplification law

The same mechanism is not specific to the fourth moment.

Let

p>2

be fixed and suppose

\boxed{ \sum_x|U_H(x)|^p \ll XH^{p-\eta} X^{o(1)}. }

Hölder gives

\boxed{ \mathcal S_X(H) \ll XH^{2-2\eta/p} X^{o(1)}. }

H=X^\alpha,

the generated fixed power is

\boxed{ \delta_p = \frac{ 2\alpha\eta }{ p }. }

The fourth moment is the first concrete even-moment case.

14. Natural conjectural scale

Montgomery and Soundararajan predict that in polynomial short intervals the centered prime count is approximately Gaussian with variance

H\log(X/H).

The corresponding fourth moment scale is

\boxed{ \mathcal M_{4,X}(H) \asymp X [ H\log(X/H) ]^2. }

Ignoring logarithms, this is

XH^2.

Thus the conjectural behavior corresponds to an $H$ -exponent improvement of essentially

\eta=2.

C4HEG asks for only:

\boxed{ \text{some fixed }\eta>0. }

It is far weaker than the full Gaussian fourth-moment conjecture.

This is plausibility calibration only.

15. Current sieve moment scale

Classical Gallagher/Klimov-type sieve moment estimates give unconditional upper bounds for uncentered short-interval prime counts of the schematic form

\sum_{m\le X} [ \psi(m+H)-\psi(m) ]^k \ll P_k \left( \frac H{\log X} \right) X(\log X)^k.

For fixed $k=4$ and polynomial

H\gg\log X,

the leading term has size

\boxed{ XH^4 }

up to a fixed constant and lower-order terms.

Thus standard raw sieve-moment control does not produce a positive $\eta$ in C4HEG.

The missing gain must come from centering and higher correlation cancellation.

Create:

O-RH-052
RAW_SIEVE_FOURTH_MOMENT_H4_FLOOR
status:
  CERTIFIED AS CURRENT-METHOD PRECISION AUDIT

This is not a universal impossibility theorem for sieve methods.

16. Four-point correlation expansion

Expanding the fourth moment gives a weighted sum of centered four-point correlations:

\boxed{ \mathcal M_{4,X}(H) = \sum_{h_1,h_2,h_3} W_H(h_1,h_2,h_3) \sum_n a_n a_{n+h_1} a_{n+h_2} a_{n+h_3}, }

where $W_H$ is a nonnegative intersection-count kernel with

0\le W_H\le H

and support on

|h_i|\ll H.

The coefficient-blind count has scale $XH^4$ .

Therefore any C4HEG proof must exploit genuine cancellation or structure in a three-dimensional average of centered four-point prime correlations.

This is the first place in the current branch where one-point prime-like countermodels are automatically excluded by the theorem statement.

17. Campaign 23 track audit

G2-1 — arithmetic low-frequency uncertainty

status:
  ONE-POINT VERSION REJECTED

countermodel:
  prime-like sparse spike + smooth drift

A higher-correlation version may still exist.

G2-2 — prime correlation with smooth self-generated drift

status:
  CURRENT AVERAGE PRIME-CORRELATION PRECISION SUBPOWER
  NO FIXED-POWER CONTRADICTION FOUND

G2-3 — nonlinear short-interval energy transfer

status:
  SURVIVOR

output:
  C4HEG

G2-4 — AP variance to additive-scale transfer

status:
  NO NEW BRIDGE

reason:
  common q=1 low-frequency mode remains the transfer bottleneck

G2-5 — direct PESC prime sampling

status:
  ROOT TARGET / OPEN

18. New canonical theorem candidate

Create:

F-RH-018
CENTERED_FOURTH_MOMENT_H_EXPONENT_GAP
abbrev:
  C4HEG
status:
  OPEN
type:
  PRIME-SIDE NONLINEAR FIXED-POWER GENERATOR

Its hypothesis contains no $X$ -power saving.

The fixed power is generated by:

H-exponent saving
  +
polynomial relation H=X^alpha
  +
Hölder
  +
residue-chain bridge.

19. New survivor

Create:

S-RH-029
CENTERED_FOUR_POINT_PRIME_CORRELATION_CANCELLATION
status:
  OPEN

A successful theorem must show a fixed $H$ -exponent gain after the full three-shift average is assembled.

It need not prove power saving for each four-point correlation separately.

20. Campaign 24

The next campaign is:

CSM_RH Campaign 24
CENTERED_FOURTH_MOMENT_ATTACK

Root target:

F-RH-010
PESC

Working candidates:

F-RH-016
MLEPG

F-RH-018
C4HEG

The scale-decorrelation route PDSD remains optional.

21. Campaign 24 tracks

M1 — averaged four-point Hardy–Littlewood

Keep the full three-dimensional shift average before absolute values.

Determine whether existing multi-correlation technology yields any fixed saving in the $H$ exponent even though it gives only logarithmic precision shiftwise.

M2 — centered Selberg/Klimov moment method

Redo the classical sieve moment expansion after exact centering by $H$ .

Identify whether the $H^4$ leading contributions cancel algebraically and where the first uncontrolled centered correlation enters.

M3 — short-interval higher uniformity

Test whether current $U^s$ / nilsequence uniformity of $\Lambda-\Lambda^\sharp$ can control enough of the four-point aggregate to reduce the $H$ exponent.

Log-power improvement alone does not count.

M4 — sieve approximant plus exact fourth-moment residual

Decompose

\Lambda-1 = (\Lambda^\sharp-1) + (\Lambda-\Lambda^\sharp)

and keep all mixed fourth-moment terms signed until the exponent ledger is complete.

M5 — direct fourth-cumulant method

Separate Gaussian pairings from the connected fourth cumulant.

C4HEG needs only an $H$ -exponent gain for the total fourth moment, not an asymptotic Gaussian law.

22. Campaign 24 rejection filters

Reject a candidate if:

R1. It uses only uncentered moment bounds.

R2. It assumes the prime $4$ -tuple conjecture.

R3. It proves only a logarithmic saving over $XH^4$ .

R4. It inserts an $X^{-\delta}$ hypothesis directly.

R5. It treats the conjectural Gaussian fourth moment as proved.

R6. It takes absolute values of every centered four-point correlation before the proposed cancellation.

23. External calibration

Relevant literature:

Montgomery and Soundararajan, Primes in short intervals, Comm. Math. Phys. 252 (2004), develop evidence and conditional moment formulae predicting an approximately Gaussian centered prime count with variance $H\log(X/H)$ on polynomial scales.
Gallagher's short-interval moment method and later Bazzanella–Languasco–Zaccagnini estimates give unconditional sieve upper bounds for uncentered moments. At fourth order and polynomial $H$ , the leading upper-bound scale remains $XH^4$ .
Chan's higher-moment work studies even and odd centered moments under RH and related strong information, underscoring that sharp higher moments are already closely tied to deep zeta/prime-correlation structure.
Current higher-uniformity theorems for the von Mangoldt function provide arbitrary logarithmic decay in short-interval structured correlations, but no fixed $H$ -exponent gap for the centered fourth moment is identified in this audit.

None proves C4HEG.

24. State transition

CSM_RH v1.14
  ->
CSM_RH v1.15

with:

Campaign 23
  CLOSED_AS_PRIME_SIDE_CANDIDATE_GENERATION_II

O-RH-051
  PRIME_LIKE_ONE_POINT_CONSTRAINTS_INSUFFICIENT
  CREATED / CERTIFIED BY COUNTERMODEL

O-RH-052
  RAW_SIEVE_FOURTH_MOMENT_H4_FLOOR
  CREATED / CERTIFIED AS CURRENT-METHOD AUDIT

F-RH-018
  C4HEG
  CREATED / OPEN

S-RH-029
  CENTERED_FOUR_POINT_PRIME_CORRELATION_CANCELLATION
  CREATED / OPEN

F-RH-017
  PDSD
  REMAINS OPEN / OPTIONAL

F-RH-016
  MLEPG
  REMAINS OPEN

F-RH-010
  PESC
  REMAINS OPEN / ROOT TARGET

Campaign 24
  CENTERED_FOURTH_MOMENT_ATTACK
  READY

25. Final status

RH = OPEN

PESC = OPEN

MLEPG = OPEN

PDSD = OPEN / OPTIONAL

C4HEG = OPEN

ONE-POINT PRIME-LIKE CONSTRAINT ROUTE = CLOSED

FOURTH-MOMENT FIXED H-EXPONENT GAP = NEW SURVIVOR

CURRENT RAW SIEVE MOMENTS = H^4 SCALE

NEXT CAMPAIGN = 24

The new theorem candidate is:

\boxed{ \sum_x \left| \sum_{r=1}^{X^\alpha} [ \Lambda(x+r)-1 ] \right|^4 \ll X \left( X^\alpha \right)^{4-\eta} (\log X)^{O(1)} }

for some fixed

0<\alpha<1, \qquad \eta>0.

Any fixed positive $\eta$ generates a fixed prime-number-theorem mean-square power through Hölder and the already certified residue-chain bridge.

CSM_RH Paper 24 — Prime-Like Spike–Drift Countermodels and Fourth-Moment $H$-Exponent Amplification

CSM_RH Paper 24

Prime-Like Spike–Drift Countermodels and Fourth-Moment HHH -Exponent Amplification

0. Trust boundary

1. Prime-like one-point data

2. Spike–drift construction

3. Exact cumulative drift

Theorem 3.1 — Prime-Like Smooth Drift

4. Correct spike density

5. One-step energy

Theorem 5.1 — Prime-Scale Jump Energy

6. Polynomial-scale lag drift

7. Long-scale energy and defect

8. One-point arithmetic uncertainty is insufficient

9. Nonlinear moment route

10. Fourth-Moment HHH -Exponent Gap

C4HEG (α,η)(\alpha,\eta)(α,η)

11. Hölder amplification

12. Global mean-square consequence

Theorem 12.1 — Fourth-Moment Exponent Amplification

13. General ppp -moment amplification law

14. Natural conjectural scale

15. Current sieve moment scale

16. Four-point correlation expansion

17. Campaign 23 track audit

G2-1 — arithmetic low-frequency uncertainty

G2-2 — prime correlation with smooth self-generated drift

G2-3 — nonlinear short-interval energy transfer

G2-4 — AP variance to additive-scale transfer

G2-5 — direct PESC prime sampling

18. New canonical theorem candidate

19. New survivor

20. Campaign 24

21. Campaign 24 tracks

M1 — averaged four-point Hardy–Littlewood

M2 — centered Selberg/Klimov moment method

M3 — short-interval higher uniformity

M4 — sieve approximant plus exact fourth-moment residual

M5 — direct fourth-cumulant method

22. Campaign 24 rejection filters

R1. It uses only uncentered moment bounds.

R2. It assumes the prime 444 -tuple conjecture.

R3. It proves only a logarithmic saving over XH4XH^4XH4.

R4. It inserts an X−δX^{-\delta}X−δ hypothesis directly.

R5. It treats the conjectural Gaussian fourth moment as proved.

R6. It takes absolute values of every centered four-point correlation before the proposed cancellation.

23. External calibration

24. State transition

25. Final status

Prime-Like Spike–Drift Countermodels and Fourth-Moment $H$ -Exponent Amplification

10. Fourth-Moment $H$ -Exponent Gap

C4HEG $(\alpha,\eta)$

13. General $p$ -moment amplification law

R2. It assumes the prime $4$ -tuple conjecture.

R3. It proves only a logarithmic saving over $XH^4$ .

R4. It inserts an $X^{-\delta}$ hypothesis directly.