CSM_RH Paper 41

Heath–Brown Carrier Audit and the Pure-Möbius Type-II Core

Project: CSM_RH
Paper: 41
Version: v0.1
Date: 2026-09-07
Parent state: CSM_RH v1.31 / Paper 40
Campaign: 40 — HEATH_BROWN_ARITHMETIC_RESONANCE_EXCLUSION
Status: arithmetic-factor carrier audit / pure-Möbius-core 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 40 proved that generic divisor-bounded large-value technology cannot supply the polynomial- $W$ Type-II mean-square saving.

Campaign 40 asks whether the actual Heath–Brown factors of the von Mangoldt function contain enough arithmetic structure to exclude the generic resonance.

The campaign reaches a structural localization rather than a fixed-power theorem.

Two opposite legal Type-II families occur inside the same Heath–Brown identity:

carrier-free smooth Type-II components;
pure-Möbius-core Type-II components with no polynomial-scale smooth complementary factor.

Thus there is no universal proof rule of the form

every Lambda Type-II component
  ->
one nontrivial Mobius carrier
plus one smooth complementary anti-resonant factor.

The remaining hard arithmetic family is the high-frequency simultaneous resonance of several short Möbius Dirichlet polynomials.

No live GLM-5.3-Flash run is claimed.

1. Heath–Brown identity

Fix an integer

L\ge1

and set

z=(2X)^{1/L}.

The Heath–Brown identity used in the higher-uniformity papers has the form

\boxed{ \Lambda(n) = \sum_{1\le j\le L} (-1)^{j-1} \binom Lj \sum_{m_1,\ldots,m_j\le z} \mu(m_1)\cdots\mu(m_j) \sum_{\substack{ m_1\cdots m_j n_1\cdots n_j=n }} \log n_1. }

After dyadic subdivision, every component is a convolution

\boxed{ f = a^{(1)}*\cdots*a^{(\ell)}, \qquad \ell\le2L, }

where each factor is a dyadic restriction of one of

1, \qquad \log, \qquad \mu.

Every Möbius factor has dyadic length

\boxed{ N_{\mu} \ll X^{1/L}. }

In the 2026 decomposition one takes

L=\left\lceil\frac{10}{\varepsilon}\right\rceil,

\boxed{ N_\mu \le X^{\varepsilon/10} }

at exponent resolution.

2. Type-II grouping in the 2026 proof

Order the dyadic lengths by

N_1\ge N_2\ge\cdots\ge N_\ell.

After excluding the Type-I and Type- $I_2$ cases, the proof has

N_1N_2 \le X^{1-\varepsilon/2}.

It then chooses an index $j_\ast\ge3$ such that

\boxed{ X^{\varepsilon/10} \le N_3\cdots N_{j_\ast} \ll X^{1/3}, }

and defines

\boxed{ \alpha = a^{(3)}*\cdots*a^{(j_\ast)}, }

with all remaining factors placed in the complementary Type-II coefficient $\beta$ .

The grouping rule is based on support lengths.

It does not preserve an arithmetic label saying that $\alpha$ or $\beta$ must contain a nontrivial Möbius block.

3. Carrier-free smooth Type-II component

Take the $j=3$ term in Heath–Brown's identity.

Restrict all Möbius variables to

m_1=m_2=m_3=1.

Then

\mu(m_1)\mu(m_2)\mu(m_3)=1.

Choose the three smooth variables on dyadic scales

n_1,n_2,n_3 \asymp X^{1/3},

with $n_1$ carrying the logarithm.

The resulting nonzero component is, up to dyadic endpoint restrictions,

\boxed{ (\log 1_{X^{1/3}}) * 1_{X^{1/3}} * 1_{X^{1/3}}. }

All Möbius factors are convolution identities supported at $1$ .

For small fixed $\varepsilon$ :

N_1 \asymp N_2 \asymp N_3 \asymp X^{1/3},

N_1 < X^{2/3-\varepsilon/2},

and

N_1N_2 \asymp X^{2/3} < X^{1-\varepsilon/2}.

Thus this lies in the Type-II branch of the 2026 grouping.

Create:

O-RH-099
HEATH_BROWN_TYPEII_NOT_UNIFORMLY_MOBIUS_CARRYING
status:
  CERTIFIED

A proof strategy which requires every Type-II component to retain a nontrivial Möbius carrier is therefore invalid.

4. Carrier-free does not mean hard by itself

The component in Section 3 is divisor-like.

Its difficulty is qualitatively different from the Möbius-bearing prime residual.

The 2026 major-arc theory obtains genuine polynomial $W$ for divisor-function Type-II problems.

Thus carrier-free smooth components are not evidence that the fixed-power route fails.

They show only that the arithmetic proof must branch according to the actual factor content.

No universal Möbius-carrier argument can cover every component.

5. Pure-Möbius Type-II core

The opposite extreme also occurs.

Take the $j=L$ term in Heath–Brown's identity.

Choose:

n_1=2,

so the logarithmic factor contributes the nonzero constant

\log2,

and choose

n_2=\cdots=n_L=1.

The product constraint is then carried almost entirely by the Möbius variables:

m_1\cdots m_L \asymp X/2.

Choose all $m_i$ on comparable dyadic scales

\boxed{ m_i \asymp (X/2)^{1/L}. }

These scales are legal because

(X/2)^{1/L} \le (2X)^{1/L}=z.

After removing the constant smooth factors, the component is essentially

\boxed{ (\mu_{M_1}*\cdots*\mu_{M_L}) \times\log2. }

6. It is genuinely Type II

Set

\lambda=\frac1L.

The largest nontrivial factors have scale

X^\lambda.

Since

L=\left\lceil\frac{10}{\varepsilon}\right\rceil,

we have

\lambda \le \frac{\varepsilon}{10}.

Therefore:

X^\lambda \ll X^{2/3-\varepsilon/2},

and

X^{2\lambda} \ll X^{1-\varepsilon/2}.

Hence the component is not forced into the Type-I or Type- $I_2$ branches.

To form the Type-II factor $\alpha$ , the length-grouping algorithm combines enough of the short Möbius blocks that their product first reaches

X^{\varepsilon/10}.

The remaining Type-II factor $\beta$ still contains the other short Möbius blocks.

For $L$ sufficiently large, both sides contain nontrivial Möbius blocks.

Create:

O-RH-100
PURE_MOBIUS_CORE_TYPEII_COMPONENT_EXISTS
status:
  CERTIFIED

7. Failure of the complementary-smooth-factor escape

Paper 40 left open the possibility that a Möbius-bearing factor could be large only when a complementary smooth factor is small.

The pure-Möbius core of Sections 5–6 shows that this cannot be the universal mechanism.

There are legal Type-II components for which the polynomial-scale factors on both sides of the Type-II split are built from Möbius blocks, while the only smooth factors are constants.

Therefore no proof may assume:

Mobius resonance
  ->
independent smooth factor supplies polynomial anti-resonance.

Create:

O-RH-101
COMPLEMENTARY_SMOOTH_ANTI_RESONANCE_NOT_UNIVERSAL
status:
  CERTIFIED

8. Dirichlet-polynomial form of the core

For dyadic Möbius blocks define

\boxed{ M_i(s) = \sum_{m\sim M_i} \frac{\mu(m)}{m^s}. }

The pure core has Dirichlet polynomial, up to a fixed nonzero scalar and harmless dyadic bookkeeping,

\boxed{ F_{\mathrm{core}}(s) = \prod_{i=1}^{L} M_i(s). }

A Type-II grouping partitions the index set into two nonempty groups:

A_{\mathrm{core}}(s) = \prod_{i\in I} M_i(s),

B_{\mathrm{core}}(s) = \prod_{i\notin I} M_i(s).

Thus simultaneous Type-II resonance becomes a high moment / multi-factor resonance problem for short Möbius Dirichlet polynomials.

9. Translated high-frequency window

In the large- $T$ major-arc application, the Type-II coefficients are twisted by

n^{iT}.

Consequently a Möbius block appears as

\sum_{m\sim M} \frac{\mu(m)}{m^{1+i(t-T)}}.

The integration variable $t$ is small relative to the major-arc center $T$ in the relevant parameter regime.

Thus the unresolved object is not the pointwise value at frequency zero.

It is a translated high-frequency window centered near

-T.

This distinction matters because the fixed-power Mertens lock of Paper 39 used the $t=0$ specialization of a pointwise theorem.

Campaign 40 does not prove that high-frequency averaged control is equivalent to fixed-power Mertens.

10. Why generic mean values still do not solve the core

For one dyadic Möbius block of length $M$ ,

\sum_{m\sim M} \frac{|\mu(m)|^2}{m^2} \asymp \frac1M

up to arithmetic constants.

The standard Dirichlet-polynomial mean-value theorem therefore gives the same diagonal scale as for generic bounded coefficients.

It does not supply an additional fixed-power suppression purely from the sign pattern of $\mu$ .

For the product core, generic mean-value technology again reaches an orthogonality-scale bound rather than the polynomially vanishing $W^{-c}$ target.

Thus arithmetic information beyond the diagonal is required.

11. No universal carrier-preserving decomposition in the current proof

The current 2026 proof applies the triangle inequality after decomposing $\Lambda$ and then treats each Type-II component separately.

Under this componentwise architecture:

smooth-only Type-II components exist;
mixed Möbius/smooth Type-II components exist;
pure-Möbius-core Type-II components exist.

Hence the proof cannot be reduced to one carrier pattern.

A future proof could reorganize or couple different Heath–Brown $j$ -levels before applying absolute values.

Campaign 40 does not provide such a coupled identity.

12. Cross- $j$ cancellation remains unaudited

The pure core is one dyadic component of one Heath–Brown $j$ -level.

The exact Heath–Brown identity contains alternating coefficients

(-1)^{j-1}\binom Lj.

It is logically possible that a proof preserving cancellation across several $j$ -levels could suppress the core before Type-II estimation.

The published higher-uniformity argument uses triangle inequalities and does not exploit such cancellation at fixed-power scale.

Therefore:

componentwise pure-Mobius obstruction:
  certified

global cross-j impossibility:
  not claimed

13. Campaign 40 track audit

HB1 — Möbius carrier audit

status:
  NO UNIVERSAL CARRIER PATTERN

smooth-only components:
  exist

pure-Mobius components:
  exist

HB2 — translated-frequency Möbius large values

status:
  LOCALIZED

frequency:
  high / translated

pointwise t=0 Mertens lock:
  not directly applicable

HB3 — simultaneous resonance exclusion

status:
  NOT PROVED

hard core:
  product of short Mobius Dirichlet polynomials

HB4 — short Möbius factor versus complementary factor

status:
  COMPLEMENTARY SMOOTH FACTOR NOT UNIVERSAL

pure-Mobius core:
  defeats this universal strategy

HB5 — integrated polynomial-W admission

status:
  NOT OBTAINED

14. Campaign 40 verdict

No polynomial- $W$ Type-II theorem is proved.

The positive result is a sharper localization than Paper 40:

\boxed{ \text{generic resonance problem} \rightsquigarrow \text{actual high-frequency multi-short-Möbius resonance core}. }

The divisor-like smooth components and mixed components must be treated separately.

The universal remaining arithmetic obstruction is not a generic divisor-bounded polynomial.

It is the possibility of spectral mass in products of genuine short Möbius Dirichlet polynomials.

15. New certified package

Create:

O-RH-099
HEATH_BROWN_TYPEII_NOT_UNIFORMLY_MOBIUS_CARRYING
CERTIFIED

O-RH-100
PURE_MOBIUS_CORE_TYPEII_COMPONENT_EXISTS
CERTIFIED

O-RH-101
COMPLEMENTARY_SMOOTH_ANTI_RESONANCE_NOT_UNIVERSAL
CERTIFIED

No new canonical frontier is created.

16. Canonical status

F-RH-010
PESC
OPEN / ROOT TARGET

F-RH-016
MLEPG
OPEN

B-RH-014
POLYNOMIAL_W_TYPEII_FIXED_POWER_AMPLIFIER
CERTIFIED COMPONENT BRIDGE

generic Type-II resonance:
  closed

actual pure-Mobius high-frequency core:
  open

17. Campaign 41

The next campaign is:

CSM_RH Campaign 41
PURE_MOBIUS_CORE_HIGH_FREQUENCY_ATTACK

This is a genuinely arithmetic campaign.

No new representation is allowed.

18. Campaign 41 tracks

PM1 — translated-window moment formula

For

M_U(s) = \sum_{m\sim U} \frac{\mu(m)}{m^s},

compute the exact mean-square and higher-moment scale on a translated interval

t\in[T-Y,T+Y].

Separate diagonal, near-diagonal, and genuinely arithmetic off-diagonal terms.

PM2 — zero-ordinate resonance test

Determine whether a zero

\rho=\beta+i\gamma

with $\beta>1/2$ forces a quantitatively large translated short-Möbius polynomial near

t=\gamma.

No coefficient-isolation claim may be made without proof.

PM3 — multi-factor simultaneous resonance

For the pure core

\prod_{i=1}^{L}M_{U_i}(1+it),

seek a polynomial bound for the spectral measure of the simultaneous-large set.

The bound must improve the generic orthogonality floor.

PM4 — high-frequency versus zero-frequency separation

Test whether excluding a neighborhood of $t=0$ genuinely weakens the arithmetic statement below fixed-power Mertens strength.

This is the central strength question.

PM5 — polynomial- $W$ admission

The only accepted output is an integrated bound strong enough to invoke B-RH-014 for the pure-Möbius core and all mixed components.

19. Campaign 41 rejection filters

Reject a candidate if:

R1. It replaces the translated window by a theorem uniform down to $t=0$ and thereby imports fixed-power Mertens.

R2. It uses generic divisor-bounded large values only.

R3. It controls only one Möbius block while repeated blocks can resonate together.

R4. It obtains only $X^{-o(1)}$ suppression.

R5. It assumes zero repulsion or a fixed zero-free strip.

R6. It ignores cross- $j$ cancellation as a logically possible alternative and claims universal impossibility.

20. External calibration

The structural facts used in this audit are explicit in the current higher-uniformity papers.

The Heath–Brown identity for $\Lambda$ contains $j$ Möbius variables and $j$ smooth variables at level $j$ .
After dyadic subdivision, every Möbius factor has length at most $X^{1/L}$ .
The 2026 Type-II grouping in Lemma 4.4 is selected by support lengths and forms
$\alpha=a^{(3)}*\cdots*a^{(j_\ast)}.$
No arithmetic carrier condition is imposed.
The proof of the major-arc theorem then applies the Type-II estimate componentwise after triangle inequalities.

These facts permit both the smooth-only and pure-Möbius Type-II examples constructed above.

21. State transition

CSM_RH v1.31
  ->
CSM_RH v1.32

with:

Campaign 40
  CLOSED_AS_HEATH_BROWN_CARRIER_AND_PURE_MOBIUS_CORE_AUDIT

O-RH-099
  HEATH_BROWN_TYPEII_NOT_UNIFORMLY_MOBIUS_CARRYING
  CREATED / CERTIFIED

O-RH-100
  PURE_MOBIUS_CORE_TYPEII_COMPONENT_EXISTS
  CREATED / CERTIFIED

O-RH-101
  COMPLEMENTARY_SMOOTH_ANTI_RESONANCE_NOT_UNIVERSAL
  CREATED / CERTIFIED

F-RH-010
  PESC
  REMAINS OPEN

F-RH-016
  MLEPG
  REMAINS OPEN

Campaign 41
  PURE_MOBIUS_CORE_HIGH_FREQUENCY_ATTACK
  READY

22. Final status

RH = OPEN

PESC = OPEN

MLEPG = OPEN

GENERIC TYPE-II LARGE-VALUE ESCAPE = CLOSED

UNIVERSAL MOBIUS-CARRIER PATTERN = FALSE

UNIVERSAL COMPLEMENTARY SMOOTH ANTI-RESONANCE = FALSE

PURE-MOBIUS TYPE-II CORE = EXISTS

HIGH-FREQUENCY MULTI-SHORT-MOBIUS RESONANCE = OPEN

POLYNOMIAL-W TYPE-II ADMISSION = NOT OBTAINED

NEXT CAMPAIGN = 41

The critical structural countercomponent is:

\boxed{ n_1=2,\quad n_2=\cdots=n_L=1,\quad m_i\asymp(X/2)^{1/L}. }

It leaves a legal Type-II component whose nontrivial factors are almost entirely Möbius blocks.

The next fixed-power question is therefore genuinely about the high-frequency spectral behavior of short Möbius Dirichlet polynomials.

CSM_RH Paper 41 — Heath–Brown Carrier Audit and the Pure-Möbius Type-II Core

CSM_RH Paper 41

Heath–Brown Carrier Audit and the Pure-Möbius Type-II Core

0. Trust boundary

1. Heath–Brown identity

2. Type-II grouping in the 2026 proof

3. Carrier-free smooth Type-II component

4. Carrier-free does not mean hard by itself

5. Pure-Möbius Type-II core

6. It is genuinely Type II

7. Failure of the complementary-smooth-factor escape

8. Dirichlet-polynomial form of the core

9. Translated high-frequency window

10. Why generic mean values still do not solve the core

11. No universal carrier-preserving decomposition in the current proof

12. Cross- jjj cancellation remains unaudited

13. Campaign 40 track audit

HB1 — Möbius carrier audit

HB2 — translated-frequency Möbius large values

HB3 — simultaneous resonance exclusion

HB4 — short Möbius factor versus complementary factor

HB5 — integrated polynomial-W admission

14. Campaign 40 verdict

15. New certified package

16. Canonical status

17. Campaign 41

18. Campaign 41 tracks

PM1 — translated-window moment formula

PM2 — zero-ordinate resonance test

PM3 — multi-factor simultaneous resonance

PM4 — high-frequency versus zero-frequency separation

PM5 — polynomial- WWW admission

19. Campaign 41 rejection filters

R1. It replaces the translated window by a theorem uniform down to t=0t=0t=0 and thereby imports fixed-power Mertens.

R2. It uses generic divisor-bounded large values only.

R3. It controls only one Möbius block while repeated blocks can resonate together.

R4. It obtains only X−o(1)X^{-o(1)}X−o(1) suppression.

R5. It assumes zero repulsion or a fixed zero-free strip.

R6. It ignores cross- jjj cancellation as a logically possible alternative and claims universal impossibility.

20. External calibration

21. State transition

22. Final status

12. Cross- $j$ cancellation remains unaudited

PM5 — polynomial- $W$ admission

R1. It replaces the translated window by a theorem uniform down to $t=0$ and thereby imports fixed-power Mertens.

R4. It obtains only $X^{-o(1)}$ suppression.

R6. It ignores cross- $j$ cancellation as a logically possible alternative and claims universal impossibility.