CSM_RH Paper 83

Principal-Character Projectors for Shifted-Prime Factor States and the Rank-One Mixed-State Barrier

Project: CSM_RH
Paper: 83
Version: v0.1
Date: 2026-09-09
Branch: MIXED_ARITHMETIC_STATE_SCREENING
Entry state: v1.73 / Paper 82 v0.1
Status: PERIODIC / SIEVE FACTOR STATES CLASSIFIED / PRINCIPAL ZETA HARD CORE IS RANK ONE / CENTERED FACTOR-ANATOMY FLUCTUATIONS ARE PRINCIPAL-ZETA-BLIND
RH_PROVED: FALSE
RH_DISPROVED: FALSE
GLOBAL_RH_CERTIFICATE: FALSE

Abstract

Paper 82 closed pure finite-degree polynomial statistics of one linear prime-error field as a source of new universal boundary coercivity.

The next proposed class was a mixed observable coupling primality to an ordinary-factorization state.

The present paper classifies the largest tractable subclass: factor states determined by congruence information, finite divisor conditions, roughness conditions over finitely many primes, or any periodic state of a shifted integer.

The conclusion is exact.

Fix:

an integer shift $a$ ;
a modulus $Q\ge1$ ;
a bounded function $g:\mathbb Z/Q\mathbb Z\to\mathbb C.$

For prime variables $n$ coprime to $Q$ , define

\boxed{ w_{g,a}(n) = g(n+a\bmod Q). }

Let

\boxed{ \mathcal P_{g,a;Q}(s) = \sum_{n\ge1} \frac{\Lambda(n)w_{g,a}(n)}{n^s}. }

The finitely many prime powers supported on primes dividing $Q$ may be separated into an entire Dirichlet polynomial and will be ignored in the spectral projector statements.

On the reduced residue group

G_Q = (\mathbb Z/Q\mathbb Z)^\times,

expand

w_{g,a}(r) = \sum_{\chi\bmod Q} \widehat g_a(\chi)\chi(r),

where

\boxed{ \widehat g_a(\chi) = \frac1{\phi(Q)} \sum_{r\in G_Q} g(r+a) \overline{\chi(r)}. }

Then

\boxed{ \mathcal P_{g,a;Q}(s) = \sum_{\chi\bmod Q} \widehat g_a(\chi) \left( -\frac{L'}{L}(s,\chi) \right) + \mathcal E_Q(s), }

where $\mathcal E_Q$ is entire in the critical strip.

The principal-character coefficient is

\boxed{ \pi_{g,a;Q} = \widehat g_a(\chi_0) = \frac1{\phi(Q)} \sum_{r\in G_Q} g(r+a). }

Since

L(s,\chi_0) = \zeta(s) \prod_{p\mid Q} \left( 1-p^{-s} \right),

one has

\boxed{ -\frac{L'}{L}(s,\chi_0) = -\frac{\zeta'}{\zeta}(s) + \mathcal A_Q(s), }

where $\mathcal A_Q$ is analytic at every nontrivial zeta zero.

Therefore:

Principal-character factor-state projector theorem

\boxed{ \mathcal P_{g,a;Q}(s) = \pi_{g,a;Q} \left( -\frac{\zeta'}{\zeta}(s) \right) + \mathcal R_{g,a;Q}(s), }

where $\mathcal R_{g,a;Q}$ contains:

nonprincipal Dirichlet- $L$ logarithmic derivatives;
finite Euler corrections;
no principal-character zeta pole.

Thus a zeta zero $\rho$ of multiplicity $m$ contributes to the principal projector with residue

\boxed{ -m\,\pi_{g,a;Q}. }

The theorem gives an immediate dichotomy.

Define the principal-centered factor state

\boxed{ g^\circ(r+a) = g(r+a)-\pi_{g,a;Q}. }

Then

\boxed{ \widehat {g^\circ}_a(\chi_0)=0. }

Hence the shifted-prime series weighted by $g^\circ$ has no principal-character $\zeta'/\zeta$ term.

So:

UNCENTERED PERIODIC FACTOR STATE:
retains a scalar copy of the original zeta hard core.

PRINCIPAL-CENTERED PERIODIC FACTOR STATE:
removes the zeta hard core and keeps only nonprincipal L-spectra.

This includes every finite divisor-sieve state.

g(m) = \sum_{d\in\mathcal D} \lambda_d \mathbf 1_{d\mid m},

then $g$ is periodic modulo

Q=\operatorname{lcm}_{d\in\mathcal D}d.

Therefore every finite Selberg-sieve weight, truncated small-prime anatomy state, finite roughness gate, and finite combination of divisibility patterns satisfies the projector theorem.

A particularly transparent example is a roughness gate.

Let

P(z) = \prod_{p\le z}p

and

g_z(m) = \mathbf 1_{(m,P(z))=1}.

For $Q=P(z)$ ,

\boxed{ \pi_{z,a} = \frac1{\phi(P(z))} \#\left\{ r\bmod P(z): (r,P(z))=1,\, (r+a,P(z))=1 \right\}. }

By the Chinese remainder theorem,

\boxed{ \pi_{z,a} = \prod_{\substack{p\le z\\p\nmid a}} \frac{p-2}{p-1}. }

For primes $p\mid a$ , the local factor is $1$ .

The parity obstruction is visible immediately: if $2\nmid a$ and $2\le z$ , the density is $0$ .

The centered roughness observable

\boxed{ g_z(n+a)-\pi_{z,a} }

therefore deletes the principal zeta boundary mode exactly.

This structural projector is consistent with current shifted-prime factor-anatomy theorems.

Ford proves a Kubilius-type model for prime factors of shifted primes and Poisson behavior for prime factors in disjoint small-prime sets.

Bharadwaj and Rodgers prove polynomial large-prime-factor correlation laws for well-distributed sequences under restricted support.

These results describe increasingly rich conditional factor anatomy.

But the present theorem shows that, for periodic/sieve-visible parts of that anatomy, the principal zeta error remains a one-dimensional scaling direction independent of the orthogonal anatomy fluctuations.

The vector form makes this explicit.

Suppose

g_1,\ldots,g_J

form a partition of unity on shifted reduced residue states:

\boxed{ \sum_{j=1}^J g_j(r+a)=1 }

for every

r\in G_Q.

Define

\pi_j = \pi_{g_j,a;Q}.

Then

\boxed{ \sum_j\pi_j=1. }

The vector of shifted-prime Dirichlet series satisfies

\boxed{ \mathbf P(s) = \boldsymbol\pi \left( -\frac{\zeta'}{\zeta}(s) \right) + \mathbf R(s). }

Thus the principal-zeta hard core lies in the rank-one direction

\boxed{ \operatorname{span}\{\boldsymbol\pi\}. }

For any vector $c\in\mathbb C^J$ :

if
$c\cdot\boldsymbol\pi=0,$
then the linear statistic $c\cdot\mathbf P$ has no principal zeta projector;
if
$c\cdot\boldsymbol\pi\ne0,$
then it contains precisely a scalar copy of the original zeta hard core.

This gives the rank-one mixed-state barrier.

A factor-state vector can contain substantial arithmetic information orthogonal to its principal direction.

But the existence of a zeta boundary zero alone only forces the principal direction.

The centered factor-anatomy fluctuations are not a new universal root forcing mechanism.

The same conclusion applies to normalized conditional distributions.

If a factor-state count is centered relative to the total prime count,

N_j(X) - \pi_j\psi(X),

then its principal zeta error cancels.

If it is instead centered relative to the deterministic main

\pi_jX,

then

\boxed{ N_j(X)-\pi_jX = \pi_j(\psi(X)-X) + \Delta_j(X), }

where

\sum_j\Delta_j(X)=0.

The first term is the rank-one root-hard direction.

The second is conditional anatomy.

Thus adding periodic factor states does not by itself create a relation which bounds the principal amplitude from the orthogonal fluctuations.

A genuinely new mixed-state route must introduce at least one of the following:

a nonperiodic complete-factorization state;
a second global arithmetic field with its own zeta-zero residue data;
a nonlinear constraint which forces the principal prime-error amplitude to interact with an orthogonal factor-state fluctuation.

The paper identifies one candidate class that survives the present projector test:

GLOBAL PRIME-ERROR / MERTENS-STATE CROSS SPECTRUM

The smoothed prime error has zero coefficients which do not depend on $\zeta'(\rho)$ , while the smoothed Möbius summatory field has coefficients proportional, for a simple zero, to

\frac1{\zeta'(\rho)}.

Thus their joint spectral state is not determined by the prime-error two-point covariance alone.

This class will be screened next.

No root frontier is opened here; the next paper must first prove a universal boundary forcing theorem for the mixed prime/Mertens state.

No RH theorem is claimed.

1. Periodic shifted factor states

Fix $Q$ and $a$ .

For

r\in G_Q,

define

w(r) = g(r+a).

Dirichlet characters form an orthonormal basis of functions on $G_Q$ .

Therefore

\boxed{ w(r) = \sum_{\chi\bmod Q} \widehat g_a(\chi)\chi(r). }

2. Shifted-prime character decomposition

Ignoring the finite set of prime powers whose prime divides $Q$ ,

\begin{aligned} \mathcal P_{g,a;Q}(s) &= \sum_n \frac{\Lambda(n)}{n^s} \sum_\chi \widehat g_a(\chi)\chi(n) \\ &= \boxed{ \sum_\chi \widehat g_a(\chi) \left( -\frac{L'}{L}(s,\chi) \right). } \end{aligned}

Restoring the exceptional prime-power terms changes the expression by a finite Dirichlet series.

Create:

B-RH-146
EVERY_PERIODIC_SHIFTED_PRIME_FACTOR_STATE_DECOMPOSES_EXACTLY_INTO_DIRICHLET_CHARACTER_LOG_DERIVATIVES
CERTIFIED

3. Principal-character projector

The principal coefficient is

\pi_{g,a;Q} = \frac1{\phi(Q)} \sum_{r\in G_Q} g(r+a).

Since

L(s,\chi_0) = \zeta(s) \prod_{p\mid Q}(1-p^{-s}),

the only globally forced zeta factor in the character decomposition is the principal-character term.

Thus:

Theorem 3.1 — Periodic factor-state zeta projector

\boxed{ \mathcal P_{g,a;Q}(s) = \pi_{g,a;Q} \left( -\frac{\zeta'}{\zeta}(s) \right) + \mathcal R_{g,a;Q}(s). }

Create:

B-RH-147
THE_PRINCIPAL_ZETA_HARD_CORE_OF_A_PERIODIC_SHIFTED_PRIME_STATE_IS_EXACTLY_ITS_REDUCED_RESIDUE_MEAN
CERTIFIED

The remainder may contain nonprincipal Dirichlet- $L$ poles; those are separate spectra and are not forced by the existence of a zeta zero alone.

4. Centering dichotomy

Define

g^\circ(r+a) = g(r+a)-\pi_{g,a;Q}.

Then

\widehat {g^\circ}_a(\chi_0)=0.

Therefore:

Corollary 4.1 — Principal-centering theorem

The shifted-prime observable weighted by $g^\circ$ contains no principal-character $\zeta'/\zeta$ term.

Create:

O-RH-183
CENTERING_A_PERIODIC_FACTOR_STATE_REMOVES_THE_PRINCIPAL_ZETA_BOUNDARY_MODE_RATHER_THAN_AMPLIFYING_IT
CERTIFIED

5. Finite divisor states are periodic

Let

g(m) = \sum_{d\in\mathcal D} \lambda_d \mathbf1_{d\mid m}

with finite $\mathcal D$ .

Then $g$ is periodic modulo

Q=\operatorname{lcm}_{d\in\mathcal D}d.

Hence Theorem 3.1 applies.

This includes:

finite Selberg-sieve weights;
finite small-prime factor-pattern indicators;
finite squarefree / roughness gates;
finite Boolean combinations of divisibility conditions.

Thus:

SMALL-FACTOR SIEVE ANATOMY
DOES NOT EVADE THE PRINCIPAL PROJECTOR.

6. Roughness gate density

Let

g_z(m) = \mathbf1_{(m,P(z))=1}.

For each prime $p\le z$ , the variable prime residue $r$ must satisfy

r\not\equiv0\pmod p.

If $p\nmid a$ , roughness of $r+a$ imposes the additional exclusion

r\not\equiv-a\pmod p.

Among the $p-1$ reduced residues, $p-2$ remain.

If $p\mid a$ , no additional reduced residue is excluded.

Therefore:

Theorem 6.1 — Shifted-prime roughness principal density

\boxed{ \pi_{z,a} = \prod_{\substack{p\le z\\p\nmid a}} \frac{p-2}{p-1}. }

Create:

B-RH-148
THE_PRINCIPAL_PROJECTOR_OF_A_FINITE_SHIFTED_PRIME_ROUGHNESS_GATE_IS_ITS_EXACT_LOCAL_SIEVE_DENSITY
CERTIFIED

7. Vector partition theorem

Suppose

\sum_{j=1}^Jg_j(r+a)=1

on $G_Q$ .

Set

\boldsymbol\pi = (\pi_1,\ldots,\pi_J).

Then

\sum_j\pi_j=1.

The vector prime series is

\boxed{ \mathbf P(s) = \boldsymbol\pi \left( -\frac{\zeta'}{\zeta}(s) \right) + \mathbf R(s). }

Thus:

Theorem 7.1 — Rank-one principal direction

The principal zeta spectrum of any finite periodic factor-state partition has rank one.

Create:

B-RH-149
THE_PRINCIPAL_ZETA_SPECTRUM_OF_A_FINITE_PERIODIC_FACTOR_STATE_VECTOR_IS_RANK_ONE
CERTIFIED

8. Orthogonal anatomy fluctuations are root-blind

Let

c\cdot\boldsymbol\pi=0.

Then

c\cdot\mathbf P(s) = c\cdot\mathbf R(s)

contains no principal zeta projector.

Therefore any statistic built purely from the factor-anatomy directions orthogonal to $\boldsymbol\pi$ is not universally forced by a zeta boundary zero.

Create:

O-RH-184
ORTHOGONAL_PERIODIC_FACTOR_ANATOMY_FLUCTUATIONS_ARE_NOT_UNIVERSALLY_FORCED_BY_A_ZETA_BOUNDARY_ZERO
CERTIFIED

9. Absolute versus conditional centering

Let $N_j(X)$ denote a factor-state prime count with main density $\pi_j$ .

Write

\boxed{ E_j(X) = N_j(X)-\pi_jX. }

Also define the conditional fluctuation

\boxed{ \Delta_j(X) = N_j(X)-\pi_j\psi(X). }

Then

\boxed{ E_j(X) = \pi_j(\psi(X)-X) + \Delta_j(X). }

Summing gives

\sum_j\Delta_j(X)=0.

This is the physical-space form of the rank-one projector.

The principal root amplitude is a common scaling direction.

The conditional factor-state fluctuation is transverse to it.

10. Why improved anatomy alone does not improve the PNT error

Suppose one proves very strong estimates for

\Delta_j(X).

Then

N_j(X) = \pi_j\psi(X) + \text{small}.

This sharpens the conditional distribution of factor states among primes.

But summing over $j$ returns the identity

\psi(X)=\psi(X).

No improved bound for

\psi(X)-X

follows.

Conversely, proving

N_j(X)-\pi_jX

smaller than the principal boundary scale for any $j$ with $\pi_j\ne0$ already removes the same zeta hard core.

Thus:

conditional anatomy can be much easier
without controlling the root;

absolute anatomy at fixed-power precision
inherits the root.

11. Current factor-anatomy literature calibration

Ford, 2025

Ford develops a Kubilius model for shifted primes and proves that prime factors in disjoint small-prime sets behave approximately like independent Poisson variables.

This is strong control of conditional factor anatomy.

It is compatible with the present projector theorem: the Poisson fluctuations live in the state distribution transverse to the total prime mass.

Bharadwaj–Rodgers, 2026

For well-distributed sequences, Bharadwaj and Rodgers obtain Poisson–Dirichlet large-factor correlations under restricted support; shifted primes are $1/2$ well-distributed.

Again this concerns conditional anatomy.

It does not change the rank-one principal zeta direction in the prime count.

12. Screening of Möbius / Liouville shifted-prime states

The centered Möbius or Liouville state of $p+a$ is not a finite periodic state.

Thus the exact projector theorem does not directly classify it.

However current theorems such as Lichtman's averaged shifted-prime Möbius cancellation control quantities of the form

\sum_{p\le X}\mu(p+h)

on average over $h$ .

Such a centered multiplicative state is not universally forced to be nonzero by a zeta boundary zero.

Therefore it cannot be a standalone root detector merely because it is arithmetically nonperiodic.

A mixed theorem would have to couple it to the principal prime-error direction through an additional exact or coercive relation.

13. First surviving mixed-state class

The present screen leaves one conceptually different class:

GLOBAL PRIME-ERROR / GLOBAL MERTENS-STATE CROSS SPECTRUM

The smoothed prime error has explicit-formula boundary coefficients which depend on the zero location and the test transform.

For a simple zero $\rho$ , the smoothed Möbius summatory field has coefficient

\boxed{ \frac{\widehat V(\rho)}{\zeta'(\rho)}. }

This derivative information is not contained in the two-point spectrum of the prime-error field.

Therefore the pair

(\text{prime error},\text{Mertens state})

passes the Paper-82 single-field novelty test.

It does not yet pass the universal root-forcing or arithmetic-upper tests.

Those are the next tasks.

14. New mixed-state entry rule

A future mixed state should be rejected if it falls into either class:

A. PERIODIC / FINITE-SIEVE STATE
   -> principal zeta scalar copy + nonprincipal remainder.

B. PRINCIPAL-CENTERED PERIODIC STATE
   -> zeta hard core removed.

A candidate advances only if:

it contains additional spectral data not determined by the prime-error $C_2$ ;
a zeta boundary zero forces a nonzero mixed signal;
the signal survives the single-boundary-pair model;
there is an explicit arithmetic quantity for which a fixed-power upper is at least conceptually distinct from PESC itself.

15. External calibration

15.1. Ford, 2025

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

Ford develops a Kubilius model for shifted primes and proves approximate independence / Poisson behavior for prime divisors in disjoint sets.

URL:

https://doi.org/10.1093/imrn/rnaf079

15.2. Bharadwaj–Rodgers, 2026

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

They obtain Poisson–Dirichlet large-factor laws at level $1$ and restricted-support correlation laws at positive level; shifted primes are $1/2$ well-distributed.

URL:

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

15.3. Lichtman

J. D. Lichtman, Averages of the Möbius function on shifted primes, Quarterly Journal of Mathematics 73 (2022), 729–750.

He proves averaged cancellation of Möbius on shifted primes with logarithmic quantitative savings.

URL:

https://doi.org/10.1093/qmath/haab054

16. State transition

Advance candidate state

v1.73 \to v1.74.

Add:

B-RH-146
EVERY_PERIODIC_SHIFTED_PRIME_FACTOR_STATE_DECOMPOSES_EXACTLY_INTO_DIRICHLET_CHARACTER_LOG_DERIVATIVES

B-RH-147
THE_PRINCIPAL_ZETA_HARD_CORE_OF_A_PERIODIC_SHIFTED_PRIME_STATE_IS_EXACTLY_ITS_REDUCED_RESIDUE_MEAN

B-RH-148
THE_PRINCIPAL_PROJECTOR_OF_A_FINITE_SHIFTED_PRIME_ROUGHNESS_GATE_IS_ITS_EXACT_LOCAL_SIEVE_DENSITY

B-RH-149
THE_PRINCIPAL_ZETA_SPECTRUM_OF_A_FINITE_PERIODIC_FACTOR_STATE_VECTOR_IS_RANK_ONE

O-RH-183
CENTERING_A_PERIODIC_FACTOR_STATE_REMOVES_THE_PRINCIPAL_ZETA_BOUNDARY_MODE_RATHER_THAN_AMPLIFYING_IT

O-RH-184
ORTHOGONAL_PERIODIC_FACTOR_ANATOMY_FLUCTUATIONS_ARE_NOT_UNIVERSALLY_FORCED_BY_A_ZETA_BOUNDARY_ZERO

Update:

PERIODIC / SMALL-FACTOR MIXED STATE ROUTE
CLOSED AS PRINCIPAL-RANK-ONE OR ROOT-BLIND.

No new root frontier is opened yet.

Next screen:

GLOBAL PRIME-ERROR / MERTENS-STATE CROSS SPECTRUM

No RH certificate is created.

17. Conclusion

Adding a finite factor state to a prime count creates a higher-dimensional arithmetic observable, but its principal zeta spectrum remains one-dimensional.

The reduced-residue mean of the state is the exact projector coefficient.

Centering removes the root mode.

Not centering retains only a scalar copy of it.

Thus small-prime divisibility anatomy, roughness gates, and finite sieve states do not create new universal boundary coercivity.

A genuinely new mixed-state route must leave the periodic/sieve category and carry additional zeta-zero data not already encoded by the prime-error two-point spectrum.

CSM_RH Paper 83 — Principal-Character Projectors for Shifted-Prime Factor States and the Rank-One Mixed-State Barrier

CSM_RH Paper 83

Principal-Character Projectors for Shifted-Prime Factor States and the Rank-One Mixed-State Barrier

Abstract

Principal-character factor-state projector theorem

1. Periodic shifted factor states

2. Shifted-prime character decomposition

3. Principal-character projector

Theorem 3.1 — Periodic factor-state zeta projector

4. Centering dichotomy

Corollary 4.1 — Principal-centering theorem

5. Finite divisor states are periodic

6. Roughness gate density

Theorem 6.1 — Shifted-prime roughness principal density

7. Vector partition theorem

Theorem 7.1 — Rank-one principal direction

8. Orthogonal anatomy fluctuations are root-blind

9. Absolute versus conditional centering

10. Why improved anatomy alone does not improve the PNT error

11. Current factor-anatomy literature calibration

Ford, 2025

Bharadwaj–Rodgers, 2026

12. Screening of Möbius / Liouville shifted-prime states

13. First surviving mixed-state class

14. New mixed-state entry rule

15. External calibration

15.1. Ford, 2025

15.2. Bharadwaj–Rodgers, 2026

15.3. Lichtman

16. State transition

17. Conclusion