CSM_RH Paper 26

Refined Singular-Series Model Closure and the Lambda-Residual Major-Arc Power Barrier

Project: CSM_RH
Paper: 26
Version: v0.1
Date: 2026-09-06
Parent state: CSM_RH v1.16 / Paper 25
Campaign: 25 — FULLY_DISTINCT_FOUR_POINT_ATTACK
Status: model/residual localization and current-method strength audit; not a proof or disproof of RH

0. Trust boundary

RH_PROVED = FALSE
RH_DISPROVED = FALSE
GLOBAL_RH_CERTIFICATE = FALSE
CSM_RH_ROOT_STATUS = OPEN

Paper 25 reduced the centered fourth-moment candidate to the fully distinct four-point aggregate.

Campaign 25 asks where the missing fixed $H$ -power actually sits.

1. Fully distinct aggregate

Let

a_n=\Lambda(n)-1.

Define

\boxed{ \mathcal A_{4,\mathrm{dist}}(X,H) = \sum_{\substack{ 1\le d_1,d_2,d_3,d_4\le H\\ d_i\text{ distinct} }} \sum_x \prod_{i=1}^{4}a_{x+d_i}. }

D4HEG asks for

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

for some fixed $0<\eta\le1$ .

2. Refined singular series

Define the centered singular series

\boxed{ \mathfrak S_0(\mathcal D) = \sum_{\mathcal Q\subseteq\{1,\ldots,k\}} (-1)^{k-|\mathcal Q|} \mathfrak S(\mathcal D_{\mathcal Q}). }

and

\boxed{ R_k(H) = \sum_{\substack{ 1\le d_1,\ldots,d_k\le H\\ d_i\text{ distinct} }} \mathfrak S_0(d_1,\ldots,d_k). }

For even $k$ , Montgomery–Soundararajan give Gaussian-pairing scale.

For $k=4$ ,

\boxed{ R_4(H) = 3(-H\log H+AH)^2 + O_\varepsilon(H^{2-1/28+\varepsilon}), }

hence

\boxed{ R_4(H)\ll H^2(\log H)^2. }

Therefore the centered Hardy–Littlewood model is not the $H^4$ barrier.

Create:

O-RH-055
REFINED_SINGULAR_SERIES_MODEL_NOT_THE_H4_BARRIER
status: CERTIFIED

3. Actual correlation error

Define

C_{\Lambda,4}(\mathcal D;X) = \sum_x \prod_{i=1}^{4}[\Lambda(x+d_i)-1].

Define

E_4(\mathcal D;X) = C_{\Lambda,4}(\mathcal D;X) - X\mathfrak S_0(\mathcal D).

Then

\boxed{ \mathcal A_{4,\mathrm{dist}}(X,H) = XR_4(H) + \mathcal E_{4,\mathrm{dist}}(X,H), }

where

\mathcal E_{4,\mathrm{dist}}(X,H) = \sum_{\mathcal D\text{ distinct}}E_4(\mathcal D;X).

Thus for $0<\eta\le1$ , a bound

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

implies D4HEG.

4. Lambda-sharp split

Use the 2026 approximant

\Lambda^\sharp(n) = \frac{P(R)}{\varphi(P(R))} 1_{(n,P(R))=1},

with

R=\exp((\log X)^{1/10}).

Set

f=\Lambda-\Lambda^\sharp, \qquad b=\Lambda^\sharp-1.

Then

\Lambda-1=f+b.

Substitution into the fully distinct four-point aggregate yields exactly sixteen sectors:

\boxed{ \mathcal A_{4,\mathrm{dist}} = \sum_{S\subseteq\{1,2,3,4\}} \mathcal T_S. }

The empty $S$ sector is pure model. Every other sector contains at least one true prime residual.

5. Current residual strength

For $H\ge X^{1/3+\varepsilon}$ , current higher-uniformity theory gives

\boxed{ \Lambda-\Lambda^\sharp: \quad H\log^{-A}X }

against nilsequences outside an exceptional set of measure $O(X\log^{-A}X)$ .

The associated Gowers-uniformity strength is subpower.

By contrast, in the same theorem,

\boxed{ d_k-d_k^\sharp: \quad HX^{-c_{k,\dots}} }

outside a power-saving exceptional set.

6. Generalized von Neumann exponent audit

Suppose a transference estimate controls a residual-containing normalized linear-forms average by

\Phi(\rho_X),

where $\rho_X$ is the relevant local residual uniformity and $\Phi(t)\ll t^c$ for some fixed $c>0$ .

\rho_X=\log^{-A}X,

then

\Phi(\rho_X)=X^{-o(1)}.

If instead

\rho_X\le X^{-c_0},

then

\Phi(\rho_X)\le X^{-cc_0} = H^{-cc_0/\alpha}

at $H=X^\alpha$ .

Therefore a power residual would generate a fixed $H$ -exponent. Current Lambda residual uniformity cannot do so through fixed algebraic transference.

7. Proof-level W split

In the 2026 proof, the major-arc parameter is chosen as

\delta=\log^{-A}X

for $\Lambda$ and $\mu$ , but

\delta=X^{-c_{k,C}\varepsilon}

for $d_k$ .

The Type-II input uses

\boxed{ W_\Lambda=\log^{A}X, }

whereas

\boxed{ W_{d_k}=X^{c_k}. }

The Type-II output saves a fixed negative power of $W$ .

Hence

W polylogarithmic
  -> log/subpower output

W polynomial
  -> fixed-power output

The prime/divisor exponent-class split is already built into the proof.

8. Why Lambda has only polylogarithmic W

The Dirichlet-polynomial bound for $\Lambda$ and $\mu$ is obtained from the Vinogradov–Korobov zero-free region.

The divisor bound is instead obtained from the divisor convolution representation.

Thus the current prime residual inherits a zero-sensitive major-arc limitation that the divisor residual does not.

Create:

O-RH-056
LAMBDA_RESIDUAL_MAJOR_ARC_W_PARAMETER_BARRIER
status: CERTIFIED AS CURRENT-METHOD AUDIT

9. d2 as comparator

The same 2026 paper proves a fixed-Fourier estimate for $d_2-d_2^\sharp$ of size

HX^{-\varepsilon/1000}

outside a power-saving exceptional set.

That proof uses a different classical route, with a fourth moment of Dirichlet $L$ -functions at the decisive stage.

This demonstrates that short-interval power discorrelation is technically possible for divisor objects.

It does not provide a prime analogue.

10. Campaign 25 verdict

pure centered singular-series model
  harmless

collision shell
  harmless

Lambda-sharp / residual decomposition
  exact

generalized von Neumann route
  current Lambda input subpower

divisor-to-prime transfer
  fails quantitatively at major-arc / Type-II W

D4HEG
  open / auxiliary nonlinear route

Create:

S-RH-030
FULLY_DISTINCT_RESIDUAL_AGGREGATE_POWER_CANCELLATION
status: OPEN

Prototype:

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

11. Campaign 26

CSM_RH Campaign 26
THREE_SHIFT_RESIDUAL_AGGREGATION_ATTACK

The question is:

can averaging the residual over three independent shifts create a fixed $H$ -power that is absent from the current one-interval residual theorem?

Tracks:

R4-1 aggregate-first transference
R4-2 Type-II W gain after shift averaging
R4-3 model-weighted residual orthogonality
R4-4 connected residual cumulant
R4-5 d2 proof surgery

Reject any candidate that assumes power uniformity of $\Lambda-\Lambda^\sharp$ , imports a fixed zero strip, obtains only subpower saving, reopens the already solved model main term, or transfers the $d_2$ theorem without re-proving the prime Dirichlet-series step.

12. State transition

CSM_RH v1.16
  ->
CSM_RH v1.17

with:

Campaign 25
  CLOSED_AS_MODEL_RESIDUAL_AND_W_PARAMETER_AUDIT

O-RH-055
  CREATED / CERTIFIED

O-RH-056
  CREATED / CERTIFIED AS CURRENT-METHOD AUDIT

F-RH-019
  D4HEG
  REMAINS OPEN / AUXILIARY

S-RH-030
  CREATED / OPEN

Campaign 26
  READY

13. Final status

RH = OPEN
PESC = OPEN
MLEPG = OPEN
C4HEG = OPEN / AUXILIARY
D4HEG = OPEN / AUXILIARY

CENTERED FOUR-POINT MODEL = HARMLESS
FULLY DISTINCT PRIME RESIDUAL = OPEN
CURRENT LAMBDA RESIDUAL UNIFORMITY = SUBPOWER
CURRENT DIVISOR RESIDUAL UNIFORMITY = FIXED POWER
PRIME/DIVISOR LOSS POINT = MAJOR-ARC TYPE-II W PARAMETER

NEXT CAMPAIGN = 26

CSM_RH Paper 26 — Refined Singular-Series Model Closure and the Lambda-Residual Major-Arc Power Barrier

CSM_RH Paper 26

Refined Singular-Series Model Closure and the Lambda-Residual Major-Arc Power Barrier

0. Trust boundary

1. Fully distinct aggregate

2. Refined singular series

3. Actual correlation error

4. Lambda-sharp split

5. Current residual strength

6. Generalized von Neumann exponent audit

7. Proof-level W split

8. Why Lambda has only polylogarithmic W

9. d2 as comparator

10. Campaign 25 verdict

11. Campaign 26

12. State transition

13. Final status