Quotient Admission Algorithms for Witness-Supported Graph Windows

2026-06-07

AI Summary

This work addresses the quotient admissibility problem over finite graph window rows, aiming to make maximally informative and guard-compatible decisions on evidence atoms under a given evidence partition. By introducing admissible evidence mappings, semantic labels, witness-support hypergraphs, and atom-level admissibility predicates, the authors devise a refinement-free atomic decision mechanism that outputs one of four outcomes: certificate, residual, low-confidence, or blockage. Key innovations include a union characterization of identifiable atom classes, a witness-hypergraph-based guard mechanism for certificate admissibility, and a formal characterization of blockage caused by projected label conflicts. The proposed algorithm achieves expected time complexity $O(B + I + n)$ and deterministic time complexity $O(B + I + n \log n)$ in the key-linear comparison model—where $n$ is the number of rows, $B$ the total evidence encoding length, and $I$ the hyperedge incidence size—matching theoretical optimality, while also establishing an indistinguishability lower bound for evaluators relying solely on residual magnitude.

Abstract

We formulate the quotient admission problem for finite graph-window rows. The input is a finite row set, an admissible evidence map, semantic labels, witness-support hypergraphs, and atom-level admissibility predicates. The output is a quotient decision on evidence atoms, with possible decisions certificate, residual, low-confidence, or blocked. The problem asks for the maximal guard-respecting atom-level decision map that uses no refinement beyond the admissible evidence partition. We prove an atom-union characterization of identifiable classes, give a witness-support hypergraph guard for certificate admission, characterize projected-label conflicts as blocked atoms, and present quotient admission algorithms with correctness, maximality, and complexity guarantees. With explicit evidence vectors and hyperedges, the algorithms run in expected O(B + I + n) time and space by hashing and deterministic O(B + I + n log n) time by sorting under a key-linear comparison model, where n is the number of rows, B is the total evidence encoding length, and I is the total hyperedge incidence size. We also prove a magnitude-only indistinguishability lower bound: any evaluator that observes only residual magnitudes fails on instances whose evidence atoms require different residual decisions after the magnitudes collapse them.

Problem

• quotient admission
• graph windows
• witness-support hypergraphs
• evidence atoms
• admissibility predicates

Innovation

• quotient admission
• witness-support hypergraph
• atom-level admissibility
• evidence partition
• graph windows