The Orientation Boundary for Step-Duplicating Recursors: Mechanized Impossibility, Escape, and Certification

Moses Rahnama

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arXiv · 2512.00081

The Orientation Boundary for Step-Duplicating Recursors: Mechanized Impossibility, Escape, and Certification

Moses Rahnama

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Abstract

We mechanize an orientation boundary for first-order rewrite systems with step-duplicating recursors, using the Right-Duplicating Recursor Schema $\mathrm{recur}(b,s,\mathrm{succ}(n)) \to \mathrm{wrap}(s,\mathrm{recur}(b,s,n))$. For a reflected scalar grammar over counter and payload coordinates with constants, sums, products, pointwise maxima, and natural scalar multiples, Lean proves $\mathrm{Orients}(e) \iff \mathrm{PayloadBlind}(e) \land \mathrm{CounterStrict}(e)$. On the counter-admissible subclass, orientation is equivalent to payload blindness. The counter projection inhabits the class; the payload-blind constant-zero expression fails counter strictness and orientation, showing the restriction is necessary. A vector extension makes payload blindness necessary whenever strict comparison forces nonincrease of a grammar-expressible scalarization. Twelve scalar and tracked-vector families form the witness-bearing barrier basis, and 80 root-level exclusions lift to context-closed rewriting under their original hypotheses. An escape trichotomy shows that every successful orienter in the stated universe must abandon wrapper-subterm sensitivity, successor transparency, or the formalized families; dependency-pair, nonlinear-polynomial, and multiset-path-order witnesses realize the escape side. For a concrete calculus, Lean certifies strong normalization and confluence of the guarded relation, termination results for the unguarded system, executable normalization and reachability procedures, derivation-length bounds, and ordinal calibrations. Three TTT2 termination proofs receive CeTA 2.36 certification. Machine-checked ledgers close the stated 76-family syntactic universe and a separate 16-row semantic universe. This is an object-level classification for a fixed terminating system and explicit measure classes, not a class-wide undecidability theorem.

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Moses Rahnama. 2025-11-26. The Orientation Boundary for Step-Duplicating Recursors: Mechanized Impossibility, Escape, and Certification. https://arxiv.org/abs/2512.00081

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