CSM_RH Paper 77
Exact Renormalized Vaughan Extraction for the Signed Shift-Prime Root Sequence
Project: CSM_RHPaper: 77Version: v0.1Date: 2026-09-09Campaign: 47 — ORDINARY_PRIME_BOUNDARY_BREAKINGTrack: C47-B-v2 — EXACT_VAUGHAN_SIGNED_ROOT_DECOMPOSITIONStatus: EXACT POWER-OUTPUT DECOMPOSITION CERTIFIED / PAPER-75 BILINEAR AXIOM SUPERSEDED AS ROOT INPUT / RENORMALIZED TYPE-II DEFECT OPENCanonical entry state: v1.67 / Paper 76 v0.1RH_PROVED: FALSERH_DISPROVED: FALSEGLOBAL_RH_CERTIFICATE: FALSE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 ) ∑ r ω H ( r ) ( Λ ( n + r ) − 1 ) . f_{N,H}(n)
=
W(n/N)
\sum_r
\omega_H(r)
\left(
\Lambda(n+r)-1
\right). f N , H ( n ) = W ( n / N ) r ∑ ω H ( r ) ( Λ ( n + r ) − 1 ) . It proved
R W ( N , H ) = ∑ n Λ ( n ) f N , H ( n ) − F ( N , H ) , \mathcal R_W(N,H)
=
\sum_n
\Lambda(n)f_{N,H}(n)
-
F(N,H), R W ( N , H ) = n ∑ Λ ( n ) f N , H ( n ) − F ( N , H ) , where
F ( N , H ) = ∑ n f N , H ( n ) , F(N,H)
=
\sum_n f_{N,H}(n), F ( N , H ) = n ∑ f N , H ( n ) , and also proved the deterministic divisor law
F d = F d + O W ( N ) . F_d
=
\frac Fd
+
O_W(N). F d = d F + 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 U,V>1 U , V > 1 with
V < N , V<N, V < N , and define
b U ( k ) = ∑ d ∣ k d ≤ U μ ( d ) , \boxed{
b_U(k)
=
\sum_{\substack{d\mid k\\d\le U}}
\mu(d),
} b U ( k ) = d ∣ k d ≤ U ∑ μ ( d ) , a U , V ( k ) = ∑ d e = k d ≤ U e ≤ V μ ( d ) Λ ( e ) . \boxed{
a_{U,V}(k)
=
\sum_{\substack{de=k\\d\le U\\e\le V}}
\mu(d)\Lambda(e).
} a U , V ( k ) = d e = k d ≤ U e ≤ V ∑ μ ( d ) Λ ( e ) . For every n > V n>V n > V
Λ ( n ) = ∑ d r = n d ≤ U μ ( d ) log r − ∑ k r = n k ≤ U V a U , V ( k ) − ∑ e k = n e > V k > U Λ ( e ) b U ( k ) . \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).
} Λ ( n ) = d r = n d ≤ U ∑ μ ( d ) log r − k r = n k ≤ U V ∑ a U , V ( k ) − e k = n e > V k > U ∑ Λ ( e ) b U ( k ) . Applying this to f = f N , H f=f_{N,H} f = f N , H
∑ n Λ ( n ) f ( n ) = T 1 − T 2 − T 3 . \sum_n\Lambda(n)f(n)
=
T_1-T_2-T_3. n ∑ Λ ( n ) f ( n ) = T 1 − T 2 − T 3 . Define
F d = ∑ d ∣ n f ( n ) , F_d=\sum_{d\mid n}f(n), F d = d ∣ n ∑ f ( n ) , and introduce the logarithmically weighted companion
G = ∑ n f ( n ) log ( n / N ) , G
=
\sum_n
f(n)\log(n/N), G = n ∑ f ( n ) log ( n / N ) , G d = ∑ d ∣ n f ( n ) log ( n / N ) . G_d
=
\sum_{d\mid n}
f(n)\log(n/N). G d = d ∣ n ∑ f ( n ) log ( n / N ) . The same bounded-variation lattice argument as in Papers 75–76 gives
F d = F d + r d , G d = G d + s d , \boxed{
F_d
=
\frac Fd+r_d,
\qquad
G_d
=
\frac Gd+s_d,
} F d = d F + r d , G d = d G + s d , with
r d , s d = O W ( N ) r_d,s_d
=
O_W(N) r d , s d = O W ( N ) uniformly, and the aggregate power bounds
∑ d ≤ D τ 5 ( d ) ( ∣ r d ∣ + ∣ s d ∣ ) ≪ W N D ( log N ) O ( 1 ) . \boxed{
\sum_{d\le D}
\tau_5(d)
\left(
|r_d|+|s_d|
\right)
\ll_W
ND(\log N)^{O(1)}.
} d ≤ D ∑ τ 5 ( d ) ( ∣ r d ∣ + ∣ s d ∣ ) ≪ W N D ( log N ) O ( 1 ) . Now define the truncated scalar coefficients
M U = ∑ d ≤ U μ ( d ) d , \boxed{
M_U
=
\sum_{d\le U}\frac{\mu(d)}d,
} M U = d ≤ U ∑ d μ ( d ) , J U = ∑ d ≤ U μ ( d ) log d d , \boxed{
J_U
=
\sum_{d\le U}
\frac{\mu(d)\log d}{d},
} J U = d ≤ U ∑ d μ ( d ) log d , L V = ∑ e ≤ V Λ ( e ) e . \boxed{
L_V
=
\sum_{e\le V}
\frac{\Lambda(e)}e.
} L V = e ≤ V ∑ e Λ ( e ) . The two Type-I terms then have an exact main-term recombination.
The first Type-I piece is
T 1 = ∑ d ≤ U μ ( d ) [ ( log N − log d ) F d + G d ] . T_1
=
\sum_{d\le U}
\mu(d)
\left[
(\log N-\log d)F_d+G_d
\right]. T 1 = d ≤ U ∑ μ ( d ) [ ( log N − log d ) F d + G d ] . The second is
T 2 = ∑ k ≤ U V a U , V ( k ) F k . T_2
=
\sum_{k\le UV}
a_{U,V}(k)F_k. T 2 = k ≤ U V ∑ a U , V ( k ) F k . Since
∑ k ≤ U V a U , V ( k ) k = M U L V , \boxed{
\sum_{k\le UV}\frac{a_{U,V}(k)}k
=
M_U L_V,
} k ≤ U V ∑ k a U , V ( k ) = M U L V , one obtains
T 1 − T 2 − F = M U , V ( F , G ; N ) + E I , \boxed{
T_1-T_2-F
=
\mathcal M_{U,V}(F,G;N)
+
\mathcal E_I,
} T 1 − T 2 − F = M U , V ( F , G ; N ) + E I , where
M U , V = F [ M U ( log N − L V ) − J U − 1 ] + G M U \boxed{
\mathcal M_{U,V}
=
F
\left[
M_U(\log N-L_V)-J_U-1
\right]
+
G M_U
} M U , V = F [ M U ( log N − L V ) − J U − 1 ] + G M U and
E I = ∑ d ≤ U μ ( d ) [ ( log N − log d ) r d + s d ] − ∑ k ≤ U V a U , V ( k ) r k . \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}
} E I = d ≤ U ∑ μ ( d ) [ ( log N − log d ) r d + s d ] − k ≤ U V ∑ a U , V ( k ) r k . The Type-II term is
T U , V I I ( f ) = ∑ e > V , k > U e k ∈ supp f Λ ( e ) b U ( k ) f ( e k ) . \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).
} T U , V I I ( f ) = e > V , k > U e k ∈ supp f ∑ Λ ( e ) b U ( k ) f ( e k ) . Therefore the root covariance has the exact decomposition
R W = M U , V − T U , V I I + E I . \boxed{
\mathcal R_W
=
\mathcal M_{U,V}
-
\mathcal T^{II}_{U,V}
+
\mathcal E_I.
} R W = M U , V − T U , V I I + E I . Equivalently, define the renormalized Vaughan defect
V U , V r e n = T U , V I I − M U , V . \boxed{
\mathcal V^{\rm ren}_{U,V}
=
\mathcal T^{II}_{U,V}
-
\mathcal M_{U,V}.
} V U , V ren = T U , V I I − M U , V . Then
R W = − V U , V r e n + E I . \boxed{
\mathcal R_W
=
-
\mathcal V^{\rm ren}_{U,V}
+
\mathcal E_I.
} R W = − V U , V ren + E I . This is the principal result of the paper.
The Type-I error is power-controlled.
Using
∣ a U , V ( k ) ∣ ≤ ∑ e ∣ k Λ ( e ) = log k , |a_{U,V}(k)|
\le
\sum_{e\mid k}\Lambda(e)
=
\log k, ∣ a U , V ( k ) ∣ ≤ e ∣ k ∑ Λ ( e ) = log k , and the aggregate divisor discrepancy,
E I ≪ W N U V ( log N ) O ( 1 ) . \boxed{
\mathcal E_I
\ll_W
NUV
(\log N)^{O(1)}.
} E I ≪ W N U V ( log N ) O ( 1 ) . Take
H = N 1 − τ , U = N u , V = N v . H=N^{1-\tau},
\qquad
U=N^u,
\qquad
V=N^v. H = N 1 − τ , U = N u , V = N v . Relative to the natural root covariance scale
N H , NH, N H , the Type-I error has saving exponent
η I = 1 − τ − u − v . \boxed{
\eta_I
=
1-\tau-u-v.
} η I = 1 − τ − u − v . Therefore, for a PESC ( κ ) (\kappa) ( κ )
u + v < 1 − τ − κ − η \boxed{
u+v
<
1-\tau-\kappa-\eta
} u + v < 1 − τ − κ − η for some fixed η > 0 \eta>0 η > 0
E I ≪ N H N − κ − η + o ( 1 ) . \boxed{
\mathcal E_I
\ll
NH
N^{-\kappa-\eta+o(1)}.
} E I ≪ N H N − κ − η + 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 ( κ ) (\kappa) ( κ )
d = κ 2 . d=\frac{\kappa}{2}. d = 2 κ . The seed gives
M U ≪ U − d + o ( 1 ) , M_U
\ll
U^{-d+o(1)}, M U ≪ U − d + o ( 1 ) , J U + 1 ≪ U − d + o ( 1 ) log U , J_U+1
\ll
U^{-d+o(1)}
\log U, J U + 1 ≪ U − d + o ( 1 ) log U , and
L V = log V − γ + O ( V − d + o ( 1 ) ) . L_V
=
\log V
-
\gamma
+
O
\left(
V^{-d+o(1)}
\right). L V = log V − γ + O ( V − d + o ( 1 ) ) . Moreover,
F , G ≪ H N 1 − d + o ( 1 ) . F,G
\ll
HN^{1-d+o(1)}. F , G ≪ H N 1 − d + o ( 1 ) . Hence
M U , V ≪ N H N − d U − d N o ( 1 ) . \boxed{
\mathcal M_{U,V}
\ll
NH
N^{-d}
U^{-d}
N^{o(1)}.
} M U , V ≪ N H N − d U − d N o ( 1 ) . If
U = N u , U=N^u, U = N u , then the scalar Type-I main has effective exponent
d ( 1 + u ) . \boxed{
d(1+u).
} d ( 1 + u ) . For every fixed
u < 1 , u<1, u < 1 , d ( 1 + u ) < 2 d = κ . d(1+u)<2d=\kappa. d ( 1 + u ) < 2 d = κ . Thus M U , V \mathcal M_{U,V} M U , V
The exact identity then forces T U , V I I \mathcal T^{II}_{U,V} T U , V I I
Therefore:
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:
F-RH-024
RENORMALIZED VAUGHAN BALANCED DEFECT POWER
Find fixed
τ , u , v , η > 0 \tau,u,v,\eta>0 τ , u , v , η > 0 such that
u + v < 1 − τ − κ − η u+v
<
1-\tau-\kappa-\eta u + v < 1 − τ − κ − η and
∣ T U , V I I ( f N , H ) − M U , V ( F , G ; N ) ∣ ≪ N H N − κ − η . \boxed{
\left|
\mathcal T^{II}_{U,V}(f_{N,H})
-
\mathcal M_{U,V}(F,G;N)
\right|
\ll
NH
N^{-\kappa-\eta}.
} T U , V I I ( f N , H ) − M U , V ( F , G ; N ) ≪ N H N − κ − η . Then the exact identity gives
R W ( N , H ) ≪ N H N − κ − η + o ( 1 ) . \boxed{
\mathcal R_W(N,H)
\ll
NH
N^{-\kappa-\eta+o(1)}.
} R W ( N , H ) ≪ N H N − κ − η + o ( 1 ) . Provided additionally
κ + η < 1 − τ , \kappa+\eta<1-\tau, κ + η < 1 − τ , 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 M U , V \mathcal M_{U,V} M U , V
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
Λ ( e ) b U ( k ) f ( e k ) \Lambda(e)b_U(k)f(ek) Λ ( e ) b U ( k ) f ( e k ) simultaneously uses:
actual prime powers through Λ ( e ) \Lambda(e) Λ ( e )
ordinary divisor truncation through b U ( k ) b_U(k) b U ( k )
ordinary product e k ek e k
ordinary additive shifts hidden in f ( e k ) f(ek) f ( e k )
The exact renormalization M U , V \mathcal M_{U,V} M U , V
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
f ( n ) = f N , H ( n ) = W ( n / N ) ∑ r ω H ( r ) ( Λ ( n + r ) − 1 ) . \boxed{
f(n)
=
f_{N,H}(n)
=
W(n/N)
\sum_r
\omega_H(r)
\left(
\Lambda(n+r)-1
\right).
} f ( n ) = f N , H ( n ) = W ( n / N ) r ∑ ω H ( r ) ( Λ ( n + r ) − 1 ) . Let
F = ∑ n f ( n ) . \boxed{
F=\sum_n f(n).
} F = n ∑ f ( n ) . Paper 76 proved
R W = ∑ n Λ ( n ) f ( n ) − F . \boxed{
\mathcal R_W
=
\sum_n\Lambda(n)f(n)-F.
} R W = n ∑ Λ ( n ) f ( n ) − F . All sums below are automatically restricted to the compact dyadic support of f f f
2. Vaughan identity
For parameters
U , V > 1 U,V>1 U , V > 1 define
b U ( k ) = ∑ d ∣ k d ≤ U μ ( d ) b_U(k)
=
\sum_{\substack{d\mid k\\d\le U}}
\mu(d) b U ( k ) = d ∣ k d ≤ U ∑ μ ( d ) and
a U , V ( k ) = ∑ d e = k d ≤ U e ≤ V μ ( d ) Λ ( e ) . a_{U,V}(k)
=
\sum_{\substack{de=k\\d\le U\\e\le V}}
\mu(d)\Lambda(e). a U , V ( k ) = d e = k d ≤ U e ≤ V ∑ μ ( d ) Λ ( e ) . For n > V n>V n > V
Λ ( n ) = ∑ d r = n d ≤ U μ ( d ) log r − ∑ k r = n k ≤ U V a U , V ( k ) − ∑ e k = n e > V k > U Λ ( e ) b U ( k ) . \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).
} Λ ( n ) = d r = n d ≤ U ∑ μ ( d ) log r − k r = n k ≤ U V ∑ a U , V ( k ) − e k = n e > V k > U ∑ Λ ( 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 ) f(n) f ( n )
Define
T 1 = ∑ d ≤ U μ ( d ) ∑ d ∣ n f ( n ) log ( n / d ) , T_1
=
\sum_{d\le U}
\mu(d)
\sum_{d\mid n}
f(n)\log(n/d), T 1 = d ≤ U ∑ μ ( d ) d ∣ n ∑ f ( n ) log ( n / d ) , T 2 = ∑ k ≤ U V a U , V ( k ) ∑ k ∣ n f ( n ) , T_2
=
\sum_{k\le UV}
a_{U,V}(k)
\sum_{k\mid n}
f(n), T 2 = k ≤ U V ∑ a U , V ( k ) k ∣ n ∑ f ( n ) , and
T 3 = ∑ e > V , k > U Λ ( e ) b U ( k ) f ( e k ) . T_3
=
\sum_{\substack{e>V,\ k>U}}
\Lambda(e)b_U(k)f(ek). T 3 = e > V , k > U ∑ Λ ( e ) b U ( k ) f ( e k ) . Then
∑ n Λ ( n ) f ( n ) = T 1 − T 2 − T 3 . \boxed{
\sum_n
\Lambda(n)f(n)
=
T_1-T_2-T_3.
} n ∑ Λ ( n ) f ( n ) = T 1 − T 2 − T 3 . Therefore
R W = T 1 − T 2 − T 3 − F . \boxed{
\mathcal R_W
=
T_1-T_2-T_3-F.
} R W = T 1 − T 2 − T 3 − F . No approximation has yet been made.
4. Log-weighted divisor law
Define
G = ∑ n f ( n ) log ( n / N ) , \boxed{
G
=
\sum_n
f(n)\log(n/N),
} G = n ∑ f ( n ) log ( n / N ) , and
G d = ∑ d ∣ n f ( n ) log ( n / N ) . \boxed{
G_d
=
\sum_{d\mid n}
f(n)\log(n/N).
} G d = d ∣ n ∑ f ( n ) log ( n / N ) . Multiplying the smooth weight W ( u ) W(u) W ( u ) log u \log u log u
The same lattice-sampling argument as Paper 76 therefore gives:
Theorem 4.1 — Log-weighted signed divisor law
G d = G d + s d , \boxed{
G_d
=
\frac Gd
+
s_d,
} G d = d G + s d , with
s d = O W ( N ) \boxed{
s_d=O_W(N)
} s d = O W ( N ) uniformly.
Moreover, for every D ≥ 1 D\ge1 D ≥ 1
∑ d ≤ D τ 5 ( d ) ( ∣ r d ∣ + ∣ s d ∣ ) ≪ W N D ( log N ) O ( 1 ) , \boxed{
\sum_{d\le D}
\tau_5(d)
\left(
|r_d|+|s_d|
\right)
\ll_W
ND(\log N)^{O(1)},
} d ≤ D ∑ τ 5 ( d ) ( ∣ r d ∣ + ∣ s d ∣ ) ≪ W N D ( log N ) O ( 1 ) , where
F d = F d + r d . F_d=\frac Fd+r_d. F d = d F + r d . Create:
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 ) , \log(n/d)
=
\log N
-
\log d
+
\log(n/N), log ( n / d ) = log N − log d + log ( n / N ) , T 1 = ∑ d ≤ U μ ( d ) [ ( log N − log d ) F d + G d ] . \begin{aligned}
T_1
&=
\sum_{d\le U}
\mu(d)
\left[
(\log N-\log d)F_d+G_d
\right].
\end{aligned} T 1 = d ≤ U ∑ μ ( d ) [ ( log N − log d ) F d + G d ] . Define
M U = ∑ d ≤ U μ ( d ) d , M_U
=
\sum_{d\le U}
\frac{\mu(d)}d, M U = d ≤ U ∑ d μ ( d ) , and
J U = ∑ d ≤ U μ ( d ) log d d . J_U
=
\sum_{d\le U}
\frac{\mu(d)\log d}{d}. J U = d ≤ U ∑ d μ ( d ) log d . Substitute the divisor laws:
T 1 = F ( log N M U − J U ) + G M U + ∑ d ≤ U μ ( d ) [ ( log N − log d ) r d + s d ] . \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}
} T 1 = F ( log N M U − J U ) + G M U + d ≤ U ∑ μ ( d ) [ ( log N − log d ) r d + s d ] . 6. Exact second Type-I main term
Since
T 2 = ∑ k ≤ U V a U , V ( k ) F k , T_2
=
\sum_{k\le UV}
a_{U,V}(k)F_k, T 2 = k ≤ U V ∑ a U , V ( k ) F k , we need the scalar sum
∑ k ≤ U V a U , V ( k ) k . \sum_{k\le UV}
\frac{a_{U,V}(k)}k. k ≤ U V ∑ k a U , V ( k ) . By definition,
∑ k ≤ U V a U , V ( k ) k = ∑ d ≤ U μ ( d ) d ∑ e ≤ V Λ ( e ) e = M U L V , \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} k ≤ U V ∑ k a U , V ( k ) = d ≤ U ∑ d μ ( d ) e ≤ V ∑ e Λ ( e ) = M U L V , where
L V = ∑ e ≤ V Λ ( e ) e . L_V
=
\sum_{e\le V}
\frac{\Lambda(e)}e. L V = e ≤ V ∑ e Λ ( e ) . Therefore:
T 2 = F M U L V + ∑ k ≤ U V a U , V ( k ) r k . \boxed{
T_2
=
F M_U L_V
+
\sum_{k\le UV}
a_{U,V}(k)r_k.
} T 2 = F M U L V + k ≤ U V ∑ a U , V ( k ) r k . 7. Renormalized exact root identity
Combine Sections 5–6 and subtract F F F
Define
M U , V = F [ M U ( log N − L V ) − J U − 1 ] + G M U . \boxed{
\mathcal M_{U,V}
=
F
\left[
M_U(\log N-L_V)
-
J_U
-
1
\right]
+
G M_U.
} M U , V = F [ M U ( log N − L V ) − J U − 1 ] + G M U . Define
E I = ∑ d ≤ U μ ( d ) [ ( log N − log d ) r d + s d ] − ∑ k ≤ U V a U , V ( k ) r k . \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}
} E I = d ≤ U ∑ μ ( d ) [ ( log N − log d ) r d + s d ] − k ≤ U V ∑ a U , V ( k ) r k . Then:
Theorem 7.1 — Exact renormalized Vaughan root decomposition
R W = M U , V − T U , V I I + E I , \boxed{
\mathcal R_W
=
\mathcal M_{U,V}
-
\mathcal T^{II}_{U,V}
+
\mathcal E_I,
} R W = M U , V − T U , V I I + E I , where
T U , V I I = ∑ e > V , k > U Λ ( e ) b U ( k ) f ( e k ) . \boxed{
\mathcal T^{II}_{U,V}
=
\sum_{\substack{e>V,\ k>U}}
\Lambda(e)b_U(k)f(ek).
} T U , V I I = e > V , k > U ∑ Λ ( e ) b U ( k ) f ( e k ) . Equivalently,
R W = − V U , V r e n + E I , \boxed{
\mathcal R_W
=
-
\mathcal V^{\rm ren}_{U,V}
+
\mathcal E_I,
} R W = − V U , V ren + E I , where
V U , V r e n = T U , V I I − M U , V . \boxed{
\mathcal V^{\rm ren}_{U,V}
=
\mathcal T^{II}_{U,V}
-
\mathcal M_{U,V}.
} V U , V ren = T U , V I I − M U , V . Create:
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 ) a_{U,V}(k) a U , V ( k )
∣ a U , V ( k ) ∣ ≤ ∑ e ∣ k e ≤ V Λ ( e ) ≤ ∑ e ∣ k Λ ( e ) = log k . \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} ∣ a U , V ( k ) ∣ ≤ e ∣ k e ≤ V ∑ Λ ( e ) ≤ e ∣ k ∑ Λ ( e ) = log k . Therefore, using Theorem 4.1,
∣ E I ∣ ≪ ( log N ) ∑ d ≤ U ( ∣ r d ∣ + ∣ s d ∣ ) + ( log N ) ∑ k ≤ U V ∣ r k ∣ ≪ N U V ( log N ) O ( 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} ∣ E I ∣ ≪ ( log N ) d ≤ U ∑ ( ∣ r d ∣ + ∣ s d ∣ ) + ( log N ) k ≤ U V ∑ ∣ r k ∣ ≪ N U V ( log N ) O ( 1 ) . Create:
B-RH-121
EXACT_VAUGHAN_TYPE_I_ERROR_RETAINS_FIXED_POWER_FROM_SHIFT_AVERAGED_DIVISOR_DISTRIBUTION
CERTIFIED
9. Exponent ledger
Set
H = N 1 − τ , H=N^{1-\tau}, H = N 1 − τ , U = N u , U=N^u, U = N u , V = N v . V=N^v. V = N v . Then
N H = N 2 − τ . NH
=
N^{2-\tau}. N H = N 2 − τ . The Type-I error satisfies
∣ E I ∣ N H ≪ N − ( 1 − τ − u − v ) + o ( 1 ) . \boxed{
\frac{
|\mathcal E_I|
}{
NH
}
\ll
N^{-(1-\tau-u-v)+o(1)}.
} N H ∣ E I ∣ ≪ N − ( 1 − τ − u − v ) + o ( 1 ) . Hence the Type-I saving exponent is
η I = 1 − τ − u − v . \boxed{
\eta_I
=
1-\tau-u-v.
} η I = 1 − τ − u − v . For a seed PESC ( κ ) (\kappa) ( κ ) η > 0 \eta>0 η > 0
u + v < 1 − τ − κ − η . \boxed{
u+v
<
1-\tau-\kappa-\eta.
} u + v < 1 − τ − κ − η . This is compatible with positive u , v u,v u , v
τ < 1 − κ − η . \tau<1-\kappa-\eta. τ < 1 − κ − η . Thus the exact extraction produces no logarithmic loss.
10. Seed size of the explicit renormalization
Assume PESC ( κ ) (\kappa) ( κ )
d = κ 2 . d=\frac{\kappa}{2}. d = 2 κ . The seed Mertens bound gives
M U ≪ U − d + o ( 1 ) . \boxed{
M_U
\ll
U^{-d+o(1)}.
} M U ≪ U − d + o ( 1 ) . Since
1 ζ ( s ) ∼ s − 1 \frac1{\zeta(s)}
\sim
s-1 ζ ( s ) 1 ∼ s − 1 at s = 1 s=1 s = 1
∑ n = 1 ∞ μ ( n ) log n n = − 1. \sum_{n=1}^{\infty}
\frac{\mu(n)\log n}{n}
=
-1. n = 1 ∑ ∞ n μ ( n ) log n = − 1. Partial summation with the seed Mertens bound gives
J U + 1 ≪ U − d + o ( 1 ) log U . \boxed{
J_U+1
\ll
U^{-d+o(1)}
\log U.
} J U + 1 ≪ U − d + o ( 1 ) log U . Likewise the seed PNT error gives
L V = log V − γ + O ( V − d + o ( 1 ) ) . \boxed{
L_V
=
\log V-\gamma
+
O
\left(
V^{-d+o(1)}
\right).
} L V = log V − γ + O ( V − d + o ( 1 ) ) . Finally,
F , G ≪ H N 1 − d + o ( 1 ) . \boxed{
F,G
\ll
HN^{1-d+o(1)}.
} F , G ≪ H N 1 − d + o ( 1 ) . Therefore:
Theorem 10.1 — Size of the explicit one-point renormalization
M U , V ≪ N H N − d U − d N o ( 1 ) . \boxed{
\mathcal M_{U,V}
\ll
NH
N^{-d}
U^{-d}
N^{o(1)}.
} M U , V ≪ N H N − d U − d N o ( 1 ) . If
U = N u , U=N^u, U = N u , the effective exponent is
d ( 1 + u ) . \boxed{
d(1+u).
} d ( 1 + u ) . 11. Why unrenormalized Type II is the wrong target
For every fixed
u < 1 , u<1, u < 1 , d ( 1 + u ) < 2 d = κ . d(1+u)
<
2d
=
\kappa. d ( 1 + u ) < 2 d = κ . 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
T U , V I I ≪ N H N − κ − η \mathcal T^{II}_{U,V}
\ll
NHN^{-\kappa-\eta} T U , V I I ≪ N H N − κ − η is structurally incompatible with the required cancellation unless the explicit main M U , V \mathcal M_{U,V} M U , V
Create:
O-RH-174
UNRENORMALIZED_VAUGHAN_TYPE_II_SMALLNESS_DISCARDS_A_NECESSARY_ONE_POINT_TRUNCATION_COUNTERTERM
CERTIFIED_AS_METHOD_BARRIER
The correct object is
T U , V I I − M U , V . \mathcal T^{II}_{U,V}
-
\mathcal M_{U,V}. T U , V I I − M U , V . 12. New arithmetic frontier F-RH-024
Open:
F-RH-024
RENORMALIZED_VAUGHAN_BALANCED_DEFECT_POWER
Given a PESC ( κ ) (\kappa) ( κ )
τ , u , v , η > 0 \tau,u,v,\eta>0 τ , u , v , η > 0 such that
u + v < 1 − τ − κ − η \boxed{
u+v
<
1-\tau-\kappa-\eta
} u + v < 1 − τ − κ − η and
∣ T U , V I I − M U , V ∣ ≪ N H N − κ − η . \boxed{
\left|
\mathcal T^{II}_{U,V}
-
\mathcal M_{U,V}
\right|
\ll
NH
N^{-\kappa-\eta}.
} T U , V I I − M U , V ≪ N H N − κ − η . Then Theorem 7.1 and Section 9 give
R W ≪ N H N − κ − η + o ( 1 ) . \boxed{
\mathcal R_W
\ll
NH
N^{-\kappa-\eta+o(1)}.
} R W ≪ N H N − κ − η + o ( 1 ) . This is a direct ordinary-prime root covariance bound.
Create:
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 ) = N H N − ( 1 − τ ) + o ( 1 ) . N\log(N/H)
=
NH
N^{-(1-\tau)+o(1)}. N log ( N / H ) = N H N − ( 1 − τ ) + o ( 1 ) . Therefore, if
κ + η < 1 − τ , \boxed{
\kappa+\eta
<
1-\tau,
} κ + η < 1 − τ , the solved local main term is smaller than the F-RH-024 target.
After standard dyadic smooth partitioning, the bound
R W ≪ N H N − κ − η \mathcal R_W
\ll
NHN^{-\kappa-\eta} R W ≪ N H N − κ − η 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:
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
Λ ( e ) b U ( k ) f ( e k ) . \Lambda(e)b_U(k)f(ek). Λ ( e ) b U ( k ) f ( e k ) . This requires ordinary:
von Mangoldt prime-power weights;
Möbius divisor truncation;
multiplication e k ek e k
additive shifts inside f ( e k ) f(ek) f ( e k )
The counterterm uses
M U , J U , L V , M_U,\quad
J_U,\quad
L_V, M U , J U , 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 μ ( d ) log r \mu(d)\log r μ ( d ) log r
a second Type-I term with the truncated convolution coefficient;
a Type-II term withΛ ( e ) ∑ d ∣ k d ≤ U μ ( d ) . \Lambda(e)
\sum_{\substack{d\mid k\\d\le U}}\mu(d). Λ ( e ) d ∣ k d ≤ U ∑ μ ( 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
γ + ν > 1. \gamma+\nu>1. γ + ν > 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
v 1.67 → v 1.68. v1.67
\to
v1.68. v 1.67 → v 1.68. Add:
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:
F-RH-024
RENORMALIZED_VAUGHAN_BALANCED_DEFECT_POWER
OPEN_ROOT_ARITHMETIC
Downgrade:
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 T U , V I I \mathcal T^{II}_{U,V} T U , V I I
Instead:
dyadically decompose e e e k k k
insert the definition
$$
f(ek)
W(ek/N)
\sum_r
\omega_H(r)
(\Lambda(ek+r)-1);
$$isolate the exact contribution corresponding to M U , V \mathcal M_{U,V} M U , V
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.