CSM_RH Paper 55

Fixed-Exponent PESC–Zero-Strip Equivalence, Boundary-Zero Principal-Arc Locking, and the No-Free-Delocking Theorem

Project: CSM_RH
Paper: 55
Version: v0.1
Date: 2026-09-08
Campaign: 45 — PESC_EXPONENT_AMPLIFICATION_OR_MLEPG_BOOTSTRAP
Tracks: EA3 / EA4 structural closure
Status: AMPLIFIER STRUCTURE COMPLETE / PRIME-SIDE ARITHMETIC GAP THEOREM OPEN
Canonical entry state: v1.45 / Paper 54 v0.1
RH_PROVED: FALSE
RH_DISPROVED: FALSE
GLOBAL_RH_CERTIFICATE: FALSE

Abstract

Paper 54 proved the seeded MLEPG amplification law

\kappa' < \min \left\{ \alpha, \delta, 2-\alpha(2-\kappa) \right\}

and showed that strict amplification requires

\alpha>\kappa, \qquad \delta>\kappa.

The present paper identifies exactly what such an amplification means in the zeta zero geometry and proves that principal-arc delocking cannot be obtained for free by projection or multiscale bookkeeping.

For every fixed

0<\kappa\le1,

PESC $(\kappa)$ is shown to be exponent-equivalent to the fixed zero-free half-plane

\boxed{ \sup_{\zeta(\rho)=0} \Re\rho \le 1-\frac{\kappa}{2}. }

The implication from PESC to the zero-free half-plane was obtained in Paper 53 by Mellin analyticity. The converse follows from the truncated explicit formula: if every nontrivial zero satisfies

\beta\le1-\frac{\kappa}{2},

then

\psi(x)-x \ll x^{1-\kappa/2}\log^2x,

and hence

J_N^\vartheta \ll N^{3-\kappa}\log^4N.

Define

\beta_* = \sup_{\zeta(\rho)=0}\Re\rho

and

\kappa_* = \sup \left\{ \kappa\in[0,1]: \operatorname{PESC}(\kappa) \text{ holds} \right\}.

Then

\boxed{ \kappa_* = 2(1-\beta_*). }

Thus the maximal PESC exponent is exactly twice the global zero-free gap from $\Re s=1$ .

The paper then analyzes the explicit-formula power mode

F_\rho(x) = -\frac{x^\rho}{\rho}.

For a fixed zero ordinate and any sublinear lag

H=N^\alpha, \qquad 0<\alpha<1,

the lag energy satisfies

\mathcal S_\rho(N,H) \asymp_\rho H^2N^{2\beta-1}.

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

this becomes

\boxed{ \mathcal S_\rho(N,H) \asymp_\rho NH^2N^{-\kappa}. }

Thus a zero on the seed strip boundary is exactly a critical-locking mode with lag exponent $\delta=\kappa$ .

Moreover, the discrete derivative of this mode is spectrally concentrated on the principal Fejer arc. If

b_{\rho,n} = F_\rho(n)-F_\rho(n-1)

on $N<n\le2N$ , then for fixed $c>0$ ,

\int_{\|\xi\|>c/H} \left| \sum_{N<n\le2N} b_{\rho,n}e(n\xi) \right|^2d\xi \ll_{\rho,c} \frac{H}{N} \sum_{N<n\le2N}|b_{\rho,n}|^2.

Hence $1-O(H/N)$ of the derivative energy lies inside the principal arc, and the Fejer-weighted principal component has order

H^2N^{2\beta-1}.

The adjacent-block defect of the same zero mode is only

O_\rho((H/N)^2)

relative to its lag energy. Therefore its cumulative dyadic contraction mass up to any terminal scale $N^\alpha$ , $\alpha<1$ , is $o(1)$ .

These facts yield two no-bypass results.

First, subtracting or projecting away a principal low-frequency component cannot prove MLEPG for the original prime sequence unless the removed component is independently bounded with exponent strictly larger than $\kappa$ . For a boundary zero mode, the removed component already has the critical size $NH^2N^{-\kappa}$ .

Second, any valid seeded MLEPG theorem with $\alpha>\kappa$ and $\delta>\kappa$ necessarily proves a strictly narrower zero-free strip. It is therefore not a weaker deterministic consequence of PESC $(\kappa)$ ; it is itself new strip-gap arithmetic.

Campaign 45 is consequently structurally complete. The remaining task is theorem generation: produce a genuinely arithmetic prime-side estimate that creates a fixed gap beyond the seed strip boundary.

No RH theorem is claimed.

1. PESC and the rightmost zeta zero

Let

E_\psi(x) = \psi(x)-x.

Paper 53 proved that for every fixed

0<\kappa\le1,

PESC $(\kappa)$ implies

\boxed{ \zeta(s)\ne0 \qquad \Re s> 1-\frac{\kappa}{2}. }

Equivalently,

\boxed{ \operatorname{PESC}(\kappa) \Longrightarrow \beta_* \le 1-\frac{\kappa}{2}. }

We now prove the converse at the same exponent resolution.

2. Fixed zero-free half-plane implies the corresponding PNT power

Assume

\boxed{ \beta_* \le \sigma_0 < 1. }

Use the standard truncated explicit formula for the symmetrized Chebyshev function:

\psi_0(x) = x - \sum_{|\gamma|\le T} \frac{x^\rho}{\rho} + O \left( \frac{x\log^2(xT)}{T} + \log x \right).

At a prime-power discontinuity,

\psi(x)-\psi_0(x) = O(\log x),

which is negligible below.

Take

T=x^2.

The Riemann-von Mangoldt zero count implies

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

Since

|x^\rho| \le x^{\sigma_0},

we obtain

\boxed{ \psi(x)-x \ll x^{\sigma_0}\log^2x. }

Therefore

\int_N^{2N} |\psi(x)-x|^2dx \ll N^{2\sigma_0+1}\log^4N.

\sigma_0 = 1-\frac{\kappa}{2},

then

2\sigma_0+1 = 3-\kappa.

Using Paper 53's prime-power and discrete-continuous transfer:

Theorem 2.1 — Fixed-exponent PESC / zero-strip equivalence

For every fixed

0<\kappa\le1,

\boxed{ \operatorname{PESC}(\kappa) \Longleftrightarrow \beta_* \le 1-\frac{\kappa}{2}. }

Equivalently,

\operatorname{PESC}(\kappa)

holds if and only if all nontrivial zeta zeros lie in

\frac{\kappa}{2} \le \Re\rho \le 1-\frac{\kappa}{2},

using functional-equation symmetry.

Create:

B-RH-055
FIXED_EXPONENT_PESC_ZERO_FREE_STRIP_EQUIVALENCE
CERTIFIED

This extends the endpoint equivalence in Paper 53 to every fixed exponent.

3. Exact maximal-exponent law

Define

\boxed{ \beta_* = \sup_{\zeta(\rho)=0} \Re\rho. }

By critical-line symmetry,

\frac12 \le \beta_* \le 1.

Define

\boxed{ \kappa_* = \sup \left\{ \kappa\in[0,1]: \operatorname{PESC}(\kappa) \text{ holds} \right\}. }

Theorem 2.1 immediately gives:

Theorem 3.1 — PESC spectral exponent law

\boxed{ \kappa_* = 2(1-\beta_*). }

Create:

B-RH-056
PESC_MAXIMAL_EXPONENT_EQUALS_TWICE_GLOBAL_ZETA_ZERO_FREE_GAP
CERTIFIED

Interpretation:

beta_* = 1
  <-> no fixed positive PESC exponent

1/2 < beta_* < 1
  <-> a maximal subendpoint PESC exponent exists

beta_* = 1/2
  <-> kappa_* = 1
  <-> RH

Thus exponent amplification and zero-strip narrowing are the same global resource measured in different coordinates.

4. Explicit-formula zero mode

Fix

\rho=\beta+i\gamma

with

0<\beta<1.

Define the natural explicit-formula mode

\boxed{ F_\rho(x) = -\frac{x^\rho}{\rho}. }

Then

F_\rho'(x) = -x^{\rho-1}.

For a lag

1\le H=o(N),

define

U_{\rho,H}(x) = F_\rho(x+H)-F_\rho(x).

Taylor expansion, uniformly for

N\le x\le2N,

gives

\boxed{ U_{\rho,H}(x) = -Hx^{\rho-1} \left[ 1+O_\rho(H/N) \right]. }

Hence:

Theorem 4.1 — Zero-mode lag-energy law

For fixed $\rho$ and $H=o(N)$ ,

\boxed{ \int_N^{2N} |U_{\rho,H}(x)|^2dx \asymp_\rho H^2N^{2\beta-1}. }

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

then

\boxed{ \int_N^{2N} |U_{\rho,H}(x)|^2dx \asymp_\rho NH^2N^{-\kappa}. }

Thus a zero on the PESC $(\kappa)$ boundary has exactly the critical lag exponent

\boxed{ \delta_{\rm zero}=\kappa. }

This is the zeta-zero version of Paper 54's abstract power-law locking countermodel.

5. Discrete derivative of a zero mode

On the dyadic integer interval define

\boxed{ b_{\rho,n} = F_\rho(n)-F_\rho(n-1), \qquad N<n\le2N. }

For fixed $\rho$ ,

b_{\rho,n} = -n^{\rho-1} \left[ 1+O_\rho(1/N) \right].

Therefore, for

\beta>\frac12,

\boxed{ \sum_{N<n\le2N} |b_{\rho,n}|^2 \asymp_\rho N^{2\beta-1}. }

At $\beta=1/2$ the same sum is $\asymp_\rho1$ , which is the limiting logarithmic case.

Define the additive Fourier transform

\boxed{ B_{\rho,N}(\xi) = \sum_{N<n\le2N} b_{\rho,n}e(n\xi). }

Parseval gives

\int_0^1 |B_{\rho,N}(\xi)|^2d\xi = \sum_{N<n\le2N}|b_{\rho,n}|^2.

6. Principal-arc concentration of a fixed zero mode

For

\|\xi\|>0,

summation by parts and the geometric-sum estimate give

\left| B_{\rho,N}(\xi) \right| \ll_\rho \frac{ N^{\beta-1} }{ \|\xi\| }.

Indeed:

the endpoint size of $b_{\rho,n}$ is $O_\rho(N^{\beta-1})$ ;
the total variation of $b_{\rho,n}$ over the dyadic block is $O_\rho(N^{\beta-1})$ ;
partial sums of $e(n\xi)$ are $O(\|\xi\|^{-1})$ .

Let

H=N^\alpha, \qquad 0<\alpha<1,

and fix an absolute $c>0$ .

Then

\begin{aligned} \int_{\|\xi\|>c/H} |B_{\rho,N}(\xi)|^2d\xi &\ll_{\rho,c} N^{2\beta-2} \int_{c/H}^{1/2} \frac{d\xi}{\xi^2} \\ &\ll_{\rho,c} HN^{2\beta-2}. \end{aligned}

Comparing with total Parseval mass:

Theorem 6.1 — Fixed-zero principal-arc concentration

For fixed $\rho$ with $\beta>1/2$ ,

\boxed{ \frac{ \displaystyle \int_{\|\xi\|>c/H} |B_{\rho,N}(\xi)|^2d\xi }{ \displaystyle \int_0^1 |B_{\rho,N}(\xi)|^2d\xi } \ll_{\rho,c} \frac{H}{N}. }

Hence, because $H=o(N)$ ,

\boxed{ \int_{\|\xi\|\le c/H} |B_{\rho,N}(\xi)|^2d\xi = \left[ 1-o(1) \right] \int_0^1 |B_{\rho,N}(\xi)|^2d\xi. }

A fixed zero mode is therefore asymptotically a principal-additive-frequency mode at every sublinear lag scale.

Create:

B-RH-057
FIXED_ZETA_ZERO_MODE_CONCENTRATES_ON_PRINCIPAL_FEJER_ARC
CERTIFIED_AS_MODEL_THEOREM

7. Principal Fejer energy of the zero mode

Let

D_H(\xi) = \sum_{r=1}^{H}e(r\xi).

For sufficiently small fixed $c_0>0$ ,

|D_H(\xi)|^2 \gg H^2

\|\xi\|\le c_0/H.

Theorem 6.1 therefore gives

\boxed{ \int_{\|\xi\|\le c_0/H} |B_{\rho,N}(\xi)|^2 |D_H(\xi)|^2d\xi \asymp_\rho H^2N^{2\beta-1} }

at exponent scale.

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

then

\boxed{ \text{principal Fejer zero-mode energy} \asymp NH^2N^{-\kappa}. }

Thus the critical-locking mode identified in Paper 54 is not merely a physical-space smooth drift. It is precisely concentrated in the principal $q=1$ Fejer arc isolated in Paper 18.

8. Adjacent-scale defect of the zero mode

Differentiate again:

F_\rho''(x) = -(\rho-1)x^{\rho-2}.

Using Taylor expansion,

U_{\rho,H}(x+H)-U_{\rho,H}(x) = -(\rho-1) H^2x^{\rho-2} \left[ 1+O_\rho(H/N) \right].

Therefore

\boxed{ \int_N^{2N} |U_{\rho,H}(x+H)-U_{\rho,H}(x)|^2dx \asymp_\rho H^4N^{2\beta-3}. }

Dividing by the lag energy from Theorem 4.1:

\boxed{ \frac{ \mathcal D_\rho(N,H) }{ \mathcal S_\rho(N,H) } \asymp_\rho \left( \frac{H}{N} \right)^2. }

Hence the local normalized contraction coefficient satisfies

\boxed{ q_\rho(H) = 1- O_\rho \left( \frac{H^2}{N^2} \right). }

For a dyadic chain

H_j=2^jH_0

with terminal scale

H_J\le N^\alpha, \qquad \alpha<1,

the cumulative contraction mass obeys

\begin{aligned} G_\rho(J) &= \sum_{j<J} -\log q_\rho(H_j) \\ &\ll_\rho \sum_{j<J} \frac{H_j^2}{N^2} \\ &\ll \frac{H_J^2}{N^2} \\ &\ll N^{2\alpha-2}. \end{aligned}

Thus:

Theorem 8.1 — Boundary-zero multiscale locking

For every fixed zero mode and every sublinear terminal scale,

\boxed{ G_\rho(J)=o(1). }

A boundary zero mode contributes no linear contraction mass.

Create:

O-RH-133
BOUNDARY_ZERO_MODE_IS_ASYMPTOTICALLY_MAXIMALLY_LOCKED_ACROSS_SUBLINEAR_DYADIC_LAGS
CERTIFIED_AS_MODEL_BARRIER

9. Seed PESC gives no principal-arc margin

Paper 54 showed that PESC $(\kappa)$ alone gives only

\mathcal S_\Lambda(N,H) \ll N^{3-\kappa+o(1)}.

H=N^\alpha,

this corresponds to the MLEPG remainder exponent

\delta_{\rm det} = \kappa+2\alpha-2.

For every

\alpha<1,

\delta_{\rm det}<\kappa.

The boundary-zero model shows a sharper reason why no deterministic principal-arc argument can cross the gap.

A legal PESC $(\kappa)$ spectral mode can sit almost entirely on the principal Fejer arc and have exact lag exponent $\delta=\kappa$ .

Therefore the missing inequality is not a Fourier-coordinate artifact.

It is a statement excluding the boundary mode itself.

10. No-free-projection theorem

Suppose a proof introduces a decomposition

b = P_Hb + (I-P_H)b,

where $P_H$ is a principal-arc, low-frequency, finite-rank, or smooth-drift projector.

Assume the residual component is shown to satisfy a strict amplifier bound.

This does not control the original lag energy unless the projected component is also bounded.

For the boundary-zero model of Sections 4–7,

P_Hb_\rho

contains

1-o(1)

of the derivative $L^2$ mass for every projector which captures the principal arc at width $\asymp H^{-1}$ .

Its Fejer lag energy remains

\asymp NH^2N^{-\kappa}.

Therefore any projected proof aiming at

NH^2N^{-\delta}, \qquad \delta>\kappa,

must independently prove an additional suppression

\boxed{ \|P_Hb\|_{\rm Fejer}^2 \ll NH^2N^{-\kappa-\eta} }

for some fixed

\eta>0.

That suppression is exactly the missing strip-gap arithmetic.

Create:

O-RH-134
PRINCIPAL_ARC_PROJECTION_CANNOT_BYPASS_BOUNDARY_MODE_WITHOUT_INDEPENDENT_POWER_SUPPRESSION
CERTIFIED

This rejects the idea that one may simply subtract the critical power-law mode and prove decorrelation of the residual.

The coefficient of the removed mode is the hard object.

11. Strict MLEPG amplification is a zero-strip improvement theorem

Assume PESC $(\kappa)$ .

Suppose MLEPG $(\alpha,\delta)$ holds with

\alpha>\kappa, \qquad \delta>\kappa.

Paper 54 gives PESC $(\kappa')$ for some fixed

\kappa'>\kappa.

Theorem 2.1 then gives

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

Therefore:

Theorem 11.1 — Seeded amplifier / strip-gap theorem

Every strict seeded MLEPG amplifier theorem is automatically a theorem creating a fixed new zero-free gap beyond the seed boundary.

Equivalently, if

\beta_* = 1-\frac{\kappa}{2},

then no MLEPG theorem in the strict-amplifier regime

\alpha>\kappa, \qquad \delta>\kappa

can hold.

Create:

B-RH-058
STRICT_SEEDED_MLEPG_AMPLIFICATION_IMPLIES_STRICT_ZERO_STRIP_NARROWING
CERTIFIED

This does not make MLEPG circular.

It calibrates its exact arithmetic strength.

12. Conversely, a strip improvement already gives the amplified PESC exponent

Suppose one proves directly that

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

for some

\kappa'>\kappa.

Theorem 2.1 immediately gives PESC $(\kappa')$ .

Thus, at the root exponent level:

\boxed{ \text{strict PESC amplification} \Longleftrightarrow \text{strict fixed zero-strip improvement}. }

MLEPG is one possible prime-side mechanism for proving such an improvement.

It is not an intermediate theorem of lower zero-strip strength.

13. EA3 verdict

Campaign 45 / EA3 asked whether a seeded PESC theorem could delock the principal Fejer arc.

The structural answer is now complete.

SEED PESC(kappa)
  permits a boundary mode

boundary mode
  is concentrated on principal Fejer arc

boundary mode
  has lag exponent delta = kappa

boundary mode
  has o(1) dyadic contraction mass

projection removal
  merely transfers the burden to coefficient suppression

strict delocking delta > kappa
  creates a genuinely narrower zero strip

Therefore close the structural audit as:

EA3S
CLOSED_AS_PRINCIPAL_ARC_DELOCKING_REQUIRES_NEW_FIXED_STRIP_GAP_ARITHMETIC

The arithmetic theorem itself remains open:

EA3A
OPEN_PRIME_SIDE_PRINCIPAL_ARC_POWER_SUPPRESSION_BEYOND_SEED_BOUNDARY

14. EA4 verdict

Paper 54 identified the contraction-mass amplifier condition

G_N(J) > \kappa\log N + o(\log N).

Theorem 8.1 shows that the boundary zero model has

G_\rho(J)=o(1)

through every sublinear lag chain.

Therefore no deterministic multiscale geometry can create the required mass from PESC $(\kappa)$ alone.

A successful EA4 theorem must again exclude or suppress the boundary mode.

Close the structural audit as:

EA4S
CLOSED_AS_LINEAR_CONTRACTION_MASS_REQUIRES_ARITHMETIC_BOUNDARY_MODE_SUPPRESSION

Keep open:

EA4A
OPEN_PRIME_SIDE_LINEAR_CONTRACTION_MASS_BEYOND_CRITICAL_LOCKING

15. Campaign 45 structural closure

Campaign 45 has now certified:

the seeded amplification law;
the optimal ideal exponent map;
the exact PESC / zero-strip exponent equivalence;
the principal-arc location of the boundary mode;
the multiscale locking of that mode;
the impossibility of projection or deterministic scale geometry as a free bypass.

Thus the amplifier architecture is complete.

The remaining problems are no longer architectural.

They are arithmetic.

Record:

CAMPAIGN_45
STRUCTURAL_AMPLIFIER_THEORY_COMPLETE
ARITHMETIC_SEED_AND_STRIP_GAP_GATES_OPEN

16. Current literature calibration

The relation between zero-free regions and PNT error terms is classical and remains an active subject.

Recent 2026 work of Broucke gives upper bounds for PNT error terms from broad classes of zero-free regions and studies near-sharpness by constructing generalized prime systems with zeros on prescribed contours.

Recent work of Johnston and Trudgian makes explicit a Pintz/Landau philosophy in the reverse direction: sufficiently strong arithmetic remainder information forces zero-free information.

These results calibrate the present theorem:

arithmetic error improvement
  <-> stronger zero exclusion

is not peculiar to CSM_RH.

The internal contribution is the exact matching of that principle to the PESC exponent, MLEPG amplifier law, and principal-Fejer critical-locking geometry.

Current Guth-Maynard large-value technology gives major zero-density and short-interval range improvements, but does not presently provide a fixed strip gap beyond a seeded boundary.

17. Next campaign

Further coordinate changes of PESC, MLEPG, or the principal arc should not count as progress unless they produce a new arithmetic inequality.

The recommended next campaign is:

CSM_RH Campaign 46
SEEDED_ARITHMETIC_STRIP_GAP_GENERATION

Root goal:

Given PESC(kappa),
prove PESC(kappa+eta)
for some fixed eta>0.

Equivalent zero goal:

Given beta_* <= 1-kappa/2,
prove beta_* <= 1-(kappa+eta)/2.

Allowed prime-side tracks:

SG1 SEEDED_PRINCIPAL_FEJER_POWER_SUPPRESSION
SG2 SEEDED_SHRINKING_THRESHOLD_SHORT_INTERVAL_L2
SG3 SEEDED_ZERO_DETECTOR_WITH_PRIME_SIDE_COERCIVITY
SG4 NEW_ARITHMETIC_CONTRACTION_THEOREM

Required output:

\boxed{ \eta>0 \text{ fixed}. }

Hard rejections:

ANOTHER_EQUIVALENT_CRITERION_WITHOUT_NEW_ESTIMATE
PROJECT_OUT_PRINCIPAL_MODE_WITHOUT_COEFFICIENT_BOUND
USE_ZERO_DENSITY_TO_IGNORE_FINITE_BOUNDARY_ZEROS
PROMOTE_SUBPOWER_TO_FIXED_POWER
ASSUME_BOUNDARY_LINE_IS_ZERO_FREE
ASSUME_PAIR_CORRELATION
ASSUME_RH

18. State transition

Advance the candidate state from

v1.45

v1.46.

Add:

B-RH-055
FIXED_EXPONENT_PESC_ZERO_FREE_STRIP_EQUIVALENCE
CERTIFIED

Add:

B-RH-056
PESC_MAXIMAL_EXPONENT_EQUALS_TWICE_GLOBAL_ZETA_ZERO_FREE_GAP
CERTIFIED

Add:

B-RH-057
FIXED_ZETA_ZERO_MODE_CONCENTRATES_ON_PRINCIPAL_FEJER_ARC
CERTIFIED_AS_MODEL_THEOREM

Add:

B-RH-058
STRICT_SEEDED_MLEPG_AMPLIFICATION_IMPLIES_STRICT_ZERO_STRIP_NARROWING
CERTIFIED

Add:

O-RH-133
BOUNDARY_ZERO_MODE_IS_ASYMPTOTICALLY_MAXIMALLY_LOCKED_ACROSS_SUBLINEAR_DYADIC_LAGS
CERTIFIED_AS_MODEL_BARRIER

Add:

O-RH-134
PRINCIPAL_ARC_PROJECTION_CANNOT_BYPASS_BOUNDARY_MODE_WITHOUT_INDEPENDENT_POWER_SUPPRESSION
CERTIFIED

Campaign 45:

STRUCTURAL_AMPLIFIER_THEORY_COMPLETE
ARITHMETIC_GATES_OPEN

No RH certificate is created.

19. Conclusion

The CSM_RH exponent has acquired an exact spectral meaning.

\boxed{ \kappa_* = 2(1-\beta_*). }

A seeded PESC exponent is therefore a measured zero-free gap.

An amplifier must enlarge that gap.

The principal Fejer arc contains precisely the kind of smooth explicit-formula mode which saturates the seed exponent. A boundary zero mode has

\boxed{ \mathcal S_\rho(N,H) \asymp NH^2N^{-\kappa} }

and

\boxed{ G_\rho(J)=o(1). }

It cannot be removed by a free projection and it cannot be destroyed by deterministic multiscale geometry.

Thus the next advance must be an actual theorem about the primes.

Not a new representation.

Not a new equivalent condition.

Not a new zero-density count which permits a finite boundary exception.

The missing object is now exact:

\boxed{ \text{a prime-side theorem which creates a fixed new strip gap}. }

CSM_RH Paper 55 — Fixed-Exponent PESC–Zero-Strip Equivalence, Boundary-Zero Principal-Arc Locking, and the No-Free-Deloc

CSM_RH Paper 55

Fixed-Exponent PESC–Zero-Strip Equivalence, Boundary-Zero Principal-Arc Locking, and the No-Free-Delocking Theorem

Abstract

1. PESC and the rightmost zeta zero

2. Fixed zero-free half-plane implies the corresponding PNT power

Theorem 2.1 — Fixed-exponent PESC / zero-strip equivalence

3. Exact maximal-exponent law

Theorem 3.1 — PESC spectral exponent law

4. Explicit-formula zero mode

Theorem 4.1 — Zero-mode lag-energy law

5. Discrete derivative of a zero mode

6. Principal-arc concentration of a fixed zero mode

Theorem 6.1 — Fixed-zero principal-arc concentration

7. Principal Fejer energy of the zero mode

8. Adjacent-scale defect of the zero mode

Theorem 8.1 — Boundary-zero multiscale locking

9. Seed PESC gives no principal-arc margin

10. No-free-projection theorem

11. Strict MLEPG amplification is a zero-strip improvement theorem

Theorem 11.1 — Seeded amplifier / strip-gap theorem

12. Conversely, a strip improvement already gives the amplified PESC exponent

13. EA3 verdict

14. EA4 verdict

15. Campaign 45 structural closure

16. Current literature calibration

17. Next campaign

18. State transition

19. Conclusion