# C-0016 Rosati purity and Loewner-kernel audit

Date: 2026-07-19 (Pacific)

Target: C-0016, the claim that every smooth projective geometrically
connected curve over a finite field has an all-order positive centered
Loewner kernel.

This target was selected by dependency. C-0016 supports O-0003 and O-0065,
but previously depended only on archived Note 412 and had no pinned primary
source or canonical verifier.

## Verdict

**ACCEPT WITH CORRECTIONS**, and strengthen the node domain from
\(\lambda>1/2\) to \(\lambda>0\).

No mathematical counterexample was found. The Rosati-to-purity implication,
the centered-factor normalization, the zero multiplicities, the
partial-fraction coefficients, the confluent diagonal, and the
diagonal-congruence factor all survive independent reconstruction.

The prior record nevertheless required seven corrections:

1. Replace "geometric Frobenius" by the unambiguous **\(q\)-power
   Frobenius endomorphism** used by the pinned source.
2. State the sourced identity in its exact order,
   \(\pi^\dagger\pi=[q]_J\). The names Verschiebung and
   \(\pi^\dagger=V\) are unnecessary for this claim and were not pinned by
   the selected source passages.
3. Do not infer that \(\mathbf Q[\pi]\) is etale merely because it lies in a
   semisimple ambient endomorphism algebra. Milne's Lemma II.1.3 proves
   directly from Rosati positivity that \(\mathbf Q[\pi]\) has no
   nilpotents and is a product of fields.
4. Distinguish the endomorphism characteristic polynomial
   \(\chi_\pi(T)=\det(T-\pi\mid V_\ell J)\) from the curve numerator
   \(P_C(u)=\det(1-u\pi\mid V_\ell J)
   =u^{2g}\chi_\pi(u^{-1})\).
5. Define the \(0/0\) diagonal of \(Q_C\) by continuity.
6. Replace the bare "Mittag--Leffler expansion" assertion by the paired
   genus-zero canonical product, including the exceptional
   \(\theta=0,\pi\) multiplicities and the proof that no additive
   logarithmic-derivative term survives.
7. Do not use the 2021 Corollary 11.4 as printed: the article first defines
   \(P(X)=\prod(X-\alpha_i)\) and later prints \(P(t)\) as the numerator.
   Instead apply Theorem 11.1 over every finite extension and sum the exact
   power-count identity.
8. Separate the Euler-series half-plane \(\operatorname{Re}z>1/2\) from
   the spectral Gram domain. The latter is valid for every positive node.

The 2008 abelian-varieties notes contain an admitted "Needs fixing" footnote
inside their curve-zeta proof. They remain sufficient for Rosati positivity
and the Frobenius modulus lemma. The 2021 *Jacobian Varieties* article
supplies the all-extension point-count theorem, but its printed Corollary
11.4 has a polynomial-convention mismatch and is explicitly not used for
the numerator identity.

## Pinned source chain

The audit retains exact PDFs and deterministic text extractions.

### Milne, Abelian Varieties, version 2.00

PDF SHA-256:

```text
f5ca4e63e5092a4b102daad1470e4cbed5fe8f82115e3a28c8881e3f67f6aaef
```

Text SHA-256:

```text
fb9fcd9960d6df085998d9a79c6784ff40af49e4edd9dce0c887722785988758
```

Load-bearing passages:

- Theorem I.14.3: the Rosati trace form is positive definite.
- Lemma II.1.2: \(\pi^\dagger\pi=[q]_A\) for any polarization.
- Lemma II.1.3 and Remark II.1.4: \(\mathbf Q[\pi]\) is a product of
  fields, is dagger-stable, and every complex embedding sends \(\pi\) to
  a number of modulus \(\sqrt q\).

### Milne, Complex Multiplication, version 0.10

PDF SHA-256:

```text
0ac337a885d72f6938f15724b897f568fe350d0d754d0b52370b8f3a01d2e2b5
```

Text SHA-256:

```text
c78451f467db58e7c3dd964658a195bc5837a349ce4ccf3d6aa4b621f65aa165
```

Propositions I.1.36 and I.1.39 independently cross-check the two algebra
steps: a positive involution forces semisimplicity, and a commutative
positive-involution algebra is a product of totally real fields with the
identity involution and CM fields with complex conjugation.

### Milne, Jacobian Varieties, 2021

PDF SHA-256:

```text
36c3f09c7462dbbd4ae1f8b81a02bd9ff84f03c5a346351d7d5d78fc3f173486
```

Text SHA-256:

```text
4f796518221e5faf5c0ddfbf0115c8ca7d182a1f93ec72e72abb79ae84eee10d
```

Load-bearing passages:

- Summary 6.11: the Jacobian has its canonical principal polarization.
- Corollary 9.6: \(H^1(J,\mathbf Z_\ell)\to H^1(C,\mathbf Z_\ell)\)
  is an isomorphism.
- Theorem 11.1, applied after every finite base extension, gives
  \[
  N_n=1+q^n-\sum_j\alpha_j^n.
  \]
- Corollary 11.4 is retained only to pin the source defect: the article
  defines the monic polynomial \(P(X)=\prod_j(X-\alpha_j)\), then prints
  \(P(t)\) as the numerator. The audit does not import that inconsistent
  notation.

## Independent proof reconstruction

Let \(J=\operatorname{Jac}(C)\), let \(\pi\) be its \(q\)-power Frobenius
endomorphism, and let \(\dagger\) be the Rosati involution from any
polarization. Milne gives

\[
\pi^\dagger\pi=[q]_J.
\]

Rosati positivity proves that \(\mathbf Q[\pi]\) is a product of fields and
that every complex embedding satisfies

\[
\sigma(\pi^\dagger)=\overline{\sigma(\pi)}.
\]

Therefore every root \(\alpha\) of \(\chi_\pi\) obeys

\[
|\alpha|^2
=\sigma(\pi)\sigma(\pi^\dagger)
=q.
\]

Write

\[
P_C(u)=\det(1-u\pi\mid V_\ell J)
=\prod_{j=1}^{2g}(1-\alpha_j u),\qquad
\beta_j=\alpha_j/\sqrt q,\qquad L=\log q.
\]

For every \(n\), apply Theorem 11.1 to
\(C/\mathbf F_{q^n}\). Its Frobenius is \(\pi^n\), hence

\[
N_n=1+q^n-\sum_j\alpha_j^n.
\]

Consequently

\[
\begin{aligned}
\log Z_C(u)
&=\sum_{n\ge1}N_n\frac{u^n}{n}\\
&=-\log(1-u)-\log(1-qu)
  +\sum_j\log(1-\alpha_j u),
\end{aligned}
\]

and therefore

\[
Z_C(u)=\frac{P_C(u)}{(1-u)(1-qu)}.
\]

The coefficients of \(P_C\) are integral. Purity therefore identifies the
complex-conjugate root of \(\alpha\), with the same multiplicity, as
\(q/\alpha\); also \(\det\pi=q^g\). Hence

\[
P_C(u)=q^g u^{2g}P_C(1/(qu)),
\]

so

\[
X_C(z)=e^{gLz}P_C(q^{-1/2}e^{-Lz})
\]

is even and \(2\pi i/L\)-periodic. The local expansions at \(0\) and
\(\pi i/L\) are therefore even, so the normalized fixed roots \(+1\) and
\(-1\) have even multiplicity and may be grouped into reciprocal pairs.

For a nonexceptional reciprocal pair
\(\beta=e^{i\theta},\bar\beta=e^{-i\theta}\),

\[
e^{Lz}(1-\beta e^{-Lz})(1-\bar\beta e^{-Lz})
=2(\cosh(Lz)-\cos\theta).
\]

Its positive zero frequencies are

\[
\frac{2\pi k+\theta}{L},\qquad
\frac{2\pi(k+1)-\theta}{L},\qquad k\ge0.
\]

The exceptional cases are load-bearing:

- \(\theta=0\): \(2(\cosh w-1)=4\sinh^2(w/2)\), so zero and all
  \(2\pi k/L\) frequencies are doubled per reciprocal pair.
- \(\theta=\pi\): \(2(\cosh w+1)=4\cosh^2(w/2)\), so all
  \((2k+1)\pi/L\) frequencies are doubled per reciprocal pair.

Let \(m_0\) be the actual multiplicity at zero and \(m_\gamma\) the actual
multiplicity at \(i\gamma\). Since there are finitely many periodic
frequency towers,

\[
\sum_{\gamma>0}\frac{m_\gamma}{\gamma^2}<\infty.
\]

The paired genus-zero product therefore converges normally:

\[
X_C(z)
=Cz^{m_0}\prod_{\gamma>0}
\left(1+\frac{z^2}{\gamma^2}\right)^{m_\gamma}.
\]

Any remaining Hadamard quotient has the form \(e^{az+b}\). Evenness forces
\(a=0\), so there is no hidden additive term in the logarithmic derivative.
Thus

\[
\frac{X_C'(z)}{X_C(z)}
=\frac{m_0}{z}
+2z\sum_{\gamma>0}
\frac{m_\gamma}{z^2+\gamma^2},
\]

and

\[
H_C(x)
=\frac{m_0}{x}
+2\sum_{\gamma>0}\frac{m_\gamma}{x+\gamma^2}.
\]

For positive \(x,y\), with the diagonal interpreted confluently,

\[
M_C(x,y)
:=-\frac{H_C(x)-H_C(y)}{x-y}
=\frac{m_0}{xy}
+2\sum_{\gamma>0}
\frac{m_\gamma}{(x+\gamma^2)(y+\gamma^2)}.
\]

This is the Gram kernel generated by

\[
\frac{\sqrt{m_0}}{x},\qquad
\frac{\sqrt{2m_\gamma}}{x+\gamma^2}.
\]

In particular,

\[
M_C(x,x)
=-H_C'(x)
=\frac{m_0}{x^2}
+2\sum_{\gamma>0}
\frac{m_\gamma}{(x+\gamma^2)^2}.
\]

Finally, for \(x_i=\lambda_i^2\),

\[
Q_C(\lambda_i,\lambda_j)
=4\lambda_i\lambda_jM_C(x_i,x_j).
\]

Hence \(Q=DMD\) with \(D=\operatorname{diag}(2\lambda_i)\), and

\[
Q_C(\lambda,\lambda)
=-4\lambda^2H_C'(\lambda^2).
\]

Every matrix order is positive semidefinite for all \(\lambda_i>0\).

## Exact curve fixtures

The verifier constructs the finite fields from pinned irreducible
polynomials, checks squarefreeness, counts projective points exactly, and
uses Newton identities to reconstruct \(P_C\).

| Curve | Exact counts | Reconstructed numerator |
|---|---|---|
| \(y^2=x^3+x+1/\mathbf F_5\) | \(N_1=9,N_2=27\) | \(1+3u+5u^2\) |
| \(y^2=x^3-x/\mathbf F_9\) | \(N_1=16\) | \((1+3u)^2\) |
| \(y^2=(1+t)(x^3-x)/\mathbf F_9,\ t^2+1=0\) | \(N_1=4\) | \((1-3u)^2\) |
| \(y^2=x^5-x/\mathbf F_3\) | \(N_1,\ldots,N_4=(4,6,28,110)\) | \(1-2u^2+9u^4\) |
| \(y^2=x^5-x/\mathbf F_5\) | \(N_1,\ldots,N_4=(6,6,126,526)\) | \((1-5u^2)^2\) |

The last fixture has normalized roots \(+1,-1\), each with multiplicity
two. Its centered numerator is

\[
X_C(z)=4\sinh^2((\log5)z),
\]

so it simultaneously tests \(m_0=2\), both exceptional angles, and every
positive frequency with multiplicity two.

For every fixture, the verifier evaluates \(M\) and \(Q\) at

\[
\lambda\in\{1/4,3/4,5/4\}
\]

using 384-bit Arb arithmetic. Every one of the seven principal minors of
each \(3\times3\) matrix is strictly positive. The node \(1/4\) explicitly
exercises the strengthened range below the old \(1/2\) boundary.

Run:

```bash
uv run --frozen python \
  scratch/proof-system--c0016-rosati-purity-loewner-audit/verify.py
```

Verifier SHA-256:

```text
87a29c5c1ee574fe376359c56d9874636a00f2834c0cea00d7a2b431d128bc16
```

Accepted JSON SHA-256:

```text
9d3828fb5ced40650e7fe693e6b791bea429106e87351d4dd0f9657a816a6da8
```

## Scope

This is a theorem about finite-field curve zeta numerators. It does not
construct a number-field polarization, transfer function-field purity to
the Riemann zeta function, or supply evidence for RH.
