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CSM_RH Paper 52 — Polylogarithmic Zero-Window Gram Reduction and the Weighted Beta-Mass Frontier

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CSM_RH Paper 52

Polylogarithmic Zero-Window Gram Reduction and the Weighted Beta-Mass Frontier

Project: CSM_RH
Paper: 52
Version: v0.1
Date: 2026-09-08
Campaign: 44 — PESC_TRIANGULAR_LAG_KERNEL_ATTACK
Track: PK5 — FIXED_POWER_PESC_ADMISSION
Subtrack: Z3 — ZERO_WINDOW_GRAM_AND_ZERO_DENSITY_CROSS_TERM_AUDIT
Status: Z3 CLOSURE CANDIDATE / Z4 NEXT
Canonical entry state: v1.42 / Paper 51 v0.1
RH_PROVED: FALSE
RH_DISPROVED: FALSE
GLOBAL_RH_CERTIFICATE: FALSE

Abstract

Paper 51 inserted the truncated zeta explicit formula into the exact centered Abel representation of the PESC root statistic and extracted the discrete zero response

Kρ,N(t)=1ρ∑m=N2N−1mρΔgN,t(m),\mathcal K_{\rho,N}(t) = \frac1\rho \sum_{m=N}^{2N-1} m^\rho \Delta g_{N,t}(m),

with exact ordinate phase

mi(γ−t)m^{i(\gamma-t)}

for

ρ=β+iγ.\rho=\beta+i\gamma.

It also proved that, for

∣t∣≤N,∣γ∣≤N,|t|\le N, \qquad |\gamma|\le N, ∣Kρ,N(t)∣≪Nβ−1[1+∣t∣∣ρ∣(1+∣γ−t∣)+1∣ρ∣(1+∣γ∣)].|\mathcal K_{\rho,N}(t)| \ll N^{\beta-1} \left[ \frac{1+|t|} {|\rho|(1+|\gamma-t|)} + \frac1{|\rho|(1+|\gamma|)} \right].

The unresolved question was whether zero-zero cross terms could themselves create a power-sized loss when the responses are summed over all zeros up to height

T=Nκ.T=N^\kappa.

This paper proves that they do not.

After dividing by the PESC energy weight ∣t∣|t|, each zero response on ∣t∣≥1|t|\ge1 is dominated by a Cauchy-type ordinate window centered at γ\gamma with coefficient

Nβ−11+∣γ∣.\frac{N^{\beta-1}}{1+|\gamma|}.

The Gram kernel of two such windows satisfies

∫Rdt(1+∣t−γ∣)(1+∣t−γ′∣)≪log⁡(2+∣γ−γ′∣)1+∣γ−γ′∣.\int_{\mathbb R} \frac{dt} {(1+|t-\gamma|)(1+|t-\gamma'|)} \ll \frac{\log(2+|\gamma-\gamma'|)} {1+|\gamma-\gamma'|}.

The Riemann-von Mangoldt local zero count implies that each unit ordinate interval contains only O(log⁡T)O(\log T) zeros, counted with multiplicity. A Schur row-sum argument then yields only

O(log⁡3T)O(\log^3T)

loss for the complete separated-window Gram operator.

The low-frequency range ∣t∣≤1|t|\le1 is treated separately. Differentiating the exact centered zero response and using the same Kusmin-Landau localization gives

sup⁡∣s∣≤1∣∂sKρ,N(s)∣≪Nβ−1(1+∣γ∣)2\sup_{|s|\le1} |\partial_s\mathcal K_{\rho,N}(s)| \ll \frac{N^{\beta-1}} {(1+|\gamma|)^2}

up to an absolute finite-low-zero adjustment. Since

Kρ,N(0)=0,\mathcal K_{\rho,N}(0)=0,

the low-frequency channel also has only a bounded Gram cost.

Consequently the entire finite-zero contribution satisfies

∫N−κ≤∣t∣≤T∣∑∣γ∣≤TKρ,N(t)∣2t2 dt≪(log⁡T)3DN(T),\boxed{ \int_{N^{-\kappa}\le|t|\le T} \frac{ \left| \sum_{|\gamma|\le T} \mathcal K_{\rho,N}(t) \right|^2 }{t^2}\,dt \ll (\log T)^3 \mathfrak D_N(T), }

where

DN(T)=∑∣γ∣≤TN2β−2(1+∣γ∣)2.\boxed{ \mathfrak D_N(T) = \sum_{|\gamma|\le T} \frac{ N^{2\beta-2} }{ (1+|\gamma|)^2 }. }

Thus all power-scale risk in Z3 is concentrated in the real parts β\beta ; ordinate clustering and separated-window interference cost only polylogarithms.

The weighted beta mass has an exact layer-cake representation. If

Zw(σ,T)=∑∣γ∣≤Tβ≥σ1(1+∣γ∣)2,\mathfrak Z_w(\sigma,T) = \sum_{\substack{|\gamma|\le T\\\beta\ge\sigma}} \frac1{(1+|\gamma|)^2},

then

DN(T)=O(N−1)+2log⁡N∫1/21N2σ−2Zw(σ,T) dσ.\boxed{ \mathfrak D_N(T) = O(N^{-1}) + 2\log N \int_{1/2}^1 N^{2\sigma-2} \mathfrak Z_w(\sigma,T)\,d\sigma. }

The uppermost ordinate shell is automatically admissible:

∑Y≤∣γ∣≤2YN2β−2(1+∣γ∣)2≪log⁡YY.\sum_{Y\le|\gamma|\le2Y} \frac{N^{2\beta-2}}{(1+|\gamma|)^2} \ll \frac{\log Y}{Y}.

In particular, with T=NκT=N^\kappa, zeros with

∣γ∣≥T(log⁡N)A|\gamma| \ge \frac{T}{(\log N)^A}

contribute only

N−κ+o(1).N^{-\kappa+o(1)}.

Classical Ingham-Huxley zero-density estimates can refine the weighted ledger, but they do not exclude even one fixed off-critical zero. Therefore zero density by itself cannot certify the endpoint κ=1\kappa=1 weighted-beta condition. Z3 is closed as an ordinate-geometry reduction, and Z4 becomes the unique remaining PK5 frontier: off-critical anti-cancellation or critical-line admission.

No RH theorem is claimed.

1. Entry from Paper 51

Paper 51 gave

LN(t)=−∑∣γ∣≤TKρ,N(t)+RN,TEF(t),\mathfrak L_N(t) = - \sum_{|\gamma|\le T} \mathcal K_{\rho,N}(t) + \mathfrak R_{N,T}^{\mathrm{EF}}(t),

where

T=NκT=N^\kappa

and

∫BN,κ∣RN,TEF(t)∣2t2 dt≪N−κ+o(1).\int_{\mathcal B_{N,\kappa}} \frac{ |\mathfrak R_{N,T}^{\mathrm{EF}}(t)|^2 }{t^2}\,dt \ll N^{-\kappa+o(1)}.

Therefore PK5 reduces to the finite-zero sum

ZN,T(t)=∑∣γ∣≤TKρ,N(t).\mathfrak Z_{N,T}(t) = \sum_{|\gamma|\le T} \mathcal K_{\rho,N}(t).

The inherited response estimate is

∣Kρ,N(t)∣≪Nβ−1[1+∣t∣∣ρ∣(1+∣γ−t∣)+1∣ρ∣(1+∣γ∣)].|\mathcal K_{\rho,N}(t)| \ll N^{\beta-1} \left[ \frac{1+|t|} {|\rho|(1+|\gamma-t|)} + \frac1{|\rho|(1+|\gamma|)} \right].

The exact centering condition is

Kρ,N(0)=0.\mathcal K_{\rho,N}(0)=0.

The task of Z3 is to price all cross-zero interactions without assuming simple zeros, pair correlation, or RH.

2. Weighted response normalization

Define

Fρ,N(t)=Kρ,N(t)tF_{\rho,N}(t) = \frac{ \mathcal K_{\rho,N}(t) }{t}

for t≠0t\ne0.

For nontrivial zeros,

∣ρ∣≍1+∣γ∣|\rho| \asymp 1+|\gamma|

uniformly up to an absolute constant, after absorbing the finite low-ordinate set.

For

∣t∣≥1,|t|\ge1,

Paper 51 therefore gives

∣Fρ,N(t)∣≪Nβ−11+∣γ∣11+∣γ−t∣+Nβ−1(1+∣γ∣)21∣t∣.\boxed{ |F_{\rho,N}(t)| \ll \frac{ N^{\beta-1} }{ 1+|\gamma| } \frac1{1+|\gamma-t|} + \frac{ N^{\beta-1} }{ (1+|\gamma|)^2 } \frac1{|t|}. }

Set

bρ=Nβ−11+∣γ∣b_\rho = \frac{ N^{\beta-1} }{ 1+|\gamma| }

and

cρ=Nβ−1(1+∣γ∣)2.c_\rho = \frac{ N^{\beta-1} }{ (1+|\gamma|)^2 }.

The first term is the localized ordinate window. The second is a rank-one integrable tail.

3. A convolution lemma for ordinate windows

Let

h(x)=11+∣x∣.h(x) = \frac1{1+|x|}.

For d≥0d\ge0, define

H(d)=∫Rh(x)h(x−d) dx.\mathcal H(d) = \int_{\mathbb R} h(x)h(x-d)\,dx.

Lemma 3.1 — Cauchy-window convolution

For all d≥0d\ge0,

H(d)≪log⁡(2+d)1+d.\boxed{ \mathcal H(d) \ll \frac{ \log(2+d) }{ 1+d }. }

Proof

For d≤2d\le2, the integral is O(1)O(1).

Assume d>2d>2 and split the real line into

(−∞,0],[0,d],[d,∞).(-\infty,0], \qquad [0,d], \qquad [d,\infty).

On the middle interval,

1(1+x)(1+d−x)=1d+2[11+x+11+d−x].\frac1{(1+x)(1+d-x)} = \frac1{d+2} \left[ \frac1{1+x} + \frac1{1+d-x} \right].

Hence

∫0ddx(1+x)(1+d−x)=2log⁡(1+d)d+2.\int_0^d \frac{dx} {(1+x)(1+d-x)} = \frac{ 2\log(1+d) }{ d+2 }.

On each exterior interval, one denominator is at least 1+d1+d at the endpoint scale, and direct integration gives the same order

O(log⁡(2+d)1+d).O\left( \frac{\log(2+d)}{1+d} \right).

Summing the three pieces proves the lemma.

□\Box

Thus the Gram interaction of two separated ordinate windows decays almost like 1/∣γ−γ′∣1/|\gamma-\gamma'|, with only one logarithm.

4. Unit-ordinate zero clusters

Let

ΓT={γ:ζ(β+iγ)=0, ∣γ∣≤T},\Gamma_T = \left\{ \gamma: \zeta(\beta+i\gamma)=0, \ |\gamma|\le T \right\},

counted with multiplicity.

For an integer jj, define the unit cluster

Cj={ρ:j≤γ<j+1}.\mathcal C_j = \left\{ \rho: j\le\gamma<j+1 \right\}.

The Riemann-von Mangoldt formula implies

#Cj≪log⁡(T+2)\boxed{ \#\mathcal C_j \ll \log(T+2) }

uniformly for

∣j∣≤T+1.|j|\le T+1.

This count includes multiplicity.

If two zeros lie in the same unit cluster, Lemma 3.1 gives an O(1)O(1) Gram interaction.

Therefore, for an arbitrary set of complex coefficients uρu_\rho supported on one cluster,

∥∑ρ∈Cjuρh( ⋅−γρ)∥22≪log⁡(T+2)∑ρ∈Cj∣uρ∣2.\left\| \sum_{\rho\in\mathcal C_j} u_\rho h(\,\cdot-\gamma_\rho) \right\|_2^2 \ll \log(T+2) \sum_{\rho\in\mathcal C_j} |u_\rho|^2.

This closes the local multiplicity problem.

Create:

B-RH-046A
PESC_UNIT_ORDINATE_ZERO_CLUSTER_GRAM_BOUND
CERTIFIED

Close:

Z3A
CLOSED_AS_UNIT_ORDINATE_CLUSTER_COST_ONLY_LOGARITHMIC

5. Separated-window Schur bound

For two zeros ρ,ρ′\rho,\rho', define

Gρ,ρ′=H(∣γ−γ′∣).G_{\rho,\rho'} = \mathcal H \left( |\gamma-\gamma'| \right).

By Lemma 3.1,

Gρ,ρ′≪log⁡(2+∣γ−γ′∣)1+∣γ−γ′∣.G_{\rho,\rho'} \ll \frac{ \log(2+|\gamma-\gamma'|) }{ 1+|\gamma-\gamma'| }.

Fix one zero ordinate γ\gamma. Group all other ordinates according to

n≤∣γ−γ′∣<n+1.n \le |\gamma-\gamma'| < n+1.

Each such shell intersects only O(1)O(1) unit ordinate intervals, each containing

O(log⁡(T+2))O(\log(T+2))

zeros.

Hence the row sum satisfies

∑∣γ′∣≤TGρ,ρ′≪log⁡(T+2)∑0≤n≤2T+2log⁡(2+n)1+n≪log⁡3(T+2).\begin{aligned} \sum_{|\gamma'|\le T} G_{\rho,\rho'} &\ll \log(T+2) \sum_{0\le n\le2T+2} \frac{\log(2+n)}{1+n} \\ &\ll \boxed{ \log^3(T+2) }. \end{aligned}

The same estimate holds for column sums.

By the Schur test:

Theorem 5.1 — Separated-window almost orthogonality

For arbitrary complex coefficients uρu_\rho,

∫R∣∑∣γ∣≤Tuρh(t−γ)∣2dt≪log⁡3(T+2)∑∣γ∣≤T∣uρ∣2.\boxed{ \int_{\mathbb R} \left| \sum_{|\gamma|\le T} u_\rho h(t-\gamma) \right|^2dt \ll \log^3(T+2) \sum_{|\gamma|\le T} |u_\rho|^2. }

No pair-correlation hypothesis is used.

No simplicity assumption is used.

Create:

B-RH-046B
PESC_SEPARATED_ZERO_WINDOW_SCHUR_ALMOST_ORTHOGONALITY
CERTIFIED

Close:

Z3B
CLOSED_AS_SEPARATED_ORDINATE_WINDOWS_WITH_POLYLOG_GRAM_COST

6. The rank-one high-frequency tail

The second term in Section 2 contributes, on ∣t∣≥1|t|\ge1,

∑ρcρ∣t∣.\sum_\rho \frac{c_\rho}{|t|}.

Therefore

∫∣t∣≥1∣∑ρcρ∣t∣∣2dt≪(∑ρcρ)2.\int_{|t|\ge1} \left| \sum_\rho \frac{c_\rho}{|t|} \right|^2dt \ll \left( \sum_\rho c_\rho \right)^2.

Now

cρ=bρ11+∣γ∣.c_\rho = b_\rho \frac1{1+|\gamma|}.

By Cauchy,

(∑ρcρ)2≤(∑ρbρ2)(∑ρ1(1+∣γ∣)2).\left( \sum_\rho c_\rho \right)^2 \le \left( \sum_\rho b_\rho^2 \right) \left( \sum_\rho \frac1{(1+|\gamma|)^2} \right).

The second zero sum converges because

Nζ(U)=O(Ulog⁡U).N_\zeta(U)=O(U\log U).

Thus

∫∣t∣≥1∣∑ρcρ∣t∣∣2dt≪∑ρbρ2.\boxed{ \int_{|t|\ge1} \left| \sum_\rho \frac{c_\rho}{|t|} \right|^2dt \ll \sum_\rho b_\rho^2. }

The nonlocalized tail is therefore harmless at power scale.

7. Refined derivative localization at low frequency

Paper 51 used the coarse centered estimate

∣Kρ,N(t)∣≪∣t∣Nβ−1∣ρ∣.|\mathcal K_{\rho,N}(t)| \ll |t| \frac{N^{\beta-1}}{|\rho|}.

For the zero ensemble, we need one additional ordinate decay.

Differentiate the exact response:

∂tKρ,N(t)=1ρ∑m=N2N−1mρΔ(∂tgN,t)(m).\partial_t \mathcal K_{\rho,N}(t) = \frac1\rho \sum_{m=N}^{2N-1} m^\rho \Delta \left( \partial_t g_{N,t} \right)(m).

Now

∂tgN,t(x)=ilog⁡2Nx(2N/x)itx.\partial_t g_{N,t}(x) = i \log\frac{2N}{x} \frac{ (2N/x)^{it} }{x}.

The centered logarithm satisfies

0≤log⁡2Nx≤log⁡20 \le \log\frac{2N}{x} \le \log2

on

N≤x≤2N.N\le x\le2N.

For ∣t∣≤1|t|\le1, write

mρΔ(∂tgN,t)(m)=i(2N)itmρ−1−itrt(m),m^\rho \Delta \left( \partial_t g_{N,t} \right)(m) = i(2N)^{it} m^{\rho-1-it} r_t(m),

where

rt(m)=log⁡2Nm−(mm+1)1+itlog⁡2Nm+1.r_t(m) = \log\frac{2N}{m} - \left( \frac{m}{m+1} \right)^{1+it} \log\frac{2N}{m+1}.

A direct derivative estimate gives

∣rt(x)∣≪x−1|r_t(x)| \ll x^{-1}

and

∣rt′(x)∣≪x−2|r_t'(x)| \ll x^{-2}

uniformly for

N≤x≤2N,∣t∣≤1.N\le x\le2N, \qquad |t|\le1.

Apply the same Kusmin-Landau plus partial-summation argument as in Paper 51 to the phase

mi(γ−t).m^{i(\gamma-t)}.

For all but the finite low-ordinate set,

∣γ−t∣≍1+∣γ∣.|\gamma-t| \asymp 1+|\gamma|.

Hence

sup⁡∣t∣≤1∣∂tKρ,N(t)∣≪Nβ−1(1+∣γ∣)2.\boxed{ \sup_{|t|\le1} \left| \partial_t \mathcal K_{\rho,N}(t) \right| \ll \frac{ N^{\beta-1} }{ (1+|\gamma|)^2 }. }

The finite exceptional low-ordinate set is absorbed into the absolute constant.

Since

Kρ,N(0)=0,\mathcal K_{\rho,N}(0)=0,

we obtain:

Theorem 7.1 — Low-frequency ensemble-ready response

For

∣t∣≤1,|t|\le1, ∣Kρ,N(t)∣∣t∣≪Nβ−1(1+∣γ∣)2.\boxed{ \frac{ |\mathcal K_{\rho,N}(t)| }{ |t| } \ll \frac{ N^{\beta-1} }{ (1+|\gamma|)^2 }. }

This additional ordinate decay is a direct consequence of centering. The logarithm is log⁡(2N/x)\log(2N/x), not log⁡x\log x, so no log⁡N\log N loss appears.

8. Low-frequency cross terms are bounded

By Theorem 7.1,

∫N−κ≤∣t∣≤1∣∑ρKρ,N(t)∣2t2 dt≪(∑ρcρ)2.\int_{N^{-\kappa}\le|t|\le1} \frac{ \left| \sum_\rho \mathcal K_{\rho,N}(t) \right|^2 }{t^2}\,dt \ll \left( \sum_\rho c_\rho \right)^2.

Section 6 already proved

(∑ρcρ)2≪∑ρbρ2.\left( \sum_\rho c_\rho \right)^2 \ll \sum_\rho b_\rho^2.

Therefore

∫N−κ≤∣t∣≤1∣∑ρKρ,N(t)∣2t2 dt≪∑ρN2β−2(1+∣γ∣)2.\boxed{ \int_{N^{-\kappa}\le|t|\le1} \frac{ \left| \sum_\rho \mathcal K_{\rho,N}(t) \right|^2 }{t^2}\,dt \ll \sum_\rho \frac{ N^{2\beta-2} }{ (1+|\gamma|)^2 }. }

The low-frequency channel is therefore not a hidden coherent rank-one power obstruction.

9. Main Z3 Gram reduction theorem

Define the weighted beta mass

DN(T)=∑∣γ∣≤TN2β−2(1+∣γ∣)2.\boxed{ \mathfrak D_N(T) = \sum_{|\gamma|\le T} \frac{ N^{2\beta-2} }{ (1+|\gamma|)^2 }. }

Combine Sections 5–8.

Theorem 9.1 — Polylogarithmic zero-ensemble Gram reduction

For

2≤T≤N,2\le T\le N, ∫T−1≤∣t∣≤T∣∑∣γ∣≤TKρ,N(t)∣2t2 dt≪log⁡3(T+2)DN(T).\boxed{ \int_{T^{-1}\le|t|\le T} \frac{ \left| \sum_{|\gamma|\le T} \mathcal K_{\rho,N}(t) \right|^2 }{t^2}\,dt \ll \log^3(T+2) \mathfrak D_N(T). }

More generally, for the Campaign 44 band

N−κ≤∣t∣≤NκN^{-\kappa} \le |t| \le N^\kappa

with

T=Nκ,T=N^\kappa, Ezeros(N,κ)≪(log⁡N)3DN(Nκ).\boxed{ \mathcal E_{\mathrm{zeros}}(N,\kappa) \ll (\log N)^3 \mathfrak D_N(N^\kappa). }

Thus zero-zero cross terms cost only No(1)N^{o(1)}.

Create:

B-RH-046
PESC_ZERO_ENSEMBLE_POLYLOG_GRAM_REDUCTION_TO_WEIGHTED_BETA_MASS
CERTIFIED

This is the central Z3 result.

10. Exact weighted zero-density layer cake

Define

Zw(σ,T)=∑∣γ∣≤Tβ≥σ1(1+∣γ∣)2.\boxed{ \mathfrak Z_w(\sigma,T) = \sum_{\substack{|\gamma|\le T\\\beta\ge\sigma}} \frac1{(1+|\gamma|)^2}. }

Zeros with

β<12\beta<\frac12

contribute only

O(N−1)O(N^{-1})

to DN(T)\mathfrak D_N(T), because

N2β−2≤N−1N^{2\beta-2}\le N^{-1}

and

∑ρ(1+∣γ∣)−2<∞.\sum_\rho (1+|\gamma|)^{-2} < \infty.

For

β≥12,\beta\ge\frac12,

the identity

N2β−2=N−1+2log⁡N∫1/2βN2σ−2 dσN^{2\beta-2} = N^{-1} + 2\log N \int_{1/2}^{\beta} N^{2\sigma-2}\,d\sigma

holds exactly.

Summing over zeros and applying Tonelli to the nonnegative integrand gives:

Theorem 10.1 — Weighted beta layer-cake identity

DN(T)=O(N−1)+2log⁡N∫1/21N2σ−2Zw(σ,T) dσ.\boxed{ \mathfrak D_N(T) = O(N^{-1}) + 2\log N \int_{1/2}^{1} N^{2\sigma-2} \mathfrak Z_w(\sigma,T)\,d\sigma. }

The implicit O(N−1)O(N^{-1}) contains the β<1/2\beta<1/2 contribution and the baseline β≥1/2\beta\ge1/2 term.

Create:

B-RH-047
PESC_WEIGHTED_BETA_MASS_EXACT_LAYER_CAKE_LEDGER
CERTIFIED

This closes the algebraic part of Z3C.

11. A sufficient weighted zero-density criterion

Theorem 9.1 immediately gives:

Corollary 11.1

If

DN(Nκ)≪N−κ+o(1),\boxed{ \mathfrak D_N(N^\kappa) \ll N^{-\kappa+o(1)}, }

then

Ezeros(N,κ)≪N−κ+o(1).\boxed{ \mathcal E_{\mathrm{zeros}}(N,\kappa) \ll N^{-\kappa+o(1)}. }

Together with the explicit-formula remainder from Paper 51, this implies the complete PK5 middle-band target.

A simple stronger sufficient condition is

β≤1−κ2\beta \le 1-\frac{\kappa}{2}

for every zero with

∣γ∣≤Nκ.|\gamma|\le N^\kappa.

Indeed,

N2β−2≤N−κ,N^{2\beta-2} \le N^{-\kappa},

and the ordinate weight is summable.

At

κ=1,\kappa=1,

this condition becomes

β≤12.\beta\le\frac12.

By functional-equation symmetry, that is exactly the critical-line condition for the truncated zero set.

This does not prove the condition. It identifies the exponent geometry of PK5.

12. Upper ordinate shells are automatically admissible

For

Y≥2,Y\ge2,

Riemann-von Mangoldt gives

#{ρ:Y≤∣γ∣≤2Y}≪Ylog⁡Y.\#\{ \rho: Y\le|\gamma|\le2Y \} \ll Y\log Y.

Since

N2β−2≤1,N^{2\beta-2}\le1,

we have:

Theorem 12.1 — High-ordinate shell bound

∑Y≤∣γ∣≤2YN2β−2(1+∣γ∣)2≪log⁡YY.\boxed{ \sum_{\substack{ Y\le|\gamma|\le2Y }} \frac{ N^{2\beta-2} }{ (1+|\gamma|)^2 } \ll \frac{\log Y}{Y}. }

Consequently,

∑∣γ∣≥YN2β−2(1+∣γ∣)2≪log⁡(Y+2)Y.\boxed{ \sum_{|\gamma|\ge Y} \frac{ N^{2\beta-2} }{ (1+|\gamma|)^2 } \ll \frac{\log(Y+2)}{Y}. }

Take

T=NκT=N^\kappa

and

Y=T(log⁡N)A.Y= \frac{ T }{ (\log N)^A }.

Then

∑Y≤∣γ∣≤TN2β−2(1+∣γ∣)2≪N−κ(log⁡N)A+1.\boxed{ \sum_{Y\le|\gamma|\le T} \frac{ N^{2\beta-2} }{ (1+|\gamma|)^2 } \ll N^{-\kappa} (\log N)^{A+1}. }

Thus the uppermost polynomial ordinate shell is fixed-power admissible without any information about β\beta beyond β<1\beta<1.

Create:

B-RH-048
PESC_TOP_ORDINATE_SHELL_FIXED_POWER_ADMISSIBILITY
CERTIFIED

Close:

Z3D
CLOSED_AS_TOP_POLYNOMIAL_ORDINATE_SHELL_AUTOMATICALLY_ADMISSIBLE

The remaining dangerous mass lies below the top ordinate shell and is controlled by the real parts.

13. Classical zero-density calibration

Classical zero-density estimates have the form

Nζ(σ,U)≪UA(σ)(1−σ)(log⁡U)B.N_\zeta(\sigma,U) \ll U^{A(\sigma)(1-\sigma)} (\log U)^B.

Two standard examples are:

For

12≤σ<34,\frac12\le\sigma<\frac34,

Ingham gives the exponent

3(1−σ)2−σ.\frac{ 3(1-\sigma) }{ 2-\sigma }.

For

34≤σ<1,\frac34\le\sigma<1,

Huxley gives

3(1−σ)3σ−1.\frac{ 3(1-\sigma) }{ 3\sigma-1 }.

These estimates imply strong rarity of zeros to the right of a fixed vertical line and are more than sufficient to make many weighted height sums converge.

However, zero-density estimates count exceptional zeros. They do not exclude a finite exceptional set.

The weighted beta criterion is sensitive to even one fixed off-critical zero.

Suppose

ρ0=β0+iγ0\rho_0=\beta_0+i\gamma_0

is fixed. Once

T>∣γ0∣,T>|\gamma_0|,

its contribution to DN(T)\mathfrak D_N(T) is

N2β0−2(1+∣γ0∣)2.\frac{ N^{2\beta_0-2} }{ (1+|\gamma_0|)^2 }.

For the endpoint

κ=1,\kappa=1,

if

β0>12,\beta_0>\frac12,

then

N2β0−2≫N−1.N^{2\beta_0-2} \gg N^{-1}.

Thus no zero-density theorem that still permits even one fixed off-critical zero can by itself certify

DN(N)≪N−1+o(1).\mathfrak D_N(N) \ll N^{-1+o(1)}.

More generally, for fixed κ\kappa a zero with

β0>1−κ2\beta_0> 1-\frac{\kappa}{2}

violates the simple weighted-beta sufficient condition.

Create:

O-RH-130
CLASSICAL_ZERO_DENSITY_ALONE_CANNOT_CERTIFY_ENDPOINT_WEIGHTED_BETA_MASS
CERTIFIED_AS_Z3C_METHOD_BARRIER

This is a method barrier, not a theorem that PK5 fails.

14. Why the classical zero-free region also remains subpower

A classical zero-free region has the qualitative form

β≤1−η(∣γ∣),\beta \le 1-\eta(|\gamma|),

where

η(U)→0\eta(U)\to0

as

U→∞.U\to\infty.

Even after the ordinate weight

(1+∣γ∣)−2(1+|\gamma|)^{-2}

is included, such a shrinking strip does not produce a uniform fixed power in NN for all intermediate subpolynomial heights.

For the elementary de la Vallee Poussin shape

η(U)≍1log⁡U,\eta(U) \asymp \frac1{\log U},

inverting the zero-free region and balancing the factors

N−2(1−β)N^{-2(1-\beta)}

and

∣γ∣−2|\gamma|^{-2}

produces a subpower exponential scale rather than

N−δN^{-\delta}

with fixed δ>0\delta>0.

Stronger classical zero-free regions improve this subpower scale, but the same structural issue remains: their distance from ℜs=1\Re s=1 tends to zero.

This is fully consistent with Paper 51 obstruction O-RH-129.

15. Z3C closure

The purpose of Z3C was not to prove RH or the PK5 endpoint directly. It was to identify exactly how zero-density information enters after cross terms are priced.

That ledger is now complete:

Ezeros≪No(1)DN(T)\boxed{ \mathcal E_{\mathrm{zeros}} \ll N^{o(1)} \mathfrak D_N(T) }

and

DN(T)=O(N−1)+2log⁡N∫1/21N2σ−2Zw(σ,T) dσ.\boxed{ \mathfrak D_N(T) = O(N^{-1}) + 2\log N \int_{1/2}^{1} N^{2\sigma-2} \mathfrak Z_w(\sigma,T)\,d\sigma. }

Classical zero-density estimates can be inserted into Zw\mathfrak Z_w, but they do not remove finite exceptional off-critical zeros.

Close:

Z3C
CLOSED_AS_EXACT_WEIGHTED_BETA_DENSITY_LEDGER_WITH_CLASSICAL_DENSITY_METHOD_BARRIER

16. Complete Z3 closure

All four Z3 subtasks have now been resolved.

Z3A CLOSED
UNIT_ORDINATE_CLUSTER_COST_ONLY_LOGARITHMIC

Z3B CLOSED
SEPARATED_ORDINATE_WINDOWS_WITH_POLYLOG_GRAM_COST

Z3C CLOSED
EXACT_WEIGHTED_BETA_DENSITY_LEDGER_WITH_CLASSICAL_DENSITY_METHOD_BARRIER

Z3D CLOSED
TOP_POLYNOMIAL_ORDINATE_SHELL_AUTOMATICALLY_ADMISSIBLE

Therefore close Z3 as:

CLOSED_AS_ZERO_ORDINATE_GEOMETRY_REDUCED_WITH_ONLY_POLYLOG_LOSS_TO_WEIGHTED_BETA_MASS

PK5 remains open.

17. The exact Z4 frontier

After Papers 51 and 52, the explicit-formula zero contribution has the power-safe upper reduction

Ezeros≪No(1)∑∣γ∣≤NκN2β−2(1+∣γ∣)2.\mathcal E_{\mathrm{zeros}} \ll N^{o(1)} \sum_{|\gamma|\le N^\kappa} \frac{ N^{2\beta-2} }{ (1+|\gamma|)^2 }.

No remaining NδN^\delta cost comes from:

  • unit zero multiplicity;
  • separated ordinate windows;
  • high-frequency rank-one tails;
  • low-frequency centering;
  • explicit-formula truncation;
  • the uppermost ordinate shell.

The only remaining power exponent is

2β−2.\boxed{ 2\beta-2. }

Thus Z4 is now exactly:

OFF_CRITICAL_ANTI_CANCELLATION_OR_CRITICAL_LINE_ADMISSION

There are two logically distinct routes.

Route Z4-A — Upper admission

Prove directly that the actual weighted beta ensemble satisfies enough cancellation or structure to force

Ezeros≪N−κ+o(1)\mathcal E_{\mathrm{zeros}} \ll N^{-\kappa+o(1)}

without assuming RH.

Route Z4-B — Converse detection

Prove that an off-critical zero with sufficiently large β\beta creates a coercive response that cannot be cancelled by the rest of the zero ensemble.

For κ=1\kappa=1, a successful global version of Z4-B would connect the PESC middle-band endpoint directly to the critical line.

Neither route is proved here.

18. Recommended Z4 attack order

The next campaign taskpack should be:

Z4A FUNCTIONAL_EQUATION_SAME_ORDINATE_PAIR_AUDIT
Z4B CONTINUOUS_RESPONSE_LIMIT_WITH_UNIFORM_DISCRETE_ERROR
Z4C FINITE_CLUSTER_LOWER_GRAM_OR_RESPONSE_LINEAR_INDEPENDENCE
Z4D HIGH_ORDINATE_CONTAMINATION_TAIL_IN_FIXED_TEST_WINDOW
Z4E OFF_CRITICAL_DOMINANT_BETA_ANTI_CANCELLATION

The first object to exploit is the functional-equation partner:

If

ρ=β+iγ\rho=\beta+i\gamma

is a zero, then

1−ρ‾=1−β+iγ1-\overline\rho = 1-\beta+i\gamma

is also a zero.

Their NN scales are

Nβ−1N^{\beta-1}

and

N−β.N^{-\beta}.

If

β>12,\beta>\frac12,

the ratio is

N2β−1.N^{2\beta-1}.

Thus the same-ordinate reflected partner is power-smaller than the right-hand zero. This does not yet prove anti-cancellation against all other zeros, but it eliminates one apparent symmetry-based cancellation mechanism.

This is the natural starting point for Paper 53.

19. External calibration

The external results used as standard inputs are:

  1. Riemann-von Mangoldt zero counting and critical-strip symmetry
    NIST Digital Library of Mathematical Functions, Section 25.10.
    https://dlmf.nist.gov/25.10

  2. Ingham-Huxley zero-density estimates
    Classical zero-density theory; see Iwaniec-Kowalski, Analytic Number Theory, Chapter 10, and the corresponding lecture reproduction:
    https://wiki.math.ntnu.no/_media/ma3001/2025h/analyticnumbertheory/ik_chapter_10.pdf

  3. Kusmin-Landau first derivative estimate
    Used already in Paper 51 for the discrete zero response and again here for the differentiated centered kernel.
    https://androma.org/theorems/9055

These are calibration and standard analytic inputs. No density hypothesis, pair-correlation conjecture, Lindelof hypothesis, or RH is assumed.

20. Campaign state transition

Advance the candidate state from

v1.42v1.42

to

v1.43.v1.43.

Add:

B-RH-046
PESC_ZERO_ENSEMBLE_POLYLOG_GRAM_REDUCTION_TO_WEIGHTED_BETA_MASS
CERTIFIED

Add:

B-RH-047
PESC_WEIGHTED_BETA_MASS_EXACT_LAYER_CAKE_LEDGER
CERTIFIED

Add:

B-RH-048
PESC_TOP_ORDINATE_SHELL_FIXED_POWER_ADMISSIBILITY
CERTIFIED

Add:

O-RH-130
CLASSICAL_ZERO_DENSITY_ALONE_CANNOT_CERTIFY_ENDPOINT_WEIGHTED_BETA_MASS
CERTIFIED_AS_Z3C_METHOD_BARRIER

Track state:

PK5 ACTIVE

Z1 CLOSED
Z2 CLOSED
Z3 CLOSED
Z4 ACTIVE

No root certificate is created.

21. Conclusion

The zero ensemble has now been separated into geometry and exponent.

The geometry is benign at fixed-power scale:

ordinate clustering+separated-window cross terms=No(1).\boxed{ \text{ordinate clustering} + \text{separated-window cross terms} = N^{o(1)}. }

The exponent is not:

N2β−2.\boxed{ N^{2\beta-2}. }

The complete Z3 reduction is

Ezeros≪(log⁡N)3∑∣γ∣≤NκN2β−2(1+∣γ∣)2.\boxed{ \mathcal E_{\mathrm{zeros}} \ll (\log N)^3 \sum_{|\gamma|\le N^\kappa} \frac{ N^{2\beta-2} }{ (1+|\gamma|)^2 }. }

Thus Campaign 44 has reached a sharply defined frontier.

The remaining question is no longer whether many zero responses overlap.

It is whether off-critical real-part mass can be admitted by a new cancellation mechanism or, conversely, whether an off-critical response can be shown to survive every allowed cancellation.

That is Z4.