Terminal Symmetry as a Carrier of Asymmetric Process Knowledge: Statewise Refinement for Anytime Verified Construction

Reasoning shortcuts are solutions of a neurosymbolic system's rules that produce correct predictions through unintended concepts. A recent framework of Takemura, Inoue, and Nishino analyzes them through an automorphism group of value relabelings and asks, as its central open question, when rules pin concepts down. We first show that the framework's key definition, one shared permutation applied at every position, does not apply as stated to any of the four heterogeneous benchmarks it was evaluated on, and that the most direct embedding, padding domains to a common size, produces confident false pathology: 90.91% of solution pairs reported unexplained on CLE4EVR, where every well-defined member of the hierarchy we introduce reports 0%, and the padded verdict's content rotates with configuration-file ordering. Re-measuring eleven rule families under fifteen pre-specified predictions (thirteen confirmed), unexplained-pair rates span 0% to 99.9999% and track provable structure: six theorems give sufficient conditions for transitivity and its failure, including a Free Slot Lemma certifying Kandinsky's pathology from syntax alone. For circuit-given rules, deciding symmetry-inertness of a coordinate is coNP-complete; nontrivial-automorphism existence is coNP-hard under randomized reductions, lies in $Σ_2^p$, is not $Σ_2^p$-complete unless PH collapses, and on monotone circuits is coNP-complete outright. In the Boolean case transitivity is classified exactly: automorphisms explain everything iff the solution set is an affine coset. Weakly supervised models place all 94 observed shortcuts at the one level the componentwise theory flags and none at the 48 it certifies transitive; twelve typed-ambiguous levels produce none, separating what symmetry permits from what optimization selects, and a dual-head control replicates the geography. All numbers trace to released artifacts.

Test-time reasoning is increasingly used as a serving-time control knob, but extra reasoning is not uniformly valuable: it can repair failed attempts, waste compute on already-correct answers, or introduce harmful answer changes. We study this as a deployment allocation problem rather than a new-verifier problem. We introduce \sevra, Selective Verification for Reasoning Allocation, a serving-layer controller that decides whether to preserve a frozen solver's initial answer or invoke active verification. Using a frozen Qwen3-4B solver, we log intervention outcomes and train recoverability-aware gates from serving-visible attempt state. On \mathfive, selective verification reaches 76.3% accuracy, compared with 75.5% for always verifying, while reducing post-generation tokens by 26.8% and harmful flips from 2.2% to 1.0%. However, an 8,192-token initial solve reaches 76.0% accuracy with 28% fewer total model tokens, showing that selective recovery is useful but not the best tested cost frontier. In frozen transfer to \gsm, the selective policy verifies only 3.0% of examples, improves accuracy from 93.4% to 94.5%, and reduces verification tokens by 91.2% relative to always verifying; again, a longer initial solve matches its accuracy with fewer realized tokens. On CommonsenseQA, always-on verification hurts, while Self-Consistency@5 improves accuracy at about five times the realized token cost. The resulting deployment rule is: tune the initial budget first, then use selective recovery when explicit checks, bounded retries, auditability, or regression-risk control matter.

Language environments such as web browsers, code terminals, and interactive simulations emit raw text rather than states, and provide none of the runtime structure that MDP analysis requires. No explicit state space, no observation-to-state mapping, no certified transitions, and no termination criterion. We introduce the State-Centric Decision Process (SDP), a runtime framework that constructs these missing inputs by having the agent build them, predicate by predicate, as it acts. At each step the agent commits to a natural-language predicate describing how the world should look, takes an action to make it true, and checks the observation against it. Predicates that pass become certified states, and the resulting trajectory carries the four objects language environments do not provide, namely a task-induced state space, an observation-to-state mapping, certified transitions, and a termination criterion. We evaluate SDP on five benchmarks spanning planning, scientific exploration, web reasoning, and multi-hop question answering. SDP achieves the best training-free results on all five, with the advantage widening as the horizon grows. The certified trajectories additionally support analyses unavailable to reactive agents, including per-predicate credit assignment, failure localization, partial-progress measurement, and modular operator replacement.