MRSR: From System Comparison to Decision Licensing — A Mathematical Framework for Verifiable Ranking and Constrained Routing of AI Agents

We present MRSR, a mathematical and operational framework that transforms the comparison of AI agent configurations into a verifiable decision path ending in either a category-scoped decision certificate or a justified abstention. The framework integrates five analytically proven theoretical papers (Parts 1–5 of the MRSR series) into a single inferential chain: D → d_eff → r_ord → C → {SWITCH, ABSTAIN} Part 1 establishes an exact Hilbert-space representation of performance differences, with sharp dimension bound dim V_min ≤ m−1 and matrix certificate K_vis = BᵀΓ†B. Part 2 introduces the ordinal dimension r_ord, proves the chain tsrank_0 ≤ r_ord ≤ d_eff, and characterizes realizable compression operators by n_frac(T) = rank(T−T²). Part 3 proves a sharp reference-sensitivity bound |Δ^π − Δ^0| ≤ 2m·TV, a C=6 instability theorem, and the semialgebraic structure of level sets. Part 4 develops a four-tier decision architecture with soundness, completeness, and uniqueness guarantees. Part 5 establishes the universal asymptotic constant c* = p_min/2 across self-concordant links (a 16× improvement over Ostrovskii–Bach), explicit design constants for the balanced C1 design, and a negative result excluding a universal Berry–Esseen constant. The accompanying experiment (partial deployment test) covers 4 code models (Qwen-Coder-1.5B/7B, DeepSeek-Coder-1.3B, CodeGemma-2B) over 164 HumanEval tasks, with 656 runs, 97 unit tests, and 7 adversarial verifier tests. The honest result: MRSR achieves 51/64 (79.7%), identical to the naive fallback policy P2, with no statistically significant accuracy improvement (McNemar exact p = 0.0625). The certificate's value lies in semantic verification (7/7 adversarial certificates rejected, including a certificate with valid SHA-256 but false mathematical claim), principled abstention (Tier 3 activates when Tier 1/2 fail), and transparency — not in raw accuracy. The current contribution is the demonstrated feasibility of certificate-guided abstention, not a claim of routing superiority. The five theoretical papers are provided as a complete mathematical resource; the experiment is deliberately partial and does not cover most predictions of Parts 2, 3, 5 (documented in Table 1 of the paper).