# Public verification report

Title: Bidirectional Inference Counts on Forests and Cycles.
Author: Alper Ferudun, Mercury Software GmbH. Date: 29 September 2026.
Six-page English unrefereed preprint.

## Accepted mathematical scope

All finite labeled forests, and all finite simple bipartite graphs of maximum
degree two, with binary units, free real biases and one shared real weight per
edge. Local conditional-MAP outputs are strict at every binary input. Tuples
are not quotiented by graph or state symmetries.

The manuscript proves forest realizability by positive vertex rescaling;
the exact incidence partition function and tree recurrence; star and path
counts; and the even-cycle subtraction formula. It also gives two trees
with the same bipartition degree lists and different bidirectional counts.
The entire source record AIM-PROBABILITY-0009 is NOT marked resolved.
Standard one-way inference counts are established prior work.

## Exact computational evidence

Two separate standard-library implementations use no floating point:

1. check_chambers.py computes characteristic polynomials by Whitney sums,
   with rational ranks certified through a prime exceeding a Hadamard bound.
   Ten arrangements collectively represent 156690 subsets. Reported counts
   match the analytic forest, path and cycle formulas.
2. check_realizations.py enumerates 95546 tuples on nine small graphs and
   builds rational shared-weight realizations. It checks 552634 local input
   values. In C4, 12472 tuples have compatible essential signs; exactly 32
   have impossible cyclic strict comparisons; 12440 are constructed and
   verified. The degree-three threshold catalog contains the standard 104
   functions; completeness of this small catalog is checked against the
   published threshold counts, not assumed from an unbounded coefficient
   search. C3 is labeled a non-RBM diagnostic throughout.

Each checker ran under Python 3.9.6 and 3.13.5; deterministic reports match
apart from runtime version. Analytic proofs establish the general results;
finite tests verify implementations and small instances only.

## Source and novelty boundary

Cueto--Morton--Sturmfels and Montufar--Morton provide the standard inference
framework. Aledo et al.'s steady-state symmetrization theorem has different
scope. The Cosnard--Goles energy paper was checked at abstract level only.
No claim is made that rescaling or symmetric threshold networks are new.
The bounded literature audit found no exact counterpart of these count
formulas in the works examined; historical priority is not certified.
The live AimPL item was inaccessible. Its frozen wording is retained but
does not establish that the workshop intended bidirectional inference.

## Artifact and review status

The standalone LaTeX source compiled successfully in the built-in editor.
The exported PDF was compiled with the existing Tectonic installation;
no TeX warning, missing glyph, undefined reference or overfull box was found.
All six rendered pages were visually checked. The title's author line
contains only Alper Ferudun; affiliation and contacts are in a footnote.
AI assistance is disclosed in normal prose.

This is a detailed originating-researcher self-audit. Independent human
review and proof-assistant certification have NOT occurred. The source ZIP
includes analytic proofs, audits and reproducibility code/output, excluding
third-party PDFs and full-text source extracts. Package hashes are recorded
in manifest.json; creation of a package alone is not publication or a DOI.