CSM_RH Paper 37

Minimal Fixed-Power Breakthrough Gate, Dyadic Observability, and the Shrinking-Threshold Admission Bridge

Project: CSM_RH
Paper: 37
Version: v0.1
Date: 2026-09-07
Parent state: CSM_RH v1.27 / Paper 36
Campaign: 36 — MINIMAL_FIXED_POWER_BREAKTHROUGH_GATE
Status: breakthrough-gate audit / no arithmetic candidate admitted; 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

Papers 17–36 generated and then audited a large family of possible surrogates:

lag energies;
fourth moments;
fully distinct correlations;
sieve-model residuals;
Selberg amplifiers;
parity bilinears;
coupled sieves;
rank-one low-frequency coordinates;
finite-rank projections;
rational/Stieltjes filters;
Müntz filters.

Campaign 36 imposes a stricter rule:

no object counts as progress merely because it has a fixed-power theorem somewhere. The theorem must observe the RH-hard prime-error mode and possess a proved deterministic bridge back to PESC or the fixed-power PNT mean square.

Under this gate, no current candidate is admitted.

The campaign nevertheless produces:

a precise dyadic-observability rejection test;
an exact counterexample showing why a known fixed-power transformed prime theorem does not qualify;
a deterministic shrinking-threshold exceptional-set bridge to MLEPG;
a single narrow next campaign.

No live GLM-5.3-Flash run is claimed.

1. Breakthrough-gate rules

A candidate theorem is admitted only if all of the following are present.

G1 — genuinely arithmetic estimate

There must be a proved estimate for the prime / von-Mangoldt sequence, not merely an approximation-theory or representation theorem.

G2 — fixed exponent

The conclusion must contain a fixed positive power of the main scale.

Allowed:

X^{-\delta}, \qquad \delta>0 \text{ fixed}.

Not sufficient:

(\log X)^{-A},

\exp[-(\log X)^c], \qquad 0<c<1,

or any

X^{-o(1)}.

G3 — no fixed-strip input

The fixed exponent may not be imported from an assumed fixed zero-free half-plane.

G4 — root observability

The observable must actually see the dyadic cumulative prime-error / principal low-frequency mode.

G5 — proved deterministic bridge

There must be a rigorous route from the estimate to one of:

F-RH-010 PESC;
F-RH-016 MLEPG;
fixed-power PNT mean-square.

2. Why fixed power alone is not enough

There exist genuine fixed-power theorems involving the von Mangoldt function which do not control the ordinary prime number theorem.

A useful test case is the quotient transform

\boxed{ (Tf)(x) = \sum_{1\le n\le x} f \left( \left\lfloor \frac{x}{n} \right\rfloor \right). }

Wei Zhang proved a fixed-power asymptotic for $f=\Lambda$ :

\boxed{ (T\Lambda)(x) = C_\Lambda x + O_\varepsilon \left( x^{7/15+1/195+\varepsilon} \right). }

The exponent is approximately

0.47179.

This is a real arithmetic fixed-power theorem.

It nevertheless fails the CSM_RH breakthrough gate.

3. Exact quotient-transform weight identity

Define

\boxed{ \Delta_m(x) = \left\lfloor \frac{x}{m} \right\rfloor - \left\lfloor \frac{x}{m+1} \right\rfloor. }

Grouping by the value

m= \left\lfloor \frac{x}{n} \right\rfloor

gives:

Theorem 3.1 — Quotient Weight Identity

\boxed{ (Tf)(x) = \sum_{m\le x} f(m)\Delta_m(x). }

The transform is therefore a sparse weighted sampler of $f$ .

4. Sparse support of the quotient weights

The number of indices with

\Delta_m(x)\ne0

is at most

\boxed{ 2\lfloor\sqrt x\rfloor+1. }

Indeed:

there are at most $\sqrt x$ possible indices $m\le\sqrt x$ ;
if $m>\sqrt x$ and $\Delta_m(x)>0$ , then $m=\lfloor x/n\rfloor$ for some $n<\sqrt x$ , giving at most another $\sqrt x$ values.

Thus a length- $x$ arithmetic sequence is sampled at only

O(\sqrt x)

positions by one quotient transform.

5. Exact dyadic blindness countermodel

Fix a scale $X$ and define

\boxed{ g_X(m) = 1_{X<m\le2X}. }

For every integer

X<x\le2X,

the $n=1$ term gives

g_X(x)=1.

For every

n\ge2,

we have

\left\lfloor \frac{x}{n} \right\rfloor \le X,

and hence the corresponding term vanishes.

Therefore:

Theorem 5.1 — Exact Dyadic Blindness

\boxed{ (Tg_X)(x)=1 \qquad (X<x\le2X). }

But the dyadic cumulative mass is

\boxed{ \sum_{X<m\le2X}g_X(m) = X. }

A transform error as strong as

O(1)

is therefore compatible with maximal linear dyadic mass.

6. Dyadic observability obstruction

Create:

O-RH-086
QUOTIENT_TRANSFORM_DYADIC_OBSERVABILITY_FAILURE
status:
  CERTIFIED

Statement:

Fixed-power control of a sparse quotient transform does not deterministically control the cumulative error on the current dyadic block.

This is sufficient to reject the quotient-PNT fixed-power theorem from the PESC breakthrough gate.

The rejection does not diminish the theorem itself.

It only distinguishes:

fixed power in a smoothed/sparse transformed observable

from

fixed power in an RH-observing prime-error observable.

7. Root-observability requirement

The quotient example motivates a general gate rule.

A candidate linear observable

\mathcal L_X(a)

must be tested against adversarial dyadic perturbations.

If there exists a family $g_X$ with

\left| \mathcal L_x(g_X) \right| = X^{o(1)}

throughout the current dyadic scale while

\left| \sum_{X<n\le2X}g_X(n) \right| = X^{1-o(1)},

then the observable has no deterministic fixed-power bridge to the ordinary cumulative prime error without additional arithmetic information.

Create:

O-RH-087
FIXED_POWER_WITHOUT_ROOT_OBSERVABILITY_NOT_ADMISSIBLE
status:
  CERTIFIED AS CAMPAIGN-GATE PRINCIPLE

8. Current direct PESC calibration

Let

B(x)=\vartheta(x)-x.

The best unconditional PNT remainder remains subpower relative to $x$ .

Current zero-free-region / zero-density refinements yield errors of the form

\boxed{ B(x) \ll x \exp [ -\omega(x) ], }

where

\omega(x) = o(\log x)

and in the Vinogradov–Korobov class is of order

(\log x)^{3/5} (\log\log x)^{-1/5}.

Hence

B(x) = x^{1-o(1)}.

Inserted into the direct PESC absolute-value bound, this remains only

\boxed{ \mathcal C_N \ll N^{3-o(1)}. }

No fixed PESC exponent is currently obtained.

9. Direct MLEPG calibration

Recall

U_H(x) = \sum_{x<n\le x+H} [ \Lambda(n)-1 ].

The lag energy is

\boxed{ \mathcal S_\Lambda(X,H) = \int_X^{2X} |U_H(x)|^2 \,dx }

up to discrete/endpoint conventions.

Current 2026 higher-uniformity technology gives, for suitable polynomial $H$ , residual estimates with arbitrary fixed logarithmic precision on almost all intervals.

For the prime residual this remains:

\boxed{ H(\log X)^{-A} }

rather than

HX^{-\eta}.

Thus the available input produces subpower rather than fixed-power MLEPG.

No candidate is admitted.

10. Guth–Maynard range improvement is not a precision exponent

The 2026 Guth–Maynard large-value theorem gives the zero-density estimate

\boxed{ N(\sigma,T) \le T^{30(1-\sigma)/13+o(1)} }

and asymptotics for primes in all short intervals at the scale

x^{17/30+o(1)}.

Combined with the exceptional-set framework, one obtains almost-all short-interval PNT down to

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

This is a major range improvement.

But:

shorter admissible H:
  range gain

polynomially shrinking relative error:
  not supplied

fixed MLEPG exponent:
  not supplied

Range and precision remain distinct coordinates.

11. Gafni–Tao exceptional-set theorem remains fixed-threshold

Let

\mathcal E_\delta(X,\theta)

denote the set on which the short-interval PNT fails by a fixed relative threshold $\delta$ .

The Gafni–Tao exceptional-set exponents are defined with

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

In the proof they explicitly:

fix delta;
choose J sufficiently large depending on delta, theta, epsilon;
let X tend to infinity with J, delta, theta, epsilon fixed.

The resulting bounds contain constants depending on

\delta,\theta,J,\varepsilon.

Therefore the theorem as stated does not authorize the substitution

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

This confirms the fixed-threshold mismatch identified in Paper 19.

12. The shrinking-threshold condition that would pass the gate

The exact missing shape can now be stated.

Let

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

Assume there exist fixed

\eta>0, \qquad c>0

such that

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

Call this a shrinking-threshold exceptional-set estimate.

This is not currently claimed as a theorem.

13. Shrinking-threshold to MLEPG bridge

On the good set,

|U_H(x)|^2 \le H^2X^{-2\eta}.

On every interval, the elementary bound gives

|U_H(x)| \ll H\log X.

Hence the bad-set contribution is

\ll X^{1-c} H^2 (\log X)^2.

Therefore:

Theorem 13.1 — Shrinking-Threshold Exceptional Set to Lag Energy

If Section 12 holds, then

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

Equivalently,

\boxed{ \mathcal S_\Lambda(X,H) \ll XH^2 X^{-\delta+o(1)}, }

where

\boxed{ \delta = \min \{ 2\eta,c \}. }

This is stronger than the second term required by MLEPG.

Create:

B-RH-012
SHRINKING_THRESHOLD_EXCEPTIONAL_SET_TO_MLEPG
status:
  CERTIFIED

14. Consequence for the global fixed strip

By Paper 17's corrected residue-chain inequality,

\sum_{n\le2N}|A(n)|^2 \ll \left( \frac NH \right)^2 \mathcal S_\Lambda(N,H) + NH^2,

where

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

With

H=N^\alpha

and Theorem 13.1, one gets a fixed global mean-square exponent whenever

0<\alpha<1

and

\eta,c>0

are fixed.

For sufficiently small fixed $\kappa$ ,

\boxed{ \kappa < \min \{ \alpha, 2\eta, c, 2-2\alpha \}. }

The Mellin-pole bridge then excludes zeros in

\Re s>1-\kappa/2.

Thus the shrinking-threshold arithmetic estimate would genuinely pass Campaign 36.

15. Why current logarithmic thresholds fail the bridge

If instead

|U_H(x)| \le H(\log X)^{-A}

outside an exceptional set of size

X(\log X)^{-A},

then the same good/bad decomposition gives only inverse powers of $\log X$ .

At polynomial $H$ these are

X^{-o(1)}.

No fixed $\delta$ in Theorem 13.1 is generated.

Thus the gate cleanly separates:

arbitrary fixed logarithmic accuracy

from

polynomially shrinking threshold accuracy.

16. Principal Fejer route

Paper 18 established that the complete triangular lag aggregate is the positive Fejer energy

\int_0^1 |S_X(\alpha)|^2 |D_H(\alpha)|^2 \,d\alpha.

On the principal arc

\|\alpha\| \le c/H,

one has

|D_H(\alpha)|^2 \gg H^2.

Therefore any fixed-power MLEPG theorem must control the principal $q=1$ low-frequency component at fixed-power strength.

Current minor-arc or non-principal-phase fixed-power exponential-sum theorems do not remove this principal contribution.

No candidate is admitted.

17. Prime-specific scale recurrence

The exact contraction identities of Papers 21–23 remain valid.

But fixed power requires cumulative contraction mass

\Omega(\log X).

Current unconditional prime information supplies at best sublinear contraction mass in the audited positive-defect routes.

No new prime-specific signed recurrence with linear contraction mass is currently proved.

No candidate is admitted.

18. Campaign 36 admission table

Candidate A — direct PESC

fixed-power arithmetic estimate:
  NO

verdict:
  REJECT / OPEN ROOT

Candidate B — direct MLEPG

fixed-power arithmetic estimate:
  NO

current precision:
  log/subpower

verdict:
  REJECT / OPEN

Candidate C — Gafni–Tao exceptional set

power-sized exceptional set:
  YES IN MANY RANGES

relative threshold:
  FIXED DELTA

polynomially shrinking threshold:
  NOT AUTHORIZED

verdict:
  REJECT

Candidate D — Guth–Maynard short intervals

range breakthrough:
  YES

fixed relative power precision:
  NO

verdict:
  REJECT FOR THIS GATE

Candidate E — quotient-transform PNT

fixed-power theorem:
  YES

root observability:
  FAILS

deterministic PESC bridge:
  NO

verdict:
  REJECT

Candidate F — principal Fejer fixed power

would pass gate:
  YES

currently proved:
  NO

Candidate G — shrinking-threshold short interval

would pass gate:
  YES

direct bridge:
  B-RH-012

currently proved:
  NO

19. Campaign 36 verdict

No candidate is admitted.

This is an intentional successful outcome of the gate.

new fixed-power arithmetic theorem admitted:
  NONE

false-positive fixed-power theorem detected:
  quotient transform

new deterministic bridge:
  B-RH-012

root target:
  unchanged

The state is not advanced by inventing another equivalent criterion.

It is advanced by making the admission boundary explicit.

20. New certified package

Create:

O-RH-086
QUOTIENT_TRANSFORM_DYADIC_OBSERVABILITY_FAILURE
CERTIFIED

O-RH-087
FIXED_POWER_WITHOUT_ROOT_OBSERVABILITY_NOT_ADMISSIBLE
CERTIFIED AS CAMPAIGN-GATE PRINCIPLE

B-RH-012
SHRINKING_THRESHOLD_EXCEPTIONAL_SET_TO_MLEPG
CERTIFIED

No new frontier is created.

21. Canonical status

F-RH-010
PESC
OPEN / ROOT TARGET

F-RH-016
MLEPG
OPEN / DIRECT THEOREM CANDIDATE

Campaign-36 admitted fixed-power theorem:
  NONE

22. Campaign 37

The next campaign is deliberately narrow:

CSM_RH Campaign 37
SHRINKING_THRESHOLD_TAIL_ATTACK

It does not create a new canonical frontier.

Its sole objective is to determine whether the Gafni–Tao / Guth–Maynard exceptional-set machinery can be quantified when the relative threshold shrinks as a power of $X$ .

23. Campaign 37 tracks

ST1 — explicit delta dependence

Re-read the exceptional-set proof and expose every dependence on the fixed threshold $\delta$ .

Track:

subdivision count;
choice of J;
explicit-formula truncation;
Markov threshold;
L2/L4 zero-sum estimates;
implied constants.

ST2 — polynomially shrinking substitution

Set

\delta=X^{-\eta}.

Allow

J=J(X)

if necessary.

Track the effect on

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

and therefore on every zero-density exponent.

ST3 — Guth–Maynard density insertion

Insert

A_0=\frac{30}{13}

and determine whether there exists any open region

(\alpha,\eta,c)

with

\eta>0, \qquad c>0

for which the shrinking-threshold exceptional-set estimate survives.

ST4 — layer-cake / second-moment optimization

If threshold-dependent tail estimates exist, integrate the full tail instead of using one threshold.

Target:

\mathcal S_\Lambda(X,H) \ll XH^2X^{-\delta_0}.

ST5 — no hidden fixed-strip source

Reject any power which ultimately comes from assuming a fixed zero-free strip.

The exponent must emerge from large-value / zero-density / exceptional-set arithmetic alone.

24. Campaign 37 hard rejects

Reject:

delta fixed while being described as shrinking;
A=A(X) inside an O_A theorem without uniformity;
J growing while T is still treated as X^(1-alpha+o(1)) without checking;
fixed zero-free strip input;
only logarithmic threshold;
power exceptional set with constant relative error;
new transformed prime observable without dyadic observability.

25. External calibration

The Campaign-36 audit uses the following current facts.

Guth–Maynard 2026 prove
$N(\sigma,T) \le T^{30(1-\sigma)/13+o(1)}$
and short-interval prime asymptotics at the $17/30$ all-interval scale.
Gafni–Tao 2026 quantify exceptional-set exponents and recover almost-all PNT for
$\theta>\frac{2}{15}.$
Their exceptional-set definitions and proof fix the relative threshold $\delta$ before taking $X\to\infty$ .
Matomäki–Radziwiłł–Shao–Tao–Teräväinen 2026 give arbitrary fixed logarithmic precision for $\Lambda-\Lambda^\sharp$ on almost all polynomial short intervals in their range.
Current PNT remainder improvements based on Vinogradov–Korobov zero-free regions remain of the class $$ x\exp[-o(\log x)]

x^{1-o(1)}. $$
Fixed-power estimates do exist for some transformed von-Mangoldt observables, such as the quotient transform, demonstrating why root observability is a necessary extra gate condition.

26. State transition

CSM_RH v1.27
  ->
CSM_RH v1.28

with:

Campaign 36
  CLOSED_WITH_NO_ADMITTED_FIXED_POWER_THEOREM

O-RH-086
  QUOTIENT_TRANSFORM_DYADIC_OBSERVABILITY_FAILURE
  CREATED / CERTIFIED

O-RH-087
  FIXED_POWER_WITHOUT_ROOT_OBSERVABILITY_NOT_ADMISSIBLE
  CREATED / CERTIFIED

B-RH-012
  SHRINKING_THRESHOLD_EXCEPTIONAL_SET_TO_MLEPG
  CREATED / CERTIFIED

F-RH-010
  PESC
  REMAINS OPEN / ROOT TARGET

F-RH-016
  MLEPG
  REMAINS OPEN

Campaign 37
  SHRINKING_THRESHOLD_TAIL_ATTACK
  READY

27. Final status

RH = OPEN

PESC = OPEN

MLEPG = OPEN

BREAKTHROUGH GATE = ACTIVE

CURRENT ADMITTED FIXED-POWER PRIME THEOREM = NONE

FIXED-POWER TRANSFORMED PRIME THEOREMS = NOT ENOUGH WITHOUT ROOT OBSERVABILITY

GUTH-MAYNARD = RANGE BREAKTHROUGH, NOT FIXED-PRECISION BREAKTHROUGH

GAFNI-TAO = POWER EXCEPTIONAL-SET TECHNOLOGY AT FIXED RELATIVE THRESHOLD

SHRINKING-THRESHOLD POWER EXCEPTIONAL SET = OPEN

NEXT CAMPAIGN = 37

The one theorem shape which now matters is:

\boxed{ \left| \{ x: |U_H(x)|>HX^{-\eta} \} \right| \ll X^{1-c}. }

If proved for any fixed positive $\eta$ and $c$ at one polynomial scale $H=X^\alpha$ , it passes the gate and feeds directly into MLEPG.

Everything else remains calibration.

CSM_RH Paper 37 — Minimal Fixed-Power Breakthrough Gate, Dyadic Observability, and the Shrinking-Threshold Admission Bridge

CSM_RH Paper 37

Minimal Fixed-Power Breakthrough Gate, Dyadic Observability, and the Shrinking-Threshold Admission Bridge

0. Trust boundary

1. Breakthrough-gate rules

G1 — genuinely arithmetic estimate

G2 — fixed exponent

G3 — no fixed-strip input

G4 — root observability

G5 — proved deterministic bridge

2. Why fixed power alone is not enough

3. Exact quotient-transform weight identity

Theorem 3.1 — Quotient Weight Identity

4. Sparse support of the quotient weights

5. Exact dyadic blindness countermodel

Theorem 5.1 — Exact Dyadic Blindness

6. Dyadic observability obstruction

7. Root-observability requirement

8. Current direct PESC calibration

9. Direct MLEPG calibration

10. Guth–Maynard range improvement is not a precision exponent

11. Gafni–Tao exceptional-set theorem remains fixed-threshold

12. The shrinking-threshold condition that would pass the gate

13. Shrinking-threshold to MLEPG bridge

Theorem 13.1 — Shrinking-Threshold Exceptional Set to Lag Energy

14. Consequence for the global fixed strip

15. Why current logarithmic thresholds fail the bridge

16. Principal Fejer route

17. Prime-specific scale recurrence

18. Campaign 36 admission table

Candidate A — direct PESC

Candidate B — direct MLEPG

Candidate C — Gafni–Tao exceptional set

Candidate D — Guth–Maynard short intervals

Candidate E — quotient-transform PNT

Candidate F — principal Fejer fixed power

Candidate G — shrinking-threshold short interval

19. Campaign 36 verdict

20. New certified package

21. Canonical status

22. Campaign 37

23. Campaign 37 tracks

ST1 — explicit delta dependence

ST2 — polynomially shrinking substitution

ST3 — Guth–Maynard density insertion

ST4 — layer-cake / second-moment optimization

ST5 — no hidden fixed-strip source

24. Campaign 37 hard rejects

25. External calibration

Current PNT remainder improvements based on Vinogradov–Korobov zero-free regions remain of the class $$ x\exp[-o(\log x)]

26. State transition

27. Final status