CSM_RH Paper 67

Absolute-Value Survival of Low-Conductor Character Harmonics and the Local-Factor Renormalization Requirement

Project: CSM_RH
Paper: 67
Version: v0.1
Date: 2026-09-09
Campaign: 46 — SEEDED_ARITHMETIC_STRIP_GAP_GENERATION
Track: PT6F — LOW-CONDUCTOR NONPRINCIPAL CHARACTER ANNIHILATION
Status: SIMPLE SHIFT-AVERAGE ANNIHILATION CLOSED / LOW-CONDUCTOR LOCAL-FACTOR RENORMALIZATION REQUIRED
Canonical entry state: v1.57 / Paper 66 v0.1
RH_PROVED: FALSE
RH_DISPROVED: FALSE
GLOBAL_RH_CERTIFICATE: FALSE

Abstract

Paper 66 proved that additive large-sieve averaging gives fixed-power suppression for every polynomial-conductor major-arc block, but leaves a low-conductor nonprincipal core.

The present paper asks whether this core can be removed simply by averaging over the shift variable.

For a nonprincipal Dirichlet character $\chi\bmod q$ , periodicity gives the strong signed cancellation

\boxed{ \sum_{h\le H} \chi(h+r) = O(q). }

However, the shifted-prime and exceptional-set problems require absolute or $L^p$ control rather than a signed mean. For every fixed $p>0$ ,

\boxed{ \sum_{h\le H} |\chi(h+r)|^p = \frac{\phi(q)}{q}H + O(q). }

Thus a fixed low-conductor character harmonic has full positive density in every absolute $L^p$ norm.

At $p=1$ this is equivalent to an exact duality obstruction:

\boxed{ \sum_{h\le H}|C(h)| = \sup_{|\varepsilon_h|\le1} \left| \sum_{h\le H} \varepsilon_h C(h) \right|. }

C(h)=A\chi(h+r),

choosing

\varepsilon_h = \overline{\chi(h+r)}

on the coprime positions turns the complete-period cancellation into a coherent sum of size

\boxed{ |A| \left( \frac{\phi(q)}{q}H + O(q) \right). }

Hence shift averaging annihilates low-conductor characters only before absolute values are introduced. It does not annihilate them in the norm geometry needed for almost-all exceptional-set estimates.

This is directly relevant to Lichtman's shifted-prime Möbius theorem, whose fundamental quantity is

\sum_{h\le H} \left| \sum_{p\le X}\mu(p+h) \right|.

Its dual weights are arbitrary bounded signs, so periodic character cancellation cannot be inserted after dualization without additional coefficient information.

The paper then shows that low-conductor character profiles are not merely artifacts of possible bad Dirichlet- $L$ zeros.

Let $\ell$ be a fixed prime and $\chi\bmod\ell$ a nonprincipal character. For

h\not\equiv0\pmod\ell,

the exact finite-field identity is

\boxed{ \sum_{a\in\mathbb F_\ell^\times} \chi(a+h) = -\chi(h). }

Therefore, even under ideal equidistribution of the ordinary primes among the nonzero residue classes modulo $\ell$ ,

\boxed{ \sum_{p\le X} \chi(p+h) = -\frac{\chi(h)}{\ell-1} \operatorname{Li}(X) + o_\ell(\operatorname{Li}(X)). }

Thus fixed low-conductor character harmonics can be legitimate local arithmetic profiles of shifted-prime observables.

They should not be blindly annihilated.

The correct low-conductor strategy must distinguish:

deterministic local factors generated by residue-class geometry; and
boundary-sensitive spectral residuals associated with twisted Möbius or von Mangoldt errors.

A projection that removes the local character space is not enough by itself. To recover an absolute-value theorem for the original observable, one must either:

subtract an explicit local main term and prove the residual small; or
retain the full character recombination before taking absolute values and prove a joint dispersion estimate.

This is analogous to Paper 47's warning that exact component recombination must occur before squaring.

The large-sieve floor of Paper 66 is therefore sharp in norm geometry. High conductor blocks are diluted by family size; fixed low-conductor harmonics have order-one operator norm and survive every $L^p$ shift average.

PT6F is closed as a simple annihilation strategy and replaced by:

PT6G
LOW-CONDUCTOR LOCAL-FACTOR RENORMALIZED DISPERSION

The desired theorem must remove explicit local arithmetic profiles while controlling the remaining spectral residual at fixed power, without requiring individual fixed strips for every nonprincipal Dirichlet $L$ -function.

No RH theorem is claimed.

1. Periodic signed cancellation

Let

\chi\bmod q

be nonprincipal.

Extend $\chi$ periodically to the integers, with

\chi(n)=0

when

(n,q)>1.

Since

\sum_{a\bmod q}\chi(a)=0,

every complete block of $q$ consecutive shifts cancels.

Therefore, uniformly in the integer shift $r$ ,

Theorem 1.1 — Signed shift annihilation

\boxed{ \sum_{h\le H} \chi(h+r) = O(q). }

For fixed $q$ , the relative signed average is therefore

O(q/H).

This is the tempting low-conductor annihilation mechanism.

2. Absolute $L^p$ mass survives

For every

p>0,

|\chi(n)|^p = 1_{(n,q)=1}.

Hence

\sum_{h\le H} |\chi(h+r)|^p

is simply the count of integers in the translated interval which are coprime to $q$ .

By periodicity,

\boxed{ \sum_{h\le H} |\chi(h+r)|^p = \frac{\phi(q)}q H + O(q). }

Thus:

Theorem 2.1 — Low-conductor $L^p$ survival

For every fixed nonprincipal character and every fixed $p>0$ ,

\boxed{ \|\chi(\,\cdot+r)\|_{\ell^p([1,H])} \asymp_q H^{1/p}. }

No power of $H$ is gained.

Create:

O-RH-155
FIXED_NONPRINCIPAL_CHARACTER_HARMONIC_SURVIVES_ALL_ABSOLUTE_SHIFT_LP_NORMS
CERTIFIED

3. $L^1$ duality destroys complete-period cancellation

For any finite complex sequence $C(h)$ ,

\boxed{ \sum_{h\le H}|C(h)| = \sup_{|\varepsilon_h|\le1} \left| \sum_{h\le H} \varepsilon_h C(h) \right|. }

Take

C(h)=A\chi(h+r).

Define

\varepsilon_h = \begin{cases} \overline{\chi(h+r)}, & (h+r,q)=1, \\ 0, & (h+r,q)>1. \end{cases}

Then

|\varepsilon_h|\le1

and

\varepsilon_hC(h) = |A| 1_{(h+r,q)=1}

up to the harmless phase of $A$ .

Therefore:

Theorem 3.1 — Dual-sign coherent recovery

\boxed{ \sum_{h\le H}|A\chi(h+r)| = |A| \left( \frac{\phi(q)}qH + O(q) \right). }

The bounded dual signs exactly reverse the residue-cycle cancellation.

Create:

O-RH-156
L1_DUALITY_REVERSES_LOW_CONDUCTOR_RESIDUE_CYCLE_CANCELLATION
CERTIFIED

4. Relevance to shifted-prime Möbius averages

Lichtman's principal shifted-prime theorem controls

\boxed{ \sum_{h\le H} \left| \sum_{p\le X}\mu(p+h) \right|. }

The absolute value is essential: the theorem asserts cancellation for almost all individual shifts, not merely in the total signed average over shifts.

By Section 3, any proof of such an $L^1$ theorem must withstand arbitrary bounded dual weights in $h$ .

Therefore:

sum over complete residue cycles:
available only before dualization.

after L1 dualization:
the test signs may align with any fixed low-q character mode.

Thus low-conductor annihilation by the raw shift average cannot prove the required almost-all result.

External calibration:

M. Lichtman, Averages of the Möbius Function on Shifted Primes, Quarterly Journal of Mathematics 73 (2022), 729–757.

5. Fixed low-modulus character profile of ideal primes

Let

\ell

be prime and let

\chi\bmod\ell

be nonprincipal.

Fix

h\not\equiv0\pmod\ell.

As $a$ runs over

\mathbb F_\ell^\times,

the value

a+h

runs over all elements of $\mathbb F_\ell$ except $h$ .

Since

\chi(0)=0

and

\sum_{b\in\mathbb F_\ell}\chi(b)=0,

we obtain:

Theorem 5.1 — Exact shifted residue character identity

\boxed{ \sum_{a\in\mathbb F_\ell^\times} \chi(a+h) = -\chi(h). }

h\equiv0\pmod\ell,

the sum is $0$ .

Create:

B-RH-086
FIXED_PRIME_MODULUS_SHIFTED_RESIDUE_CHARACTER_PROFILE
CERTIFIED

6. Consequence for uniformly distributed primes

For fixed $\ell$ , the prime number theorem in arithmetic progressions gives

\pi(X;\ell,a) = \frac{\operatorname{Li}(X)}{\ell-1} + o_\ell(\operatorname{Li}(X))

for every

a\in\mathbb F_\ell^\times.

Therefore

\begin{aligned} \sum_{\substack{p\le X\\p\ne\ell}} \chi(p+h) &= \sum_{a\in\mathbb F_\ell^\times} \chi(a+h) \pi(X;\ell,a) \\ &= -\frac{\chi(h)}{\ell-1} \operatorname{Li}(X) + o_\ell(\operatorname{Li}(X)). \end{aligned}

Thus:

Theorem 6.1 — Prime-equidistribution does not annihilate the shifted character mode

For

h\not\equiv0\pmod\ell,

\boxed{ \sum_{p\le X} \chi(p+h) = -\frac{\chi(h)}{\ell-1} \operatorname{Li}(X) + o_\ell(\operatorname{Li}(X)). }

A fixed low-conductor character harmonic can therefore be a genuine local main profile even under perfect prime equidistribution.

This does not assert such a main term for the Möbius-shifted-prime correlation. It calibrates the local character geometry which appears after additive decomposition.

7. Local profiles versus spectral residuals

Sections 5–6 show that the low-conductor character space contains two logically different phenomena.

Local arithmetic profile

This arises from the deterministic distribution of admissible prime residue classes.

It may have full main-term size.

Spectral residual

This arises from cancellation errors in twisted Möbius or twisted von Mangoldt sums and can be sensitive to zeros of nonprincipal Dirichlet $L$ -functions.

A successful PT6F/PT6G theorem must not confuse the two.

Blindly forcing every nonprincipal coefficient to be small would contradict legitimate local structure in other shifted-prime observables.

8. Projection before absolute value

Let

\mathcal V_Q

be the finite-dimensional span of low-conductor character harmonics

\chi(h+r), \qquad q\le Q.

One may project an observable as

C = P_QC + (I-P_Q)C.

High-conductor large-sieve methods may control the residual.

However:

\|P_QC\|_1

is not small merely because every nonprincipal basis vector has signed mean zero.

Theorem 2.1 shows that every such basis vector has full $L^1$ mass.

Therefore a projection strategy must separately identify and subtract the actual local main profile, or prove a coefficient bound for the boundary-sensitive part.

This is the low-conductor analogue of Paper 55's principal-arc projection barrier.

9. The low-conductor large-sieve floor is sharp

Paper 66 proved that polynomial conductor blocks gain

R^{-1}

in major-arc $L^2$ mass.

For fixed $R$ , this becomes only an order-one constant.

The fixed-character mode

C(h)=A\chi(h)

shows that no universal power of $H$ can arise from shift-family dimension alone:

\|C\|_2^2 = |A|^2 \left( \frac{\phi(q)}qH+O(q) \right).

Thus:

Corollary 9.1

The low-conductor floor O-RH-154 is attained in absolute norm geometry by fixed character harmonics.

Family dilution is intrinsically a high-conductor phenomenon.

10. Why exceptional-set geometry has the same barrier

Suppose

|A|>T

for a threshold $T>0$ .

Then the character mode

C(h)=A\chi(h+r)

satisfies

|C(h)|>T

for

\boxed{ \frac{\phi(q)}qH+O(q) }

shifts.

Therefore a single fixed-character coefficient above threshold creates a positive-density exceptional set.

It cannot be hidden by residue-cycle sign cancellation.

Thus any exceptional-set upper theorem must control the coefficient amplitude itself or remove it as an explicit local factor.

11. Closure of simple PT6F annihilation

The original PT6F target was:

LOW-CONDUCTOR NONPRINCIPAL CHARACTER ANNIHILATION

The simple shift-average interpretation is now rejected.

Record:

PT6F-SIGNED
CLOSED:
SIGNED SHIFT AVERAGE ANNIHILATES FIXED CHARACTERS.

PT6F-ABSOLUTE
CLOSED AS NONVIABLE:
ABSOLUTE / EXCEPTIONAL NORMS RETAIN FULL FIXED-CHARACTER MASS.

The remaining low-conductor task must be renormalized rather than annihilating.

12. New track: PT6G

Open:

PT6G
LOW-CONDUCTOR LOCAL-FACTOR RENORMALIZED DISPERSION

Required architecture:

identify the exact finite low- $q$ local arithmetic profile of the chosen ordinary-prime observable;
subtract or preserve that profile before absolute values are taken;
form a residual which still detects the zeta-boundary packet but not deterministic local biases;
prove fixed-power dispersion of that residual;
combine with Paper 66's polynomial-conductor large-sieve control.

The theorem must not merely project onto a low-character complement and discard the projected term.

A full recovery ledger is required.

13. Why this is not yet an amplifier theorem

The present paper does not prove that the residual in PT6G is fixed-power small.

It proves only that:

high-conductor channels are already averaged away;
fixed low-conductor harmonics cannot be averaged away in absolute norms;
some such harmonics are genuine local profiles rather than spectral errors.

The coefficient of the boundary-sensitive residual remains the hard quantity.

Thus no PESC exponent improvement is claimed.

14. External calibration

14.1. Lichtman shifted-prime theorem

M. Lichtman, Averages of the Möbius Function on Shifted Primes, Quarterly Journal of Mathematics 73 (2022), 729–757.

The main theorem controls

\sum_{h\le H} \left| \sum_{p\le X}\mu(p+h) \right|,

so the absolute shift norm is fundamental, not incidental.

URL:

https://academic.oup.com/qjmath/article/73/2/729/6446139

14.2. Large-sieve duality

The duality principle behind large-sieve estimates equates the operator norm of a family transform with that of its adjoint. The elementary $L^1$ duality used here is the corresponding endpoint fact for absolute shift sums.

No deep large-sieve input is required for Theorems 2.1–3.1.

15. State transition

Advance the candidate state from

v1.57

v1.58.

Add:

B-RH-086
FIXED_PRIME_MODULUS_SHIFTED_RESIDUE_CHARACTER_PROFILE
CERTIFIED

Add:

O-RH-155
FIXED_NONPRINCIPAL_CHARACTER_HARMONIC_SURVIVES_ALL_ABSOLUTE_SHIFT_LP_NORMS
CERTIFIED

Add:

O-RH-156
L1_DUALITY_REVERSES_LOW_CONDUCTOR_RESIDUE_CYCLE_CANCELLATION
CERTIFIED

Update:

F-RH-019/HIGH
CLOSED

F-RH-019/LOW SIMPLE ANNIHILATION
CLOSED AS NONVIABLE

Open:

PT6G
LOW-CONDUCTOR LOCAL-FACTOR RENORMALIZED DISPERSION

No RH certificate is created.

16. Conclusion

Shift averaging does annihilate a fixed nonprincipal character in the signed sense:

\sum_{h\le H}\chi(h+r)=O(q).

But the CSM_RH amplifier requires absolute / exceptional information.

There,

\sum_{h\le H}|\chi(h+r)|^p \asymp H

for every fixed $p>0$ .

The cancellation disappears.

Moreover, fixed low-conductor character profiles can arise from perfectly ordinary local prime residue geometry.

Thus the low-conductor problem is not solved by averaging harder.

It requires a local-factor renormalization followed by a genuinely spectral residual estimate.

That is PT6G.

CSM_RH Paper 67 — Absolute-Value Survival of Low-Conductor Character Harmonics and the Local-Factor Renormalization Requ

CSM_RH Paper 67

Absolute-Value Survival of Low-Conductor Character Harmonics and the Local-Factor Renormalization Requirement

Abstract

1. Periodic signed cancellation

Theorem 1.1 — Signed shift annihilation

2. Absolute LpL^pLp mass survives

Theorem 2.1 — Low-conductor LpL^pLp survival

3. L1L^1L1 duality destroys complete-period cancellation

Theorem 3.1 — Dual-sign coherent recovery

4. Relevance to shifted-prime Möbius averages

5. Fixed low-modulus character profile of ideal primes

Theorem 5.1 — Exact shifted residue character identity

6. Consequence for uniformly distributed primes

Theorem 6.1 — Prime-equidistribution does not annihilate the shifted character mode

7. Local profiles versus spectral residuals

Local arithmetic profile

Spectral residual

8. Projection before absolute value

9. The low-conductor large-sieve floor is sharp

Corollary 9.1

10. Why exceptional-set geometry has the same barrier

11. Closure of simple PT6F annihilation

12. New track: PT6G

13. Why this is not yet an amplifier theorem

14. External calibration

14.1. Lichtman shifted-prime theorem

14.2. Large-sieve duality

15. State transition

16. Conclusion

2. Absolute $L^p$ mass survives

Theorem 2.1 — Low-conductor $L^p$ survival

3. $L^1$ duality destroys complete-period cancellation