Optimal Quotient Rigidity for Affine-Permission Representations of Paths, with a Constructive Converse and a Tree Obstruction

This deposit contains two companion research manuscripts on exact affine-permission (QA) representations of Boolean relations over the binary field. An input selects one affine-or-empty permission per coordinate and prescribes a linear quotient of a shared state; acceptance is nonempty intersection with an arbitrary global filter. The measured resource is total state dimension, not running time or filter-description size. The main manuscript establishes the matching-number dimension formula for forest vertex-cover relations and a necessary rigidity theorem for optimal path representations: the entire deterministic quotient rowspace must lie in one common member of an explicit staircase family (or the alternating coordinate space for odd paths). The supplementary manuscript gives explicit constructions realizing every subspace permitted by the path theorem, completing the minimum-dimension classification. It also exhibits connected trees with nested prescribed quotient spaces for which passing to the smaller space increases the minimum state dimension, and gives resource-explicit comparisons with binary literal-labelled span programs for the affine-filter subclass. Contents: (1) Optimal Quotient Rigidity for Affine-Permission Representations of Paths; (2) A Constructive Converse for Paths and a Quotient Obstruction on Trees. The supplement depends on the main manuscript's forest-dimension and path-necessity results; both are included in this deposit. Status and scope: AI-assisted research preprints with internal mathematical and computational checks. No independent external peer review, proof-assistant verification, or definitive novelty determination is claimed. Finite tests are distinguished from the written general proofs. The work does not resolve P versus NP, provide a polynomial-time SAT algorithm, or establish a general circuit lower bound. Code and computational evidence: This deposit contains the two manuscripts, not the complete computational evidence or an executable capsule. The historical checking suite and full finite-audit evidence are retained separately by the author. The GitHub repository remains private. To request access to the repository or supporting verification files for scholarly review, contact Baran Dindar at [email protected], identifying the requested materials and, for repository access, your GitHub username. Access is subject to the author's approval; no automatic or public access to these separate materials is granted by this deposit. The license on this record applies to the two deposited PDFs and does not grant a license to the separately retained code or evidence.