# 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

```text
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 $X^{-\delta}$.

# 1. Short-interval error

Let

$$
a_n=\Lambda(n)-1
$$

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

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

and

$$
\boxed{
\mathcal S_X(H)=\sum_{x\in\mathbb Z}|U_H(x)|^2.
}
$$

# 2. Exact two-scale identity

Since

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

define

$$
\mathcal C_X(H)
=
\Re\sum_xU_H(x)\overline{U_H(x+H)}.
$$

Then

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

Cauchy gives

$$
-\mathcal S_X(H)\le\mathcal C_X(H)\le\mathcal S_X(H),
$$

hence

$$
\boxed{
\mathcal S_X(2H)\le4\mathcal S_X(H).
}
$$

# 3. Normalized energy

Set

$$
R_X(H)=\frac{\mathcal S_X(H)}{XH^2}.
$$

Then

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

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

# 4. One good scale

If for some fixed $0<\varepsilon\le1$,

$$
\boxed{
\mathcal C_X(H)\le(1-\varepsilon)\mathcal S_X(H),
}
$$

then

$$
\boxed{
R_X(2H)
\le
\left(1-\frac{\varepsilon}{2}\right)R_X(H).
}
$$

Equivalently,

$$
\mathcal S_X(2H)\le(4-2\varepsilon)\mathcal S_X(H).
$$

So any constant improvement over the trivial doubling constant $4$ yields a fixed contraction of normalized lag energy.

# 5. Positive-density scales

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

$$
H_0=\lceil(\log X)^B\rceil,
\qquad
H_j=2^jH_0,
$$

and let $J$ be maximal with $H_J\le X^\alpha$. Then

$$
J=\frac{\alpha}{\log2}\log X+O(\log\log X).
$$

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

$$
\boxed{
\mathcal C_X(H_j)
\le
(1-\varepsilon)\mathcal S_X(H_j),
}
$$

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

# 6. Initial scale

The trivial estimate

$$
|U_{H_0}(x)|\ll H_0\log X
$$

gives

$$
R_X(H_0)\ll(\log X)^2.
$$

# 7. Fixed power generated by iteration

Let

$$
q=1-\frac{\varepsilon}{2}<1.
$$

Then

$$
R_X(H_J)
\ll
(\log X)^2 q^{dJ+O(1)}.
$$

Therefore

$$
\boxed{
R_X(H_J)
\ll
X^{-\delta_{\rm sc}+o(1)},
}
$$

where

$$
\boxed{
\delta_{\rm sc}
=
\alpha d\log_2
\left(
\frac1{1-\varepsilon/2}
\right)
>0.
}
$$

Equivalently,

$$
\boxed{
\mathcal S_X(H_J)
\ll
XH_J^2X^{-\delta_{\rm sc}+o(1)}.
}
$$

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

# 8. Bridge to MLEPG and PESC

Set

$$
\delta_M=\min\{\alpha,\delta_{\rm sc}\}.
$$

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

$$
\mathcal S_X(H_J)
\ll
XH_J(\log X)^{O(1)}
+
XH_J^2X^{-\delta_M+o(1)}.
$$

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

$$
\int_X^{2X}|\psi(x)-x|^2dx
\ll
X^{3-\kappa+o(1)}
$$

for every sufficiently small

$$
0<\kappa<
\min\{
\alpha,\delta_M,2-2\alpha
\}.
$$

Paper 20's Mellin pole recovery then implies

$$
\boxed{
\zeta(s)\neq0
\qquad
\Re s>1-\frac{\kappa}{2}.
}
$$

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

# 9. Spectral form

Paper 18 gives

$$
\mathcal S_X(H)
=
\int_0^1|S_X(\xi)|^2|D_H(\xi)|^2d\xi.
$$

Since

$$
D_{2H}(\xi)
=
D_H(\xi)[1+e(H\xi)],
$$

we have

$$
|D_{2H}(\xi)|^2
=
4\cos^2(\pi H\xi)|D_H(\xi)|^2.
$$

Hence

$$
\boxed{
4\mathcal S_X(H)-\mathcal S_X(2H)
=
4\int_0^1
|S_X(\xi)|^2|D_H(\xi)|^2
\sin^2(\pi H\xi)d\xi.
}
$$

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 $X^{-\delta}$ improvement at one mesoscopic scale.

PDSD assumes only a constant comparison between adjacent scales. No $X^{-\delta}$ 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

$$
\mathcal S_X(H)
\asymp
XH\log(X/H).
$$

It predicts

$$
\frac{\mathcal S_X(2H)}{\mathcal S_X(H)}
\sim
2\frac{\log(X/2H)}{\log(X/H)},
$$

well below the universal barrier $4$.

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 $\mathcal S_X(H)$, but do not compare $\mathcal S_X(2H)$ with $\mathcal S_X(H)$ by a fixed factor on a positive density of scales.

Thus PDSD is open.

# 13. Campaign 20 audit

```text
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:

```text
F-RH-017
POSITIVE_DENSITY_DYADIC_LAG_DECORRELATION
PDSD
status: OPEN
```

Create:

```text
O-RH-046
DYADIC_SCALE_LOCKING_BARRIER
status: CERTIFIED
```

Create:

```text
S-RH-028
ADJACENT_SHORT_INTERVAL_ERROR_DECORRELATION
status: OPEN
```

# 14. Campaign 21

```text
CSM_RH Campaign 21
DYADIC_PRIME_ERROR_DECORRELATION_ATTACK
```

Tracks:

```text
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 $X$ -power, assumes RH/fixed strip, uses conjectural variance as theorem, proves only one isolated good scale, or yields only $O(\log\log X)$ cumulative contraction mass.

# 15. Final status

```text
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
```