# CSM_RH Paper 77 ## Exact Renormalized Vaughan Extraction for the Signed Shift-Prime Root Sequence **Project:** CSM_RH **Paper:** 77 **Version:** v0.1 **Date:** 2026-09-09 **Campaign:** 47 — `ORDINARY_PRIME_BOUNDARY_BREAKING` **Track:** C47-B-v2 — `EXACT_VAUGHAN_SIGNED_ROOT_DECOMPOSITION` **Status:** EXACT POWER-OUTPUT DECOMPOSITION CERTIFIED / PAPER-75 BILINEAR AXIOM SUPERSEDED AS ROOT INPUT / RENORMALIZED TYPE-II DEFECT OPEN **Canonical entry state:** v1.67 / Paper 76 v0.1 **RH_PROVED:** FALSE **RH_DISPROVED:** FALSE **GLOBAL_RH_CERTIFICATE:** FALSE --- # Abstract Paper 76 corrected the first Campaign-47 root bridge and identified the signed root sequence $$ f_{N,H}(n) = W(n/N) \sum_r \omega_H(r) \left( \Lambda(n+r)-1 \right). $$ It proved $$ \mathcal R_W(N,H) = \sum_n \Lambda(n)f_{N,H}(n) - F(N,H), $$ where $$ F(N,H) = \sum_n f_{N,H}(n), $$ and also proved the deterministic divisor law $$ F_d = \frac Fd + O_W(N). $$ The present paper applies Vaughan's identity **directly to this signed root sequence**, retains every Type-I main term exactly, and identifies the unique renormalized Type-II quantity which remains. Let $$ U,V>1 $$ with $$ VV$, Vaughan's identity is $$ \boxed{ \Lambda(n) = \sum_{\substack{dr=n\\d\le U}} \mu(d)\log r - \sum_{\substack{kr=n\\k\le UV}} a_{U,V}(k) - \sum_{\substack{ek=n\\e>V\\k>U}} \Lambda(e)b_U(k). } $$ Applying this to $f=f_{N,H}$ gives $$ \sum_n\Lambda(n)f(n) = T_1-T_2-T_3. $$ Define $$ F_d=\sum_{d\mid n}f(n), $$ and introduce the logarithmically weighted companion $$ G = \sum_n f(n)\log(n/N), $$ $$ G_d = \sum_{d\mid n} f(n)\log(n/N). $$ The same bounded-variation lattice argument as in Papers 75–76 gives $$ \boxed{ F_d = \frac Fd+r_d, \qquad G_d = \frac Gd+s_d, } $$ with $$ r_d,s_d = O_W(N) $$ uniformly, and the aggregate power bounds $$ \boxed{ \sum_{d\le D} \tau_5(d) \left( |r_d|+|s_d| \right) \ll_W ND(\log N)^{O(1)}. } $$ Now define the truncated scalar coefficients $$ \boxed{ M_U = \sum_{d\le U}\frac{\mu(d)}d, } $$ $$ \boxed{ J_U = \sum_{d\le U} \frac{\mu(d)\log d}{d}, } $$ $$ \boxed{ L_V = \sum_{e\le V} \frac{\Lambda(e)}e. } $$ The two Type-I terms then have an exact main-term recombination. The first Type-I piece is $$ T_1 = \sum_{d\le U} \mu(d) \left[ (\log N-\log d)F_d+G_d \right]. $$ The second is $$ T_2 = \sum_{k\le UV} a_{U,V}(k)F_k. $$ Since $$ \boxed{ \sum_{k\le UV}\frac{a_{U,V}(k)}k = M_U L_V, } $$ one obtains $$ \boxed{ T_1-T_2-F = \mathcal M_{U,V}(F,G;N) + \mathcal E_I, } $$ where $$ \boxed{ \mathcal M_{U,V} = F \left[ M_U(\log N-L_V)-J_U-1 \right] + G M_U } $$ and $$ \boxed{ \begin{aligned} \mathcal E_I &= \sum_{d\le U} \mu(d) \left[ (\log N-\log d)r_d+s_d \right] \\ &\quad - \sum_{k\le UV} a_{U,V}(k)r_k. \end{aligned} } $$ The Type-II term is $$ \boxed{ \mathcal T^{II}_{U,V}(f) = \sum_{\substack{e>V,\ k>U\\ek\in\operatorname{supp}f}} \Lambda(e)b_U(k)f(ek). } $$ Therefore the root covariance has the exact decomposition $$ \boxed{ \mathcal R_W = \mathcal M_{U,V} - \mathcal T^{II}_{U,V} + \mathcal E_I. } $$ Equivalently, define the **renormalized Vaughan defect** $$ \boxed{ \mathcal V^{\rm ren}_{U,V} = \mathcal T^{II}_{U,V} - \mathcal M_{U,V}. } $$ Then $$ \boxed{ \mathcal R_W = - \mathcal V^{\rm ren}_{U,V} + \mathcal E_I. } $$ This is the principal result of the paper. The Type-I error is power-controlled. Using $$ |a_{U,V}(k)| \le \sum_{e\mid k}\Lambda(e) = \log k, $$ and the aggregate divisor discrepancy, $$ \boxed{ \mathcal E_I \ll_W NUV (\log N)^{O(1)}. } $$ Take $$ H=N^{1-\tau}, \qquad U=N^u, \qquad V=N^v. $$ Relative to the natural root covariance scale $$ NH, $$ the Type-I error has saving exponent $$ \boxed{ \eta_I = 1-\tau-u-v. } $$ Therefore, for a PESC $(\kappa)$ seed, if $$ \boxed{ u+v < 1-\tau-\kappa-\eta } $$ for some fixed $\eta>0$, then $$ \boxed{ \mathcal E_I \ll NH N^{-\kappa-\eta+o(1)}. } $$ This proves that **exact Vaughan extraction itself preserves fixed power**. The generic logarithmic extraction floor of the 1998 asymptotic sieve is absent. But the paper also identifies why an unrenormalized Type-II estimate is the wrong target. Under PESC $(\kappa)$, put $$ d=\frac{\kappa}{2}. $$ The seed gives $$ M_U \ll U^{-d+o(1)}, $$ $$ J_U+1 \ll U^{-d+o(1)} \log U, $$ and $$ L_V = \log V - \gamma + O \left( V^{-d+o(1)} \right). $$ Moreover, $$ F,G \ll HN^{1-d+o(1)}. $$ Hence $$ \boxed{ \mathcal M_{U,V} \ll NH N^{-d} U^{-d} N^{o(1)}. } $$ If $$ U=N^u, $$ then the scalar Type-I main has effective exponent $$ \boxed{ d(1+u). } $$ For every fixed $$ u<1, $$ $$ d(1+u)<2d=\kappa. $$ Thus $\mathcal M_{U,V}$ can be parametrically larger than the root-critical covariance scale. The exact identity then forces $\mathcal T^{II}_{U,V}$ to carry a matching one-point truncation component. Therefore: ```text DO NOT PROVE T^{II}_{U,V} IS SMALL. PROVE T^{II}_{U,V} - M_{U,V} IS SMALL. ``` This supersedes the root use of Paper 75's F-RH-023. The new arithmetic frontier is: ```text F-RH-024 RENORMALIZED VAUGHAN BALANCED DEFECT POWER ``` Find fixed $$ \tau,u,v,\eta>0 $$ such that $$ u+v < 1-\tau-\kappa-\eta $$ and $$ \boxed{ \left| \mathcal T^{II}_{U,V}(f_{N,H}) - \mathcal M_{U,V}(F,G;N) \right| \ll NH N^{-\kappa-\eta}. } $$ Then the exact identity gives $$ \boxed{ \mathcal R_W(N,H) \ll NH N^{-\kappa-\eta+o(1)}. } $$ Provided additionally $$ \kappa+\eta<1-\tau, $$ the already-solved local Hardy–Littlewood variance is smaller, so the weighted pair energy has a strict fixed-power excess beyond the seed. Thus F-RH-024 is a direct root bootstrap inequality. It has no generic sieve extraction loss and no shifted-Möbius coercivity bridge. Its arithmetic content is very specific: > the ordinary-factorization Type-II term must reproduce the explicit one-point Möbius truncation main $\mathcal M_{U,V}$, with a covariance defect smaller by a fixed power. This is a second-order parity-cancellation theorem rather than a first-order prime-producing theorem. The pseudo-prime and Beurling calibrations remain effective. The term $$ \Lambda(e)b_U(k)f(ek) $$ simultaneously uses: - actual prime powers through $\Lambda(e)$ ; - ordinary divisor truncation through $b_U(k)$ ; - ordinary product $ek$ ; - ordinary additive shifts hidden in $f(ek)$. The exact renormalization $\mathcal M_{U,V}$ additionally uses the ordinary Möbius and von Mangoldt Dirichlet coefficients. Thus F-RH-024 is not implied by the positive integer-lattice pseudo-prime properties of Paper 64 and is not intrinsic to a generic Beurling prime system. Campaign 47 therefore passes C47-B-v2. The next task is no longer extraction. It is to attack F-RH-024 itself. No RH theorem is claimed. --- # 1. Signed root sequence Recall $$ \boxed{ f(n) = f_{N,H}(n) = W(n/N) \sum_r \omega_H(r) \left( \Lambda(n+r)-1 \right). } $$ Let $$ \boxed{ F=\sum_n f(n). } $$ Paper 76 proved $$ \boxed{ \mathcal R_W = \sum_n\Lambda(n)f(n)-F. } $$ All sums below are automatically restricted to the compact dyadic support of $f$. --- # 2. Vaughan identity For parameters $$ U,V>1 $$ define $$ b_U(k) = \sum_{\substack{d\mid k\\d\le U}} \mu(d) $$ and $$ a_{U,V}(k) = \sum_{\substack{de=k\\d\le U\\e\le V}} \mu(d)\Lambda(e). $$ For $n>V$, $$ \boxed{ \Lambda(n) = \sum_{\substack{dr=n\\d\le U}} \mu(d)\log r - \sum_{\substack{kr=n\\k\le UV}} a_{U,V}(k) - \sum_{\substack{ek=n\\e>V\\k>U}} \Lambda(e)b_U(k). } $$ This is the standard Vaughan identity obtained by comparing the coefficients in the truncated Dirichlet-series decomposition. External calibration: Encyclopedia of Mathematics, `Vaughan identity`. --- # 3. Exact decomposition of the root prime-detection sum Multiply Vaughan's identity by $f(n)$ and sum. Define $$ T_1 = \sum_{d\le U} \mu(d) \sum_{d\mid n} f(n)\log(n/d), $$ $$ T_2 = \sum_{k\le UV} a_{U,V}(k) \sum_{k\mid n} f(n), $$ and $$ T_3 = \sum_{\substack{e>V,\ k>U}} \Lambda(e)b_U(k)f(ek). $$ Then $$ \boxed{ \sum_n \Lambda(n)f(n) = T_1-T_2-T_3. } $$ Therefore $$ \boxed{ \mathcal R_W = T_1-T_2-T_3-F. } $$ No approximation has yet been made. --- # 4. Log-weighted divisor law Define $$ \boxed{ G = \sum_n f(n)\log(n/N), } $$ and $$ \boxed{ G_d = \sum_{d\mid n} f(n)\log(n/N). } $$ Multiplying the smooth weight $W(u)$ by $\log u$ preserves compact support and bounded variation. The same lattice-sampling argument as Paper 76 therefore gives: ## Theorem 4.1 — Log-weighted signed divisor law $$ \boxed{ G_d = \frac Gd + s_d, } $$ with $$ \boxed{ s_d=O_W(N) } $$ uniformly. Moreover, for every $D\ge1$, $$ \boxed{ \sum_{d\le D} \tau_5(d) \left( |r_d|+|s_d| \right) \ll_W ND(\log N)^{O(1)}, } $$ where $$ F_d=\frac Fd+r_d. $$ Create: ```text B-RH-119 LOG_WEIGHTED_SIGNED_SHIFT_ERROR_SEQUENCE_RETAINS_DETERMINISTIC_DIVISOR_DISTRIBUTION CERTIFIED ``` --- # 5. Exact first Type-I main term Since $$ \log(n/d) = \log N - \log d + \log(n/N), $$ $$ \begin{aligned} T_1 &= \sum_{d\le U} \mu(d) \left[ (\log N-\log d)F_d+G_d \right]. \end{aligned} $$ Define $$ M_U = \sum_{d\le U} \frac{\mu(d)}d, $$ and $$ J_U = \sum_{d\le U} \frac{\mu(d)\log d}{d}. $$ Substitute the divisor laws: $$ \boxed{ \begin{aligned} T_1 &= F \left( \log N\,M_U-J_U \right) + G M_U \\ &\quad + \sum_{d\le U} \mu(d) \left[ (\log N-\log d)r_d+s_d \right]. \end{aligned} } $$ --- # 6. Exact second Type-I main term Since $$ T_2 = \sum_{k\le UV} a_{U,V}(k)F_k, $$ we need the scalar sum $$ \sum_{k\le UV} \frac{a_{U,V}(k)}k. $$ By definition, $$ \begin{aligned} \sum_{k\le UV} \frac{a_{U,V}(k)}k &= \sum_{d\le U} \frac{\mu(d)}d \sum_{e\le V} \frac{\Lambda(e)}e \\ &= \boxed{ M_U L_V, } \end{aligned} $$ where $$ L_V = \sum_{e\le V} \frac{\Lambda(e)}e. $$ Therefore: $$ \boxed{ T_2 = F M_U L_V + \sum_{k\le UV} a_{U,V}(k)r_k. } $$ --- # 7. Renormalized exact root identity Combine Sections 5–6 and subtract $F$. Define $$ \boxed{ \mathcal M_{U,V} = F \left[ M_U(\log N-L_V) - J_U - 1 \right] + G M_U. } $$ Define $$ \boxed{ \begin{aligned} \mathcal E_I &= \sum_{d\le U} \mu(d) \left[ (\log N-\log d)r_d+s_d \right] \\ &\quad - \sum_{k\le UV} a_{U,V}(k)r_k. \end{aligned} } $$ Then: ## Theorem 7.1 — Exact renormalized Vaughan root decomposition $$ \boxed{ \mathcal R_W = \mathcal M_{U,V} - \mathcal T^{II}_{U,V} + \mathcal E_I, } $$ where $$ \boxed{ \mathcal T^{II}_{U,V} = \sum_{\substack{e>V,\ k>U}} \Lambda(e)b_U(k)f(ek). } $$ Equivalently, $$ \boxed{ \mathcal R_W = - \mathcal V^{\rm ren}_{U,V} + \mathcal E_I, } $$ where $$ \boxed{ \mathcal V^{\rm ren}_{U,V} = \mathcal T^{II}_{U,V} - \mathcal M_{U,V}. } $$ Create: ```text B-RH-120 EXACT_RENORMALIZED_VAUGHAN_DECOMPOSITION_OF_THE_SIGNED_ROOT_COVARIANCE CERTIFIED ``` --- # 8. Power bound for the Type-I error The coefficient $a_{U,V}(k)$ satisfies $$ \begin{aligned} |a_{U,V}(k)| &\le \sum_{\substack{e\mid k\\e\le V}} \Lambda(e) \\ &\le \sum_{e\mid k} \Lambda(e) \\ &= \boxed{ \log k. } \end{aligned} $$ Therefore, using Theorem 4.1, $$ \begin{aligned} |\mathcal E_I| &\ll (\log N) \sum_{d\le U} \left( |r_d|+|s_d| \right) + (\log N) \sum_{k\le UV} |r_k| \\ &\ll \boxed{ NUV (\log N)^{O(1)}. } \end{aligned} $$ Create: ```text B-RH-121 EXACT_VAUGHAN_TYPE_I_ERROR_RETAINS_FIXED_POWER_FROM_SHIFT_AVERAGED_DIVISOR_DISTRIBUTION CERTIFIED ``` --- # 9. Exponent ledger Set $$ H=N^{1-\tau}, $$ $$ U=N^u, $$ $$ V=N^v. $$ Then $$ NH = N^{2-\tau}. $$ The Type-I error satisfies $$ \boxed{ \frac{ |\mathcal E_I| }{ NH } \ll N^{-(1-\tau-u-v)+o(1)}. } $$ Hence the Type-I saving exponent is $$ \boxed{ \eta_I = 1-\tau-u-v. } $$ For a seed PESC $(\kappa)$ and desired excess $\eta>0$, Type I is supercritical provided $$ \boxed{ u+v < 1-\tau-\kappa-\eta. } $$ This is compatible with positive $u,v$ whenever $$ \tau<1-\kappa-\eta. $$ Thus the exact extraction produces no logarithmic loss. --- # 10. Seed size of the explicit renormalization Assume PESC $(\kappa)$ and put $$ d=\frac{\kappa}{2}. $$ The seed Mertens bound gives $$ \boxed{ M_U \ll U^{-d+o(1)}. } $$ Since $$ \frac1{\zeta(s)} \sim s-1 $$ at $s=1$, $$ \sum_{n=1}^{\infty} \frac{\mu(n)\log n}{n} = -1. $$ Partial summation with the seed Mertens bound gives $$ \boxed{ J_U+1 \ll U^{-d+o(1)} \log U. } $$ Likewise the seed PNT error gives $$ \boxed{ L_V = \log V-\gamma + O \left( V^{-d+o(1)} \right). } $$ Finally, $$ \boxed{ F,G \ll HN^{1-d+o(1)}. } $$ Therefore: ## Theorem 10.1 — Size of the explicit one-point renormalization $$ \boxed{ \mathcal M_{U,V} \ll NH N^{-d} U^{-d} N^{o(1)}. } $$ If $$ U=N^u, $$ the effective exponent is $$ \boxed{ d(1+u). } $$ --- # 11. Why unrenormalized Type II is the wrong target For every fixed $$ u<1, $$ $$ d(1+u) < 2d = \kappa. $$ Hence the explicit Type-I main may be larger than the desired root-critical scale. The exact identity then implies that the Type-II term must contain a matching component. Therefore a conjecture such as $$ \mathcal T^{II}_{U,V} \ll NHN^{-\kappa-\eta} $$ is structurally incompatible with the required cancellation unless the explicit main $\mathcal M_{U,V}$ is independently negligible, which it is not at the seed level. Create: ```text O-RH-174 UNRENORMALIZED_VAUGHAN_TYPE_II_SMALLNESS_DISCARDS_A_NECESSARY_ONE_POINT_TRUNCATION_COUNTERTERM CERTIFIED_AS_METHOD_BARRIER ``` The correct object is $$ \mathcal T^{II}_{U,V} - \mathcal M_{U,V}. $$ --- # 12. New arithmetic frontier F-RH-024 Open: ```text F-RH-024 RENORMALIZED_VAUGHAN_BALANCED_DEFECT_POWER ``` Given a PESC $(\kappa)$ seed, find fixed $$ \tau,u,v,\eta>0 $$ such that $$ \boxed{ u+v < 1-\tau-\kappa-\eta } $$ and $$ \boxed{ \left| \mathcal T^{II}_{U,V} - \mathcal M_{U,V} \right| \ll NH N^{-\kappa-\eta}. } $$ Then Theorem 7.1 and Section 9 give $$ \boxed{ \mathcal R_W \ll NH N^{-\kappa-\eta+o(1)}. } $$ This is a direct ordinary-prime root covariance bound. Create: ```text B-RH-122 F_RH_024_PLUS_THE_DETERMINISTIC_TYPE_I_POWER_GIVES_A_DIRECT_ROOT_COVARIANCE_EXCESS CERTIFIED_CONDITIONAL_BRIDGE ``` --- # 13. Connection to F-RH-022 The triangular kernel is the normalized Fejér lag kernel. The local diagonal and modified-singular-series contribution has scale $$ N\log(N/H) = NH N^{-(1-\tau)+o(1)}. $$ Therefore, if $$ \boxed{ \kappa+\eta < 1-\tau, } $$ the solved local main term is smaller than the F-RH-024 target. After standard dyadic smooth partitioning, the bound $$ \mathcal R_W \ll NHN^{-\kappa-\eta} $$ therefore supplies the same fixed-power excess required by the weighted version of F-RH-022. No shifted-Möbius or central-coercivity bridge is needed. --- # 14. Status of F-RH-023 Paper 75's F-RH-023 remains a meaningful Friedlander–Iwaniec parity statement. However, the exact root extraction has now identified a different arithmetic quantity. Update: ```text F-RH-023 OPEN_AUXILIARY_PARITY_CANDIDATE NO_LONGER_PREFERRED_ROOT_INPUT ``` The preferred Campaign-47 arithmetic frontier becomes F-RH-024. This is not merely a change of notation. F-RH-024 includes the explicit counterterm required by the exact Vaughan identity. --- # 15. Pseudo-prime / Beurling discrimination The balanced term contains $$ \Lambda(e)b_U(k)f(ek). $$ This requires ordinary: - von Mangoldt prime-power weights; - Möbius divisor truncation; - multiplication $ek$ ; - additive shifts inside $f(ek)$. The counterterm uses $$ M_U,\quad J_U,\quad L_V, $$ which are ordinary Möbius and von Mangoldt Dirichlet coefficients. The positive pseudo-prime model of Paper 64 does not determine these quantities from its preserved properties. A generic Beurling prime system does not identify its generalized multiplicative semigroup with the ordinary additive lattice in the required way. Thus the new frontier remains inside the intended ordinary-prime intersection. --- # 16. External calibration ## 16.1. Vaughan identity The Encyclopedia of Mathematics gives the exact decomposition into: - a first Type-I term with $\mu(d)\log r$ ; - a second Type-I term with the truncated convolution coefficient; - a Type-II term with $$ \Lambda(e) \sum_{\substack{d\mid k\\d\le U}}\mu(d). $$ URL: https://encyclopediaofmath.org/wiki/Vaughan_identity ## 16.2. Ford–Maynard prime-producing sieves Ford and Maynard formulate Type I / Type II comparison estimates and record that Vaughan's identity yields an asymptotic when $$ \gamma+\nu>1. $$ Their more general theory shows that sufficiently strong Type-I/II information can be transferred to the prime sum without an intrinsic logarithmic extraction floor. URL: https://www.ford126.web.illinois.edu/wwwpapers/prime-producing-sieves.pdf This is used only as extraction calibration; the present theorem is an exact direct computation on the signed root sequence. --- # 17. State transition Advance candidate state $$ v1.67 \to v1.68. $$ Add: ```text B-RH-119 LOG_WEIGHTED_SIGNED_SHIFT_ERROR_SEQUENCE_RETAINS_DETERMINISTIC_DIVISOR_DISTRIBUTION B-RH-120 EXACT_RENORMALIZED_VAUGHAN_DECOMPOSITION_OF_THE_SIGNED_ROOT_COVARIANCE B-RH-121 EXACT_VAUGHAN_TYPE_I_ERROR_RETAINS_FIXED_POWER_FROM_SHIFT_AVERAGED_DIVISOR_DISTRIBUTION B-RH-122 F_RH_024_PLUS_THE_DETERMINISTIC_TYPE_I_POWER_GIVES_A_DIRECT_ROOT_COVARIANCE_EXCESS O-RH-174 UNRENORMALIZED_VAUGHAN_TYPE_II_SMALLNESS_DISCARDS_A_NECESSARY_ONE_POINT_TRUNCATION_COUNTERTERM ``` Open: ```text F-RH-024 RENORMALIZED_VAUGHAN_BALANCED_DEFECT_POWER OPEN_ROOT_ARITHMETIC ``` Downgrade: ```text F-RH-023 AUXILIARY_PARITY_CANDIDATE ``` No RH certificate is created. --- # 18. Recommended next action C47-B-v2 is complete. The next round should attack F-RH-024 itself. Do not estimate $\mathcal T^{II}_{U,V}$ absolutely. Instead: 1. dyadically decompose $e$ and $k$ ; 2. insert the definition $$ f(ek) = W(ek/N) \sum_r \omega_H(r) (\Lambda(ek+r)-1); $$ 3. isolate the exact contribution corresponding to $\mathcal M_{U,V}$ ; 4. test whether the remaining balanced covariance has a large-sieve, dispersion, or Möbius-parity fixed-power saving. This is now a concrete arithmetic theorem with an exact root bridge. --- # 19. Conclusion The Campaign-47 extraction problem has been solved at the structural level. Vaughan's identity preserves the fixed-power divisor information generated by shift averaging. The price is not logarithmic. The real issue is that the balanced Type-II term contains an explicit one-point truncation component which must be subtracted. After that renormalization, one single arithmetic quantity remains. That quantity is F-RH-024. Campaign 47 now has its first root-sufficient fixed-power inequality.