CSM_RH Paper 18

Fejer Positivity, Principal-Arc Necessity, and the Modern $2/15$ Short-Interval Precision Floor

Project: CSM_RH
Paper: 18
Version: v0.1
Date: 2026-09-05
Parent state: CSM_RH v1.8 / Paper 17
Campaign: 17 — TRIANGULAR_SIGNED_ERROR_POWER_ATTACK
Status: mechanism correction / spectral bottleneck 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

Paper 17 introduced the direct theorem candidate MLEPG and suggested that a fixed power might arise from signed cancellation across the triangular shift kernel.

The present paper corrects that mechanism interpretation.

The exact triangular aggregate is a positive Fejer-weighted spectral energy.

Therefore the missing fixed power cannot be obtained merely by "preserving the signs of the shifts".

It requires genuine spectral deconcentration, including on the principal $q=1$ frequency arc.

No live GLM-5.3-Flash run is claimed.

1. Centered von Mangoldt sequence

Let

a_n = \Lambda(n)-1.

For a finite interval

1\le n\le X,

extend $a_n$ by zero outside this interval.

Define the exponential sum

\boxed{ S_X(\alpha) = \sum_{1\le n\le X} a_n e(n\alpha). }

For an integer lag

1\le H<X,

define the full zero-extended lag energy

\boxed{ \widetilde{\mathcal S}_\Lambda(X,H) = \sum_{x\in\mathbb Z} \left| \sum_{x<n\le x+H} a_n \right|^2. }

Only finitely many $x$ contribute.

2. Exact Fejer identity

Define

D_H(\alpha) = \sum_{r=1}^{H} e(r\alpha).

Then:

Theorem 2.1 — Exact Fejer Lag-Energy Identity

\boxed{ \widetilde{\mathcal S}_\Lambda(X,H) = \int_0^1 |S_X(\alpha)|^2 |D_H(\alpha)|^2 \,d\alpha. }

Proof

Write

\sum_{x<n\le x+H}a_n = \sum_{r=1}^{H} a_{x+r}.

The left side is the $\ell^2$ norm of the convolution of the finite sequence $a$ with the length- $H$ interval indicator.

Discrete Fourier Plancherel gives exactly the stated identity.

$\square$

Thus the triangular correlation kernel is positive definite.

3. Triangular shift expansion

Expanding the square gives the equivalent form

\boxed{ \widetilde{\mathcal S}_\Lambda(X,H) = \sum_{|h|<H} (H-|h|) \sum_n a_n a_{n+h}, }

with zero-extension understood.

The individual off-diagonal correlations may have either sign.

But their triangular signed sum is not an arbitrary signed quantity.

By Theorem 2.1 it is exactly a positive spectral energy.

This corrects the interpretation of the survivor introduced in Paper 17.

4. Fejer positivity obstruction

Create:

O-RH-039
FEJER_POSITIVITY_NO_SHIFT_SIGN_BYPASS
status:
  CERTIFIED

Statement:

The complete triangular shift aggregate in MLEPG is a positive Fejer-weighted Fourier energy. A proof cannot obtain the required fixed power merely by avoiding absolute values over shifts; it must prove genuine deconcentration of the centered prime exponential sum in the spectral regions emphasized by the Fejer kernel.

This does not prohibit using signed correlation identities inside a proof.

It prohibits treating "signed $h$ -cancellation" by itself as independent theorem authority.

5. Principal Fejer arc

The Dirichlet kernel satisfies

|D_H(\alpha)| = \left| \frac{ \sin(\pi H\alpha) }{ \sin(\pi\alpha) } \right|.

Fix any sufficiently small absolute constant

0<c_0<\frac14.

For

\|\alpha\| \le \frac{c_0}{H},

we have

\boxed{ |D_H(\alpha)|^2 \gg H^2. }

Therefore:

Theorem 5.1 — Principal-Arc Necessary Bound

\boxed{ H^2 \int_{\|\alpha\|\le c_0/H} |S_X(\alpha)|^2 \,d\alpha \ll \widetilde{\mathcal S}_\Lambda(X,H). }

Consequently, any MLEPG bound

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

forces

\boxed{ \int_{\|\alpha\|\le c_0/H} |S_X(\alpha)|^2 \,d\alpha \ll \frac{X}{H} (\log X)^{O(1)} + X^{1-\delta+o(1)}. }

The principal frequency arc is therefore an unavoidable fixed-power subproblem.

6. Principal $q=1$ spectral bottleneck

The interval

\|\alpha\| \le c_0/H

is the central additive frequency arc.

It is the $q=1$ major-arc component.

Create:

O-RH-040
PRINCIPAL_FEJER_ARC_FIXED_POWER_NECESSITY
status:
  CERTIFIED

Statement:

Any fixed-power MLEPG theorem at scale $H$ must improve the centered prime exponential-sum $L^2$ mass on the principal $q=1$ arc at the corresponding fixed-power scale.

Therefore a power saving on strongly minor arcs alone cannot close MLEPG.

This reproduces, in the direct lag-energy branch, the principal-component bottleneck seen repeatedly in earlier CSM_RH campaigns.

7. Why the MRT averaging step does not lose the shift signs

Matomäki–Radziwiłł–Tao use the Hardy–Littlewood circle method for

\sum_n \Lambda(n)\Lambda(n+h).

After averaging in $h$ and applying Plancherel, their analysis is reduced to positive local Fourier-energy estimates of the schematic form

\boxed{ \int_{\beta-1/H}^{\beta+1/H} |S(\alpha)|^2 \,d\alpha \ll_A X\log^{-A}X }

for minor-arc centers $\beta$ .

Thus the average-over-shifts stage already converts the correlation problem to a positive local $L^2$ problem.

There is no unused global sign cancellation in $h$ left at that stage.

The loss to logarithmic precision occurs in the strength of the available local spectral / Dirichlet-polynomial estimates, not because one first took absolute values of every shift correlation.

8. MRT strongly minor arcs

For strongly minor arcs, the MRT route converts the exponential-sum local energy into Dirichlet-polynomial mean values.

Their most delicate pieces are the Type $d_3$ and Type $d_4$ components arising after Heath-Brown decomposition.

The Type $d_3$ component uses:

Jutila short-interval mean values;
Hölder-type decomposition;
return to physical space;
oscillatory sums;
Robert-Sargos estimates;
van der Corput processing.

This machinery is responsible for reaching the shift threshold

H \ge X^{8/33+\varepsilon}.

It supplies arbitrary logarithmic savings, not a fixed $X$ -power for the von Mangoldt pair correlation.

Improving this machinery could improve nonprincipal spectral control.

But Theorem 5.1 shows that it cannot by itself bypass the principal arc.

9. Updated short-interval scale: $2/15$

The natural first scale in Paper 17 was chosen from the 2019 average prime-pair threshold

\frac8{33}.

Current 2026 short-interval technology provides a smaller exponent for the prime number theorem in almost all short intervals.

The Guth–Maynard zero-density estimate implies the almost-all short-interval PNT for every fixed

\boxed{ \alpha>\frac2{15}. }

The 2026 Matomäki–Radziwiłł–Shao–Tao–Teräväinen paper records the quantitative form:

for every fixed

A>0

and suitable $H$ in this regime, outside an exceptional set of measure

O \left( X\log^{-A}X \right),

one has

\boxed{ \left| \sum_{x<n\le x+H} [ \Lambda(n)-1 ] \right| \le H\log^{-A}X. }

Thus the current range obstruction is below the Paper 17 scale.

The precision obstruction remains.

10. Current short-interval mean-square consequence

Use the quantitative almost-all statement in Section 9.

On the good set,

\left| \sum_{x<n\le x+H}a_n \right|^2 \le H^2 \log^{-2A}X.

On the exceptional set, Brun–Titchmarsh / trivial Chebyshev-type bounds give

\left| \sum_{x<n\le x+H}a_n \right| \ll H(\log X)^{O(1)}

in the polynomial short-interval regime.

Since $A$ is arbitrary, after increasing $A$ we obtain:

Proposition 10.1 — Current Log-Power Lag-Energy Bound

For every fixed

\alpha>\frac2{15}

and every fixed

B>0,

with

H=X^\alpha,

current technology yields schematically

\boxed{ \widetilde{\mathcal S}_\Lambda(X,H) \ll_B XH^2 \log^{-B}X + \text{standard boundary terms}. }

This is

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

It is not a fixed-power improvement.

11. Modern precision floor

MLEPG needs, for some fixed

\delta>0,

a bound of the form

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

Current short-interval technology supplies

XH^2\log^{-B}X

for arbitrary fixed $B$ , but no fixed

X^{-\delta}.

Create:

O-RH-041
MODERN_SHORT_INTERVAL_LOG_TO_POWER_PRECISION_FLOOR
status:
  CERTIFIED AS CURRENT-TECHNOLOGY AUDIT

The modern bottleneck is therefore not primarily the available exponent $\alpha$ .

It is the distinction

\boxed{ \log^{-B}X \quad\text{versus}\quad X^{-\delta}. }

12. Two-scale comparison

There are now two natural scales relevant to the direct theorem candidate.

Prime-pair/circle-method scale

\alpha > \frac8{33}.

At this scale, averaged Hardy–Littlewood pair correlations are known for almost all shifts with arbitrary logarithmic savings.

Almost-all short-interval PNT scale

\alpha > \frac2{15}.

At this smaller scale, the short-interval PNT is known for almost all intervals, again with subpower/logarithmic quantitative precision.

Neither route supplies a fixed lag-energy power.

The smaller scale improves the possible exponent cap in a future residue-chain theorem only if the fixed-power precision problem is also solved.

13. Exceptional-set power alone is insufficient

Suppose one proves that only

X^{1-\eta}

intervals are exceptional for a fixed relative-error threshold.

This does not by itself imply MLEPG.

The error on the nonexceptional intervals must also shrink with $X$ at fixed-power scale, or a direct $L^2$ argument must replace the pointwise good/bad decomposition.

Indeed, if on the good set one knows only

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

for fixed $\varepsilon>0$ , then the good-set contribution remains

\asymp \varepsilon^2 XH^2,

which has no fixed $X$ -power gain.

Thus modern exceptional-set improvements do not automatically solve the MLEPG precision problem.

14. Two-axis fixed-power criterion

A sufficient pointwise/exceptions formulation is the following.

Suppose for fixed

\eta_1,\eta_2>0

one has:

Good intervals

Outside an exceptional set $\mathcal E$ ,

\boxed{ |\Delta_HA(x)| \ll H X^{-\eta_1+o(1)}. }

Exceptional set

\boxed{ |\mathcal E| \ll X^{1-\eta_2+o(1)}. }

Then, using the trivial polynomial bound on exceptional intervals,

\boxed{ \widetilde{\mathcal S}_\Lambda(X,H) \ll XH^2 X^{-\min(2\eta_1,\eta_2)+o(1)}. }

Thus a fixed-power lag-energy theorem needs fixed-power accuracy in an $L^2$ sense.

A power-saving exceptional-set exponent with only fixed relative accuracy is insufficient.

15. Strength calibration by a single off-axis zero

Let

\rho = \beta+i\gamma

be a fixed nontrivial zero.

For

H=o(X)

and eventually

|\gamma|H/X\ll1,

the explicit-formula contribution of this zero to the short-interval error has local amplitude of order

H X^{\beta-1}

up to oscillation and smoothing issues.

Its squared contribution over an $X$ -sized range is therefore of natural scale

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

The MLEPG error budget

XH^2X^{-\delta} = H^2X^{1-\delta}

is incompatible at exponential scale with

\beta > 1-\frac{\delta}{2},

once a coefficient-recovery / cancellation-isolation argument is supplied.

This agrees with the earlier fixed-strip strength audits.

This subsection is a strength calibration, not a new proof of zero isolation for the lag energy.

16. Status of triangular signed cancellation

Paper 17 created:

S-RH-025
TRIANGULAR_SIGNED_PRIME_PAIR_ERROR_CANCELLATION

The current correction is:

status:
  DEMOTED AS INDEPENDENT MECHANISM

reason:
  the complete triangular aggregate is a positive Fejer energy

Signed prime-pair identities may still be used internally.

But they must produce a positive spectral-energy improvement in the end.

17. Surviving mechanism: principal spectral deconcentration

Create:

S-RH-026
PRINCIPAL_FEJER_ARC_POWER_DECONCENTRATION
status:
  OPEN

Prototype:

for

H=X^\alpha

and some fixed

\delta>0,

prove

\boxed{ \int_{\|\alpha'\|\le c/H} |S_X(\alpha')|^2 \,d\alpha' \ll \frac{X}{H} (\log X)^{O(1)} + X^{1-\delta+o(1)}. }

This is a proof-mechanism sublemma.

It is not promoted as a new root frontier.

MLEPG remains the direct theorem candidate and PESC remains the root target.

18. Campaign 17 verdict

SHIFT-SIGN BYPASS
  REJECTED

FEJER POSITIVITY
  CERTIFIED

MRT AVERAGING STEP
  ALREADY POSITIVE-L2

STRONGLY MINOR ARC IMPROVEMENT
  POSSIBLE TOOL / NOT SUFFICIENT ALONE

PRINCIPAL q=1 ARC
  NECESSARY FIXED-POWER SUBPROBLEM

CURRENT ALMOST-ALL SHORT-INTERVAL RANGE
  alpha > 2/15

CURRENT QUANTITATIVE PRECISION
  arbitrary log-power

FIXED X-POWER
  NOT OBTAINED

MLEPG
  REMAINS OPEN

19. Campaign 18

The next campaign is:

CSM_RH Campaign 18
PRINCIPAL_FEJER_ARC_POWER_ATTACK

It remains theorem-generation work.

The root target does not change.

The campaign asks whether any current zero-density, large-value, Dirichlet-polynomial, or explicit-formula technique can prove a fixed-power principal-arc estimate without assuming a fixed zero strip.

20. Campaign 18 tracks

P1 — Guth–Maynard large-value upgrade

Insert the 2026 large-value estimates into a quantitative $L^2$ short-interval calculation rather than merely a density-zero exceptional-set statement.

Test the strongest exponent actually obtainable.

P2 — Gafni–Tao exceptional-set optimization

Use the quantitative zero-density-to-exceptional-set machine with a shrinking error threshold.

Determine whether the optimization remains subpower or could produce a fixed $X$ -power in $L^2$ .

P3 — Principal Dirichlet-polynomial mean square

Work directly with the $q=1$ local Fourier mass and Mellin/Dirichlet-polynomial transforms.

Identify the first large-value estimate which would have to improve by a fixed power.

P4 — Explicit-formula zero packet

Smooth the lag energy and compute the contribution of one fixed off-axis zero.

Use this only as a strength/countermodel audit unless a genuine coefficient-isolation argument is proved.

21. Campaign 18 rejection filters

Reject a candidate if:

R1. It improves only strongly minor arcs.

R2. It obtains only logarithmic or stretched-logarithmic savings.

R3. It assumes a fixed zero-free half-plane.

R4. It invokes exceptional-set density without also controlling the shrinking good-interval error in $L^2$ .

R5. It treats the Fejer-weighted energy as a signed object after Theorem 2.1.

R6. It claims that lowering the short-interval exponent alone creates a fixed-power gain.

22. External calibration

Current literature relevant to this campaign includes:

Matomäki–Radziwiłł–Tao, Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges, Proc. Lond. Math. Soc. 118 (2019). Their shift averaging plus Plancherel reduces the minor-arc problem to positive local Fourier $L^2$ estimates; the Type $d_3$ and Type $d_4$ Dirichlet-polynomial analysis yields arbitrary logarithmic savings and the $8/33$ threshold.
Guth–Maynard, New large value estimates for Dirichlet polynomials, Annals of Mathematics 203 (2026), 623–675. They prove
$N(\sigma,T) \le T^{30(1-\sigma)/13+o(1)}$
and derive improved prime-distribution consequences.
Matomäki–Radziwiłł–Shao–Tao–Teräväinen, Higher uniformity of arithmetic functions in short intervals II. Almost all intervals, Inventiones Mathematicae 244 (2026), 967–1091. They record the quantitative almost-all short-interval PNT with arbitrary logarithmic accuracy; the exponent $1/6$ is improved to $2/15$ using Guth–Maynard.
Gafni–Tao, On the number of exceptional intervals to the prime number theorem in short intervals, Essential Number Theory 5 (2026), 221–241. They quantify the translation from zero-density estimates to exceptional-set bounds.

None of these results currently supplies the fixed $X$ -power MLEPG requires.

23. State transition

The canonical transition is:

CSM_RH v1.8
  ->
CSM_RH v1.9

with:

Campaign 17
  CLOSED_AS_FEJER_POSITIVITY_AND_PRECISION_AUDIT

O-RH-039
  FEJER_POSITIVITY_NO_SHIFT_SIGN_BYPASS
  CREATED / CERTIFIED

O-RH-040
  PRINCIPAL_FEJER_ARC_FIXED_POWER_NECESSITY
  CREATED / CERTIFIED

O-RH-041
  MODERN_SHORT_INTERVAL_LOG_TO_POWER_PRECISION_FLOOR
  CREATED / CERTIFIED AS CURRENT-TECHNOLOGY AUDIT

S-RH-025
  TRIANGULAR_SIGNED_PRIME_PAIR_ERROR_CANCELLATION
  DEMOTED AS INDEPENDENT MECHANISM

S-RH-026
  PRINCIPAL_FEJER_ARC_POWER_DECONCENTRATION
  CREATED / OPEN

F-RH-016
  MLEPG
  REMAINS OPEN / DIRECT THEOREM CANDIDATE

F-RH-010
  PESC
  REMAINS OPEN / ROOT TARGET

Campaign 18
  PRINCIPAL_FEJER_ARC_POWER_ATTACK
  READY

24. Final status

RH = OPEN

PESC = OPEN / ROOT TARGET

MLEPG = OPEN / DIRECT THEOREM CANDIDATE

TRIANGULAR SHIFT-SIGN BYPASS = CLOSED

FEJER ENERGY = POSITIVE

PRINCIPAL q=1 ARC = NECESSARY

CURRENT SHORT-INTERVAL RANGE = alpha > 2/15

CURRENT PRECISION = LOG-POWER / SUBPOWER

FIXED X-POWER = OPEN

NEXT CAMPAIGN = 18

The central unresolved mechanism has become:

\boxed{ \text{fixed-power deconcentration of centered prime spectral mass on the principal Fejer arc}. }

Improving shift range or strongly minor-arc technology alone is not enough.

CSM_RH Paper 18 — Fejer Positivity, Principal-Arc Necessity, and the Modern $2_15$ Short-Interval Precision Floor

CSM_RH Paper 18

Fejer Positivity, Principal-Arc Necessity, and the Modern 2/152/152/15 Short-Interval Precision Floor

0. Trust boundary

1. Centered von Mangoldt sequence

2. Exact Fejer identity

Theorem 2.1 — Exact Fejer Lag-Energy Identity

Proof

3. Triangular shift expansion

4. Fejer positivity obstruction

5. Principal Fejer arc

Theorem 5.1 — Principal-Arc Necessary Bound

6. Principal q=1q=1q=1 spectral bottleneck

7. Why the MRT averaging step does not lose the shift signs

8. MRT strongly minor arcs

9. Updated short-interval scale: 2/152/152/15

10. Current short-interval mean-square consequence

Proposition 10.1 — Current Log-Power Lag-Energy Bound

11. Modern precision floor

12. Two-scale comparison

Prime-pair/circle-method scale

Almost-all short-interval PNT scale

13. Exceptional-set power alone is insufficient

14. Two-axis fixed-power criterion

Good intervals

Exceptional set

15. Strength calibration by a single off-axis zero

16. Status of triangular signed cancellation

17. Surviving mechanism: principal spectral deconcentration

18. Campaign 17 verdict

19. Campaign 18

20. Campaign 18 tracks

P1 — Guth–Maynard large-value upgrade

P2 — Gafni–Tao exceptional-set optimization

P3 — Principal Dirichlet-polynomial mean square

P4 — Explicit-formula zero packet

21. Campaign 18 rejection filters

R1. It improves only strongly minor arcs.

R2. It obtains only logarithmic or stretched-logarithmic savings.

R3. It assumes a fixed zero-free half-plane.

R4. It invokes exceptional-set density without also controlling the shrinking good-interval error in L2L^2L2.

R5. It treats the Fejer-weighted energy as a signed object after Theorem 2.1.

R6. It claims that lowering the short-interval exponent alone creates a fixed-power gain.

22. External calibration

23. State transition

24. Final status

Fejer Positivity, Principal-Arc Necessity, and the Modern $2/15$ Short-Interval Precision Floor

6. Principal $q=1$ spectral bottleneck

9. Updated short-interval scale: $2/15$

R4. It invokes exceptional-set density without also controlling the shrinking good-interval error in $L^2$ .