CSM_RH Paper 86

Polynomial-Threshold Exceptional Sets, the Gafni–Tao Moment Transfer, and the Boundary Phase Transition at ν = d

Project: CSM_RH
Paper: 86
Version: v0.1
Date: 2026-09-09
Canonical root frontier: F-RH-017-v3
Entry state: v1.76 / Paper 85 v0.1
Status: RETURN TO EXCEPTIONAL-SET ROOT / POLYNOMIAL-THRESHOLD MOMENT TRANSFER AUDITED / FIXED-MOMENT MARKOV ROUTE PROVED TO STOP EXACTLY AT ν=d / THRESHOLD-NATIVE ARITHMETIC REQUIRED
RH_PROVED: FALSE
RH_DISPROVED: FALSE
GLOBAL_RH_CERTIFICATE: FALSE

Abstract

Paper 85 showed that the first robust escape from finite common-power analytic constructions is a scale-adaptive threshold.

This returned the campaign to the canonical exceptional-set frontier F-RH-017-v3.

The present paper asks whether the strongest modern exceptional-interval machinery can supply that frontier.

The main external calibration is the 2025–2026 work of Gafni and Tao on exceptional intervals for the prime number theorem in short intervals.

For

H=X^\theta, \qquad \theta=1-\tau,

they study the fixed-relative-threshold exceptional set

\left\{ x\in[X,2X]: |\psi(x+H_x)-\psi(x)-H_x| \ge \delta H_x \right\},

with fixed $\delta>0$ .

Their explicit-formula method combines:

zero-density estimates;
second moments of zero-band contributions;
fourth moments controlled by zero additive energy.

For a zero band with left endpoint $\sigma$ , their exponent functions are

\boxed{ \mu_{2,\sigma}(\theta) = \tau(1-\sigma)A(\sigma) + 2\sigma-1, }

and

\boxed{ \mu_{4,\sigma}(\theta) = \tau(1-\sigma)A^*(\sigma) + 4\sigma-3. }

They also note that fixed higher even moments lead formally to

\boxed{ \mu_{2k,\sigma}(\theta) = \tau(1-\sigma)A^{(2k)}(\sigma) + 2k\sigma-2k+1, }

where $A^{(2k)}$ is the corresponding $2k$ -zero additive-energy exponent.

F-RH-017-v3 is quantitatively different.

It requires a polynomially shrinking relative threshold

\boxed{ |\Delta_H(x)| > H X^{-\nu}, }

where

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

Under a PESC $(\kappa)$ seed write

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

The strict root gate is

\boxed{ \nu>d, \qquad c>\min(d,\tau), }

for an exceptional-set bound

\boxed{ |\mathcal E_\nu(X,H)| \ll X^{1-c+o(1)}. }

The first result of the present paper is the polynomial-threshold Markov law.

Suppose a zero-band contribution $S_I$ satisfies a $2k$ -moment estimate of the standard explicit-formula form

\boxed{ \frac1X \int_X^{2X} |S_I(x)|^{2k}dx \ll X^{ \mathcal Z_{2k} + 2k\theta + 2k\sigma_+ - 2k + o(1) }, }

where $\mathcal Z_{2k}\ge0$ is the cost of counting / correlating zeros in that band.

Applying Markov at threshold

H X^{-\nu} = X^{\theta-\nu}

gives

\boxed{ |\{ x: |S_I(x)|\ge H X^{-\nu} \}| \ll X^{ \mathcal Z_{2k} + 2k\sigma_+ - 2k + 1 + 2k\nu + o(1) }. }

For a narrow band centered at a possible zero real part $\beta$ this becomes

\boxed{ \mu_{2k}^{(\nu)}(\beta) \ge 1 - 2k(1-\beta) + 2k\nu. }

The inequality is written with $\ge$ because every zero-density / additive-energy cost is nonnegative.

Therefore:

Universal threshold phase transition

\boxed{ \mu_{2k}^{(\nu)}(\beta) \ge 1 + 2k \left( \nu-(1-\beta) \right). }

This yields three regimes.

Super-boundary threshold

\nu<1-\beta,

then a fixed moment can in principle give a power-saving exceptional set.

Boundary threshold

\nu=1-\beta,

then the best possible baseline exponent is

1.

No power density saving remains.

Sub-boundary threshold

\nu>1-\beta,

then

\boxed{ \mu_{2k}^{(\nu)}(\beta)>1. }

A fixed-moment Markov argument cannot even certify a density-zero exceptional set from that band.

At a saturated PESC $(\kappa)$ boundary,

\beta=1-d,

this becomes

\boxed{ \mu_{2k}^{(\nu)} \ge 1+2k(\nu-d). }

F-RH-017-v3 requires precisely the third regime:

\boxed{ \nu>d. }

Thus the canonical exceptional-set frontier begins strictly beyond the reach of every fixed even-moment / Markov argument compatible with a possible seed-boundary zero.

Higher moments do not help.

They increase the slope

2k(\nu-d)

but do not move the transition point.

Even a hypothetical perfect higher zero-additive-energy theorem

\mathcal Z_{2k}=0

would not change this conclusion.

The hard-interval explicit formula makes the obstruction stronger.

To approximate

\Delta_H(x)

to precision

H X^{-\nu},

the usual truncated explicit formula needs

\boxed{ T \gtrsim \frac{X}{H} X^\nu (\log X)^{O(1)} = X^{\tau+\nu+o(1)}. }

Thus the Gafni–Tao second- and fourth-moment exponent functions acquire the threshold-aware forms

\boxed{ \mu_{2,\sigma}^{(\nu)} = (\tau+\nu)(1-\sigma)A(\sigma) + 2\sigma-1 + 2\nu, }

and

\boxed{ \mu_{4,\sigma}^{(\nu)} = (\tau+\nu)(1-\sigma)A^*(\sigma) + 4\sigma-3 + 4\nu. }

At $\sigma=1-d$ ,

\boxed{ \mu_{2,1-d}^{(\nu)} = 1 + 2(\nu-d) + (\tau+\nu)dA(1-d), }

and

\boxed{ \mu_{4,1-d}^{(\nu)} = 1 + 4(\nu-d) + (\tau+\nu)dA^*(1-d). }

The density terms are nonnegative, so the universal baseline barrier already decides the sign.

A smooth interval kernel can reduce or remove the explicit truncation-height penalty, but it cannot remove the baseline

1+2k(\nu-d).

Hence the phase transition is not an artifact of hard cutoffs.

This gives the key campaign conclusion:

GAFNI–TAO / ZERO-DENSITY / ZERO-ADDITIVE-ENERGY
EXCEPTIONAL-SET MACHINERY:

excellent for fixed relative thresholds;

not a direct route to F-RH-017-v3
once the threshold exponent satisfies ν>d.

The reason is structural, not merely quantitative.

A possible boundary zero itself contributes at relative size

X^{-d}.

A threshold

X^{-\nu}, \qquad \nu>d,

lies below that amplitude.

A global moment bound which remains compatible with that zero necessarily sees its contribution before Markov is applied.

F-RH-017-v3 asks for something different:

prove directly that ordinary-prime arithmetic prevents the boundary-sized mode from occupying the exceptional mass forced by the zeta explicit formula.

This is a threshold-native arithmetic problem, not a moment problem.

The paper also separates two notions of “almost all”.

Modern multiplicative-function and almost-prime results can sometimes produce power-saving exceptional sets.

For example, Matomäki proves an $O(X/h)$ exceptional set for suitable $P_2$ counts in intervals of length $h\log X$ .

This demonstrates that power-saving exceptional-set mechanisms are arithmetically possible.

But the theorem is a sieve / almost-prime statement at a coarse relative threshold.

It does not provide the polynomially shrinking relative prime-count accuracy required by F-RH-017-v3.

The surviving root question is therefore sharpened.

A successful proof must avoid the chain

explicit formula
→ fixed L^{2k} norm
→ Markov.

Candidate mechanisms must instead be threshold-native, such as:

a stopping-time or density-increment theorem;
a lower-tail prime-deficiency sieve with genuinely parity-breaking input;
an inertia / persistence theorem coupled to ordinary-prime arithmetic;
a scale-adaptive arithmetic state whose complexity is controlled before averaging.

No new frontier is opened.

F-RH-017-v3 remains the canonical root.

No RH theorem is claimed.

1. Exceptional-set notation

Let

H=X^{1-\tau}.

Define

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

For a threshold exponent $\nu>0$ define

\boxed{ \mathcal E_\nu(X,H) = \left\{ x\in[X,2X]: |\Delta_H(x)| > H X^{-\nu} \right\}. }

The target is

\boxed{ |\mathcal E_\nu(X,H)| \ll X^{1-c+o(1)}. }

2. Canonical F-RH-017-v3 gate

Under PESC $(\kappa)$ put

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

Paper 61 established that a strict amplifier follows if

\boxed{ \nu>d, }

and

\boxed{ c>\min(d,\tau). }

This is the gate to be tested against known exceptional-set technology.

3. Gafni–Tao fixed-threshold framework

Gafni and Tao define fixed-relative-threshold exceptional sets

|\Delta(x,x^\theta)| \ge \delta x^\theta

for fixed

\delta>0.

Their general zero-density exponent is

\boxed{ \mu_{2,\sigma}(\theta) = (1-\theta)(1-\sigma)A(\sigma) + 2\sigma-1. }

Their refined fourth-moment exponent is

\boxed{ \mu_{4,\sigma}(\theta) = (1-\theta)(1-\sigma)A^*(\sigma) + 4\sigma-3. }

They prove an exceptional exponent by taking the supremum over relevant zero bands and the minimum of these two moment bounds.

Create:

B-RH-161
GAFNI_TAO_EXPRESS_EXCEPTIONAL_SET_EXPONENTS_DIRECTLY_IN_TERMS_OF_ZERO_DENSITY_AND_ZERO_ADDITIVE_ENERGY
CERTIFIED_EXTERNAL

4. General moment-to-threshold lemma

Let $p=2k$ .

Suppose

\frac1X \int_X^{2X} |S(x)|^pdx \ll X^{M_p+o(1)}.

Let the threshold be

X^{\theta-\nu}.

Markov gives

\begin{aligned} |\{ x: |S(x)|\ge X^{\theta-\nu} \}| &\le X \frac{ X^{M_p+o(1)} }{ X^{p(\theta-\nu)} }. \end{aligned}

Therefore:

Lemma 4.1 — Polynomial-threshold Markov penalty

\boxed{ \operatorname{exc\ exponent} = 1 + M_p - p\theta + p\nu. }

Relative to a fixed-threshold calculation, a threshold $X^{-\nu}$ costs exactly

\boxed{ p\nu }

in exceptional-set exponent.

Create:

B-RH-162
A_POLYNOMIALLY_SHRINKING_RELATIVE_THRESHOLD_COSTS_P_NU_IN_ANY_PTH_MOMENT_MARKOV_EXCEPTIONAL_BOUND
CERTIFIED

5. Best-case zero-band baseline

The standard explicit-formula $p$ th moment of a narrow band around real part $\beta$ contains the deterministic amplitude factor

X^{p\theta+p\beta-p}.

Ignore all zero-counting costs.

This is the most optimistic possible scenario.

Then

M_p = p\theta+p\beta-p.

Lemma 4.1 gives

\boxed{ \mu_p^{(\nu)}(\beta) \ge 1 + p\beta - p + p\nu. }

Equivalently:

Theorem 5.1 — Universal fixed-moment threshold barrier

\boxed{ \mu_p^{(\nu)}(\beta) \ge 1 + p \left( \nu-(1-\beta) \right). }

Create:

B-RH-163
EVEN_WITH_ZERO_DENSITY_COST_SET_TO_ZERO_A_FIXED_MOMENT_MARKOV_ARGUMENT_HAS_A_PHASE_TRANSITION_AT_NU_EQUALS_ONE_MINUS_BETA
CERTIFIED

6. Boundary phase transition

Let

\delta_\beta=1-\beta.

Then:

If $\nu<\delta_\beta$

1+p(\nu-\delta_\beta)<1.

A power-saving exceptional exponent is not ruled out by the baseline.

If $\nu=\delta_\beta$

\boxed{ \mu_p^{(\nu)}\ge1. }

If $\nu>\delta_\beta$

\boxed{ \mu_p^{(\nu)}>1. }

Thus the threshold phase transition is exactly

\boxed{ \nu=1-\beta. }

7. Saturated PESC boundary

\beta=1-d,

Theorem 5.1 becomes

\boxed{ \mu_{2k}^{(\nu)} \ge 1 + 2k(\nu-d). }

Hence:

Corollary 7.1 — F-RH moment incompatibility

\nu>d,

no fixed even-moment / Markov argument compatible with a possible boundary zero can prove

|\mathcal E_\nu| \ll X^{1-c}

for any fixed

c>0.

Create:

O-RH-188
F_RH_017_V3_LIES_STRICTLY_BEYOND_THE_FIXED_MOMENT_MARKOV_PHASE_TRANSITION
CERTIFIED

8. Higher moments do not move the transition

Replacing $p=2$ by

p=4,6,8,\ldots

changes

1+p(\nu-d)

only by multiplying the distance from the transition.

The zero of the expression remains

\boxed{ \nu=d. }

Thus even hypothetical optimal higher zero additive-energy estimates cannot move the threshold.

This strengthens Paper 81's moment-degree neutrality:

higher moments do not merely fail to improve the exponent;
at the adaptive exceptional threshold they all fail at exactly the same ν=d wall.

9. Hard-window truncation height

For a hard interval of length $H$ , the truncated explicit formula has error of schematic size

\boxed{ \frac{ X(\log X)^{O(1)} }{ T }. }

To make this smaller than

H X^{-\nu},

one needs

\boxed{ T \ge \frac{X}{H} X^\nu (\log X)^{O(1)}. }

Since

H=X^{1-\tau},

\boxed{ T = X^{\tau+\nu+o(1)} }

is the natural threshold-aware height.

Create:

B-RH-164
A_HARD_INTERVAL_EXPLICIT_FORMULA_AT_RELATIVE_PRECISION_X_MINUS_NU_REQUIRES_ZERO_HEIGHT_X_TO_TAU_PLUS_NU_UP_TO_SUBPOWER_FACTORS
CERTIFIED

10. Threshold-aware Gafni–Tao second moment

The Gafni–Tao second moment counts zeros to height $T$ .

Replacing

T=X^{\tau+o(1)}

T=X^{\tau+\nu+o(1)}

gives the threshold-aware density cost

\boxed{ (\tau+\nu)(1-\sigma)A(\sigma). }

After Markov:

\boxed{ \mu_{2,\sigma}^{(\nu)} = (\tau+\nu)(1-\sigma)A(\sigma) + 2\sigma-1 + 2\nu. }

At $\sigma=1-d$ ,

\boxed{ \mu_{2,1-d}^{(\nu)} = 1 + 2(\nu-d) + (\tau+\nu)dA(1-d). }

11. Threshold-aware fourth moment

Similarly,

\boxed{ \mu_{4,\sigma}^{(\nu)} = (\tau+\nu)(1-\sigma)A^*(\sigma) + 4\sigma-3 + 4\nu. }

At $\sigma=1-d$ ,

\boxed{ \mu_{4,1-d}^{(\nu)} = 1 + 4(\nu-d) + (\tau+\nu)dA^*(1-d). }

The extra zero-energy terms cannot improve the baseline because they are nonnegative.

Create:

B-RH-165
THE_THRESHOLD_AWARE_GAFNI_TAO_L2_AND_L4_EXPONENTS_RETAIN_THE_UNIVERSAL_NU_EQUALS_D_PHASE_TRANSITION
CERTIFIED_AS_EXPONENT_TRANSFER

12. Smooth kernels do not remove the baseline wall

A compact smooth interval kernel may suppress high zero ordinates rapidly.

This can improve the truncation-height bookkeeping in Sections 9–11.

But the response of a zero at real part $\beta$ still has relative power

X^{-(1-\beta)}.

The moment threshold penalty is still

p\nu.

Therefore the best-case baseline theorem in Section 5 remains.

The $\nu=d$ wall is not a hard-cutoff artifact.

13. Why fixed-relative exceptional results remain useful but insufficient

For fixed

\delta>0,

the threshold exponent is

\nu=0.

A boundary zero at

\beta=1-d

then has baseline exceptional exponent

1-2kd.

Power-saving exceptional-set estimates are therefore compatible with fixed-relative thresholds.

This explains why zero-density and additive-energy machinery can be highly effective for the classical “almost all short intervals” problem.

But F-RH-017-v3 asks for a threshold polynomially smaller than the possible boundary amplitude.

That is a different problem.

14. Almost-prime exceptional sets as a calibration

Matomäki proves that for suitable $P_2$ -counting problems in intervals of length

h\log X,

all but

O(X/h)

starting points satisfy the desired lower bound.

Thus genuinely power-saving exceptional sets are possible in short-interval sieve problems.

This is useful calibration.

However:

the counted objects are almost primes, not primes;
the conclusion is at a coarse relative lower threshold;
it does not give polynomially shrinking relative accuracy for the prime count.

The result therefore illustrates that the obstacle is not “power exceptional sets” in general.

The obstacle is the threshold-native, parity-sensitive accuracy required for primes.

External source:

Kaisa Matomäki, Almost primes in almost all very short intervals, JLMS 106 (2022).

15. What a successful F-RH-017 proof must avoid

The route

\boxed{ \text{explicit formula} \to L^{2k} \to \text{Markov} }

cannot cross the frontier.

A successful argument must instead produce exceptional-set control before reducing the arithmetic to a fixed global moment.

Possible architectures include:

15.1. Stopping-time / density increment

Detect the first scale at which a prime deficiency or excess develops and exploit structure conditioned on that stopping event.

15.2. Threshold-native parity-breaking sieve

Prove that prime deficiency by

H X^{-\nu}

forces a structured factorization event whose total mass can be power bounded.

15.3. Inertia plus arithmetic persistence contradiction

Use the deterministic persistence of a threshold exceedance together with an arithmetic theorem that forbids too many persistent deficient intervals.

15.4. Growing-complexity adaptive state

Allow scale resolution to depend on $X$ , but maintain an explicit entropy / union-loss ledger.

No such theorem is certified here.

16. Relation to the canonical boundary forcing

Paper 61 showed that a boundary zero forces exceptional mass at the critical scale

X^{1-\min(d,\tau)-o(1)}

for thresholds beyond the seed.

Therefore F-RH-017-v3 requires

c>\min(d,\tau)

to contradict that forcing.

Paper 86 adds:

fixed moments cannot even reach c>0 once ν>d.

Thus the exceptional-mass side of F-RH-017 is genuinely stronger than every fixed moment-Markov consequence.

17. Current best exceptional-interval calibration

Gafni and Tao give the modern explicit bridge from:

zero density $A(\sigma)$ ;
zero additive energy $A^*(\sigma)$ ;

to the fixed-relative exceptional exponent $\mu(\theta)$ .

Their 2026 paper incorporates the recent Guth–Maynard and Tao–Trudgian–Yang density technology and provides the strongest current systematic bounds of this form.

They explicitly note that higher $2k$ moments could be incorporated through higher zero additive energies, but no known unconditional estimates there improve the existing $L^2/L^4$ inputs.

Paper 86 shows that, for F-RH-017-v3, this lack of higher-energy estimates is not the decisive obstruction.

The baseline Markov phase transition already blocks all fixed $k$ .

18. State transition

Advance candidate state

v1.76 \to v1.77.

Add:

B-RH-161
GAFNI_TAO_EXPRESS_EXCEPTIONAL_SET_EXPONENTS_DIRECTLY_IN_TERMS_OF_ZERO_DENSITY_AND_ZERO_ADDITIVE_ENERGY

B-RH-162
A_POLYNOMIALLY_SHRINKING_RELATIVE_THRESHOLD_COSTS_P_NU_IN_ANY_PTH_MOMENT_MARKOV_EXCEPTIONAL_BOUND

B-RH-163
EVEN_WITH_ZERO_DENSITY_COST_SET_TO_ZERO_A_FIXED_MOMENT_MARKOV_ARGUMENT_HAS_A_PHASE_TRANSITION_AT_NU_EQUALS_ONE_MINUS_BETA

B-RH-164
A_HARD_INTERVAL_EXPLICIT_FORMULA_AT_RELATIVE_PRECISION_X_MINUS_NU_REQUIRES_ZERO_HEIGHT_X_TO_TAU_PLUS_NU_UP_TO_SUBPOWER_FACTORS

B-RH-165
THE_THRESHOLD_AWARE_GAFNI_TAO_L2_AND_L4_EXPONENTS_RETAIN_THE_UNIVERSAL_NU_EQUALS_D_PHASE_TRANSITION

O-RH-188
F_RH_017_V3_LIES_STRICTLY_BEYOND_THE_FIXED_MOMENT_MARKOV_PHASE_TRANSITION

Canonical root:

F-RH-017-v3
ACTIVE

No RH certificate is created.

19. Recommended next action

The next round should not compute another moment.

Split the exceptional set into:

\boxed{ \mathcal E_\nu^+ = \{ \Delta_H>HX^{-\nu} \}, }

and

\boxed{ \mathcal E_\nu^- = \{ \Delta_H<-HX^{-\nu} \}. }

Then audit them separately.

The immediate question is:

Can an ordinary-prime upper-bound sieve
control the tiny relative excess tail E_nu^+
at polynomial threshold?

If not, where exactly does sharp-factorial-moment information fail?

For the deficiency tail E_nu^-,
what parity-breaking lower-tail input would be required?

This one-sided decomposition may identify which half of F-RH-017 is truly root-hard.

20. Conclusion

Modern exceptional-interval methods are extremely strong at fixed relative error.

But the CSM_RH root threshold shrinks polynomially.

At a possible PESC boundary zero, the threshold crosses below the natural zero-mode amplitude exactly when $\nu>d$ .

Every fixed moment-Markov method undergoes a phase transition at that same point.

Higher moments and better zero additive-energy estimates cannot move the transition.

Therefore F-RH-017-v3 is genuinely threshold-native.

The campaign is now back at the correct root frontier with a sharper description of what cannot prove it.

CSM_RH Paper 86 — Polynomial-Threshold Exceptional Sets, the Gafni–Tao Moment Transfer, and the Boundary Phase Transitio

CSM_RH Paper 86

Polynomial-Threshold Exceptional Sets, the Gafni–Tao Moment Transfer, and the Boundary Phase Transition at ν = d

Abstract

Universal threshold phase transition

Super-boundary threshold

Boundary threshold

Sub-boundary threshold

1. Exceptional-set notation

2. Canonical F-RH-017-v3 gate

3. Gafni–Tao fixed-threshold framework

4. General moment-to-threshold lemma

Lemma 4.1 — Polynomial-threshold Markov penalty

5. Best-case zero-band baseline

Theorem 5.1 — Universal fixed-moment threshold barrier

6. Boundary phase transition

If ν<δβ\nu<\delta_\betaν<δβ​

If ν=δβ\nu=\delta_\betaν=δβ​

If ν>δβ\nu>\delta_\betaν>δβ​

7. Saturated PESC boundary

Corollary 7.1 — F-RH moment incompatibility

8. Higher moments do not move the transition

9. Hard-window truncation height

10. Threshold-aware Gafni–Tao second moment

11. Threshold-aware fourth moment

12. Smooth kernels do not remove the baseline wall

13. Why fixed-relative exceptional results remain useful but insufficient

14. Almost-prime exceptional sets as a calibration

15. What a successful F-RH-017 proof must avoid

15.1. Stopping-time / density increment

15.2. Threshold-native parity-breaking sieve

15.3. Inertia plus arithmetic persistence contradiction

15.4. Growing-complexity adaptive state

16. Relation to the canonical boundary forcing

17. Current best exceptional-interval calibration

18. State transition

19. Recommended next action

20. Conclusion

If $\nu<\delta_\beta$

If $\nu=\delta_\beta$

If $\nu>\delta_\beta$