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CSM_RH Paper 80 — Cutoff-Flow Spectral Charge Conservation, Empty-Divisor Hard-Core Localization, and Closure of the Buc

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CSM_RH Paper 80

Cutoff-Flow Spectral Charge Conservation, Empty-Divisor Hard-Core Localization, and Closure of the Buchstab-Telescoping Route

Project: CSM_RH
Paper: 80
Version: v0.1
Date: 2026-09-09
Campaign: 47 — ORDINARY_PRIME_BOUNDARY_BREAKING
Entry state: v1.70 / Paper 79 v0.1
Status: BUCHSTAB CUTOFF-FLOW AUDITED / EMPTY-DIVISOR HARD CORE CERTIFIED / F-RH-024 AND F-RH-025 REDUCED TO ROOT COVARIANCE PLUS POWER-SMALL ENTIRE CORRECTION
RH_PROVED: FALSE
RH_DISPROVED: FALSE
GLOBAL_RH_CERTIFICATE: FALSE

Abstract

Paper 79 showed that the rightmost-zero resonance of the renormalized Vaughan coefficient is distributed across the logarithmic prime-factor scale

e≍Nθ.e\asymp N^\theta.

It then proposed decomposing the truncated Möbius coefficient

bU(k)=∑d∣kd≤Uμ(d)b_U(k) = \sum_{\substack{d\mid k\\d\le U}} \mu(d)

by Buchstab / divisor-size increments and testing whether cross-scale arithmetic telescoping could generate an additional fixed-power cancellation.

The present paper shows that this route cannot create a new spectral amplifier.

Define

MU(s)=∑d≤Uμ(d)dsM_U(s) = \sum_{d\le U} \frac{\mu(d)}{d^s}

and

BU(s)=∑k>UbU(k)ks=ζ(s)MU(s)−1.\boxed{ B_U(s) = \sum_{k>U} \frac{b_U(k)}{k^s} = \zeta(s)M_U(s)-1. }

For any

U1<U2,U_1<U_2,

one has the exact cutoff-flow identity

BU2(s)−BU1(s)=ζ(s)∑U1<d≤U2μ(d)ds.\boxed{ B_{U_2}(s)-B_{U_1}(s) = \zeta(s) \sum_{U_1<d\le U_2} \frac{\mu(d)}{d^s}. }

Therefore, at every nontrivial zeta zero ρ\rho,

BU2(ρ)−BU1(ρ)=0.\boxed{ B_{U_2}(\rho)-B_{U_1}(\rho)=0. }

Since

BU(ρ)=−1B_U(\rho)=-1

for one, hence every, finite U>1U>1,

BU(ρ)≡−1\boxed{ B_U(\rho)\equiv-1 }

along the entire truncated Möbius cutoff flow.

This is a spectral charge-conservation law.

It has a sharper depth decomposition.

Write

MU(s)=1+RU(s),M_U(s) = 1+R_U(s),

where

RU(s)=∑2≤d≤Uμ(d)ds.R_U(s) = \sum_{2\le d\le U} \frac{\mu(d)}{d^s}.

Then

BU(s)=(ζ(s)−1)+ζ(s)RU(s).\boxed{ B_U(s) = \left( \zeta(s)-1 \right) + \zeta(s)R_U(s). }

At a zeta zero ρ\rho,

ζ(ρ)−1=−1,\zeta(\rho)-1=-1,

while

ζ(ρ)RU(ρ)=0.\zeta(\rho)R_U(\rho)=0.

Thus the universal value

BU(ρ)=−1B_U(\rho)=-1

is carried entirely by the empty-divisor branch

d=1.d=1.

Every nonempty Möbius-divisor correction vanishes at the zero.

Now let

AV(s)=−ζ′(s)ζ(s)−LV(s),A_V(s) = -\frac{\zeta'(s)}{\zeta(s)} - L_V(s),

where

LV(s)=∑e≤VΛ(e)es.L_V(s) = \sum_{e\le V} \frac{\Lambda(e)}{e^s}.

The Vaughan balanced coefficient has Dirichlet series

CU,V(s)=AV(s)BU(s).C_{U,V}(s) = A_V(s)B_U(s).

The empty-divisor decomposition gives

CU,V(s)=AV(s)(ζ(s)−1)+(−ζ′(s)−LV(s)ζ(s))RU(s).\boxed{ C_{U,V}(s) = A_V(s) \left( \zeta(s)-1 \right) + \left( -\zeta'(s)-L_V(s)\zeta(s) \right) R_U(s). }

The second term is analytic at every nontrivial zeta zero.

Hence:

all nontrivial zero poles of the Vaughan balanced coefficient are already present in the d=1 branch.\boxed{ \text{all nontrivial zero poles of the Vaughan balanced coefficient are already present in the }d=1\text{ branch.} }

Changing UU, splitting UU into Buchstab shells, or recursively decomposing nonempty divisors can only modify the zero-pole-free regular sector.

This is the divisor-depth analogue of Paper 79's factor-scale resonance theorem:

prime-factor scale:
rightmost resonance is distributed across log e;

Möbius-divisor depth:
rightmost spectral charge is localized at the empty divisor d=1.

The paper then strengthens the conclusion using Paper 78's exact spectral split.

Recall

HU,V(s)=ζ′(s)ζ(s)+EU,V(s),H_{U,V}(s) = \frac{\zeta'(s)}{\zeta(s)} + E_{U,V}(s),

where EU,VE_{U,V} is analytic at every zeta zero and has residue +1+1 at s=1s=1.

Define

RU,V(s)=EU,V(s)−ζ(s).\boxed{ R_{U,V}(s) = E_{U,V}(s)-\zeta(s). }

Because the +1+1 residue at s=1s=1 is cancelled by ζ(s)\zeta(s) and EU,VE_{U,V} has no zero poles,

RU,V(s) has an entire analytic continuation.\boxed{ R_{U,V}(s) \text{ has an entire analytic continuation.} }

Coefficientwise,

hU,V(n)=−(Λ(n)−1)+rU,V(n),\boxed{ h_{U,V}(n) = -\left( \Lambda(n)-1 \right) + r_{U,V}(n), }

where the Dirichlet series of rU,Vr_{U,V} has entire continuation.

The explicit entire correction is

RU,V(s)=ζ′(s)(MU−MU(s))+LV(s)(1−ζ(s)MU(s))+(MULV+JU)ζ(s).\boxed{ \begin{aligned} R_{U,V}(s) &= \zeta'(s) \left( M_U-M_U(s) \right) \\ &\quad + L_V(s) \left( 1-\zeta(s)M_U(s) \right) \\ &\quad + \left( M_UL_V+J_U \right) \zeta(s). \end{aligned} }

Paper 77's exact extraction identity becomes

∑nfN,H(n)rU,V(n)=EI,\boxed{ \sum_n f_{N,H}(n) r_{U,V}(n) = \mathcal E_I, }

with

EI≪NUV(log⁡N)O(1).\boxed{ \mathcal E_I \ll NUV(\log N)^{O(1)}. }

Therefore

VU,Vren=−RW+EI.\boxed{ \mathcal V^{\rm ren}_{U,V} = -\mathcal R_W + \mathcal E_I. }

This is not merely a formal bookkeeping identity.

It is a hard-core decomposition:

  • the entire correction is already Type-I power-controlled;
  • the only non-entire zero-pole sector is exactly the negative original prime-error coefficient.

Consequently F-RH-024 is not an independent new root inequality.

Once the Type-I parameter gate makes EI\mathcal E_I supercritical, a fixed-power estimate for

VU,Vren\mathcal V^{\rm ren}_{U,V}

is equivalent at exponent resolution to the same fixed-power estimate for the original root covariance

RW.\mathcal R_W.

Likewise, F-RH-025 cannot obtain a new gain by Buchstab telescoping of bUb_U.

The nonempty divisor shells live in the entire / zero-pole-free correction sector and their accumulated correlation is exactly part of the already-controlled Type-I error.

The remaining hard term is the empty-divisor prime-error branch.

There is also an exact squarefree complementary-divisor identity.

If k>1k>1 is squarefree, then

∑d∣kμ(d)=0.\sum_{d\mid k}\mu(d)=0.

Thus

bU(k)=−μ(k)∑q∣kq<k/Uμ(q).\boxed{ b_U(k) = - \mu(k) \sum_{\substack{q\mid k\\q<k/U}} \mu(q). }

Moving the cutoff from small divisors to complementary divisors therefore does not eliminate parity.

It transports it and multiplies by the global sign μ(k)\mu(k).

This is the combinatorial analogue of cutoff-flow spectral conservation.

The campaign consequence is a structural closure of the first Campaign-47 route:

F-RH-023:
auxiliary parity statement only;

F-RH-024:
root covariance plus known entire correction;

F-RH-025:
Buchstab / factor-scale reformulation of the same root hard core.

Campaign 47 produced valuable exact structure:

  • deterministic divisor laws after shift averaging;
  • a power-preserving Vaughan extraction;
  • the exact counterterm;
  • the parameter-invariant zero-pole hard core.

But it did not generate a strict exponent amplifier.

Any next campaign should therefore adopt a stronger entry test:

Before developing a new linear arithmetic transform, compute its zeta-zero pole projector.

Reject the candidate as a new amplifier if its hard coefficient is $$ c(\Lambda-1)+\text{zero-pole-free correction}, $$ with c≠0c\ne0.

A genuinely new route must be nonlinear, multi-copy, or otherwise use ordinary-prime information which is not spectrally equivalent to a single prime-error coefficient plus an entire correction.

No RH theorem is claimed.

1. Truncated Möbius cutoff flow

Define

bU(k)=∑d∣kd≤Uμ(d).\boxed{ b_U(k) = \sum_{\substack{d\mid k\\d\le U}} \mu(d). }

For 1<k≤U1<k\le U,

bU(k)=∑d∣kμ(d)=0.b_U(k) = \sum_{d\mid k}\mu(d) = 0.

Also

bU(1)=1.b_U(1)=1.

Therefore

∑k>UbU(k)ks=∑k≥1bU(k)ks−1=ζ(s)MU(s)−1.\begin{aligned} \sum_{k>U} \frac{b_U(k)}{k^s} &= \sum_{k\ge1} \frac{b_U(k)}{k^s} - 1 \\ &= \boxed{ \zeta(s)M_U(s)-1. } \end{aligned}

This is valid initially for ℜs>1\Re s>1 and then by meromorphic continuation.

2. Cutoff increment identity

Let

U1<U2.U_1<U_2.

Then

MU2(s)−MU1(s)=∑U1<d≤U2μ(d)ds.M_{U_2}(s)-M_{U_1}(s) = \sum_{U_1<d\le U_2} \frac{\mu(d)}{d^s}.

Hence:

Theorem 2.1 — Möbius cutoff-flow identity

BU2(s)−BU1(s)=ζ(s)∑U1<d≤U2μ(d)ds.\boxed{ B_{U_2}(s)-B_{U_1}(s) = \zeta(s) \sum_{U_1<d\le U_2} \frac{\mu(d)}{d^s}. }

Create:

B-RH-131
TRUNCATED_MOBIUS_CUTOFF_INCREMENTS_CARRY_AN_EXPLICIT_ZETA_FACTOR
CERTIFIED

3. Spectral charge conservation

Let ρ\rho be any nontrivial zero of ζ\zeta.

Theorem 2.1 gives

BU2(ρ)−BU1(ρ)=0.\boxed{ B_{U_2}(\rho)-B_{U_1}(\rho)=0. }

Since

BU(ρ)=ζ(ρ)MU(ρ)−1=−1,B_U(\rho) = \zeta(\rho)M_U(\rho)-1 = -1,

for every finite UU,

Theorem 3.1 — Cutoff-flow zero-charge conservation

BU(ρ)≡−1.\boxed{ B_U(\rho)\equiv-1. }

Create:

B-RH-132
THE_VAUGHAN_COFACTOR_ZERO_VALUE_IS_A_CONSERVED_QUANTITY_ALONG_THE_MOBIUS_CUTOFF_FLOW
CERTIFIED

The value does not drift, diffuse, or telescope with UU.

4. Empty-divisor localization

Write

MU(s)=1+RU(s),M_U(s) = 1+R_U(s), RU(s)=∑2≤d≤Uμ(d)d−s.R_U(s) = \sum_{2\le d\le U} \mu(d)d^{-s}.

Then

BU(s)=ζ(s)−1+ζ(s)RU(s).\boxed{ B_U(s) = \zeta(s)-1 + \zeta(s)R_U(s). }

At a zeta zero,

(ζ(ρ)−1)=−1,\boxed{ \left( \zeta(\rho)-1 \right) =-1, } ζ(ρ)RU(ρ)=0.\boxed{ \zeta(\rho)R_U(\rho)=0. }

Thus:

Theorem 4.1 — Empty-divisor hard-core localization

The entire zero value BU(ρ)=−1B_U(\rho)=-1 is carried by the divisor d=1d=1 branch.

Every nonempty Möbius-divisor branch has zero spectral charge at ρ\rho.

Create:

B-RH-133
THE_UNIVERSAL_VAUGHAN_ZERO_CHARGE_IS_LOCALIZED_AT_THE_EMPTY_DIVISOR_BRANCH
CERTIFIED

5. Balanced coefficient depth decomposition

Define

AV(s)=−ζ′(s)ζ(s)−LV(s).A_V(s) = -\frac{\zeta'(s)}{\zeta(s)} - L_V(s).

Then

CU,V(s)=AV(s)BU(s).C_{U,V}(s) = A_V(s)B_U(s).

Use Theorem 4.1:

CU,V(s)=AV(s)(ζ(s)−1)+AV(s)ζ(s)RU(s).\boxed{ \begin{aligned} C_{U,V}(s) &= A_V(s) \left( \zeta(s)-1 \right) \\ &\quad + A_V(s)\zeta(s)R_U(s). \end{aligned} }

But

AV(s)ζ(s)=−ζ′(s)−LV(s)ζ(s),A_V(s)\zeta(s) = -\zeta'(s) - L_V(s)\zeta(s),

which is analytic at every nontrivial zero.

Hence:

Theorem 5.1 — All zero poles occur at divisor depth zero

CU,V(s)=AV(s)(ζ(s)−1)+zero-pole-free divisor-depth correction.\boxed{ C_{U,V}(s) = A_V(s)(\zeta(s)-1) + \text{zero-pole-free divisor-depth correction}. }

Create:

B-RH-134
ALL_NONTRIVIAL_ZERO_POLES_OF_THE_VAUGHAN_BALANCED_COEFFICIENT_ALREADY_OCCUR_IN_THE_D_EQUALS_ONE_BRANCH
CERTIFIED

6. Buchstab-shell increments cannot alter the zero residue

A shell increment

ΔU1,U2M(s)=∑U1<d≤U2μ(d)ds\Delta_{U_1,U_2}M(s) = \sum_{U_1<d\le U_2} \frac{\mu(d)}{d^s}

changes the balanced coefficient by

ΔC(s)=(−ζ′(s)−LV(s)ζ(s))ΔU1,U2M(s).\boxed{ \Delta C(s) = \left( -\zeta'(s)-L_V(s)\zeta(s) \right) \Delta_{U_1,U_2}M(s). }

This is analytic at every zeta zero.

Therefore every divisor-size shell, and every finite recursive grouping of such shells, has zero nontrivial-zero pole residue.

Create:

O-RH-178
BUCHSTAB_OR_DIVISOR_SIZE_SHELLS_CANNOT_CHANGE_THE_UNIVERSAL_RIGHTMOST_ZERO_RESIDUE
CERTIFIED

This is the precise obstruction to F-RH-025's proposed cross-scale telescoping mechanism.

7. Squarefree complementary-divisor duality

Assume k>1k>1 is squarefree.

Then

∑d∣kμ(d)=0.\sum_{d\mid k}\mu(d)=0.

Therefore

bU(k)=−∑d∣kd>Uμ(d).b_U(k) = - \sum_{\substack{d\mid k\\d>U}} \mu(d).

Set

q=k/d.q=k/d.

Because kk is squarefree,

μ(d)=μ(k)μ(q).\mu(d) = \mu(k)\mu(q).

The condition d>Ud>U is

q<k/U.q<k/U.

Hence:

Theorem 7.1 — Squarefree complementary parity transport

bU(k)=−μ(k)∑q∣kq<k/Uμ(q).\boxed{ b_U(k) = - \mu(k) \sum_{\substack{q\mid k\\q<k/U}} \mu(q). }

Create:

B-RH-135
TRUNCATED_MOBIUS_PARITY_DEFECT_IS_TRANSPORTED_TO_THE_COMPLEMENTARY_DIVISOR_SCALE_ON_SQUAREFREE_INTEGERS
CERTIFIED

Moving the cutoff does not destroy parity; it relocates it.

8. Entire correction theorem

Paper 78 gave

HU,V(s)=ζ′(s)ζ(s)+EU,V(s),H_{U,V}(s) = \frac{\zeta'(s)}{\zeta(s)} + E_{U,V}(s),

with

EU,V(s)=ζ′(s)(MU−MU(s))+LV(s)(1−ζ(s)MU(s))+(MULV+JU+1)ζ(s).\boxed{ \begin{aligned} E_{U,V}(s) &= \zeta'(s) \left( M_U-M_U(s) \right) \\ &\quad + L_V(s) \left( 1-\zeta(s)M_U(s) \right) \\ &\quad + \left( M_UL_V+J_U+1 \right) \zeta(s). \end{aligned} }

Define

RU,V(s)=EU,V(s)−ζ(s).\boxed{ R_{U,V}(s) = E_{U,V}(s)-\zeta(s). }

Then

RU,V(s)=ζ′(s)(MU−MU(s))+LV(s)(1−ζ(s)MU(s))+(MULV+JU)ζ(s).\boxed{ \begin{aligned} R_{U,V}(s) &= \zeta'(s) \left( M_U-M_U(s) \right) \\ &\quad + L_V(s) \left( 1-\zeta(s)M_U(s) \right) \\ &\quad + \left( M_UL_V+J_U \right) \zeta(s). \end{aligned} }

At every zeta zero, all terms are regular.

At s=1s=1, the simple-pole residues cancel:

  • the first term contributes −JU-J_U ;
  • the second contributes −MULV-M_UL_V ;
  • the third contributes MULV+JUM_UL_V+J_U.

Thus:

Theorem 8.1 — Entire Type-I correction

RU,V(s)\boxed{ R_{U,V}(s) }

admits an entire analytic continuation.

Create:

B-RH-136
THE_RENORMALIZED_VAUGHAN_REGULAR_SECTOR_MINUS_ITS_MEAN_HAS_ENTIRE_CONTINUATION
CERTIFIED

9. Coefficient hard-core decomposition

Let

RU,V(s)=∑n≥1rU,V(n)nsR_{U,V}(s) = \sum_{n\ge1} \frac{r_{U,V}(n)}{n^s}

initially in its half-plane of Dirichlet convergence.

Since

HU,V=ζ′ζ+ζ+RU,V,H_{U,V} = \frac{\zeta'}{\zeta} + \zeta + R_{U,V},

and

ζ′ζ=−∑nΛ(n)ns,\frac{\zeta'}{\zeta} = -\sum_n \frac{\Lambda(n)}{n^s},

one has:

Theorem 9.1 — Prime-error hard-core decomposition

hU,V(n)=−(Λ(n)−1)+rU,V(n).\boxed{ h_{U,V}(n) = -\left( \Lambda(n)-1 \right) + r_{U,V}(n). }

The correction rU,Vr_{U,V} has entire Dirichlet-series continuation.

Create:

B-RH-137
RENORMALIZED_VAUGHAN_COEFFICIENT_EQUALS_NEGATIVE_PRIME_ERROR_PLUS_AN_ENTIRE_CORRECTION
CERTIFIED

10. Correlation of the entire correction is already power-controlled

Paper 77 gave

VU,Vren=−RW+EI.\mathcal V^{\rm ren}_{U,V} = -\mathcal R_W + \mathcal E_I.

On the other hand, Theorem 9.1 gives

VU,Vren=∑nf(n)hU,V(n)=−RW+∑nf(n)rU,V(n).\begin{aligned} \mathcal V^{\rm ren}_{U,V} &= \sum_n f(n)h_{U,V}(n) \\ &= -\mathcal R_W + \sum_n f(n)r_{U,V}(n). \end{aligned}

Therefore:

Theorem 10.1 — Entire correction correlation identity

∑nfN,H(n)rU,V(n)=EI.\boxed{ \sum_n f_{N,H}(n)r_{U,V}(n) = \mathcal E_I. }

Since

EI≪NUV(log⁡N)O(1),\boxed{ \mathcal E_I \ll NUV(\log N)^{O(1)}, }

the zero-pole-free correction is already controlled at the deterministic Type-I exponent.

Create:

B-RH-138
THE_ENTIRE_VAUGHAN_CORRECTION_PAIRS_WITH_THE_SIGNED_ROOT_SEQUENCE_ONLY_THROUGH_THE_ALREADY_CONTROLLED_TYPE_I_ERROR
CERTIFIED

11. F-RH-024 equivalence at exponent resolution

Assume

H=N1−τ,H=N^{1-\tau}, U=Nu,U=N^u, V=Nv,V=N^v,

and choose parameters such that

1−τ−u−v>κ+η.\boxed{ 1-\tau-u-v > \kappa+\eta. }

Then

EI≪NHN−κ−η+o(1).\mathcal E_I \ll NHN^{-\kappa-\eta+o(1)}.

Hence

VU,Vren=−RW+opower(NHN−κ).\boxed{ \mathcal V^{\rm ren}_{U,V} = -\mathcal R_W + o_{\rm power} \left( NHN^{-\kappa} \right). }

Therefore:

Corollary 11.1 — F-RH-024 root equivalence

At every parameter choice for which the Type-I correction is supercritical,

VU,Vren≪NHN−κ−η\boxed{ \mathcal V^{\rm ren}_{U,V} \ll NHN^{-\kappa-\eta} }

is equivalent at fixed-power resolution to

RW≪NHN−κ−η.\boxed{ \mathcal R_W \ll NHN^{-\kappa-\eta}. }

F-RH-024 is therefore not an independent amplifier input.

Update:

F-RH-024
CLOSED_AS_ROOT_EQUIVALENT_REPRESENTATION

12. Consequence for F-RH-025

F-RH-025 proposed gaining cancellation from factor-scale / Buchstab decomposition of the Vaughan cofactor.

Theorems 3.1, 5.1 and 6 show:

  • the zero charge is invariant under cutoff motion;
  • the hard pole is already in the empty-divisor branch;
  • every nonempty divisor shell is zero-pole-free.

Theorem 10.1 further shows that the full zero-pole-free correction contributes only the already-controlled Type-I error.

Therefore:

Corollary 12.1 — Buchstab telescoping route closure

Cross-scale telescoping of the nonempty Möbius-divisor anatomy cannot generate a new fixed-power root gain within this exact Vaughan architecture.

Update:

F-RH-025
CLOSED_AS_NONAMPLIFYING_BUCHSTAB_REORGANIZATION

The distributed prime-factor resonance of Paper 79 remains correct, but it is carried by the depth-zero prime-error hard core.

13. Relation to classical parity theory

Friedlander and Iwaniec's asymptotic sieve breaks the classical parity obstruction by adding a Möbius-sensitive bilinear hypothesis.

Ford and Maynard's later general theory proves that substantial Type-II information is genuinely necessary for arbitrary nonnegative prime-producing sequences.

These results remain important calibrations.

The present conclusion is narrower:

for the specific Campaign-47 signed root sequence and its exact Vaughan renormalization, the nonempty truncated-Möbius anatomy contributes only to a zero-pole-free correction whose root pairing has already been estimated by Type-I divisor discrepancy.

Thus classical parity sensitivity does not automatically become a new PESC amplifier in this specialized construction.

External calibration:

  • Friedlander–Iwaniec, Asymptotic sieve for primes;
  • Ford–Maynard, On the theory of prime producing sieves.

14. Campaign-47 structural verdict

Campaign 47 began with a stronger entry rule than Campaign 46:

explicit q=1 fixed-power inequality first;
framework second.

It produced:

  1. deterministic shift-averaged divisor distribution;
  2. the signed root sequence;
  3. a power-preserving exact Vaughan extraction;
  4. the coefficientwise renormalization;
  5. universal zero-pole preservation;
  6. distributed prime-factor resonance;
  7. cutoff-flow zero-charge conservation;
  8. empty-divisor hard-core localization;
  9. an entire correction whose root correlation is already Type-I small.

The final exact decomposition is

renormalized Vaughan defect=−original root covariance+power-small entire correction.\boxed{ \text{renormalized Vaughan defect} = -\text{original root covariance} + \text{power-small entire correction}. }

Hence the Campaign-47 Vaughan/Buchstab route does not supply an independent fixed-power arithmetic inequality.

Record:

CAMPAIGN_47_VAUGHAN_BUCHSTAB_BRANCH
STRUCTURALLY_CLOSED_AT_EMPTY_DIVISOR_HARD_CORE

This does not close the root problem itself.

15. Entry rule for a future campaign

Any new linear arithmetic transform g(n)g(n) should first be tested spectrally.

Let

G(s)=∑ng(n)n−s.G(s) = \sum_n g(n)n^{-s}.

If one can write

G(s)=c ζ′(s)ζ(s)+E(s),\boxed{ G(s) = c\, \frac{\zeta'(s)}{\zeta(s)} + E(s), }

with

c≠0c\ne0

and EE free of nontrivial zeta-zero poles, then gg carries the same one-point prime-error hard core.

Such a transform should be rejected as a new amplifier representation unless it introduces a genuinely new nonlinear inequality.

A future campaign should therefore prioritize:

nonlinear / multi-copy ordinary-prime boundary breaking

rather than another linear convolution or truncated-factorization transform.

16. External calibration

16.1. Friedlander–Iwaniec

J. Friedlander and H. Iwaniec, Asymptotic sieve for primes, Annals of Mathematics 148 (1998), 1041–1065.

They introduce a Möbius-sensitive bilinear axiom specifically to break the sieve parity problem.

URL:

https://arxiv.org/abs/math/9811186

16.2. Ford–Maynard

K. Ford and J. Maynard, On the theory of prime producing sieves, 2024.

They establish a general Type-I / Type-II framework and prove that a substantial Type-II range is always necessary to guarantee a nontrivial lower bound for primes in an arbitrary nonnegative sequence.

URL:

https://arxiv.org/abs/2407.14368

These results justify treating parity-sensitive bilinear information as genuine arithmetic input, while the present paper shows that the specific truncated-Möbius anatomy of Campaign 47 does not escape the root hard core.

17. State transition

Advance candidate state

v1.70→v1.71.v1.70 \to v1.71.

Add:

B-RH-131
TRUNCATED_MOBIUS_CUTOFF_INCREMENTS_CARRY_AN_EXPLICIT_ZETA_FACTOR

B-RH-132
THE_VAUGHAN_COFACTOR_ZERO_VALUE_IS_A_CONSERVED_QUANTITY_ALONG_THE_MOBIUS_CUTOFF_FLOW

B-RH-133
THE_UNIVERSAL_VAUGHAN_ZERO_CHARGE_IS_LOCALIZED_AT_THE_EMPTY_DIVISOR_BRANCH

B-RH-134
ALL_NONTRIVIAL_ZERO_POLES_OF_THE_VAUGHAN_BALANCED_COEFFICIENT_ALREADY_OCCUR_IN_THE_D_EQUALS_ONE_BRANCH

B-RH-135
TRUNCATED_MOBIUS_PARITY_DEFECT_IS_TRANSPORTED_TO_THE_COMPLEMENTARY_DIVISOR_SCALE_ON_SQUAREFREE_INTEGERS

B-RH-136
THE_RENORMALIZED_VAUGHAN_REGULAR_SECTOR_MINUS_ITS_MEAN_HAS_ENTIRE_CONTINUATION

B-RH-137
RENORMALIZED_VAUGHAN_COEFFICIENT_EQUALS_NEGATIVE_PRIME_ERROR_PLUS_AN_ENTIRE_CORRECTION

B-RH-138
THE_ENTIRE_VAUGHAN_CORRECTION_PAIRS_WITH_THE_SIGNED_ROOT_SEQUENCE_ONLY_THROUGH_THE_ALREADY_CONTROLLED_TYPE_I_ERROR

O-RH-178
BUCHSTAB_OR_DIVISOR_SIZE_SHELLS_CANNOT_CHANGE_THE_UNIVERSAL_RIGHTMOST_ZERO_RESIDUE

Update:

F-RH-024
CLOSED_AS_ROOT_EQUIVALENT_REPRESENTATION

F-RH-025
CLOSED_AS_NONAMPLIFYING_BUCHSTAB_REORGANIZATION

No RH certificate is created.

18. Conclusion

The truncated Möbius cutoff has a conserved zeta-zero charge.

That charge is not distributed among the nonempty divisor shells.

It sits entirely in the empty-divisor branch.

All Buchstab cutoff increments are zero-pole-free.

After the exact Vaughan renormalization, the remaining coefficient is exactly the negative prime error plus an entire correction whose pairing with the signed root sequence is already power-controlled.

Therefore the Campaign-47 Vaughan/Buchstab route has returned exactly to the original root covariance.

The next genuinely new route must be nonlinear or multi-copy.