CSM_RH Paper 14

One-Sided Möbius Parity, Inverse-Zeta Coefficient Strength, and Endogenous Dilate Quasi-Orthogonality

Project: CSM_RH
Paper: 14
Version: v0.1
Date: 2026-09-05
Parent state: CSM_RH v1.4 / Paper 13
Campaign: 13 — ENDOGENOUS_MOBIUS_BILINEAR_STRENGTH_AUDIT
Status: Möbius-strength / second-moment mechanism audit; not a proof or disproof of RH

0. Trust boundary

This paper does not prove or disprove the Riemann Hypothesis.

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

The main conclusions are:

mu(mn) under the outer absolute value
  becomes one-sided Möbius cancellation in n

the inner parity coefficient
  carries an explicit inverse-zeta Dirichlet-series factor

coefficient-only fixed-power cancellation
  is zero-sensitive and may import zero-strip strength

generic Cauchy closure
  makes EMBF downstream of prime-error energy

a genuinely new route remains:
  diagonal-scale second-moment quasi-orthogonality across multiplicative dilates

No live GLM-5.3-Flash run is claimed.

1. Canonical EMBF object

Let

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

Let

\gamma(n,C)=\sum_{\substack{d\mid n\\ d\le C}}\mu(d).

For

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

define

\boxed{ \mathfrak B_{N,L,C} = \sum_m \left| \sum_{\substack{L<n\le2L\\ mn<2N}} \gamma(n,C)\mu(mn)w_N(mn)B(mn-1) \right|. }

2. Exact Möbius factorization

For all positive integers $m,n$ ,

\boxed{ \mu(mn)=\mu(m)\mu(n)\mathbf1_{(m,n)=1}. }

Hence:

Theorem 2.1 — One-Sided Parity Factorization

\boxed{ \mathfrak B_{N,L,C} = \sum_{\substack{m\\ \mu(m)\neq0}} \left| \sum_{\substack{L<n\le2L\\ mn<2N\\ (n,m)=1}} \gamma(n,C)\mu(n)w_N(mn)B(mn-1) \right|. }

The sign of $\mu(m)$ disappears because of the outer absolute value. The parity source is therefore one-sided in the inner $n$ variable.

O-RH-030
OUTER_ABSOLUTE_ONE_SIDED_PARITY
status: CERTIFIED

3. The parity coefficient

Define

\boxed{a_C(n)=\mu(n)\gamma(n,C).}

For $d\mid n$ ,

\mu(n)\mu(d) = \mu^2(d)\mu(n/d)\mathbf1_{(d,n/d)=1}.

Therefore:

Lemma 3.1

\boxed{ a_C(n) = \sum_{\substack{dr=n\\ d\le C\\ (d,r)=1}} \mu^2(d)\mu(r). }

4. Dirichlet series of the parity coefficient

For $\Re s>1$ ,

A_C(s)=\sum_{n\ge1}\frac{a_C(n)}{n^s}.

Then

\boxed{ A_C(s)=\frac{P_C(s)}{\zeta(s)}, }

where

\boxed{ P_C(s) = \sum_{d\le C} \frac{\mu^2(d)}{d^s} \prod_{p\mid d}(1-p^{-s})^{-1}. }

For $C=1$ ,

P_1(s)=1, \qquad A_1(s)=\frac1{\zeta(s)}.

Thus the parity coefficient has an explicit inverse-zeta backbone.

5. Coprime-restricted coefficient

For squarefree $m$ , define

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

For $\Re s>1$ ,

\boxed{ \sum_{n\ge1}\frac{a_{C,m}(n)}{n^s} = \frac{P_{C,m}(s)}{\zeta(s)}, }

with

\boxed{ P_{C,m}(s) = \sum_{\substack{d\le C\\ (d,m)=1}} \frac{\mu^2(d)}{d^s} \prod_{p\mid dm}(1-p^{-s})^{-1}. }

The finite Euler modifier changes. The inverse-zeta factor remains.

6. Coefficient partial sums and zero strength

Let

A_C(x)=\sum_{n\le x}a_C(n).

Suppose for fixed $C$ and fixed $\theta<1$ , for every $\varepsilon>0$ ,

A_C(x)=O_\varepsilon(x^{\theta+\varepsilon}).

Then the Dirichlet series is holomorphic in $\Re s>\theta$ . Hence every zeta zero $\rho$ in this half-plane must satisfy

\boxed{P_C(\rho)=0.}

For $C=1$ , this becomes

\boxed{\zeta(s)\neq0\qquad\Re s>\theta.}

O-RH-031
PARITY_COEFFICIENT_INVERSE_ZETA_STRENGTH
status: CERTIFIED AS STRENGTH AUDIT

This is a strength warning for coefficient-only proof routes; it is not an equivalence theorem for EMBF itself.

7. Inner one-variable transform

Define

F_m(n)=w_N(mn)B(mn-1),

and

S_m = \sum_{\substack{L<n\le2L\\mn<2N}} a_{C,m}(n)F_m(n).

Then

\boxed{\mathfrak B_{N,L,C}=\sum_{\mu(m)\neq0}|S_m|.}

8. Variation of the endogenous dilate profile

Assume $mL\ll N$ . Chebyshev gives $B(x)=O(x)$ and $w_N\le N$ .

As $n$ runs over $[L,2L]$ , the intervals $[mn,m(n+1))$ are disjoint. Using

|B(y)-B(x)|\le \vartheta(y)-\vartheta(x)+(y-x)

and Chebyshev, the total variation of $B(mn-1)$ is $O(N)$ . The total variation of $w_N(mn)$ is also $O(N)$ .

Hence:

Lemma 8.1

\boxed{\operatorname{Var}_{L<n\le2L}F_m(n)=O(N^2).}

Endpoint sizes are also $O(N^2)$ .

9. Coefficient-only Abel route

Let

M_{C,m}(t;L) = \sum_{\substack{L<n\le t\\ (n,m)=1}}a_C(n).

Discrete partial summation gives:

Proposition 9.1

\boxed{ |S_m| \ll N^2 \max_{L<t\le2L}|M_{C,m}(t;L)|. }

Consequently, if uniformly in relevant $m$ ,

\max_{L<t\le2L}|M_{C,m}(t;L)| \ll L^{1-\delta+o(1)},

then

\boxed{ \mathfrak B_{N,L,C} \ll N^3L^{-\delta+o(1)}. }

If $L=N^\alpha$ ,

\boxed{\mathfrak B_{N,L,C}\ll N^{3-\alpha\delta+o(1)}.}

At $\alpha=1/2$ , the gain is $\delta/2$ .

A uniform coefficient theorem strong enough to include $C=1,m=1$ is a fixed-power Mertens theorem and hence fixed-zero-strip strength.

10. Generic second-moment closure

Define

\boxed{ \mathcal Q_{N,L,C} = \sum_{\mu(m)\neq0}|S_m|^2. }

Let $M\asymp N/L$ be the number of outer $m$ values. Then

\boxed{ \mathfrak B_{N,L,C} \le M^{1/2}\mathcal Q_{N,L,C}^{1/2}. }

Since $|a_{C,m}(n)|\le\tau(n)=N^{o(1)}$ , Cauchy in $n$ , divisor multiplicity, and $w_N^2\le N^2$ yield:

Theorem 10.1 — Generic Energy Fallback

Let

\mathcal E_B(2N)=\sum_{k<2N}|B(k-1)|^2.

Then

\boxed{ \mathfrak B_{N,L,C} \ll N^{3/2+o(1)}\mathcal E_B(2N)^{1/2}. }

If PODEE $(\kappa)$ holds uniformly on dyadic scales up to $N$ , then

\boxed{ \mathfrak B_{N,L,C} \ll N^{3-\kappa/2+o(1)}. }

Thus generic second-moment closure is downstream of the target energy.

O-RH-032
EMBF_GENERIC_ENERGY_FALLBACK
status: CERTIFIED

11. Second-moment expansion

Expand

\mathcal Q_{N,L,C} = \sum_m\sum_{n_1,n_2} a_{C,m}(n_1)a_{C,m}(n_2) F_m(n_1)F_m(n_2).

The diagonal part $n_1=n_2$ has scale

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

The completely uncontrolled full second moment may be as large as $N^{11/2+o(1)}$ in a balanced window. Therefore proving diagonal-scale behavior is a genuine fixed-power gain.

12. Endogenous Möbius-Dilate Quasi-Orthogonality

Assume the balanced window

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

Define the target:

EMDQO

\boxed{\mathcal Q_{N,L,C}\ll N^{5+o(1)}.}

Then $M=N^{1/2+o(1)}$ , so

\begin{aligned} \mathfrak B_{N,L,C} &\le M^{1/2}\mathcal Q_{N,L,C}^{1/2}\\ &\ll N^{1/4+o(1)}N^{5/2+o(1)}\\ &=N^{11/4+o(1)}. \end{aligned}

Hence:

Theorem 12.1 — Diagonal Scale Gives a Fixed Power

\boxed{ \operatorname{EMDQO} \Longrightarrow \mathfrak B_{N,L,C} \ll N^{3-1/4+o(1)}. }

No fixed-power Mertens estimate appears in the statement.

13. General window law

If $L=N^{\alpha+o(1)}$ and the same diagonal-scale estimate $\mathcal Q\ll N^{5+o(1)}$ holds, then

\boxed{ \mathfrak B_{N,L,C} \ll N^{3-\alpha/2+o(1)}. }

Thus the generated fixed power is

\boxed{\kappa=\alpha/2.}

14. What EMDQO asks

The off-diagonal piece is

\sum_m\sum_{n_1\neq n_2} a_{C,m}(n_1)a_{C,m}(n_2) w_N(mn_1)w_N(mn_2) B(mn_1-1)B(mn_2-1).

EMDQO therefore asks whether distinct multiplicative dilates of the prime-error profile are quasi-orthogonal on average over $m$ after parity weighting.

This is not ordinary Möbius randomness against an external bounded test function. The profile is generated by the same prime sequence.

15. Known Möbius uniformity calibration

Davenport-type Möbius exponential-sum estimates give savings stronger than every fixed power of $\log N$ against additive phases.

Modern Gowers-uniformity and ergodic results likewise provide strong logarithmic decay for Möbius against structured external systems.

These are substantial results, but they do not supply a fixed $N$ -power for the endogenous multiplicative profile $B(mn-1)$ .

Recent 2026 work on multiple Möbius sums obtains nontrivial bounds in several multivariable settings while emphasizing that binary analogues remain substantially harder in that setting.

No cited result proves EMDQO.

16. EMBF status correction

Paper 13 promoted EMBF as a mechanism frontier. The present audit refines it:

F-RH-012 EMBF
  DEMOTED TO AUXILIARY PARITY-SENSITIVE CONDITION

Reasons:

coefficient-only route
  zero-sensitive

generic second moment
  downstream of PODEE

standalone EMBF
  not known to imply PESC at required precision

17. New canonical mechanism frontier

Create:

F-RH-013
ENDOGENOUS_MOBIUS_DILATE_QUASI_ORTHOGONALITY
abbrev: EMDQO
status: OPEN

Canonical balanced target:

\boxed{ \sum_m \left| \sum_{\substack{L<n\le2L\\mn<2N}} a_{C,m}(n)w_N(mn)B(mn-1) \right|^2 \ll N^{5+o(1)}, \qquad L=N^{1/2+o(1)}. }

18. Campaign 13 verdict

TWO-SIDED MOBIUS CANCELLATION
  NO

INNER N-MOBIUS CANCELLATION
  YES / ONLY PARITY SIGN SOURCE

COEFFICIENT-ONLY FIXED POWER
  ZERO-SENSITIVE / STRENGTH-LOADED

GENERIC CAUCHY SECOND MOMENT
  DOWNSTREAM OF PODEE

DIAGONAL-SCALE SECOND MOMENT
  GENUINE FIXED-POWER SUFFICIENT MECHANISM

EMBF
  AUXILIARY

EMDQO
  NEW CANONICAL MECHANISM FRONTIER

No fixed-power EMDQO theorem is proved.

19. Campaign 14

The next campaign is:

CSM_RH Campaign 14
EMDQO_OFF_DIAGONAL_CORRELATION_AUDIT

Write

\mathcal Q = \mathcal Q_{\rm diag} + \mathcal Q_{\rm off}.

The diagonal already satisfies

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

Therefore the next task is to determine whether one can prove

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

in signed total, or another estimate sufficient for the full second moment.

20. Campaign 14 required questions

Q1
Can the m-sum be interpreted as a multiplicative correlation kernel in n1/n2?

Q2
Does dispersion reintroduce PODEE without gaining a power?

Q3
Can gamma(n,C) and Möbius signs force off-diagonal cancellation?

Q4
What happens near n1=n2?

Q5
Can known large-sieve technology see the endogenous B(mn) profile?

Q6
Do current Möbius uniformity estimates give only log-power decay?

Q7
What exact new off-diagonal theorem is required if known tools fail?

21. State transition

CSM_RH v1.4
  ->
CSM_RH v1.5

with:

Campaign 13
  CLOSED_AS_MOBIUS_STRENGTH_AND_SECOND_MOMENT_AUDIT

O-RH-030
  OUTER_ABSOLUTE_ONE_SIDED_PARITY
  CREATED / CERTIFIED

O-RH-031
  PARITY_COEFFICIENT_INVERSE_ZETA_STRENGTH
  CREATED / CERTIFIED AS STRENGTH AUDIT

O-RH-032
  EMBF_GENERIC_ENERGY_FALLBACK
  CREATED / CERTIFIED

F-RH-012
  EMBF
  DEMOTED TO AUXILIARY CONDITION

F-RH-013
  EMDQO
  CREATED / OPEN

Campaign 14
  EMDQO_OFF_DIAGONAL_CORRELATION_AUDIT
  READY

22. Final status

RH = OPEN

PESC = OPEN

EMBF = AUXILIARY

COEFFICIENT-ONLY EMBF FIXED POWER
= ZERO-SENSITIVE

GENERIC EMBF SECOND MOMENT
= DOWNSTREAM OF PODEE

EMDQO DIAGONAL-SCALE SECOND MOMENT
= OPEN / GENUINE FIXED-POWER MECHANISM

BALANCED EMDQO SUCCESS
= kappa 1/4 parity-breaking gain

NEXT CAMPAIGN = 14

The new concrete target is

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

At the current closure state, this is the first parity-sensitive second-moment theorem which could produce a fixed power without simply assuming a fixed-power Mertens estimate or using PESC itself as input.

CSM_RH Paper 14 — One-Sided Möbius Parity, Inverse-Zeta Coefficient Strength, and Endogenous Dilate Quasi-Orthogonality

CSM_RH Paper 14

One-Sided Möbius Parity, Inverse-Zeta Coefficient Strength, and Endogenous Dilate Quasi-Orthogonality

0. Trust boundary

1. Canonical EMBF object

2. Exact Möbius factorization

Theorem 2.1 — One-Sided Parity Factorization

3. The parity coefficient

Lemma 3.1

4. Dirichlet series of the parity coefficient

5. Coprime-restricted coefficient

6. Coefficient partial sums and zero strength

7. Inner one-variable transform

8. Variation of the endogenous dilate profile

Lemma 8.1

9. Coefficient-only Abel route

Proposition 9.1

10. Generic second-moment closure

Theorem 10.1 — Generic Energy Fallback

11. Second-moment expansion

12. Endogenous Möbius-Dilate Quasi-Orthogonality

EMDQO

Theorem 12.1 — Diagonal Scale Gives a Fixed Power

13. General window law

14. What EMDQO asks

15. Known Möbius uniformity calibration

16. EMBF status correction

17. New canonical mechanism frontier

18. Campaign 13 verdict

19. Campaign 14

20. Campaign 14 required questions

21. State transition

22. Final status