CSM_RH Paper 43

Liouville Sign Collapse, Determinant Fibres, and Polynomial Erosion of the Low-Variation Shift Shell

Project: CSM_RH
Paper: 43
Version: 0.1
Date: 2026-09-07
Campaign: 42 — OSCILLATORY_MOBIUS_CONVOLUTION_SHIFT_ATTACK
Canonical state transition: v1.33 to v1.34

0. Trust boundary

This paper continues directly from CSM_RH Paper 42.

The inherited pure-core coefficient is

c_X(n) = \sum_{\substack{m_1\cdots m_L=n\\m_i\sim U_i}} \mu(m_1)\cdots\mu(m_L),

with

\prod_{i=1}^{L}U_i\asymp X,

and the remaining translated-window obstruction is

\mathfrak R_{\mathrm{osc}}(Q,Y) = Y \sum_{X/Q<h\lesssim X^{o(1)}X/Y} \sum_m \frac{c_X(m)c_X(m+h)}{m(m+h)} e^{iQ\log(1+h/m)} \widehat\Phi\!\left(Y\log(1+h/m)\right).

The large-major-arc geometry inherited from Paper 42 gives

\frac{Y}{Q} \ll X^{3w/10-\varepsilon/2},

where

W=X^w, \qquad 0<w\le\frac{\varepsilon}{1000}.

The polynomial- $W$ admission target remains

\boxed{ |\mathfrak R_{\mathrm{osc}}(Q,Y)| \ll X^{-3w/10+o(1)}. }

No claim of RH, no fixed zero-free strip, and no pointwise fixed-power Mertens estimate is made.

This paper performs three tasks:

close Campaign 42 track OM1 algebraically;
sharpen OM2 by removing a much larger low-variation shift shell;
identify the exact remaining arithmetic object as a structured weighted Liouville correlation on determinant fibres.

1. Exact Liouville sign collapse

Let

\lambda(n)=(-1)^{\Omega(n)}

be the Liouville function.

For every integer $m\ge1$ ,

\boxed{ \mu(m)=\lambda(m)\mu^2(m). }

Indeed, if $m$ is not squarefree then both sides vanish, while if $m$ is squarefree then

\mu(m)=(-1)^{\Omega(m)}=\lambda(m).

Define the nonnegative restricted squarefree-factorisation weight

\boxed{ r_X(n) = \sum_{\substack{m_1\cdots m_L=n\\m_i\sim U_i}} \mu^2(m_1)\cdots\mu^2(m_L). }

Since $\lambda$ is completely multiplicative,

\lambda(m_1)\cdots\lambda(m_L) = \lambda(m_1\cdots m_L) = \lambda(n).

Therefore every nonzero summand in $c_X(n)$ has the same sign.

Theorem 1.1 — Pure-core Liouville factorisation

For every $n$ ,

\boxed{ c_X(n)=\lambda(n)r_X(n), \qquad r_X(n)\ge0. }

Proof

Using $\mu(m)=\lambda(m)\mu^2(m)$ ,

\begin{aligned} c_X(n) &= \sum_{m_1\cdots m_L=n} \prod_{i=1}^{L}\lambda(m_i)\mu^2(m_i)\\ &= \lambda(n) \sum_{m_1\cdots m_L=n} \prod_{i=1}^{L}\mu^2(m_i)\\ &= \lambda(n)r_X(n). \end{aligned}

The dyadic restrictions are unchanged throughout.
$\square$

Create:

B-RH-017
PURE_MOBIUS_CORE_LIOUVILLE_SIGN_FACTORISATION
status:
  CERTIFIED EXACT IDENTITY

This materially changes the interpretation of the pure core.

It is not an internally cancelling high-order Möbius convolution. Its complete sign is carried by one Liouville value, while the convolution multiplicity is nonnegative.

2. Grouped Type-II coefficients inherit the same sign rigidity

Let $I\sqcup J=\{1,\ldots,L\}$ be a Type-II partition.

Define

\alpha_I(u) = \sum_{\substack{\prod_{i\in I}m_i=u\\m_i\sim U_i}} \prod_{i\in I}\mu(m_i),

and

\beta_J(v) = \sum_{\substack{\prod_{j\in J}m_j=v\\m_j\sim U_j}} \prod_{j\in J}\mu(m_j).

Then

c_X=\alpha_I*\beta_J.

Define the nonnegative weights

r_I(u) = \sum_{\substack{\prod_{i\in I}m_i=u\\m_i\sim U_i}} \prod_{i\in I}\mu^2(m_i),

r_J(v) = \sum_{\substack{\prod_{j\in J}m_j=v\\m_j\sim U_j}} \prod_{j\in J}\mu^2(m_j).

Exactly as above,

\boxed{ \alpha_I(u)=\lambda(u)r_I(u), \qquad \beta_J(v)=\lambda(v)r_J(v). }

Hence

\lambda(u)\lambda(v)=\lambda(uv),

so the Type-II grouping does not create new independent sign carriers.

The entire pure component remains a single Liouville sign multiplied by a structured nonnegative factorisation multiplicity.

3. Exact weighted-Liouville form of the hard shell

The correlation coefficient becomes

\boxed{ c_X(m)c_X(m+h) = \lambda(m)\lambda(m+h) r_X(m)r_X(m+h). }

Thus

\boxed{ \begin{aligned} \mathfrak R_{\mathrm{osc}}(Q,Y) = Y \sum_h\sum_m &\frac{\lambda(m)\lambda(m+h) r_X(m)r_X(m+h)}{m(m+h)}\\ &\times e^{iQ\log(1+h/m)} \widehat\Phi\!\left(Y\log(1+h/m)\right), \end{aligned} }

with the inherited shift restrictions.

This is a weighted, locally Cesaro-scale, oscillatory two-point Liouville correlation.

Three distinctions are essential:

$r_X$ is not identically $1$ ;
the $m$ -average is localized to one dyadic scale, rather than logarithmically averaging over many scales;
the phase depends on both $m$ and the growing shift $h$ .

Therefore known logarithmically averaged Chowla results do not directly prove the required bound.

4. Type-II determinant expansion

Choose a Type-II split with

u\sim M, \qquad v\sim N, \qquad MN\asymp X, \qquad M\le N.

Write

c_X(n) = \sum_{uv=n}\alpha(u)\beta(v).

For a fixed shift $h$ ,

\sum_m c_X(m)c_X(m+h)\mathcal K(m,h)

expands into quadruples

(u,v,u',v')

satisfying

\boxed{ u'v'-uv=h, }

where $u,u'\sim M$ , $v,v'\sim N$ , and $\mathcal K$ contains the denominator, translated phase, and Gaussian cutoff.

The additive shift of products is therefore a determinant-type incidence equation.

5. GCD fibre parametrisation

Fix $u,u'$ and set

g=(u,u'), \qquad u=ga, \qquad u'=gb, \qquad (a,b)=1.

The equation

u'v'-uv=h

becomes

\boxed{ bv'-av=k, \qquad k=\frac{h}{g}. }

Hence a necessary condition is

\boxed{g\mid h.}

If one solution $(v_0,v_0')$ exists, all integral solutions are

\boxed{ v=v_0+bt, \qquad v'=v_0'+at, \qquad t\in\mathbb Z. }

Since

a,b\asymp\frac{M}{g},

the number of points of this affine lattice fibre inside $v,v'\asymp N$ is

\boxed{ O\!\left(1+\frac{Ng}{M}\right). }

Create:

B-RH-018
TYPEII_SHIFTED_PRODUCT_DETERMINANT_FIBRE_PARAMETRISATION
status:
  CERTIFIED EXACT REDUCTION

6. Algebraic incidence reaches only the orthogonality floor

The number of pairs $u,u'\sim M$ with $(u,u')=g$ is at most

\ll \left(\frac{M}{g}\right)^2

up to harmless divisor losses.

Therefore the number of quadruples at a fixed $h$ is bounded by

\begin{aligned} \mathcal N_h &\ll X^{o(1)} \sum_{g\mid h} \left(\frac{M}{g}\right)^2 \left(1+\frac{Ng}{M}\right)\\ &\ll X^{o(1)} \left( M^2\sum_{g\mid h}\frac1{g^2} + MN\sum_{g\mid h}\frac1g \right). \end{aligned}

Since

M^2\le MN\asymp X,

and

\sum_{g\mid h}\frac1g \le h^{o(1)},

we obtain

Theorem 6.1 — Determinant incidence floor

\boxed{ \mathcal N_h \ll X^{1+o(1)}. }

After inserting divisor-bounded grouped coefficients and the factor

\frac1{m(m+h)}\asymp X^{-2},

this reproduces the fixed-shift absolute scale

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

This is the same orthogonality floor already visible in Paper 42.

Thus product-equality plus additive-shift algebra does not, by itself, provide the fixed $X$ -power needed by B-RH-014.

Create:

O-RH-107
DETERMINANT_INCIDENCE_ALGEBRA_STOPS_AT_ORTHOGONALITY_FLOOR
status:
  CERTIFIED

This closes Campaign 42 track OM1 as a structural reduction rather than a successful fixed-power estimate.

7. Phase geometry on one determinant fibre

Along a fibre

v=v_0+bt, \qquad v'=v_0'+at,

the translated phase is

\psi(t) = Q\log\frac{u'v'}{uv} = Q\log\frac{b}{a} + Q\log\frac{v'}{v}.

Differentiate with respect to the real interpolation variable $t$ .

Since

bv'-av=k=\frac hg,

we have

\begin{aligned} \psi'(t) &= Q\left(\frac{a}{v'}-\frac{b}{v}\right)\\ &= -Q\frac{k}{vv'}. \end{aligned}

Hence

Theorem 7.1 — Exact first derivative on determinant fibres

\boxed{ \psi'(t) = -\frac{Qh}{g\,v(t)v'(t)}. }

On $v,v'\asymp N$ ,

\boxed{ |\psi'(t)| \asymp \frac{Qh}{gN^2}. }

A long fibre has length

\asymp \frac{Ng}{M}.

Therefore its total phase variation is

\boxed{ \operatorname{Var}_{\mathrm{fibre}}\psi \asymp \frac{Qh}{gN^2} \frac{Ng}{M} \asymp \frac{Qh}{MN} \asymp \frac{Qh}{X}. }

The GCD parameter cancels from the total variation.

This is an important geometric rigidity: every genuinely long determinant fibre sees essentially the same normalized oscillation parameter

\boxed{V(h)=\frac{Qh}{X}.}

8. The original $X/Q$ cutoff was not maximal

Paper 42 declared

h\le\frac{X}{Q}

harmless by absolute values.

The same argument works much farther.

For any $H_0$ in the Gaussian-localized range,

\begin{aligned} \left| Y\sum_{1\le h\le H_0} \sum_m \frac{c_X(m)c_X(m+h)}{m(m+h)} e^{iQ\log(1+h/m)} \widehat\Phi(\cdots) \right| &\ll YH_0X^{-1+o(1)}. \end{aligned}

Equivalently, if

H_0=\frac{X}{Q}V_0,

then

\boxed{ |\mathfrak R_{V\le V_0}| \ll \frac{Y}{Q}V_0X^{o(1)}. }

This gives a direct conversion between phase-variation threshold and trivial total mass.

9. Polynomial erosion of the low-variation shell

Set

\boxed{ V_\star = X^{\varepsilon/2-61w/100}. }

Because

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

we have

\frac{\varepsilon}{2}-\frac{61w}{100} \ge \left(\frac12-\frac{61}{100000}\right)\varepsilon >0.

Define

\boxed{ h_\star = \frac{X}{Q}V_\star. }

Then, using the inherited bound

\frac{Y}{Q} \ll X^{3w/10-\varepsilon/2},

we obtain

\begin{aligned} |\mathfrak R_{h\le h_\star}| &\ll \frac{Y}{Q} X^{\varepsilon/2-61w/100+o(1)}\\ &\ll X^{3w/10-61w/100+o(1)}\\ &= X^{-31w/100+o(1)}. \end{aligned}

The target is

X^{-3w/10+o(1)} = X^{-30w/100+o(1)}.

Hence the enlarged shell has an extra margin

\boxed{X^{-w/100+o(1)}.}

Theorem 9.1 — Polynomial low-variation shell erosion

The entire range

\boxed{ 1\le h\le \frac{X}{Q} X^{\varepsilon/2-61w/100} }

is harmless for polynomial- $W$ admission.

Create:

O-RH-108
POLYNOMIAL_LOW_VARIATION_SHIFT_SHELL_HARMLESS
status:
  CERTIFIED

This strictly strengthens O-RH-103.

10. The new residual shell has polynomial phase variation

The residual range now begins at

h>h_\star.

Therefore

\boxed{ \frac{Qh}{X} > X^{\varepsilon/2-61w/100}. }

Since the Gaussian cutoff still allows shifts up to

h\lesssim X^{o(1)}\frac{X}{Y},

and

\frac{Q}{Y} \gtrsim X^{\varepsilon/2-3w/10},

the residual shell has room of at least

\boxed{ X^{31w/100-o(1)} }

in multiplicative shift scale at the extremal inherited geometry.

Thus the new unresolved object is not a barely oscillatory shell.

It is a polynomially oscillatory shell.

Create:

O-RH-109
RESIDUAL_SHIFT_SHELL_HAS_POLYNOMIAL_PHASE_VARIATION
status:
  CERTIFIED

11. Why polynomial phase variation does not yet solve the problem

If the arithmetic weights on a determinant fibre were smooth or absent, the derivative identity from Section 7 would make classical oscillatory-sum methods immediately relevant.

But the exact fibre weight inherits Liouville signs and structured squarefree-factorisation weights.

At the atomic level, fixing all variables except one factor on each side produces affine forms

A_0+B_0t, \qquad A_1+B_1t,

and a weight of the schematic form

\boxed{ \lambda(A_0+B_0t) \lambda(A_1+B_1t) \times \text{structured nonnegative multiplicity}. }

Therefore a derivative test cannot simply discard the arithmetic amplitude.

The remaining problem is a hybrid of:

two-linear-form Liouville decorrelation;
polynomially varying phase;
determinant-fibre arithmetic;
restricted squarefree-factorisation weights.

The phase is now strong enough to matter, but a theorem coupling it to the Liouville carrier is still required.

12. Calibration against two-point Chowla technology

The exact sign collapse makes ordinary Liouville correlation a direct calibration rather than merely an analogy.

Tao's two-point logarithmically averaged Chowla theorem proves cancellation after logarithmic averaging for two affine-linear Liouville forms, but it does not prove the ordinary Cesaro two-point Chowla conjecture.

Pilatte obtained a fixed power of the logarithm for the logarithmically weighted shift-one correlation.

Guo's August 2026 result gives quantitative logarithmic cancellation uniformly for shifts up to a small power of $\log x$ and explicitly states that it does not prove ordinary Cesaro two-point Chowla.

Our residual object is stronger in several ways:

\boxed{ \text{one dyadic scale} + \text{structured weights} + \text{polynomial shift range} + \text{oscillatory phase}. }

The phase is additional information, but current logarithmic Chowla estimates cannot be inserted as a black box to produce

X^{-3w/10}.

Create:

O-RH-110
CURRENT_LOG_CHOWLA_THEOREMS_DO_NOT_CLOSE_WEIGHTED_OSCILLATORY_LIOUVILLE_CORE
status:
  CERTIFIED AS CURRENT-LITERATURE CALIBRATION

This is a scope statement, not a universal impossibility theorem.

13. Campaign 42 track audit

OM1 — balanced Möbius-convolution correlation algebra

Status: CLOSED_AS_LIOUVILLE_WEIGHTED_DETERMINANT_REDUCTION

Outputs:

c_X(n)=\lambda(n)r_X(n),

and

u'v'-uv=h

with exact GCD fibre parametrisation.

Pure incidence counting stops at the orthogonality floor.

OM2 — high-frequency phase geometry

Status: ADVANCED_TO_POLYNOMIAL_VARIATION_RESIDUAL

The harmless shell extends to

h\le \frac{X}{Q} X^{\varepsilon/2-61w/100},

so every residual long fibre has

\operatorname{Var}\psi \gtrsim X^{\varepsilon/2-61w/100}.

OM3 — averaged shift cancellation

Status: NOT_CLOSED_BY_CURRENT_CHOWLA_INPUT

Known logarithmic/average technology does not yield the required uniform fixed $X$ -power for the weighted dyadic object.

OM4 — Ramaré extraction

Status: STILL_OPEN

The Liouville sign collapse makes small-prime extraction more natural, but extraction must be performed without converting the problem back into an uncontrolled two-linear-form parity problem.

OM5 — cross- $j$ coupling

Status: STILL_OPEN

The exact Heath-Brown identity contains pointwise inclusion-exclusion across $j$ -levels. Any successful use must preserve those cross terms before absolute values and before componentwise mean-square bounds.

14. New canonical core

After removing the enlarged harmless shell, define

\boxed{ \begin{aligned} \mathfrak R_{\lambda}(Q,Y) = Y \sum_{h_\star<h\lesssim X^{o(1)}X/Y} \sum_m &\frac{\lambda(m)\lambda(m+h) r_X(m)r_X(m+h)}{m(m+h)}\\ &\times e^{iQ\log(1+h/m)} \widehat\Phi\!\left(Y\log(1+h/m)\right). \end{aligned} }

Then

\boxed{ \mathfrak R_{\mathrm{osc}}(Q,Y) = \mathfrak R_{\lambda}(Q,Y) + O\!\left(X^{-31w/100+o(1)}\right). }

Hence polynomial- $W$ admission reduces to proving

\boxed{ |\mathfrak R_{\lambda}(Q,Y)| \ll X^{-3w/10+o(1)}. }

This is not declared a new root frontier.

It is the sharpened internal form of the existing F-RH-010 / B-RH-014 route.

15. Campaign 42 verdict

Campaign 42 does not prove the required fixed-power estimate.

It does, however, achieve two irreversible reductions:

the high-order Möbius sign structure collapses exactly to Liouville;
all low and moderately varying phase shells can be removed by absolute values until the remaining phase variation is polynomial in $X$ .

Thus the obstruction is now more specific than in Paper 42:

\boxed{ \text{weighted Liouville two-point decorrelation} + \text{polynomial determinant-fibre oscillation}. }

Campaign 42 is therefore closed as

CLOSED_AS_WEIGHTED_LIOUVILLE_POLYNOMIAL_PHASE_LOCALIZATION

with no RH promotion.

16. Campaign 43

The next campaign is

CSM_RH Campaign 43
WEIGHTED_LIOUVILLE_POLYNOMIAL_PHASE_ATTACK

The target remains B-RH-014 through the sharpened core $\mathfrak R_\lambda(Q,Y)$ .

No new observable is introduced.

17. Campaign 43 tracks

WL1 — phase-first determinant-fibre dispersion

Apply Cauchy or van der Corput only after preserving the determinant-fibre phase.

The goal is to turn the polynomial lower bound

\frac{Qh}{X} \ge X^{\varepsilon/2-61w/100}

into a fixed-power gain without replacing Liouville coefficients by generic bounded coefficients.

WL2 — Liouville-aware Ramaré extraction

Exploit complete multiplicativity of $\lambda$ and extract a prime from exactly one side when it does not divide $h$ .

Track the common divisor

(m,m+h)=(m,h)

separately so that common-prime signs cancel before extraction.

WL3 — structured-weight decoupling

Use

r_X = (\mu^2 1_{U_1})*\cdots*(\mu^2 1_{U_L})

to determine whether the nonnegative multiplicity weight can be separated from the Liouville sign at acceptable exponent cost.

WL4 — cross- $j$ pre-square recombination

Audit whether the alternating Heath-Brown $j$ -sum can be recombined before translated mean-square expansion so that the pure Liouville carrier cancels against adjacent $j$ -levels.

WL5 — root-recoupling comparison

If WL4 collapses back to the full prime-error coefficient, compare the resulting object directly with F-RH-010 rather than pretending a new component theorem has been obtained.

18. Rejection filters for Campaign 43

Reject a candidate proof if any of the following occurs.

R1. Generic-coefficient phase cancellation

The argument applies an oscillatory derivative bound after replacing Liouville-weighted coefficients by arbitrary bounded coefficients.

R2. Logarithmic-to-power promotion

A bound of size $\log^{-A}X$ , $o(1)$ , or $\exp(-c\log^\alpha X)$ is treated as $X^{-\delta}$ for fixed $\delta>0$ .

R3. Scale averaging mismatch

A logarithmically averaged Chowla theorem over many scales is used as a uniform theorem on one prescribed dyadic scale without an explicit transfer argument.

R4. Weight deletion

The factor $r_X(m)r_X(m+h)$ is discarded without a positive majorant whose exponent ledger is verified.

R5. Hidden fixed-strip input

A pointwise estimate equivalent to a fixed zero-free strip is inserted through a Möbius or Liouville Dirichlet polynomial estimate.

R6. Cross- $j$ cancellation after big- $O$

Separate component bounds are estimated absolutely and then claimed to cancel.

19. External calibration

The following sources calibrate the scope of the claims in this paper.

K. Matomäki, M. Radziwiłł, X. Shao, T. Tao, J. Teräväinen, Higher uniformity of arithmetic functions in short intervals II. Almost all intervals, Inventiones Mathematicae 244 (2026), 967–1091, DOI 10.1007/s00222-026-01408-6. The relevant Type-II target is Lemma 3.5, especially equations (3.9)–(3.13).
T. Tao, The logarithmically averaged Chowla and Elliott conjectures for two-point correlations, Forum of Mathematics, Pi 4 (2016), arXiv:1509.05422. This proves logarithmically averaged two-point cancellation, not ordinary Cesaro two-point Chowla.
C. Pilatte, Improved bounds for the two-point logarithmic Chowla conjecture, arXiv:2310.19357. This obtains a fixed power of the logarithm for the logarithmically weighted shift-one problem.
J. Guo, Quantitative Logarithmic Chowla Correlations Uniformly over Growing Shifts, arXiv:2608.23500 (2026). The result remains logarithmically weighted and its stated shift range is polylogarithmic.

No cited theorem is claimed to prove the weighted dyadic fixed- $X$ -power estimate required here.

20. State transition

The canonical state advances from v1.33 to v1.34.

New bridge identities:

B-RH-017
PURE_MOBIUS_CORE_LIOUVILLE_SIGN_FACTORISATION
CERTIFIED

B-RH-018
TYPEII_SHIFTED_PRODUCT_DETERMINANT_FIBRE_PARAMETRISATION
CERTIFIED

New obstructions/refinements:

O-RH-107
DETERMINANT_INCIDENCE_ALGEBRA_STOPS_AT_ORTHOGONALITY_FLOOR
CERTIFIED

O-RH-108
POLYNOMIAL_LOW_VARIATION_SHIFT_SHELL_HARMLESS
CERTIFIED

O-RH-109
RESIDUAL_SHIFT_SHELL_HAS_POLYNOMIAL_PHASE_VARIATION
CERTIFIED

O-RH-110
CURRENT_LOG_CHOWLA_THEOREMS_DO_NOT_CLOSE_WEIGHTED_OSCILLATORY_LIOUVILLE_CORE
CERTIFIED AS CURRENT-LITERATURE CALIBRATION

Root status remains

F-RH-010  PRIME_ERROR_SELF_CORRELATION  OPEN
F-RH-016  MESOSCOPIC_LAG_ENERGY_POWER_GAIN  OPEN
RH_PROVED  FALSE
RH_DISPROVED  FALSE
GLOBAL_RH_CERTIFICATE  FALSE

21. Final status

The campaign has not solved RH and has not proved polynomial- $W$ Type-II admission.

The accepted advance is

\boxed{ \text{pure Möbius convolution} \longrightarrow \lambda(n)\times\text{nonnegative structured weight} }

combined with

\boxed{ \text{residual shifts} \Longrightarrow \frac{Qh}{X} \ge X^{\varepsilon/2-61w/100}. }

Thus the next valid attack is no longer generic Möbius correlation algebra.

It is a Liouville-aware polynomial-phase dispersion problem, with Ramaré extraction and cross- $j$ recombination retained as live alternatives.

CSM_RH Paper 43 — Liouville Sign Collapse, Determinant Fibres, and Polynomial Erosion of the Low-Variation Shift Shell

CSM_RH Paper 43

Liouville Sign Collapse, Determinant Fibres, and Polynomial Erosion of the Low-Variation Shift Shell

0. Trust boundary

1. Exact Liouville sign collapse

Theorem 1.1 — Pure-core Liouville factorisation

Proof

2. Grouped Type-II coefficients inherit the same sign rigidity

3. Exact weighted-Liouville form of the hard shell

4. Type-II determinant expansion

5. GCD fibre parametrisation

6. Algebraic incidence reaches only the orthogonality floor

Theorem 6.1 — Determinant incidence floor

7. Phase geometry on one determinant fibre

Theorem 7.1 — Exact first derivative on determinant fibres

8. The original X/QX/QX/Q cutoff was not maximal

9. Polynomial erosion of the low-variation shell

Theorem 9.1 — Polynomial low-variation shell erosion

10. The new residual shell has polynomial phase variation

11. Why polynomial phase variation does not yet solve the problem

12. Calibration against two-point Chowla technology

13. Campaign 42 track audit

OM1 — balanced Möbius-convolution correlation algebra

OM2 — high-frequency phase geometry

OM3 — averaged shift cancellation

OM4 — Ramaré extraction

OM5 — cross- jjj coupling

14. New canonical core

15. Campaign 42 verdict

16. Campaign 43

17. Campaign 43 tracks

WL1 — phase-first determinant-fibre dispersion

WL2 — Liouville-aware Ramaré extraction

WL3 — structured-weight decoupling

WL4 — cross- jjj pre-square recombination

WL5 — root-recoupling comparison

18. Rejection filters for Campaign 43

R1. Generic-coefficient phase cancellation

R2. Logarithmic-to-power promotion

R3. Scale averaging mismatch

R4. Weight deletion

R5. Hidden fixed-strip input

R6. Cross- jjj cancellation after big- OOO

19. External calibration

20. State transition

21. Final status

8. The original $X/Q$ cutoff was not maximal

OM5 — cross- $j$ coupling

WL4 — cross- $j$ pre-square recombination

R6. Cross- $j$ cancellation after big- $O$