CSM_RH Paper 19

Zero-Density Exponent Law, Shrinking-Strip Barrier, and Closure Back to the Fixed-Zero-Strip Core

Project: CSM_RH
Paper: 19
Version: v0.1
Date: 2026-09-05
Parent state: CSM_RH v1.9 / Paper 18
Campaign: 18 — PRINCIPAL_FEJER_ARC_POWER_ATTACK
Status: current large-value / zero-density closure audit; not a proof or disproof of RH

0. Trust boundary

This paper does not prove or disprove the Riemann Hypothesis.

Canonical root state:

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

Campaign 18 asks whether current large-value, zero-density, exceptional-set, and explicit-formula techniques can upgrade the principal Fejer arc from subpower precision to a fixed power.

The answer for the currently audited absolute zero-density schema is negative.

The main result is an exponent law which separates:

range threshold
from
precision threshold

and shows why improving zero density alone does not create a fixed power while the zero-free gap still shrinks with height.

No live GLM-5.3-Flash run is claimed.

1. Direct theorem candidate

Paper 17 introduced MLEPG.

Let

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

Let

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

Define

\mathcal S_\Lambda(X,H) = \sum_x |A(x+H)-A(x)|^2

with an appropriate finite or smoothed range around $X$ .

MLEPG asks for

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

for one fixed

\delta>0.

Paper 18 proved that the principal Fejer arc is a necessary subproblem.

2. Guth–Maynard already prove the subpower $L^2$ analogue

In the proof of their almost-all short-interval prime theorem, Guth and Maynard use the explicit formula and reduce the problem to an $L^2$ estimate for a zero packet.

Their equation (13.4) has the form

\boxed{ \int_X^{3X} \left| \sum_{|\rho|<T} x^\rho \frac{ (1+\Delta)^\rho-1 }{ \rho } \right|^2 dx \ll \Delta^2X^3 \exp \left( -c(\log X)^{1/4} \right), }

at the scale used in their proof.

Since

H\asymp \Delta X,

the natural normalization is

\Delta^2X^3 = XH^2.

Thus the current large-value method already proves the structural analogue

\boxed{ XH^2 \times X^{-o(1)}. }

The missing MLEPG input is exactly the replacement

X^{-o(1)} \longrightarrow X^{-\delta}.

3. Abstract zero-density input

Assume an upper bound

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

uniformly at the exponent level, where

A_0>0

is fixed.

For Guth–Maynard,

\boxed{ A_0=\frac{30}{13}. }

Take

H=X^\alpha

and the explicit-formula truncation height

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

This is the natural short-interval scale

T\asymp X/H

up to subpower factors.

4. Zero-density exponent law

A zero with real part $\sigma$ contributes at squared $L^2$ exponent scale

H^2X^{2\sigma-1}

relative to an $X$ -length integration range.

Equivalently, relative to the baseline

XH^2,

the single-zero exponent factor is

X^{-2(1-\sigma)}.

The density bound permits approximately

T^{A_0(1-\sigma)+o(1)} = X^{A_0(1-\alpha)(1-\sigma)+o(1)}

zeros at that real-part scale.

Therefore the absolute zero-density contribution has exponent ratio

\boxed{ X^{-c_{A_0}(\alpha)(1-\sigma)+o(1)}, }

where

\boxed{ c_{A_0}(\alpha) = 2 - A_0(1-\alpha). }

This is the canonical density-to- $L^2$ exponent law for the present audit.

5. Range threshold

The density schema yields decay in the real-part variable only if

c_{A_0}(\alpha)>0.

Thus:

Theorem 5.1 — Zero-Density Range Threshold

\boxed{ \alpha > 1-\frac2{A_0}. }

For

A_0=\frac{30}{13},

we obtain

\begin{aligned} 1-\frac2{A_0} &= 1-\frac{26}{30} \\ &= \boxed{ \frac2{15}. } \end{aligned}

This recovers the Guth–Maynard almost-all short-interval threshold at exponent level.

The threshold is therefore a density-balance phenomenon.

6. Shrinking zero-free boundary

Let the available zero-free region have the form

\boxed{ \Re\rho \le 1-\eta(T), }

where

\eta(T)>0

and

\eta(T)\to0

T\to\infty.

The Vinogradov–Korobov region is of this type.

The worst permitted real part in the density exponent law is then

\sigma = 1-\eta(T).

Substituting into Section 4 gives the best exponent ratio available from the absolute density schema:

\boxed{ X^{-c_{A_0}(\alpha)\eta(T)+o(1)}. }

Because

\eta(T)\to0,

this is

\boxed{ X^{-o(1)}. }

It is not a fixed power.

7. Shrinking-Strip Barrier Theorem

Theorem 7.1

Fix

A_0<\infty

and

\alpha > 1-\frac2{A_0}.

Consider an explicit-formula $L^2$ proof schema which:

bounds zero contributions by absolute values or positive moments at each real-part scale;
uses only the density estimate $N(\sigma,T) \ll T^{A_0(1-\sigma)+o(1)};$
uses a zero-free boundary $\Re\rho \le 1-\eta(T)$ with $\eta(T)\to0.$

Then the zero-density exponent law supplies at best a subpower factor

\boxed{ X^{-o(1)} }

over the baseline $XH^2$ .

It cannot by itself produce

X^{-\delta}

for fixed

\delta>0.

This theorem is about the specified proof schema.

It is not a universal impossibility theorem for all uses of zeros.

8. Improving zero density alone does not solve the precision problem

Suppose the density hypothesis were available in the ideal form

A_0=2.

Then the range threshold becomes

\alpha>0.

Thus almost-all short-interval coverage could, at the density-balance level, extend to every fixed positive power scale.

But the precision factor would still be

X^{-2\alpha\eta(T)+o(1)} = X^{-o(1)}

whenever

\eta(T)\to0.

Therefore:

Corollary 8.1

Even an optimal density exponent does not, by itself, give MLEPG fixed-power precision unless one also obtains:

a fixed zero-free gap,
or
new cancellation / coefficient isolation beyond the absolute density schema.

This cleanly separates the density problem from the fixed-strip problem.

9. Guth–Maynard proof location

Guth and Maynard prove

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

and use the Vinogradov–Korobov zero-free region in their prime-distribution corollaries.

In their almost-all short-interval proof they choose a truncation scale corresponding to the interval length and reduce the problem to the zero-packet $L^2$ estimate described in Section 2.

The final saving is subpower.

The exponent law in this paper explains structurally why the new density exponent changes the admissible short-interval range to $2/15$ but does not generate a fixed $X$ -power.

10. Gafni–Tao exceptional-set framework

Gafni and Tao define, for fixed relative-error tolerance

\delta>0,

an exceptional set of intervals where the short-interval PNT fails at that tolerance.

They then define exceptional-set exponents and derive explicit bounds from zero-density information.

This is strong quantitative information about the number of bad intervals.

However the parameter $\delta$ in the theorem framework is fixed while $X$ tends to infinity.

By diagonalizing over fixed values

\delta=1,\frac12,\frac13,\ldots,

one obtains an $o(H)$ error outside a density-zero set.

One does not obtain a polynomially shrinking threshold

HX^{-\eta}.

11. Fixed-threshold versus shrinking-threshold mismatch

Suppose for each fixed

\varepsilon>0

one proves

|\Delta_HA(x)| \le \varepsilon H

outside an exceptional set of power-saving size.

Then the good-set contribution to the lag energy is still

\boxed{ \varepsilon^2XH^2. }

For fixed $\varepsilon$ , this has no fixed $X$ -power saving.

Taking $\varepsilon$ arbitrarily small after the theorem is proved does not create a bound

X^{-\delta}.

To obtain MLEPG through a good/bad decomposition, one needs a threshold which shrinks quantitatively with $X$ , for example

\boxed{ |\Delta_HA(x)| \le HX^{-\eta_1} }

on the good set.

Create:

O-RH-043
FIXED_THRESHOLD_EXCEPTIONAL_SET_MISMATCH
status:
  CERTIFIED

12. Guth–Maynard and Gafni–Tao combined

The two technologies address different axes.

Guth–Maynard

Improves:

zero-density exponent;
large-value estimates;
range of short intervals;
subpower quantitative error.

Gafni–Tao

Improves:

quantification of exceptional-set size;
translation from zero-density / zero additive energy to bad-interval counts.

Neither currently supplies:

\boxed{ \text{fixed-power shrinking error threshold in }L^2. }

Thus their combination does not currently prove MLEPG.

13. Density blindness to a fixed off-axis zero

For every fixed

\beta<1,

a density bound of the form

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

permits finitely many zeros with real part $\beta$ .

A single such zero is negligible for the asymptotic count $N(\sigma,T)$ .

Therefore density information alone cannot exclude a fixed off-axis zero.

But a fixed zero with real part

\beta

has short-interval explicit-formula amplitude at the natural scale

H X^{\beta-1}

and squared integrated scale

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

Thus fixed-power principal-arc / lag-energy control must ultimately be sensitive to individual persistent off-axis zero modes.

Create:

O-RH-044
ZERO_DENSITY_SINGLE_ZERO_BLINDNESS
status:
  CERTIFIED AS STRENGTH AUDIT

This is not a claim that one zero automatically gives a rigorous lower bound without a coefficient-recovery argument.

It records the information deficit of density estimates.

14. Closure back to the fixed-zero-strip core

Paper 17 gives the deterministic chain

\operatorname{MLEPG}(\alpha,\delta) \Longrightarrow \sum_{n\le2N} |\psi(n)-n|^2 \ll N^{3-\kappa+o(1)}

for some

\kappa>0.

At exponent level, a fixed zero

\rho = \beta+i\gamma

contributes the scale

N^{2\beta+1}

to the PNT mean-square explicit formula.

Thus the bound

N^{3-\kappa+o(1)}

has the fixed-strip strength

\boxed{ \beta \le 1-\frac{\kappa}{2}. }

This is the same exponent law already encountered in the earlier ZPPF / principal zeta-packet campaigns.

Therefore the direct PESC branch has closed a loop:

PESC
  ->
MLEPG candidate
  ->
principal Fejer arc
  ->
explicit formula / zero density
  ->
fixed-zero-strip strength

The loop is not circular as a proof.

It is a closure-space strength identification.

15. New obstruction: density / strip decoupling

Create:

O-RH-042
ZERO_DENSITY_SHRINKING_STRIP_EXPONENT_BARRIER
status:
  CERTIFIED FOR ABSOLUTE EXPLICIT-FORMULA SCHEMA

Statement:

A finite zero-density exponent determines the short-interval range threshold through $2-A_0(1-\alpha)$ , but a shrinking zero-free boundary forces the resulting power gain to shrink to zero. Density improvement alone therefore cannot supply MLEPG fixed-power precision within the audited absolute explicit-formula schema.

16. Status of Campaign 18 tracks

P1 — Guth–Maynard large-value upgrade

status:
  CURRENT METHOD ALREADY PRODUCES L2 SUBPOWER

fixed power:
  NO

barrier:
  shrinking zero-free boundary in absolute density summation

P2 — Gafni–Tao exceptional-set optimization

status:
  STRONG EXCEPTIONAL-SET COUNTING

shrinking polynomial good-error threshold:
  NOT PROVIDED BY CURRENT FRAMEWORK

fixed-power L2:
  NO

P3 — principal Dirichlet-polynomial mean square

status:
  CURRENT LARGE-VALUE INPUT IMPROVES DENSITY/RANGE

single-zero-sensitive fixed power:
  NOT IDENTIFIED

P4 — smooth explicit-formula zero packet

status:
  STRENGTH AUDIT CONFIRMS FIXED-STRIP SCALE

new coefficient-isolation theorem:
  NOT PROVED

17. What a genuinely new input must do

Any successful next mechanism must be sensitive to a single persistent near- $1$ zero.

Density-only information is insufficient.

The new input must do at least one of:

S1 — fixed zero-free gap

Directly prove

\Re\rho \le 1-\eta_0

for some fixed

\eta_0>0.

S2 — coefficient recovery / isolation

Show that a persistent off-axis zero contributes a noncancellable amount to the principal Fejer / lag-energy observable.

Then MLEPG would exclude it.

S3 — phase-sensitive zero-packet coercivity

Use the structure of the zero packet beyond absolute counting to obtain a fixed-power lower/upper incompatibility.

S4 — new prime-side theorem independent of zero density

Prove MLEPG directly by arithmetic methods whose fixed power is not obtained by summing a density estimate.

18. New survivor

Create:

S-RH-027
SINGLE_ZERO_SENSITIVE_PRINCIPAL_ARC_MECHANISM
status:
  OPEN

Definition:

a theorem mechanism which can detect or exclude the contribution of one persistent off-axis zero in the principal Fejer / short-interval $L^2$ observable, rather than merely bound how many such zeros may exist.

This is a mechanism requirement.

It is not a new target.

19. Campaign 18 verdict

GUTH-MAYNARD ZERO DENSITY
  RANGE BREAKTHROUGH

GUTH-MAYNARD PRINCIPAL L2
  SUBPOWER

GAFNI-TAO EXCEPTIONAL SET
  POWER-SIZED BAD-SET INFORMATION POSSIBLE

GAFNI-TAO GOOD ERROR THRESHOLD
  FIXED / o(1), NOT POLYNOMIALLY SHRINKING

ABSOLUTE ZERO-DENSITY SCHEMA
  CANNOT CREATE FIXED POWER WITH SHRINKING ZERO-FREE GAP

DENSITY HYPOTHESIS ALONE
  STILL NOT ENOUGH FOR FIXED POWER

MLEPG
  OPEN

PESC
  OPEN

No fixed-power theorem is proved.

20. Campaign 19

The next campaign is:

CSM_RH Campaign 19
SINGLE_ZERO_SENSITIVE_THEOREM_GENERATION

The root target remains PESC.

The working theorem candidate remains MLEPG.

The new design constraint is:

every candidate mechanism must be capable of distinguishing
"no near-1 zero"
from
"one persistent near-1 zero".

Density-only candidates are rejected automatically.

21. Campaign 19 tracks

Z1 — multiscale coefficient recovery

Use principal Fejer energies over a family of scales $H$ to recover a fixed zero mode.

A valid proof must quantify the inverse map and control all other zeros.

Z2 — smooth Mellin packet coercivity

Replace the hard interval kernel by a smooth positive packet and test whether Mellin coefficients of individual zeros can be isolated with a fixed exponent.

Z3 — zero-pair Gram positivity with gauge control

Build a canonical zero-mode Gram form whose diagonal from one off-axis orbit cannot be removed by representation changes or uncontrolled cross terms.

Paper 07's Gram-gauge obstruction must be respected.

Z4 — prime-side multiscale contraction

Avoid zeros entirely and prove a fixed-power recurrence for lag energies across scales.

The recurrence must have linear cumulative contraction mass.

22. Campaign 19 rejection filters

Reject a candidate if:

R1. It uses only a zero-density count.

R2. It uses a shrinking zero-free region and calls the resulting subpower a fixed power.

R3. It controls exceptional-set size without a polynomially shrinking good-error threshold.

R4. It assumes a fixed zero-free strip.

R5. It ignores cancellation among zero modes in a coefficient-recovery claim.

R6. It creates a gauge-dependent Gram gap.

R7. It merely restates PESC / MLEPG.

23. External calibration

The present audit uses the following current results.

Guth–Maynard, New large value estimates for Dirichlet polynomials, Annals of Mathematics 203 (2026), prove
$N(\sigma,T) \le T^{30(1-\sigma)/13+o(1)}$
and obtain almost-all prime asymptotics for short intervals beginning at exponent $2/15+\varepsilon$ . Their proof reduces the almost-all result to an explicit-formula $L^2$ zero-packet estimate and obtains subpower exponential decay.
Gafni–Tao, On the number of exceptional intervals to the prime number theorem in short intervals, Essential Number Theory 5 (2026), develop explicit bounds on exceptional-set exponents from zero-density and zero-additive-energy estimates. Their exceptional-set definitions use fixed relative-error tolerances.
Matomäki–Radziwiłł–Shao–Tao–Teräväinen, Higher uniformity of arithmetic functions in short intervals II. Almost all intervals, Inventiones Mathematicae 244 (2026), record the current almost-all short-interval PNT range and arbitrary logarithmic quantitative accuracy.

None supplies a fixed $X$ -power principal-arc estimate.

24. State transition

The canonical transition is:

CSM_RH v1.9
  ->
CSM_RH v1.10

with:

Campaign 18
  CLOSED_AS_ZERO_DENSITY_AND_EXCEPTIONAL_SET_PRECISION_AUDIT

O-RH-042
  ZERO_DENSITY_SHRINKING_STRIP_EXPONENT_BARRIER
  CREATED / CERTIFIED FOR ABSOLUTE EXPLICIT-FORMULA SCHEMA

O-RH-043
  FIXED_THRESHOLD_EXCEPTIONAL_SET_MISMATCH
  CREATED / CERTIFIED

O-RH-044
  ZERO_DENSITY_SINGLE_ZERO_BLINDNESS
  CREATED / CERTIFIED AS STRENGTH AUDIT

S-RH-027
  SINGLE_ZERO_SENSITIVE_PRINCIPAL_ARC_MECHANISM
  CREATED / OPEN

F-RH-016
  MLEPG
  REMAINS OPEN

F-RH-010
  PESC
  REMAINS OPEN / ROOT TARGET

Campaign 19
  SINGLE_ZERO_SENSITIVE_THEOREM_GENERATION
  READY

25. Final status

RH = OPEN

PESC = OPEN

MLEPG = OPEN

CURRENT ZERO-DENSITY RANGE = STRONG

CURRENT ZERO-DENSITY PRECISION = SUBPOWER

2/15 = RANGE THRESHOLD, NOT FIXED-POWER THRESHOLD

GAFNI-TAO = EXCEPTIONAL-SET TOOL, NOT SHRINKING-THRESHOLD POWER L2

DENSITY-ONLY SCHEMA = SINGLE-ZERO BLIND

NEXT REQUIREMENT = SINGLE-ZERO-SENSITIVE MECHANISM

NEXT CAMPAIGN = 19

The key exponent identity is

\boxed{ c_{A_0}(\alpha) = 2 - A_0(1-\alpha). }

It controls whether zero-density information can gain anything at a short-interval scale.

But the actual fixed-power exponent available from a shrinking zero-free boundary is only

\boxed{ c_{A_0}(\alpha)\eta(T), }

which tends to zero.

The next theorem-generation round must therefore add information which is qualitatively stronger than density.

CSM_RH Paper 19 — Zero-Density Exponent Law, Shrinking-Strip Barrier, and Closure Back to the Fixed-Zero-Strip Core

CSM_RH Paper 19

Zero-Density Exponent Law, Shrinking-Strip Barrier, and Closure Back to the Fixed-Zero-Strip Core

0. Trust boundary

1. Direct theorem candidate

2. Guth–Maynard already prove the subpower L2L^2L2 analogue

3. Abstract zero-density input

4. Zero-density exponent law

5. Range threshold

Theorem 5.1 — Zero-Density Range Threshold

6. Shrinking zero-free boundary

7. Shrinking-Strip Barrier Theorem

Theorem 7.1

8. Improving zero density alone does not solve the precision problem

Corollary 8.1

9. Guth–Maynard proof location

10. Gafni–Tao exceptional-set framework

11. Fixed-threshold versus shrinking-threshold mismatch

12. Guth–Maynard and Gafni–Tao combined

Guth–Maynard

Gafni–Tao

13. Density blindness to a fixed off-axis zero

14. Closure back to the fixed-zero-strip core

15. New obstruction: density / strip decoupling

16. Status of Campaign 18 tracks

P1 — Guth–Maynard large-value upgrade

P2 — Gafni–Tao exceptional-set optimization

P3 — principal Dirichlet-polynomial mean square

P4 — smooth explicit-formula zero packet

17. What a genuinely new input must do

S1 — fixed zero-free gap

S2 — coefficient recovery / isolation

S3 — phase-sensitive zero-packet coercivity

S4 — new prime-side theorem independent of zero density

18. New survivor

19. Campaign 18 verdict

20. Campaign 19

21. Campaign 19 tracks

Z1 — multiscale coefficient recovery

Z2 — smooth Mellin packet coercivity

Z3 — zero-pair Gram positivity with gauge control

Z4 — prime-side multiscale contraction

22. Campaign 19 rejection filters

R1. It uses only a zero-density count.

R2. It uses a shrinking zero-free region and calls the resulting subpower a fixed power.

R3. It controls exceptional-set size without a polynomially shrinking good-error threshold.

R4. It assumes a fixed zero-free strip.

R5. It ignores cancellation among zero modes in a coefficient-recovery claim.

R6. It creates a gauge-dependent Gram gap.

R7. It merely restates PESC / MLEPG.

23. External calibration

24. State transition

25. Final status

2. Guth–Maynard already prove the subpower $L^2$ analogue