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CSM_RH Paper 40 — Averaged Type-II Large Values, the Orthogonality Floor, and the Single-Resonance Obstruction

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CSM_RH Paper 40

Averaged Type-II Large Values, the Orthogonality Floor, and the Single-Resonance Obstruction

Project: CSM_RH
Paper: 40
Version: v0.1
Date: 2026-09-07
Parent state: CSM_RH v1.30 / Paper 39
Campaign: 39 — AVERAGED_TYPEII_POLYNOMIAL_W_ATTACK
Status: averaged large-value replacement audit; 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

Paper 39 proved that the current pointwise route to polynomial WW is fixed-strip locked.

Campaign 39 asks whether modern Dirichlet-polynomial large-value estimates can replace the pointwise hypothesis in the Type-II amplifier.

The result is negative at the generic bounded-coefficient level.

The Guth–Maynard theorem improves the nontrivial large-value terms, but retains the classical orthogonality term

N2V−2.N^2V^{-2}.

After normalization to the 1-line, this becomes a P−2P^{-2} level-set floor.

That floor yields at best logarithmic control after layer-cake integration and cannot generate a vanishing fixed power.

More decisively, the divisor-bounded coefficient class of Lemma 3.5 contains explicit phase-aligned examples with a single constant-size resonant frequency. Such one-frequency resonance already forces an O(1)O(1) contribution to the product mean square, whereas polynomial WW requires that mean square to tend to zero like a fixed power.

Therefore generic large-value technology cannot replace the arithmetic pointwise/nonresonance input.

A genuinely arithmetic simultaneous-resonance exclusion for the actual Heath–Brown factors remains open.

No live GLM-5.3-Flash run is claimed.

1. Type-II integrated target

Recall

A(s)=∑m∼Ma(m)m−s,A(s) = \sum_{m\sim M} a(m)m^{-s},

and

B(s)=∑N/3<n≤3Nb(n)n−s,B(s) = \sum_{N/3<n\le3N} b(n)n^{-s},

where

MN≍X.MN\asymp X.

The Type-II proof reduces to estimates such as

∫W≤∣t∣≤T∣A(1+it)B(1+it)∣2 dt≪THXlog⁡O(1)XW3/10\boxed{ \int_{W\le|t|\le T} |A(1+it)B(1+it)|^2 \,dt \ll \frac{TH}{X} \frac{\log^{O(1)}X}{W^{3/10}} }

or the stronger sufficient estimate

∫W≤∣t∣≤X2/3∣A(1+it)B(1+it)∣2 dt≪log⁡O(1)XW3/10.\boxed{ \int_{W\le|t|\le X^{2/3}} |A(1+it)B(1+it)|^2 \,dt \ll \frac{\log^{O(1)}X}{W^{3/10}}. }

When

W=Xw,w>0,W=X^w, \qquad w>0,

the right side is

X−3w/10+o(1).X^{-3w/10+o(1)}.

Thus the desired integrated product must itself vanish at a fixed polynomial rate.

2. Product polynomial normalization

Write

F(s)=A(s)B(s).F(s)=A(s)B(s).

After partitioning its support into O(1)O(1) dyadic blocks, one obtains a length- ≍X\asymp X polynomial

F(1+it)=1XDF(t)⋅Xo(1),F(1+it) = \frac1X D_F(t) \cdot X^{o(1)},

where

DF(t)=∑ℓ≍Xcℓℓ−itD_F(t) = \sum_{\ell\asymp X} c_\ell \ell^{-it}

and

∣cℓ∣≤d2(ℓ)O(1)=Xo(1).|c_\ell| \le d_2(\ell)^{O(1)} = X^{o(1)}.

After division by an Xo(1)X^{o(1)} factor, Guth–Maynard's bounded-coefficient large-value theorem applies at exponent resolution.

3. Guth–Maynard large-value theorem

For a length- LL Dirichlet polynomial

D(t)=∑L<n≤2Lbnnit,∣bn∣≤1,D(t) = \sum_{L<n\le2L} b_n n^{it}, \qquad |b_n|\le1,

suppose that at RR one-separated points

∣D(tr)∣≥V.|D(t_r)|\ge V.

Guth–Maynard prove

R≤To(1)(L2V−2+L18/5V−4+TL12/5V−4).\boxed{ R \le T^{o(1)} \left( L^2V^{-2} + L^{18/5}V^{-4} + TL^{12/5}V^{-4} \right). }

The first term is the classical orthogonality term.

The new theorem improves the large-value geometry in the critical intermediate-amplitude region through the remaining terms.

4. Normalized product level sets

For the product polynomial, set

L=X.L=X.

A normalized threshold

∣F(1+it)∣≥P|F(1+it)|\ge P

corresponds, up to Xo(1)X^{o(1)}, to

V=XP.V=XP.

Therefore the Guth–Maynard estimate becomes

Theorem 4.1 — Product Large-Value Ledger

RF(P)≪Xo(1)(P−2+X−2/5P−4+TX−8/5P−4).\boxed{ R_F(P) \ll X^{o(1)} \left( P^{-2} + X^{-2/5}P^{-4} + TX^{-8/5}P^{-4} \right). }

For the Type-II range

T≤X2/3,T\le X^{2/3},

the last term is at most

X−14/15P−4.X^{-14/15}P^{-4}.

Hence the persistent first term is

RF(P)≪Xo(1)P−2+smaller large-value terms.\boxed{ R_F(P) \ll X^{o(1)}P^{-2} +\text{smaller large-value terms}. }

Create:

B-RH-015
GUTH_MAYNARD_PRODUCT_LEVEL_SET_NORMALIZATION
status:
  CERTIFIED

5. Orthogonality floor

The term

P−2P^{-2}

is scale invariant for the L2L^2 layer cake.

Indeed, schematically,

∫∣F(1+it)∣2dt=2∫0∞Pmeas⁡{∣F∣>P} dP.\int |F(1+it)|^2dt = 2 \int_0^\infty P \operatorname{meas} \{ |F|>P \} \,dP.

If the available general level-set estimate has the form

meas⁡{∣F∣>P}≪Xo(1)P−2,\operatorname{meas} \{ |F|>P \} \ll X^{o(1)}P^{-2},

then its contribution to the layer cake is

Xo(1)∫dPP.\boxed{ X^{o(1)} \int \frac{dP}{P}. }

This is logarithmic / subpower.

It is not

X−cwX^{-cw}

for any fixed c,w>0c,w>0.

Create:

O-RH-096
GENERIC_LARGE_VALUE_ORTHOGONALITY_FLOOR
status:
  CERTIFIED

Statement:

A generic Dirichlet-polynomial large-value theorem whose leading normalized term is P−2P^{-2} cannot by itself imply the polynomially vanishing Type-II product mean square required for polynomial WW.

6. Sharpness of the first large-value term

Guth–Maynard discuss general bounded-coefficient constructions with

≫L2−2σ\gg L^{2-2\sigma}

one-separated points at amplitude

Lσ.L^\sigma.

This is precisely the scale represented by

L2V−2.L^2V^{-2}.

Thus the orthogonality term is not merely an artifact of their proof.

It reflects a real generic bounded-coefficient phenomenon.

Therefore replacing the first term by a fixed-power improvement is impossible without exploiting additional arithmetic structure.

7. Explicit one-frequency resonance

There is an even simpler obstruction tailored directly to Lemma 3.5.

Fix

t0t_0

in the relevant integration interval and choose

a(m)=mit0,b(n)=nit0.\boxed{ a(m)=m^{it_0}, \qquad b(n)=n^{it_0}. }

These coefficients satisfy

∣a(m)∣=∣b(n)∣=1.|a(m)|=|b(n)|=1.

Then

A(1+it)=∑m∼Mm−i(t−t0)m,A(1+it) = \sum_{m\sim M} \frac{ m^{-i(t-t_0)} }{m},

and

B(1+it)=∑N/3<n≤3Nn−i(t−t0)n.B(1+it) = \sum_{N/3<n\le3N} \frac{ n^{-i(t-t_0)} }{n}.

At

t=t0,t=t_0,

both are positive harmonic sums of constant size.

8. Fixed-width resonance interval

Let

s=t−t0.s=t-t_0.

For m∈[M,2M]m\in[M,2M],

m−is=M−is(mM)−is.m^{-is} = M^{-is} \left( \frac mM \right)^{-is}.

If

∣s∣≤π6log⁡2,|s| \le \frac{\pi}{6\log2},

then

cos⁡(slog⁡mM)≥32.\cos \left( s\log\frac mM \right) \ge \frac{\sqrt3}{2}.

Hence

∣A(1+i(t0+s))∣≫1.\boxed{ |A(1+i(t_0+s))| \gg1. }

For n∈[N/3,3N]n\in[N/3,3N], write

n−is=N−is(n/N)−is.n^{-is} = N^{-is} (n/N)^{-is}.

If

∣s∣≤π6log⁡3,|s| \le \frac{\pi}{6\log3},

then

∣B(1+i(t0+s))∣≫1.\boxed{ |B(1+i(t_0+s))| \gg1. }

Thus on one fixed-width interval around t0t_0,

∣A(1+it)B(1+it)∣≫1.\boxed{ |A(1+it)B(1+it)| \gg1. }

9. Resonance contradicts a generic polynomial- WW integrated theorem

Choose

t0t_0

so that the fixed-width interval of Section 8 lies inside

W≤∣t∣≤X2/3.W\le|t|\le X^{2/3}.

This is possible whenever

1≪W≪X2/3.1\ll W\ll X^{2/3}.

Then

Theorem 9.1 — Single-Resonance Lower Bound

For the legal coefficients of Section 7,

∫W≤∣t∣≤X2/3∣A(1+it)B(1+it)∣2dt≫1.\boxed{ \int_{W\le|t|\le X^{2/3}} |A(1+it)B(1+it)|^2dt \gg1. }

But a polynomial- WW Type-II theorem would require

∫∣AB∣2≪W−3/10log⁡O(1)X=X−3w/10+o(1)→0.\boxed{ \int |AB|^2 \ll W^{-3/10}\log^{O(1)}X = X^{-3w/10+o(1)} \to0. }

Contradiction.

Create:

O-RH-097
SINGLE_RESONANT_FREQUENCY_OBSTRUCTS_GENERIC_POLYNOMIAL_W_MEAN_SQUARE
status:
  CERTIFIED

10. Consequence for generic averaged replacement

Theorems 4.1 and 9.1 show that no statement of the form

divisor-bounded coefficients
+
generic Dirichlet-polynomial large-value theorem
->
polynomial-W integrated Type-II saving

can hold.

The pointwise smallness in Lemma 3.5 is not merely a convenient sufficient condition.

It excludes resonances which are legal in the ambient coefficient class.

11. Weak pointwise logarithmic control is still insufficient

Suppose one supplements large-value theory by an arbitrary fixed logarithmic pointwise estimate

∣A(1+it)∣, ∣B(1+it)∣≤log⁡−KX\boxed{ |A(1+it)|,\, |B(1+it)| \le \log^{-K}X }

for fixed KK.

A single peak at the maximal allowed scale contributes only a negative power of log⁡X\log X to the product mean square.

But

log⁡−CX=X−o(1)≫X−cw.\log^{-C}X = X^{-o(1)} \gg X^{-cw}.

Thus a finite or subpower number of subpower-sized resonant peaks is still incompatible with the required fixed-power integrated bound.

A fixed-power conclusion requires polynomial control of the total resonance mass, not merely arbitrary fixed log-power peak suppression.

12. Large-value count quantization

Large-value theorems count one-separated points.

Such a count is an integer.

A bound which says

R≤Xo(1)R\le X^{o(1)}

does not distinguish between:

no resonance;
one resonance;
a subpower number of resonances.

But the polynomially vanishing Type-II target is sensitive even to one sufficiently broad subpower-amplitude resonance.

Create:

O-RH-098
LARGE_VALUE_COUNT_QUANTIZATION_BARRIER
status:
  CERTIFIED AS FIXED-POWER TRANSFERENCE AUDIT

This is another formulation of why improving the number of large values is not enough when the target integral itself must tend to zero polynomially.

13. Applying Guth–Maynard separately to the two factors

One might attempt to control simultaneous large values through

min⁡{RA(U),RB(V)}.\min \{ R_A(U),R_B(V) \}.

For the normalized factor AA, the first Guth–Maynard term gives

RA(U)≪Xo(1)U−2+⋯\boxed{ R_A(U) \ll X^{o(1)}U^{-2} +\cdots }

and likewise

RB(V)≪Xo(1)V−2+⋯ .\boxed{ R_B(V) \ll X^{o(1)}V^{-2} +\cdots. }

The same orthogonality floor remains.

Moreover the phase-aligned example makes the two resonances occur at the same t0t_0.

Thus independent large-value estimates do not solve simultaneous resonance.

14. Applying the theorem to the product

Treating

F=ABF=AB

as one Dirichlet polynomial improves bookkeeping but not the exponent class.

The first term becomes

P−2P^{-2}

and the one-frequency example still survives.

Therefore neither:

factorwise large-value control

nor

product large-value control

gives the required polynomial mean-square saving generically.

15. Why Guth–Maynard still matters

This negative conclusion does not diminish the Guth–Maynard theorem.

Their improvement in the critical regime near amplitude

V=L3/4V=L^{3/4}

is strong enough to improve zero-density estimates and prime short-interval ranges.

It is simply a different quantitative task.

The current CSM_RH Type-II target asks for an integrated quantity which tends to zero by a fixed power.

That requires arithmetic cancellation beyond the generic bounded-coefficient orthogonality floor.

16. Actual Heath–Brown factors are not generic

The obstruction above applies to the ambient divisor-bounded coefficient class.

The Type-II factors arising from the actual decomposition of Λ\Lambda and μ\mu have additional structure.

Heath–Brown's identity writes the relevant components as convolutions of factors which are:

1 on a dyadic block;
log n on a dyadic block;
mu(n) on a short dyadic block.

In particular, Möbius-bearing factors occur on scales at most a small fixed power of XX in the chosen identity.

Therefore the generic resonance example does not prove that the actual arithmetic factors possess such resonances.

It only proves that general large-value theory cannot rule them out.

17. New surviving arithmetic question

The remaining route is:

prove that the actual Möbius-bearing Heath–Brown factors and their complementary factors cannot be simultaneously resonant on a set with enough spectral mass to defeat polynomial WW.

This is stronger than generic large-value theory and weaker in form than demanding pointwise fixed-power Mertens for the full Möbius polynomial.

Whether it is genuinely lower-strength is open.

No theorem is claimed here.

18. Campaign 39 track audit

AV1 — large-value replacement of pointwise input

status:
  FAILS GENERICALLY

reason:
  P^(-2) orthogonality floor

AV2 — Guth–Maynard large-value technology

status:
  IMPROVES CRITICAL LARGE-VALUE TERMS

does not remove:
  orthogonality floor

AV3 — low-frequency excision

status:
  DOES NOT FIX GENERIC RESONANCE

resonance can be placed at any legal t0 > W

AV4 — product large-value geometry

status:
  GENERIC PRODUCT THEOREM STILL HAS P^(-2) FLOOR

single simultaneous resonance:
  legal

AV5 — polynomial-W integrated bridge

status:
  NOT OBTAINED GENERICALLY

requires:
  arithmetic resonance exclusion

19. Campaign 39 verdict

No polynomial- WW integrated theorem is proved.

The campaign closes the purely generic averaged escape:

pointwise fixed-power input:
  too strong / fixed-strip locked

generic averaged large values:
  too weak / resonance floor

remaining possibility:
  arithmetic simultaneous-resonance exclusion

The component bridge B-RH-014 remains valid.

20. New certified package

Create:

B-RH-015
GUTH_MAYNARD_PRODUCT_LEVEL_SET_NORMALIZATION
CERTIFIED

O-RH-096
GENERIC_LARGE_VALUE_ORTHOGONALITY_FLOOR
CERTIFIED

O-RH-097
SINGLE_RESONANT_FREQUENCY_OBSTRUCTS_GENERIC_POLYNOMIAL_W_MEAN_SQUARE
CERTIFIED

O-RH-098
LARGE_VALUE_COUNT_QUANTIZATION_BARRIER
CERTIFIED

No new canonical frontier is created.

21. Canonical root status

F-RH-010
PESC
OPEN / ROOT TARGET

F-RH-016
MLEPG
OPEN / DIRECT THEOREM CANDIDATE

B-RH-014
POLYNOMIAL_W_TYPEII_FIXED_POWER_AMPLIFIER
CERTIFIED COMPONENT BRIDGE

generic large-value replacement:
  CLOSED

22. Campaign 40

The next campaign is:

CSM_RH Campaign 40
HEATH_BROWN_ARITHMETIC_RESONANCE_EXCLUSION

This is a genuinely arithmetic campaign.

No generic bounded-coefficient theorem is admissible as the main step.

23. Campaign 40 tracks

HB1 — Möbius carrier audit

For every Type-II component arising from the chosen Heath–Brown identity for Λ−Λ♯\Lambda-\Lambda^\sharp, identify which grouped factor contains a Möbius-bearing short variable.

Determine whether at least one factor always retains explicit Möbius structure after grouping.

HB2 — translated-frequency Möbius large values

The large- TT major-arc application twists the factors by niTn^{iT}.

Study the actual frequency window seen by each Möbius factor and whether its short length relative to ∣T∣|T| permits fixed-power averaged cancellation without a global Mertens theorem.

HB3 — simultaneous resonance exclusion

Define the set where both Type-II factors are larger than polynomial thresholds.

Seek a bound on the spectral measure of the simultaneous-large set stronger than the generic P−2P^{-2} floor.

HB4 — short Möbius factor versus complementary factor

Exploit that Möbius factors in the fixed Heath–Brown identity have length at most a small power of XX.

Test whether large values of such a factor force a complementary factor into a regime controlled by mean-value or large-value estimates.

HB5 — integrated polynomial- WW admission

The only accepted output is the exact integrated estimate needed by Lemma 3.5 with

W=XwW=X^w

for some fixed w>0w>0.

If obtained, invoke B-RH-014.

24. Campaign 40 rejection filters

Reject a candidate if:

R1. It treats the actual factors as arbitrary divisor-bounded coefficients.

R2. It assumes pointwise fixed-power Mertens.

R3. It assumes a fixed zero-free strip.

R4. It only reduces the number of resonant frequencies to Xo(1)X^{o(1)}.

R5. It obtains only logarithmic total resonance mass.

R6. It ignores an allowed Heath–Brown component with no certified Möbius carrier.

25. External calibration

The current audit uses:

  1. Matomäki–Radziwiłł–Shao–Tao–Teräväinen, Higher uniformity of arithmetic functions in short intervals II, Lemmas 3.4 and 3.5. The Type-II product mean square is proved using a Baker–Harman–Pintz parallelogram lemma plus pointwise smallness of one factor.

  2. Guth–Maynard, New large value estimates for Dirichlet polynomials, Theorem 1.1:

    R≤To(1)(N2V−2+N18/5V−4+TN12/5V−4).R \le T^{o(1)} \left( N^2V^{-2} + N^{18/5}V^{-4} + TN^{12/5}V^{-4} \right).

    The first term is the generic orthogonality floor.

  3. The higher-uniformity paper's Heath–Brown decomposition expresses Λ\Lambda components using dyadic factors of type 11, log⁡\log, and short μ\mu.

The next campaign uses the third item as arithmetic structure rather than discarding it into the generic divisor-bounded class.

26. State transition

CSM_RH v1.30
  ->
CSM_RH v1.31

with:

Campaign 39
  CLOSED_AS_GENERIC_AVERAGED_TYPEII_LARGE_VALUE_AUDIT

B-RH-015
  GUTH_MAYNARD_PRODUCT_LEVEL_SET_NORMALIZATION
  CREATED / CERTIFIED

O-RH-096
  GENERIC_LARGE_VALUE_ORTHOGONALITY_FLOOR
  CREATED / CERTIFIED

O-RH-097
  SINGLE_RESONANT_FREQUENCY_OBSTRUCTS_GENERIC_POLYNOMIAL_W_MEAN_SQUARE
  CREATED / CERTIFIED

O-RH-098
  LARGE_VALUE_COUNT_QUANTIZATION_BARRIER
  CREATED / CERTIFIED

F-RH-010
  PESC
  REMAINS OPEN

F-RH-016
  MLEPG
  REMAINS OPEN

Campaign 40
  HEATH_BROWN_ARITHMETIC_RESONANCE_EXCLUSION
  READY

27. Final status

RH = OPEN

PESC = OPEN

MLEPG = OPEN

POLYNOMIAL-W TYPE-II AMPLIFIER = VALID

POINTWISE POLYNOMIAL-W INPUT = FIXED-STRIP LOCKED

GENERIC AVERAGED LARGE-VALUE REPLACEMENT = CLOSED

GUTH-MAYNARD IMPROVEMENT = REAL BUT RETAINS ORTHOGONALITY FLOOR

SINGLE GENERIC RESONANCE = ENOUGH TO KILL VANISHING PRODUCT MEAN SQUARE

ARITHMETIC HEATH-BROWN RESONANCE EXCLUSION = OPEN

NEXT CAMPAIGN = 40

The decisive generic obstruction is

RF(P)≪Xo(1)(P−2+⋯ ),\boxed{ R_F(P) \ll X^{o(1)} \left( P^{-2} +\cdots \right), }

together with the legal phase-aligned example

a(m)=mit0,b(n)=nit0,\boxed{ a(m)=m^{it_0}, \qquad b(n)=n^{it_0}, }

for which

∫∣A(1+it)B(1+it)∣2dt≫1.\boxed{ \int |A(1+it)B(1+it)|^2dt \gg1. }

The breakthrough route, if it exists, must use the arithmetic identity of the actual prime factors rather than generic large-value geometry.