# 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: ```text 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. If $$ g(m) = \sum_{d\in\mathcal D} \lambda_d \mathbf 1_{d\mid m}, $$ 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: 1. a nonperiodic complete-factorization state; 2. a second global arithmetic field with its own zeta-zero residue data; 3. 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: ```text 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: ```text 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: ```text 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: ```text 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: ```text 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: ```text 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: ```text 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: ```text 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: ```text 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: ```text 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: ```text 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: 1. it contains additional spectral data not determined by the prime-error $C_2$ ; 2. a zeta boundary zero forces a nonzero mixed signal; 3. the signal survives the single-boundary-pair model; 4. 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: ```text 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: ```text PERIODIC / SMALL-FACTOR MIXED STATE ROUTE CLOSED AS PRINCIPAL-RANK-ONE OR ROOT-BLIND. ``` No new root frontier is opened yet. Next screen: ```text 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.