CSM_RH Paper 58

Euler-Product Positivity Walls, Higher-Derivative Sign Loss, and the Heath-Brown Joint-Covariance Requirement

Project: CSM_RH
Paper: 58
Version: v0.1
Date: 2026-09-08
Campaign: 46 — SEEDED_ARITHMETIC_STRIP_GAP_GENERATION
Track: SG4 — NEW_ARITHMETIC_CONTRACTION_THEOREM
Status: STANDARD POSITIVITY / LINEAR-FACTORIZATION ROUTES CLOSED AS NON-AMPLIFYING; DIRECT ARITHMETIC FRONTIERS REMAIN OPEN
Canonical entry state: v1.48 / Paper 57 v0.1
RH_PROVED: FALSE
RH_DISPROVED: FALSE
GLOBAL_RH_CERTIFICATE: FALSE

Abstract

Campaign 46 seeks a genuine arithmetic theorem which upgrades a seed exponent

\operatorname{PESC}(\kappa)

\operatorname{PESC}(\kappa+\eta)

for some fixed

\eta>0.

Papers 55–57 showed that this is exactly a fixed zero-strip improvement problem and that principal-arc projection, standard moment-Markov methods, and classical Turan-Gallagher single-zero detectors do not generate the missing exponent.

The present paper audits the remaining standard candidates based on Euler-product positivity, higher logarithmic derivatives, prime support positivity, and Heath-Brown factorization.

First, positivity in the absolute-convergence half-plane has an intrinsic logarithmic-distance wall. Let

F(s) = -\frac{\zeta'(s)}{\zeta(s)}.

For

\sigma>1,

the Dirichlet coefficients of $F$ are nonnegative:

F(s) = \sum_{n\ge1} \frac{\Lambda(n)}{n^s}.

A nonnegative trigonometric polynomial with nonnegative cosine coefficients therefore produces the classical positivity inequality

\sum_{j=0}^{J} a_j \Re F(\sigma+ijt) \ge0.

\rho=\beta+i\gamma

is the target zero and $t=\gamma$ , the $j=1$ term contains the negative zero pole

-\frac{a_1}{\sigma-\beta}.

However, the same explicit formula contains an archimedean contribution of order

\frac12 \left( \sum_{j\ge1}a_j \right) \log|\gamma|.

In the standard one-target zero-dropping positivity architecture, a contradiction is therefore possible only if

\boxed{ 1-\beta \ll \frac1{\log|\gamma|}. }

A zero at a fixed seeded distance

1-\beta = \frac{\kappa}{2}

is asymptotically invisible to this scale comparison.

This explains why modern positivity-based improvements continue to sharpen constants in logarithmic zero-free regions rather than create fixed strips. In 2026, Bellotti, Trudgian and Yang improved the classical-shaped region to

\sigma \ge 1-\frac1{4.896\log t},

but the shape remains logarithmic.

Second, one cannot remove the logarithmic wall simply by taking higher derivatives of the logarithmic derivative. Define

F_m(s) = (-1)^m \frac{d^m}{ds^m} F(s) = \sum_{n\ge1} \frac{ \Lambda(n)(\log n)^m }{ n^s },

which again has nonnegative Dirichlet coefficients for $\Re s>1$ .

A target zero pole is amplified to order

m! (s-\rho)^{-m-1}.

But for every

m\ge1,

the zero-side kernel

\Re (s-\rho)^{-m-1}

changes sign as the ordinate separation varies. Thus the crucial one-sign Poisson-kernel property of $m=0$ is lost. Other zeros can no longer be discarded in a positivity upper bound. Higher derivatives amplify the target pole and simultaneously destroy the zero-side positivity that made the classical argument possible.

Third, raw positivity of the prime measure is far below the seeded boundary scale. A boundary zero mode contributes a short-interval error of relative size

N^{-\kappa/2}.

Since

\psi(x+H)-\psi(x) = H+U_H(x)

and

N^{-\kappa/2}=o(1),

the positivity condition

\psi(x+H)-\psi(x)\ge0

is automatically compatible with such a mode. Prime support positivity therefore supplies no fixed-power boundary suppression.

Fourth, the exact Heath-Brown factorization route is already algebraically calibrated by Paper 47. The complete alternating cross- $j$ recombination returns the original von Mangoldt root object. If the lag vectors of the finitely many factorization components are each controlled only at seed scale

NH^2N^{-\kappa+o(1)},

componentwise triangle or Cauchy estimates can recover at best the same seed scale. A strict amplifier requires a genuinely joint covariance deficit of fixed-power size, for example

\boxed{ \left\| \sum_j c_jX_j \right\|_2^2 \le N^{-\eta} \sum_j|c_j|^2\|X_j\|_2^2. }

Such a theorem is not supplied by the identity itself. Paper 47's exact recoupling and Papers 43–46's component barriers show that alternating signs, Ramaré extraction, structured-weight smallness, and componentwise Type-II estimates do not create this fixed-power joint deficit for free.

Finally, the paper calibrates positivity inside the critical strip. Recent work on the real part of the logarithmic derivative of the Riemann xi-function shows that positivity near the critical line may remain compatible with hypothetical off-critical zeros outside local neighborhoods of those zeros. Thus positivity of the xi log-derivative is not, by itself, a global off-critical exclusion principle.

The conclusion is a closure theorem for the standard SG4 candidates:

Euler-product coefficient positivity:
non-amplifying at fixed-strip scale.

Higher log-derivative positivity:
zero-side sign structure fails.

Raw prime-measure positivity:
boundary error is too small relative to the positive main term.

Existing Heath-Brown factorization:
requires a new fixed-power joint cross-j covariance theorem.

No new equivalent criterion is needed. The preferred direct arithmetic frontier remains Paper 56's F-RH-017:

\boxed{ \#\left\{ x: |U_H(x)|>HN^{-\nu} \right\} \ll N^{1-c}, \qquad \nu>\frac{\kappa}{2}, \qquad c>2(1-\alpha). }

Campaign 46 has therefore exhausted its standard structural mechanisms. The next progress must be a genuinely new prime-side excess estimate.

No RH theorem is claimed.

1. Entry state

Assume

\operatorname{PESC}(\kappa)

for fixed

0<\kappa<1.

The inherited facts are:

\boxed{ \beta_* \le 1-\frac{\kappa}{2} }

and

\boxed{ |\psi(x)-x| \ll x^{1-\kappa/2+o(1)}. }

Define

\boxed{ d_{\rm seed} = \frac{\kappa}{2}. }

A hypothetical zero on the seed boundary has

\beta = 1-d_{\rm seed}.

Campaign 46 asks for an arithmetic mechanism proving a fixed additional distance from $\Re s=1$ .

2. Absolute-convergence positivity

For

\Re s>1,

define

\boxed{ F(s) = -\frac{\zeta'(s)}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\Lambda(n)}{n^s}. }

The coefficients are nonnegative.

Let

P(\theta) = a_0 + \sum_{j=1}^{J} a_j\cos(j\theta)

satisfy

P(\theta)\ge0

for all real $\theta$ , with

a_j\ge0.

Then for every

\sigma>1

and real $t$ ,

\boxed{ \sum_{j=0}^{J} a_j \Re F(\sigma+ijt) = \sum_{n\ge1} \frac{ \Lambda(n) }{ n^\sigma } P(t\log n) \ge0. }

This is the basic Euler-product positivity mechanism used in the classical zero-free-region method.

3. One-target positivity ledger

Let

\rho=\beta+i\gamma

be a hypothetical target zero, with

|\gamma|

large.

Set

t=\gamma

and

\sigma=1+\eta, \qquad \eta>0.

Write

d=1-\beta.

Then

\sigma-\beta = \eta+d.

The completed-zeta explicit formula gives, uniformly in the relevant right half-plane,

\Re F(\sigma+i\tau) = \frac12\log(|\tau|+3) - \sum_\rho \frac{ \sigma-\Re\rho }{ (\sigma-\Re\rho)^2+(\tau-\Im\rho)^2 } + O_\sigma(1)

for nonzero $\tau$ , with the pole at $1$ treated separately at $\tau=0$ .

In the $j=1$ term, the target zero contributes

\boxed{ -\frac1{\eta+d}. }

Dropping all other negative zero contributions yields the standard one-target upper bound.

At $j=0$ ,

F(1+\eta) = \frac1{\eta} + O(1).

For $j\ge1$ ,

\Re F(1+\eta+ij\gamma) \le \frac12 \log(|j\gamma|+3) + O_\eta(1),

except that the target negative pole is retained for $j=1$ .

Hence

\begin{aligned} 0 &\le \sum_{j=0}^{J} a_j \Re F(1+\eta+ij\gamma) \\ &\le \frac{a_0}{\eta} + \frac12 \left( \sum_{j\ge1}a_j \right) \log|\gamma| - \frac{a_1}{\eta+d} + O_{P,\eta} \left( 1+\log(J+2) \right). \end{aligned}

This is the classical positivity budget in the form relevant to the seed audit.

4. The logarithmic-distance wall

Set

A_+ = \sum_{j\ge1}a_j.

Since

a_1\le A_+,

the target zero term satisfies

\frac{a_1}{\eta+d} \le \frac{A_+}{d}.

d \gg \frac1{\log|\gamma|},

then

\frac{A_+}{d} \ll A_+\log|\gamma|.

Thus the target pole is no larger than the unavoidable archimedean scale retained in the standard zero-dropping positivity ledger.

Therefore:

Theorem 4.1 — Positive-coefficient Euler-product logarithmic wall

Within the standard one-target isolation architecture based on:

the nonnegative Dirichlet coefficients of $-\zeta'/\zeta$ ;
a nonnegative trigonometric polynomial with nonnegative cosine coefficients; and
discarding all non-target zero contributions by sign,

the method can exclude zeros only at horizontal distance

\boxed{ 1-\beta = O\left( \frac1{\log|\gamma|} \right). }

It cannot by itself create a fixed zero-free strip.

Create:

O-RH-139
POSITIVE_COEFFICIENT_EULER_PRODUCT_ZERO_DROPPING_HAS_INTRINSIC_LOGARITHMIC_DISTANCE_WALL
CERTIFIED_AS_METHOD_BARRIER

This is a statement about the standard positivity architecture, not about every possible use of the Euler product.

5. Seeded boundary is far outside the positivity detection scale

Under PESC $(\kappa)$ , the active boundary distance is

d_{\rm seed} = \frac{\kappa}{2},

which is fixed as

|\gamma|\to\infty.

Therefore

d_{\rm seed} \log|\gamma| \to\infty.

The boundary zero contribution in the absolute-line positivity ledger is only

O_\kappa(1),

while the natural archimedean budget is

\asymp \log|\gamma|.

Hence the classical Euler-product positivity mechanism is asymptotically less sensitive than the seed.

This is consistent with the modern state of the art.

Bellotti, Trudgian and Yang proved in 2026 the explicit region

\boxed{ \zeta(\sigma+it)\ne0 \quad \text{for} \quad \sigma \ge 1- \frac1{4.896\log t}, \qquad t\ge3. }

The constant is substantially improved, but the asymptotic distance scale remains

1/\log t.

Thus modern refinements of this positivity family improve constants inside the same logarithmic geometry rather than supplying a fixed-strip bootstrap.

6. Why higher logarithmic derivatives do not repair the wall

Define

\boxed{ F_m(s) = (-1)^m \frac{d^m}{ds^m} F(s). }

For

\Re s>1,

\boxed{ F_m(s) = \sum_{n\ge1} \frac{ \Lambda(n)(\log n)^m }{ n^s }, }

so the Dirichlet coefficients remain nonnegative.

Near a zero $\rho$ ,

F_m(s)

contains the amplified pole

\boxed{ -\frac{ m! }{ (s-\rho)^{m+1} }. }

This looks promising because the target zero pole becomes high order.

However, the zero-side sign structure changes.

For

s-\rho = a+ib, \qquad a>0,

\boxed{ \Re \frac1{(a+ib)^{m+1}} = \frac{ \cos \left( (m+1)\arctan(b/a) \right) }{ (a^2+b^2)^{(m+1)/2} }. }

When

m=0,

the cosine is positive for every real $b$ .

When

m\ge1,

the cosine changes sign as $b/a$ varies.

For example, take

\arctan(b/a) = \frac{\pi}{m+2}.

Then

\cos \left( \frac{(m+1)\pi}{m+2} \right) <0.

Thus:

Theorem 6.1 — Higher-derivative zero-side sign-loss theorem

For every

m\ge1,

the real zero kernel associated with $F_m$ is not one-signed in the half-plane to the right of the zeros.

Therefore the classical step

keep the target zero
drop all other zero contributions by sign

is no longer valid.

Create:

O-RH-140
HIGHER_LOG_DERIVATIVE_POLE_AMPLIFICATION_DESTROYS_ZERO_SIDE_ONE_SIGN_STRUCTURE
CERTIFIED

Higher derivatives amplify the target pole but simultaneously remove the positivity mechanism needed to isolate it.

7. Relation to Turan detectors

One can attempt to recover target isolation from higher derivatives using Turan power-sum selection.

That is exactly the architecture audited in Paper 57.

The result was:

target isolation is recovered,
but at a fixed distance-exponent loss C,
and the classical detector cannot amplify the seed when C>2.

Thus the two routes fit together:

m = 0:
one-sign zero kernel,
but logarithmic-distance wall.

m >= 1:
strong pole amplification,
but zero-side sign is lost.

Turan recovery:
restores target sensitivity,
but pays a non-amplifying detector constant.

There is no missing free derivative trick between Papers 57 and 58.

8. Raw prime-measure positivity is boundary-blind

Let

H=N^\alpha

and

U_H(x) = \psi(x+H)-\psi(x)-H.

Since $\Lambda(n)\ge0$ ,

\boxed{ \psi(x+H)-\psi(x) \ge0. }

Equivalently,

U_H(x)\ge-H.

A boundary zero under PESC $(\kappa)$ contributes at scale

\boxed{ |U_H(x)| \asymp H N^{-\kappa/2} }

in the critical model.

Since

N^{-\kappa/2} =o(1),

HN^{-\kappa/2} \ll H.

Therefore the positivity lower bound

U_H(x)\ge-H

is compatible with the entire boundary-mode oscillation.

Create:

O-RH-141
RAW_PRIME_MEASURE_POSITIVITY_IS_BLIND_TO_SEEDED_BOUNDARY_MODE
CERTIFIED_AS_SCALE_BARRIER

Prime positivity controls order-one relative negativity.

The required strip amplifier lives at a vanishing relative scale.

9. Positivity of the xi logarithmic derivative does not give global exclusion

The completed zeta function satisfies

\boxed{ \frac{\xi'}{\xi}(s) = \sum_\rho \frac1{s-\rho} }

in the standard paired sense.

\Re s>\beta_*,

every individual Poisson kernel has positive real part, so

\Re \frac{\xi'}{\xi}(s) >0.

This is simply positivity to the right of the complete zero set.

It contains no mechanism for moving $\beta_*$ .

Recent work by Grigutis and Turcinskas studies positivity of

\Re \xi'/\xi

near the critical line. Their 2026 analysis explicitly considers hypothetical off-critical zeros and finds that positivity may persist away from relatively small neighborhoods of those zeros.

This is external evidence for the internal conclusion:

xi-log-derivative positivity
is compatible with off-critical zeros
and is not itself a global strip-gap theorem.

10. Heath-Brown exact recoupling is unchanged by seeding

Paper 47 certified the exact cross- $j$ recombination of the Heath-Brown identity.

After all $j$ -levels are retained before squaring and the inherited output range is respected,

\boxed{ \sum_j c_j H_j = \Lambda }

on the relevant root range.

The translated lag operator is linear.

Therefore, if

X_j

denotes the lag vector of component $H_j$ ,

\boxed{ X_\Lambda = \sum_j c_jX_j. }

The seed PESC exponent does not change this identity.

It only changes the scale against which the component and joint estimates are judged.

11. Componentwise estimates cannot create a joint fixed power for free

Assume there are only

N^{o(1)}

effective $j$ -levels / dyadic component cells after the fixed Heath-Brown order and dyadic partition are accounted for.

Suppose every component is controlled only at the seed lag scale:

\boxed{ \|X_j\|_2^2 \ll NH^2N^{-\kappa+o(1)}. }

Then triangle inequality gives

\begin{aligned} \|X_\Lambda\|_2 &\le \sum_j |c_j| \|X_j\|_2 \\ &\ll N^{o(1)} \left( NH^2N^{-\kappa} \right)^{1/2}. \end{aligned}

Hence

\boxed{ \|X_\Lambda\|_2^2 \ll NH^2N^{-\kappa+o(1)}. }

No fixed improvement occurs.

Therefore:

Theorem 11.1 — Seed-scale componentwise Heath-Brown no-amplification theorem

A finite or $N^{o(1)}$ Heath-Brown decomposition whose components are controlled only through separate seed-scale norm bounds cannot yield a strict MLEPG exponent

\delta>\kappa.

Create:

O-RH-142
COMPONENTWISE_HEATH_BROWN_SEED_SCALE_BOUNDS_CANNOT_CREATE_FIXED_POWER_AMPLIFICATION
CERTIFIED

This is independent of the detailed form of the component bounds.

12. The exact missing Heath-Brown theorem is joint

Expand

\boxed{ \|X_\Lambda\|_2^2 = \sum_{j,k} c_j \overline{c_k} \langle X_j,X_k \rangle. }

A strict improvement at the component natural scale would follow from a genuinely joint inequality such as

\boxed{ \left\| \sum_j c_jX_j \right\|_2^2 \le N^{-\eta} \sum_j |c_j|^2 \|X_j\|_2^2 }

for some fixed

\eta>0,

or from another estimate of equivalent fixed-power strength.

This is a covariance theorem.

It is not contained in the Heath-Brown identity.

Paper 47 already proved that the exact alternating signs restore the original $\Lambda$ object rather than automatically creating a power saving.

Papers 43–46 further certified that:

isolated Liouville components;
Ramaré extraction;
prime-harmonic leverage;
structured-weight $L^2$ smallness; and
factor-by-factor determinant geometry

do not supply the missing fixed power.

Thus SG4B is not "apply Heath-Brown again."

It is:

prove a new seeded joint cross-j covariance theorem.

No such theorem is certified.

13. Boundary-mode interpretation of the joint covariance problem

Suppose the root error contains a seed-boundary mode at exponent

\beta = 1-\frac{\kappa}{2}.

Paper 55 showed that its lag energy is

NH^2N^{-\kappa}

and is concentrated on the principal Fejer arc.

Since the complete Heath-Brown sum reconstructs $\Lambda$ , the component covariance matrix must reconstruct that boundary mode as well.

Therefore a fixed-power joint covariance deficit would imply that the complete reconstructed boundary mode is smaller than the seed scale.

By Papers 54–55, that is already a zero-strip improvement.

Hence:

Corollary 13.1 — Joint covariance improvement is genuine strip-gap arithmetic

Any seeded Heath-Brown joint inequality strong enough to yield

\delta>\kappa

is itself a new fixed zero-strip theorem.

It cannot be justified solely by the formal alternating signs of the identity.

This is a calibration, not a circularity objection.

14. SG4A verdict: prime support and positivity

The original SG4A candidate was:

PRIME_SUPPORT_AND_POSITIVITY_VERSUS_SMOOTH_BOUNDARY_MODE

Verdict:

STANDARD POSITIVITY ROUTE CLOSED AS NON-AMPLIFYING.

Reasons:

raw prime-measure positivity only controls the relative scale $1$ ;
the boundary signal has relative scale $N^{-\kappa/2}$ ;
Euler-product positivity in $\Re s>1$ has only logarithmic zero-distance resolution;
near-critical xi positivity is compatible with hypothetical off-critical zeros away from local neighborhoods.

A new positivity theorem would need quantitative information beyond coefficient nonnegativity.

15. SG4B verdict: seeded Heath-Brown bilinear route

The original SG4B candidate was:

SEEDED_HEATH_BROWN_BILINEAR_MAJOR_ARC_INCOMPATIBILITY

Verdict:

EXISTING COMPONENTWISE ROUTE CLOSED.

The missing theorem is explicitly joint:

\boxed{ \text{fixed-power cross-j covariance deficit}. }

The seed does not manufacture it.

Record:

SG4B-JOINT
OPEN

as a possible but high-cost arithmetic subfrontier.

16. SG4C verdict: nonlinear Euler-product boundary detector

The original SG4C candidate was:

NONLINEAR_EULER_PRODUCT_BOUNDARY_MODE_DETECTOR

The standard candidates are exhausted:

first log derivative:
logarithmic-distance wall

higher log derivatives:
zero-side sign loss

Turan derivative selection:
Paper 57 detector saturation

xi log-derivative positivity:
compatible with hypothetical off-line zeros

Therefore close the standard SG4C architecture as:

CLOSED_AS_NO_STANDARD_EULER_POSITIVITY_OR_DERIVATIVE_ROUTE_CROSSES_THE_SEEDED_FIXED_BOUNDARY

A genuinely nonlinear Euler-product theorem remains logically possible but would require new information not contained in ordinary coefficient positivity.

17. SG4D status

SG4D was:

MULTISCALE_PRIME_FACTOR_COHERENCE_BREAKING

No certified theorem currently separates the von Mangoldt sequence from the critical boundary mode at a fixed-power level.

This remains the only genuinely open structural-arithmetic direction inside SG4.

However, it is not yet as concrete as F-RH-017.

Therefore Campaign 46 should not create a new canonical frontier merely to rename the gap.

The preferred direct target remains:

F-RH-017
SEEDED_SUPERCRITICAL_SHRINKING_THRESHOLD_EXCEPTIONAL_SET

because it has explicit parameters and a certified amplification map.

The secondary detector target remains:

F-RH-018
SEEDED_TURAN_GALLAGHER_DETECTOR_EXCESS_POWER

but is quantitatively more expensive.

18. Current external calibration

18.1. 2026 Heath-Brown-inspired zero-free region

Chiara Bellotti, Tim Trudgian and Andrew Yang, Zero-free regions inspired by work of Heath-Brown, arXiv:2603.21490.

They prove

\zeta(\sigma+it)\ne0

for

t\ge3, \qquad \sigma \ge 1-\frac1{4.896\log t}.

The result substantially improves the constant in the classical logarithmic region but does not alter its $1/\log t$ shape.

URL:

https://arxiv.org/abs/2603.21490

18.2. Positivity of the xi log derivative

Andrius Grigutis and Lukas Turcinskas, Note on the positivity of the real part of the log-derivative of the Riemann xi-function near the critical line, Lithuanian Mathematical Journal, 2026.

The paper studies positivity near the critical line and explicitly considers scenarios with hypothetical off-critical zeros. Positivity can persist away from local neighborhoods of such zeros.

URL:

https://arxiv.org/abs/2509.18963

These results are used only as calibration of method scope.

19. Campaign 46 structural status

After Papers 56–58:

SG1
principal-arc standard structure:
CLOSED
arithmetic suppression:
OPEN

SG2
seeded shrinking-threshold bridge:
CERTIFIED
F-RH-017:
OPEN

SG3
standard Turan-Gallagher / Mellin detector:
CLOSED AS NON-AMPLIFYING
F-RH-018:
OPEN SECONDARY

SG4A
standard positivity:
CLOSED AS NON-AMPLIFYING

SG4B
componentwise Heath-Brown:
CLOSED
joint covariance theorem:
OPEN HIGH-COST

SG4C
standard Euler-product derivative positivity:
CLOSED

SG4D
new nonlinear prime coherence:
OPEN / UNSPECIFIED

The campaign has reached the point where further formal architecture is unlikely to reduce the gap.

20. State transition

Advance the candidate state from

v1.48

v1.49.

Add:

O-RH-139
POSITIVE_COEFFICIENT_EULER_PRODUCT_ZERO_DROPPING_HAS_INTRINSIC_LOGARITHMIC_DISTANCE_WALL
CERTIFIED_AS_METHOD_BARRIER

Add:

O-RH-140
HIGHER_LOG_DERIVATIVE_POLE_AMPLIFICATION_DESTROYS_ZERO_SIDE_ONE_SIGN_STRUCTURE
CERTIFIED

Add:

O-RH-141
RAW_PRIME_MEASURE_POSITIVITY_IS_BLIND_TO_SEEDED_BOUNDARY_MODE
CERTIFIED_AS_SCALE_BARRIER

Add:

O-RH-142
COMPONENTWISE_HEATH_BROWN_SEED_SCALE_BOUNDS_CANNOT_CREATE_FIXED_POWER_AMPLIFICATION
CERTIFIED

No new root theorem and no RH certificate is created.

21. Recommended next action

The most economical remaining theorem is still F-RH-017.

Choose

H=N^{1-\tau}

with fixed small $\tau>0$ .

Prove

\boxed{ \#\left\{ x\in[N,2N]: |\psi(x+H)-\psi(x)-H| > HN^{-\nu} \right\} \ll N^{1-c} }

with

\boxed{ \nu>\frac{\kappa}{2}, \qquad c>2\tau. }

Paper 56 then converts this directly into

\operatorname{PESC}(\kappa+\eta)

for an explicit fixed $\eta>0$ .

This target is weaker and more concrete than:

full natural Selberg variance;
a large-constant Turan detector excess;
a new joint Heath-Brown covariance theorem; or
an unspecified nonlinear Euler-product principle.

The next mathematical campaign should therefore attack F-RH-017 directly rather than generating another representation.

22. Conclusion

The standard positivity idea has now been separated into its real strengths and its real limits.

Positive von Mangoldt coefficients are powerful enough to prove nonvanishing near $\Re s=1$ .

They are not fine enough to see a seeded boundary at fixed distance.

Higher derivatives increase pole sensitivity but lose one-sign zero geometry.

Raw positivity of prime counts is blind to a vanishing relative boundary oscillation.

Heath-Brown factorization exposes arithmetic structure but, after exact cross- $j$ recoupling, does not create a fixed covariance deficit by formal algebra alone.

Thus Campaign 46 has exhausted its standard structural mechanisms.

The missing theorem is no longer hidden.

It is a genuinely new prime-side excess estimate beyond the critical seed scale.

Among the currently certified formulations, F-RH-017 remains the cheapest direct target.

CSM_RH Paper 58 — Euler-Product Positivity Walls, Higher-Derivative Sign Loss, and the Heath-Brown Joint-Covariance Requ

CSM_RH Paper 58

Euler-Product Positivity Walls, Higher-Derivative Sign Loss, and the Heath-Brown Joint-Covariance Requirement

Abstract

1. Entry state

2. Absolute-convergence positivity

3. One-target positivity ledger

4. The logarithmic-distance wall

Theorem 4.1 — Positive-coefficient Euler-product logarithmic wall

5. Seeded boundary is far outside the positivity detection scale

6. Why higher logarithmic derivatives do not repair the wall

Theorem 6.1 — Higher-derivative zero-side sign-loss theorem

7. Relation to Turan detectors

8. Raw prime-measure positivity is boundary-blind

9. Positivity of the xi logarithmic derivative does not give global exclusion

10. Heath-Brown exact recoupling is unchanged by seeding

11. Componentwise estimates cannot create a joint fixed power for free

Theorem 11.1 — Seed-scale componentwise Heath-Brown no-amplification theorem

12. The exact missing Heath-Brown theorem is joint

13. Boundary-mode interpretation of the joint covariance problem

Corollary 13.1 — Joint covariance improvement is genuine strip-gap arithmetic

14. SG4A verdict: prime support and positivity

15. SG4B verdict: seeded Heath-Brown bilinear route

16. SG4C verdict: nonlinear Euler-product boundary detector

17. SG4D status

18. Current external calibration

18.1. 2026 Heath-Brown-inspired zero-free region

18.2. Positivity of the xi log derivative

19. Campaign 46 structural status

20. State transition

21. Recommended next action

22. Conclusion