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lm-003879 · 2026-09

CSM_RH Paper 21 — Dyadic Lag-Energy Decorrelation and a Prime-Side Linear Contraction-Mass Candidate

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CSM_RH Paper 21

Dyadic Lag-Energy Decorrelation and a Prime-Side Linear Contraction-Mass Candidate

Project: CSM_RH
Paper: 21
Version: v0.1
Date: 2026-09-06
Parent: CSM_RH v1.11 / Paper 20
Campaign: 20 — PRIME_SIDE_FIXED_POWER_LEMMA_GENERATION

0. Trust boundary

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

This paper introduces a prime-side fixed-power generator whose hypothesis contains no factor X−δX^{-\delta}.

1. Short-interval error

Let

an=Λ(n)−1a_n=\Lambda(n)-1

for 1≤n≤X1\le n\le X, extended by zero outside this range. Define

UH(x)=∑r=1Hax+r,U_H(x)=\sum_{r=1}^{H}a_{x+r},

and

SX(H)=∑x∈Z∣UH(x)∣2.\boxed{ \mathcal S_X(H)=\sum_{x\in\mathbb Z}|U_H(x)|^2. }

2. Exact two-scale identity

Since

U2H(x)=UH(x)+UH(x+H),U_{2H}(x)=U_H(x)+U_H(x+H),

define

CX(H)=ℜ∑xUH(x)UH(x+H)‾.\mathcal C_X(H) = \Re\sum_xU_H(x)\overline{U_H(x+H)}.

Then

SX(2H)=2SX(H)+2CX(H).\boxed{ \mathcal S_X(2H)=2\mathcal S_X(H)+2\mathcal C_X(H). }

Cauchy gives

−SX(H)≤CX(H)≤SX(H),-\mathcal S_X(H)\le\mathcal C_X(H)\le\mathcal S_X(H),

hence

SX(2H)≤4SX(H).\boxed{ \mathcal S_X(2H)\le4\mathcal S_X(H). }

3. Normalized energy

Set

RX(H)=SX(H)XH2.R_X(H)=\frac{\mathcal S_X(H)}{XH^2}.

Then

RX(2H)RX(H)=12(1+CX(H)SX(H))=SX(2H)4SX(H).\boxed{ \frac{R_X(2H)}{R_X(H)} = \frac12 \left( 1+\frac{\mathcal C_X(H)}{\mathcal S_X(H)} \right) = \frac{\mathcal S_X(2H)}{4\mathcal S_X(H)}. }

Thus the universal bound is RX(2H)≤RX(H)R_X(2H)\le R_X(H).

4. One good scale

If for some fixed 0<ε≤10<\varepsilon\le1,

CX(H)≤(1−ε)SX(H),\boxed{ \mathcal C_X(H)\le(1-\varepsilon)\mathcal S_X(H), }

then

RX(2H)≤(1−ε2)RX(H).\boxed{ R_X(2H) \le \left(1-\frac{\varepsilon}{2}\right)R_X(H). }

Equivalently,

SX(2H)≤(4−2ε)SX(H).\mathcal S_X(2H)\le(4-2\varepsilon)\mathcal S_X(H).

So any constant improvement over the trivial doubling constant 44 yields a fixed contraction of normalized lag energy.

5. Positive-density scales

Fix B>0B>0 and 0<α<2/30<\alpha<2/3. Let

H0=⌈(log⁡X)B⌉,Hj=2jH0,H_0=\lceil(\log X)^B\rceil, \qquad H_j=2^jH_0,

and let JJ be maximal with HJ≤XαH_J\le X^\alpha. Then

J=αlog⁡2log⁡X+O(log⁡log⁡X).J=\frac{\alpha}{\log2}\log X+O(\log\log X).

Define PDSD (d,ε;α,B)(d,\varepsilon;\alpha,B) by requiring that at least dJ+O(1)dJ+O(1) scales satisfy

CX(Hj)≤(1−ε)SX(Hj),\boxed{ \mathcal C_X(H_j) \le (1-\varepsilon)\mathcal S_X(H_j), }

for fixed d,ε>0d,\varepsilon>0. All other scales may use only the trivial bound.

6. Initial scale

The trivial estimate

∣UH0(x)∣≪H0log⁡X|U_{H_0}(x)|\ll H_0\log X

gives

RX(H0)≪(log⁡X)2.R_X(H_0)\ll(\log X)^2.

7. Fixed power generated by iteration

Let

q=1−ε2<1.q=1-\frac{\varepsilon}{2}<1.

Then

RX(HJ)≪(log⁡X)2qdJ+O(1).R_X(H_J) \ll (\log X)^2 q^{dJ+O(1)}.

Therefore

RX(HJ)≪X−δsc+o(1),\boxed{ R_X(H_J) \ll X^{-\delta_{\rm sc}+o(1)}, }

where

δsc=αdlog⁡2(11−ε/2)>0.\boxed{ \delta_{\rm sc} = \alpha d\log_2 \left( \frac1{1-\varepsilon/2} \right) >0. }

Equivalently,

SX(HJ)≪XHJ2X−δsc+o(1).\boxed{ \mathcal S_X(H_J) \ll XH_J^2X^{-\delta_{\rm sc}+o(1)}. }

This is a genuine fixed power generated dynamically from Ω(log⁡X)\Omega(\log X) constant contractions.

8. Bridge to MLEPG and PESC

Set

δM=min⁡{α,δsc}.\delta_M=\min\{\alpha,\delta_{\rm sc}\}.

Then the preceding estimate is stronger than the MLEPG-compatible form

SX(HJ)≪XHJ(log⁡X)O(1)+XHJ2X−δM+o(1).\mathcal S_X(H_J) \ll XH_J(\log X)^{O(1)} + XH_J^2X^{-\delta_M+o(1)}.

Paper 17's residue-chain theorem gives a dyadic PNT mean-square bound

∫X2X∣ψ(x)−x∣2dx≪X3−κ+o(1)\int_X^{2X}|\psi(x)-x|^2dx \ll X^{3-\kappa+o(1)}

for every sufficiently small

0<κ<min⁡{α,δM,2−2α}.0<\kappa< \min\{ \alpha,\delta_M,2-2\alpha \}.

Paper 20's Mellin pole recovery then implies

ζ(s)≠0ℜs>1−κ2.\boxed{ \zeta(s)\neq0 \qquad \Re s>1-\frac{\kappa}{2}. }

Thus PDSD has certified fixed-zero-strip authority without assuming a zero-side theorem.

9. Spectral form

Paper 18 gives

SX(H)=∫01∣SX(ξ)∣2∣DH(ξ)∣2dξ.\mathcal S_X(H) = \int_0^1|S_X(\xi)|^2|D_H(\xi)|^2d\xi.

Since

D2H(ξ)=DH(ξ)[1+e(Hξ)],D_{2H}(\xi) = D_H(\xi)[1+e(H\xi)],

we have

∣D2H(ξ)∣2=4cos⁡2(πHξ)∣DH(ξ)∣2.|D_{2H}(\xi)|^2 = 4\cos^2(\pi H\xi)|D_H(\xi)|^2.

Hence

4SX(H)−SX(2H)=4∫01∣SX(ξ)∣2∣DH(ξ)∣2sin⁡2(πHξ)dξ.\boxed{ 4\mathcal S_X(H)-\mathcal S_X(2H) = 4\int_0^1 |S_X(\xi)|^2|D_H(\xi)|^2 \sin^2(\pi H\xi)d\xi. }

A good scale is therefore a fixed amount of spectral escape from the dyadic resonance set.

10. Why PDSD is not MLEPG renamed

MLEPG assumes an X−δX^{-\delta} improvement at one mesoscopic scale.

PDSD assumes only a constant comparison between adjacent scales. No X−δX^{-\delta} occurs in its hypothesis.

The exponent is produced by multiplication over logarithmically many scales.

This is the main reason PDSD survives Campaign 20.

11. Plausibility calibration

The Montgomery–Soundararajan short-interval variance prediction is

SX(H)≍XHlog⁡(X/H).\mathcal S_X(H) \asymp XH\log(X/H).

It predicts

SX(2H)SX(H)∼2log⁡(X/2H)log⁡(X/H),\frac{\mathcal S_X(2H)}{\mathcal S_X(H)} \sim 2\frac{\log(X/2H)}{\log(X/H)},

well below the universal barrier 44.

Recent conditional work on joint prime counts in multiple short intervals also finds weak negative correlations between disjoint intervals.

These are plausibility calibrations only, not proof inputs.

12. Current unconditional status

Current almost-all short-interval PNT results give strong subpower control of SX(H)\mathcal S_X(H), but do not compare SX(2H)\mathcal S_X(2H) with SX(H)\mathcal S_X(H) by a fixed factor on a positive density of scales.

Thus PDSD is open.

13. Campaign 20 audit

A1 principal-arc arithmetic power estimate
  valid but same fixed-power bottleneck as MLEPG

A2 shrinking-threshold short-interval theorem
  sufficient but unnecessarily strong

A3 direct MLEPG
  remains open

A4 prime-side scale contraction
  SURVIVOR -> PDSD

A5 direct PESC
  root target

Create:

F-RH-017
POSITIVE_DENSITY_DYADIC_LAG_DECORRELATION
PDSD
status: OPEN

Create:

O-RH-046
DYADIC_SCALE_LOCKING_BARRIER
status: CERTIFIED

Create:

S-RH-028
ADJACENT_SHORT_INTERVAL_ERROR_DECORRELATION
status: OPEN

14. Campaign 21

CSM_RH Campaign 21
DYADIC_PRIME_ERROR_DECORRELATION_ATTACK

Tracks:

D1 average Hardy-Littlewood covariance
D2 spectral octave escape
D3 sieve variance lower bound
D4 positive-density scales
D5 contradiction from long critical locking

Reject any candidate that inserts a fixed XX -power, assumes RH/fixed strip, uses conjectural variance as theorem, proves only one isolated good scale, or yields only O(log⁡log⁡X)O(\log\log X) cumulative contraction mass.

15. Final status

RH = OPEN
PESC = OPEN
MLEPG = OPEN
PDSD = OPEN

FIXED POWER IN PDSD HYPOTHESIS = NONE
FIXED POWER GENERATED BY SCALE ITERATION = YES
CURRENT UNCONDITIONAL PDSD PROOF = NONE

NEXT CAMPAIGN = 21