# C-0016 witness pin

Audit source:
`scratch/proof-system--c0016-rosati-purity-loewner-audit/`

## Primary sources

The witness retains three official J.S. Milne source sets.

### Abelian Varieties, version 2.00 (2008)

Official source:
`https://www.jmilne.org/math/CourseNotes/AV.pdf`

```text
milne-abelian-varieties-v2.pdf
sha256 f5ca4e63e5092a4b102daad1470e4cbed5fe8f82115e3a28c8881e3f67f6aaef

milne-abelian-varieties-v2.txt
sha256 fb9fcd9960d6df085998d9a79c6784ff40af49e4edd9dce0c887722785988758
```

Pinned passages: Theorem I.14.3, Lemmas II.1.2--1.3, and Remark II.1.4.
They supply Rosati positivity, the exact identity
\(\pi^\dagger\pi=[q]\), semisimplicity of \(\mathbf Q[\pi]\), and the
modulus of every complex characteristic root.

### Complex Multiplication, version 0.10 (2020)

Official source:
`https://www.jmilne.org/math/CourseNotes/CM.pdf`

```text
milne-complex-multiplication.pdf
sha256 0ac337a885d72f6938f15724b897f568fe350d0d754d0b52370b8f3a01d2e2b5

milne-complex-multiplication.txt
sha256 c78451f467db58e7c3dd964658a195bc5837a349ce4ccf3d6aa4b621f65aa165
```

Pinned passages: Propositions I.1.36 and I.1.39, retained as an independent
cross-check of the positive-involution algebra classification.

### Jacobian Varieties (2021)

Official source:
`https://www.jmilne.org/math/xnotes/JVs.pdf`

```text
milne-jacobian-varieties.pdf
sha256 36c3f09c7462dbbd4ae1f8b81a02bd9ff84f03c5a346351d7d5d78fc3f173486

milne-jacobian-varieties.txt
sha256 4f796518221e5faf5c0ddfbf0115c8ca7d182a1f93ec72e72abb79ae84eee10d
```

Pinned passages: Summary 6.11, Corollary 9.6, Theorem 11.1, and Corollary
11.4. The first three supply the canonical Jacobian polarization,
curve/Jacobian first cohomology, and the all-extension point-count formula.
Corollary 11.4 is retained to pin, not import, its convention mismatch: the
article first defines \(P(X)=\prod_i(X-\alpha_i)\), then prints \(P(t)\)
as the zeta numerator.

The exact numerator is instead derived from Theorem 11.1 applied to every
\(\mathbf F_{q^n}\):

\[
N_n=1+q^n-\sum_i\alpha_i^n
\]

implies, by formal logarithmic summation,

\[
Z_C(t)
=\frac{\prod_i(1-\alpha_i t)}{(1-t)(1-qt)}.
\]

The 2008 abelian-varieties proof passage carrying an admitted correction
footnote is likewise not used for this seam.

## What the verifier checks

1. Every source and deterministic text-extraction hash, ordered theorem
   anchors, and the 2021 article's monic-\(P(X)\)/printed-\(P(t)\)
   convention mismatch.
2. The exact power-sum-to-zeta logarithmic-derivative identity deriving
   \(\prod_i(1-\alpha_i t)\) from the all-extension point counts.
3. The normalized reciprocal-pair factor and the exact
   \(\theta=0,\pi\) double-zero cases.
4. The zero-frequency coefficient, the factor two on positive frequencies,
   the confluent divided difference, and
   \(Q=DMD\) with \(D=\operatorname{diag}(2\lambda_i)\).
5. Exact finite-field construction, squarefreeness, point counts, and Newton
   reconstruction for five honest genus-one and genus-two curves.
6. The repeated self-reciprocal fixture
   \(y^2=x^5-x/\mathbf F_5\), whose numerator is
   \((1-5u^2)^2\) and whose centered function is
   \(4\sinh^2((\log5)z)\).
7. Every principal minor of the direct \(3\times3\) negative-Loewner and
   \(Q\) matrices at \(\lambda=\{1/4,3/4,5/4\}\), using 384-bit Arb
   arithmetic. The first node exercises the sharpened domain below
   \(1/2\).

Dependencies: Python 3.14, SymPy 1.14, and python-flint 0.9.0/FLINT 3.6.0.

Run:

```bash
uv run --frozen python canon/witnesses/C-0016/verify.py
```

Pinned verifier SHA-256:

```text
87a29c5c1ee574fe376359c56d9874636a00f2834c0cea00d7a2b431d128bc16
```

Accepted JSON SHA-256:

```text
9d3828fb5ced40650e7fe693e6b791bea429106e87351d4dd0f9657a816a6da8
```

The conclusion is finite-field only. It has no implication for the Riemann
zeta function.
