CSM_RH Paper 38

Polynomial Shrinking Thresholds, Explicit-Formula Height Tax, and the Right-Edge Subpower Barrier

Project: CSM_RH
Paper: 38
Version: v0.1
Date: 2026-09-07
Parent state: CSM_RH v1.28 / Paper 37
Campaign: 37 — SHRINKING_THRESHOLD_TAIL_ATTACK
Status: explicit dependence audit of the Gafni–Tao exceptional-set architecture; not a proof or disproof of RH

0. Trust boundary

This paper does not prove or disprove the Riemann Hypothesis.

RH_PROVED = FALSE
RH_DISPROVED = FALSE
GLOBAL_RH_CERTIFICATE = FALSE
CSM_RH_ROOT_STATUS = OPEN

Paper 37 certified that a polynomial shrinking-threshold exceptional-set theorem would imply MLEPG and hence a fixed zero-free strip.

Campaign 37 asks whether the current Gafni–Tao exceptional-set proof can already be pushed to such a threshold by exposing all parameter dependence.

The answer is negative for the audited architecture.

There are three independent reasons:

the theorem is proved with $\delta$ and $J$ fixed before $X\to\infty$ ;
a polynomial threshold forces the explicit-formula height to increase by a fixed power of $X$ ;
the right-edge zero packet is controlled only at stretched-log / subpower strength.

The third obstruction is decisive.

No new fixed-power theorem is admitted.

1. Shrinking-threshold admission target

Let

H=X^\theta

with fixed

0<\theta<1.

Define

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

The Paper-37 admission target is:

\boxed{ \left| \left\{ x\in[X,2X]: |U_H(x)|>HX^{-\eta} \right\} \right| \ll X^{1-c} }

for some fixed

\eta>0, \qquad c>0.

By Paper 37 this implies

\mathcal S_\Lambda(X,H) \ll XH^2X^{-\min(2\eta,c)+o(1)},

and therefore MLEPG.

2. Fixed-parameter structure of Gafni–Tao

The Gafni–Tao proof begins by fixing

0<\delta<1.

It then chooses a natural number $J$ sufficiently large depending on

\delta,\theta,\varepsilon,

and proves its exceptional-set estimate as

X\to\infty

with

J,\delta,\theta,\varepsilon

held fixed.

Therefore the published theorem does not by itself authorize the substitution

\boxed{ \delta=X^{-\eta}. }

Create:

O-RH-088
GAFNI_TAO_FIXED_PARAMETER_NONUNIFORMITY
status:
  CERTIFIED AS THEOREM-SCOPE AUDIT

This is a theorem-scope statement, not a claim that the proof cannot be uniformized by new work.

3. Spatial localization cost

To replace the varying interval length $x^\theta$ by the multiplicatively normalized length $x/\tau$ , the proof localizes $x$ to intervals of the form

[X,(1+\delta/J)X],

where

\tau=X^{1-\theta}.

Covering $[X,2X]$ requires

\boxed{ O(J/\delta) }

such intervals.

For fixed $\delta,J$ this is a harmless constant.

\delta=X^{-\eta}

and $J$ grows polynomially, this becomes an exponent-level cost.

4. Explicit-formula height tax

The truncated explicit formula used in the proof has error

\boxed{ O \left( \frac{ X(\log X)^2 }{ T } \right). }

To resolve the shrinking threshold

\delta X^\theta,

it is necessary that

\frac{ X(\log X)^2 }{ T } \ll \delta X^\theta.

Therefore:

Theorem 4.1 — Polynomial Threshold Truncation Law

\boxed{ T \gtrsim \delta^{-1} X^{1-\theta} (\log X)^2. }

In particular, if

\delta=X^{-\eta},

then

\boxed{ T \ge X^{1-\theta+\eta+o(1)}. }

Create:

B-RH-013
SHRINKING_THRESHOLD_EXPLICIT_FORMULA_HEIGHT_LAW
status:
  CERTIFIED

The original proof chooses

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

Thus, within that parameterization, polynomial shrinking forces at least

\boxed{ J\gtrsim X^\eta. }

5. Zero-density exponent tax

Suppose generously that one has the uniform zero-density bound

N(\sigma,T) \le T^{A_0(1-\sigma)+o(1)}

with

A_0=\frac{30}{13}.

Under the polynomial truncation height

T=X^{1-\theta+\eta+o(1)},

the density contribution becomes

\boxed{ X^{ A_0(1-\theta+\eta)(1-\sigma) +o(1) }. }

Thus the zero-density part itself pays a fixed exponent tax

\boxed{ A_0\eta(1-\sigma). }

This is not yet the decisive obstruction, because for $\sigma$ bounded away from $1$ and $\theta$ above the almost-all threshold by a fixed margin, sufficiently small $\eta$ can still leave exponent room.

The fatal region is $\sigma\to1$ .

6. Right-edge lemma

Gafni–Tao isolate a fixed strip

I\subset[1-\eta_0,1].

Using the Vinogradov–Korobov zero-free region and a near-one zero-density estimate, they prove

\boxed{ \sup_{X\le x\le2X} |S_I(x)| \ll_\theta \exp \left[ -c_\theta \frac{ (\log X)^{1/3} }{ (\log\log X)^{1/3} } \right] \frac{X}{\tau}. }

Since

\frac{X}{\tau}=X^\theta,

the available relative suppression is

\boxed{ \Delta_{\mathrm{edge}}(X) = \exp \left[ -c_\theta \frac{ (\log X)^{1/3} }{ (\log\log X)^{1/3} } \right]. }

This is

X^{-o(1)}.

7. Polynomial threshold comparison

For any fixed

\eta>0,

the desired relative threshold is

\delta_X=X^{-\eta}.

But

\boxed{ \Delta_{\mathrm{edge}}(X) \gg X^{-\eta} }

as $X\to\infty$ .

Indeed,

\eta\log X \gg \frac{ (\log X)^{1/3} }{ (\log\log X)^{1/3} }.

Therefore the right-edge estimate is asymptotically much larger than the desired polynomial threshold.

Create:

O-RH-089
GAFNI_TAO_RIGHT_EDGE_SUBPOWER_FLOOR
status:
  CERTIFIED AS CURRENT-PROOF BARRIER

The existing Lemma 2.1 cannot dispose of the right-edge zero packet at threshold $X^{-\eta}$ .

8. This failure occurs before Markov

The right-edge treatment is an $L^\infty$ disposal step.

It occurs before the proof invokes the $L^2$ and $L^4$ Markov bounds for the remaining zero strips.

Therefore improving bookkeeping in the later Markov step does not repair this failure.

The proof already lacks sufficient amplitude suppression on the near-one zero packet.

9. Why the second moment cannot repair the edge

Away from the right edge, the second-moment exponent has the form

(1-\theta) (1-\sigma) A(\sigma) + 2\theta + 2\sigma - 2.

After the shrinking-threshold height tax, the natural generous replacement is

\boxed{ (1-\theta+\eta) (1-\sigma) A(\sigma) + 2\theta + 2\sigma - 2. }

Markov at threshold $X^{\theta-\eta}$ introduces an additional cost

X^{2\eta}.

Even ignoring localization and binning costs, the exceptional-set exponent tends to

\boxed{ 1+2\eta }

\sigma\to1.

The zero-density term is nonnegative and vanishes with $1-\sigma$ .

Hence no uniform power saving can emerge from the second moment arbitrarily close to $\sigma=1$ .

10. Fourth and higher moments have the same edge geometry

The fourth-moment exponent has the form

(1-\theta) (1-\sigma) A^\ast(\sigma) + 4\theta + 4\sigma - 4.

At polynomial threshold $X^{\theta-\eta}$ , Markov contributes $X^{4\eta}$ .

\sigma\to1,

the resulting exceptional-set exponent tends to

\boxed{ 1+4\eta. }

The same phenomenon persists for a hypothetical $2k$ -th moment of the standard zero packet.

The baseline exponent is

2k\sigma-2k = -2k(1-\sigma),

while the shrinking threshold costs

X^{2k\eta}.

1-\sigma<\eta,

then even the best possible contribution obtained by discarding the nonnegative zero-density term cannot compensate the threshold cost.

Create:

O-RH-090
MOMENT_ORDER_CANNOT_CURE_POLYNOMIAL_THRESHOLD_AT_SHRINKING_EDGE
status:
  CERTIFIED AS STANDARD_MARKOV-PACKET STRENGTH AUDIT

This concerns the standard moment-plus-Markov zero-packet architecture.

It is not a universal impossibility theorem for every high-moment method.

11. The zero-free-region scale

The Vinogradov–Korobov region only removes zeros in a shrinking neighborhood

1-\sigma \asymp (\log X)^{-2/3} (\log\log X)^{-1/3}

at the present height scale, up to constants and the harmless effect of subpower height changes.

This width tends to zero.

For every fixed

\eta>0,

eventually

1-\sigma<\eta

throughout part of the remaining admissible edge layer.

Thus the moment-order audit in Section 10 is consistent with the explicit $L^\infty$ barrier of Section 7.

12. Maximum natural shrinking class of the current edge input

The right-edge bound is compatible with thresholds no smaller than a stretched-log class such as

\boxed{ \delta_X \gtrsim \exp \left[ -C \frac{ (\log X)^{1/3} }{ (\log\log X)^{1/3} } \right]. }

For such a subpower threshold,

\delta_X^{-1}=X^{o(1)},

so the necessary truncation height can remain

T=X^{1-\theta+o(1)}.

This explains structurally why the existing architecture naturally lives in a subpower-precision class.

This section is an architecture calibration, not a new uniform theorem, because the published proof keeps $\delta$ fixed.

13. Interior zero-density region is not the primary failure

To localize the obstruction, suppose hypothetically that all zeros with

\sigma>1-\eta_0

were absent for one fixed

\eta_0>0.

Then the remaining $\sigma$ -range is bounded away from $1$ .

With Guth–Maynard

A_0=\frac{30}{13},

and

\theta>\frac{2}{15}

by a fixed margin, sufficiently small shrinking exponent $\eta$ leaves room in the second-moment exponent ledger.

For example, at the most basic $\sigma=1/2$ density point, ignoring localization losses, the condition is

\boxed{ \frac{A_0}{2} (1-\theta+\eta) + 2\eta < 1. }

This has a positive solution $\eta$ whenever

\theta>\frac{2}{15}.

Thus the critical issue is not merely the $30/13$ density coefficient.

It is the absence of a fixed right-edge gap.

14. Current-proof J coupling adds further costs

In the published proof the same $J$ controls:

spatial localization;
zero-strip subdivision;
explicit-formula truncation height.

If one formally sets

J=X^j, \qquad \delta=X^{-\eta},

then:

truncation requirement

j\ge\eta;

number of spatial intervals

J/\delta = X^{j+\eta};

number of zero strips

J = X^j;

pigeonhole threshold per zero strip

\frac{\delta}{J} = X^{-(\eta+j)}.

These costs were constants in the fixed-parameter proof.

They become exponent-level losses when $J$ and $\delta^{-1}$ grow polynomially.

Even without the right-edge failure, the proof would require a new uniform re-optimization.

15. Generous current-architecture L2 ledger

If one keeps the original $J$ coupling and chooses the minimal polynomial scale

j=\eta,

then a generous global $A_0$ second-moment calculation produces the schematic exceptional exponent

\boxed{ \Xi_2(\sigma) = 2\sigma-1 + A_0 (1-\theta+\eta) (1-\sigma) + 7\eta }

after accounting for:

the threshold $\delta/J$ ;
the $J$ zero strips;
the $J/\delta$ spatial blocks.

For $\sigma=1/2$ ,

\boxed{ \Xi_2(1/2) = \frac{A_0}{2} (1-\theta+\eta) + 7\eta. }

This can remain below $1$ for very small $\eta$ when $\theta$ is above $2/15$ by a fixed margin.

But

\boxed{ \lim_{\sigma\to1} \Xi_2(\sigma) = 1+7\eta>1. }

Again the right edge is fatal.

The precise coefficient $7$ is architecture-specific and is not claimed to be optimal.

16. Campaign 37 track audit

ST1 — fixed-delta dependence

status:
  NONUNIFORM IN PUBLISHED THEOREM

delta, J:
  fixed before X -> infinity

ST2 — explicit-formula truncation

status:
  POLYNOMIAL HEIGHT TAX

delta = X^(-eta)
  forces T >= X^(1-theta+eta+o(1))

ST3 — Guth–Maynard density insertion

status:
  INTERIOR REGION CAN RETAIN EXPONENT ROOM FOR SMALL eta

global A0:
  30/13

decisive failure:
  not interior density

ST4 — right-edge zero packet

status:
  FATAL FOR CURRENT PROOF

available suppression:
  X^(-o(1))

required suppression:
  X^(-eta)

ST5 — higher moments

status:
  STANDARD MARKOV MOMENTS DO NOT CURE SHRINKING EDGE

reason:
  threshold penalty survives as sigma -> 1

17. Campaign 37 verdict

No polynomial shrinking-threshold theorem is obtained.

The Paper-37 admission target remains valid and open.

The current Gafni–Tao architecture is classified as:

fixed threshold:
  valid

power exceptional sets:
  valid

subpower shrinking threshold:
  structurally compatible with edge scale, but not a published uniform theorem

polynomial shrinking threshold:
  not supported

first decisive obstruction:
  right-edge X^(-o(1)) amplitude floor

No theorem passes the Campaign-36 breakthrough gate.

18. New certified package

Create:

B-RH-013
SHRINKING_THRESHOLD_EXPLICIT_FORMULA_HEIGHT_LAW
CERTIFIED

O-RH-088
GAFNI_TAO_FIXED_PARAMETER_NONUNIFORMITY
CERTIFIED AS THEOREM-SCOPE AUDIT

O-RH-089
GAFNI_TAO_RIGHT_EDGE_SUBPOWER_FLOOR
CERTIFIED AS CURRENT-PROOF BARRIER

O-RH-090
MOMENT_ORDER_CANNOT_CURE_POLYNOMIAL_THRESHOLD_AT_SHRINKING_EDGE
CERTIFIED AS STANDARD MARKOV-PACKET STRENGTH AUDIT

No new frontier is created.

19. Canonical root status

F-RH-010
PESC
OPEN / ROOT TARGET

F-RH-016
MLEPG
OPEN / DIRECT THEOREM CANDIDATE

B-RH-012
SHRINKING_THRESHOLD_EXCEPTIONAL_SET_TO_MLEPG
CERTIFIED BRIDGE

polynomial shrinking-threshold theorem:
  OPEN / NOT PROVED

20. Campaign 38

The zero-density exceptional-set route is now localized to the same right-edge subpower barrier already seen in other explicit-formula arguments.

The next campaign therefore switches to a genuinely different arithmetic source:

CSM_RH Campaign 38
GROWING_ACCURACY_LAMBDA_RESIDUAL_ATTACK

The target is the 2026 higher-uniformity theorem for

f=\Lambda-\Lambda^\sharp.

The question is:

can its arbitrary fixed log-power accuracy be uniformized to an accuracy parameter growing with $X$ strongly enough to produce $HX^{-\eta}$ ?

21. Campaign 38 tracks

GA1 — accuracy parameter scaling

The current theorem gives

H\log^{-A}X

for every fixed $A$ .

Set formally

A \asymp \eta \frac{\log X}{\log\log X}

and audit every proof constant.

GA2 — W-parameter transition

The 2026 proof uses a prime major-arc parameter of the form

W_\Lambda=\log^{C A}X.

At growing $A$ , compute the exact point at which

W_\Lambda

becomes polynomial in $X$ .

GA3 — Vinogradov–Korobov dependency

Track where the prime Dirichlet-polynomial input uses the shrinking zero-free region.

Determine whether that step alone prevents polynomial $W_\Lambda$ .

GA4 — exceptional-set parameter uniformity

The almost-all theorem also gives a logarithmic exceptional set.

Audit whether making $A$ grow causes the exceptional-set constant or auxiliary complexity to become exponent-level.

GA5 — direct bridge to Paper 37 target

If a uniform theorem of the form

|U_H(x)| \ll HX^{-\eta}

outside

O(X^{1-c})

intervals emerges, immediately invoke B-RH-012.

No new observable is allowed.

22. Campaign 38 rejection filters

Reject a candidate if:

R1. $A$ is treated as growing while a theorem assumes it fixed.

R2. A hidden $O_A(1)$ constant is ignored.

R3. $W_\Lambda$ becomes polynomial without re-proving the prime Dirichlet-polynomial estimate.

R4. The proof imports a fixed zero-free strip.

R5. The final exceptional set remains only logarithmically small.

R6. The output is $X^{-o(1)}$ rather than $X^{-\eta}$ .

23. External calibration

The audited source is Gafni–Tao, On the number of exceptional intervals to the prime number theorem in short intervals, published in Essential Number Theory 5 (2026), 221–241.

The proof:

fixes $\delta$ and $J$ before letting $X\to\infty$ ;
uses the truncation height $T=J(\log X)^2X^{1-\theta};$
disposes of the right-edge zero packet with a Vinogradov–Korobov stretched-log bound;
controls the remaining zero strips by $L^2$ and $L^4$ moments and Markov's inequality.

The Guth–Maynard zero-density theorem gives the global bound

A(\sigma)\le\frac{30}{13},

which yields the all-interval threshold $17/30$ and almost-all threshold $2/15$ .

Neither theorem currently provides the polynomial shrinking-threshold statement required by B-RH-012.

24. State transition

CSM_RH v1.28
  ->
CSM_RH v1.29

with:

Campaign 37
  CLOSED_AS_SHRINKING_THRESHOLD_GAFNI_TAO_DEPENDENCE_AUDIT

B-RH-013
  SHRINKING_THRESHOLD_EXPLICIT_FORMULA_HEIGHT_LAW
  CREATED / CERTIFIED

O-RH-088
  GAFNI_TAO_FIXED_PARAMETER_NONUNIFORMITY
  CREATED / CERTIFIED

O-RH-089
  GAFNI_TAO_RIGHT_EDGE_SUBPOWER_FLOOR
  CREATED / CERTIFIED

O-RH-090
  MOMENT_ORDER_CANNOT_CURE_POLYNOMIAL_THRESHOLD_AT_SHRINKING_EDGE
  CREATED / CERTIFIED

F-RH-010
  PESC
  REMAINS OPEN

F-RH-016
  MLEPG
  REMAINS OPEN

Campaign 38
  GROWING_ACCURACY_LAMBDA_RESIDUAL_ATTACK
  READY

25. Final status

RH = OPEN

PESC = OPEN

MLEPG = OPEN

POLYNOMIAL SHRINKING-THRESHOLD EXCEPTIONAL SET = NOT PROVED

GAFNI-TAO FIXED-DELTA THEOREM = NOT UNIFORM IN delta = X^(-eta)

EXPLICIT-FORMULA HEIGHT = PAYS +eta EXPONENT

RIGHT-EDGE ZERO PACKET = X^(-o(1)) FLOOR

STANDARD L2/L4/HIGHER-MOMENT MARKOV = CANNOT CURE SHRINKING EDGE

INTERIOR GUTH-MAYNARD DENSITY = NOT THE PRIMARY BARRIER

NEXT CAMPAIGN = 38

The decisive comparison is

\boxed{ \exp \left[ -c \frac{ (\log X)^{1/3} }{ (\log\log X)^{1/3} } \right] \gg X^{-\eta} }

for every fixed $\eta>0$ .

The present exceptional-set proof reaches the right edge at subpower precision.

The breakthrough gate requires polynomial precision.

CSM_RH Paper 38 — Polynomial Shrinking Thresholds, Explicit-Formula Height Tax, and the Right-Edge Subpower Barrier

CSM_RH Paper 38

Polynomial Shrinking Thresholds, Explicit-Formula Height Tax, and the Right-Edge Subpower Barrier

0. Trust boundary

1. Shrinking-threshold admission target

2. Fixed-parameter structure of Gafni–Tao

3. Spatial localization cost

4. Explicit-formula height tax

Theorem 4.1 — Polynomial Threshold Truncation Law

5. Zero-density exponent tax

6. Right-edge lemma

7. Polynomial threshold comparison

8. This failure occurs before Markov

9. Why the second moment cannot repair the edge

10. Fourth and higher moments have the same edge geometry

11. The zero-free-region scale

12. Maximum natural shrinking class of the current edge input

13. Interior zero-density region is not the primary failure

14. Current-proof J coupling adds further costs

truncation requirement

number of spatial intervals

number of zero strips

pigeonhole threshold per zero strip

15. Generous current-architecture L2 ledger

16. Campaign 37 track audit

ST1 — fixed-delta dependence

ST2 — explicit-formula truncation

ST3 — Guth–Maynard density insertion

ST4 — right-edge zero packet

ST5 — higher moments

17. Campaign 37 verdict

18. New certified package

19. Canonical root status

20. Campaign 38

21. Campaign 38 tracks

GA1 — accuracy parameter scaling

GA2 — W-parameter transition

GA3 — Vinogradov–Korobov dependency

GA4 — exceptional-set parameter uniformity

GA5 — direct bridge to Paper 37 target

22. Campaign 38 rejection filters

R1. AAA is treated as growing while a theorem assumes it fixed.

R2. A hidden OA(1)O_A(1)OA​(1) constant is ignored.

R3. WΛW_\LambdaWΛ​ becomes polynomial without re-proving the prime Dirichlet-polynomial estimate.

R4. The proof imports a fixed zero-free strip.

R5. The final exceptional set remains only logarithmically small.

R6. The output is X−o(1)X^{-o(1)}X−o(1) rather than X−ηX^{-\eta}X−η.

23. External calibration

24. State transition

25. Final status

R1. $A$ is treated as growing while a theorem assumes it fixed.

R2. A hidden $O_A(1)$ constant is ignored.

R3. $W_\Lambda$ becomes polynomial without re-proving the prime Dirichlet-polynomial estimate.

R6. The output is $X^{-o(1)}$ rather than $X^{-\eta}$ .