CSM_RH Paper 79

Factor-Scale Resonance Flatness and the Distributed Vaughan Parity Wall

Project: CSM_RH
Paper: 79
Version: v0.1
Date: 2026-09-09
Campaign: 47 — ORDINARY_PRIME_BOUNDARY_BREAKING
Active frontier: F-RH-024 — RENORMALIZED_VAUGHAN_BALANCED_DEFECT_POWER
Status: DYADIC FACTOR-SCALE ZERO RESPONSE LOCALIZED / EDGE-LOCALIZATION HYPOTHESIS REJECTED / DISTRIBUTED LOG-SCALE PARITY WALL CERTIFIED
Canonical entry state: v1.69 / Paper 78 v0.1
RH_PROVED: FALSE
RH_DISPROVED: FALSE
GLOBAL_RH_CERTIFICATE: FALSE

Abstract

Paper 78 proved that the renormalized Vaughan coefficient preserves every nontrivial zeta-zero pole with universal residue.

The next question was whether this universal rightmost-zero mass is concentrated in a narrow polynomial factor range

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

or distributed across the full Vaughan factor strip.

The present paper answers this in the smooth Mellin-resonance sense.

Fix

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

For a multiplicative prime scale $E$ define the prime-error block

\boxed{ P_E(s) = \sum_{n\ge1} \Lambda(n)\psi(n/E)n^{-s} - \int_0^\infty \psi(x/E)x^{-s}dx. }

Let

\widehat\psi(w) = \int_0^\infty \psi(u)u^{w-1}du.

The smooth explicit formula gives

\boxed{ P_E(s) = - \sum_{\rho'} E^{\rho'-s} \widehat\psi(\rho'-s) + \mathcal T_{\psi,E}(s), }

where $\mathcal T_{\psi,E}$ is the trivial-zero / gamma-factor contribution and is negligible at positive large $E$ for the fixed spectral points considered below.

Let

\rho=\Theta+i\gamma

be a nontrivial zeta zero of multiplicity $m_\rho$ .

At the resonant spectral point

s=\rho,

the contribution of the zero $\rho$ itself is

\boxed{ -m_\rho\widehat\psi(0), }

where

\widehat\psi(0) = \int_1^2 \psi(u)\frac{du}{u}.

This term is exactly independent of $E$ .

Thus, after the prime main is removed correctly, a rightmost zero does not become larger at the largest $e$ -scale and does not concentrate at a Vaughan edge.

Its self-resonance is flat per logarithmic factor scale.

Now insert the Vaughan cofactor tail

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

At every nontrivial zeta zero,

\boxed{ B_U(\rho)=-1, }

for every $U>1$ .

Define the smooth $e$ -block of the balanced prime-error coefficient by

\boxed{ \mathscr C_{E,U}(s) = P_E(s)B_U(s). }

Then the self-zero resonance of $\mathscr C_{E,U}$ at $\rho$ is

\boxed{ +m_\rho\widehat\psi(0), }

again independent of both $E$ and $U$ .

This is the central theorem of the paper.

E=e^t,

the other rightmost zeros

\rho'=\Theta+i\gamma'

contribute phases

e^{i(\gamma'-\gamma)t}.

Hence the boundary response is an almost-periodic function of the factor scale $t=\log E$ .

Long averaging in $t$ diagonalizes distinct ordinates:

\boxed{ \lim_{T\to\infty} \frac1T \int_0^T \left| \sum_{\Re\rho'=\Theta} m_{\rho'} e^{i(\gamma'-\gamma)t} \widehat\psi(\rho'-\rho) \right|^2dt = \sum_{\Re\rho'=\Theta} m_{\rho'}^2 \left| \widehat\psi(\rho'-\rho) \right|^2 >0. }

No linear-independence conjecture is required.

Therefore off-diagonal zeros cannot make the rightmost-zero resonance disappear on almost all factor scales.

For Campaign 47, take

U=N^u, \qquad V=N^v.

The Vaughan balanced range contains

V<e<N/U.

Its logarithmic length is

\boxed{ \log\frac{N/U}{V} = (1-u-v)\log N. }

Thus, whenever

u+v<1,

the number of unit-log or dyadic factor scales tends to infinity with $N$ .

Each scale carries the same self-zero resonance after the prime main is removed.

The nontrivial zero pole of Paper 78 is therefore the accumulated resonance of a growing family of logarithmic factor blocks.

It is not supported on a fixed finite set of dyadic blocks.

It is not concentrated near

e\asymp N/U.

It is a distributed parity wall across the entire admissible logarithmic Vaughan strip.

This resolves the dyadic-localization question opened in Paper 78.

The result also corrects an overly narrow literature calibration in Paper 78.

Ford's 2025 shifted-prime Kubilius theorem gives a particularly strong total-variation approximation for small prime factors.

But 2026 work of Bharadwaj and Rodgers goes further: for a sequence with level of distribution $\sigma>0$ , correlation functions of polynomial-scale large prime factors match the Poisson–Dirichlet model against test functions supported in

\boxed{ y_1+\cdots+y_k<\sigma. }

They prove that shifted primes are $1/2$ well-distributed.

Thus current anatomy theory does reach polynomial factor exponents.

However, the support condition is exactly the remaining barrier for the present problem.

A complementary factorization

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

has total logarithmic size

\boxed{ \theta+(1-\theta)=1. }

No theorem with level

\sigma<1

controls the complete complementary factor pair through the restricted-support correlation statement.

Even the shift-averaged nonnegative sequence of Paper 75 has deterministic level only up to

\boxed{ \sigma<1-\tau, }

because its divisor remainder ratio is

N^{\sigma-(1-\tau)+o(1)}.

Hence its current large-factor correlation information still has support strictly below total factor mass $1$ .

The distributed Vaughan resonance lives exactly on that full complementary-factor boundary.

This produces a new, precise scope statement:

current anatomy theorems can see
proper subcollections of polynomial prime factors;

F-RH-024 requires coherent information
across a complete complementary factorization
whose normalized log sizes sum to 1.

The next Campaign-47 arithmetic target should therefore not single out one $\theta$ .

Instead, it should seek a factor-scale averaged renormalized defect theorem whose cancellation is uniform over a positive-length $\theta$ interval and remains coherent after summing all such intervals.

Open:

F-RH-025
DISTRIBUTED VAUGHAN FACTOR-SCALE DEFECT POWER

A candidate form is:

partition the $e$ -variable by a smooth multiplicative partition

1= \int \psi(e/e^t)\,dt

V<e<N/U.

Let

\mathcal V_t^{\rm ren}(N,H;U,V)

denote the corresponding locally renormalized Vaughan block.

Find fixed

\eta>0

such that the accumulated defect satisfies

\boxed{ \left| \int_{\log V}^{\log(N/U)} \mathcal V_t^{\rm ren} \,dt \right| \ll NH N^{-\kappa-\eta}, }

while the local Type-I renormalizations are retained exactly.

This is essentially F-RH-024 with its factor-scale geometry made explicit.

The new information is that no fixed $\theta$ block can be declared the unique hard block.

The boundary zero is coherently replicated across the full logarithmic factor strip.

No RH theorem is claimed.

1. Smooth prime-error block

Fix

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

Define

\boxed{ P_E(s) = \sum_{n\ge1} \Lambda(n) \psi(n/E)n^{-s} - \int_0^\infty \psi(x/E)x^{-s}dx. }

The continuous term is

\boxed{ E^{1-s}\widehat\psi(1-s). }

This is the correct blockwise removal of the prime main.

Using $\Lambda(n)-1$ directly gives the same fixed-power resonance after the usual Euler-summation adjustment between the integer sum of $1$ and the continuous integral.

2. Smooth explicit formula

For fixed $s$ and large $E$ , the explicit formula gives

\boxed{ P_E(s) = - \sum_{\rho'} E^{\rho'-s} \widehat\psi(\rho'-s) + \mathcal T_{\psi,E}(s). }

The smooth compact support gives rapid decay of

\widehat\psi(\sigma+i\tau)

in $|\tau|$ .

Thus the zero sum is absolutely manageable at every fixed spectral point after standard grouping.

3. Self-zero resonance is factor-scale invariant

Let

\rho=\Theta+i\gamma

be a zero of multiplicity $m_\rho$ .

Set

s=\rho.

Its own contribution in Section 2 is

\boxed{ -m_\rho E^{\rho-\rho} \widehat\psi(0) = -m_\rho\widehat\psi(0). }

No factor of $E$ remains.

Create:

B-RH-128
A_RIGHTMOST_ZETA_ZERO_HAS_SCALE_INVARIANT_SELF_RESONANCE_IN_EACH_SMOOTH_LOGARITHMIC_PRIME_FACTOR_BLOCK
CERTIFIED

This is the factor-scale analogue of Paper 73's log- $N$ resonance.

4. Vaughan cofactor tail does not alter the resonance amplitude

Recall

B_U(s) = \zeta(s)M_U(s)-1.

At every nontrivial zeta zero,

\boxed{ B_U(\rho)=-1. }

Therefore

\mathscr C_{E,U}(s) = P_E(s)B_U(s)

has self-zero response

\boxed{ +m_\rho\widehat\psi(0). }

This is independent of $U$ .

Create:

B-RH-129
VAUGHAN_COFACTOR_TAIL_TRANSPORTS_THE_BLOCKWISE_ZERO_RESONANCE_WITH_UNIVERSAL_UNIT_GAIN
CERTIFIED

5. Factor-scale almost periodicity

Let

E=e^t.

At $s=\rho=\Theta+i\gamma$ , the rightmost-zero part is

\boxed{ \sum_{\Re\rho'=\Theta} m_{\rho'} e^{i(\gamma'-\gamma)t} \widehat\psi(\rho'-\rho). }

This is an absolutely summable almost-periodic Fourier series because the Mellin transform of $\psi$ decays rapidly.

Long $t$ -averaging gives:

Theorem 5.1 — Factor-scale zero orthogonality

\boxed{ \begin{aligned} & \lim_{T\to\infty} \frac1T \int_0^T \left| \sum_{\Re\rho'=\Theta} m_{\rho'} e^{i(\gamma'-\gamma)t} \widehat\psi(\rho'-\rho) \right|^2dt \\ &\qquad = \sum_{\Re\rho'=\Theta} m_{\rho'}^2 \left| \widehat\psi(\rho'-\rho) \right|^2. \end{aligned} }

The right side is positive if

\widehat\psi(0)\ne0.

Create:

B-RH-130
RIGHTMOST_ZERO_RESPONSES_DIAGONALIZE_IN_LOGARITHMIC_FACTOR_SCALE
CERTIFIED

6. Why naive edge localization was wrong

If one evaluates the uncentered prime block

\sum\Lambda(n)\psi(n/E)n^{-\rho},

its main term has size

E^{1-\rho}.

This grows like

E^{1-\Theta}.

Looking only at that quantity falsely suggests domination by the largest $E$ .

But this is the prime main, not the zero resonance.

After the continuous prime main is removed, the self-zero term becomes

-m_\rho\widehat\psi(0),

independent of $E$ .

Therefore:

C-RH-006
UNRENORMALIZED_E_POWER_GROWTH_MUST_NOT_BE_USED_TO_LOCALIZE_THE_RIGHTMOST_ZERO_TO_THE_VAUGHAN_EDGE

7. Accumulation across the Vaughan factor strip

Let

U=N^u, \qquad V=N^v.

The balanced coefficient uses

V<e<N/U.

Thus the factor-scale interval is

\boxed{ v\log N < \log e < (1-u)\log N. }

Its length is

\boxed{ (1-u-v)\log N. }

A smooth partition into unit-log blocks therefore contains

\asymp (1-u-v)\log N

blocks.

The self-zero resonance has the same size in each block.

Hence the simple pole of the global renormalized coefficient is naturally interpreted as the accumulation of equal-strength logarithmic-scale resonances.

Create:

O-RH-176
THE_UNIVERSAL_ZETA_ZERO_POLE_IS_DISTRIBUTED_ACROSS_A_GROWING_NUMBER_OF_VAUGHAN_LOG_FACTOR_SCALES
CERTIFIED

No fixed finite set of logarithmic blocks carries the pole.

8. No persistent factor-scale cancellation

Theorem 5.1 shows that other rightmost zeros may oscillate against a chosen zero as $E$ changes.

But their long factor-scale cross terms vanish.

The self-zero diagonal remains.

Therefore the rightmost-zero resonance cannot be canceled on almost all factor scales by generic zero-phase interference.

This is a second distributed-parity obstruction:

factor-scale averaging does not remove
the rightmost-zero diagonal;
it reveals it.

9. 2026 large-prime-factor anatomy correction

Paper 78 emphasized Ford's small-factor Kubilius regime.

Current literature is stronger.

Bharadwaj and Rodgers prove that if an arithmetic sequence has level of distribution $\sigma>0$ , then the correlation functions of normalized logarithmic prime factors match those of the Poisson–Dirichlet process against test functions supported in

\boxed{ y_1+\cdots+y_k<\sigma. }

They also prove that shifted primes are

\boxed{ \sigma=\frac12 }

well-distributed.

Therefore modern anatomy theory does see polynomial-size prime factors.

Create correction:

C-RH-007
PAPER78_SMALL_FACTOR_ONLY_DESCRIPTION_OF_CURRENT_SHIFTED_PRIME_ANATOMY_WAS_TOO_NARROW

The correct statement is support-limited polynomial anatomy.

10. Why the complete complementary factorization remains outside current anatomy

For the Vaughan factorization

n=ek,

write

e\asymp N^\theta,

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

The normalized logarithmic sizes sum to

\boxed{ 1. }

Bharadwaj–Rodgers correlation transference at level $\sigma$ requires total support strictly below $\sigma$ .

For shifted primes,

\sigma=\frac12.

Thus it cannot encode a full complementary factorization.

Even if one applies the general theorem to the shift-averaged nonnegative sequence from Paper 75, its deterministic divisor estimate gives level of distribution only for every

\boxed{ \sigma<1-\tau. }

Indeed, summing the $O(N)$ divisor error to

D=N^\sigma

costs

N^{1+\sigma},

while the total mass is

NH=N^{2-\tau}.

The relative error is

N^{\sigma-(1-\tau)}.

Hence $\sigma=1$ is not reached for any fixed $\tau>0$ .

The complementary factor pair still has total size $1$ .

Create:

O-RH-177
CURRENT_LEVEL_OF_DISTRIBUTION_FACTOR_ANATOMY_DOES_NOT_CONTROL_A_COMPLETE_COMPLEMENTARY_FACTOR_PAIR_OF_TOTAL_LOG_SIZE_ONE
CERTIFIED_AS_SCOPE_BARRIER

11. Consequence for dyadic F-RH-024

The factor-scale question now has a definite answer.

EDGE LOCALIZATION:
REJECTED AFTER PROPER PRIME-MAIN RENORMALIZATION.

SINGLE CRITICAL THETA:
NOT IDENTIFIED.

DISTRIBUTED LOG-SCALE RESONANCE:
CERTIFIED.

Every positive-length subinterval of the admissible $\theta$ strip contains a number of resonant logarithmic blocks proportional to its length.

The root-hard spectrum is therefore distributed through the factorization scale.

12. New explicit geometry frontier F-RH-025

Open:

F-RH-025
DISTRIBUTED_VAUGHAN_FACTOR_SCALE_DEFECT_POWER

Let

\psi_t(e) = \psi(e/e^t)

form a smooth multiplicative partition over

V<e<N/U.

Decompose the exact renormalized Vaughan defect into

\boxed{ \mathcal V^{\rm ren}_{U,V} = \int_{\log V}^{\log(N/U)} \mathcal V_t^{\rm ren} \,dt + \text{endpoint terms}. }

The target is not that each block be absolutely tiny.

The target is a fixed-power estimate for the full accumulated defect:

\boxed{ \left| \int \mathcal V_t^{\rm ren}dt \right| \ll NH N^{-\kappa-\eta}, }

with the local prime main and Type-I counterterms retained exactly.

Paper 79 warns that any proof must overcome a coherent rightmost-zero resonance which is present across the entire factor-scale interval.

13. What anatomy information might still help

The Bharadwaj–Rodgers theorem may still control proper subcollections of the factorization.

For example, prime factors with total normalized log size below

\sigma

can have Poisson–Dirichlet correlation behavior.

This could be useful for:

the factorization of a small cofactor;
excluding exceptional anatomy classes;
controlling portions of $b_U(k)$ built from prime factors whose total size lies below the level.

But it does not determine the joint law of the complete pair $(e,k)$ whose total normalized size is $1$ .

Thus anatomy may become a supporting tool for F-RH-025, not a complete theorem.

14. State transition

Advance candidate state

v1.69 \to v1.70.

Add:

B-RH-128
A_RIGHTMOST_ZETA_ZERO_HAS_SCALE_INVARIANT_SELF_RESONANCE_IN_EACH_SMOOTH_LOGARITHMIC_PRIME_FACTOR_BLOCK

B-RH-129
VAUGHAN_COFACTOR_TAIL_TRANSPORTS_THE_BLOCKWISE_ZERO_RESONANCE_WITH_UNIVERSAL_UNIT_GAIN

B-RH-130
RIGHTMOST_ZERO_RESPONSES_DIAGONALIZE_IN_LOGARITHMIC_FACTOR_SCALE

O-RH-176
THE_UNIVERSAL_ZETA_ZERO_POLE_IS_DISTRIBUTED_ACROSS_A_GROWING_NUMBER_OF_VAUGHAN_LOG_FACTOR_SCALES

O-RH-177
CURRENT_LEVEL_OF_DISTRIBUTION_FACTOR_ANATOMY_DOES_NOT_CONTROL_A_COMPLETE_COMPLEMENTARY_FACTOR_PAIR_OF_TOTAL_LOG_SIZE_ONE

C-RH-006
UNRENORMALIZED_E_POWER_GROWTH_MUST_NOT_BE_USED_TO_LOCALIZE_THE_RIGHTMOST_ZERO_TO_THE_VAUGHAN_EDGE

C-RH-007
PAPER78_SMALL_FACTOR_ONLY_DESCRIPTION_OF_CURRENT_SHIFTED_PRIME_ANATOMY_WAS_TOO_NARROW

Open:

F-RH-025
DISTRIBUTED_VAUGHAN_FACTOR_SCALE_DEFECT_POWER
OPEN_GEOMETRIC_ROOT_FRONTIER

F-RH-024 remains the algebraic root frontier.

No RH certificate is created.

15. External calibration

15.1. Bharadwaj–Rodgers, 2026

A. Bharadwaj and B. Rodgers, Large prime factors of well-distributed sequences, Canadian Mathematical Bulletin, published online 17 April 2026.

They prove:

level $1$ implies full Poisson–Dirichlet convergence of large prime factors;
positive level $\sigma$ implies convergence of large-prime-factor correlation functions against test functions supported in $y_1+\cdots+y_k<\sigma;$
shifted primes have level $1/2$ .

URL:

https://www.cambridge.org/core/journals/canadian-mathematical-bulletin/article/large-prime-factors-of-welldistributed-sequences/270043D7BAB4CDA2A601A061FB482AA0

15.2. Ford, 2025

K. Ford, Poisson Approximation of Prime Divisors of Shifted Primes, International Mathematics Research Notices 2025.

Ford gives a strong total-variation shifted-prime Kubilius theorem and a transference principle for factor anatomy.

The 2026 Bharadwaj–Rodgers theorem provides the more appropriate polynomial-factor support calibration used in this paper.

16. Recommended next action

The next round should not choose a single $\theta$ block.

Instead, use the distributed resonance theorem to ask whether the ordinary-factorization coefficient

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

has a factor-scale martingale / Buchstab / Poisson–Dirichlet decomposition whose cross-scale increments telescope against the equal-strength prime-error resonances.

The test is strict:

any proposed scale decomposition must reproduce the universal unit zero residue when all scales are summed;
a claimed fixed-power gain must come from arithmetic cancellation between factor-anatomy classes, not from deleting or centering the resonant diagonal.

If no such cross-scale arithmetic cancellation exists, F-RH-025 is another RH-equivalent distributed parity wall.

17. Conclusion

The rightmost-zero mass in the renormalized Vaughan coefficient is not an edge effect.

After the prime main is removed correctly, each logarithmic prime-factor scale carries the same self-zero resonance.

The global pole is the accumulation of these resonances across a factor-scale interval of length proportional to $\log N$ .

Modern large-factor anatomy reaches polynomial scales, but only under a support-sum condition strictly below the available level of distribution.

A complete complementary Vaughan factorization lies on total log mass $1$ and remains outside that regime.

Campaign 47 has therefore reached a distributed, not localized, parity geometry.

CSM_RH Paper 79 — Factor-Scale Resonance Flatness and the Distributed Vaughan Parity Wall

CSM_RH Paper 79

Factor-Scale Resonance Flatness and the Distributed Vaughan Parity Wall

Abstract

1. Smooth prime-error block

2. Smooth explicit formula

3. Self-zero resonance is factor-scale invariant

4. Vaughan cofactor tail does not alter the resonance amplitude

5. Factor-scale almost periodicity

Theorem 5.1 — Factor-scale zero orthogonality

6. Why naive edge localization was wrong

7. Accumulation across the Vaughan factor strip

8. No persistent factor-scale cancellation

9. 2026 large-prime-factor anatomy correction

10. Why the complete complementary factorization remains outside current anatomy

11. Consequence for dyadic F-RH-024

12. New explicit geometry frontier F-RH-025

13. What anatomy information might still help

14. State transition

15. External calibration

15.1. Bharadwaj–Rodgers, 2026

15.2. Ford, 2025

16. Recommended next action

17. Conclusion