CSM_RH Paper 78

Universal Zero-Pole Preservation in the Renormalized Vaughan Coefficient and the Polynomial-Factor Anatomy Barrier

Project: CSM_RH
Paper: 78
Version: v0.1
Date: 2026-09-09
Campaign: 47 — ORDINARY_PRIME_BOUNDARY_BREAKING
Active frontier: F-RH-024 — RENORMALIZED_VAUGHAN_BALANCED_DEFECT_POWER
Status: RENORMALIZED COEFFICIENT ZERO SPECTRUM AUDITED / PARAMETER-INVARIANT ROOT-HARD CORE CERTIFIED / GENERIC TITCHMARSH AND SMALL-FACTOR ANATOMY ROUTES INSUFFICIENT
Canonical entry state: v1.68 / Paper 77 v0.1
RH_PROVED: FALSE
RH_DISPROVED: FALSE
GLOBAL_RH_CERTIFICATE: FALSE

Abstract

Paper 77 derived the exact root identity

\mathcal R_W = - \mathcal V^{\rm ren}_{U,V} + \mathcal E_I,

where

\mathcal V^{\rm ren}_{U,V} = \mathcal T^{II}_{U,V} - \mathcal M_{U,V}

and

\mathcal E_I \ll NUV(\log N)^{O(1)}.

It therefore opened F-RH-024: prove a fixed-power upper bound for the renormalized Vaughan defect.

The present paper determines the exact Dirichlet-series spectrum of the renormalized coefficient.

Define

M_U(s) = \sum_{d\le U} \frac{\mu(d)}{d^s},

L_V(s) = \sum_{e\le V} \frac{\Lambda(e)}{e^s},

and

b_U(k) = \sum_{\substack{d\mid k\\d\le U}} \mu(d).

Let

\boxed{ c_{U,V}(n) = \sum_{\substack{ek=n\\e>V\\k>U}} \Lambda(e)b_U(k). }

Then

\boxed{ \mathcal T^{II}_{U,V} = \sum_n f_{N,H}(n)c_{U,V}(n). }

Since for $U>1$

\sum_{k>U} \frac{b_U(k)}{k^s} = \zeta(s)M_U(s)-1,

and

\sum_{e>V} \frac{\Lambda(e)}{e^s} = -\frac{\zeta'(s)}{\zeta(s)} - L_V(s),

the Dirichlet series of the balanced coefficient is

\boxed{ C_{U,V}(s) = \left( -\frac{\zeta'(s)}{\zeta(s)} - L_V(s) \right) \left( \zeta(s)M_U(s)-1 \right). }

Write

M_U=M_U(1),

J_U = \sum_{d\le U} \frac{\mu(d)\log d}{d},

and

L_V=L_V(1).

Paper 77's scalar counterterm can be written coefficientwise as

\boxed{ q_{U,V}(n) = M_U\log n - M_UL_V - J_U - 1. }

Indeed,

\boxed{ \mathcal M_{U,V} = \sum_n f_{N,H}(n)q_{U,V}(n). }

Its Dirichlet series is

\boxed{ Q_{U,V}(s) = -M_U\zeta'(s) - \left( M_UL_V+J_U+1 \right) \zeta(s). }

Therefore the renormalized coefficient

\boxed{ h_{U,V}(n) = c_{U,V}(n)-q_{U,V}(n) }

has Dirichlet series

\boxed{ H_{U,V}(s) = C_{U,V}(s)-Q_{U,V}(s). }

The first main theorem is that the renormalization removes exactly the pole at $s=1$ .

Near

s=1,

C_{U,V}(s) = \frac{M_U}{(s-1)^2} + \frac{ -J_U-1-M_UL_V }{ s-1 } + O(1),

while $Q_{U,V}$ has precisely the same principal part.

Thus

\boxed{ H_{U,V}(s) \text{ is holomorphic at }s=1. }

But no nontrivial zeta-zero pole is removed.

If $\rho$ is a zero of $\zeta$ of multiplicity $m$ , then

\zeta(\rho)=0

and

\zeta(s)M_U(s)-1 = -1+O(s-\rho).

Since

-\frac{\zeta'}{\zeta}(s) = -\frac{m}{s-\rho} + O(1),

one obtains

\boxed{ \operatorname*{Res}_{s=\rho} H_{U,V}(s) = +m. }

This residue is independent of $U$ and $V$ .

Equivalently,

\boxed{ H_{U,V}(s) = \frac{\zeta'(s)}{\zeta(s)} + E_{U,V}(s), }

where $E_{U,V}$ has no pole at any nontrivial zero of $\zeta$ .

An explicit formula is

\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} }

Thus, coefficientwise,

\boxed{ h_{U,V} = -\Lambda + e_{U,V}, }

where $e_{U,V}$ is a Type-I prime approximant whose Dirichlet series has no nontrivial zeta-zero poles.

The $s=1$ residue of $E_{U,V}$ is $+1$ , so $e_{U,V}$ has the correct first-order mean to approximate the constant coefficient $1$ .

This gives a conceptual form of Paper 77's exact identity:

renormalized Vaughan balanced coefficient
=
zero-pole-free Type-I prime approximant
minus the actual von Mangoldt coefficient.

The RH-sensitive part is therefore parameter-invariant.

For any two parameter choices

(U,V), \qquad (U',V'),

\boxed{ H_{U,V}(s) - H_{U',V'}(s) }

has no nontrivial zeta-zero poles.

All choices carry the same universal

\zeta'/\zeta

hard core.

Hence optimizing Vaughan parameters may improve the Type-I error budget, but it cannot remove or weaken the boundary zero spectrum.

The second main result is a central cross-Gram calibration.

Let

W\in C_c^\infty(1,2),

and define the smoothed renormalized coefficient transform

\boxed{ \mathcal H_{U,V;W}(N,y) = \sum_n h_{U,V}(n) W(n/N) e(ny/N). }

Let the smoothed prime-error transform be

\boxed{ \mathcal S_W(N,y) = \sum_n \Lambda(n) W(n/N) e(ny/N) - N \int W(u)e(yu)du. }

Paper 73 gives the boundary response

\mathcal S_W(N,y) = -\sum_\rho m_\rho N^\rho G_\rho(y) + \text{lower terms},

where

G_\rho(y) = \int_1^2 W(u)u^{\rho-1}e(yu)du.

The residue theorem above gives the opposite response

\boxed{ \mathcal H_{U,V;W}(N,y) = +\sum_\rho m_\rho N^\rho G_\rho(y) + \text{lower terms}. }

Therefore the rightmost-zero diagonal in their central cross-energy is negative.

Assume the rightmost abscissa

\Theta = \sup_\rho\Re\rho

is attained.

Set

H=N^{1-\tau}.

Define

\boxed{ \mathfrak X_{U,V}(N,H;c) = \frac{H}{N} \int_{-c}^{c} \mathcal H_{U,V;W}(N,y) \overline{ \mathcal S_W(N,y) } dy. }

Then logarithmic-scale orthogonality gives

\boxed{ \lim_{T\to\infty} \frac1T \int_0^T \frac{ \mathfrak X_{U,V}(e^t,e^{(1-\tau)t};c) }{ e^{(1-\tau)t} e^{(2\Theta-1)t} } dt = - \sum_{\Re\rho=\Theta} m_\rho^2 \|G_\rho\|_{L^2(-c,c)}^2 <0. }

Thus the renormalized balanced coefficient is anti-aligned with the prime error on the rightmost zero diagonal.

Off-diagonal zero pairs cannot remove this contribution on all logarithmic scales.

The fixed-power size is

\boxed{ NH N^{-2(1-\Theta)}. }

At a saturated PESC $(\kappa)$ boundary,

\Theta = 1-\frac{\kappa}{2},

this is exactly

\boxed{ NH N^{-\kappa}. }

Therefore the spectral sector of F-RH-024 is genuinely root-critical.

This does not make F-RH-024 useless. Its value is that $h_{U,V}$ has a highly specific ordinary-factorization realization.

But it closes two naive routes.

Generic Titchmarsh/divisor-shift transfer

Power-saving Titchmarsh-type theorems apply to coefficients such as $\tau$ , $\tau_k$ , automorphic coefficients, or other dense multiplicative sequences whose Dirichlet series do not carry a universal $\zeta'/\zeta$ pole at every zeta zero.

The renormalized coefficient $h_{U,V}$ is not in that spectral class.

A power-saving theorem for its shifted-prime correlation would already be RH-level horizontal information.

Existing shifted-prime anatomy transfer

Ford's shifted-prime Kubilius model gives a strong total-variation approximation for the prime factors of $p+a$ below a cutoff $y$ .

The approximation becomes asymptotically sharp in the regime

y=x^{o(1)}.

The paper explicitly notes that the distribution of the large prime factors of shifted primes is not well understood.

F-RH-024 requires balanced polynomial factorization:

e\asymp N^\theta, \qquad k\asymp N^{1-\theta},

with $\theta$ bounded away from $0$ and $1$ in the genuinely Type-II blocks.

Thus current anatomy transference does not reach the required factor range.

The Campaign-47 conclusion is:

F-RH-024 remains the preferred root arithmetic frontier.

Its s=1 main is fully renormalized.

Its nontrivial zero spectrum is not renormalized at all.

Vaughan parameter optimization cannot remove the hard spectrum.

Generic prime-times-divisor power-saving theorems do not apply automatically.

Current shifted-prime factor-anatomy theorems control only the small-factor regime, not balanced polynomial factors.

The next task is therefore to dyadically localize the ordinary-factorization source of the universal zero pole and determine which polynomial factor blocks carry the rightmost-zero mass.

That is a concrete arithmetic question.

No RH theorem is claimed.

1. Balanced coefficient Dirichlet series

Define

M_U(s) = \sum_{d\le U} \mu(d)d^{-s}.

The truncated divisor coefficient

b_U(k) = \sum_{\substack{d\mid k\\d\le U}} \mu(d)

has Dirichlet series

\sum_{k\ge1} b_U(k)k^{-s} = \zeta(s)M_U(s).

1<k\le U,

then every divisor of $k$ is at most $U$ , so

b_U(k) = \sum_{d\mid k}\mu(d) = 0.

Also

b_U(1)=1.

Hence for $U>1$ ,

\boxed{ \sum_{k>U} b_U(k)k^{-s} = \zeta(s)M_U(s)-1. }

Likewise,

\boxed{ \sum_{e>V} \Lambda(e)e^{-s} = -\frac{\zeta'(s)}{\zeta(s)} - L_V(s). }

Multiplying gives Theorem 1.1:

\boxed{ C_{U,V}(s) = \left( -\frac{\zeta'}{\zeta}-L_V \right) \left( \zeta M_U-1 \right). }

2. Coefficient form of the Paper-77 counterterm

Paper 77 defined

\mathcal M_{U,V} = F \left[ M_U(\log N-L_V)-J_U-1 \right] + G M_U,

where

G = \sum_n f(n)\log(n/N).

Since

G = \sum_n f(n) \left( \log n-\log N \right),

one obtains

\begin{aligned} \mathcal M_{U,V} &= \sum_n f(n) \left[ M_U\log n - M_UL_V - J_U - 1 \right]. \end{aligned}

Therefore:

Theorem 2.1 — Coefficientwise renormalization

\boxed{ q_{U,V}(n) = M_U\log n - M_UL_V - J_U - 1. }

Create:

B-RH-123
PAPER77_SCALAR_COUNTERTERM_IS_A_COEFFICWISE_LOG_AFFINE_RENORMALIZATION
CERTIFIED

3. Dirichlet series of the counterterm

Since

\sum_{n\ge1} \frac{\log n}{n^s} = -\zeta'(s),

\sum_{n\ge1} n^{-s} = \zeta(s),

the Dirichlet series of $q_{U,V}$ is

\boxed{ Q_{U,V}(s) = -M_U\zeta'(s) - \left( M_UL_V+J_U+1 \right) \zeta(s). }

4. Exact cancellation at $s=1$

Let

t=s-1.

Use

\zeta(s) = \frac1t + \gamma + O(t),

-\frac{\zeta'}{\zeta}(s) = \frac1t - \gamma + O(t),

and

M_U(s) = M_U - J_U t + O(t^2).

Then

\zeta(s)M_U(s)-1 = \frac{M_U}{t} + \gamma M_U - J_U - 1 + O(t).

Also

-\frac{\zeta'}{\zeta}(s)-L_V(s) = \frac1t - \gamma - L_V + O(t).

Therefore

\boxed{ C_{U,V}(s) = \frac{M_U}{t^2} + \frac{ -J_U-1-M_UL_V }{t} + O(1). }

But $Q_{U,V}$ has exactly the same two polar coefficients.

Hence:

Theorem 4.1 — Complete $s=1$ renormalization

\boxed{ H_{U,V}(s) = C_{U,V}(s)-Q_{U,V}(s) }

is holomorphic at $s=1$ .

Create:

B-RH-124
RENORMALIZED_VAUGHAN_COEFFICIENT_REMOVES_THE_COMPLETE_S1_PRINCIPAL_PART
CERTIFIED

5. Universal nontrivial-zero residue

Let $\rho$ be a zeta zero of multiplicity $m$ .

Then

-\frac{\zeta'}{\zeta}(s) = -\frac{m}{s-\rho} + O(1).

At the same point,

\zeta(s)M_U(s)-1 = -1+O(s-\rho).

Thus

C_{U,V}(s) = \frac{m}{s-\rho} + O(1).

The counterterm series $Q_{U,V}$ is analytic at $\rho$ .

Therefore:

Theorem 5.1 — Parameter-independent zero-pole preservation

\boxed{ \operatorname*{Res}_{s=\rho} H_{U,V}(s) = m. }

This does not depend on $U$ or $V$ .

Create:

B-RH-125
RENORMALIZED_VAUGHAN_COEFFICIENT_PRESERVES_EVERY_NONTRIVIAL_ZETA_ZERO_WITH_UNIVERSAL_OPPOSITE_PRIME_RESIDUE
CERTIFIED

6. Zero-pole-free Type-I approximant

Expand $C_{U,V}$ :

\begin{aligned} C_{U,V} &= -\zeta'M_U(s) + \frac{\zeta'}{\zeta} \\ &\quad - L_V(s)\zeta M_U(s) + L_V(s). \end{aligned}

Subtract $Q_{U,V}$ .

Then

\boxed{ H_{U,V} = \frac{\zeta'}{\zeta} + E_{U,V}, }

where

\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} }

At every nontrivial zeta zero, $E_{U,V}$ is analytic.

Since

\frac{\zeta'}{\zeta}(s) = -\sum_n \frac{\Lambda(n)}{n^s},

the coefficient identity is

\boxed{ h_{U,V} = -\Lambda + e_{U,V}. }

The series $E_{U,V}$ has residue $+1$ at $s=1$ , so $e_{U,V}$ is a first-order prime approximant with mean $1$ .

This gives the spectral meaning of the Type-I recombination in Paper 77.

7. Parameter-invariant hard core

Let two Vaughan choices be

(U,V)

and

(U',V').

Then

\begin{aligned} H_{U,V}(s)-H_{U',V'}(s) = E_{U,V}(s)-E_{U',V'}(s). \end{aligned}

The right side has no nontrivial zeta-zero poles.

Therefore:

Corollary 7.1 — Vaughan parameter changes are zero-pole-free

\boxed{ H_{U,V} - H_{U',V'} }

contains no rightmost-zero singularity.

Create:

B-RH-126
ALL_VAUGHAN_PARAMETER_CHOICES_SHARE_THE_SAME_PARAMETER_INVARIANT_ZETA_PRIME_HARD_CORE
CERTIFIED

Parameter optimization can improve the deterministic Type-I error but cannot spectrally weaken the root component.

8. Exact form of F-RH-024

Define

h_{U,V}(n) = c_{U,V}(n)-q_{U,V}(n).

Then

\boxed{ \mathcal V^{\rm ren}_{U,V} = \sum_n f_{N,H}(n)h_{U,V}(n). }

Using

h_{U,V} = -\Lambda+e_{U,V},

\begin{aligned} \mathcal V^{\rm ren}_{U,V} &= -\sum_n \Lambda(n)f(n) + \sum_n e_{U,V}(n)f(n). \end{aligned}

Paper 77's Type-I computation is exactly

\boxed{ \sum_n e_{U,V}(n)f(n) = F+\mathcal E_I. }

Therefore

\boxed{ \mathcal V^{\rm ren}_{U,V} = -\mathcal R_W + \mathcal E_I. }

This recovers the exact root bridge and shows where the universal pole enters.

9. Smoothed zero response

Fix

W\in C_c^\infty(1,2).

Let

G_\rho(y) = \int_1^2 W(u)u^{\rho-1}e(yu)du.

The prime error has rightmost-zero response

\boxed{ \mathcal S_W(N,y) = -\sum_\rho m_\rho N^\rho G_\rho(y) + \text{lower terms}. }

The universal residue theorem gives

\boxed{ \mathcal H_{U,V;W}(N,y) = +\sum_\rho m_\rho N^\rho G_\rho(y) + \text{lower terms}. }

Thus the two transforms contain the same boundary functions with opposite sign.

10. Negative rightmost-zero cross Gram

Assume the rightmost zero abscissa

\Theta = \sup_\rho\Re\rho

is attained.

Let

H=N^{1-\tau}.

Define

\mathfrak X_{U,V}(N,H;c) = \frac{H}{N} \int_{-c}^{c} \mathcal H_{U,V;W}(N,y) \overline{ \mathcal S_W(N,y) } dy.

Smoothness of $W$ gives rapid decay of $G_\rho$ in $|\Im\rho|$ , so logarithmic-scale Hilbert orthogonality applies.

Therefore:

Theorem 10.1 — Renormalized Vaughan / prime-error log-scale anti-alignment

\boxed{ \begin{aligned} & \lim_{T\to\infty} \frac1T \int_0^T \frac{ \mathfrak X_{U,V} \left( e^t,e^{(1-\tau)t};c \right) }{ e^{(1-\tau)t} e^{(2\Theta-1)t} } dt \\ &\qquad = - \sum_{\substack{\rho:\Re\rho=\Theta}} m_\rho^2 \|G_\rho\|_{L^2(-c,c)}^2 <0. \end{aligned} }

Create:

B-RH-127
RIGHTMOST_ZERO_MODES_OF_THE_RENORMALIZED_VAUGHAN_COEFFICIENT_AND_PRIME_ERROR_ARE_LOG_SCALE_ANTI_ALIGNED
CERTIFIED_IF_RIGHTMOST_ABSCISSA_IS_ATTAINED

Thus the boundary diagonal is not removed by the Vaughan renormalization.

11. Critical exponent

The cross-Gram boundary scale is

H N^{2\Theta-1}.

Relative to the natural root scale

NH,

the saving exponent is

\boxed{ 2(1-\Theta). }

At a saturated PESC $(\kappa)$ boundary,

\Theta = 1-\frac{\kappa}{2},

this equals

\boxed{ \kappa. }

Therefore F-RH-024 asks for a strict fixed-power improvement over an explicitly present boundary diagonal.

That is genuine horizontal zero-strip progress.

12. Titchmarsh/divisor-shift calibration

Power-saving shifted-convolution theorems are available for many dense divisor-like coefficients.

For example, Drappeau obtains power-saving errors in Titchmarsh-divisor problems using Kloosterman-sum dispersion; in the prime-times-divisor case a fixed-power error is obtained under GRH for Dirichlet $L$ -functions.

Uniform Titchmarsh variants also exploit divisor or automorphic structure.

These results demonstrate that a factorized coefficient can be analytically easier than a second prime.

But $h_{U,V}$ is spectrally different from $\tau$ or a generic divisor coefficient:

H_{U,V}(s)

contains a universal pole at every nontrivial zeta zero.

Therefore a direct transfer of existing prime-times-divisor power-saving theorems is not justified.

External calibration:

Drappeau, Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method;
Assing–Blomer–Li, Uniform Titchmarsh divisor problems.

13. Shifted-prime factor-anatomy calibration

Ford's shifted-prime Kubilius model proves a total-variation approximation between the small prime factors of shifted primes and an independent probabilistic model.

Theorem 1 gives

d_{\rm TV} \ll e^{-\alpha u\log u} + (\log x)^{-A},

where

u= \frac{\log x}{\log y}.

For the error to tend rapidly to zero from the first term, one needs

u\to\infty,

equivalently

\boxed{ y=x^{o(1)}. }

The paper explicitly states that the distribution of the large prime factors of shifted primes is not well understood.

F-RH-024 instead contains balanced polynomial factors

e\asymp N^\theta,

k\asymp N^{1-\theta}.

Thus current shifted-prime anatomy results do not control the decisive balanced range.

Create:

O-RH-175
CURRENT_SHIFTED_PRIME_KUBILIUS_TRANSFERENCE_CONTROLS_SMALL_FACTORS_BUT_NOT_THE_POLYNOMIAL_BALANCED_FACTORS_REQUIRED_BY_F_RH_024
CERTIFIED_EXTERNAL_SCOPE_BARRIER

14. Consequence for Campaign 47

Paper 77 solved the extraction problem.

Paper 78 now shows that extraction does not spectrally soften the root.

The next arithmetic theorem must therefore exploit the specific coefficient realization

c_{U,V}(n) = \sum_{\substack{ek=n\\e>V\\k>U}} \Lambda(e)b_U(k),

not merely its mean, Fourier norm, or generic divisor-boundedness.

In particular:

DO NOT:
  optimize U,V hoping to remove the zero pole;
  apply a generic prime-times-divisor theorem;
  replace balanced polynomial factorization by small-factor anatomy.

DO:
  dyadically decompose e and k;
  derive the exact dyadic portion of the counterterm;
  identify which balanced factor ranges carry the universal zero-pole mass;
  search for ordinary-factorization cancellation specifically in those ranges.

15. State transition

Advance candidate state

v1.68 \to v1.69.

Add:

B-RH-123
PAPER77_SCALAR_COUNTERTERM_IS_A_COEFFICIENTWISE_LOG_AFFINE_RENORMALIZATION

B-RH-124
RENORMALIZED_VAUGHAN_COEFFICIENT_REMOVES_THE_COMPLETE_S1_PRINCIPAL_PART

B-RH-125
RENORMALIZED_VAUGHAN_COEFFICIENT_PRESERVES_EVERY_NONTRIVIAL_ZETA_ZERO_WITH_UNIVERSAL_OPPOSITE_PRIME_RESIDUE

B-RH-126
ALL_VAUGHAN_PARAMETER_CHOICES_SHARE_THE_SAME_PARAMETER_INVARIANT_ZETA_PRIME_HARD_CORE

B-RH-127
RIGHTMOST_ZERO_MODES_OF_THE_RENORMALIZED_VAUGHAN_COEFFICIENT_AND_PRIME_ERROR_ARE_LOG_SCALE_ANTI_ALIGNED

O-RH-175
CURRENT_SHIFTED_PRIME_KUBILIUS_TRANSFERENCE_CONTROLS_SMALL_FACTORS_BUT_NOT_THE_POLYNOMIAL_BALANCED_FACTORS_REQUIRED_BY_F_RH_024

F-RH-024 remains open and preferred.

No RH certificate is created.

16. Recommended next action

The next round should derive a dyadic decomposition

e\asymp E, \qquad k\asymp K, \qquad EK\asymp N,

c_{U,V}(n)

and a matching dyadic partition of the coefficientwise counterterm

q_{U,V}(n).

The goal is to identify the smallest set of polynomial factor exponents

\theta=\frac{\log E}{\log N}

which can carry the universal rightmost-zero pole.

If the pole mass can be forced into a restricted balanced range, that range becomes the next explicit Campaign-47 arithmetic target.

If every polynomial range can independently carry it, the factorization route reaches a stronger distributed parity wall.

17. Conclusion

Vaughan renormalization removes the complete main pole at $s=1$ .

It removes none of the nontrivial zeta-zero poles.

Every parameter choice shares the same $\zeta'/\zeta$ hard core.

The balanced coefficient is therefore not a generic divisor coefficient; it is a composite-supported arithmetic realization of the opposite prime zero spectrum.

This explains both the promise and the difficulty of F-RH-024.

The next progress must come from the ordinary polynomial factorization itself.

CSM_RH Paper 78 — Universal Zero-Pole Preservation in the Renormalized Vaughan Coefficient and the Polynomial-Factor Ana

CSM_RH Paper 78

Universal Zero-Pole Preservation in the Renormalized Vaughan Coefficient and the Polynomial-Factor Anatomy Barrier

Abstract

Generic Titchmarsh/divisor-shift transfer

Existing shifted-prime anatomy transfer

1. Balanced coefficient Dirichlet series

2. Coefficient form of the Paper-77 counterterm

Theorem 2.1 — Coefficientwise renormalization

3. Dirichlet series of the counterterm

4. Exact cancellation at s=1s=1s=1

Theorem 4.1 — Complete s=1s=1s=1 renormalization

5. Universal nontrivial-zero residue

Theorem 5.1 — Parameter-independent zero-pole preservation

6. Zero-pole-free Type-I approximant

7. Parameter-invariant hard core

Corollary 7.1 — Vaughan parameter changes are zero-pole-free

8. Exact form of F-RH-024

9. Smoothed zero response

10. Negative rightmost-zero cross Gram

Theorem 10.1 — Renormalized Vaughan / prime-error log-scale anti-alignment

11. Critical exponent

12. Titchmarsh/divisor-shift calibration

13. Shifted-prime factor-anatomy calibration

14. Consequence for Campaign 47

15. State transition

16. Recommended next action

17. Conclusion

4. Exact cancellation at $s=1$

Theorem 4.1 — Complete $s=1$ renormalization