CSM_RH Paper 39

Growing Accuracy, Polynomial $W$ , and the Möbius Pointwise Fixed-Strip Lock

Project: CSM_RH
Paper: 39
Version: v0.1
Date: 2026-09-07
Parent state: CSM_RH v1.29 / Paper 38
Campaign: 38 — GROWING_ACCURACY_LAMBDA_RESIDUAL_ATTACK
Status: growing-accuracy proof-dependence 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

Campaign 38 asks whether the 2026 higher-uniformity theorem

\Lambda-\Lambda^\sharp : \qquad H\log^{-A}X

for every fixed $A>0$ can be uniformized by allowing

A=A(X)

to grow strongly enough that the logarithmic saving becomes a fixed power.

The answer for the published pointwise-major-arc architecture is negative.

The surprising feature is that the Type-II amplifier itself would convert a polynomial parameter $W$ into a genuine fixed power.

The obstruction occurs earlier:

the Dirichlet-polynomial input used to authorize that $W$ is proved only for fixed logarithmic accuracy, and a growing-accuracy version of its Möbius component already implies a fixed-power Mertens bound and hence a fixed zero-free strip.

Thus the current proof cannot be bootstrapped from arbitrary fixed log powers to a fixed $X$ -power without upgrading one of its foundational arithmetic inputs to target-level strength.

No live GLM-5.3-Flash run is claimed.

1. Current higher-uniformity theorem

For

X^{1/3+\varepsilon} \le H \le X^{1-\varepsilon},

the 2026 theorem gives, for every fixed

A>0,

\boxed{ \left| \sum_{x<n\le x+H} ( \Lambda(n)-\Lambda^\sharp(n) ) F(g(n)\Gamma) \right|^\ast \le H\log^{-A}X }

outside an exceptional set of measure

\boxed{ O_A \left( \delta^{-O(1)} X\log^{-A}X \right). }

For the constant nilsequence needed by the present campaign, the nilmanifold-complexity issue can be discarded.

The remaining dependence on $A$ is still essential.

2. Fixed power requires growing accuracy

Suppose we want

\boxed{ \log^{-A(X)}X = X^{-\eta+o(1)} }

for one fixed

\eta>0.

Taking logarithms gives

A(X)\log\log X = \eta\log X+o(\log X).

Therefore:

Theorem 2.1 — Growing-Accuracy Scale

\boxed{ A(X) = \eta \frac{ \log X }{ \log\log X } + o \left( \frac{\log X}{\log\log X} \right). }

Create:

O-RH-091
FIXED_POWER_ACCURACY_REQUIRES_GROWING_A
status:
  CERTIFIED

3. The major-arc $W$ parameter

In the proof of Theorem 3.1(i), after the type-I/type-II decomposition, the prime and Möbius type-II terms are treated using Lemma 3.5 with

\boxed{ W=\log^{100A}X. }

At the growing-accuracy scale of Section 2:

Theorem 3.1 — Polynomial- $W$ Transition

\boxed{ W = X^{100\eta+o(1)}. }

Thus a fixed $X$ -power target forces the proof's major-arc parameter to become polynomial.

This is not merely a hidden constant issue.

It changes the exponent class of the proof.

4. Type-II lemma is formally a fixed-power amplifier

Lemma 3.5 assumes

1\le W\le X^{\varepsilon/1000}

and suitable pointwise Dirichlet-polynomial hypotheses.

Its output has the form

\boxed{ \text{Type-II short sum} \ll \frac{H}{W^{1/10}} }

outside an exceptional set of measure

\boxed{ O \left( X \frac{ \log^{O(1)}X }{ W^{1/10} } \right). }

Therefore:

Theorem 4.1 — Polynomial- $W$ Type-II Amplifier

W=X^{w+o(1)}

with

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

and the hypotheses of Lemma 3.5 hold uniformly at that $W$ , then

\boxed{ \text{Type-II short sum} \ll HX^{-w/10+o(1)} }

outside a set of measure

\boxed{ X^{1-w/10+o(1)}. }

Create:

B-RH-014
POLYNOMIAL_W_TYPEII_FIXED_POWER_AMPLIFIER
status:
  CERTIFIED

This is a component bridge, not a complete theorem for $\Lambda-\Lambda^\sharp$ .

5. Formal admissible $\eta$ window

For

W=X^{100\eta+o(1)},

the structural restriction

W\le X^{\varepsilon/1000}

requires

100\eta \le \frac{\varepsilon}{1000}.

Hence:

\boxed{ \eta \le \frac{\varepsilon}{100000}. }

This interval is extremely small but positive.

Therefore the abstract Type-II parameter range does not by itself rule out a fixed exponent.

The problem is the input used to verify the Type-II hypotheses.

6. Pointwise Dirichlet-polynomial input

Lemma 3.2 of the 2026 paper defines

W_\mu=0,

\boxed{ W_\Lambda=\log^A X, }

and obtains pointwise maximal Dirichlet-polynomial bounds with constants depending on the fixed parameter $A$ .

The proof explicitly reduces the $\mu$ and $\Lambda$ cases to Lemma 3.9(ii)–(iii) of the prequel, which is an application of the Vinogradov–Korobov zero-free region.

The theorem is therefore not stated uniformly for

A=A(X)\to\infty.

7. The foundational Möbius estimate

The prequel's Lemma 3.9(ii) states, for fixed

A>0,

that for

0<\alpha\le1,

characters of modulus

q\le\log^A X,

and all subintervals

I\subset[X^\alpha,2X^\alpha],

one has a bound of the schematic form

\boxed{ \left| \sum_{\ell\in I} \frac{ \mu(r\ell)\chi(\ell) }{ \ell^{1/2+it} } \right| \ll_{\alpha,A} \frac{ X^{\alpha/2} }{ \log^A X }. }

This is uniform in

|t|\le X

but not in a growing $A$ .

The case

r=1, \qquad q=1, \qquad t=0, \qquad \alpha=1

already contains the obstruction.

8. Growing $A$ would give a weighted fixed-power Möbius theorem

Assume hypothetically that the estimate of Section 7 were valid for

A(X) = \eta \frac{\log X}{\log\log X}

with an implied constant of size

X^{o(1)}.

Then for every

I\subset[X,2X],

\boxed{ \left| \sum_{n\in I} \frac{\mu(n)}{\sqrt n} \right| \ll X^{1/2-\eta+o(1)}. }

This is already a fixed-power arithmetic estimate.

9. Partial summation converts it to fixed-power Mertens

Let

S(y) = \sum_{X<n\le y} \frac{\mu(n)}{\sqrt n}, \qquad X\le y\le2X.

The hypothetical growing-accuracy estimate gives

\sup_{X\le y\le2X}|S(y)| \ll X^{1/2-\eta+o(1)}.

Discrete partial summation yields

\sum_{X<n\le2X}\mu(n) = \sqrt{2X}\,S(2X) - \sum_{X<n<2X} S(n) [ \sqrt{n+1}-\sqrt n ].

Hence

\boxed{ \left| \sum_{X<n\le2X}\mu(n) \right| \ll X^{1-\eta+o(1)}. }

Summing dyadic blocks gives

Theorem 9.1 — Growing-Accuracy Möbius Lock

The uniform growing- $A$ extension of Lemma 3.9(ii) would imply

\boxed{ M(X) = \sum_{n\le X}\mu(n) \ll X^{1-\eta+o(1)}. }

Create:

O-RH-092
GROWING_ACCURACY_MOBIUS_INPUT_IMPLIES_FIXED_POWER_MERTENS
status:
  CERTIFIED

10. Fixed-power Mertens has fixed-strip strength

By partial summation,

\sum_{n=1}^\infty \frac{\mu(n)}{n^s}

converges and defines a holomorphic function for

\Re s>1-\eta.

For

\Re s>1,

this function equals

\frac1{\zeta(s)}.

Analytic continuation therefore forces

\boxed{ \zeta(s)\ne0 \qquad \text{for } \Re s>1-\eta. }

Thus the growing-accuracy Möbius input required by the present pointwise route is already a fixed-zero-strip theorem.

Create:

O-RH-093
POINTWISE_GROWING_ACCURACY_ROUTE_FIXED_STRIP_LOCK
status:
  CERTIFIED

This is the central closure of Campaign 38.

11. Why arbitrary fixed $A$ does not evade the lock

For every fixed $A$ ,

\log^{-A}X = X^{-o(1)}.

This is compatible with all currently known zero-free regions.

But the transition

A \asymp \frac{\log X}{\log\log X}

changes

\log^{-A}X

into a fixed power.

The fixed- $A$ theorem cannot be diagonalized across $A$ without controlling the dependence of all implied constants and, more importantly, without proving the stronger arithmetic statement in Sections 8–10.

Arbitrary fixed logarithmic accuracy is therefore not a hidden fixed-power theorem.

12. Exceptional-set constants also become unauthorised

The 2026 theorem gives

O_A \left( X\log^{-A}X \right)

exceptional measure.

At growing

A=A(X),

the unspecified constant

C(A)

must also be controlled.

To conclude

X^{1-c}

one needs at least

C(A(X)) = X^{o(1)}

relative to the intended exponent budget.

No such growing- $A$ uniformity is supplied by the theorem.

Create:

O-RH-094
FIXED_A_EXCEPTIONAL_CONSTANT_NONUNIFORMITY
status:
  CERTIFIED AS THEOREM-SCOPE AUDIT

This is secondary to the Möbius fixed-strip lock.

13. The prime Dirichlet-polynomial input has the same qualitative issue

The prequel's Lemma 3.9(iii) gives, for fixed $A$ , a prime Dirichlet-polynomial estimate for characters of modulus

q\le\log^A X

and derives it by contour integration using the classical zero-free region for Dirichlet $L$ -functions.

At growing

A\asymp \eta \frac{\log X}{\log\log X},

the formal modulus range becomes polynomial:

q\le X^{\eta+o(1)}.

A fixed-power version would therefore require a qualitatively stronger prime/Dirichlet- $L$ input.

The present campaign does not need this second lock, because the Möbius input alone already reaches fixed-strip strength.

14. Current proof architecture is circular at fixed-power precision

The logical chain of the published pointwise major-arc route is:

fixed-A zero-free-region Dirichlet input
  ->
W = log^(100 A) X
  ->
Type-II log-power amplifier
  ->
H log^(-A) X

Attempting

A ~ eta log X / log log X

changes it to

fixed-power Möbius / prime Dirichlet input
  ->
W = X^(100 eta)
  ->
Type-II fixed-power amplifier
  ->
H X^(-eta')

The amplifier works.

The input is already breakthrough-strength.

Create:

O-RH-095
CURRENT_POINTWISE_HIGHER_UNIFORMITY_FIXED_POWER_CIRCULARITY
status:
  CERTIFIED AS CURRENT-METHOD CLOSURE

15. A possible non-circular escape

Lemma 3.5 ultimately proves an integrated estimate of the form

\int |A(1+it)B(1+it)|^2 \,dt \ll \frac{ \log^{O(1)}X }{ W^{3/10} }.

Its published proof obtains this by imposing pointwise bounds on one or both Dirichlet-polynomial factors.

Pointwise control at

t=0

is what exposes the fixed-power Mertens lock.

But an integrated product estimate need not logically require a pointwise fixed-power bound at every $t$ .

A small set of bad frequencies could, in principle, be tolerated if a sufficiently strong large-value theorem controls their measure and contribution.

This is a genuinely different arithmetic possibility.

It is not solved in Campaign 38.

16. Campaign 38 track audit

GA1 — accuracy parameter scaling

status:
  EXACT

A ~ eta log X / log log X

GA2 — W-parameter transition

status:
  POLYNOMIAL

W = log^(100 A) X
  -> X^(100 eta+o(1))

GA3 — Vinogradov–Korobov dependency

status:
  FATAL FOR CURRENT POINTWISE ROUTE

growing-A Möbius input:
  fixed-power Mertens
  -> fixed zero strip

GA4 — exceptional-set parameter uniformity

status:
  NOT PROVIDED FOR GROWING A

secondary obstruction:
  O_A constants

GA5 — direct bridge to shrinking-threshold target

status:
  NOT REACHED

reason:
  foundational pointwise input already target-strength

17. Campaign 38 verdict

No fixed-power residual theorem is proved.

The important positive result is the localization:

polynomial W:
  would be useful

Type-II amplifier:
  already capable of fixed power

published pointwise input:
  only fixed-A

growing pointwise input:
  already fixed-strip strength

Thus current higher-uniformity technology does not hide a fixed-power theorem behind the phrase "for every fixed $A$ ."

18. New certified package

Create:

B-RH-014
POLYNOMIAL_W_TYPEII_FIXED_POWER_AMPLIFIER
CERTIFIED

O-RH-091
FIXED_POWER_ACCURACY_REQUIRES_GROWING_A
CERTIFIED

O-RH-092
GROWING_ACCURACY_MOBIUS_INPUT_IMPLIES_FIXED_POWER_MERTENS
CERTIFIED

O-RH-093
POINTWISE_GROWING_ACCURACY_ROUTE_FIXED_STRIP_LOCK
CERTIFIED

O-RH-094
FIXED_A_EXCEPTIONAL_CONSTANT_NONUNIFORMITY
CERTIFIED AS THEOREM-SCOPE AUDIT

O-RH-095
CURRENT_POINTWISE_HIGHER_UNIFORMITY_FIXED_POWER_CIRCULARITY
CERTIFIED AS CURRENT-METHOD CLOSURE

No new canonical frontier is created.

19. Canonical root status

F-RH-010
PESC
OPEN / ROOT TARGET

F-RH-016
MLEPG
OPEN / DIRECT THEOREM CANDIDATE

B-RH-012
SHRINKING_THRESHOLD_EXCEPTIONAL_SET_TO_MLEPG
CERTIFIED

B-RH-014
POLYNOMIAL_W_TYPEII_FIXED_POWER_AMPLIFIER
CERTIFIED COMPONENT BRIDGE

20. Campaign 39

The pointwise growing-accuracy route is closed.

The next campaign is:

CSM_RH Campaign 39
AVERAGED_TYPEII_POLYNOMIAL_W_ATTACK

The target is to remove the pointwise Dirichlet-polynomial hypothesis from the fixed-power amplifier.

This is a genuinely arithmetic theorem-generation campaign.

21. Campaign 39 tracks

AV1 — large-value replacement of pointwise input

Replace

\sup_t|A(1+it)| \ll W^{-1/3}

by a distribution estimate for the set

\{ t: |A(1+it)|>V \}.

Insert the distribution directly into the $|AB|^2$ integral.

AV2 — Guth–Maynard large-value technology

Audit whether modern large-value estimates for Dirichlet polynomials can support

W=X^w

at some fixed

w>0

without a pointwise Mertens theorem.

AV3 — low-frequency excision

The point $t=0$ is measure zero in the Parseval integral.

Quantify the contribution of a neighborhood of zero instead of controlling it pointwise.

A valid theorem must prevent a polynomially wide bad-frequency block.

AV4 — product large-value geometry

Use the Heath–Brown type-II factorization.

It may be enough that at each frequency at least one factor is small, or that simultaneous large values are rare.

This must be proved quantitatively.

AV5 — polynomial- $W$ integrated bridge

The admission target is an estimate of the form

\boxed{ \int_{|t|\le T} |A(1+it)B(1+it)|^2dt \ll T X^{-cw+o(1)} }

or the exact scale required by Lemma 3.5, with

W=X^w.

If obtained, invoke B-RH-014.

22. Campaign 39 rejection filters

Reject a candidate if:

R1. It assumes pointwise fixed-power Mertens.

R2. It assumes a fixed zero-free strip.

R3. It merely takes $A=A(X)$ inside a fixed- $A$ theorem.

R4. It controls only one Dirichlet factor while simultaneous large values remain unbounded.

R5. The large-value exceptional set is only logarithmically small when a polynomial $W$ is required.

R6. The final integrated saving is $X^{-o(1)}$ .

23. External calibration

The audit uses two parts of the Matomäki–Radziwiłł–Shao–Tao–Teräväinen architecture.

In the 2026 paper, Theorem 1.1 gives $H\log^{-A}X$ discorrelation for every fixed $A$ , and the proof of Theorem 3.1 uses
$W=\log^{100A}X$
for the $\mu,\Lambda$ type-II terms.
Lemma 3.5 converts $W$ into the output factor $W^{-1/10}$ .
Lemma 3.2 is reduced to the prequel's Lemma 3.9(ii)–(iii), which is proved using the classical zero-free region. The Möbius estimate in Lemma 3.9(ii), if extended uniformly to growing $A$ , would already imply a fixed-power Mertens bound.

The next campaign asks whether modern large-value technology can replace this pointwise step by an averaged one.

24. State transition

CSM_RH v1.29
  ->
CSM_RH v1.30

with:

Campaign 38
  CLOSED_AS_GROWING_ACCURACY_POINTWISE_INPUT_AUDIT

B-RH-014
  POLYNOMIAL_W_TYPEII_FIXED_POWER_AMPLIFIER
  CREATED / CERTIFIED

O-RH-091
  FIXED_POWER_ACCURACY_REQUIRES_GROWING_A
  CREATED / CERTIFIED

O-RH-092
  GROWING_ACCURACY_MOBIUS_INPUT_IMPLIES_FIXED_POWER_MERTENS
  CREATED / CERTIFIED

O-RH-093
  POINTWISE_GROWING_ACCURACY_ROUTE_FIXED_STRIP_LOCK
  CREATED / CERTIFIED

O-RH-094
  FIXED_A_EXCEPTIONAL_CONSTANT_NONUNIFORMITY
  CREATED / CERTIFIED

O-RH-095
  CURRENT_POINTWISE_HIGHER_UNIFORMITY_FIXED_POWER_CIRCULARITY
  CREATED / CERTIFIED

F-RH-010
  PESC
  REMAINS OPEN

F-RH-016
  MLEPG
  REMAINS OPEN

Campaign 39
  AVERAGED_TYPEII_POLYNOMIAL_W_ATTACK
  READY

25. Final status

RH = OPEN

PESC = OPEN

MLEPG = OPEN

ARBITRARY FIXED LOG ACCURACY = REAL

GROWING A = REQUIRED FOR FIXED POWER

GROWING A -> POLYNOMIAL W

POLYNOMIAL W TYPE-II AMPLIFIER = VALID

POINTWISE DIRICHLET INPUT AT GROWING A = FIXED-STRIP STRENGTH

FIXED-A THEOREM CANNOT BE DIAGONALIZED TO FIXED POWER

AVERAGED LARGE-VALUE REPLACEMENT = OPEN

NEXT CAMPAIGN = 39

The decisive scale conversion is

\boxed{ A \sim \eta \frac{\log X}{\log\log X} \quad\Longrightarrow\quad W=\log^{100A}X = X^{100\eta+o(1)}. }

The decisive obstruction is:

\boxed{ \text{uniform growing-}A\text{ Möbius input} \Longrightarrow M(X)\ll X^{1-\eta+o(1)} \Longrightarrow \zeta(s)\ne0 \text{ for } \Re s>1-\eta. }

The current pointwise route therefore reaches the desired fixed power only after importing fixed-strip-strength arithmetic.

CSM_RH Paper 39 — Growing Accuracy, Polynomial $W$, and the Möbius Pointwise Fixed-Strip Lock

CSM_RH Paper 39

Growing Accuracy, Polynomial WWW, and the Möbius Pointwise Fixed-Strip Lock

0. Trust boundary

1. Current higher-uniformity theorem

2. Fixed power requires growing accuracy

Theorem 2.1 — Growing-Accuracy Scale

3. The major-arc WWW parameter

Theorem 3.1 — Polynomial- WWW Transition

4. Type-II lemma is formally a fixed-power amplifier

Theorem 4.1 — Polynomial- WWW Type-II Amplifier

5. Formal admissible η\etaη window

6. Pointwise Dirichlet-polynomial input

7. The foundational Möbius estimate

8. Growing AAA would give a weighted fixed-power Möbius theorem

9. Partial summation converts it to fixed-power Mertens

Theorem 9.1 — Growing-Accuracy Möbius Lock

10. Fixed-power Mertens has fixed-strip strength

11. Why arbitrary fixed AAA does not evade the lock

12. Exceptional-set constants also become unauthorised

13. The prime Dirichlet-polynomial input has the same qualitative issue

14. Current proof architecture is circular at fixed-power precision

15. A possible non-circular escape

16. Campaign 38 track audit

GA1 — accuracy parameter scaling

GA2 — W-parameter transition

GA3 — Vinogradov–Korobov dependency

GA4 — exceptional-set parameter uniformity

GA5 — direct bridge to shrinking-threshold target

17. Campaign 38 verdict

18. New certified package

19. Canonical root status

20. Campaign 39

21. Campaign 39 tracks

AV1 — large-value replacement of pointwise input

AV2 — Guth–Maynard large-value technology

AV3 — low-frequency excision

AV4 — product large-value geometry

AV5 — polynomial- WWW integrated bridge

22. Campaign 39 rejection filters

R1. It assumes pointwise fixed-power Mertens.

R2. It assumes a fixed zero-free strip.

R3. It merely takes A=A(X)A=A(X)A=A(X) inside a fixed- AAA theorem.

R4. It controls only one Dirichlet factor while simultaneous large values remain unbounded.

R5. The large-value exceptional set is only logarithmically small when a polynomial WWW is required.

R6. The final integrated saving is X−o(1)X^{-o(1)}X−o(1).

23. External calibration

24. State transition

25. Final status

Growing Accuracy, Polynomial $W$ , and the Möbius Pointwise Fixed-Strip Lock

3. The major-arc $W$ parameter

Theorem 3.1 — Polynomial- $W$ Transition

Theorem 4.1 — Polynomial- $W$ Type-II Amplifier

5. Formal admissible $\eta$ window

8. Growing $A$ would give a weighted fixed-power Möbius theorem

11. Why arbitrary fixed $A$ does not evade the lock

AV5 — polynomial- $W$ integrated bridge

R3. It merely takes $A=A(X)$ inside a fixed- $A$ theorem.

R5. The large-value exceptional set is only logarithmically small when a polynomial $W$ is required.

R6. The final integrated saving is $X^{-o(1)}$ .