CSM_RH Paper 25

Collision Elimination, Raw-Sieve Centering Debt, and the Fully Distinct Four-Point Core

Project: CSM_RH
Paper: 25
Version: v0.1
Date: 2026-09-06
Parent state: CSM_RH v1.15 / Paper 24
Campaign: 24 — CENTERED_FOURTH_MOMENT_ATTACK
Status: exact fourth-moment reduction / current-technology 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 24 introduced C4HEG:

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

for some fixed $\eta>0$ at a polynomial lag $H=X^\alpha$ .

Campaign 24 asks whether this is already contained in known sieve/higher-uniformity technology and, if not, which part of the fourth moment is genuinely new.

The answer is:

collision patterns:
  harmless at H^3 scale

fully distinct four-point sector:
  only H^4-sized combinatorial sector

raw Gallagher/Klimov moment bounds:
  uncentered; no legal centered cancellation

2026 higher uniformity:
  powerful but log/subpower for Lambda

fixed H-exponent gain:
  still open

No live GLM-5.3-Flash run is claimed.

1. Centered fourth short-interval moment

Let

a_n = \Lambda(n)-1

on a finite prime block and extend by zero outside.

Define

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

Define

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

Since $a_n$ is real, no complex-conjugation convention is needed here.

Expand:

\boxed{ \mathcal M_{4,X}(H) = \sum_{1\le r_1,r_2,r_3,r_4\le H} \sum_x a_{x+r_1} a_{x+r_2} a_{x+r_3} a_{x+r_4}. }

2. Partition by offset collisions

The ordered quadruple of offsets has one of the five multiplicity partitions:

[4]
[3,1]
[2,2]
[2,1,1]
[1,1,1,1]

Write

\mathcal M_4 = \mathcal Q_{4} + \mathcal Q_{31} + \mathcal Q_{22} + \mathcal Q_{211} + \mathcal Q_{1111}.

The exact pieces are:

All four equal

\boxed{ \mathcal Q_4 = \sum_{r} \sum_x a_{x+r}^4. }

Three plus one

\boxed{ \mathcal Q_{31} = 4 \sum_{r\ne s} \sum_x a_{x+r}^3a_{x+s}. }

Two plus two

\boxed{ \mathcal Q_{22} = 6 \sum_{r<s} \sum_x a_{x+r}^2a_{x+s}^2. }

Two plus one plus one

\boxed{ \mathcal Q_{211} = 12 \sum_r \sum_{\substack{s<t\\s,t\ne r}} \sum_x a_{x+r}^2a_{x+s}a_{x+t}. }

Fully distinct

\boxed{ \mathcal Q_{1111} = 24 \sum_{r_1<r_2<r_3<r_4} \sum_x a_{x+r_1} a_{x+r_2} a_{x+r_3} a_{x+r_4}. }

Define

\boxed{ \mathcal A_{4,\mathrm{dist}}(X,H) = \mathcal Q_{1111}. }

3. Collision bound

On the relevant range,

|a_n| \ll \log X.

Every collision partition has at most three free offset parameters.

There are $O(X)$ contributing translations $x$ .

Therefore:

Theorem 3.1 — Collision Sector Bound

\boxed{ | \mathcal Q_{4} | + | \mathcal Q_{31} | + | \mathcal Q_{22} | + | \mathcal Q_{211} | \ll XH^3 (\log X)^4. }

No prime-correlation theorem is needed.

The estimate is purely combinatorial.

4. Fully distinct reduction

Let

0<\eta\le1.

Suppose

\boxed{ \mathcal A_{4,\mathrm{dist}}(X,H) \ll XH^{4-\eta} (\log X)^C }

for some fixed $C$ .

Then Theorem 3.1 gives

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

Therefore:

Theorem 4.1 — Distinct Four-Point Gain Implies C4HEG

For any fixed $0<\eta\le1$ ,

\boxed{ \operatorname{D4HEG}(\alpha,\eta) \Longrightarrow \operatorname{C4HEG}(\alpha,\eta), }

where D4HEG is the one-sided upper bound on the fully distinct centered aggregate.

A separate power theorem for each four-point shift tuple is not required.

Only the complete signed three-dimensional shift aggregate needs the $H$ -exponent gain.

5. Automatic lower bound on the distinct aggregate

Because

\mathcal M_{4,X}(H)\ge0,

Theorem 3.1 also gives

\boxed{ \mathcal A_{4,\mathrm{dist}}(X,H) \ge - O \left( XH^3(\log X)^4 \right). }

Thus for $0<\eta\le1$ , a one-sided upper bound

\mathcal A_{4,\mathrm{dist}} \ll XH^{4-\eta}\operatorname{polylog}X

already controls the distinct sector at the required scale.

Large negative cancellation is not a separate danger.

6. Centered four-point correlation

For four distinct offsets

r_1,r_2,r_3,r_4,

define

\boxed{ C_a(r_1,r_2,r_3,r_4) = \sum_x \prod_{i=1}^{4} [ \Lambda(x+r_i)-1 ]. }

Then

\boxed{ \mathcal A_{4,\mathrm{dist}} = 24 \sum_{r_1<r_2<r_3<r_4} C_a(r_1,r_2,r_3,r_4). }

Expansion by subsets gives the exact inclusion-exclusion identity

\boxed{ C_a(\mathbf r) = \sum_{S\subseteq\{1,2,3,4\}} (-1)^{4-|S|} \sum_x \prod_{i\in S} \Lambda(x+r_i). }

The empty product is interpreted as $1$ .

Thus centering couples the one-, two-, three-, and four-point prime correlations with exact signed coefficients.

7. Why raw sieve moments do not prove the centered result

Classical Gallagher/Klimov sieve moment bounds control the uncentered discrete moment

\widetilde J_k(X,H) = \sum_{m\le X} [ \psi(m+H)-\psi(m) ]^k

at the schematic scale

\boxed{ \widetilde J_k(X,H) \le \left[ P_k \left( \frac{H}{\log X} \right) + \varepsilon \right] X\log^kX. }

For

P_k(y) = \sum_{r=1}^{k} \left\{ \begin{matrix} k\\r \end{matrix} \right\} 2^r r!y^r,

the fourth-order polynomial is

\boxed{ P_4(y) = 2y + 56y^2 + 288y^3 + 384y^4. }

Hence at polynomial $H$ the raw upper-bound scale contains an $XH^4$ leading term.

8. Centering requires coherent signed main terms

The centered moment is

\sum_m [ Y_m-H ]^4,

where

Y_m = \psi(m+H)-\psi(m).

Exactly,

\boxed{ \sum_m(Y_m-H)^4 = J_4 - 4HJ_3 + 6H^2J_2 - 4H^3J_1 + XH^4 }

up to harmless endpoint conventions.

To cancel the $H^4$ and $H^3$ scales, one needs compatible asymptotic main terms for the separate moments $J_r$ .

Independent one-sided upper bounds do not permit this cancellation.

The Gallagher/Klimov constants in $P_r$ are sieve-majorant constants, not a coherent signed probability law.

Create:

O-RH-053
UNCENTERED_SIEVE_MOMENT_CENTERING_DEBT
status:
  CERTIFIED

Statement:

An upper bound for each uncentered moment cannot be inserted into the alternating binomial formula for the centered moment as though the upper-bound main coefficients cancelled.

9. Conditional higher-moment calibration

Montgomery and Soundararajan predict an approximately Gaussian centered short-interval prime count with variance

H\log(X/H)

at polynomial scales.

Their higher-moment formulas, under strong Hardy–Littlewood input, predict for the fourth centered moment a scale

\boxed{ X [ H\log(X/H) ]^2. }

Chan's higher-moment work studies equivalences between higher even short-interval moments under RH and related strong information.

These results show that the expected centered scale is much smaller than $XH^4$ .

They do not provide an unconditional C4HEG theorem.

10. 2026 higher-uniformity calibration

Current higher-uniformity theory gives for the von Mangoldt function:

\Lambda-\Lambda^\sharp

is highly discorrelated from nilsequences on almost all short intervals, with arbitrary logarithmic accuracy in the stated ranges.

It also gives asymptotically small short-interval Gowers norms and, as an application, $\ell$ -point Hardy–Littlewood correlations with one averaging shift variable.

For the prime case the quantitative gain remains logarithmic/subpower rather than a fixed power in $X$ or $H$ .

Therefore using the generalized von Neumann theorem on the fully distinct sector can at current strength yield at best a subpower relative error unless a new power-saving uniformity estimate is supplied.

11. One-shift versus three-shift geometry

The explicit 2026 Hardy–Littlewood application has one averaging parameter:

\sum_{n\le X} \Lambda(n) \Lambda(n+h) \cdots \Lambda(n+(\ell-1)h).

The fully distinct fourth-moment aggregate has three independent relative shift degrees of freedom:

r_2-r_1, \qquad r_3-r_1, \qquad r_4-r_1.

The underlying Gowers-uniformity machinery can address richer finite-complexity systems, but its current quantitative prime saving remains subpower.

Thus increasing shift dimension does not manufacture an $H^{-\eta}$ gain from the existing log-power estimates.

12. Contrast with divisor functions

The 2026 higher-uniformity paper proves, for $d_2$ against a fixed linear phase in a suitable short-interval range, a genuine power-saving estimate of the form

X^{-c\varepsilon}.

The corresponding general $\Lambda$ discorrelation results in that work remain at arbitrary logarithmic saving.

This contrast is useful calibration:

power-saving short-interval discorrelation:
  reachable for some divisor problems

same type of fixed power for Lambda:
  not presently supplied

The prime-specific obstruction is therefore visible inside the current technology itself.

13. Current four-point core

After collision elimination, the first genuinely unresolved nonlinear arithmetic object is

\boxed{ \mathcal A_{4,\mathrm{dist}}(X,H) = 24 \sum_{r_1<r_2<r_3<r_4\le H} \sum_x \prod_{i=1}^{4} [ \Lambda(x+r_i)-1 ]. }

A sufficient theorem is:

D4HEG $(\alpha,\eta)$

For

H=X^\alpha

and some fixed

0<\eta\le1,

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

This contains no fixed $X$ -power in its statement.

At polynomial $H$ it generates one.

14. Strength chain

D4HEG gives:

\boxed{ \operatorname{D4HEG} \Longrightarrow \operatorname{C4HEG} }

by Theorem 4.1.

Paper 24 gives:

\boxed{ \operatorname{C4HEG} \Longrightarrow \operatorname{MLEPG} }

through Hölder at one polynomial scale.

Paper 17 gives:

\boxed{ \operatorname{MLEPG} \Longrightarrow \text{fixed-power PNT mean square}. }

Paper 20 gives:

\boxed{ \text{fixed-power PNT mean square} \Longrightarrow \text{fixed zeta zero strip}. }

No arrow in this chain is claimed to prove the input D4HEG.

15. Campaign 24 track audit

M1 — averaged four-point Hardy–Littlewood

status:
  CURRENT ONE-SHIFT THEOREMS / GOWERS INPUT GIVE SUBPOWER PRECISION

fixed H-exponent gain:
  NOT IDENTIFIED

M2 — centered Selberg/Klimov moment method

status:
  RAW MOMENTS CONTROLLED
  CENTERING CANCELLATION NOT CERTIFIED FROM UPPER BOUNDS

M3 — short-interval higher uniformity

status:
  STRONG STRUCTURAL CONTROL
  PRIME QUANTITATIVE SAVING LOG/SUBPOWER

M4 — sieve approximant plus residual

status:
  MODEL COMPUTABLE
  RESIDUAL FOURTH-MOMENT POWER FIDELITY OPEN

M5 — fourth cumulant

status:
  USEFUL ORGANIZATION
  CONNECTED FULLY DISTINCT SECTOR REMAINS OPEN

16. New obstruction: collision shell is not the barrier

Create:

O-RH-054
FOURTH_MOMENT_COLLISION_SHELL_HARMLESS
status:
  CERTIFIED

Statement:

All fourth-moment offset configurations containing a collision contribute at most $XH^3(\log X)^4$ . For any target gain $0<\eta\le1$ , the fixed $H$ -exponent obstruction lies entirely in the fully distinct sector.

17. New mechanism subfrontier

Create:

F-RH-019
FULLY_DISTINCT_CENTERED_FOUR_POINT_H_EXPONENT_GAP
abbrev:
  D4HEG
status:
  OPEN
type:
  MECHANISM SUBFRONTIER

Root target remains:

F-RH-010
PESC

Direct theorem candidate remains:

F-RH-016
MLEPG

Nonlinear theorem candidate remains:

F-RH-018
C4HEG

D4HEG is the first irreducible four-point sublemma for C4HEG.

18. Campaign 25

The next campaign is:

CSM_RH Campaign 25
FULLY_DISTINCT_FOUR_POINT_ATTACK

The target does not move beyond D4HEG as a sublemma.

19. Campaign 25 tracks

F1 — exact $\Lambda^\sharp$ / residual decomposition

Write

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

inside the fully distinct aggregate.

Compute the pure model term exactly enough to identify its $H$ exponent.

Audit every mixed residual term without early absolute-value leakage.

F2 — three-dimensional generalized von Neumann

Use short-interval $U^s$ uniformity to control the fully distinct three-shift average.

Determine the exact quantitative dependence needed to turn log-power uniformity into an $H^{-\eta}$ saving.

F3 — singular-series centered average

Average the four-point singular series over all distinct triples of relative shifts and determine which lower-order one-, two-, and three-point model terms cancel after exact centering.

F4 — connected fourth cumulant

Define the connected prime four-point function after subtracting all pairings and lower-order singular-series contributions.

Test whether C4HEG reduces to an $H$ -exponent bound on that connected component.

F5 — divisor/primes technology transfer

Use the power-saving $d_2$ short-interval discorrelation theorem as a model proof.

Locate the exact input which fails when $d_2$ is replaced by $\Lambda$ .

20. Campaign 25 rejection filters

Reject a candidate if:

R1. It applies uncentered moment upper bounds inside a signed centered identity.

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

R3. It gets only $\log^{-A}X$ or $H^{-o(1)}$ .

R4. It proves a fixed power only for collision configurations.

R5. It takes absolute values of all distinct four-point correlations before using the three-dimensional average.

R6. It treats one-shift Hardy–Littlewood as though it directly controlled all three independent shift variables.

21. State transition

CSM_RH v1.15
  ->
CSM_RH v1.16

with:

Campaign 24
  CLOSED_AS_COLLISION_REDUCTION_AND_CURRENT_TECHNOLOGY_AUDIT

O-RH-053
  UNCENTERED_SIEVE_MOMENT_CENTERING_DEBT
  CREATED / CERTIFIED

O-RH-054
  FOURTH_MOMENT_COLLISION_SHELL_HARMLESS
  CREATED / CERTIFIED

F-RH-018
  C4HEG
  REMAINS OPEN

F-RH-019
  D4HEG
  CREATED / OPEN / MECHANISM SUBFRONTIER

Campaign 25
  FULLY_DISTINCT_FOUR_POINT_ATTACK
  READY

22. Final status

RH = OPEN

PESC = OPEN

MLEPG = OPEN

C4HEG = OPEN

D4HEG = OPEN

COLLISION FOURTH-MOMENT SECTOR = CLOSED / HARMLESS

RAW SIEVE MOMENT -> CENTERED MOMENT = INVALID WITHOUT COHERENT MAIN TERMS

2026 PRIME HIGHER UNIFORMITY = SUBPOWER QUANTITATIVE STRENGTH

FULLY DISTINCT THREE-SHIFT AVERAGE = CURRENT FOUR-POINT CORE

NEXT CAMPAIGN = 25

The concrete open sublemma is:

\boxed{ \sum_{1\le r_1<r_2<r_3<r_4\le X^\alpha} \sum_x \prod_{i=1}^{4} [ \Lambda(x+r_i)-1 ] \ll X \left( X^\alpha \right)^{4-\eta} (\log X)^{O(1)} }

for some fixed $\alpha\in(0,1)$ and some fixed $\eta\in(0,1]$ .

A fixed $H$ -exponent gain in this fully distinct aggregate is enough to generate a fixed prime-side power through the already certified chain.

CSM_RH Paper 25 — Collision Elimination, Raw-Sieve Centering Debt, and the Fully Distinct Four-Point Core

CSM_RH Paper 25

Collision Elimination, Raw-Sieve Centering Debt, and the Fully Distinct Four-Point Core

0. Trust boundary

1. Centered fourth short-interval moment

2. Partition by offset collisions

All four equal

Three plus one

Two plus two

Two plus one plus one

Fully distinct

3. Collision bound

Theorem 3.1 — Collision Sector Bound

4. Fully distinct reduction

Theorem 4.1 — Distinct Four-Point Gain Implies C4HEG

5. Automatic lower bound on the distinct aggregate

6. Centered four-point correlation

7. Why raw sieve moments do not prove the centered result

8. Centering requires coherent signed main terms

9. Conditional higher-moment calibration

10. 2026 higher-uniformity calibration

11. One-shift versus three-shift geometry

12. Contrast with divisor functions

13. Current four-point core

D4HEG (α,η)(\alpha,\eta)(α,η)

14. Strength chain

15. Campaign 24 track audit

M1 — averaged four-point Hardy–Littlewood

M2 — centered Selberg/Klimov moment method

M3 — short-interval higher uniformity

M4 — sieve approximant plus residual

M5 — fourth cumulant

16. New obstruction: collision shell is not the barrier

17. New mechanism subfrontier

18. Campaign 25

19. Campaign 25 tracks

F1 — exact Λ♯\Lambda^\sharpΛ♯ / residual decomposition

F2 — three-dimensional generalized von Neumann

F3 — singular-series centered average

F4 — connected fourth cumulant

F5 — divisor/primes technology transfer

20. Campaign 25 rejection filters

R1. It applies uncentered moment upper bounds inside a signed centered identity.

R2. It assumes the prime 444 -tuple conjecture.

R3. It gets only log⁡−AX\log^{-A}Xlog−AX or H−o(1)H^{-o(1)}H−o(1).

R4. It proves a fixed power only for collision configurations.

R5. It takes absolute values of all distinct four-point correlations before using the three-dimensional average.

R6. It treats one-shift Hardy–Littlewood as though it directly controlled all three independent shift variables.

21. State transition

22. Final status

D4HEG $(\alpha,\eta)$

F1 — exact $\Lambda^\sharp$ / residual decomposition

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

R3. It gets only $\log^{-A}X$ or $H^{-o(1)}$ .