CSM_RH Paper 66

Large-Sieve Power Suppression of High-Conductor Major Arcs and the Low-Conductor Character Core

Project: CSM_RH
Paper: 66
Version: v0.1
Date: 2026-09-09
Campaign: 46 — SEEDED_ARITHMETIC_STRIP_GAP_GENERATION
Frontier: F-RH-019 — AVERAGED_CHARACTER_MAJOR_ARC_POWER_WITHOUT_INDIVIDUAL_FAMILY_STRIPS
Status: HIGH-CONDUCTOR AVERAGING CERTIFIED / LOW-CONDUCTOR CORE REMAINS OPEN
Canonical entry state: v1.56 / Paper 65 v0.1
RH_PROVED: FALSE
RH_DISPROVED: FALSE
GLOBAL_RH_CERTIFICATE: FALSE

Abstract

Paper 65 showed that additive rational frequencies introduce all Dirichlet-character channels and that a PESC seed for the Riemann zeta function controls only the principal channels at fixed-power resolution.

The present paper proves that individual fixed strips are nevertheless unnecessary for polynomially large conductors.

Let

A(\alpha) = \sum_{M<n\le M+H} a_n e(n\alpha)

and let

\mathfrak M_R(K)

be the union of arcs

\left| \alpha-\frac aq \right| \le \frac{c_0}{RK}, \qquad R<q\le2R, \qquad (a,q)=1.

For each fixed offset $\beta$ , the rational points $a/q+\beta$ with $q\sim R$ are $\gg R^{-2}$ separated. The additive large sieve therefore gives

\sum_{R<q\le2R} \sum_{(a,q)=1} \left| A\left( \frac aq+\beta \right) \right|^2 \ll (H+R^2) \sum|a_n|^2.

Integrating over the common arc offset yields the dyadic major-arc theorem

\boxed{ \int_{\mathfrak M_R(K)} |A(\alpha)|^2\,d\alpha \ll \frac{H+R^2}{RK} \sum|a_n|^2. }

K\asymp H

and

R^2\le H,

then

\boxed{ \int_{\mathfrak M_R(K)} |A(\alpha)|^2\,d\alpha \ll \frac1R \sum|a_n|^2. }

Thus every denominator block

R=X^u, \qquad u>0

carries a genuine fixed-power factor

X^{-u}.

This conclusion is completely independent of zero-free regions for the individual nonprincipal Dirichlet $L$ -functions.

In character language, the saving is the combined effect of:

Gauss-sum normalization in the additive-to-multiplicative transform;
rational-arc width;
family orthogonality / large-sieve dispersion.

Hence F-RH-019 succeeds on all polynomial-conductor major arcs.

The same calculation also identifies its limit.

R=X^{o(1)},

the factor

R^{-1}

is only subpower. In particular, for a fixed modulus $q$ , a nonprincipal character channel retains fixed positive weight in the major-arc norm.

The PESC seed controls the principal character because

L(s,\chi_0) = \zeta(s) \times \text{finite Euler factors},

but supplies no fixed zero-free half-plane for a fixed nonprincipal

L(s,\chi).

The classical zero-free region for such a channel still shrinks like a reciprocal logarithm of conductor-height, so its twisted Möbius / prime sums are not known to have a common fixed-power bound.

Therefore family averaging does not eliminate the character barrier. It localizes it:

\boxed{ \text{all polynomial conductors} \quad\longrightarrow\quad \text{fixed-power averaged control}, }

while

\boxed{ \text{subpolynomial and fixed conductors} \quad\longrightarrow\quad \text{unresolved low-conductor spectral core}. }

A complementary log-free zero-density calculation confirms this geometry. Jutila's near-one estimate

\sum_{q\le Q} \sum_{\chi\bmod q}^{*} N(\sigma,T,\chi) \ll (Q^2T)^{(2+o(1))(1-\sigma)}

implies that, in a dyadic conductor block

R=X^u

and height

T=X^v,

the conductor-weighted count of channels with a zero to the right of

\sigma=1-\delta

has schematic exponent

\boxed{ -u+2\delta(2u+v). }

For

\delta<\frac14

this becomes power-small once

u> \frac{2\delta v}{1-4\delta}.

Again, zero-density averaging becomes effective only after a positive conductor exponent is present. It does not resolve the $u=0$ core.

The paper therefore partially closes F-RH-019:

HIGH-CONDUCTOR MAJOR-ARC AVERAGING:
CERTIFIED AT FIXED POWER.

LOW-CONDUCTOR NONPRINCIPAL CORE:
OPEN.

The next useful theorem must either annihilate the low-conductor nonprincipal channels in the ordinary shifted-prime observable or prove a fixed-power averaged estimate for them without requiring individual Dirichlet- $L$ strips.

No RH theorem is claimed.

1. Dyadic rational major arcs

Let

A(\alpha) = \sum_{M<n\le M+H} a_n e(n\alpha),

where

e(x)=e^{2\pi ix}.

Fix

R\ge1, \qquad K\ge1.

Define the dyadic major-arc set

\boxed{ \mathfrak M_R(K) = \bigcup_{\substack{R<q\le2R\\(a,q)=1}} \left\{ \alpha: \left| \alpha-\frac aq \right| \le \frac{c_0}{RK} \right\}, }

where $c_0>0$ is a sufficiently small absolute constant.

The exact value of $c_0$ is immaterial for exponent bookkeeping.

2. Farey separation

\frac aq\ne\frac{a'}{q'}

with

R<q,q'\le2R,

then

\left| \frac aq-\frac{a'}{q'} \right| \ge \frac1{qq'} \ge \boxed{ \frac1{4R^2}. }

Thus, for every fixed real $\beta$ , the translated set

\left\{ \frac aq+\beta: R<q\le2R, (a,q)=1 \right\}

has the same separation.

3. Additive large-sieve input

The additive large sieve gives, for a $\delta$ -separated set of frequencies,

\sum_r \left| \sum_{M<n\le M+H} a_ne(n\alpha_r) \right|^2 \ll \left( H+\delta^{-1} \right) \sum|a_n|^2.

With

\delta^{-1}\ll R^2,

we obtain:

Theorem 3.1 — Dyadic rational-point large sieve

For every real $\beta$ ,

\boxed{ \sum_{R<q\le2R} \sum_{(a,q)=1} \left| A\left( \frac aq+\beta \right) \right|^2 \ll (H+R^2) \sum|a_n|^2. }

The implicit constant is absolute.

External calibration: this is the standard additive large-sieve theorem applied to Farey fractions.

4. Integrated dyadic major-arc theorem

Integrate Theorem 3.1 over

|\beta| \le \frac{c_0}{RK}.

By nonnegativity,

\begin{aligned} \int_{\mathfrak M_R(K)} |A(\alpha)|^2d\alpha &\le \int_{|\beta|\le c_0/(RK)} \sum_{q\sim R} \sum_{(a,q)=1} \left| A\left( \frac aq+\beta \right) \right|^2 d\beta. \end{aligned}

Therefore:

Theorem 4.1 — High-conductor major-arc $L^2$ suppression

\boxed{ \int_{\mathfrak M_R(K)} |A(\alpha)|^2d\alpha \ll \frac{ H+R^2 }{ RK } \sum|a_n|^2. }

Create:

B-RH-084
DYADIC_HIGH_CONDUCTOR_ADDITIVE_MAJOR_ARC_L2_SUPPRESSION
CERTIFIED

No Dirichlet- $L$ zero-free information enters this theorem.

5. Polynomial-conductor fixed power

Take

K\asymp H.

R^2\le H,

then

H+R^2\ll H.

Hence:

Corollary 5.1

\boxed{ \int_{\mathfrak M_R(H)} |A(\alpha)|^2d\alpha \ll R^{-1} \sum|a_n|^2. }

R=X^u, \qquad u>0,

then

\boxed{ \int_{\mathfrak M_R(H)} |A|^2 \ll X^{-u} \sum|a_n|^2. }

This is a genuine fixed-power family average.

Create:

B-RH-085
POLYNOMIAL_CONDUCTOR_MAJOR_ARCS_HAVE_ZERO_FREE_REGION_INDEPENDENT_POWER_SAVING
CERTIFIED

6. Character interpretation

For a rational frequency

\frac aq

with $(a,q)=1$ , the additive phase on integers coprime to $q$ admits a Dirichlet-character expansion.

Schematically,

e(an/q) = \frac1{\phi(q)} \sum_{\chi\bmod q} \tau(\overline\chi) \chi(a)\chi(n)

with the usual primitive / induced-character bookkeeping.

For primitive characters,

|\tau(\chi)|^2=q.

After summing over reduced residues $a$ and using character orthogonality, the total squared additive mass becomes a weighted character mean square.

The factor $R^{-1}$ in Corollary 5.1 is therefore the additive formulation of:

Gauss normalization
+
character orthogonality
+
arc-width dilution.

Thus the result is precisely the kind of averaged-character mechanism sought in F-RH-019.

7. The low-conductor floor

The gain in Corollary 5.1 is

R^{-1}.

R=X^{o(1)},

then

R^{-1}=X^{-o(1)}.

For fixed $R$ it is merely a constant.

Therefore:

Theorem 7.1 — Large-sieve conductor floor

Large-sieve averaging alone does not provide a fixed power on the union of fixed or subpolynomial denominator major arcs.

Create:

O-RH-154
AVERAGED_CHARACTER_LARGE_SIEVE_HAS_A_LOW_CONDUCTOR_FIXED_POWER_FLOOR
CERTIFIED

This is not a defect of the large sieve. It is the correct scaling of the family size.

8. Principal channels are already seeded

Let

\chi_0\bmod q

be principal.

Paper 65 certified

L(s,\chi_0) = \zeta(s) \prod_{p\mid q} (1-p^{-s}),

so PESC $(\kappa)$ gives principal-character fixed-power cancellation at exponent resolution.

Hence the low-conductor obstruction is entirely nonprincipal.

We may record:

LOW CONDUCTOR PRINCIPAL:
CONTROLLED BY ZETA SEED.

LOW CONDUCTOR NONPRINCIPAL:
NOT CONTROLLED AT FIXED POWER.

9. Why a fixed nonprincipal channel is not automatically harmless

For a fixed nonprincipal character $\chi$ , the best general zero-free region has logarithmically shrinking width as the height grows.

Thus current unconditional theory gives strong subpower cancellation in twisted Mertens / prime sums, but not a uniform fixed exponent which can be inserted into F-RH-017-v3.

If a sequence of zeros of $L(s,\chi)$ approached $\Re s=1$ within the classical allowed region, its twisted low-frequency channel would remain only subpower-suppressed.

The zeta seed says nothing about this possibility.

Thus the low-conductor floor is spectral, not merely combinatorial.

10. Zero-density calibration

Let

N(\sigma,T,\chi)

count zeros of

L(s,\chi)

with

\Re\rho\ge\sigma, \qquad |\Im\rho|\le T.

A classical near-one zero-density theorem of Jutila gives, for

\frac45\le\sigma\le1,

\boxed{ \sum_{q\le Q} \sum_{\chi\bmod q}^{*} N(\sigma,T,\chi) \ll (Q^2T)^{(2+o(1))(1-\sigma)}. }

Let

\sigma=1-\delta,

R=X^u, \qquad T=X^v.

In a dyadic conductor block $q\sim R$ , the number of bad primitive character-zero incidences is bounded at exponent level by

X^{2\delta(2u+v)+o(1)}.

The additive major-arc character weight / arc-width supplies an effective conductor factor of order $R^{-1}$ .

Therefore the weighted bad-channel count has schematic exponent

\boxed{ -u+2\delta(2u+v). }

\delta<\frac14

and

\boxed{ u> \frac{ 2\delta v }{ 1-4\delta }, }

this quantity is power-small.

This is only a spectral-count calibration, not a complete major-arc amplitude theorem.

Its purpose is to show that log-free zero density agrees with the large-sieve geometry: averaging becomes effective after a positive conductor exponent appears.

11. Why zero density still leaves $u=0$

u=0,

the preceding exponent becomes

2\delta v>0.

No conductor-weight saving remains.

Thus zero-density estimates allow many zero incidences over growing heights even for a fixed small conductor family.

They do not prove that a fixed nonprincipal $L$ -function has a fixed zero-free half-plane.

Hence zero-density averaging does not remove the low-conductor core.

12. Consequence for shifted-prime Möbius major arcs

Lichtman's current major-arc proof takes

q\le W

and controls every character individually.

F-RH-019 proposed replacing this worst-case treatment by an averaged norm.

Theorem 5.1 proves that this replacement is effective for every dyadic range

q\asymp X^u, \qquad u>0.

Therefore a redesigned shifted-prime proof need not solve the entire polynomial conductor family.

It only needs a separate treatment of the low-conductor region.

This is a real reduction of the family barrier.

13. Why summing the dyadic bounds does not finish the problem

Summing

R^{-1}

over dyadic

1\le R\le X^w

gives a series dominated by the first few denominator blocks.

Thus the total major-arc bound is controlled by low conductors.

Choosing a fixed lower cutoff

R\ge X^{u_0}

gives the clean power

X^{-u_0}.

But the omitted range

q<X^{u_0}

still contains nonprincipal character channels with no seed fixed strip.

Letting

u_0\to0

destroys the fixed power.

This is the exact remaining F-RH-019 gap.

14. Partial closure of F-RH-019

Record:

F-RH-019/HIGH
AVERAGED POLYNOMIAL-CONDUCTOR MAJOR-ARC POWER
CLOSED / CERTIFIED

F-RH-019/LOW
LOW-CONDUCTOR NONPRINCIPAL MAJOR-ARC POWER
OPEN

The original frontier is therefore partially closed.

It should no longer be described as a full polynomial Dirichlet- $L$ family problem.

The unresolved part is localized near conductor exponent zero.

15. Next arithmetic target

The next target is:

PT6F
LOW-CONDUCTOR NONPRINCIPAL CHARACTER ANNIHILATION

Desired outcome:

a fixed-power estimate for the contribution of

q\le X^{u_0}

for some small fixed $u_0>0$ which:

does not assume a fixed zero-free strip for every nonprincipal $L(s,\chi)$ ;
exploits the shifted-prime / ordinary-integer observable;
combines with B-RH-085 on $q\ge X^{u_0}$ .

Possible mechanisms:

algebraic cancellation of nonprincipal low-conductor channels in the prime-shift average;
a dispersion identity in which principal terms survive but nonprincipal low- $q$ terms enter quadratically and can be large-sieved;
a signed average over shifts or residue classes which kills fixed low-conductor characters before absolute values are taken;
a new low-conductor bilinear theorem.

No theorem is claimed.

16. External calibration

16.1. Additive and multiplicative large sieve

The additive large sieve over Farey fractions gives

\sum_{q\le Q} \sum_{(a,q)=1} \left| \sum a_ne(an/q) \right|^2 \ll (Q^2+H) \sum|a_n|^2.

The multiplicative form gives

\sum_{q\le Q} \frac{q}{\phi(q)} \sum_{\chi\bmod q}^{*} \left| \sum a_n\chi(n) \right|^2 \ll (Q^2+H) \sum|a_n|^2.

These are standard large-sieve inequalities.

16.2. Jutila zero density

For

4/5\le\sigma\le1,

Jutila proved

\sum_{q\le Q} \sum_{\chi\bmod q}^{*} N(\sigma,T,\chi) \ll_\varepsilon (Q^2T)^{(2+\varepsilon)(1-\sigma)}.

This is used only for family-size calibration.

17. State transition

Advance the candidate state from

v1.56

v1.57.

Add:

B-RH-084
DYADIC_HIGH_CONDUCTOR_ADDITIVE_MAJOR_ARC_L2_SUPPRESSION
CERTIFIED

Add:

B-RH-085
POLYNOMIAL_CONDUCTOR_MAJOR_ARCS_HAVE_ZERO_FREE_REGION_INDEPENDENT_POWER_SAVING
CERTIFIED

Add:

O-RH-154
AVERAGED_CHARACTER_LARGE_SIEVE_HAS_A_LOW_CONDUCTOR_FIXED_POWER_FLOOR
CERTIFIED

Update F-RH-019:

HIGH-CONDUCTOR PART CLOSED
LOW-CONDUCTOR CORE OPEN

Open:

PT6F
LOW-CONDUCTOR NONPRINCIPAL CHARACTER ANNIHILATION

No RH certificate is created.

18. Conclusion

Averaging over character channels works.

For polynomial conductor blocks it works extremely cleanly:

\boxed{ q\asymp X^u \quad\Rightarrow\quad X^{-u} \text{ major-arc }L^2\text{ saving}. }

No individual Dirichlet- $L$ zero-free region is required.

Thus the family barrier of Paper 65 was too broad.

The real unresolved set is the low-conductor core.

There, the family is too small for large-sieve dilution to create a fixed power, while the zeta seed controls only the principal characters.

The next problem is therefore finite-/subpolynomial-conductor nonprincipal annihilation, not polynomial-family GRH.

CSM_RH Paper 66 — Large-Sieve Power Suppression of High-Conductor Major Arcs and the Low-Conductor Character Core

CSM_RH Paper 66

Large-Sieve Power Suppression of High-Conductor Major Arcs and the Low-Conductor Character Core

Abstract

1. Dyadic rational major arcs

2. Farey separation

3. Additive large-sieve input

Theorem 3.1 — Dyadic rational-point large sieve

4. Integrated dyadic major-arc theorem

Theorem 4.1 — High-conductor major-arc L2L^2L2 suppression

5. Polynomial-conductor fixed power

Corollary 5.1

6. Character interpretation

7. The low-conductor floor

Theorem 7.1 — Large-sieve conductor floor

8. Principal channels are already seeded

9. Why a fixed nonprincipal channel is not automatically harmless

10. Zero-density calibration

11. Why zero density still leaves u=0u=0u=0

12. Consequence for shifted-prime Möbius major arcs

13. Why summing the dyadic bounds does not finish the problem

14. Partial closure of F-RH-019

15. Next arithmetic target

16. External calibration

16.1. Additive and multiplicative large sieve

16.2. Jutila zero density

17. State transition

18. Conclusion

Theorem 4.1 — High-conductor major-arc $L^2$ suppression

11. Why zero density still leaves $u=0$