← Archive
lm-003875 · 2026-09

CSM_RH Paper 17 — Mesoscopic Lag-Energy Power Gain and the First Direct PESC Theorem Candidate

en · md Raw source ↗ Metadata ↗
下載 MD 檔 ⬇

CSM_RH Paper 17

Mesoscopic Lag-Energy Power Gain and the First Direct PESC Theorem Candidate

Project: CSM_RH
Paper: 17
Version: v0.1
Date: 2026-09-05
Parent state: CSM_RH v1.7 / Paper 16
Campaign: 16 — DIRECT_PESC_THEOREM_GENERATION
Status: direct theorem-candidate generation / deterministic bridge certification; 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 16 changes research mode.

The canonical target no longer moves:

F-RH-010
PESC

The goal is to generate one concrete lemma which:

  1. is not PESC merely renamed;
  2. does not assume a fixed zero strip;
  3. contains a genuine fixed-power source;
  4. has a certified deterministic bridge back to PESC / PODEE.

The first candidate surviving those filters is a mesoscopic lag-energy estimate.

No live GLM-5.3-Flash run is claimed.

1. Work first with the von Mangoldt error

Define

an=Λ(n)−1.a_n = \Lambda(n)-1.

Define

A(x)=∑n≤xan=ψ(x)−xA(x) = \sum_{n\le x}a_n = \psi(x)-x

up to the harmless integer convention.

Paper 10 already established that for every fixed

0<κ<12,0<\kappa<\frac12,

prime-power stripping transfers a dyadic fixed-power mean-square estimate between ψ\psi and ϑ\vartheta at exponent level.

Therefore a first fixed-strip theorem may be proved on the Λ\Lambda / ψ\psi side.

2. Mesoscopic lag energy

For an integer lag

1≤H<N,1\le H<N,

define

SΛ(N,H)=∑0≤x<2N−H∣A(x+H)−A(x)∣2.\boxed{ \mathcal S_\Lambda(N,H) = \sum_{0\le x<2N-H} \left| A(x+H)-A(x) \right|^2. }

Since

A(x+H)−A(x)=∑x<n≤x+H[Λ(n)−1],A(x+H)-A(x) = \sum_{x<n\le x+H} [ \Lambda(n)-1 ],

this is a global discrete Selberg-type increment energy at one fixed additive scale.

3. Deterministic residue-chain inequality

The key bridge does not use any arithmetic.

Let

X≥H≥1X\ge H\ge1

be integers.

Let

u0,u1,…,uXu_0,u_1,\ldots,u_X

be any complex sequence.

For

0≤n≤X−H,0\le n\le X-H,

define

dH(n)=un+H−un.d_H(n) = u_{n+H}-u_n.

Let

M=⌈XH⌉.M = \left\lceil \frac XH \right\rceil.

Then:

Theorem 3.1 — Residue-Chain Energy Inequality

∑n=0X∣un∣2≤2(M+1)∑r=0H−1∣ur∣2+2M(M+1)∑n=0X−H∣dH(n)∣2.\boxed{ \sum_{n=0}^{X}|u_n|^2 \le 2(M+1) \sum_{r=0}^{H-1}|u_r|^2 + 2M(M+1) \sum_{n=0}^{X-H}|d_H(n)|^2. }

Proof

Fix one residue

0≤r<H.0\le r<H.

Write every index in that residue class as

r+kH.r+kH.

For each admissible kk,

ur+kH=ur+∑j=0k−1dH(r+jH).u_{r+kH} = u_r + \sum_{j=0}^{k-1} d_H(r+jH).

Hence

∣ur+kH∣2≤2∣ur∣2+2k∑j=0k−1∣dH(r+jH)∣2.|u_{r+kH}|^2 \le 2|u_r|^2 + 2k \sum_{j=0}^{k-1} |d_H(r+jH)|^2.

There are at most

M+1M+1

indices in each residue class, and

k≤M.k\le M.

Summing first in kk and then in rr gives the stated inequality.

□\square

4. Arithmetic specialization

Apply Theorem 3.1 with

un=A(n)u_n=A(n)

and

X=2N.X=2N.

Chebyshev bounds give

A(r)=O(r),A(r)=O(r),

hence

∑r<H∣A(r)∣2≪H3.\sum_{r<H}|A(r)|^2 \ll H^3.

Therefore:

Corollary 4.1 — Lag-to-Global Energy Bridge

∑n≤2N∣A(n)∣2≪(NH)2SΛ(N,H)+NH2.\boxed{ \sum_{n\le2N}|A(n)|^2 \ll \left( \frac NH \right)^2 \mathcal S_\Lambda(N,H) + NH^2. }

All logarithmic refinements of the initial term are irrelevant at fixed exponent level.

This is the certified bridge for Campaign 16.

5. Candidate theorem: Mesoscopic Lag-Energy Power Gain

Fix constants

0<α<230<\alpha<\frac23

and

δ>0.\delta>0.

Set

H=Nα+o(1).H=N^{\alpha+o(1)}.

Define:

MLEPG (α,δ)(\alpha,\delta)

SΛ(N,H)≪NH(log⁡N)O(1)+NH2N−δ+o(1).\boxed{ \mathcal S_\Lambda(N,H) \ll NH(\log N)^{O(1)} + NH^2N^{-\delta+o(1)}. }

The first term is the expected diagonal / prime-variance scale.

The second term permits a much larger remainder than the conjectural main scale when

δ<α.\delta<\alpha.

Thus MLEPG does not require a full asymptotic formula for the Selberg integral.

It asks only for a fixed-power improvement over the completely unresolved scale

NH2.NH^2.

6. Fixed-power consequence

Insert MLEPG into Corollary 4.1.

We obtain

∑n≤2N∣A(n)∣2≪N2H2[NH(log⁡N)O(1)+NH2N−δ+o(1)]+NH2≪N3−α+o(1)+N3−δ+o(1)+N1+2α+o(1).\begin{aligned} \sum_{n\le2N}|A(n)|^2 &\ll \frac{N^2}{H^2} \left[ NH(\log N)^{O(1)} + NH^2N^{-\delta+o(1)} \right] + NH^2 \\ &\ll N^{3-\alpha+o(1)} + N^{3-\delta+o(1)} + N^{1+2\alpha+o(1)}. \end{aligned}

For

0<α<23,0<\alpha<\frac23,

and sufficiently small fixed

κ<min⁡{α,δ},\kappa < \min \{ \alpha, \delta \},

the initial residue term is harmless.

Therefore:

Theorem 6.1 — MLEPG Implies a Fixed PNT Mean-Square Power

If MLEPG (α,δ)(\alpha,\delta) holds for fixed

0<α<23,δ>0,0<\alpha<\frac23, \qquad \delta>0,

then

∑n≤2N∣ψ(n)−n∣2≪N3−κ+o(1)\boxed{ \sum_{n\le2N} | \psi(n)-n |^2 \ll N^{3-\kappa+o(1)} }

for some fixed

κ>0.\kappa>0.

One may take any sufficiently small

κ<min⁡{α,δ,2−2α}.\kappa < \min \{ \alpha, \delta, 2-2\alpha \}.

For

α<23,\alpha<\frac23,

the third quantity is not the active restriction when κ\kappa is chosen small.

By the established CSM_RH bridges, this gives PODEE / PESC at a fixed exponent below the prime-power barrier, and hence a fixed zeta zero strip.

No such MLEPG theorem is proved here.

7. Why MLEPG is not PESC merely renamed

PESC controls the absolute cumulative error through

∑nwN(n)[Λ(n)−1][ψ(n−1)−(n−1)].\sum_n w_N(n) [ \Lambda(n)-1 ] [ \psi(n-1)-(n-1) ].

MLEPG controls only one additive difference scale:

A(x+H)−A(x).A(x+H)-A(x).

A sequence may have a large slowly varying low-frequency component while having much smaller increments at one scale.

Thus the two statements are not identical by definition.

The residue-chain theorem supplies the nontrivial deterministic bridge.

This is exactly the type of candidate Campaign 16 permits.

8. Correlation expansion of the lag energy

Let

an=Λ(n)−1.a_n=\Lambda(n)-1.

Ignoring only endpoint truncation notation, the standard expansion is

SΛ(N,H)=∑∣h∣<H(H−∣h∣)∑nanan+h+boundary terms.\boxed{ \mathcal S_\Lambda(N,H) = \sum_{|h|<H} (H-|h|) \sum_n a_n a_{n+h} + \text{boundary terms}. }

The kernel

H−∣h∣H-|h|

is the discrete Fejér / triangular kernel.

Thus MLEPG is equivalent to obtaining a fixed-power saving in one signed triangular average of centered prime-pair correlations, together with standard diagonal and boundary estimates.

9. Hardy–Littlewood decomposition

For nonzero shifts define schematically

∑X<n≤2XΛ(n)Λ(n+h)=S(h)X+EX(h).\sum_{X<n\le2X} \Lambda(n)\Lambda(n+h) = \mathfrak S(h)X + E_X(h).

The triangular sum splits into:

  1. the diagonal;
  2. the singular-series deterministic average;
  3. the weighted correlation-error aggregate.

The first two are not the new Campaign 16 obstruction.

The prime-pair singular series has the precise Cesàro expansion

∑h≤H(H−h)S(h)=12H2−12Hlog⁡H+O(H)+ES(H),\sum_{h\le H} (H-h) \mathfrak S(h) = \frac12H^2 - \frac12H\log H + O(H) + E_{\mathfrak S}(H),

with a much sharper known unconditional estimate for the error than the H2H^2 scale.

Thus the fixed-power question localizes to the actual prime-pair correlation errors.

10. Current averaged Hardy–Littlewood input

Matomäki, Radziwiłł, and Tao prove that for

H≥X8/33+ε,H \ge X^{8/33+\varepsilon},

the Hardy–Littlewood prime-pair asymptotic holds for all but a logarithmically small proportion of shifts, with individual good-shift error

O(Xlog⁡−AX)O \left( X\log^{-A}X \right)

for every fixed A>0A>0.

The exceptional set also has arbitrary logarithmic density saving.

After crude control of exceptional shifts, this yields an L1L^1 error budget

∑∣h∣≤H∣EX(h)∣≪XHlog⁡−AX\boxed{ \sum_{|h|\le H} |E_X(h)| \ll XH\log^{-A}X }

after changing AA.

11. The triangular-kernel precision loss

MLEPG does not sum EX(h)E_X(h) with unit weights.

It carries the triangular weight

H−∣h∣.H-|h|.

Direct absolute transfer from the current averaged Hardy–Littlewood theorem gives only

∑∣h∣<H(H−∣h∣)∣EX(h)∣≪XH2log⁡−AX.\boxed{ \sum_{|h|<H} (H-|h|) |E_X(h)| \ll XH^2 \log^{-A}X. }

This is the exact current precision floor.

After the residue-chain bridge,

(NH)2⋅NH2log⁡−AN=N3log⁡−AN.\left( \frac NH \right)^2 \cdot NH^2 \log^{-A}N = N^3 \log^{-A}N.

Hence current average-Hardy–Littlewood technology produces only subpower global energy through this route.

This matches the Campaign 15 classification.

12. The new fixed-power request is weaker than shiftwise power saving

A much stronger theorem would be:

EX(h)≪X1−δE_X(h) \ll X^{1-\delta}

for every or almost every shift.

Campaign 16 does not ask for that.

It asks only for cancellation in the one aggregate

T(X,H)=∑∣h∣<H(H−∣h∣)EX(h).\boxed{ \mathcal T(X,H) = \sum_{|h|<H} (H-|h|) E_X(h). }

A sufficient new estimate is

∣T(X,H)∣≪XH2X−δ+o(1).\boxed{ |\mathcal T(X,H)| \ll XH^2X^{-\delta+o(1)}. }

Individual shifts may remain much larger.

Therefore the candidate theorem is genuinely a signed-average power theorem, not a uniform Hardy–Littlewood power theorem.

13. Candidate interface: Triangular Signed Hardy–Littlewood Power Gain

Create the proof-interface name:

TSHLPG
TRIANGULAR_SIGNED_HARDY_LITTLEWOOD_POWER_GAIN

For

H=Xα+o(1),H=X^{\alpha+o(1)},

TSHLPG (α,δ)(\alpha,\delta) is the assertion that the endpoint-compatible triangular aggregate of prime-pair errors satisfies

∣∑∣h∣<H(H−∣h∣)EX(h)∣≪XH2X−δ+o(1).\boxed{ \left| \sum_{|h|<H} (H-|h|) E_X(h) \right| \ll XH^2X^{-\delta+o(1)}. }

After standard diagonal, singular-series, and dyadic assembly, TSHLPG implies MLEPG.

TSHLPG is a proof interface.

MLEPG is the cleaner theorem candidate because it avoids dependence on a particular Hardy–Littlewood centering convention.

14. Natural first scale

The strongest classical long-shift theorem for ΛΛ\Lambda\Lambda begins at

α=833+ε.\alpha = \frac8{33} + \varepsilon.

Therefore the first natural candidate scale is

H=N8/33+ε.\boxed{ H=N^{8/33+\varepsilon}. }

At this scale, any fixed

δ>0\delta>0

in MLEPG or TSHLPG would produce a fixed global power.

If

δ<833,\delta<\frac8{33},

the resulting exponent gain is essentially

κ<δ.\kappa<\delta.

If

δ≥833,\delta\ge\frac8{33},

the lag length itself becomes the bottleneck and one obtains essentially

κ<833.\kappa<\frac8{33}.

15. Candidate strength classification

MLEPG is breakthrough-strength.

It is not a low-strength theorem.

But it passes the Campaign 16 design filters:

NOT merely a definition of PESC
  PASS

fixed exponent enters explicitly
  PASS

does not assume a fixed zero strip
  PASS

has a certified bridge to PESC
  PASS

current theorem comparison available
  PASS

current best gives only log-power analogue
  PASS

Status:

SURVIVOR

16. Rejected candidate: shiftwise Hardy–Littlewood power saving

Candidate:

EX(h)≪X1−δE_X(h) \ll X^{1-\delta}

for almost all shifts.

Status:

REJECTED AS UNNECESSARILY STRONG

Reason:

TSHLPG needs only one signed triangular aggregate to save a power.

Demanding a power for individual shifts adds proof obligations not required by PESC.

17. Rejected candidate: Fejér-frequency power theorem as a new target

By Fourier expansion, the lag energy can be written with a Fejér kernel against a prime exponential-sum energy.

This is useful analytically.

But defining the Fourier form as another canonical frontier would only re-represent MLEPG.

Status:

REJECTED AS REPRESENTATION ONLY

It may be used inside the proof.

18. Rejected candidate: another positive prime-pair variance

Replacing the signed triangular correlation error by

∑h∣EX(h)∣2\sum_h |E_X(h)|^2

creates a stronger positive gate.

Campaign 15 already classified such variance gates as stronger than the minimal signed target.

Status:

REJECTED AS STRONGER POSITIVE SURROGATE

19. New canonical theorem candidate

Create:

F-RH-016
MESOSCOPIC_LAG_ENERGY_POWER_GAIN
abbrev:
  MLEPG
status:
  OPEN
type:
  DIRECT THEOREM CANDIDATE

This is the first new frontier created after the representation-generation stop rule.

It is permitted because it is a concrete sufficient lemma with a proved deterministic bridge, not an equivalent restatement declared canonical for its own sake.

PESC remains the root target.

20. New obstruction: triangular kernel amplification

Create:

O-RH-038
TRIANGULAR_KERNEL_LOG_TO_POWER_GAP
status:
  CERTIFIED

Statement:

Almost-all per-shift Hardy–Littlewood errors of size Xlog⁡−AXX\log^{-A}X become XH2log⁡−AXXH^2\log^{-A}X after direct triangular weighting. The residue-chain bridge then returns only N3log⁡−ANN^3\log^{-A}N. A fixed power must therefore arise from signed cancellation across shifts or from a stronger local theorem.

21. New survivor

Create:

S-RH-025
TRIANGULAR_SIGNED_PRIME_PAIR_ERROR_CANCELLATION
status:
  OPEN

This is the principal proof mechanism for MLEPG.

It asks for power cancellation across shifts before absolute values are taken.

22. Campaign 16 verdict

Campaign 16 generated several candidates.

The result is:

SHIFTWISE POWER HL
  too strong

FOURIER / FEJER REFORMULATION
  representation only

POSITIVE PRIME-PAIR VARIANCE
  stronger surrogate

MESOSCOPIC LAG-ENERGY POWER GAIN
  survives

TRIANGULAR SIGNED ERROR CANCELLATION
  first concrete attack mechanism

No fixed-power theorem is proved.

23. Campaign 17

The next campaign is:

CSM_RH Campaign 17
TRIANGULAR_SIGNED_ERROR_POWER_ATTACK

The canonical root target remains PESC.

The working theorem candidate is MLEPG.

The proof interface is TSHLPG.

24. Campaign 17 attack directions

T1 — Circle-method cancellation across shifts

Do not estimate each shift independently.

Insert the triangular kernel before major/minor arc separation and test whether the Fejér localization creates a fixed-power minor-arc gain unavailable shiftwise.

T2 — Correlated exceptional-set cancellation

Current MRT exceptional shifts are controlled absolutely.

Test whether their signed triangular contribution has stronger cancellation than their cardinality bound.

T3 — Major-arc residual cancellation

The singular-series main term is already understood.

Audit whether the residual major-arc errors possess cancellation across hh after the triangular kernel is inserted.

T4 — Multiplicative-frequency averaging

MRT convert additive-frequency local energy into Dirichlet-polynomial mean values.

Test whether keeping the complete triangular shift average before Cauchy–Schwarz exposes an additional mean-value gain.

T5 — Hybrid short-interval theorem

Seek MLEPG directly as a Selberg-increment energy theorem without resolving individual pair correlations.

25. Campaign 17 rejection filters

Reject a candidate if:

R1. It takes absolute values in hh before the proposed new cancellation.

R2. It simply assumes power-saving Hardy–Littlewood for almost every shift.

R3. It uses PODEE/PESC as an input to estimate the lag energy.

R4. It replaces N−δN^{-\delta} by log⁡−AN\log^{-A}N.

R5. It counts the singular-series Cesàro expansion as the missing theorem.

R6. It hides a fixed zero-free half-plane inside an exponential-sum input.

26. External calibration

Current literature gives the correct location of the new gap.

  1. Matomäki–Radziwiłł–Tao prove averaged Hardy–Littlewood for Λ(n)Λ(n+h)\Lambda(n)\Lambda(n+h) in shift windows of length at least X8/33+εX^{8/33+\varepsilon}, with arbitrary logarithmic savings over the trivial error for almost all shifts.

  2. Their proof obtains only logarithmic savings in the relevant prime-correlation minor-arc estimates; power savings are available or expected in some divisor-function cases but not supplied for the von Mangoldt pair correlation.

  3. The prime-pair singular-series Cesàro mean has the expansion

12H2−12Hlog⁡H+O(H)+E(H),\frac12H^2 - \frac12H\log H + O(H) + E(H),

and Vaughan's unconditional estimate places E(H)E(H) far below the H2H^2 scale.

  1. Selberg-integral theory identifies the same triangular correlation kernel as the natural variance of primes in short intervals.

Thus the missing fixed power is localized to the actual prime-correlation residual, not to deterministic singular-series averaging.

27. State transition

The canonical transition is:

CSM_RH v1.7
  ->
CSM_RH v1.8

with:

Campaign 16
  CLOSED_AS_FIRST_DIRECT_THEOREM_GENERATION_ROUND

F-RH-010
  PESC
  REMAINS ROOT TARGET / OPEN

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

O-RH-038
  TRIANGULAR_KERNEL_LOG_TO_POWER_GAP
  CREATED / CERTIFIED

S-RH-025
  TRIANGULAR_SIGNED_PRIME_PAIR_ERROR_CANCELLATION
  CREATED / OPEN

Campaign 17
  TRIANGULAR_SIGNED_ERROR_POWER_ATTACK
  READY

28. Final status

RH = OPEN

PESC = OPEN / ROOT TARGET

REPRESENTATION GENERATION = STOPPED

FIRST DIRECT NEW LEMMA CANDIDATE = MLEPG

CURRENT AVERAGE HL ANALOGUE = LOG-POWER ONLY

FIXED-POWER SOURCE NEEDED = SIGNED TRIANGULAR ERROR CANCELLATION

MLEPG PROVED = NO

NEXT CAMPAIGN = 17

The concrete new theorem candidate is:

SΛ(N,Nα)≪N1+α+o(1)+N1+2α−δ+o(1)\boxed{ \mathcal S_\Lambda(N,N^\alpha) \ll N^{1+\alpha+o(1)} + N^{1+2\alpha-\delta+o(1)} }

for some fixed

0<α<23,δ>0.0<\alpha<\frac23, \qquad \delta>0.

A natural first scale is

α=833+ε.\alpha = \frac8{33} + \varepsilon.

The deterministic residue-chain theorem then converts any such fixed δ\delta into a fixed global prime-number-theorem mean-square power and hence back into the canonical PESC branch.