# 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:

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

If

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

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:

1. deterministic local factors generated by residue-class geometry; and
2. 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:

```text
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:

```text
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:

```text
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:

```text
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

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

If

$$
h\equiv0\pmod\ell,
$$

the sum is $0$.

Create:

```text
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:

```text
LOW-CONDUCTOR NONPRINCIPAL CHARACTER ANNIHILATION
```

The simple shift-average interpretation is now rejected.

Record:

```text
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:

```text
PT6G
LOW-CONDUCTOR LOCAL-FACTOR RENORMALIZED DISPERSION
```

Required architecture:

1. identify the exact finite low- $q$ local arithmetic profile of the chosen ordinary-prime observable;
2. subtract or preserve that profile before absolute values are taken;
3. form a residual which still detects the zeta-boundary packet but not deterministic local biases;
4. prove fixed-power dispersion of that residual;
5. 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

$$
v1.57
$$

to

$$
v1.58.
$$

Add:

```text
B-RH-086
FIXED_PRIME_MODULUS_SHIFTED_RESIDUE_CHARACTER_PROFILE
CERTIFIED
```

Add:

```text
O-RH-155
FIXED_NONPRINCIPAL_CHARACTER_HARMONIC_SURVIVES_ALL_ABSOLUTE_SHIFT_LP_NORMS
CERTIFIED
```

Add:

```text
O-RH-156
L1_DUALITY_REVERSES_LOW_CONDUCTOR_RESIDUE_CYCLE_CANCELLATION
CERTIFIED
```

Update:

```text
F-RH-019/HIGH
CLOSED

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

Open:

```text
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.