# Verification report

Complete self-audited proof of exact counting and exact uniform sampling for successful pressing sequences on all labeled bicolored graphs of underlying maximum degree at most two. Includes full sequences, singular paths/cycles, disconnected unions, empty graphs and zero-count inputs. One complete restricted-class theorem and one auxiliary published2023 path-counting answer; no closure of the general-graph AMR-030-0046 record or rapid-mixing question.

Written proof: binary unit-pivot congruence reduces rank by one; induced-subset restriction preserves legal moves; successful orders have rank length; choosing the last vertex yields the determinant-gated full-order recurrence. Paths give two independent intervals and binomial shuffles. Path nullity<=1 and cycle nullity<=2 enumerate all possible unpressed sets. Component ranks give the final multinomial. Exact mixed-radix unranking provides a bijection to the sequence set. Counting has deterministic polynomial bit complexity; fair-bit uniform sampling has EXPECTED polynomial complexity. The implementation uses a Pascal table for binomial coefficients.

Independent forward checker: literal closed-neighborhood complementation on8191 colored paths through order12,2040 colored cycles through order10, and300 seeded unions. Both full and successful conventions give21062 matching count comparisons. Exhaustive unranking through order7 validates38426 distinct legal successful sequences. The known coloring-average identity is also checked. These finite tests support, and do not replace, the universal written proof.
Checker SHA-256: 0e7dd9844480c80a3f1619fc2108331ef428d1cb15146f2676582cf1ddd5b8c3.
Certificate SHA-256: 960c13de5a3f6c1a9f1c235a8ee577111f7ea6544f644a60fadb6d7f944dce17.

Sources: Cooper-Davis's binary matrix formulation, Bixby-Flint-Miklos path connectivity, Miklos2023 section4.8, Cooper-Whitlatch unique pressability, Cooper-Gartland-Whitlatch V-posets, and Bailey et al.2024 different genome operations. Original auxiliary question PDF23/printed71 was visually inspected. No matching exact path/cycle counting algorithm was found in the bounded search; absolute priority is not certified. No independent human review or proof-assistant verification is claimed. No gap was found in the originating-system audit.

Six-page English PDF compiled with the built-in editor and Tectonic, with all pages visually inspected. Only the name is prominent; affiliation/contact is a footnote; AI assistance is disclosed in prose.