CSM_RH Paper 68

Dimension-Two Shifted-Prime Möbius Local Factors, Seeded Signed Power Cancellation, and the Absoluteization Energy Frontier

Project: CSM_RH
Paper: 68
Version: v0.1
Date: 2026-09-09
Campaign: 46 — SEEDED_ARITHMETIC_STRIP_GAP_GENERATION
Track: PT6G — LOW-CONDUCTOR LOCAL-FACTOR RENORMALIZED DISPERSION
Status: LOCAL FACTOR RENORMALIZATION CERTIFIED / SIGNED FIXED-POWER CANCELLATION CERTIFIED / ABSOLUTEIZATION-PARITY ENERGY OPEN
Canonical entry state: v1.58 / Paper 67 v0.1
RH_PROVED: FALSE
RH_DISPROVED: FALSE
GLOBAL_RH_CERTIFICATE: FALSE

Abstract

Paper 67 showed that fixed low-conductor character harmonics survive absolute shift norms. The present paper computes the exact local arithmetic profile of

C_X(h) = \sum_{p\le X}\mu(p+h),

and shows that the low-conductor character modes do admit a natural multiplicative renormalization.

For a prime $\ell$ , define

\mu_\ell(n) = \begin{cases} 1,&v_\ell(n)=0,\\ -1,&v_\ell(n)=1,\\ 0,&v_\ell(n)\ge2. \end{cases}

If $\ell\nmid h$ , averaging over reduced residue classes modulo $\ell^2$ gives

\boxed{ \kappa_\ell(h) = \frac{\ell^2-3\ell+1}{\ell(\ell-1)}. }

If $\ell\mid h$ , then

\boxed{ \kappa_\ell(h)=1. }

For a finite set of primes $S$ , with

Q_S=\prod_{\ell\in S}\ell^2,

and

\mu_S(n)=\prod_{\ell\in S}\mu_\ell(n),

CRT gives the exact identity

\boxed{ \frac1{\phi(Q_S)} \sum_{\substack{a\bmod Q_S\\(a,Q_S)=1}} \mu_S(a+h) = \prod_{\ell\in S}\kappa_\ell(h). }

For $\ell\nmid h$ ,

\kappa_\ell(h) = \left(1-\frac1\ell\right)^2 \left(1+O(\ell^{-2})\right),

so for fixed nonzero $h$ ,

\boxed{ \prod_{\substack{\ell\le z\\\ell\nmid h}} \kappa_\ell(h) = \frac{C(h)+o_h(1)}{(\log z)^2}, }

with $C(h)\ne0$ .

Even more cleanly,

\boxed{ \frac1\ell \sum_{h\bmod\ell}\kappa_\ell(h) = \left(1-\frac1\ell\right)^2, }

and therefore a complete shift average satisfies

\boxed{ \mathbb E_h \prod_{\ell\le z}\kappa_\ell(h) \sim \frac{e^{-2\gamma}}{(\log z)^2}. }

Thus the exact low-conductor local profile is a dimension-two sieve factor. This renormalizes the deterministic local characters, but it only gives logarithmic decay. Even a polynomial cutoff $z=X^\theta$ produces only $(\log X)^{-2}$ , never a fixed power $X^{-\eta}$ .

The PESC seed already does better in the signed shift mean. Put

d=\frac{\kappa}{2},

M(y)=\sum_{n\le y}\mu(n)\ll y^{1-d+o(1)}.

For

H=X^{1-\tau},

\sum_{h\le H}C_X(h) = \sum_{p\le X} \left( M(p+H)-M(p) \right),

and hence, when $\tau<d$ ,

\boxed{ \left| \sum_{h\le H}C_X(h) \right| \ll H\pi(X)X^{-(d-\tau)+o(1)}. }

So the seed has already crossed the log-to-power barrier before absolute values.

The unresolved gap is exactly

\boxed{ \left|\sum_hC_X(h)\right| \quad\longrightarrow\quad \sum_h|C_X(h)|. }

A sufficient second-moment theorem is

\boxed{ E_2(X,H) = \sum_{h\le H} |C_X(h)|^2 \ll H\pi(X)^2X^{-2\eta}. }

By Cauchy this implies

\sum_{h\le H}|C_X(h)| \ll H\pi(X)X^{-\eta}.

Expanding the energy gives

\boxed{ E_2(X,H) = \sum_{p_1,p_2\le X} \sum_{h\le H} \mu(p_1+h)\mu(p_2+h). }

The diagonal is only $O(H\pi(X))$ , harmless for every fixed $\eta<1/2$ . The hard term is the off-diagonal prime-pair averaged two-point Chowla energy.

This motivates the auxiliary frontier

F-RH-020
PRIME_PAIR_AVERAGED_CHOWLA_ABSOLUTEIZATION_ENERGY

with an explicit warning: no certified bridge from F-RH-020 to the root frontier F-RH-017-v3 has yet been established.

No RH theorem is claimed.

1. Exact one-prime local factor

Fix a prime $\ell$ .

If $\ell\mid h$ , then every reduced residue $a\bmod\ell^2$ has

a+h\not\equiv0\pmod\ell,

\kappa_\ell(h)=1.

Assume now $\ell\nmid h$ .

There are

\phi(\ell^2)=\ell(\ell-1)

reduced residues modulo $\ell^2$ .

Exactly $\ell(\ell-2)$ have $v_\ell(a+h)=0$ .

Exactly $\ell-1$ have $v_\ell(a+h)=1$ .

Exactly one has $v_\ell(a+h)\ge2$ .

Therefore

\boxed{ \kappa_\ell(h) = \frac{\ell(\ell-2)-(\ell-1)}{\ell(\ell-1)} = \frac{\ell^2-3\ell+1}{\ell(\ell-1)}. }

Record:

B-RH-087
EXACT_SHIFTED_PRIME_MOBIUS_ONE_PRIME_LOCAL_FACTOR
CERTIFIED

2. Finite CRT recombination

Let $S$ be finite and define

Q_S=\prod_{\ell\in S}\ell^2.

Since reduced residues modulo $Q_S$ are the CRT product of reduced residues modulo each $\ell^2$ , and $\mu_S$ factors prime by prime,

\boxed{ \frac1{\phi(Q_S)} \sum_{\substack{a\bmod Q_S\\(a,Q_S)=1}} \mu_S(a+h) = \prod_{\ell\in S}\kappa_\ell(h). }

Record:

B-RH-088
FINITE_LOW_CONDUCTOR_CHARACTER_PROFILES_RECOMBINE_TO_A_LOCAL_EULER_PRODUCT
CERTIFIED

For fixed $S$ , the prime number theorem in arithmetic progressions transfers this local residue profile to the prime average as $X\to\infty$ .

3. Sieve dimension two

For $\ell\nmid h$ , set

f_\ell = \frac{\ell^2-3\ell+1}{\ell(\ell-1)}.

Then

\frac{f_\ell}{(1-1/\ell)^2} = \frac{\ell(\ell^2-3\ell+1)}{(\ell-1)^3} = 1-\frac2{\ell^2}+O(\ell^{-3}).

The ratio product therefore converges absolutely.

Mertens' product theorem gives

\prod_{\ell\le z}\left(1-\frac1\ell\right) \sim \frac{e^{-\gamma}}{\log z}.

Hence for fixed nonzero $h$ ,

\boxed{ \prod_{\substack{\ell\le z\\\ell\nmid h}} \kappa_\ell(h) = \frac{C(h)+o_h(1)}{(\log z)^2}. }

Record:

B-RH-089
SHIFTED_PRIME_MOBIUS_LOCAL_PROFILE_HAS_SIEVE_DIMENSION_TWO
CERTIFIED

4. Exact shift-averaged local product

Among $h\bmod\ell$ , one class has $\ell\mid h$ and contributes $1$ ; the other $\ell-1$ classes contribute $f_\ell$ .

Thus

\begin{aligned} \frac1\ell\sum_{h\bmod\ell}\kappa_\ell(h) &= \frac1\ell+\frac{\ell-1}{\ell}f_\ell \\ &= \boxed{ \left(1-\frac1\ell\right)^2. } \end{aligned}

By CRT,

\boxed{ \mathbb E_h \prod_{\ell\le z}\kappa_\ell(h) = \prod_{\ell\le z} \left(1-\frac1\ell\right)^2 \sim \frac{e^{-2\gamma}}{(\log z)^2}. }

This is the exact dimension-two local factor seen by the shift average.

5. Local-factor logarithmic ceiling

Even if one could impose the complete local profile through

z=X^\theta,

one obtains only

\boxed{ (\log X)^{-2}. }

No fixed $\eta>0$ satisfies

(\log X)^{-2}\ll X^{-\eta}.

Thus:

O-RH-157
LOW_CONDUCTOR_LOCAL_FACTOR_RENORMALIZATION_HAS_ONLY_LOGARITHMIC_DIMENSION_TWO_DECAY
CERTIFIED

This is the local-congruence version of the classical sieve parity limitation.

6. Seeded signed shifted-prime power

Assume PESC $(\kappa)$ and set

d=\frac{\kappa}{2}.

Then

M(y)\ll y^{1-d+o(1)}.

Let

C_X(h)=\sum_{p\le X}\mu(p+h), \qquad H=X^{1-\tau}.

Interchanging sums,

\boxed{ \sum_{h\le H}C_X(h) = \sum_{p\le X} \left( M(p+H)-M(p) \right). }

Since $p+H\le2X$ ,

|M(p+H)-M(p)| \ll X^{1-d+o(1)}.

Therefore, if $\tau<d$ ,

\boxed{ \left| \sum_{h\le H}C_X(h) \right| \ll H\pi(X)X^{-(d-\tau)+o(1)}. }

Record:

B-RH-090
PESC_SEED_GIVES_FIXED_POWER_SIGNED_SHIFTED_PRIME_MOBIUS_CANCELLATION
CERTIFIED

7. The absoluteization gap

Lichtman's theorem controls

\sum_{h\le H}|C_X(h)|.

The seed gives only

\left|\sum_{h\le H}C_X(h)\right|.

Paper 67 proved that no deterministic signed-to-absolute conversion exists even for a fixed character harmonic.

Thus:

O-RH-158
PESC_SEED_CROSSES_SIGNED_LOG_TO_POWER_BUT_NOT_THE_SHIFT_ABSOLUTEIZATION_GAP
CERTIFIED

The missing information is global sign energy.

8. A sufficient second-moment theorem

By Cauchy-Schwarz,

\sum_{h\le H}|C_X(h)| \le H^{1/2} E_2(X,H)^{1/2},

where

\boxed{ E_2(X,H) = \sum_{h\le H}|C_X(h)|^2. }

Therefore

E_2(X,H) \ll H\pi(X)^2X^{-2\eta}

implies

\sum_{h\le H}|C_X(h)| \ll H\pi(X)X^{-\eta}.

9. Prime-pair averaged Chowla expansion

Expand:

\boxed{ E_2(X,H) = \sum_{p_1,p_2\le X} \sum_{h\le H} \mu(p_1+h)\mu(p_2+h). }

The diagonal satisfies

\boxed{ E_2^{\rm diag} \le H\pi(X). }

For any fixed $\eta<1/2$ ,

H\pi(X) = o\left( H\pi(X)^2X^{-2\eta} \right).

Hence the hard object is

\boxed{ E_2^{\rm off} = \sum_{\substack{p_1,p_2\le X\\p_1\ne p_2}} \sum_{h\le H} \mu(p_1+h)\mu(p_2+h). }

This is a prime-pair averaged two-point Chowla problem.

10. Auxiliary frontier

Open:

F-RH-020
PRIME_PAIR_AVERAGED_CHOWLA_ABSOLUTEIZATION_ENERGY
OPEN_AUXILIARY

Target:

\boxed{ E_2(X,H) \ll H\pi(X)^2X^{-2\eta} }

for some fixed $\eta>0$ .

Important scope:

NO_CERTIFIED_ROOT_BRIDGE_TO_F-RH-017-v3.

The canonical RH frontier remains F-RH-017-v3.

F-RH-020 is a parity diagnostic: it identifies a concrete ordinary-arithmetic theorem that would cross the signed-to-absolute gap in the shifted-prime auxiliary model.

11. Relation to current literature

Lichtman proves

\sum_{h\le H} \left| \sum_{p\le X}\mu(p+h) \right| \ll H\pi(X)(\log X)^{-1/3+\delta}

for polynomial shift ranges, and proves higher averaged Hardy-Littlewood-Chowla correlations with logarithmic savings.

His proof uses sieve restriction to typical factorizations, Fourier decoupling, and major/minor arc estimates. The local-factor calculation above explains why purely local congruence data naturally live at logarithmic scale.

Friedlander and Iwaniec's asymptotic sieve gives complementary calibration: the classical parity problem is broken only after adding extra bilinear information.

Thus the remaining fixed-power target is naturally a parity-energy theorem, not another local-factor computation.

12. PT6G status

PT6G-LOCAL:
CLOSED / EXACT EULER PRODUCT CERTIFIED.

PT6G-LOCAL-DECAY:
CLOSED AS NON-AMPLIFYING / DIMENSION-TWO LOG DECAY.

PT6G-SIGNED:
CLOSED / SEED FIXED POWER CERTIFIED.

PT6G-ABSOLUTE:
OPEN / GLOBAL PARITY ENERGY REQUIRED.

13. State transition

Advance candidate state

v1.58 \to v1.59.

Add:

B-RH-087
EXACT_SHIFTED_PRIME_MOBIUS_ONE_PRIME_LOCAL_FACTOR

B-RH-088
FINITE_LOW_CONDUCTOR_CHARACTER_PROFILES_RECOMBINE_TO_A_LOCAL_EULER_PRODUCT

B-RH-089
SHIFTED_PRIME_MOBIUS_LOCAL_PROFILE_HAS_SIEVE_DIMENSION_TWO

B-RH-090
PESC_SEED_GIVES_FIXED_POWER_SIGNED_SHIFTED_PRIME_MOBIUS_CANCELLATION

O-RH-157
LOW_CONDUCTOR_LOCAL_FACTOR_RENORMALIZATION_HAS_ONLY_LOGARITHMIC_DIMENSION_TWO_DECAY

O-RH-158
PESC_SEED_CROSSES_SIGNED_LOG_TO_POWER_BUT_NOT_THE_SHIFT_ABSOLUTEIZATION_GAP

Open F-RH-020 as auxiliary only.

No RH certificate is created.

14. Conclusion

The low-conductor local characters can be recombined exactly.

Their total deterministic effect is a dimension-two Euler product with logarithmic decay.

The seed already gives fixed-power cancellation in the signed shifted-prime mean.

Therefore the only unresolved part of this auxiliary route is absoluteization: a global parity/sign-energy theorem.

The simplest sufficient formulation is the prime-pair averaged two-point Chowla energy F-RH-020.

The root CSM_RH frontier remains F-RH-017-v3.

CSM_RH Paper 68 — Dimension-Two Shifted-Prime Möbius Local Factors, Seeded Signed Power Cancellation, and the Absoluteiz

CSM_RH Paper 68

Dimension-Two Shifted-Prime Möbius Local Factors, Seeded Signed Power Cancellation, and the Absoluteization Energy Frontier

Abstract

1. Exact one-prime local factor

2. Finite CRT recombination

3. Sieve dimension two

4. Exact shift-averaged local product

5. Local-factor logarithmic ceiling

6. Seeded signed shifted-prime power

7. The absoluteization gap

8. A sufficient second-moment theorem

9. Prime-pair averaged Chowla expansion

10. Auxiliary frontier

11. Relation to current literature

12. PT6G status

13. State transition

14. Conclusion