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CSM_RH Paper 46 — Natural-Energy Saturation, Bounded Mark Multiplicity, and the Convolution-Decoupling Cycle

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CSM_RH Paper 46

Natural-Energy Saturation, Bounded Mark Multiplicity, and the Convolution-Decoupling Cycle

Project: CSM_RH
Paper: 46
Version: 0.1
Date: 2026-09-07
Campaign: 43 — WEIGHTED_LIOUVILLE_POLYNOMIAL_PHASE_ATTACK
Track: WL3 — STRUCTURED_WEIGHT_DECOUPLING
Canonical state transition: v1.36 to v1.37

0. Trust boundary

This paper continues directly from CSM_RH Paper 45.

The inherited residual core is

Rλ(Q,Y)=Y∑h⋆<h≲Xo(1)X/Y∑mrX(m)rX(m+h)m(m+h)×fQ(m)‾fQ(m+h)Φ^ ⁣(Ylog⁡(1+h/m)),\boxed{ \begin{aligned} \mathfrak R_{\lambda}(Q,Y) = Y \sum_{h_\star<h\lesssim X^{o(1)}X/Y} \sum_m &\frac{r_X(m)r_X(m+h)}{m(m+h)}\\ &\times \overline{f_Q(m)}f_Q(m+h) \widehat\Phi\!\left(Y\log(1+h/m)\right), \end{aligned} }

where

fQ(n)=λ(n)niQ,f_Q(n)=\lambda(n)n^{iQ}, W=Xw,0<w≤ε1000,W=X^w, \qquad 0<w\le\frac{\varepsilon}{1000}, h⋆=XQXε/2−61w/100,h_\star = \frac{X}{Q} X^{\varepsilon/2-61w/100},

and the admission target remains

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

Paper 45 proved an exact factorisation-space Ramaré marking identity and removed the large common-prime and repeated-large-prime branches at the stronger scale

X−31w/100+o(1).X^{-31w/100+o(1)}.

It also isolated two hard coefficient families:

  1. a rough marked affine prime-Liouville branch;
  2. a no-large-prime smooth-carrier branch.

WL3 asks whether the exact convolution structure of

rX=(μ21U1)∗⋯∗(μ21UL)r_X = (\mu^2 1_{U_1})*\cdots*(\mu^2 1_{U_L})

contains an L2L^2, dispersion, or factor-decoupling reserve large enough to supply a fixed power of XX.

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

1. Factorisation-space notation

As before,

L=⌈10ε⌉,L=\left\lceil\frac{10}{\varepsilon}\right\rceil, mi∼Ui,Ui=X1/L+o(1),m_i\sim U_i, \qquad U_i=X^{1/L+o(1)},

and

∏i=1LUi≍X.\prod_{i=1}^{L}U_i\asymp X.

Define

gi(n)=μ2(n)1n∼Ui.g_i(n) = \mu^2(n)1_{n\sim U_i}.

Then

rX=g1∗⋯∗gL.\boxed{ r_X = g_1*\cdots*g_L. }

The corresponding signed coefficient is

cX(n)=λ(n)rX(n),c_X(n) = \lambda(n)r_X(n),

and, by Paper 43,

cX=(μ1U1)∗⋯∗(μ1UL).\boxed{ c_X = (\mu 1_{U_1})*\cdots*(\mu 1_{U_L}). }

Define the polynomial prime threshold

ρ⋆=31w100,\rho_\star = \frac{31w}{100}, P⋆=Xρ⋆.P_\star = X^{\rho_\star}.

Split the structured weight into

rXsm(n)=rX(n)1P+(n)<P⋆,\boxed{ r_X^{\mathrm{sm}}(n) = r_X(n)1_{P^+(n)<P_\star}, }

and

rXrough(n)=rX(n)1P+(n)≥P⋆.\boxed{ r_X^{\mathrm{rough}}(n) = r_X(n)1_{P^+(n)\ge P_\star}. }

At exponent resolution the distinction between strict and non-strict endpoint conventions is harmless.

2. Total mass of the structured weight

For every fixed dyadic interval,

∑m∼Uiμ2(m)≍Ui.\sum_{m\sim U_i}\mu^2(m) \asymp U_i.

Therefore

∑nrX(n)=∏i=1L∑mi∼Uiμ2(mi)≍X.\boxed{ \sum_n r_X(n) = \prod_{i=1}^{L} \sum_{m_i\sim U_i}\mu^2(m_i) \asymp X. }

The support of rXr_X lies in a fixed multiplicative enlargement of the dyadic block n≍Xn\asymp X. Hence the number of possible support integers is

OL(X).O_L(X).

Cauchy-Schwarz gives

(∑nrX(n))2≤OL(X)∑nrX(n)2.\left(\sum_n r_X(n)\right)^2 \le O_L(X) \sum_n r_X(n)^2.

Consequently,

∑nrX(n)2≫LX.\boxed{ \sum_n r_X(n)^2 \gg_L X. }

On the other hand,

rX(n)≤dL(n),r_X(n) \le d_L(n),

and the fixed- LL divisor second moment gives

∑n≍XdL(n)2≪LX(log⁡X)OL(1).\sum_{n\asymp X}d_L(n)^2 \ll_L X(\log X)^{O_L(1)}.

Thus:

Theorem 2.1 — Natural L2L^2 scale of the pure-core structured weight

∑nrX(n)2=X1+o(1).\boxed{ \sum_n r_X(n)^2 = X^{1+o(1)}. }

Equivalently,

∥rX∥2=X1/2+o(1).\boxed{ \|r_X\|_2 = X^{1/2+o(1)}. }

For the normalized Dirichlet coefficient

aX(n)=rX(n)n,a_X(n) = \frac{r_X(n)}{n},

this becomes

∑n∣aX(n)∣2=X−1+o(1).\boxed{ \sum_n|a_X(n)|^2 = X^{-1+o(1)}. }

Create:

B-RH-025
PURE_CORE_STRUCTURED_WEIGHT_NATURAL_L2_SCALE
CERTIFIED

The structured weight is not an L2L^2 -small perturbation.

3. Smooth carrier has full exponent energy

Paper 45 constructed a fixed integer K=K(ε,w)K=K(\varepsilon,w) and a legal smooth factorisation subfamily for which

P+(n)<P⋆P^+(n)<P_\star

and

∑nrXsm(n)≫ε,wX(log⁡X)Cε,w.\sum_n r_X^{\mathrm{sm}}(n) \gg_{\varepsilon,w} \frac{X}{(\log X)^{C_{\varepsilon,w}}}.

Since the support still has OL(X)O_L(X) possible integers,

(∑nrXsm(n))2≤OL(X)∑n(rXsm(n))2.\left( \sum_n r_X^{\mathrm{sm}}(n) \right)^2 \le O_L(X) \sum_n \left(r_X^{\mathrm{sm}}(n)\right)^2.

Hence

Theorem 3.1 — Smooth-carrier full exponent energy

∑n(rXsm(n))2≫ε,wX(log⁡X)2Cε,w=X1−o(1).\boxed{ \sum_n \left(r_X^{\mathrm{sm}}(n)\right)^2 \gg_{\varepsilon,w} \frac{X}{(\log X)^{2C_{\varepsilon,w}}} = X^{1-o(1)}. }

The upper bound from Theorem 2.1 gives

∑n(rXsm(n))2=X1+o(1)\boxed{ \sum_n \left(r_X^{\mathrm{sm}}(n)\right)^2 = X^{1+o(1)} }

at exponent resolution.

Thus the no-large-prime branch cannot be discarded through coefficient sparsity or coefficient energy.

4. A sign-coherent smooth subweight also has full exponent energy

Let

sX(n)s_X(n)

count only the legal tuples in the explicit Paper 45 construction in which each factor mim_i is a product of exactly KK primes from its prescribed small-prime interval.

Every such tuple has

Ω(m1⋯mL)=KL\Omega(m_1\cdots m_L) = KL

counted with multiplicity. Therefore on the support of sXs_X,

λ(n)=(−1)KL.\boxed{ \lambda(n) = (-1)^{KL}. }

Also,

0≤sX(n)≤rXsm(n),0\le s_X(n)\le r_X^{\mathrm{sm}}(n),

and Paper 45 gives

∑nsX(n)≫ε,wX(log⁡X)Cε,w.\sum_n s_X(n) \gg_{\varepsilon,w} \frac{X}{(\log X)^{C_{\varepsilon,w}}}.

The same Cauchy-Schwarz argument yields

Theorem 4.1 — Sign-coherent smooth subweight has full exponent energy

∑nsX(n)2≫ε,wX(log⁡X)2Cε,w.\boxed{ \sum_n s_X(n)^2 \gg_{\varepsilon,w} \frac{X}{(\log X)^{2C_{\varepsilon,w}}}. }

Therefore any WL3 mechanism that first replaces the Liouville sign by an absolute value, or expects the nonnegative factorisation weight itself to create a fixed-power cancellation, has no coefficient-side power reserve on this explicit subfamily.

This theorem does not assert a nonzero shifted two-point correlation for sXs_X. It is a method-scope statement about positivity and one-variable coefficient decoupling.

5. Rough carrier is also not power-sparse

The rough branch admits an elementary full-exponent construction independent of the smooth construction.

Fix one factor position ii.

Choose a small fixed η>0\eta>0. Take primes

P⋆<p≤(1+η)P⋆.P_\star<p\le(1+\eta)P_\star.

Since

ρ⋆<1L,\rho_\star<\frac1L,

we have

Uip=X1/L−ρ⋆+o(1)→∞.\frac{U_i}{p} = X^{1/L-\rho_\star+o(1)} \to\infty.

For each such prime choose a squarefree integer qiq_i from a fixed relative interval of scale

Uip\frac{U_i}{p}

such that

pqi∼Uipq_i\sim U_i

and

p∤qi.p\nmid q_i.

For all j≠ij\ne i, choose arbitrary squarefree

mj∼Uj,m_j\sim U_j,

and impose

p∤mj.p\nmid m_j.

The latter condition removes only a negligible proportion because p→∞p\to\infty.

The prime number theorem on the fixed relative prime interval gives

#{p: P⋆<p≤(1+η)P⋆}≍P⋆log⁡X.\#\{p:\ P_\star<p\le(1+\eta)P_\star\} \asymp \frac{P_\star}{\log X}.

For each such prime there are

≫Uip\gg \frac{U_i}{p}

legal choices of qiq_i and

≫∏j≠iUj\gg \prod_{j\ne i}U_j

choices for the remaining factors.

Therefore the number of legal tuples in this one-large-prime family is

≫P⋆log⁡XUiP⋆∏j≠iUj=Xlog⁡X.\gg \frac{P_\star}{\log X} \frac{U_i}{P_\star} \prod_{j\ne i}U_j = \frac{X}{\log X}.

Every tuple has

P+(n)>P⋆P^+(n)>P_\star

and the selected prime occurs exactly once.

Thus

Theorem 5.1 — Rough non-repeated carrier has full exponent mass

∑nrXrough(n)≫ε,wXlog⁡X.\boxed{ \sum_n r_X^{\mathrm{rough}}(n) \gg_{\varepsilon,w} \frac{X}{\log X}. }

Consequently,

Corollary 5.2 — Rough carrier has full exponent energy

∑n(rXrough(n))2≫ε,wX(log⁡X)2=X1−o(1).\boxed{ \sum_n \left(r_X^{\mathrm{rough}}(n)\right)^2 \gg_{\varepsilon,w} \frac{X}{(\log X)^2} = X^{1-o(1)}. }

The rough and smooth branches therefore both survive at full XX -exponent in coefficient energy.

Create:

B-RH-026
SMOOTH_AND_ROUGH_CARRIERS_BOTH_HAVE_FULL_EXPONENT_WEIGHT_ENERGY
CERTIFIED

6. Polynomial-prime marking has bounded fibre multiplicity

Let

P⋆={p: p>P⋆}.\mathcal P_\star = \{p:\ p>P_\star\}.

For every

n≍X,n\asymp X,

we have

P⋆ΩP⋆(n)≤n≪X.P_\star^{\Omega_{\mathcal P_\star}(n)} \le n \ll X.

Hence

ΩP⋆(n)≤log⁡(CX)log⁡P⋆=1ρ⋆+o(1).\boxed{ \Omega_{\mathcal P_\star}(n) \le \frac{\log(CX)}{\log P_\star} = \frac1{\rho_\star} + o(1). }

Thus a coefficient at scale XX carries only

Ow(1)O_w(1)

prime occurrences above the fixed polynomial threshold.

The exact Paper 45 marking decomposition is

rX(n)=∑p∈P⋆∑i=1Lwi,p(n)r_X(n) = \sum_{p\in\mathcal P_\star} \sum_{i=1}^{L} w_{i,p}(n)

on the rough branch, where

wi,p(n)=rX;i,p−(n/p)ΩP⋆(n)w_{i,p}(n) = \frac{ r_{X;i,p}^{-}(n/p) }{ \Omega_{\mathcal P_\star}(n) }

when the denominator is nonzero.

For fixed nn, the number of nonzero pairs (i,p)(i,p) is at most

N⋆=L(1ρ⋆+2)=Oε,w(1).\boxed{ N_\star = L \left( \frac1{\rho_\star}+2 \right) = O_{\varepsilon,w}(1). }

Pointwise Cauchy-Schwarz gives

rX(n)2≤N⋆∑i,pwi,p(n)2.r_X(n)^2 \le N_\star \sum_{i,p} w_{i,p}(n)^2.

Since the wi,pw_{i,p} are nonnegative,

∑i,pwi,p(n)2≤rX(n)2.\sum_{i,p} w_{i,p}(n)^2 \le r_X(n)^2.

Therefore:

Theorem 6.1 — Marked L2L^2 energy stability

On the rough branch,

1N⋆∑n(rXrough(n))2≤∑i,p∑nwi,p(n)2≤∑n(rXrough(n))2.\boxed{ \frac1{N_\star} \sum_n \left(r_X^{\mathrm{rough}}(n)\right)^2 \le \sum_{i,p}\sum_n w_{i,p}(n)^2 \le \sum_n \left(r_X^{\mathrm{rough}}(n)\right)^2. }

The exact polynomial-prime marking decomposition changes the total coefficient energy by at most a constant depending on ε\varepsilon and ww.

Create:

B-RH-027
POLYNOMIAL_PRIME_MARKING_BOUNDED_MULTIPLICITY_ENERGY_STABILITY
CERTIFIED

This is the L2L^2 counterpart of the prime-harmonic leverage barrier from Paper 45.

7. Fixed exponent prime bands carry full marked energy

The bounded-multiplicity statement can be strengthened on any genuine polynomial sub-band.

Choose fixed exponents

ρ⋆<ρ2<1L.\rho_\star<\rho_2<\frac1L.

Let

P†={p: Xρ⋆<p≤Xρ2}.\mathcal P^\dagger = \{p:\ X^{\rho_\star}<p\le X^{\rho_2}\}.

Fix a factor position ii.

For every

p∈P†,p\in\mathcal P^\dagger,

the one-prime-removed weight rX;i,p−(a)r_{X;i,p}^{-}(a) has total mass

∑arX;i,p−(a)≍ε,wXp.\sum_a r_{X;i,p}^{-}(a) \asymp_{\varepsilon,w} \frac{X}{p}.

Its support has

OL(X/p)O_L(X/p)

possible integers.

Hence

∑a(rX;i,p−(a))2≫ε,wXp.\sum_a \left(r_{X;i,p}^{-}(a)\right)^2 \gg_{\varepsilon,w} \frac{X}{p}.

Summing over the fixed exponent band and using Mertens' theorem,

∑Xρ⋆<p≤Xρ21p=log⁡ρ2ρ⋆+o(1).\sum_{X^{\rho_\star}<p\le X^{\rho_2}}\frac1p = \log\frac{\rho_2}{\rho_\star}+o(1).

Therefore:

Theorem 7.1 — Polynomial marked band has natural total energy

∑p∈P†∑a(rX;i,p−(a))2≫ε,w,ρ2X.\boxed{ \sum_{p\in\mathcal P^\dagger} \sum_a \left(r_{X;i,p}^{-}(a)\right)^2 \gg_{\varepsilon,w,\rho_2} X. }

The corresponding divisor-moment upper bound is

X1+o(1).X^{1+o(1)}.

Thus the raw marked family itself carries natural XX -scale energy. There is no hidden P⋆−cP_\star^{-c} coefficient-energy gain after summing over polynomially many prime locations.

8. The multiplicity-leverage closure

Paper 45 proved the prime-harmonic tradeoff:

fixed polynomial prime threshold⟹bounded prime-harmonic mass,\text{fixed polynomial prime threshold} \Longrightarrow \text{bounded prime-harmonic mass},

whereas

diverging harmonic mass⟹P=Xo(1).\text{diverging harmonic mass} \Longrightarrow P=X^{o(1)}.

Theorem 6.1 now adds:

fixed polynomial prime threshold⟹bounded mark multiplicity.\text{fixed polynomial prime threshold} \Longrightarrow \text{bounded mark multiplicity}.

Therefore the two natural decoupling resources cannot both grow.

If one keeps

P=XρP=X^{\rho}

with fixed

ρ>0,\rho>0,

then:

  1. prime-size leverage is polynomial;
  2. prime-harmonic mass is O(1)O(1) ;
  3. mark multiplicity per coefficient is O(1)O(1) ;
  4. marked L2L^2 energy remains at the natural exponent.

If one lowers the threshold until the number of usable marks or the harmonic mass grows, then

P=Xo(1),P=X^{o(1)},

and any gain based only on a fixed power of PP is

X−o(1).X^{-o(1)}.

Create:

O-RH-117
POLYNOMIAL_PRIME_MARKING_HAS_NO_GROWING_ORTHOGONAL_MULTIPLICITY
CERTIFIED AS MECHANISM-SCOPE BARRIER

This closes the possibility that WL2 merely needed an L2L^2 reinterpretation of the same polynomial prime marking.

9. What a pure coefficient-energy proof would need

Let BB denote either the smooth or rough coefficient branch and define

EB=∑nrB(n)2.E_B = \sum_n r_B(n)^2.

The normalized coefficient is

aB(n)=rB(n)n.a_B(n) = \frac{r_B(n)}{n}.

Since

n≍X,n\asymp X, ∑n∣aB(n)∣2≍EBX2.\sum_n|a_B(n)|^2 \asymp \frac{E_B}{X^2}.

A generic Dirichlet-polynomial mean-value or large-sieve inequality at length XX has coefficient-energy scale

(Y+X)∑n∣aB(n)∣2.(Y+X) \sum_n|a_B(n)|^2.

Because

Y≤X1+o(1)Y\le X^{1+o(1)}

in the present application, this is

≪Xo(1)EBX.\ll X^{o(1)} \frac{E_B}{X}.

To obtain the admission target purely from coefficient energy would require

EB≪X1−3w/10+o(1).E_B \ll X^{1-3w/10+o(1)}.

But Theorems 3.1 and 5.2 give

Esm≥X1−o(1),E_{\mathrm{sm}} \ge X^{1-o(1)},

and

Erough≥X1−o(1).E_{\mathrm{rough}} \ge X^{1-o(1)}.

Thus no argument whose only quantitative reserve is small coefficient L2L^2 energy can provide the required fixed power.

Create:

O-RH-118
STRUCTURED_WEIGHT_L2_ENERGY_HAS_NO_FIXED_POWER_RESERVE
CERTIFIED AS COEFFICIENT-ENERGY METHOD BARRIER

This does not rule out an operator estimate that uses genuine Liouville arithmetic. It rules out obtaining the target by treating rXr_X or its marked descendants as unusually low-energy weights.

10. Exact factor bipartitions

The convolution can be split across any nonempty proper subset of factor positions.

Let

I⊔J={1,…,L}.I\sqcup J = \{1,\ldots,L\}.

Define

RI=∗i∈Igi,R_I = \mathop{*}_{i\in I}g_i, RJ=∗j∈Jgj.R_J = \mathop{*}_{j\in J}g_j.

Then

rX=RI∗RJ.r_X = R_I*R_J.

For the signed coefficient define

CI=∗i∈I(μ1Ui),C_I = \mathop{*}_{i\in I} (\mu1_{U_i}), CJ=∗j∈J(μ1Uj),C_J = \mathop{*}_{j\in J} (\mu1_{U_j}),

so

cX=CI∗CJ.c_X = C_I*C_J.

Let the two product scales be

R=∏i∈IUi,R = \prod_{i\in I}U_i, S=∏j∈JUj,S = \prod_{j\in J}U_j,

with

RS≍X.RS\asymp X.

Without loss of generality orient the split so that

R≤S.R\le S.

Then

R≤X1/2+o(1).R\le X^{1/2+o(1)}.

11. A factor split returns a determinant fibre

Expand a fixed shifted product.

Ignoring the harmless smooth kernel and the m−1(m+h)−1m^{-1}(m+h)^{-1} normalization for the moment, the signed correlation contains

∑a,a′∼Rb,b′∼Sa′b′−ab=hCI(a)CI(a′)CJ(b)CJ(b′).\sum_{\substack{ a,a'\sim R\\ b,b'\sim S\\ a'b'-ab=h }} C_I(a)C_I(a') C_J(b)C_J(b').

Thus every factor bipartition returns the determinant incidence equation

a′b′−ab=h.\boxed{ a'b'-ab=h. }

Fix

a,a′∼Ra,a'\sim R

and let

d=(a,a′).d=(a,a').

Solutions exist only if

d∣h.d\mid h.

When they exist, the (b,b′)(b,b') solutions form one affine lattice fibre with step sizes

adanda′d.\frac{a}{d} \qquad\text{and}\qquad \frac{a'}{d}.

Because

b,b′∼S,b,b'\sim S,

the number of solutions on that fibre is

O(1+SdR).\boxed{ O\left( 1+ \frac{Sd}{R} \right). }

Summing over the outer variables gives

∑a,a′∼R(1+S(a,a′)R).\sum_{a,a'\sim R} \left( 1+\frac{S(a,a')}{R} \right).

The classical gcd average satisfies

∑a,a′∼R(a,a′)≪R2log⁡R.\sum_{a,a'\sim R}(a,a') \ll R^2\log R.

Therefore

#{a,a′,b,b′:a′b′−ab=h}≪R2+RSlog⁡R.\boxed{ \#\{ a,a',b,b': a'b'-ab=h \} \ll R^2 + RS\log R. }

Since

R2≤RS≍X,R^2\le RS\asymp X,

we obtain

Theorem 11.1 — Factorisation-bipartition fibre-volume conservation

For every factor bipartition,

#{a,a′,b,b′:a′b′−ab=h}≪X1+o(1)\boxed{ \#\{ a,a',b,b': a'b'-ab=h \} \ll X^{1+o(1)} }

uniformly in the inherited shift range.

The same bound survives the structured convolution multiplicities because fixed-order divisor weights are Xo(1)X^{o(1)} on this support.

Create:

B-RH-028
FACTORISATION_BIPARTITION_DETERMINANT_FIBRE_VOLUME_CONSERVATION
CERTIFIED

This generalizes the one-coordinate determinant fibre already visible in Papers 43–45.

12. Why the apparent long fibre gives no geometric fixed power

For a generic coprime pair

(a,a′)=1,(a,a')=1,

a single fibre has length approximately

SR=XR2.\frac{S}{R} = \frac{X}{R^2}.

There are approximately

R2R^2

outer pairs.

Thus the leading geometric volume is

R2⋅XR2=X.\boxed{ R^2 \cdot \frac{X}{R^2} = X. }

The gcd average changes this only by a logarithm.

Hence splitting off a short convolution coordinate does create a long affine parameter, but the number of fibres grows by the reciprocal amount. At exponent resolution the two effects cancel exactly.

After restoring the normalization

1m(m+h)≍X−2,\frac1{m(m+h)} \asymp X^{-2},

one fixed shift has absolute geometric scale

X−1+o(1).X^{-1+o(1)}.

The number of relevant shifts is

Xo(1)XY.X^{o(1)}\frac{X}{Y}.

Multiplying by the outer factor YY returns

Xo(1).X^{o(1)}.

The desired bound is instead

X−3w/10+o(1).X^{-3w/10+o(1)}.

Thus the absolute-value determinant geometry still lands at the old orthogonality floor.

Create:

O-RH-119
ABSOLUTE_FACTOR_DECOUPLING_RETURNS_ORTHOGONALITY_FLOOR
CERTIFIED AS GEOMETRIC METHOD BARRIER

13. Preserving the signs returns genuine Mobius arithmetic

The signed factor split is not identical to the unsigned count.

For example, splitting off one factor position gives

cX=(μ1Ui)∗C−i.c_X = (\mu1_{U_i})*C_{-i}.

The fixed-shift correlation then contains sums of the form

∑a,a′∼Uib,b′∼X/Uia′b′−ab=hμ(a)μ(a′)C−i(b)C−i(b′).\sum_{\substack{ a,a'\sim U_i\\ b,b'\sim X/U_i\\ a'b'-ab=h }} \mu(a)\mu(a') C_{-i}(b)C_{-i}(b').

Therefore a successful improvement beyond Theorem 11.1 must exploit cancellation in the exposed Mobius variables or in a higher structured coupling.

If one takes absolute values or applies Cauchy-Schwarz until the Mobius signs disappear, Theorems 2.1–12.1 return the natural energy/incidence scale.

If one keeps the signs, the problem has not been decoupled into a purely weight-theoretic estimate. It has returned to a short-Mobius or affine-Liouville correlation problem.

Create:

O-RH-120
SIGNED_FACTOR_DECOUPLING_RETURNS_MOBIUS_PARITY_NOT_WEIGHT_SMALLNESS
CERTIFIED AS STRUCTURAL REDUCTION

This is not a theorem that all future factor-splitting arguments fail. It identifies the exact point at which WL3 stops being a structured-weight problem and becomes an arithmetic-cancellation problem again.

14. Current quantitative literature calibration

The present conclusion is compatible with the strongest nearby unconditional results.

Matomaki and Teravainen use Ramaré extraction to obtain cancellation of the Mobius function in all intervals of length xθx^\theta for θ>0.55\theta>0.55. The normalized conclusion is qualitative o(1)o(1), not a fixed power of the ambient variable.

Helfgott and Radziwill obtain strong expansion for a divisibility-by-primes graph. Its quantitative resource is the harmonic prime mass

L=∑p∈P1p,\mathscr L = \sum_{p\in\mathbf P}\frac1p,

and the resulting Liouville correlation gain is logarithmic.

Recent quantitative Gowers-uniformity and polynomial-pattern results give arbitrary powers of logarithm for broad Mobius and Liouville polynomial averages. This is a major strengthening of qualitative cancellation, but it is still not a fixed power of XX.

The 2026 growing-shift logarithmic Chowla results likewise give power-logarithmic cancellation, not the single-dyadic, structured-weight estimate

X−3w/10X^{-3w/10}

required here.

Accordingly, WL3 does not import any external theorem whose precision is weaker than the Campaign 43 admission ledger.

15. WL3 required-check audit

Required check 1 — separate smooth and rough carriers

Completed.

The smooth branch has

Esm=X1+o(1)E_{\mathrm{sm}} = X^{1+o(1)}

at exponent resolution.

The rough branch has

Erough=X1+o(1)E_{\mathrm{rough}} = X^{1+o(1)}

at exponent resolution.

Neither branch is power-negligible by coefficient mass or energy.

Required check 2 — preserve the exact convolution

Completed.

All energy identities are derived from

rX=g1∗⋯∗gLr_X = g_1*\cdots*g_L

without replacing rXr_X by a generic coefficient until an explicitly labelled divisor-moment upper bound.

Required check 3 — audit L2L^2 energy of rXr_X and marked weights

Completed.

∑rX2=X1+o(1).\sum r_X^2 = X^{1+o(1)}.

Polynomial-prime marking preserves rough-branch energy up to Oε,w(1)O_{\varepsilon,w}(1).

A fixed polynomial prime sub-band carries natural total marked energy

X1+o(1).X^{1+o(1)}.

Required check 4 — sign-coherent smooth subfamily

Completed.

There is an explicit nonnegative subweight sXs_X with

λ(n)=(−1)KL\lambda(n)=(-1)^{KL}

on its support and

∑sX(n)2≥X1−o(1).\sum s_X(n)^2 \ge X^{1-o(1)}.

Thus positivity or absolute-value decoupling cannot extract a fixed power from the smooth coefficient itself.

Required check 5 — no external XX -scale averaging

Satisfied.

Every estimate remains on the inherited single dyadic XX -scale.

Required check 6 — fixed-power ledger

Satisfied.

All coefficient-energy, mark-multiplicity, and determinant-volume gains are at most constants or logarithms at exponent resolution.

No step promotes

X−o(1)X^{-o(1)}

to

X−δ.X^{-\delta}.

Required check 7 — create a new long averaging variable

Audited.

A factor split creates affine fibres of length approximately

X/R2,X/R^2,

but also approximately R2R^2 outer fibres. The total geometric volume remains

X1+o(1).X^{1+o(1)}.

No fixed-power gain is obtained without using genuine Mobius/Liouville cancellation.

16. WL3 rejection list

The following routes are rejected inside WL3.

R1. Treat rXr_X as L2L^2 -small

Rejected by Theorem 2.1.

R2. Discard the smooth carrier as sparse

Rejected by Theorems 3.1 and 4.1.

R3. Assume the rough carrier is sparse after removing common and repeated large primes

Rejected at coefficient-energy level by Theorem 5.1 and Corollary 5.2.

R4. Expect polynomial-prime marking to create polynomially many orthogonal pieces per coefficient

Rejected by Theorem 6.1.

R5. Sum the marked pieces and claim a P⋆−cP_\star^{-c} energy gain

Rejected by Theorem 7.1.

R6. Split one or several convolution factors and count determinant fibres absolutely

Rejected by Theorems 11.1 and 12.1.

R7. Remove the exposed Mobius signs and still claim the signed problem was solved

Rejected by Section 13.

R8. Use power-logarithmic literature as a fixed- XX -power theorem

Rejected by the admission ledger.

17. WL3 closure

WL3 has produced a complete coefficient-side and factorisation-side audit.

The exact convolution structure is useful for organizing the residual problem, but it does not contain a hidden fixed-power reserve of any of the following forms:

  1. low total L2L^2 energy;
  2. power-sparse smooth support;
  3. power-sparse rough support;
  4. growing orthogonal multiplicity from polynomial-prime marks;
  5. absolute determinant-fibre volume reduction.

The only remaining possible gain after a legal factor split must use arithmetic cancellation that survives the full structured coupling.

Thus WL3 closes as

WL3
CLOSED_AS_NATURAL_L2_SATURATION_BOUNDED_MARK_MULTIPLICITY_AND_DETERMINANT_CYCLE_BARRIER

Campaign 43 remains active.

No fixed-power theorem is proved.

18. Campaign 43 continuation

The next live track is WL4:

CROSS_J_PRE_SQUARE_RECOMBINATION

This route returns to the exact Heath-Brown identity before componentwise triangle inequalities and before componentwise Type-II squaring.

The Heath-Brown levels carry alternating coefficients

(−1)j−1(Lj).(-1)^{j-1}\binom Lj.

Paper 41 explicitly left open the possibility that cross- jj recombination suppresses the pure-Mobius core before componentwise estimation.

WL4 must therefore determine whether the Paper 41 pure-core component is an artifact of estimating each jj -level separately or whether it survives any legal pre-square recombination.

Required WL4 checks:

  1. reconstruct the relevant Heath-Brown jj -levels before componentwise absolute values;
  2. identify which dyadic pure-Mobius components at different jj have overlapping product support;
  3. compute cross- jj terms at the translated frequency rather than assuming cancellation from alternating signs;
  4. preserve the single dyadic XX -scale and the major-arc (Q,Y)(Q,Y) geometry;
  5. quantify any overlap or cancellation with a fixed XX -power ledger;
  6. reject cancellation that appears only after replacing unequal dyadic components by identical formal symbols;
  7. if recombination reconstructs a simpler Lambda-level object, compare it directly with the root frontier FF -RH-010 rather than hiding the recoupling.

19. State transition

The canonical state advances

CSM_RH v1.36
  ->
CSM_RH v1.37

Create:

B-RH-025
PURE_CORE_STRUCTURED_WEIGHT_NATURAL_L2_SCALE
CERTIFIED

B-RH-026
SMOOTH_AND_ROUGH_CARRIERS_BOTH_HAVE_FULL_EXPONENT_WEIGHT_ENERGY
CERTIFIED

B-RH-027
POLYNOMIAL_PRIME_MARKING_BOUNDED_MULTIPLICITY_ENERGY_STABILITY
CERTIFIED

B-RH-028
FACTORISATION_BIPARTITION_DETERMINANT_FIBRE_VOLUME_CONSERVATION
CERTIFIED

Create:

O-RH-117
POLYNOMIAL_PRIME_MARKING_HAS_NO_GROWING_ORTHOGONAL_MULTIPLICITY
CERTIFIED AS MECHANISM-SCOPE BARRIER

O-RH-118
STRUCTURED_WEIGHT_L2_ENERGY_HAS_NO_FIXED_POWER_RESERVE
CERTIFIED AS COEFFICIENT-ENERGY METHOD BARRIER

O-RH-119
ABSOLUTE_FACTOR_DECOUPLING_RETURNS_ORTHOGONALITY_FLOOR
CERTIFIED AS GEOMETRIC METHOD BARRIER

O-RH-120
SIGNED_FACTOR_DECOUPLING_RETURNS_MOBIUS_PARITY_NOT_WEIGHT_SMALLNESS
CERTIFIED AS STRUCTURAL REDUCTION

Campaign 43:

WL1:
  CLOSED_AS_ARCHIMEDEAN_GAUGE_AND_EVEN_ORDER_LIOUVILLE_BARRIER

WL2:
  CLOSED_AS_EXACT_MARKED_EXTRACTION_WITH_PRIME_HARMONIC_AND_SMOOTH_CARRIER_BARRIERS

WL3:
  CLOSED_AS_NATURAL_L2_SATURATION_BOUNDED_MARK_MULTIPLICITY_AND_DETERMINANT_CYCLE_BARRIER

WL4:
  NEXT

WL5:
  OPEN

No root promotion occurs.

RH_PROVED=false\boxed{ \mathrm{RH\_PROVED} = \mathrm{false} } RH_DISPROVED=false\boxed{ \mathrm{RH\_DISPROVED} = \mathrm{false} } GLOBAL_RH_CERTIFICATE=false\boxed{ \mathrm{GLOBAL\_RH\_CERTIFICATE} = \mathrm{false} }

20. References and external calibration

  1. K. Matomaki and J. Teravainen, On the Mobius function in all short intervals, J. Eur. Math. Soc. 25 (2023), 1207-1225, arXiv:1911.09076. Ramaré prime extraction produces a strong structural factorisation, while the normalized short-interval conclusion is qualitative rather than a fixed power of the ambient scale.

  2. H. A. Helfgott and M. Radziwill, Expansion, divisibility and parity, arXiv:2103.06853. The prime-divisibility graph is controlled by the harmonic mass L=∑1/p\mathscr L=\sum 1/p and yields logarithmic-scale Liouville correlation improvements.

  3. L. Matthiesen, Quantitative asymptotics for polynomial patterns in the primes, Mathematika (2026). The Mobius/Liouville polynomial-pattern estimates save arbitrary powers of logarithm; this remains weaker than the fixed- XX -power admission target in Campaign 43.

  4. J. Guo, Logarithmic Chowla Correlations Across All Shift Scales, arXiv:2608.23500, version current in September 2026. The result gives power-logarithmic logarithmically weighted two-point Liouville cancellation across shifts, not the present single-dyadic structured-weight fixed- XX -power estimate.

  5. O. Gorodetsky, The variance of integers without small prime factors in short intervals, Math. Z. 308 (2024), Article 59. Its smooth-number analysis explicitly involves squarefree smooth counts Ψμ2(x,y)\Psi_{\mu^2}(x,y), consistent with the fact that fixed-power smoothness thresholds need not create power-sparse coefficient families.

21. Final status block

CSM_RH PAPER 46

WL3 STRUCTURED WEIGHT DECOUPLING = CLOSED

TOTAL r_X L2 ENERGY = X^(1+o(1))

SMOOTH CARRIER ENERGY = X^(1+o(1)) AT EXPONENT RESOLUTION

ROUGH CARRIER ENERGY = X^(1+o(1)) AT EXPONENT RESOLUTION

POLYNOMIAL PRIME MARK MULTIPLICITY = O_{epsilon,w}(1)

MARKED ENERGY = NATURAL SCALE

FACTOR SPLIT = DETERMINANT FIBRE VOLUME X^(1+o(1))

ABSOLUTE / ENERGY-ONLY DECOUPLING = NO FIXED X POWER

SIGNED DECOUPLING = RETURNS TO MOBIUS / LIOUVILLE ARITHMETIC

CAMPAIGN 43 = ACTIVE

NEXT = WL4 CROSS_J_PRE_SQUARE_RECOMBINATION

RH = OPEN