CSM_RH Paper 70

Boundary-Mode Concentration in the Shift Mean, Centering Attenuation, and Closure of the Original Hankel-Coercivity Guess

Project: CSM_RH
Paper: 70
Version: v0.1
Date: 2026-09-09
Campaign: 46 — SEEDED_ARITHMETIC_STRIP_GAP_GENERATION
Auxiliary tracks: F-RH-020 / F-RH-021
Canonical root frontier: F-RH-017-v3
Status: ORIGINAL CENTERED HANKEL COERCIVITY TARGET CORRECTED / SIGNED PRIME-SAMPLING BRIDGE REMAINS UNPROVED / AUXILIARY ROUTE DE-PRIORITIZED
Canonical entry state: v1.60 / Paper 69 v0.1
RH_PROVED: FALSE
RH_DISPROVED: FALSE
GLOBAL_RH_CERTIFICATE: FALSE

Abstract

Paper 69 proposed a possible root bridge:

if a zeta zero lies on the seed boundary

\rho=1-d+i\gamma, \qquad d=\frac{\kappa}{2},

then perhaps the centered shifted-prime Möbius Hankel variance satisfies

V_2(X,H) \gg H\pi(X)^2X^{-2d-o(1)}.

The present paper tests this conjecture on the canonical smooth Mertens boundary mode and shows that the target is too strong.

Let

\beta=\Re\rho=1-d

and define the synthetic Mertens mode

M_\rho(x)=x^\rho.

Its discrete density is

\boxed{ m_\rho(n) = n^\rho-(n-1)^\rho. }

Sample this density at the actual rational primes:

\boxed{ C_{\rho,X}(h) = \sum_{p\le X} m_\rho(p+h). }

The prime number theorem implies

\boxed{ C_{\rho,X}(0) = \frac{X^\rho}{\log X} \left( 1+o_\rho(1) \right). }

More importantly, if

h\to\infty, \qquad h=o(X),

then

\boxed{ C_{\rho,X}(h) - C_{\rho,X}(0) = - \frac{h^\rho}{\log h} \left( 1+o_\rho(1) \right). }

Thus the boundary mode is sampled by the primes as

\boxed{ C_{\rho,X}(h) = A_\rho(X) - B_\rho(h) + \text{lower order}, }

with

A_\rho(X) \asymp \frac{X^\rho}{\log X},

B_\rho(h) \asymp \frac{h^\rho}{\log h}.

For a near-macroscopic shift range

H=X^{1-\tau},

the $X$ -scale term dominates the $H$ -scale variation by the fixed factor

X^{\beta\tau+o(1)}.

Consequently,

\boxed{ \sum_{h\le H} C_{\rho,X}(h) = H \frac{X^\rho}{\log X} \left( 1+o_\rho(1) \right). }

The uncentered shift mean therefore retains the critical boundary amplitude:

\boxed{ H \left| \frac1H \sum_{h\le H}C_{\rho,X}(h) \right|^2 \asymp H\pi(X)^2 X^{-2d}. }

But centering removes precisely this dominant component.

Let

\overline C_{\rho,X} = \frac1H \sum_{h\le H} C_{\rho,X}(h).

Then

\boxed{ V_{2,\rho}(X,H) := \sum_{h\le H} |C_{\rho,X}(h)-\overline C_{\rho,X}|^2 \asymp_\rho \frac{ H^{2\beta+1} }{ \log^2H }. }

Since

\pi(X) \sim \frac{X}{\log X},

this becomes

\boxed{ V_{2,\rho}(X,H) = H\pi(X)^2 X^{-2[d+\tau(1-d)]+o(1)}. }

Define the centered effective exponent

\boxed{ d_{\rm cent} = d+\tau(1-d). }

Then

d_{\rm cent}>d

for every fixed $\tau>0$ .

Thus the smooth boundary mode does not support Paper 69's centered lower target

V_2 \gg H\pi(X)^2X^{-2d-o(1)}.

Centering itself removes the critical boundary signal and leaves a strictly smaller variation.

The exponent

d+\tau(1-d)

is not new to the CSM_RH chain. It is exactly the anchor exponent which appeared in the corrected $L^1$ residue-chain theorem of Paper 61. The same geometry is reappearing in a different language:

a multiplicative boundary mode is nearly constant
across a sublinear additive shift window;
subtracting the mean differentiates the mode
and gains a factor determined by H/X.

Therefore the original F-RH-021 is closed.

A more plausible bridge is the uncentered signed mean:

F-RH-021R
SIGNED SHIFTED-PRIME BOUNDARY-SAMPLING COERCIVITY

Desired theorem for the actual Möbius function:

\rho=1-d+i\gamma

is a zeta boundary zero, then along an unbounded sequence of $X$ ,

\boxed{ \left| \sum_{h\le H} \sum_{p\le X} \mu(p+h) \right| \gg_\rho H\pi(X) X^{-d-o(1)}. }

The smooth boundary mode proves that this is the correct scaling.

If F-RH-021R were certified, then any F-RH-020 energy upper theorem with exponent

\eta>d

would imply by Cauchy-Schwarz

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

contradicting the boundary lower bound.

However, F-RH-021R is presently unproved.

The reason is structural. Zeta zeros are Mellin-multiplicative singularities. The map

\mu \mapsto \left( h\mapsto \sum_{p\le X}\mu(p+h) \right)

is an additive prime-sampling transform and has no known Mellin-diagonal pole-transport identity.

Recent work of Pintz proves that Mertens oscillation itself strongly reflects the largest zeta-zero term. This supports the existence of the boundary mode before sampling, but does not supply coercivity after additive prime sampling.

Current shifted-prime Möbius theorems prove averaged cancellation in the opposite direction. They do not prove that a hypothetical zeta boundary zero forces a large shifted-prime Möbius correlation.

Accordingly:

F-RH-020:
remains a useful parity-energy auxiliary problem.

F-RH-021:
closed in its centered form.

F-RH-021R:
open high-cost bridge candidate, not certified.

canonical root attack:
return to F-RH-017-v3.

No RH theorem is claimed.

1. Synthetic boundary mode

Fix

\rho=\beta+i\gamma, \qquad 0<\beta<1.

For the CSM_RH boundary calibration,

\beta=1-d.

Define

\boxed{ M_\rho(x)=x^\rho }

and its discrete increment

\boxed{ m_\rho(n) = n^\rho-(n-1)^\rho. }

Taylor expansion gives

\boxed{ m_\rho(n) = \rho n^{\rho-1} + O_\rho(n^{\beta-2}). }

The remainder is absolutely summable over primes.

2. Prime sampling of the smooth mode

Define

\boxed{ C_{\rho,X}(h) = \sum_{p\le X} m_\rho(p+h). }

Using Section 1,

C_{\rho,X}(h) = \rho \sum_{p\le X} (p+h)^{\rho-1} + O_\rho(1+h^{\beta-1}).

For $h=0$ , partial summation with the prime number theorem gives

\sum_{p\le X} p^{\rho-1} = \frac{ X^\rho }{ \rho\log X } \left( 1+o_\rho(1) \right).

Hence:

Theorem 2.1 — Critical prime-sampled constant mode

\boxed{ C_{\rho,X}(0) = \frac{ X^\rho }{ \log X } \left( 1+o_\rho(1) \right). }

Create:

B-RH-096
SMOOTH_MERTENS_BOUNDARY_MODE_HAS_CRITICAL_PRIME_SAMPLED_SHIFT_CONSTANT_COMPONENT
CERTIFIED_MODEL_THEOREM

3. Shift variation asymptotic

Consider

C_{\rho,X}(h)-C_{\rho,X}(0).

At the principal smooth level,

\rho \sum_{p\le X} \left[ (p+h)^{\rho-1} - p^{\rho-1} \right].

The prime number theorem reduces this, with an error smaller than the displayed main term, to

\rho \int_2^X \frac{ (t+h)^{\rho-1} - t^{\rho-1} }{ \log t }\,dt.

Scale

t=hu.

Then

\frac{\log h}{h^\rho} \int_2^X \frac{ (t+h)^{\rho-1} - t^{\rho-1} }{ \log t }\,dt

tends to

\int_0^\infty \left[ (1+u)^{\rho-1} - u^{\rho-1} \right]du.

But

\begin{aligned} \int_0^R \left[ (1+u)^{\rho-1} - u^{\rho-1} \right]du &= \frac{ (R+1)^\rho-R^\rho-1 }{ \rho } \\ &\longrightarrow -\frac1\rho. \end{aligned}

Therefore:

Theorem 3.1 — Boundary-mode shift variation

h\to\infty, \qquad h=o(X),

then

\boxed{ C_{\rho,X}(h) - C_{\rho,X}(0) = - \frac{ h^\rho }{ \log h } \left( 1+o_\rho(1) \right). }

Create:

B-RH-097
PRIME_SAMPLED_SMOOTH_BOUNDARY_MODE_VARIES_ONLY_AT_THE_H_SCALE
CERTIFIED_MODEL_THEOREM

4. Signed shift mean retains the critical boundary scale

Let

H=X^{1-\tau}, \qquad 0<\tau<1.

Sum Theorem 3.1 over

h\le H.

Standard power summation gives

\sum_{h\le H} \frac{h^\rho}{\log h} = \frac{ H^{\rho+1} }{ (\rho+1)\log H } \left( 1+o_\rho(1) \right).

This has magnitude

H^{\beta+1}/\log H.

The constant-mode contribution has magnitude

H X^\beta/\log X.

Their ratio is

\boxed{ \left( \frac HX \right)^\beta \frac{\log X}{\log H} = X^{-\beta\tau+o(1)} \to0. }

Hence:

Theorem 4.1 — Signed mean boundary retention

\boxed{ \sum_{h\le H} C_{\rho,X}(h) = H \frac{ X^\rho }{ \log X } \left( 1+o_\rho(1) \right). }

Since

\pi(X)\sim X/\log X,

\boxed{ \left| \sum_{h\le H} C_{\rho,X}(h) \right| \asymp_\rho H\pi(X)X^{-(1-\beta)}. }

At the PESC boundary,

1-\beta=d.

Thus the smooth model supports F-RH-021R at exactly the critical exponent $d$ .

5. Centered variance attenuates the boundary mode

Define

\overline C_{\rho,X} = \frac1H \sum_{h\le H} C_{\rho,X}(h).

The $X$ -dependent constant cancels from

C_{\rho,X}(h)-\overline C_{\rho,X}.

Thus the leading centered profile is

- \frac{h^\rho}{\log h} + \frac1H \sum_{j\le H} \frac{j^\rho}{\log j}.

Rescale

h=Hu.

Then the variance is asymptotic to

\frac{ H^{2\beta+1} }{ \log^2 H }

times the positive constant

\boxed{ \mathfrak c_\rho = \int_0^1 \left| u^\rho - \frac1{\rho+1} \right|^2du. }

The constant simplifies to

\boxed{ \mathfrak c_\rho = \frac1{2\beta+1} - \frac1{|\rho+1|^2} >0. }

Hence:

Theorem 5.1 — Centered smooth-mode Hankel scale

\boxed{ V_{2,\rho}(X,H) \sim_\rho \mathfrak c_\rho \frac{ H^{2\beta+1} }{ \log^2H }. }

Create:

B-RH-098
CENTERING_ATTENUATES_THE_SMOOTH_BOUNDARY_MODE_TO_H_SCALE_VARIANCE
CERTIFIED_MODEL_THEOREM

6. Effective centered exponent

Theorem 6.1 — Centered effective exponent

\beta=1-d

and

H=X^{1-\tau},

H^{2\beta+1} = H X^{2\beta(1-\tau)}.

Using

H\pi(X)^2 = H X^2 \log^{-2}X (1+o(1)),

we obtain

\boxed{ V_{2,\rho}(X,H) = H\pi(X)^2 X^{-2d_{\rm cent}+o(1)}, }

where

\boxed{ d_{\rm cent} = d+\tau(1-d). }

Thus

\boxed{ d_{\rm cent}>d. }

The centered boundary signal is strictly smaller than the uncentered critical signal.

7. Reappearance of the Paper-61 anchor exponent

Paper 61's corrected $L^1$ exceptional amplifier had the anchor term

\boxed{ d+\tau(1-d). }

Theorem 6.1 produces exactly the same exponent.

This is not a coincidence.

A multiplicative mode

x^{1-d+i\gamma}

changes only by a relative amount controlled by

H/X

over a sublinear additive shift window.

Subtracting its shift mean removes the zeroth-order part.

What remains is its sublinear variation.

Thus the same exponent appears in:

residue-chain anchoring;
long-scale comparison;
centered prime-sampled Mertens variance.

This is a common critical-locking geometry.

8. Correction to F-RH-021

Paper 69 proposed:

F-RH-021
BOUNDARY MERTENS MODE PRIME-SAMPLING HANKEL COERCIVITY

with desired lower scale

V_2 \gg H\pi(X)^2X^{-2d-o(1)}.

The smooth boundary mode itself has only

H\pi(X)^2 X^{-2[d+\tau(1-d)]+o(1)}.

Therefore the old target is not supported even by the canonical residue mode.

Record:

C-RH-004
PAPER69_CENTERED_F_RH_021_CRITICAL_LOWER_SCALE_CORRECTED

and

O-RH-160
CENTERED_HANKEL_CRITICAL_LOWER_BOUND_X_MINUS_2D_IS_TOO_STRONG_FOR_THE_SMOOTH_BOUNDARY_MODE
CERTIFIED_MODEL_BARRIER

Close the original F-RH-021 formulation.

9. Revised bridge candidate F-RH-021R

The smooth mode shows that the critical signal survives in the uncentered signed mean.

Open only as a high-cost auxiliary bridge candidate:

F-RH-021R
SIGNED_SHIFTED_PRIME_BOUNDARY_SAMPLING_COERCIVITY

Desired theorem for the actual Möbius function:

\rho=1-d+i\gamma

is a zeta zero on the seed boundary, then along an unbounded sequence of $X$ ,

\boxed{ \left| \sum_{h\le H} \sum_{p\le X} \mu(p+h) \right| \gg_\rho H\pi(X) X^{-d-o(1)}. }

The smooth model proves only that this scale is natural.

It does not prove the theorem for $\mu$ .

10. How F-RH-020 would become a root theorem if F-RH-021R held

Suppose F-RH-020 gave

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

with

\eta>d.

Then Cauchy gives

\begin{aligned} \left| \sum_{h\le H}C_X(h) \right| &\le H^{1/2} E_2(X,H)^{1/2} \\ &\ll \boxed{ H\pi(X)X^{-\eta}. } \end{aligned}

If F-RH-021R also held, this would contradict a boundary zero.

Thus:

F-RH-020 with eta>d
+
F-RH-021R
= genuine root strip improvement.

But the bridge remains unproved.

11. Why zeta zeros do not automatically transport through additive prime sampling

The Mellin transform of $M(x)$ is directly tied to

1/\zeta(s).

A zero of zeta therefore creates a Mellin singularity and forces Mertens oscillation.

By contrast,

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

is an additive correlation.

For fixed $h$ , the Dirichlet series

\sum_p \frac{\mu(p+h)}{p^s}

has no known Euler-product or Mellin-diagonal identity in terms of $\zeta$ .

Likewise, the shift-averaged transform

\sum_{h\le H}\sum_{p\le X}\mu(p+h)

is not a Dirichlet convolution observable.

Therefore there is no formal pole-transport theorem analogous to

\Lambda=\mu*\log.

Create:

O-RH-161
ZETA_MELLIN_POLE_HAS_NO_FORMAL_TRANSPORT_IDENTITY_TO_ADDITIVE_SHIFTED_PRIME_MOBIUS_SAMPLING
CERTIFIED_AS_BRIDGE_BARRIER

This is a statement about the current structural identities, not a proof that F-RH-021R is false.

12. External Mertens oscillation calibration

Recent work of Pintz studies the oscillation of

M(x)=\sum_{n\le x}\mu(n)

and shows that the average modulus of $M(x)$ closely tracks the largest zeta-zero term.

This strengthens the calibration that a rightmost zeta zero genuinely produces a large Mertens boundary component.

However, the theorem concerns $M(x)$ itself.

It does not prove that the same component survives an additive prime-sampling operator with a quantitative lower bound.

Thus the present bridge gap is real:

Mellin boundary forcing:
strongly understood.

additive prime-sampling coercivity:
not established.

13. Relation to shifted-prime Möbius upper theorems

Lichtman proves strong cancellation of

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

on average over shifts.

This is an upper theorem.

It does not give a converse saying that a zeta zero near a fixed boundary forces this correlation to be large.

Hence existing shifted-prime technology and existing Mertens oscillation technology point in opposite directions but do not currently meet in a coercive equivalence.

14. Strategic status of F-RH-020

F-RH-020 remains mathematically meaningful:

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

would be a strong fixed-power parity theorem.

But without F-RH-021R it has no certified root implication.

Moreover, the exponent required for a root contradiction would be

\eta>d,

which is stronger than the subcritical energy exponents previously considered.

Therefore:

F-RH-020:
retain as auxiliary parity frontier,
de-prioritize as RH root route.

F-RH-021 original centered bridge:
close.

F-RH-021R:
open high-cost bridge candidate,
not canonical.

F-RH-017-v3:
resume as canonical root frontier.

15. Recommended return to the root problem

The direct root observable

U_H(x) = \psi(x+H)-\psi(x)-H

has one decisive advantage over the prime–Möbius auxiliary observable:

a zeta boundary zero is known to act on it directly through the explicit formula.

Papers 55, 60 and 61 already certified the critical boundary amplitude and exceptional-set forcing geometry there.

Thus the next root attack should return to:

\boxed{ \text{F-RH-017-v3} }

and seek genuinely prime-side low-frequency suppression of

U_H

rather than first passing through shifted Möbius sampling.

This avoids the unproved additive pole-transport bridge.

16. External calibration

16.1. Pintz 2026 Mertens oscillation

J. Pintz, Oscillation of partial sums of the Möbius function and zeros of Riemann's zeta function, arXiv:2608.24878, August 2026.

The paper relates the average modulus of $M(x)$ closely to the largest zeta-zero contribution.

URL:

https://arxiv.org/abs/2608.24878

16.2. Lichtman shifted-prime Möbius theorem

J. D. Lichtman, Averages of the Möbius Function on Shifted Primes, Quarterly Journal of Mathematics 73 (2022), 729–757.

The paper proves averaged shifted-prime Möbius cancellation, but no converse zero-to-correlation lower theorem.

URL:

https://academic.oup.com/qjmath/article/73/2/729/6446139

17. State transition

Advance candidate state

v1.60 \to v1.61.

Add:

B-RH-096
SMOOTH_MERTENS_BOUNDARY_MODE_HAS_CRITICAL_PRIME_SAMPLED_SHIFT_CONSTANT_COMPONENT
CERTIFIED_MODEL_THEOREM

B-RH-097
PRIME_SAMPLED_SMOOTH_BOUNDARY_MODE_VARIES_ONLY_AT_THE_H_SCALE
CERTIFIED_MODEL_THEOREM

B-RH-098
CENTERING_ATTENUATES_THE_SMOOTH_BOUNDARY_MODE_TO_H_SCALE_VARIANCE
CERTIFIED_MODEL_THEOREM

O-RH-160
CENTERED_HANKEL_CRITICAL_LOWER_BOUND_X_MINUS_2D_IS_TOO_STRONG_FOR_THE_SMOOTH_BOUNDARY_MODE
CERTIFIED_MODEL_BARRIER

O-RH-161
ZETA_MELLIN_POLE_HAS_NO_FORMAL_TRANSPORT_IDENTITY_TO_ADDITIVE_SHIFTED_PRIME_MOBIUS_SAMPLING
CERTIFIED_AS_BRIDGE_BARRIER

C-RH-004
PAPER69_CENTERED_F_RH_021_CRITICAL_LOWER_SCALE_CORRECTED

Close:

F-RH-021
CLOSED_IN_ORIGINAL_CENTERED_FORM

Open only as noncanonical bridge candidate:

F-RH-021R
SIGNED_SHIFTED_PRIME_BOUNDARY_SAMPLING_COERCIVITY
OPEN_HIGH_COST

Canonical next target:

F-RH-017-v3

No RH certificate is created.

18. Conclusion

The smooth Mertens boundary mode survives prime sampling, but almost entirely as a shift-constant component.

The critical signal has relative size

X^{-d}.

Centering removes it.

The residual centered variance has the strictly smaller scale

X^{-2[d+\tau(1-d)]}.

Therefore the centered Hankel variance is the wrong place to demand critical boundary coercivity.

A root bridge would have to preserve and control the signed shift mean.

No theorem currently transports a zeta Mellin pole through that additive prime-sampling operator.

The shifted-prime Möbius energy route remains an interesting parity problem, but it is no longer the shortest certified path to the RH frontier.

Campaign 46 should return to F-RH-017-v3.

CSM_RH Paper 70 — Boundary-Mode Concentration in the Shift Mean, Centering Attenuation, and Closure of the Original Hank

CSM_RH Paper 70

Boundary-Mode Concentration in the Shift Mean, Centering Attenuation, and Closure of the Original Hankel-Coercivity Guess

Abstract

1. Synthetic boundary mode

2. Prime sampling of the smooth mode

Theorem 2.1 — Critical prime-sampled constant mode

3. Shift variation asymptotic

Theorem 3.1 — Boundary-mode shift variation

4. Signed shift mean retains the critical boundary scale

Theorem 4.1 — Signed mean boundary retention

5. Centered variance attenuates the boundary mode

Theorem 5.1 — Centered smooth-mode Hankel scale

6. Effective centered exponent

Theorem 6.1 — Centered effective exponent

7. Reappearance of the Paper-61 anchor exponent

8. Correction to F-RH-021

9. Revised bridge candidate F-RH-021R

10. How F-RH-020 would become a root theorem if F-RH-021R held

11. Why zeta zeros do not automatically transport through additive prime sampling

12. External Mertens oscillation calibration

13. Relation to shifted-prime Möbius upper theorems

14. Strategic status of F-RH-020

15. Recommended return to the root problem

16. External calibration

16.1. Pintz 2026 Mertens oscillation

16.2. Lichtman shifted-prime Möbius theorem

17. State transition

18. Conclusion