# Verification report

Date: 2026-09-02

## Verdict

`PASS` for the manuscript's stated negative answer. This is an internal,
unrefereed verification record and is not peer review.

## Claim boundary

For every rational prime `p`, the manuscript compares

\[
f_0(z)=z^2+1, \qquad f_1(z)=z^2-p^{-6}
\]

over `Q_p`. Both maps have infinite postcritical orbit and hence, by the
polynomial case of Pink's theorem, the same full arithmetic and geometric
profinite image `Aut(T_2)`. Their non-Archimedean Julia sets have different
topological types: good reduction gives the Gauss singleton for `f_0`, while
the two base-field inverse branches of `f_1` give a Cantor set. The result is
therefore a counterexample to reconstruction from the bare profinite group
image. It does not claim reconstruction failure from the entire valued cover
tower or from a discrete iterated-monodromy correspondence carrying additional
self-similar data.

## Proof audit

- The exact polynomial theorem in Pink was checked, avoiding the collision
  exception belonging to the more general rational-map sequel.
- The critical orbit of `0` under `z^2+1` is strictly increasing after its
  first step; the valuations for `z^2-p^{-6}` double negatively at every step.
- The exponent six makes the base-field square-root argument valid also at
  `p=2`, where the Cantor threshold is stricter.
- Each inverse branch is defined over `Q_p`, maps the relevant ball strictly
  into itself, and is contracting; the Cantor conclusion is therefore not
  obtained only after scalar extension.
- Equality is equality with the full rooted-tree automorphism group after an
  ordinary choice of tree labels, not merely an unsupported abstract-group
  analogy.
- Singleton and Cantor topological types rule out reconstruction without a
  coordinate-choice ambiguity.

The complete implication and attempted-falsification audit is preserved in
`problems/AIM-DYNAMICAL_SYSTEMS-0005/proof_audit.md`.

## Exact reproducibility

From the repository root:

```sh
python3 scripts/check_aim_dynamical_systems_0005.py
python3 problems/AIM-DYNAMICAL_SYSTEMS-0005/reproducibility/independent_julia_galois_check.py
```

Both use only the Python standard library and exact integer or `Fraction`
arithmetic. The first enumerates the finite binary-tree groups through height
four; the second independently checks sign ranks, conjugacy orbits through
level nine, critical valuations, and the residue-characteristic-two threshold.
Finite checks certify explicit hypotheses and mechanisms; they do not replace
Pink's theorem or the published Julia-set classification.

## Build and visual QA

- Compiled PDF: `main.pdf`
- Pages: 5
- Manifest: `manifest.json`
- Undefined citations/references: none in the retained build log
- Rendered pages visually inspected: yes
- Observed clipping or formula-layout defect: none

## Public release and license

The author approved public release on 2026-09-02. This paper, its source files, and this verification report are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/