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lm-003885 · 2026-09

CSM_RH Paper 27 — Three-Shift Aggregation Collapse and the Residual Local-Moment Barrier

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CSM_RH Paper 27

Three-Shift Aggregation Collapse and the Residual Local-Moment Barrier

Project: CSM_RH
Paper: 27
Version: v0.1
Date: 2026-09-07
Parent state: CSM_RH v1.17 / Paper 26
Campaign: 26 — THREE_SHIFT_RESIDUAL_AGGREGATION_ATTACK
Status: aggregation-collapse theorem / no-free-dimension 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 26 localized the current four-point quantitative loss to the residual

f=Λ−Λ♯.f=\Lambda-\Lambda^\sharp.

Campaign 26 asks whether averaging over the three independent shift variables of the fully distinct four-point family can create a fixed HH -power which is absent from the one-interval residual theorem.

The answer is negative at the purely aggregation/transference level:

after the full shift sum is assembled, the three-dimensional family collapses, modulo the already harmless collision shell, to one-dimensional local mixed moments of the residual and model block sums.

No live GLM-5.3-Flash run is claimed.

1. Residual / model decomposition

Let

f(n)=Λ(n)−Λ♯(n),f(n)=\Lambda(n)-\Lambda^\sharp(n),

and

b(n)=Λ♯(n)−1.b(n)=\Lambda^\sharp(n)-1.

Then

Λ(n)−1=f(n)+b(n).\Lambda(n)-1=f(n)+b(n).

For interval length HH, define

FH(x)=∑r=1Hf(x+r)\boxed{F_H(x)=\sum_{r=1}^{H}f(x+r)}

and

BH(x)=∑r=1Hb(x+r).\boxed{B_H(x)=\sum_{r=1}^{H}b(x+r)}.

2. Mixed fully distinct sectors

For 0≤m≤40\le m\le4, define

Tmdist(X,H)=∑S⊆{1,2,3,4}∣S∣=m∑x∑1≤d1,d2,d3,d4≤Hdi distinct∏i∈Sf(x+di)∏i∉Sb(x+di).\boxed{ \mathcal T_m^{\mathrm{dist}}(X,H) = \sum_{\substack{S\subseteq\{1,2,3,4\}\\|S|=m}} \sum_x \sum_{\substack{1\le d_1,d_2,d_3,d_4\le H\\d_i\text{ distinct}}} \prod_{i\in S}f(x+d_i) \prod_{i\notin S}b(x+d_i). }

The full distinct aggregate is

A4,dist=∑m=04Tmdist.\mathcal A_{4,\mathrm{dist}} = \sum_{m=0}^{4}\mathcal T_m^{\mathrm{dist}}.

3. Unrestricted shift factorization

Remove the distinctness condition. For fixed SS with ∣S∣=m|S|=m,

∑d1,…,d4≤H∏i∈Sf(x+di)∏i∉Sb(x+di)=FH(x)mBH(x)4−m.\sum_{d_1,\ldots,d_4\le H} \prod_{i\in S}f(x+d_i) \prod_{i\notin S}b(x+d_i) = F_H(x)^mB_H(x)^{4-m}.

Summing over the (4m)\binom4m subsets gives

Tmall=(4m)∑xFH(x)mBH(x)4−m.\boxed{ \mathcal T_m^{\mathrm{all}} = \binom4m\sum_xF_H(x)^mB_H(x)^{4-m}. }

Thus the apparent three independent relative-shift directions disappear completely after the aggregate is formed.

4. Distinctness correction

The difference between unrestricted and fully distinct tuples is supported on offset collisions. There are O(H3)O(H^3) such ordered quadruples. Since ff and bb are pointwise polylogarithmically bounded on the relevant block,

Tmdist=(4m)∑xFH(x)mBH(x)4−m+O ⁣(XH3(log⁡X)O(1)).\boxed{ \mathcal T_m^{\mathrm{dist}} = \binom4m\sum_xF_H(x)^mB_H(x)^{4-m} + O\!\left(XH^3(\log X)^{O(1)}\right). }

Create:

B-RH-003
THREE_SHIFT_TO_LOCAL_MOMENT_COLLAPSE
status:
  CERTIFIED

5. Full aggregate collapse

Summing over mm,

A4,dist(X,H)=∑x[FH(x)+BH(x)]4+O ⁣(XH3(log⁡X)O(1)).\boxed{ \mathcal A_{4,\mathrm{dist}}(X,H) = \sum_x[F_H(x)+B_H(x)]^4 + O\!\left(XH^3(\log X)^{O(1)}\right). }

Since

FH(x)+BH(x)=∑r=1H[Λ(x+r)−1],F_H(x)+B_H(x) = \sum_{r=1}^{H}[\Lambda(x+r)-1],

the fully distinct aggregate is, modulo the collision shell, the original centered local fourth moment.

6. No free three-dimensional averaging gain

After complete aggregation the three shift variables are not independent oscillatory resources. They create powers of local block sums. The remaining analytic variable is xx.

Create:

O-RH-057
THREE_SHIFT_AGGREGATION_NO_FREE_DIMENSION_GAIN
status:
  CERTIFIED

A fixed power must therefore arise from a theorem controlling the xx -distribution of the local residual/model block sums.

7. Current residual block-sum theorem

For H≥X1/3+εH\ge X^{1/3+\varepsilon}, current 2026 higher-uniformity theory implies, using the constant nilsequence test, that for every fixed A>0A>0,

∣FH(x)∣≪Hlog⁡−AX\boxed{|F_H(x)|\ll H\log^{-A}X}

outside an exceptional set of measure

OA(Xlog⁡−AX).\boxed{O_A(X\log^{-A}X)}.

On every interval one has the trivial polylogarithmic bound

∣FH(x)∣≪H(log⁡X)O(1).|F_H(x)|\ll H(\log X)^{O(1)}.

8. Current residual moments

For any fixed 1≤p≤41\le p\le4, split into good and exceptional intervals. Choosing the theorem parameter AA sufficiently large relative to any prescribed B>0B>0 gives

∑x≍X∣FH(x)∣p≪BXHplog⁡−BX.\boxed{ \sum_{x\asymp X}|F_H(x)|^p \ll_B XH^p\log^{-B}X. }

In particular,

∑x≍X∣FH(x)∣4≪BXH4log⁡−BX.\boxed{ \sum_{x\asymp X}|F_H(x)|^4 \ll_B XH^4\log^{-B}X. }

This is strong subpower control, but at polynomial HH it is still H4−o(1)H^{4-o(1)} rather than H4−ηH^{4-\eta}.

9. Mixed sectors with current technology

Using the crude pointwise model bound

∣BH(x)∣≪H(log⁡X)O(1),|B_H(x)|\ll H(\log X)^{O(1)},

one obtains for every residual-containing mixed sector

Tmdist≪BXH4log⁡−BX+O ⁣(XH3(log⁡X)O(1)),1≤m≤4.\boxed{ \mathcal T_m^{\mathrm{dist}} \ll_B XH^4\log^{-B}X + O\!\left(XH^3(\log X)^{O(1)}\right), \qquad 1\le m\le4. }

Thus current one-interval residual technology remains in exponent class H4−o(1)H^{4-o(1)}.

10. Pure residual sector

For m=4m=4,

T4all=∑xFH(x)4≥0.\boxed{ \mathcal T_4^{\mathrm{all}} = \sum_xF_H(x)^4 \ge0. }

So the pure residual sector has no internal signed xx -cancellation. Any sectorwise proof must improve the residual fourth moment itself.

11. Cross-sector cancellation is the original problem

If one instead allows cancellation among

F4,4F3B,6F2B2,4FB3,B4,F^4, \quad 4F^3B, \quad 6F^2B^2, \quad 4FB^3, \quad B^4,

the exact sum is

(F+B)4.\boxed{(F+B)^4.}

Therefore cross-sector cancellation is simply the original centered fourth-moment problem. The sixteen-sector decomposition does not itself create an exponent.

12. Type-II WW parameter after shift aggregation

Paper 26 identified the current prime residual scale

WΛ=log⁡AX.W_\Lambda=\log^A X.

The collapse theorem shows that a complete three-shift sum does not algebraically introduce a new multiplicative parameter. After aggregation, the residual reappears through FH(x)F_H(x).

Therefore a polynomial effective WW cannot be obtained merely by reordering the additive shift sums.

Create:

O-RH-058
SHIFT_AVERAGING_DOES_NOT_AMPLIFY_W_LAMBDA
status:
  CERTIFIED AS STRUCTURAL AUDIT

13. d2d_2 proof surgery verdict

The 2026 fixed-Fourier theorem for d2−d2♯d_2-d_2^\sharp obtains a power saving using a different classical argument and a fourth moment of Dirichlet LL -functions. The present aggregation collapse does not create an analogous prime Dirichlet-series structure.

Replacing the divisor residual by Λ−Λ♯\Lambda-\Lambda^\sharp returns to the zero-sensitive prime Dirichlet-polynomial problem identified in Paper 26.

No prime fixed-power transfer is obtained.

14. Campaign 26 verdict

R4-1 aggregate-first transference
  COLLAPSES TO LOCAL MIXED MOMENTS

R4-2 Type-II W gain after shift averaging
  NO ALGEBRAIC W AMPLIFICATION

R4-3 model-weighted residual orthogonality
  REDUCED TO ONE-DIMENSIONAL x-CORRELATIONS

R4-4 connected residual cumulant
  REORGANIZATION ONLY

R4-5 d2 proof surgery
  NO PRIME POWER TRANSFER FOUND

Create:

O-RH-059
LAMBDA_RESIDUAL_LOCAL_FOURTH_MOMENT_BARRIER
status:
  CERTIFIED AS CURRENT-TECHNOLOGY AUDIT

15. Status of the four-point branch

F-RH-018 C4HEG
  OPEN / AUXILIARY

F-RH-019 D4HEG
  OPEN / AUXILIARY

S-RH-030
  CLOSED AS A FREE-AGGREGATION MECHANISM

The four-point route remains mathematically valid, but it is not currently shorter than PESC/MLEPG.

16. Campaign 27

CSM_RH Campaign 27
DIRECT_PESC_ATTACK_II

Allowed tracks:

P2-1 signed prime sampling with structured drift subtraction
P2-2 PESC after sieve-model subtraction
P2-3 bilinear residual/model cross term
P2-4 scale-coupled signed recurrence
P2-5 direct arithmetic drift exclusion

Reject any candidate that introduces another positive moment frontier, merely rephrases MLEPG, uses zero density/fixed strip as the power source, achieves only logarithmic precision, or creates another high-dimensional shift aggregate that collapses to a local moment.

17. State transition

CSM_RH v1.17
  ->
CSM_RH v1.18

with:

Campaign 26
  CLOSED_AS_THREE_SHIFT_AGGREGATION_COLLAPSE_AUDIT

B-RH-003
  THREE_SHIFT_TO_LOCAL_MOMENT_COLLAPSE
  CREATED / CERTIFIED

O-RH-057
  THREE_SHIFT_AGGREGATION_NO_FREE_DIMENSION_GAIN
  CREATED / CERTIFIED

O-RH-058
  SHIFT_AVERAGING_DOES_NOT_AMPLIFY_W_LAMBDA
  CREATED / CERTIFIED AS STRUCTURAL AUDIT

O-RH-059
  LAMBDA_RESIDUAL_LOCAL_FOURTH_MOMENT_BARRIER
  CREATED / CERTIFIED AS CURRENT-TECHNOLOGY AUDIT

S-RH-030
  CLOSED AS FREE-AGGREGATION MECHANISM

Campaign 27
  DIRECT_PESC_ATTACK_II
  READY

18. Final status

RH = OPEN
PESC = OPEN / ROOT TARGET
MLEPG = OPEN / DIRECT THEOREM CANDIDATE
C4HEG = OPEN / AUXILIARY
D4HEG = OPEN / AUXILIARY
THREE-SHIFT FREE AGGREGATION GAIN = CLOSED
CURRENT RESIDUAL FOURTH MOMENT = H^4 X^{-o(1)}
POLYNOMIAL H-EXPONENT GAIN = OPEN
FOUR-POINT ROUTE = NOT CURRENTLY SHORTER THAN PESC/MLEPG
NEXT CAMPAIGN = 27