# CSM_RH Paper 15
## Off-Diagonal Chowla Lift, Diagonal-Scale Precision, and Parity-Bridge Authority Correction

**Project:** `CSM_RH`  
**Paper:** `15`  
**Version:** `v0.1`  
**Date:** `2026-09-05`  
**Parent state:** `CSM_RH v1.5 / Paper 14`  
**Campaign:** `14 — EMDQO_OFF_DIAGONAL_CORRELATION_AUDIT`  
**Status:** off-diagonal reduction / bridge-authority audit; not a proof or disproof of RH

---

# 0. Trust boundary

This paper does not prove or disprove the Riemann Hypothesis.

Canonical root state:

```text
RH_PROVED = FALSE
RH_DISPROVED = FALSE
GLOBAL_RH_CERTIFICATE = FALSE
CSM_RH_ROOT_STATUS = OPEN
```

This paper performs two corrections.

First, it expands the EMDQO off-diagonal exactly and identifies its $C=1$ slice as an endogenous weighted binary Chowla problem.

Second, it corrects the promotion semantics of EMDQO:

> EMDQO at diagonal scale gives a fixed-power estimate for the auxiliary parity-breaking bilinear condition EMBF, but EMBF alone is not a certified bridge to PESC. A power-accurate coupled sieve assembly is still required.

No live GLM-5.3-Flash run is claimed.

---

# 1. EMDQO setup

Let

$$
B(x)=\vartheta(x)-x.
$$

Let

$$
a_C(n)=\mu(n)\gamma(n,C),
$$

and for squarefree $m$ define

$$
a_{C,m}(n)
=
a_C(n)\mathbf1_{(m,n)=1}.
$$

Let

$$
F_m(n)
=
w_N(mn)B(mn-1).
$$

For a bilinear window

$$
L<n\le2L,
\qquad
mn<2N,
$$

define

$$
S_m
=
\sum_n
a_{C,m}(n)F_m(n).
$$

Then

$$
\boxed{
\mathcal Q_{N,L,C}
=
\sum_{\mu^2(m)=1}
|S_m|^2.
}
$$

The balanced EMDQO target from Paper 14 is

$$
\boxed{
\mathcal Q_{N,L,C}
\ll
N^{5+o(1)},
\qquad
L=N^{1/2+o(1)}.
}
$$

---

# 2. Exact diagonal / off-diagonal split

Expanding the square gives

$$
\mathcal Q
=
\mathcal Q_{\rm diag}
+
\mathcal Q_{\rm off},
$$

where

$$
\boxed{
\mathcal Q_{\rm diag}
=
\sum_m
\sum_n
|a_{C,m}(n)|^2
|F_m(n)|^2
}
$$

and

$$
\boxed{
\mathcal Q_{\rm off}
=
\sum_m
\sum_{\substack{
n_1,n_2\\
n_1\neq n_2
}}
a_{C,m}(n_1)
a_{C,m}(n_2)
F_m(n_1)
F_m(n_2).
}
$$

At balanced scale, the generic diagonal estimate is

$$
\boxed{
\mathcal Q_{\rm diag}
\ll
N^{5+o(1)}.
}
$$

Therefore EMDQO is precisely an off-diagonal problem.

---

# 3. Kernel representation

For $n_1,n_2$ in the bilinear window define

$$
\boxed{
K_{N,L,C}(n_1,n_2)
=
\sum_{\substack{
m\\
\mu^2(m)=1\\
(m,n_1n_2)=1\\
m\max(n_1,n_2)<2N
}}
w_N(mn_1)
w_N(mn_2)
B(mn_1-1)
B(mn_2-1).
}
$$

Then:

## Theorem 3.1 — Exact Off-Diagonal Kernel Form

$$
\boxed{
\mathcal Q_{\rm off}
=
\sum_{\substack{
n_1,n_2\\
n_1\neq n_2
}}
a_C(n_1)a_C(n_2)
K_{N,L,C}(n_1,n_2).
}
$$

This is an exact identity.

The kernel is endogenous because it contains the prime-generated error $B$.

---

# 4. Shift form

Write

$$
n_2=n+h,
\qquad
h\ge1.
$$

Define

$$
K_{N,L,C}(n,h)
=
K_{N,L,C}(n,n+h).
$$

Then, because the quantities are real,

## Theorem 4.1 — Weighted Shift Correlation Form

$$
\boxed{
\mathcal Q_{\rm off}
=
2
\sum_{h\ge1}
\sum_n
a_C(n)a_C(n+h)
K_{N,L,C}(n,h),
}
$$

with the natural restrictions

$$
L<n,n+h\le2L.
$$

Thus the off-diagonal is an average of two-point correlations of the parity coefficient $a_C$, but against a highly structured endogenous kernel.

---

# 5. Minimal parity slice $C=1$

For

$$
C=1,
$$

we have

$$
\gamma(n,1)=1,
$$

so

$$
a_1(n)=\mu(n).
$$

Hence:

## Corollary 5.1 — Endogenous Weighted Binary Chowla Slice

$$
\boxed{
\mathcal Q_{\rm off}^{(C=1)}
=
2
\sum_{h\ge1}
\sum_n
\mu(n)\mu(n+h)
K_{N,L,1}(n,h).
}
$$

The coefficient structure is exactly binary Möbius correlation.

The weight is not external.

It is built from multiplicative dilates of the prime-counting error.

This is the cleanest strength diagnostic for the EMDQO off-diagonal.

---

# 6. Precision budget at balanced scale

Take

$$
L=N^{1/2+o(1)}.
$$

Then the number of outer $m$ values is

$$
M=N^{1/2+o(1)}.
$$

The number of $(m,n)$ diagonal pairs is

$$
ML
=
N^{1+o(1)}.
$$

Since

$$
|F_m(n)|
\ll
N^2,
$$

the diagonal scale is

$$
N^{1+o(1)}N^4
=
N^{5+o(1)}.
$$

The number of off-diagonal triples $(m,n_1,n_2)$ is

$$
ML^2
=
N^{3/2+o(1)}.
$$

The coefficient-blind scale is therefore

$$
N^{11/2+o(1)}.
$$

Thus EMDQO requires a net gain of

$$
\boxed{
N^{-1/2+o(1)}
}
$$

over coefficient-blind off-diagonal treatment.

Equivalently, one needs square-root-in-window cancellation at the global second-moment level.

---

# 7. Existing averaged Chowla strength calibration

For ordinary Liouville or Möbius correlations, modern results establish:

```text
logarithmically averaged fixed-shift cancellation;
averaged-shift Cesaro cancellation;
higher short-interval uniformity on average;
quantitative logarithmic improvements.
```

But the ordinary fixed-shift Cesaro two-point Chowla conjecture remains open.

More importantly for EMDQO, the known averaged-shift estimates do not provide the required fixed $L^{-1/2}$ power at this precision level.

They also do not come with uniformity for kernels of the form

$$
K_{N,L,1}(n,h),
$$

which are generated by the prime-error process itself.

Therefore current Chowla technology does not close EMDQO.

This is a literature-strength statement, not a no-go theorem.

---

# 8. Endogenous-kernel transfer debt

An estimate such as

$$
\sum_{h\le H}
\left|
\sum_{n\le X}
\mu(n)\mu(n+h)
\right|
=
o(HX)
$$

does not automatically imply cancellation in

$$
\sum_h
\sum_n
\mu(n)\mu(n+h)
K(n,h).
$$

To transfer it one needs quantitative control on the kernel class.

The EMDQO kernel depends on

$$
B(mn-1)B(m(n+h)-1),
$$

so its complexity is itself tied to prime distribution.

Create:

```text
O-RH-033
ENDOGENOUS_WEIGHTED_CHOWLA_TRANSFER_DEBT
status:
  CERTIFIED
```

Statement:

> Unweighted or externally weighted Chowla estimates cannot be promoted to EMDQO without a uniform theorem for the endogenous prime-error kernel class.

---

# 9. Near-diagonal region

For small $h$, the two prime-error arguments are

$$
mn-1
$$

and

$$
m(n+h)-1,
$$

whose separation is

$$
mh.
$$

At balanced scale,

$$
m=N^{1/2+o(1)}.
$$

Therefore even the first nonzero shift produces prime-error samples separated by roughly $N^{1/2}$.

This is not an infinitesimal perturbation on the prime scale.

At the same time, the two samples are strongly geometrically related through a common multiplier $m$.

No current theorem identified in this audit gives fixed-power quasi-orthogonality for these two endogenous dilates uniformly across the required range.

The near-diagonal cannot simply be discarded.

---

# 10. Large-sieve / dispersion audit

A standard large-sieve gain normally comes from an explicit family of separated additive or multiplicative phases.

The kernel form in Theorem 4.1 has no such free phase parameter.

Any dispersion step must first manufacture a transform of

$$
B(mn-1)
$$

and then control the resulting transformed coefficients.

If the transform estimate is closed by Cauchy using the prime-error energy itself, Paper 14's generic energy fallback returns.

Thus:

```text
GENERIC DISPERSION
  does not yet create independent EMDQO authority

GENERIC LARGE SIEVE
  lacks a canonical separated phase family

MELLIN / DIRICHLET DIAGONALIZATION
  risks reintroducing inverse-zeta zero sensitivity
```

No fixed-power theorem is obtained.

---

# 11. Möbius-uniformity audit

Modern Möbius-uniformity theory gives strong cancellation against many structured external sequences.

The EMDQO weight is different.

For fixed $m$,

$$
n
\mapsto
w_N(mn)B(mn-1)
$$

is generated by the prime sequence itself.

It is neither a fixed smooth phase nor a fixed nilsequence independent of the Möbius function.

Therefore existing external-test uniformity theorems do not directly apply.

The known quantitative savings relevant to Chowla-type problems are also logarithmic or qualitative at the needed level, not a fixed $N$ -power for this endogenous kernel.

---

# 12. Bridge-authority correction

Paper 14 proved that EMDQO at diagonal scale implies

$$
\mathfrak B_{N,L,C}
\ll
N^{11/4+o(1)}
$$

in the balanced window.

This is a genuine fixed-power improvement for the auxiliary parity-breaking bilinear condition EMBF.

However Paper 13 had already proved:

```text
STANDARD ASYMPTOTIC SIEVE
  does not convert EMBF to PESC at fixed-power precision.

POSITIVITY LIFT
  incurs an exponent-3 baseline.

SEPARATE NONNEGATIVE SIEVE ERRORS
  do not have coupled cancellation authority.
```

Therefore:

## Theorem 12.1 — EMDQO Bridge-Authority Boundary

At the present certified state,

$$
\boxed{
\operatorname{EMDQO}
\Longrightarrow
\text{fixed-power EMBF}
}
$$

is certified, but

$$
\boxed{
\operatorname{EMDQO}
\Longrightarrow
\operatorname{PESC}
}
$$

is not certified.

A separate power-accurate coupled sieve assembly theorem is required.

This corrects any earlier wording which treated the EMDQO $\kappa=1/4$ gain as a direct RH-branch power gain.

---

# 13. New bridge frontier

Create:

```text
F-RH-014
POWER_ACCURATE_COUPLED_PARITY_SIEVE_ASSEMBLY
abbrev:
  PACPSA
status:
  OPEN
```

A valid PACPSA theorem must combine:

```text
power-accurate divisor-distribution control;
parity-breaking bilinear input;
coupled signed treatment of the positivity lifts;
fixed-power final prime-detection error;
target fidelity to PESC.
```

EMDQO may supply the parity-breaking sublemma.

It does not supply the assembly.

---

# 14. EMDQO status

Paper 14 promoted EMDQO as the canonical parity-sensitive second-moment mechanism frontier.

After the current audit:

```text
F-RH-013
EMDQO

status:
  OPEN / AUXILIARY PARITY SUBFRONTIER

standalone PESC authority:
  NO

fixed-power EMBF authority:
  YES
```

This is a demotion in theorem authority, not a rejection of the estimate.

---

# 15. Direct PESC shift interface

The canonical PESC target itself has an exact all-shift form.

Recall

$$
c_n
=
\log n\mathbf1_{\mathbb P}(n)-1.
$$

Then

$$
\mathcal C_N^\vartheta
=
\sum_{m<n<2N}
w_N(n)c_mc_n.
$$

Writing

$$
n=m+h,
$$

we obtain:

## Theorem 15.1 — Prime-Only All-Shift Interface

$$
\boxed{
\mathcal C_N^\vartheta
=
\sum_{h=1}^{2N-2}
\sum_{m<2N-h}
w_N(m+h)c_mc_{m+h}.
}
$$

This is the direct all-shift centered prime-pair aggregate.

It contains no auxiliary Möbius layer.

It remains exponent-equivalent to PODEE.

---

# 16. Why return to the direct target after Campaign 14

The EMDQO route now requires two new theorems:

1. endogenous weighted binary Chowla / off-diagonal quasi-orthogonality;
2. PACPSA to transfer the parity input back to PESC.

The direct PESC route requires one new fixed-power prime-pair aggregate theorem.

Therefore CSM_RH should not claim that EMDQO is a shorter route merely because its second moment has a clean diagonal scale.

The closure graph now contains a genuine path-length comparison.

---

# 17. Campaign 14 verdict

```text
EMDQO OFF-DIAGONAL
  EXACTLY REDUCED TO ENDOGENOUS WEIGHTED TWO-POINT PARITY CORRELATION

C=1 SLICE
  BINARY MOBIUS / CHOWLA-TYPE COEFFICIENTS

EXISTING AVERAGED CHOWLA
  INSUFFICIENT PRECISION / INSUFFICIENT ENDOGENOUS KERNEL UNIFORMITY

GENERIC DISPERSION / LARGE SIEVE
  NO INDEPENDENT FIXED POWER FOUND

EMDQO -> EMBF
  CERTIFIED

EMDQO -> PESC
  NOT CERTIFIED

PACPSA
  NEW OPEN BRIDGE FRONTIER

DIRECT PESC
  STILL SHORTEST CERTIFIED TARGET PATH
```

No fixed-power theorem is proved.

---

# 18. New obstruction: parity-sublemma bridge debt

Create:

```text
O-RH-034
PARITY_SUBLEMMA_BRIDGE_DEBT
status:
  CERTIFIED
```

Statement:

> A fixed-power parity-breaking bilinear estimate is not automatically a fixed-power prime-detection theorem for the signed endogenous target. The coupled sieve assembly must be proved at the same exponent resolution.

---

# 19. New survivor

Create:

```text
S-RH-024
ENDOGENOUS_WEIGHTED_BINARY_CHOWLA
status:
  OPEN
```

Canonical $C=1$ prototype:

$$
\boxed{
\sum_{h}
\sum_n
\mu(n)\mu(n+h)
K_{N,L,1}(n,h)
\ll
N^{5+o(1)}
}
$$

after the diagonal normalization appropriate to EMDQO.

This is not ordinary Chowla.

It is a prime-error-weighted all-shift version.

---

# 20. Campaign 15

The next campaign is:

```text
CSM_RH Campaign 15
DIRECT_ALL_SHIFT_PRIME_CORRELATION_AUDIT
```

The campaign returns to the shortest certified target path:

```text
PESC
```

using Theorem 15.1.

Its task is to compare the all-shift prime-only aggregate against:

```text
average Hardy-Littlewood prime-pair theorems;
Selberg-integral / correlation identities;
Barban-Davenport-Halberstam-type mean squares;
dispersion identities;
known averaged-shift prime-correlation estimates.
```

The goal is to determine whether any existing average-prime-pair technology reaches a fixed $N$ -power for the complete signed aggregate.

---

# 21. Campaign 15 required questions

```text
Q1
What is the exact weighted shift correlation R_N(h)?

Q2
Which shift ranges dominate the PESC aggregate?

Q3
Can known average prime-pair results control the signed sum before absolute values?

Q4
Does singular-series subtraction reappear automatically or is it unnecessary?

Q5
Does a Selberg-integral identity reduce the target or only restate the same energy?

Q6
What fixed-power error would an averaged Hardy-Littlewood theorem need?

Q7
What is the first new prime-pair theorem if known results fall short?
```

---

# 22. Campaign 15 rejection filters

Reject a candidate if:

## R1. It takes absolute values in every shift before using signed aggregation.

## R2. It inserts the Hardy-Littlewood singular series and counts the model as a proof.

## R3. It uses PODEE/PESC itself to control the average correlation.

## R4. It proves only logarithmic improvement.

## R5. It assumes uniform fixed-shift Hardy-Littlewood asymptotics.

## R6. It confuses an average over shifts with the exact endpoint-weighted signed aggregate.

---

# 23. External calibration

Current literature provides the following calibration.

1. Matomäki–Radziwiłł–Tao proved averaged forms of Chowla's conjecture with qualitative cancellation and quantitative decay of logarithmic type.

2. Matomäki–Radziwiłł–Tao–Teräväinen–Ziegler proved higher short-interval uniformity on average and new averaged Chowla consequences.

3. The ordinary fixed-shift two-point Chowla conjecture remains open; recent quantitative advances are predominantly logarithmically averaged or averaged over shifts.

4. Banks–Shparlinski (2026) obtain nontrivial estimates for several multi-variable Möbius sums while explicitly noting that binary analogues remain difficult in their setting.

These results show that the EMDQO off-diagonal sits beyond standard currently available binary Möbius-correlation technology at the required precision.

None proves EMDQO or PESC.

---

# 24. State transition

The canonical transition is:

```text
CSM_RH v1.5
  ->
CSM_RH v1.6
```

with:

```text
Campaign 14
  CLOSED_AS_OFF_DIAGONAL_AND_BRIDGE_AUTHORITY_AUDIT

O-RH-033
  ENDOGENOUS_WEIGHTED_CHOWLA_TRANSFER_DEBT
  CREATED / CERTIFIED

O-RH-034
  PARITY_SUBLEMMA_BRIDGE_DEBT
  CREATED / CERTIFIED

F-RH-013
  EMDQO
  DEMOTED TO AUXILIARY PARITY SUBFRONTIER

F-RH-014
  PACPSA
  CREATED / OPEN

S-RH-024
  ENDOGENOUS_WEIGHTED_BINARY_CHOWLA
  CREATED / OPEN

F-RH-010
  PESC
  REMAINS OPEN / SHORTEST CERTIFIED TARGET PATH

Campaign 15
  DIRECT_ALL_SHIFT_PRIME_CORRELATION_AUDIT
  READY
```

---

# 25. Final status

```text
RH = OPEN

PESC = OPEN

EMDQO = OPEN / AUXILIARY

EMDQO OFF-DIAGONAL = WEIGHTED BINARY CHOWLA-TYPE

CURRENT CHOWLA TECHNOLOGY = INSUFFICIENT FOR REQUIRED FIXED POWER

EMDQO -> EMBF = CERTIFIED

EMDQO -> PESC = NOT CERTIFIED

PACPSA = OPEN

DIRECT PESC PATH = SHORTER

NEXT CAMPAIGN = 15
```

The direct canonical target remains

$$
\boxed{
\left|
\sum_{h=1}^{2N-2}
\sum_{m<2N-h}
w_N(m+h)
c_m
c_{m+h}
\right|
\ll
N^{3-\kappa+o(1)}
}
$$

for one fixed

$$
0<\kappa<\frac12.
$$

The next task is to audit this all-shift prime-pair aggregate directly against the strongest available average-correlation technology.