CSM_RH Paper 59

Seeded $L^p$ Residue-Chain Amplification, $L^1$ Optimality, and the Minimal Exceptional-Set Strip-Gap Gate

Project: CSM_RH
Paper: 59
Version: v0.1
Date: 2026-09-08
Campaign: 46 — SEEDED_ARITHMETIC_STRIP_GAP_GENERATION
Direct frontier: F-RH-017
Status: STRONGER DETERMINISTIC EXCEPTIONAL-SET AMPLIFIER CERTIFIED / SUPERCRITICAL ARITHMETIC THRESHOLD STILL OPEN
Canonical entry state: v1.49 / Paper 58 v0.1
RH_PROVED: FALSE
RH_DISPROVED: FALSE
GLOBAL_RH_CERTIFICATE: FALSE

Abstract

Paper 56 converted a seeded shrinking-threshold exceptional-set theorem into a lag $L^2$ estimate and then into a PESC exponent improvement. That route required

c>2(1-\alpha)

when

H=N^\alpha

and

|\{x:|U_H(x)|>HN^{-\nu}\}| \ll N^{1-c}.

The present paper proves that this is not the optimal deterministic use of the exceptional set.

Let

A(n)=\psi(n)-n, \qquad U_H(n)=A(n+H)-A(n).

Assume PESC $(\kappa)$ and write

d=\frac{\kappa}{2}.

Paper 55 gives the pointwise seed envelope

|A(n)| \ll n^{1-d+o(1)}.

For every fixed $p\ge1$ , a residue-chain inequality gives

\sum_{n\le2N}|A(n)|^p \ll_p \frac{N}{H} \sum_{r\le H}|A(r)|^p + \left( \frac{N}{H} \right)^p \sum_{n\le2N-H}|U_H(n)|^p.

H=N^\alpha

and

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

then the seed pointwise envelope on the exceptional set yields

\sum_{n\le2N}|A(n)|^p \ll N^{1+p(1-d_p')+o(1)}

for every

\boxed{ d_p' < \Psi_p(d;\alpha,\nu,c) := \min \left\{ \nu,\, d+\alpha-1+\frac{c}{p},\, 1-\alpha(1-d) \right\}. }

A dyadic $L^p$ Mellin argument then excludes all zeta zeros with

\Re\rho>1-d_p'.

Using Paper 55's fixed-exponent PESC/zero-strip equivalence, this gives PESC $(\kappa')$ for every

\boxed{ \kappa' < 2\Psi_p \left( \frac{\kappa}{2}; \alpha,\nu,c \right). }

Among all $p\ge1$ , the only $p$ -dependent term is $c/p$ . Therefore:

\boxed{ p=1 }

is the optimal member of the entire $L^p$ residue-chain family.

The resulting $L^1$ amplification law is

\boxed{ \kappa' < \Phi_1(\kappa;\alpha,\nu,c) := \min \left\{ 2\nu,\, \kappa+2c+2\alpha-2,\, 2-2\alpha+\alpha\kappa \right\}. }

Strict amplification occurs exactly when

\boxed{ \nu>\frac{\kappa}{2}, \qquad c>1-\alpha, \qquad \alpha<1. }

Thus the exceptional-set requirement from Paper 56 is cut in half.

For the near-macroscopic parametrization

H=N^{1-\tau},

the strict gate is simply

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

The output gain is

\boxed{ \eta < \min \left\{ 2\nu-\kappa,\, 2(c-\tau),\, (2-\kappa)\tau \right\}. }

If the threshold is not the bottleneck and $c$ is fixed, the optimal scale is

\boxed{ \tau_* = \frac{2c}{4-\kappa}, \qquad \alpha_* = 1-\frac{2c}{4-\kappa}, }

and the one-step gain is

\boxed{ \eta_* = \frac{ 2c(2-\kappa) }{ 4-\kappa }. }

This is twice the exceptional-set gain produced by the $L^2$ bridge of Paper 56 for the same $c$ .

The paper also audits the current Gafni–Tao exceptional-set architecture against a polynomially shrinking threshold. Their published theorem fixes the relative threshold $\delta>0$ and a sufficiently large integer $J=J(\delta,\theta,\varepsilon)$ before taking $X\to\infty$ . In their explicit-formula reduction the truncation error is $O(X^\theta/J)$ , so replacing

\delta

X^{-\nu}

requires at least

J\gg X^\nu.

Thus the published theorem cannot be invoked by direct substitution.

More importantly, even after rebuilding the proof with a polynomial truncation height, the standard $L^2$ and $L^4$ zero-packet Markov exponents are supercritical at the seeded boundary whenever

\nu>\frac{\kappa}{2}.

Hence the fixed-threshold parameter issue is not the sole obstruction. The boundary mode remains the genuine wall.

The direct Campaign-46 frontier is therefore strengthened:

F-RH-017-v2
SEEDED_SUPERCRITICAL_SHRINKING_THRESHOLD_EXCEPTIONAL_SET

with the minimal deterministic requirements

\boxed{ H=N^{1-\tau}, \qquad \nu>\frac{\kappa}{2}, \qquad c>\tau. }

No RH theorem is claimed.

1. Seed notation

Assume PESC $(\kappa)$ for fixed

0<\kappa<1.

Set

\boxed{ d=\frac{\kappa}{2}. }

Paper 55 gives

\boxed{ \beta_*\le1-d }

and the pointwise PNT error estimate

\boxed{ |A(x)| = |\psi(x)-x| \ll x^{1-d+o(1)}. }

Let

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

For integers $n$ define

\boxed{ U_H(n) = A(n+H)-A(n). }

2. A general residue-chain $L^p$ inequality

Fix

1\le p<\infty.

Write every integer $n\le2N$ as

n=r+kH,

with

1\le r\le H

and

0\le k\le M, \qquad M\ll\frac{N}{H}.

Along each residue chain,

A(r+kH) = A(r) + \sum_{j=0}^{k-1} U_H(r+jH).

Using

|x+y|^p \le 2^{p-1} \left( |x|^p+|y|^p \right)

and

\left| \sum_{j<k}u_j \right|^p \le k^{p-1} \sum_{j<k}|u_j|^p,

we get

|A(r+kH)|^p \ll_p |A(r)|^p + k^{p-1} \sum_{j<k} |U_H(r+jH)|^p.

Summing $k\le M$ gives

\sum_{k\le M} |A(r+kH)|^p \ll_p M|A(r)|^p + M^p \sum_{j<M} |U_H(r+jH)|^p.

Summing over $r$ proves:

Theorem 2.1 — Residue-chain $L^p$ inequality

\boxed{ \sum_{n\le2N}|A(n)|^p \ll_p \frac{N}{H} \sum_{r\le H}|A(r)|^p + \left( \frac{N}{H} \right)^p \sum_{n\le2N-H}|U_H(n)|^p. }

Create:

B-RH-062
SEEDED_RESIDUE_CHAIN_LP_INEQUALITY
CERTIFIED

3. Seeded anchor in $L^p$

The seed pointwise bound gives

\sum_{r\le H}|A(r)|^p \ll H^{1+p(1-d)+o(1)}.

Therefore the residue-chain anchor is

\boxed{ \frac{N}{H} \sum_{r\le H}|A(r)|^p \ll N H^{p(1-d)+o(1)}. }

H=N^\alpha,

this is

\boxed{ N^{1+\alpha p(1-d)+o(1)}. }

4. Exceptional-set $L^p$ lag bound

Assume

\boxed{ |\mathcal E| \ll N^{1-c}, }

where

\mathcal E = \left\{ n\in[N,2N]: |U_H(n)|>HN^{-\nu} \right\}.

Good set

On the complement,

|U_H(n)|^p \le H^pN^{-p\nu}.

Hence

\boxed{ \sum_{n\notin\mathcal E}|U_H(n)|^p \ll N H^p N^{-p\nu}. }

Bad set

The seed pointwise envelope gives

|U_H(n)| \le |A(n+H)|+|A(n)| \ll N^{1-d+o(1)}.

Therefore

\boxed{ \sum_{n\in\mathcal E}|U_H(n)|^p \ll N^{1-c+p(1-d)+o(1)}. }

Thus

\boxed{ \sum_{n\le2N-H}|U_H(n)|^p \ll NH^pN^{-p\nu} + N^{1-c+p(1-d)+o(1)}. }

5. Global seeded $L^p$ error exponent

Multiply the lag terms by

(N/H)^p.

The good-set term becomes

\boxed{ N^{p+1-p\nu}. }

The bad-set term becomes

\boxed{ N^{1-c+p(2-d-\alpha)+o(1)}. }

The anchor term is

\boxed{ N^{1+\alpha p(1-d)+o(1)}. }

We compare these with the target form

\boxed{ N^{1+p(1-d')+o(1)}. }

The three constraints are:

d'<\nu,

d' < d+\alpha-1+\frac{c}{p},

and

d' < 1-\alpha(1-d).

Therefore:

Theorem 5.1 — Seeded exceptional-set $L^p$ exponent

For every

d_p' < \boxed{ \Psi_p(d;\alpha,\nu,c) = \min \left\{ \nu,\, d+\alpha-1+\frac{c}{p},\, 1-\alpha(1-d) \right\}, }

one has

\boxed{ \sum_{n\le2N}|A(n)|^p \ll N^{1+p(1-d_p')+o(1)}. }

Create:

B-RH-063
SEEDED_EXCEPTIONAL_SET_TO_GLOBAL_LP_PNT_ERROR_GAIN
CERTIFIED

6. Global $L^p$ error implies a zero-free strip

The discrete / continuous difference between

A(n)

and

\psi(x)-x

inside a unit interval is bounded by $1$ , so the same exponent holds for

\int_N^{2N} |\psi(x)-x|^pdx.

Suppose

\int_X^{2X} |\psi(x)-x|^pdx \ll X^{1+p(1-d')+\varepsilon}.

Let

s=\sigma+it.

For $p>1$ , Hölder gives

\begin{aligned} \int_X^{2X} |\psi(x)-x| x^{-\sigma-1}dx &\le \left( \int_X^{2X} |\psi(x)-x|^pdx \right)^{1/p} \\ &\quad\times \left( \int_X^{2X} x^{-q(\sigma+1)}dx \right)^{1/q} \\ &\ll X^{1-d'-\sigma+\varepsilon/p}, \end{aligned}

where

1/p+1/q=1.

For $p=1$ the same bound follows directly.

Thus the dyadic Mellin sum converges locally uniformly for

\boxed{ \Re s>1-d'. }

By Paper 53's Mellin identity,

-\frac{\zeta'(s)}{\zeta(s)} - \frac{s}{s-1}

is analytic there.

Therefore:

Theorem 6.1 — $L^p$ Mellin strip theorem

If the dyadic $L^p$ exponent $d'$ holds, then

\boxed{ \zeta(s)\ne0 \quad \text{for} \quad \Re s>1-d'. }

By Paper 55, this is equivalent at exponent scale to PESC $(\kappa')$ for every

\boxed{ \kappa'<2d'. }

Combining with Theorem 5.1:

Theorem 6.2 — Seeded $L^p$ exceptional-set PESC amplifier

\boxed{ \kappa' < 2 \Psi_p \left( \frac{\kappa}{2}; \alpha,\nu,c \right). }

Create:

B-RH-064
SEEDED_LP_EXCEPTIONAL_SET_TO_ZERO_STRIP_AND_PESC_AMPLIFICATION
CERTIFIED

7. $L^1$ is optimal in the entire $L^p$ family

The function

\Psi_p = \min \left\{ \nu,\, d+\alpha-1+\frac{c}{p},\, 1-\alpha(1-d) \right\}

depends on $p$ only through

c/p.

For

p\ge1,

\frac{c}{p} \le c.

Hence:

Theorem 7.1 — $L^1$ optimality

Among all fixed

p\ge1,

the strongest deterministic zero-strip output from the seeded residue-chain / exceptional-set architecture is attained at

\boxed{ p=1. }

Create:

B-RH-065
L1_IS_OPTIMAL_SEEDED_EXCEPTIONAL_SET_RESIDUE_CHAIN_EXPONENT_CONVERTER
CERTIFIED

This is the key improvement over Paper 56.

8. The optimal $L^1$ amplification law

Set

p=1.

Then

d_1' < \min \left\{ \nu,\, d+\alpha-1+c,\, 1-\alpha(1-d) \right\}.

Returning to

\kappa=2d,

we obtain:

Theorem 8.1 — Seeded $L^1$ shrinking-threshold amplification law

Every

\boxed{ \kappa' < \Phi_1(\kappa;\alpha,\nu,c) }

is admissible, where

\boxed{ \Phi_1 = \min \left\{ 2\nu,\, \kappa+2c+2\alpha-2,\, 2-2\alpha+\alpha\kappa \right\}. }

Create:

B-RH-066
SEEDED_L1_SHRINKING_THRESHOLD_PESC_AMPLIFICATION_LAW
CERTIFIED

9. Exact strict-amplifier gate

We ask when

\Phi_1>\kappa.

The three inequalities are

2\nu>\kappa,

\kappa+2c+2\alpha-2>\kappa,

and

2-2\alpha+\alpha\kappa>\kappa.

For

\alpha<1,

the last is automatic:

2-2\alpha+\alpha\kappa-\kappa = (1-\alpha)(2-\kappa) >0.

Thus:

Corollary 9.1 — Minimal deterministic exceptional-set gate

Strict amplification occurs exactly when

\boxed{ \nu>\frac{\kappa}{2}, \qquad c>1-\alpha, \qquad \alpha<1. }

Compared with Paper 56,

c>2(1-\alpha)

has been improved to

\boxed{ c>1-\alpha. }

The boundary threshold

\nu>\kappa/2

cannot be weakened.

10. Near-macroscopic formulation

Write

\boxed{ \alpha=1-\tau, \qquad \tau>0. }

Then

\Phi_1 = \min \left\{ 2\nu,\, \kappa+2(c-\tau),\, \kappa+(2-\kappa)\tau \right\}.

Thus the exponent gain

\eta=\kappa'-\kappa

may be any fixed number satisfying

\boxed{ \eta < \min \left\{ 2\nu-\kappa,\, 2(c-\tau),\, (2-\kappa)\tau \right\}. }

The minimal gate becomes

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

This is the strengthened direct frontier.

11. Optimal use of an exceptional exponent

Assume

2\nu

is not the active bottleneck.

For fixed $c$ , maximize

\min \left\{ \kappa+2c+2\alpha-2,\, 2-2\alpha+\alpha\kappa \right\}.

Balance the two terms:

\kappa+2c+2\alpha-2 = 2-2\alpha+\alpha\kappa.

This gives

\boxed{ \alpha_* = 1- \frac{2c}{4-\kappa}. }

Equivalently,

\boxed{ \tau_* = \frac{2c}{4-\kappa}. }

The optimized output exponent is

\boxed{ \kappa_* ' = \kappa + \frac{ 2c(2-\kappa) }{ 4-\kappa }. }

Thus the optimized one-step gain is

\boxed{ \eta_* = \frac{ 2c(2-\kappa) }{ 4-\kappa }. }

This is exactly twice the modest- $c$ exceptional-set gain produced by the $L^2$ conversion in Paper 56.

To keep the threshold term non-bottlenecking, it suffices that

\boxed{ \nu \ge \frac{\kappa}{2} + \frac{ c(2-\kappa) }{ 4-\kappa }. }

If the formula attempts to produce $\kappa'>1$ , the assumptions are themselves incompatible with the known critical-line zeros; in applications one stops at any fixed $\kappa'<1$ .

12. General $L^p$ optimization

For completeness, if $p\ge1$ is fixed and the threshold is not active, balance

d+\alpha-1+\frac{c}{p}

with

1-\alpha(1-d).

This gives

\boxed{ \alpha_{p,*} = 1- \frac{ 2c }{ p(4-\kappa) }. }

The PESC exponent gain is

\boxed{ \eta_{p,*} = \frac{ 2c(2-\kappa) }{ p(4-\kappa) }. }

This decreases exactly like $1/p$ .

Thus the optimality of $p=1$ is quantitative, not merely qualitative.

13. Comparison with Paper 56

Paper 56 used:

exceptional set
-> lag L2 energy
-> MLEPG
-> seeded residue-chain L2
-> PESC

The present paper uses:

exceptional set
-> lag Lp moment
-> residue-chain Lp directly
-> global PNT-error Lp
-> Mellin analyticity
-> zero strip
-> PESC

The direct Mellin route avoids the extra $L^2$ cost of squaring the exceptional-set pointwise envelope.

Therefore Paper 56 remains correct but is not optimal as an exceptional-set exponent converter.

The improved canonical direct target should use the $L^1$ bridge.

14. F-RH-017 v2

Replace the old preferred direct target by the stronger version:

F-RH-017-v2
SEEDED_SUPERCRITICAL_SHRINKING_THRESHOLD_EXCEPTIONAL_SET

At scale

\boxed{ H=N^{1-\tau}, }

prove

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

with

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

Then a strict fixed PESC exponent improvement follows.

This is the cheapest currently certified exceptional-set amplifier.

15. Audit of the Gafni-Tao fixed-threshold proof

Gafni and Tao define their exceptional set using a fixed relative threshold

\delta>0.

In the proof they explicitly:

fix $\delta$ ;
choose a natural number $J$ sufficiently large depending on $\delta,\theta,\varepsilon$ ;
hold $\delta,J,\theta,\varepsilon$ fixed as $X\to\infty$ .

Their explicit-formula truncation height is

T = J(\log^2X)X^{1-\theta},

and the truncation error is

O\left( \frac{X^\theta}{J} \right).

To compare this with a polynomial threshold

\delta_X = X^{-\nu},

one would need

\boxed{ J \gg X^\nu }

already at the truncation step.

Thus the published theorem cannot be used by the formal substitution

\delta=X^{-\nu}.

Its asymptotic notation permits constants depending on fixed $\delta$ and $J$ .

This is a theorem-scope statement, not a criticism of the result.

16. Fixed- $H$ removes one technical polynomial cost, but not the wall

The Gafni-Tao proof also subdivides

[X,2X]

into

O_\delta(1)

multiplicative intervals in order to replace the varying length

x^\theta

by a nearly fixed multiplicative increment.

For the CSM_RH frontier we may instead formulate the interval length as a fixed dyadic

H=N^\alpha.

This removes the need for that subdivision and therefore avoids a polynomial covering factor when $\delta$ shrinks.

This is a useful technical simplification.

However it does not resolve the seeded boundary threshold.

The explicit formula still requires a polynomial height

T \gtrsim N^{1-\alpha+\nu}.

More importantly, a zero on

\beta=1-\kappa/2

contributes relative size

N^{-\kappa/2}.

No finite positive moment argument can prove a density-small exceptional set below that amplitude while the boundary zero remains admissible.

17. Polynomial-threshold $L^2$ packet exponent

For comparison with the current exceptional-set technology, take a narrow zero packet around real part

\sigma

and let

q = 1-\alpha+\nu

be the minimal explicit-formula height exponent.

The standard $L^2$ zero-packet estimate has average-square exponent

2\alpha+2\sigma-2 + qA(\sigma)(1-\sigma).

Markov at threshold

N^{\alpha-\nu}

gives exceptional-measure exponent

\boxed{ \mu_{2,\sigma}^{\rm poly} = 1 + 2 \left( \nu-(1-\sigma) \right) + qA(\sigma)(1-\sigma). }

At the seed boundary

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

\nu>\frac{\kappa}{2},

then

\boxed{ \mu_{2,\sigma}^{\rm poly}>1 }

even before any positive zero-density cost is used.

Thus $L^2$ Markov cannot cross the boundary.

18. Polynomial-threshold $L^4$ packet exponent

Similarly the standard fourth-moment estimate gives

\boxed{ \mu_{4,\sigma}^{\rm poly} = 1 + 4 \left( \nu-(1-\sigma) \right) + qA^*(\sigma)(1-\sigma). }

At the seed boundary and supercritical threshold,

\boxed{ \mu_{4,\sigma}^{\rm poly}>1. }

The same baseline obstruction occurs at every finite even moment:

\boxed{ 1 + 2r \left( \nu-\frac{\kappa}{2} \right) }

before nonnegative density / additive-energy costs.

Therefore the failure of direct polynomial-threshold uniformization is structural, not merely caused by the fixed- $\delta$ quantifiers in the published theorem.

This refines Paper 56's moment barrier using the current Gafni-Tao parameter ledger.

19. Current technology verdict

Current zero-density and additive-zero-energy technology is highly effective for:

\boxed{ \nu<\frac{\kappa}{2} }

after a seed strip is present, because the boundary packet then lies below the threshold.

It cannot, through a standard finite positive moment + Markov argument, enter

\boxed{ \nu>\frac{\kappa}{2}. }

But the new $L^1$ deterministic bridge substantially lowers the amount of exceptional-set rarity required once a supercritical arithmetic theorem is found.

This is the exact division of labor:

analytic seed:
moves the right edge to 1-kappa/2

current density/moment technology:
controls subcritical thresholds

new arithmetic theorem:
must cross nu=kappa/2

L1 residue-chain bridge:
converts even modest c>tau into a fixed strip gain

20. New minimal arithmetic target

Choose a small fixed

\tau>0

and put

H=N^{1-\tau}.

The minimal direct target is now:

\boxed{ \#\left\{ n\in[N,2N]: |U_H(n)| > HN^{-\kappa/2-\varepsilon} \right\} \ll N^{1-c} }

for some fixed

\varepsilon>0, \qquad c>\tau.

Then Theorem 10.1 gives the explicit exponent gain

\boxed{ \eta < \min \left\{ 2\varepsilon,\, 2(c-\tau),\, (2-\kappa)\tau \right\}. }

This is the sharpest currently certified direct Campaign-46 target.

21. State transition

Advance the candidate state from

v1.49

v1.50.

Add:

B-RH-062
SEEDED_RESIDUE_CHAIN_LP_INEQUALITY
CERTIFIED

Add:

B-RH-063
SEEDED_EXCEPTIONAL_SET_TO_GLOBAL_LP_PNT_ERROR_GAIN
CERTIFIED

Add:

B-RH-064
SEEDED_LP_EXCEPTIONAL_SET_TO_ZERO_STRIP_AND_PESC_AMPLIFICATION
CERTIFIED

Add:

B-RH-065
L1_IS_OPTIMAL_SEEDED_EXCEPTIONAL_SET_RESIDUE_CHAIN_EXPONENT_CONVERTER
CERTIFIED

Add:

B-RH-066
SEEDED_L1_SHRINKING_THRESHOLD_PESC_AMPLIFICATION_LAW
CERTIFIED

Update preferred frontier:

F-RH-017-v2
nu > kappa/2
c > 1-alpha

No RH certificate is created.

22. Conclusion

The exceptional-set route has become cheaper.

The earlier $L^2$ conversion paid the square of the exceptional-set envelope.

The $L^1$ residue-chain / Mellin route pays it only once.

For

H=N^{1-\tau},

the strict exceptional-set gate is now

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

The threshold wall did not move.

The exception-count wall did.

This matters because any future arithmetic theorem only needs a modest polynomial rarity of supercritical failures, not the stronger rarity previously required.

The current Gafni-Tao machinery cannot simply be parameter-substituted into this regime, and its finite-moment architecture remains critically blocked at the boundary.

Thus the next breakthrough target is as small and explicit as the present chain can make it:

\boxed{ \text{beat the boundary amplitude on almost all } N^{1-\tau}\text{-intervals, with only }N^{1-c} \text{ exceptions and }c>\tau. }

CSM_RH Paper 59 — Seeded $L^p$ Residue-Chain Amplification, $L^1$ Optimality, and the Minimal Exceptional-Set Strip-Gap

CSM_RH Paper 59

Seeded LpL^pLp Residue-Chain Amplification, L1L^1L1 Optimality, and the Minimal Exceptional-Set Strip-Gap Gate

Abstract

1. Seed notation

2. A general residue-chain LpL^pLp inequality

Theorem 2.1 — Residue-chain LpL^pLp inequality

3. Seeded anchor in LpL^pLp

4. Exceptional-set LpL^pLp lag bound

Good set

Bad set

5. Global seeded LpL^pLp error exponent

Theorem 5.1 — Seeded exceptional-set LpL^pLp exponent

6. Global LpL^pLp error implies a zero-free strip

Theorem 6.1 — LpL^pLp Mellin strip theorem

Theorem 6.2 — Seeded LpL^pLp exceptional-set PESC amplifier

7. L1L^1L1 is optimal in the entire LpL^pLp family

Theorem 7.1 — L1L^1L1 optimality

8. The optimal L1L^1L1 amplification law

Theorem 8.1 — Seeded L1L^1L1 shrinking-threshold amplification law

9. Exact strict-amplifier gate

Corollary 9.1 — Minimal deterministic exceptional-set gate

10. Near-macroscopic formulation

11. Optimal use of an exceptional exponent

12. General LpL^pLp optimization

13. Comparison with Paper 56

14. F-RH-017 v2

15. Audit of the Gafni-Tao fixed-threshold proof

16. Fixed- HHH removes one technical polynomial cost, but not the wall

17. Polynomial-threshold L2L^2L2 packet exponent

18. Polynomial-threshold L4L^4L4 packet exponent

19. Current technology verdict

20. New minimal arithmetic target

21. State transition

22. Conclusion

Seeded $L^p$ Residue-Chain Amplification, $L^1$ Optimality, and the Minimal Exceptional-Set Strip-Gap Gate

2. A general residue-chain $L^p$ inequality

Theorem 2.1 — Residue-chain $L^p$ inequality

3. Seeded anchor in $L^p$

4. Exceptional-set $L^p$ lag bound

5. Global seeded $L^p$ error exponent

Theorem 5.1 — Seeded exceptional-set $L^p$ exponent

6. Global $L^p$ error implies a zero-free strip

Theorem 6.1 — $L^p$ Mellin strip theorem

Theorem 6.2 — Seeded $L^p$ exceptional-set PESC amplifier

7. $L^1$ is optimal in the entire $L^p$ family

Theorem 7.1 — $L^1$ optimality

8. The optimal $L^1$ amplification law

Theorem 8.1 — Seeded $L^1$ shrinking-threshold amplification law

12. General $L^p$ optimization

16. Fixed- $H$ removes one technical polynomial cost, but not the wall

17. Polynomial-threshold $L^2$ packet exponent

18. Polynomial-threshold $L^4$ packet exponent