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CSM_RH Paper 42 — Translated Pure-Möbius Mean Square, Harmless Diagonal_Ultra-Short Shells, and the Oscillatory Off-Diagonal Core

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CSM_RH Paper 42

Translated Pure-Möbius Mean Square, Harmless Diagonal/Ultra-Short Shells, and the Oscillatory Off-Diagonal Core

Project: CSM_RH
Paper: 42
Version: v0.1
Date: 2026-09-07
Parent state: CSM_RH v1.32 / Paper 41
Campaign: 41 — PURE_MOBIUS_CORE_HIGH_FREQUENCY_ATTACK
Status: translated-window arithmetic localization; 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

Paper 41 isolated a legal Type-II pure-Möbius core in the Heath–Brown decomposition.

Campaign 41 asks whether translated high-frequency mean squares of that core can produce the polynomial- WW saving required by the Type-II amplifier without importing pointwise fixed-power Mertens.

The campaign does not prove such a saving.

It does, however, localize the difficulty sharply:

  1. the diagonal of the translated mean square is already polynomially harmless;
  2. the ultra-short shift shell is harmless by the separation between the high-frequency center and the local Parseval window;
  3. the remaining obstruction is an oscillatory off-diagonal shifted-correlation problem for the balanced Möbius-convolution coefficients;
  4. current Chowla-type technology does not provide the required fixed XX -power for this object;
  5. no standard zero-detection theorem forces one individual dyadic short Möbius block to be large at every off-line zero ordinate.

Thus the high-frequency route is neither solved nor reduced to the pointwise Mertens lock.

No live GLM-5.3-Flash run is claimed.

1. Pure-Möbius core coefficient

Let

Mi(s)=∑m∼Uiμ(m)ms,1≤i≤L,M_i(s) = \sum_{m\sim U_i} \frac{\mu(m)}{m^s}, \qquad 1\le i\le L,

where

∏i=1LUi≍X.\prod_{i=1}^{L}U_i\asymp X.

Define

FX(s)=∏i=1LMi(s)=∑ℓ≍XcX(ℓ)ℓs.\boxed{ F_X(s) = \prod_{i=1}^{L}M_i(s) = \sum_{\ell\asymp X} \frac{c_X(\ell)}{\ell^s}. }

The coefficient is the restricted balanced Möbius convolution

cX(ℓ)=∑m1⋯mL=ℓmi∼Uiμ(m1)⋯μ(mL).\boxed{ c_X(\ell) = \sum_{\substack{ m_1\cdots m_L=\ell\\ m_i\sim U_i }} \mu(m_1)\cdots\mu(m_L). }

For fixed LL,

∣cX(ℓ)∣≤dL(ℓ).\boxed{ |c_X(\ell)| \le d_L(\ell). }

Hence

∑ℓ≍X∣cX(ℓ)∣2≪LX(log⁡X)OL(1).\boxed{ \sum_{\ell\asymp X} |c_X(\ell)|^2 \ll_L X(\log X)^{O_L(1)}. }

2. The actual translated window

In the large-major-arc application, the Heath–Brown factors are twisted by a major-arc frequency.

Let the absolute center frequency be

Q>0.Q>0.

Let the local Parseval half-width be

Y.Y.

The large- TT branch supplies

Q≳XHXε/2.\boxed{ Q \gtrsim \frac{X}{H} X^{\varepsilon/2}. }

Inside the proof of Lemma 3.5, after the reduction corresponding to equation (3.11), it is enough to consider

Y≲XHW3/10.\boxed{ Y \lesssim \frac{X}{H} W^{3/10}. }

For polynomial

W=Xw,W=X^w,

with

0<w≤ε1000,0<w\le\frac{\varepsilon}{1000},

we therefore have

YQ≪X3w/10−ε/2.\boxed{ \frac{Y}{Q} \ll X^{3w/10-\varepsilon/2}. }

The translated window is strongly separated from frequency zero.

3. Gaussian majorant

Define

Φ(u)=exp⁡(1−u22).\boxed{ \Phi(u) = \exp \left( \frac{1-u^2}{2} \right). }

Then

Φ(u)≥1\Phi(u)\ge1

for

∣u∣≤1.|u|\le1.

Its Fourier transform, with convention

Φ^(ξ)=∫RΦ(u)e−iuξ du,\widehat\Phi(\xi) = \int_{\mathbb R} \Phi(u)e^{-iu\xi} \,du,

is

Φ^(ξ)=e1/22πe−ξ2/2.\boxed{ \widehat\Phi(\xi) = e^{1/2} \sqrt{2\pi} e^{-\xi^2/2}. }

Thus the sharp-window mean square is bounded by the smooth mean square

∫Q−YQ+Y∣FX(1+it)∣2dt≤IΦ(Q,Y),\boxed{ \int_{Q-Y}^{Q+Y} |F_X(1+it)|^2dt \le \mathcal I_\Phi(Q,Y), }

where

IΦ(Q,Y)=∫RΦ(t−QY)∣FX(1+it)∣2 dt.\boxed{ \mathcal I_\Phi(Q,Y) = \int_{\mathbb R} \Phi \left( \frac{t-Q}{Y} \right) |F_X(1+it)|^2 \,dt. }

4. Exact translated-window kernel identity

Expanding the square and evaluating the Fourier integral gives:

Theorem 4.1 — Gaussian Translated Mean-Square Identity

IΦ(Q,Y)=Y∑m,ncX(m)cX(n)‾mne−iQlog⁡(m/n)Φ^(Ylog⁡(m/n)).\boxed{ \mathcal I_\Phi(Q,Y) = Y \sum_{m,n} \frac{ c_X(m)\overline{c_X(n)} }{ mn } e^{-iQ\log(m/n)} \widehat\Phi \left( Y\log(m/n) \right). }

The diagonal is

DΦ(Y)=YΦ^(0)∑n∣cX(n)∣2n2.\boxed{ \mathcal D_\Phi(Y) = Y\widehat\Phi(0) \sum_n \frac{ |c_X(n)|^2 }{ n^2 }. }

The off-diagonal is

OΦ(Q,Y)=Y∑m≠ncX(m)cX(n)‾mne−iQlog⁡(m/n)Φ^(Ylog⁡(m/n)).\boxed{ \mathcal O_\Phi(Q,Y) = Y \sum_{m\ne n} \frac{ c_X(m)\overline{c_X(n)} }{ mn } e^{-iQ\log(m/n)} \widehat\Phi \left( Y\log(m/n) \right). }

Create:

B-RH-016
TRANSLATED_PURE_CORE_MEAN_SQUARE_IDENTITY
status:
  CERTIFIED

5. Diagonal scale

Since

n≍Xn\asymp X

on the support and

∑∣cX(n)∣2≪X(log⁡X)OL(1),\sum|c_X(n)|^2 \ll X(\log X)^{O_L(1)},

we obtain

Theorem 5.1 — Pure-Core Diagonal Bound

DΦ(Y)≪YX(log⁡X)OL(1).\boxed{ \mathcal D_\Phi(Y) \ll \frac{Y}{X} (\log X)^{O_L(1)}. }

Using

Y≲XHW3/10,Y \lesssim \frac{X}{H} W^{3/10},

this becomes

DΦ(Y)≪W3/10HXo(1).\boxed{ \mathcal D_\Phi(Y) \ll \frac{ W^{3/10} }{ H } X^{o(1)}. }

For

H≥X1/3+εH\ge X^{1/3+\varepsilon}

and

W=Xw,w≤ε1000,W=X^w, \qquad w\le\frac{\varepsilon}{1000},

the diagonal is far below the required Type-II scale

W−3/10.W^{-3/10}.

Create:

O-RH-102
PURE_MOBIUS_CORE_DIAGONAL_HARMLESS
status:
  CERTIFIED

The diagonal is not the fixed-power obstruction.

6. Shift form of the off-diagonal

Write

n=m+h.n=m+h.

Then

OΦ(Q,Y)=2Yℜ∑h≥1∑mcX(m)cX(m+h)‾m(m+h)eiQlog⁡(1+h/m)Φ^(Ylog⁡(1+h/m)),\boxed{ \mathcal O_\Phi(Q,Y) = 2Y \Re \sum_{h\ge1} \sum_m \frac{ c_X(m)\overline{c_X(m+h)} }{ m(m+h) } e^{iQ\log(1+h/m)} \widehat\Phi \left( Y\log(1+h/m) \right), }

with the support restrictions implicit.

Thus the hard object is an additively shifted correlation of the restricted Möbius-convolution coefficient, twisted by a multiplicative high-frequency phase.

7. Gaussian localization in shift

For

m≍Xm\asymp X

and

h=o(X),h=o(X), log⁡(1+h/m)≍h/X.\log(1+h/m) \asymp h/X.

Because

Φ^(ξ)=OA((1+∣ξ∣)−A)\widehat\Phi(\xi) = O_A((1+|\xi|)^{-A})

for every fixed A>0A>0,

the contribution from

h≥Xo(1)XYh \ge X^{o(1)} \frac{X}{Y}

is negligible after taking the Xo(1)X^{o(1)} divisor-bound loss into account.

Thus the smooth-window off-diagonal is effectively localized to

1≤h≲Xo(1)XY.\boxed{ 1\le h \lesssim X^{o(1)} \frac{X}{Y}. }

At the Type-II scale,

XY≳HW3/10.\frac{X}{Y} \gtrsim \frac{H}{W^{3/10}}.

8. Ultra-short shifts are harmless without Möbius cancellation

Consider

1≤h≤XQ.1\le h\le\frac{X}{Q}.

Using only

∣cX(n)∣≤dL(n),|c_X(n)| \le d_L(n),

Cauchy–Schwarz and divisor moments give, uniformly in hh,

∑m∣cX(m)cX(m+h)∣m(m+h)≪X−1+o(1).\boxed{ \sum_m \frac{ |c_X(m)c_X(m+h)| }{ m(m+h) } \ll X^{-1+o(1)}. }

Also

∣Φ^∣≪1.|\widehat\Phi|\ll1.

Therefore the entire ultra-short shell contributes

Theorem 8.1 — High-Frequency Ultra-Short Shell Bound

∣Oultra∣≪YQXo(1).\boxed{ |\mathcal O_{\mathrm{ultra}}| \ll \frac{Y}{Q} X^{o(1)}. }

By Section 2,

∣Oultra∣≪X3w/10−ε/2+o(1).\boxed{ |\mathcal O_{\mathrm{ultra}}| \ll X^{3w/10-\varepsilon/2+o(1)}. }

Since

w≤ε1000,w\le\frac{\varepsilon}{1000},

this is much smaller than

W−3/10=X−3w/10.W^{-3/10} = X^{-3w/10}.

Create:

O-RH-103
HIGH_FREQUENCY_ULTRASHORT_SHIFT_SHELL_HARMLESS
status:
  CERTIFIED

This is a genuine benefit of the translated high-frequency geometry.

9. Remaining hard shell

After Sections 7–8, the unresolved shifts satisfy

XQ<h≲Xo(1)XY.\boxed{ \frac{X}{Q} < h \lesssim X^{o(1)} \frac{X}{Y}. }

For these shifts, the phase

ϕQ,h(m)=Qlog⁡(1+h/m)\boxed{ \phi_{Q,h}(m) = Q\log(1+h/m) }

has order-one or larger total variation across a dyadic mm -block.

Indeed,

ϕQ,h′(m)=−Qhm(m+h),\phi'_{Q,h}(m) = -\frac{ Qh }{ m(m+h) },

so over a block of length ≍X\asymp X the total phase variation is

≍QhX.\asymp \frac{Qh}{X}.

At the lower edge

h=X/Q,h=X/Q,

this is already order one.

Thus the unresolved shell is genuinely oscillatory.

10. Oscillatory shifted-correlation core

Define

Rosc(Q,Y)=Y∑XQ<h≲Xo(1)X/Y∑mcX(m)cX(m+h)‾m(m+h)eiQlog⁡(1+h/m)Φ^(Ylog⁡(1+h/m)).\boxed{ \mathfrak R_{\mathrm{osc}}(Q,Y) = Y \sum_{\frac{X}{Q}<h\lesssim X^{o(1)}X/Y} \sum_m \frac{ c_X(m)\overline{c_X(m+h)} }{ m(m+h) } e^{iQ\log(1+h/m)} \widehat\Phi \left( Y\log(1+h/m) \right). }

Modulo the harmless diagonal, ultra-short shell, and Gaussian tail:

IΦ(Q,Y)=2ℜRosc(Q,Y)+harmless terms.\boxed{ \mathcal I_\Phi(Q,Y) = 2\Re \mathfrak R_{\mathrm{osc}}(Q,Y) + \text{harmless terms}. }

Create:

O-RH-104
PURE_CORE_HIGH_FREQUENCY_HARDNESS_IS_OSCILLATORY_SHIFTED_CORRELATION
status:
  CERTIFIED AS EXACT LOCALIZATION

No new canonical frontier is created.

11. Polynomial- WW admission shape

To invoke B-RH-014 for the pure core, it is sufficient to prove

∣Rosc(Q,Y)∣≪W−3/10Xo(1)\boxed{ | \mathfrak R_{\mathrm{osc}}(Q,Y) | \ll W^{-3/10} X^{o(1)} }

uniformly over the Type-II parameter range.

For

W=Xw,W=X^w,

this is a genuine fixed-power correlation theorem.

The phase oscillation is available.

The arithmetic cancellation is not presently known.

12. Relation to ordinary Möbius correlations

The coefficient cXc_X is not the Möbius function itself.

It is a balanced restricted convolution of several short Möbius factors.

Thus even a theorem for

∑nμ(n)μ(n+h)\sum_n \mu(n)\mu(n+h)

does not transfer automatically.

Nevertheless the standard two-point Chowla problem provides a lower-complexity calibration.

Ordinary unweighted two-point Chowla remains open.

Known averaged results produce qualitative or logarithmic savings over shifts, not a fixed XX -power uniformly of the form required in Section 11.

Recent 2026 progress on full-range logarithmically weighted Liouville correlations also remains power-logarithmic rather than polynomial in XX.

Therefore current correlation technology does not supply the required core estimate.

Create:

O-RH-105
CURRENT_CHOWLA_PRECISION_INSUFFICIENT_FOR_PURE_CORE_FIXED_POWER
status:
  CERTIFIED AS CURRENT-LITERATURE CALIBRATION

13. Average-Chowla comparison

A representative averaged-Chowla estimate has the shape

∑h≤H∣∑n≤Xμ(n)μ(n+h)∣=o(XH)\sum_{h\le H} \left| \sum_{n\le X} \mu(n)\mu(n+h) \right| = o(XH)

or obtains powers of log⁡X\log X in quantitatively strengthened settings.

Such an estimate does not imply

X−δX^{-\delta}

suppression for the weighted oscillatory aggregate in Section 10.

The gap is quantitative, not merely notational.

14. Zero-ordinate resonance test

One might hope that a zero

ρ=β+iγ\rho=\beta+i\gamma

with

β>12\beta>\frac12

would force one short dyadic Möbius polynomial

∑m∼Uμ(m)m1+iγ\sum_{m\sim U} \frac{\mu(m)}{m^{1+i\gamma}}

to be large.

No such standard deterministic statement is available.

Classical zero-detection methods instead construct a composite polynomial by multiplying a short Möbius truncation by a smoothed approximation to ζ(s)\zeta(s).

The resulting zero-detecting polynomial has different coefficients and a wider length range.

Thus:

single short Mobius block:
  not a certified zero detector

composite zero-detecting polynomial:
  classical and effective

Create:

O-RH-106
NO_CERTIFIED_SINGLE_DYADIC_MOBIUS_ZERO_DETECTOR
status:
  CERTIFIED AS METHOD-SCOPE AUDIT

This does not prove that no such theorem can exist.

15. High frequency is not automatically zero-frequency Mertens

The pure-core hard shell is evaluated in a translated frequency window.

The pointwise Mertens lock of Paper 39 arose by specializing a theorem at

t=0.t=0.

The present fixed-power admission target concerns a weighted mean square centered at

Q≫Y.Q \gg Y.

Campaign 41 does not prove a deterministic implication

translated pure-core mean square⟹fixed-power Mertens.\text{translated pure-core mean square} \Longrightarrow \text{fixed-power Mertens}.

Therefore the high-frequency route remains logically distinct from the already closed pointwise route.

16. But frequency translation alone is not arithmetic cancellation

Replacing

tt

by

t−Qt-Q

is equivalent to twisting the coefficients by

niQ.n^{iQ}.

The twist

μ(n)niQ\mu(n)n^{iQ}

remains a bounded multiplicative function.

Generic mean-value theorems depend mainly on coefficient magnitudes and therefore do not gain a fixed power merely from this translation.

The gain in Section 8 comes from geometric shell size.

The hard shell still needs arithmetic cancellation.

17. Campaign 41 track audit

PM1 — translated-window moment formula

status:
  EXACT

result:
  B-RH-016

PM2 — zero-ordinate resonance test

status:
  NO SINGLE-BLOCK ZERO-DETECTOR THEOREM FOUND

standard zero detection:
  uses composite polynomial

PM3 — multi-factor simultaneous resonance

status:
  REDUCED TO OSCILLATORY SHIFTED CORRELATION OF c_X

fixed-power estimate:
  OPEN

PM4 — high-frequency versus zero-frequency separation

status:
  LOGICALLY DISTINCT

ultra-short shifts:
  polynomially harmless from Q >> Y

remaining shell:
  arithmetic

PM5 — polynomial-W admission

status:
  NOT OBTAINED

18. Campaign 41 verdict

No polynomial- WW theorem is proved.

The important localization is:

pure-core diagonal:
  harmless

Gaussian far-shift tail:
  harmless

ultra-short h <= X/Q:
  harmless

remaining X/Q < h <= X^(o(1)) X/Y:
  hard oscillatory shifted-correlation shell

The hard shell is a genuinely arithmetic object and is not presently controlled at fixed XX -power precision.

19. New certified package

Create:

B-RH-016
TRANSLATED_PURE_CORE_MEAN_SQUARE_IDENTITY
CERTIFIED

O-RH-102
PURE_MOBIUS_CORE_DIAGONAL_HARMLESS
CERTIFIED

O-RH-103
HIGH_FREQUENCY_ULTRASHORT_SHIFT_SHELL_HARMLESS
CERTIFIED

O-RH-104
PURE_CORE_HIGH_FREQUENCY_HARDNESS_IS_OSCILLATORY_SHIFTED_CORRELATION
CERTIFIED

O-RH-105
CURRENT_CHOWLA_PRECISION_INSUFFICIENT_FOR_PURE_CORE_FIXED_POWER
CERTIFIED AS CURRENT-LITERATURE CALIBRATION

O-RH-106
NO_CERTIFIED_SINGLE_DYADIC_MOBIUS_ZERO_DETECTOR
CERTIFIED AS METHOD-SCOPE AUDIT

No new root frontier is created.

20. Canonical status

F-RH-010
PESC
OPEN

F-RH-016
MLEPG
OPEN

B-RH-014
POLYNOMIAL_W_TYPEII_FIXED_POWER_AMPLIFIER
CERTIFIED

pure-Mobius high-frequency hard shell:
  OPEN

21. Campaign 42

The next campaign is:

CSM_RH Campaign 42
OSCILLATORY_MOBIUS_CONVOLUTION_SHIFT_ATTACK

The target is exactly the hard shell from Section 10.

No new observable is allowed.

22. Campaign 42 tracks

OM1 — balanced Möbius-convolution correlation algebra

Expand

cX(m)cX(m+h)c_X(m)c_X(m+h)

back into the underlying short Möbius variables.

Determine whether product equality plus additive shift forces a usable bilinear or multilinear Diophantine structure.

OM2 — phase geometry

Exploit

Qlog⁡(1+h/m)Q\log(1+h/m)

on the shell

X/Q<h≲Xo(1)X/Y.X/Q<h\lesssim X^{o(1)}X/Y.

Quantify curvature and derivative scales.

OM3 — averaged shift cancellation

Test current averaged Chowla technology after the convolution expansion.

A valid result must produce a fixed XX -power, not merely o(1)o(1) or log savings.

OM4 — Ramaré extraction

Audit whether extracting one small prime factor from one Möbius variable breaks the simultaneous resonance in the same spirit as the all-short-interval Möbius theorem.

The exponent ledger must survive the extra variable.

OM5 — cross-j coupling

If the single pure-core component remains hard, test whether keeping the alternating Heath–Brown jj -sum before absolute values cancels the oscillatory core.

No cancellation between separate big- OO estimates is allowed.

23. Campaign 42 rejection filters

Reject a candidate if:

R1. It replaces the oscillatory shell by ordinary Chowla without tracking the phase.

R2. It obtains only logarithmic or qualitative cancellation.

R3. It assumes pointwise fixed-power Mertens.

R4. It assumes a fixed zero-free strip.

R5. It uses generic bounded coefficients.

R6. It discards the alternating cross- jj structure and then claims universal impossibility.

24. External calibration

The current structural calibration is:

  1. the 2026 higher-uniformity paper uses Heath–Brown decomposition and componentwise Type-II estimates;
  2. the actual Type-II proof requires the product mean square in Lemma 3.5;
  3. standard zero-detection does not use a single short dyadic Möbius block as a canonical zero detector;
  4. averaged Chowla results give qualitative/logarithmic cancellation over shifts;
  5. as of September 2026, recent logarithmically weighted Liouville correlation progress remains in a power-of-logarithm precision class.

These facts leave the oscillatory pure-core shell open without falsely identifying it with a solved or fixed-strip-equivalent theorem.

25. State transition

CSM_RH v1.32
  ->
CSM_RH v1.33

with:

Campaign 41
  CLOSED_AS_TRANSLATED_WINDOW_OFFDIAGONAL_LOCALIZATION

B-RH-016
  TRANSLATED_PURE_CORE_MEAN_SQUARE_IDENTITY
  CREATED / CERTIFIED

O-RH-102
  PURE_MOBIUS_CORE_DIAGONAL_HARMLESS
  CREATED / CERTIFIED

O-RH-103
  HIGH_FREQUENCY_ULTRASHORT_SHIFT_SHELL_HARMLESS
  CREATED / CERTIFIED

O-RH-104
  PURE_CORE_HIGH_FREQUENCY_HARDNESS_IS_OSCILLATORY_SHIFTED_CORRELATION
  CREATED / CERTIFIED

O-RH-105
  CURRENT_CHOWLA_PRECISION_INSUFFICIENT_FOR_PURE_CORE_FIXED_POWER
  CREATED / CERTIFIED

O-RH-106
  NO_CERTIFIED_SINGLE_DYADIC_MOBIUS_ZERO_DETECTOR
  CREATED / CERTIFIED

F-RH-010
  PESC
  REMAINS OPEN

F-RH-016
  MLEPG
  REMAINS OPEN

Campaign 42
  OSCILLATORY_MOBIUS_CONVOLUTION_SHIFT_ATTACK
  READY

26. Final status

RH = OPEN

PESC = OPEN

MLEPG = OPEN

PURE-MOBIUS DIAGONAL = HARMLESS

ULTRA-SHORT HIGH-FREQUENCY SHIFT SHELL = HARMLESS

SINGLE SHORT MOBIUS ZERO-DETECTOR LOCK = NOT ESTABLISHED

OSCILLATORY MOBIUS-CONVOLUTION OFF-DIAGONAL = OPEN

CURRENT CHOWLA PRECISION = INSUFFICIENT FOR FIXED X POWER

POLYNOMIAL-W TYPE-II ADMISSION = NOT OBTAINED

NEXT CAMPAIGN = 42

The decisive reduction is:

pure-Mobius polynomial-W problem⇝Rosc(Q,Y)\boxed{ \text{pure-Mobius polynomial-W problem} \rightsquigarrow \mathfrak R_{\mathrm{osc}}(Q,Y) }

with

XQ<h≲Xo(1)XY.\boxed{ \frac{X}{Q} < h \lesssim X^{o(1)} \frac{X}{Y}. }

Everything outside this oscillatory shifted-correlation shell is already below the required fixed-power scale.