What was already known
Kesten–Stigum already gives reconstruction above threshold. Sly and Mossel–Sly–Sohn established sharpness, including equality, for sufficiently large regular branching or Poisson mean; some small-degree cases were known.
AI involvement
OpenAI credits an internal model with producing the manuscript collection and is the named author. Session-level autonomy is not documented.
OpenAI institutional authorship and its release account support central internal-model work. Session-level independence and a named human-origin theorem step are not documented.
Importance
Resolves the all-degree three-state symmetric threshold in both tree families.
Limitations
The negative channel is included down to λ=−1/2. The exact-arithmetic certificates address finite proof inequalities; the linked Lean result covers only named parameter regions and does not establish the full negative-channel theorem. No complete human proof review is documented.
Technical verification
The manuscript provides exact-arithmetic certificate cases for finite inequalities. A related Lean module proves nonreconstruction only for nonnegative or restricted negative parameters, plus supercritical reconstruction. Neither supplies a documented end-to-end check of the full negative-channel, all-degree theorem; the certificate suite was not rerun in this review.
Human understanding
No named human explanation of the central all-degree negative-channel argument is documented. Few is the evidence default.
Correction and publication history
If correct, resolves the all-degree three-state symmetric threshold in both tree families.
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