CSM_RH Paper 51

Rigorous Truncated Zero Insertion, Exact Discrete Zero-Ordinate Response, and RH-Conditional Middle-Band Admission

Project: CSM_RH
Paper: 51
Version: v0.1
Date: 2026-09-08
Campaign: 44 — PESC_TRIANGULAR_LAG_KERNEL_ATTACK
Track: PK5 — FIXED_POWER_PESC_ADMISSION
Subtracks: Z1 / Z2, with Z3–Z4 frontier audit
Status: Z1 CLOSED / Z2 CLOSED / PK5 STILL OPEN UNCONDITIONALLY
Canonical entry state: v1.41 / Paper 50 v0.1
RH_PROVED: FALSE
RH_DISPROVED: FALSE
GLOBAL_RH_CERTIFICATE: FALSE

Abstract

Paper 50 converted the centered prime PESC statistic into the exact centered Abel transform

\mathfrak L_N(t) = \sum_{m=N}^{2N-1} (\psi(m)-m) \Delta g_{N,t}(m),

where

g_{N,t}(x) = \frac{(2N/x)^{it}-1}{x}, \qquad \Delta g_{N,t}(m) = g_{N,t}(m)-g_{N,t}(m+1).

The fixed-power target is

\mathcal E_\Lambda(N,\kappa) = \int_{N^{-\kappa}\le |t|\le N^\kappa} \frac{|\mathfrak L_N(t)|^2}{t^2}\,dt \ll N^{-\kappa+o(1)}

for fixed

0<\kappa\le1.

The present paper first inserts a standard truncated von Mangoldt explicit formula at the exact integer arguments occurring in the Abel sum. Taking the zero height

T=N^\kappa,

the explicit-formula remainder is shown to contribute only

O\!\left( N^{-\kappa}\log^4 N \right)

to the centered middle-band energy. Thus the truncation ledger itself is fixed-power admissible.

Second, for each nontrivial zero

\rho=\beta+i\gamma,

an exact discrete response kernel is extracted:

\mathcal K_{\rho,N}(t) = \frac1\rho \sum_{m=N}^{2N-1} m^\rho \Delta g_{N,t}(m).

It has the exact centered frequency representation

\boxed{ \mathcal K_{\rho,N}(t) = (2N)^{it}A_{\rho,N}(t)-A_{\rho,N}(0), }

where

\boxed{ A_{\rho,N}(t) = \frac1\rho \sum_{m=N}^{2N-1} m^{\rho-1-it} \left[ 1- \left( \frac{m}{m+1} \right)^{1+it} \right]. }

The oscillatory phase is therefore exactly

m^{i(\gamma-t)}.

The alignment of the translated Mellin frequency $t$ with the zero ordinate $\gamma$ is not a continuous approximation; it is already present in the finite discrete kernel.

Using the Kusmin–Landau first derivative estimate and partial summation, the paper proves the uniform localization bound

|\mathcal K_{\rho,N}(t)| \ll N^{\beta-1} \left[ \frac{1+|t|} {|\rho|(1+|\gamma-t|)} + \frac1{|\rho|(1+|\gamma|)} \right]

throughout

|t|\le N, \qquad |\gamma|\le N.

For $|t|\le1$ , exact centering improves this to

|\mathcal K_{\rho,N}(t)| \ll |t| \frac{N^{\beta-1}}{|\rho|}.

As a calibration theorem, RH implies the full PK5 middle-band target. Indeed, under RH every $\beta=1/2$ ; Riemann–von Mangoldt zero counting and the response localization give a finite-zero contribution

O(N^{-1}\log^4 N)

in energy, while the explicit-formula truncation contributes

O(N^{-\kappa}\log^4 N).

Thus RH implies

\mathcal E_\Lambda(N,\kappa) \ll N^{-\kappa+o(1)}

for every fixed $0<\kappa\le1$ .

This is only a one-way conditional calibration. No converse and no RH proof is obtained. Unconditionally, classical zero-free regions combined with absolute zero-response summation do not supply a fixed power near the top of the polynomial zero window. The remaining PK5 problem is therefore the cross-zero / near-ordinate structure and the exclusion or detection of off-critical mass.

1. Canonical entry object

Paper 50 certified

\boxed{ \mathfrak L_N(t) = \sum_{m=N}^{2N-1} E_\psi(m) \Delta g_{N,t}(m), }

where

E_\psi(m)=\psi(m)-m

and

g_{N,t}(x) = \frac{(2N/x)^{it}-1}{x}.

The target is

\boxed{ \mathcal E_\Lambda(N,\kappa) = \int_{\mathcal B_{N,\kappa}} \frac{|\mathfrak L_N(t)|^2}{t^2}\,dt \ll N^{-\kappa+o(1)}, }

with

\mathcal B_{N,\kappa} = \left\{ t\in\mathbb R: N^{-\kappa}\le |t|\le N^\kappa \right\}.

Paper 50 left the following sequence:

Z1 RIGOROUS_TRUNCATED_EXPLICIT_FORMULA_INSERTION
Z2 CENTERED_ZERO_RESPONSE_KERNEL_EXTRACTION
Z3 ZERO_WINDOW_CROSS_TERM_AND_NEAR_ORDINATE_AUDIT
Z4 OFF_CRITICAL_ANTI_CANCELLATION_OR_CRITICAL_LINE_ADMISSION

The present paper closes Z1 and Z2.

2. Truncated explicit formula

We use the standard truncated von Mangoldt explicit formula in the form

\boxed{ \psi(x) = x - \sum_{|\gamma|\le T} \frac{x^\rho}{\rho} + R_\psi(x,T), }

where the sum runs over nontrivial zeros

\rho=\beta+i\gamma

with multiplicity and

\boxed{ R_\psi(x,T) \ll \frac{x\log^2(xT)}{T} + \log x }

for

x,T\ge2.

The constant and trivial-zero terms are absorbed into the displayed error at the resolution needed here.

For

N\le m\le2N,

this becomes

E_\psi(m) = - \sum_{|\gamma|\le T} \frac{m^\rho}{\rho} + R_\psi(m,T),

with

|R_\psi(m,T)| \ll \frac{N\log^2(NT)}{T} + \log N.

Since the zero sum is finite, it can be interchanged with the finite Abel sum without any convergence issue.

3. Total variation of the centered Abel kernel

Differentiate

g_{N,t}(x) = (2N)^{it}x^{-1-it}-x^{-1}.

Then

g_{N,t}'(x) = x^{-2} \left[ 1- (1+it) \left( \frac{2N}{x} \right)^{it} \right].

For

N\le x\le2N,

write

\theta = t\log\frac{2N}{x}.

If $|t|\le1$ ,

\left| 1-(1+it)e^{i\theta} \right| \le |1-e^{i\theta}|+|t| \ll |t|.

If $|t|\ge1$ ,

\left| 1-(1+it)e^{i\theta} \right| \le 1+|1+it| \ll |t|.

Therefore, for all real $t$ ,

\boxed{ |g_{N,t}'(x)| \ll \frac{|t|}{N^2} }

on the complete dyadic interval.

By the fundamental theorem of calculus,

|\Delta g_{N,t}(m)| \le \int_m^{m+1}|g_{N,t}'(x)|\,dx,

and hence

\boxed{ \sum_{m=N}^{2N-1} |\Delta g_{N,t}(m)| \ll \frac{|t|}{N}. }

This is the key reason the ordinary pointwise explicit-formula remainder becomes power-admissible after insertion into the centered Abel kernel.

4. Z1 theorem: polynomial-height truncation is admissible

Insert the truncated explicit formula into Paper 50:

\mathfrak L_N(t) = - \sum_{|\gamma|\le T} \mathcal K_{\rho,N}(t) + \mathfrak R_{N,T}^{\mathrm{EF}}(t),

where

\boxed{ \mathcal K_{\rho,N}(t) = \frac1\rho \sum_{m=N}^{2N-1} m^\rho \Delta g_{N,t}(m) }

and

\mathfrak R_{N,T}^{\mathrm{EF}}(t) = \sum_{m=N}^{2N-1} R_\psi(m,T) \Delta g_{N,t}(m).

Using Section 3,

|\mathfrak R_{N,T}^{\mathrm{EF}}(t)| \ll |t| \left[ \frac{\log^2(NT)}{T} + \frac{\log N}{N} \right].

Define the remainder energy

\mathcal E_{\mathrm{EF}}(N,\kappa,T) = \int_{\mathcal B_{N,\kappa}} \frac{ |\mathfrak R_{N,T}^{\mathrm{EF}}(t)|^2 }{t^2}\,dt.

Since the length of the two-sided band is

O(N^\kappa),

we obtain

\mathcal E_{\mathrm{EF}}(N,\kappa,T) \ll N^\kappa \left[ \frac{\log^4(NT)}{T^2} + \frac{\log^2N}{N^2} \right].

Take

\boxed{ T=N^\kappa. }

For fixed

0<\kappa\le1,

the first term becomes

N^{-\kappa}\log^4N,

while

N^{\kappa-2}\log^2N \le N^{-\kappa}\log^2N.

Therefore:

Theorem 4.1 — Middle-band admissible explicit-formula truncation

For every fixed $0<\kappa\le1$ , taking $T=N^\kappa$ gives

\boxed{ \mathcal E_{\mathrm{EF}} \left( N,\kappa,N^\kappa \right) \ll N^{-\kappa}\log^4N = N^{-\kappa+o(1)}. }

Thus Z1 is closed.

Create:

B-RH-044
PESC_CENTERED_ABEL_POLYNOMIAL_HEIGHT_EXPLICIT_FORMULA_WITH_ADMISSIBLE_REMAINDER
CERTIFIED

Close:

Z1
CLOSED_AS_POLYNOMIAL_HEIGHT_EXPLICIT_FORMULA_WITH_MIDDLE_BAND_ADMISSIBLE_REMAINDER

No hypothesis on the zero real parts has been used.

5. Exact discrete response of one zero

The zero response is

\mathcal K_{\rho,N}(t) = \frac1\rho \sum_{m=N}^{2N-1} m^\rho \Delta g_{N,t}(m).

Expand

g_{N,t}(m) = (2N)^{it}m^{-1-it}-m^{-1}.

Then

\Delta g_{N,t}(m) = (2N)^{it} \left[ m^{-1-it}-(m+1)^{-1-it} \right] - \left[ m^{-1}-(m+1)^{-1} \right].

Define

\boxed{ A_{\rho,N}(t) = \frac1\rho \sum_{m=N}^{2N-1} m^\rho \left[ m^{-1-it}-(m+1)^{-1-it} \right]. }

Then:

Theorem 5.1 — Exact centered zero response

For every nontrivial zero $\rho$ and every real $t$ ,

\boxed{ \mathcal K_{\rho,N}(t) = (2N)^{it}A_{\rho,N}(t)-A_{\rho,N}(0). }

Moreover,

m^\rho \left[ m^{-1-it}-(m+1)^{-1-it} \right] = m^{\rho-1-it} \left[ 1- \left( \frac{m}{m+1} \right)^{1+it} \right].

Hence

\boxed{ A_{\rho,N}(t) = \frac1\rho \sum_{m=N}^{2N-1} m^{\rho-1-it} q_t(m), }

where

q_t(m) = 1- \left( \frac{m}{m+1} \right)^{1+it}.

\rho=\beta+i\gamma,

the oscillatory factor is exactly

\boxed{ m^{i(\gamma-t)}. }

Thus the translated frequency variable $t$ is exactly centered on the zero ordinate $\gamma$ in the finite discrete response.

The centering is also exact:

\boxed{ \mathcal K_{\rho,N}(0)=0. }

Create:

B-RH-045
PESC_EXACT_DISCRETE_CENTERED_ZERO_ORDINATE_RESPONSE_KERNEL
CERTIFIED

6. Bounds for the local response amplitude

For real $x\ge N$ , define

q_t(x) = 1- \left( \frac{x}{x+1} \right)^{1+it}.

Write

\ell_x = \log\left(1+\frac1x\right).

Then

q_t(x) = 1- e^{-(1+it)\ell_x}.

|t|\le N

and

x\ge N,

then

|(1+it)\ell_x| \ll1.

Therefore

\boxed{ |q_t(x)| \ll \frac{1+|t|}{x}. }

Differentiating gives

q_t'(x) = - \left( \frac{x}{x+1} \right)^{1+it} \frac{1+it}{x(x+1)}

up to the immaterial sign convention, so

\boxed{ |q_t'(x)| \ll \frac{1+|t|}{x^2}. }

Now define the slowly varying coefficient

b_{\beta,t}(x) = x^{\beta-1}q_t(x).

Since

0<\beta<1,

we have uniformly on $[N,2N]$ ,

|b_{\beta,t}(x)| \ll (1+|t|)N^{\beta-2}

and

|b_{\beta,t}'(x)| \ll (1+|t|)N^{\beta-3}.

Thus its total variation is

\boxed{ \operatorname{Var}_{[N,2N]} b_{\beta,t} \ll (1+|t|)N^{\beta-2}. }

7. Kusmin–Landau localization in $\gamma-t$

Let

u=\gamma-t.

For

1\le |u|\le2N,

consider the phase

f_u(x) = \frac{u}{2\pi}\log x.

Then

f_u'(x) = \frac{u}{2\pi x}

is monotone and, on

N\le x\le2N,

satisfies

\frac{|u|}{4\pi N} \le |f_u'(x)| \le \frac1\pi < \frac12.

Therefore the distance of $f_u'(x)$ to the nearest integer is at least

\frac{|u|}{4\pi N}.

The Kusmin–Landau first derivative estimate yields

\sup_{N\le M\le2N} \left| \sum_{m=N}^{M} m^{iu} \right| \ll \frac{N}{|u|}.

For $|u|<1$ , the trivial estimate is $O(N)$ .

Combining the two regimes,

\boxed{ \sup_{N\le M\le2N} \left| \sum_{m=N}^{M} m^{iu} \right| \ll \frac{N}{1+|u|}. }

Partial summation with Section 6 therefore gives

\left| \sum_{m=N}^{2N-1} m^{\rho-1-it}q_t(m) \right| \ll \frac{ (1+|t|)N^{\beta-1} }{ 1+|\gamma-t| }.

Thus:

Theorem 7.1 — Exact discrete zero-frequency localization

Whenever

|t|\le N, \qquad |\gamma|\le N,

we have

\boxed{ |A_{\rho,N}(t)| \ll \frac{ (1+|t|)N^{\beta-1} }{ |\rho|(1+|\gamma-t|) }. }

Consequently,

\boxed{ |\mathcal K_{\rho,N}(t)| \ll N^{\beta-1} \left[ \frac{1+|t|} {|\rho|(1+|\gamma-t|)} + \frac1{|\rho|(1+|\gamma|)} \right]. }

This is the rigorous discrete analogue of the continuous single-zero response calibration in Paper 50.

8. Additional low-frequency gain from exact centering

For

|t|\le1,

differentiate

g_{N,t}(x)

with respect to $t$ :

\partial_t g_{N,t}(x) = i \log\frac{2N}{x} \frac{(2N/x)^{it}}{x}.

N\le x\le2N

and

|t|\le1,

one has

\left| \partial_x\partial_t g_{N,t}(x) \right| \ll N^{-2}.

Therefore

\left| \partial_t \Delta g_{N,t}(m) \right| \ll N^{-2}.

It follows that

\left| \partial_t \mathcal K_{\rho,N}(t) \right| \ll \frac1{|\rho|} \sum_{m=N}^{2N-1} m^\beta N^{-2} \ll \frac{N^{\beta-1}}{|\rho|}.

Since

\mathcal K_{\rho,N}(0)=0,

the mean value theorem gives:

Theorem 8.1 — Centered low-frequency zero response

For

|t|\le1,

\boxed{ |\mathcal K_{\rho,N}(t)| \ll |t| \frac{N^{\beta-1}}{|\rho|}. }

This removes the apparent $t^{-2}$ singularity in the energy near the bottom of the middle band.

Close:

Z2
CLOSED_AS_EXACT_DISCRETE_CENTERED_ZERO_ORDINATE_RESPONSE_WITH_KUSMIN_LANDAU_LOCALIZATION

9. Zero counting inputs

Let

N_\zeta(T)

denote the number of nontrivial zeta zeros with

0<\gamma\le T

counted with multiplicity.

The Riemann–von Mangoldt formula implies

N_\zeta(T) = \frac{T}{2\pi} \log\frac{T}{2\pi} - \frac{T}{2\pi} + O(\log T).

In particular,

N_\zeta(T)=O(T\log T).

It also gives the unit-interval bound

\boxed{ N_\zeta(U+1)-N_\zeta(U) \ll \log(U+2). }

By partial summation,

\boxed{ \sum_{0<\gamma\le T} \frac1\gamma \ll \log^2(T+2) }

and

\boxed{ \sum_{\gamma} \frac1{(1+|\gamma|)^2} \ll1. }

Only these coarse zero-counting facts are required for the conditional theorem below.

10. RH-conditional pointwise control of the finite zero sum

Assume RH in this section only.

Then every nontrivial zero has

\beta=\frac12.

Set

T=N^\kappa, \qquad 0<\kappa\le1,

and define

\mathfrak Z_{N,T}(t) = \sum_{|\gamma|\le T} \mathcal K_{\rho,N}(t).

10.1. The range $|t|\le1$

By Theorem 8.1,

|\mathfrak Z_{N,T}(t)| \ll |t|N^{-1/2} \sum_{|\gamma|\le T} \frac1{|\rho|}.

The first nontrivial zero has positive ordinate bounded away from zero, so

|\rho|\asymp1+|\gamma|.

Using Section 9,

\boxed{ |\mathfrak Z_{N,T}(t)| \ll |t|N^{-1/2}\log^2(T+2). }

10.2. The range $1\le |t|\le T$

By Theorem 7.1,

|\mathfrak Z_{N,T}(t)| \ll N^{-1/2} \sum_{|\gamma|\le T} \left[ \frac{1+|t|} {(1+|\gamma|)(1+|\gamma-t|)} + \frac1{(1+|\gamma|)^2} \right].

The second sum is $O(1)$ .

For the first sum, decompose the ordinates into unit intervals.

If $|\gamma|\le |t|/2$ , then the factor involving $|\gamma-t|$ reduces the summand to $O((1+|\gamma|)^{-1})$ .
If $|t|/2<\gamma<2|t|$ , the factor $(1+|t|)/(1+\gamma)$ is $O(1)$ , while unit intervals at distance $j$ from $t$ contribute $O(\log(T+2)/(1+j))$ .
If $\gamma\ge2|t|$ , the summand is $O(|t|/\gamma^2)$ .
Negative ordinates satisfy the same or a stronger bound because $|\gamma-t|\asymp |t|+|\gamma|$ .

Using the unit-interval zero count,

\boxed{ \sum_{|\gamma|\le T} \frac{1+|t|} {(1+|\gamma|)(1+|\gamma-t|)} \ll \log^2(T+2). }

Therefore

\boxed{ |\mathfrak Z_{N,T}(t)| \ll N^{-1/2}\log^2(T+2) }

for

1\le|t|\le T.

11. RH implies the PK5 middle-band target

Still assuming RH, split the finite-zero energy at $|t|=1$ .

For

N^{-\kappa}\le|t|\le1,

Section 10 gives

\frac{ |\mathfrak Z_{N,T}(t)|^2 }{t^2} \ll N^{-1}\log^4T.

Hence this range contributes

O(N^{-1}\log^4T).

For

1\le|t|\le T,

|\mathfrak Z_{N,T}(t)|^2 \ll N^{-1}\log^4T,

and

\int_1^T\frac{dt}{t^2}\le1.

Thus the second range also contributes

O(N^{-1}\log^4T).

Therefore

\boxed{ \int_{\mathcal B_{N,\kappa}} \frac{ |\mathfrak Z_{N,T}(t)|^2 }{t^2}\,dt \ll N^{-1}\log^4N. }

The explicit-formula remainder from Theorem 4.1 contributes

O(N^{-\kappa}\log^4N).

Since

0<\kappa\le1,

we have

N^{-1}\le N^{-\kappa}.

Using

|F+G|^2\le2|F|^2+2|G|^2,

we obtain:

Theorem 11.1 — RH-conditional PK5 admission

Assume RH. Then for every fixed

0<\kappa\le1,

\boxed{ \mathcal E_\Lambda(N,\kappa) \ll N^{-\kappa}\log^4N = N^{-\kappa+o(1)}. }

Thus the exact centered Mellin / Abel PESC middle-band target is compatible with RH at the full fixed-power scale required by Campaign 44.

This is a conditional theorem only.

Record:

RH_CONDITIONAL_CALIBRATION
PK5_MIDDLE_BAND_ADMISSION
PASS

Do not create an RH certificate.

12. What Theorem 11.1 does and does not establish

Theorem 11.1 proves the one-way implication

\boxed{ \mathrm{RH} \Longrightarrow \mathrm{PK5\ middle\text{-}band\ admission}. }

It does not prove the converse

\mathrm{PK5\ admission} \Longrightarrow \mathrm{RH}.

It also does not prove PK5 unconditionally.

The reason is now explicit. Without RH, each zero response carries the scale

N^{\beta-1}.

The frequency localization controls

\gamma-t,

but it does not create a negative power in $N$ when $\beta$ is close to $1$ .

13. Classical zero-free regions are not a fixed-power substitute

The classical zero-free region has the qualitative shape

\beta \le 1-\frac{c}{\log(|\gamma|+2)}

at large height.

At the top of the polynomial window

|\gamma| \asymp T = N^\kappa,

this gives only

N^{\beta-1} \le N^{-c/\log T} = e^{-c/\kappa},

which is a constant-scale saving rather than

N^{-\delta}

for a fixed $\delta>0$ .

Near

t\asymp\gamma,

the factor

(1+|t|)/(1+|\gamma|)

is also of constant size.

Hence absolute summation of the zero responses combined only with the classical zero-free region cannot yield the required fixed power.

Create:

O-RH-129
CLASSICAL_ZERO_FREE_REGION_PLUS_ABSOLUTE_ZERO_RESPONSE_SUMMATION_HAS_NO_FIXED_POWER
CERTIFIED_AS_PK5_METHOD_BARRIER

This obstruction does not exclude cancellation, zero-density amplification, orthogonality, or a more global anti-cancellation argument. It excludes only the naive route.

14. Z3 is now the exact frontier

After Z1 and Z2, the transform is

\boxed{ \mathfrak L_N(t) = - \sum_{|\gamma|\le N^\kappa} \mathcal K_{\rho,N}(t) + \mathfrak R_{N,N^\kappa}^{\mathrm{EF}}(t), }

with

\int_{\mathcal B_{N,\kappa}} \frac{ |\mathfrak R_{N,N^\kappa}^{\mathrm{EF}}(t)|^2 }{t^2}\,dt \ll N^{-\kappa+o(1)}.

Therefore the only nontrivial fixed-power object remaining is the zero-window sum itself.

Z3 should now ask:

how large can clusters with $|\gamma-t|\lesssim1$ be in the weighted response norm?
can zero-density estimates reduce the contribution of $\beta>1/2+\delta$ to a power-admissible level?
can cross terms between separated ordinate windows be controlled by an almost-orthogonality inequality?
does the exact centering create a positive or coercive Gram structure after pairing conjugate zeros?
if an off-critical zero exists, can its response be prevented from being cancelled by the remaining zero ensemble?

The first three are primarily upper-bound questions.

The fifth is the anti-cancellation problem required for a converse or RH implication.

15. Updated PK5 task order

The next taskpack is:

PK5/Z1 CLOSED
POLYNOMIAL_HEIGHT_EXPLICIT_FORMULA_WITH_MIDDLE_BAND_ADMISSIBLE_REMAINDER

PK5/Z2 CLOSED
EXACT_DISCRETE_CENTERED_ZERO_ORDINATE_RESPONSE_WITH_KUSMIN_LANDAU_LOCALIZATION

PK5/Z3 ACTIVE
ZERO_WINDOW_GRAM_AND_ZERO_DENSITY_CROSS_TERM_AUDIT

PK5/Z4 OPEN
OFF_CRITICAL_ANTI_CANCELLATION_OR_CRITICAL_LINE_ADMISSION

Recommended Z3 decomposition:

Z3A UNIT_ORDINATE_CLUSTER_GRAM_BOUND
Z3B SEPARATED_WINDOW_ALMOST_ORTHOGONALITY
Z3C ZERO_DENSITY_WEIGHTED_BETA_LEDGER
Z3D HIGH_ORDINATE_TAIL_AT_FIXED_WINDOW

Hard rejections:

DROP_ZERO_ZERO_CROSS_TERMS
ASSUME_SIMPLE_ZEROS
ASSUME_PAIR_CORRELATION
ASSUME_RH
PROMOTE_ZERO_DENSITY_LOG_GAIN_TO_FIXED_POWER
IGNORE_MULTIPLICITY
REPLACE_EXACT_DISCRETE_RESPONSE_BY_CONTINUOUS_MODEL_WITHOUT_ERROR_LEDGER

16. External calibration and sources

The following external results are used as standard inputs or calibration.

16.1. Truncated von Mangoldt explicit formula

A standard form is

\psi(x) = x - \sum_{|\gamma|\le T} \frac{x^\rho}{\rho} + O\left( \frac{x\log^2(xT)}{T} + \log x \right).

See, for example:

Dimitris Koukoulopoulos, The Distribution of Prime Numbers, AMS Graduate Studies in Mathematics, chapter on the explicit formula.
K. Kedlaya, analytic number theory notes, von Mangoldt explicit formula.

16.2. Zero distribution

The Riemann–von Mangoldt formula gives

N_\zeta(T) = \frac{T}{2\pi}\log\frac{T}{2\pi} - \frac{T}{2\pi} + O(\log T).

See NIST DLMF Section 25.10 and standard zeta-function texts.

16.3. RH and the Chebyshev error

NIST DLMF Section 25.16 records the classical equivalence

\mathrm{RH} \Longleftrightarrow \psi(x) = x+O(x^{1/2+\varepsilon})

for every $\varepsilon>0$ .

The present conditional theorem does not use this pointwise equivalence as its proof. Instead it uses the zero-location statement $\beta=1/2$ directly in the exact response kernel.

16.4. Kusmin–Landau

The first derivative estimate for exponential sums is classical. In the form used here, if the derivative of a real phase is monotone and stays a distance $\lambda$ from the integers, the exponential sum is $O(\lambda^{-1})$ .

17. Campaign state transition

Advance the candidate research state from

v1.41

v1.42.

Campaign 44 remains active.

Add:

B-RH-044
PESC_CENTERED_ABEL_POLYNOMIAL_HEIGHT_EXPLICIT_FORMULA_WITH_ADMISSIBLE_REMAINDER
CERTIFIED

Add:

B-RH-045
PESC_EXACT_DISCRETE_CENTERED_ZERO_ORDINATE_RESPONSE_KERNEL
CERTIFIED

Add:

O-RH-129
CLASSICAL_ZERO_FREE_REGION_PLUS_ABSOLUTE_ZERO_RESPONSE_SUMMATION_HAS_NO_FIXED_POWER
CERTIFIED_AS_PK5_METHOD_BARRIER

Track state:

PK5 ACTIVE

Z1 CLOSED
Z2 CLOSED
Z3 ACTIVE
Z4 OPEN

No root certificate is created.

18. Conclusion

The zero layer has now been entered rigorously.

The chain is

\boxed{ \begin{aligned} \text{centered PESC root energy} &\longleftrightarrow \text{centered Mellin middle band} \\ &\longleftrightarrow \text{centered von Mangoldt transform} \\ &= \text{centered Abel transform of }\psi-m \\ &= - \sum_{|\gamma|\le N^\kappa} \mathcal K_{\rho,N} + \text{power-admissible remainder}. \end{aligned} }

The response of each zero is finite, discrete, centered, and exactly ordinate-aligned:

m^{i(\gamma-t)}.

Under RH the entire zero ensemble satisfies the required middle-band fixed-power estimate using only coarse zero counting and Kusmin–Landau localization.

Unconditionally, the truncation and single-zero geometry are no longer the bottleneck.

The bottleneck is now the ensemble:

\boxed{ \text{zero-window Gram structure} + \text{zero-density weighting} + \text{off-critical anti-cancellation}. }

That is the exact mathematical content of PK5/Z3–Z4.

CSM_RH Paper 51 — Rigorous Truncated Zero Insertion, Exact Discrete Zero-Ordinate Response, and RH-Conditional Middle-Ba

CSM_RH Paper 51

Rigorous Truncated Zero Insertion, Exact Discrete Zero-Ordinate Response, and RH-Conditional Middle-Band Admission

Abstract

1. Canonical entry object

2. Truncated explicit formula

3. Total variation of the centered Abel kernel

4. Z1 theorem: polynomial-height truncation is admissible

Theorem 4.1 — Middle-band admissible explicit-formula truncation

5. Exact discrete response of one zero

Theorem 5.1 — Exact centered zero response

6. Bounds for the local response amplitude

7. Kusmin–Landau localization in γ−t\gamma-tγ−t

Theorem 7.1 — Exact discrete zero-frequency localization

8. Additional low-frequency gain from exact centering

Theorem 8.1 — Centered low-frequency zero response

9. Zero counting inputs

10. RH-conditional pointwise control of the finite zero sum

10.1. The range ∣t∣≤1|t|\le1∣t∣≤1

10.2. The range 1≤∣t∣≤T1\le |t|\le T1≤∣t∣≤T

11. RH implies the PK5 middle-band target

Theorem 11.1 — RH-conditional PK5 admission

12. What Theorem 11.1 does and does not establish

13. Classical zero-free regions are not a fixed-power substitute

14. Z3 is now the exact frontier

15. Updated PK5 task order

16. External calibration and sources

16.1. Truncated von Mangoldt explicit formula

16.2. Zero distribution

16.3. RH and the Chebyshev error

16.4. Kusmin–Landau

17. Campaign state transition

18. Conclusion

7. Kusmin–Landau localization in $\gamma-t$

10.1. The range $|t|\le1$

10.2. The range $1\le |t|\le T$