The Orientation Boundary for Step-Duplicating Recursors: Mechanized Impossibility, Escape, and Certification
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866918515968049152 |
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| author | Rahnama, Moses |
| author_facet | Rahnama, Moses |
| contents | We formalize the orientation boundary for first-order step-duplicating recursors, centered on the Right-Duplicating Recursor Schema (RDRS), $\mathrm{recur}(b,s,\mathrm{succ}(n))\to\mathrm{wrap}(s,\mathrm{recur}(b,s,n))$. In Lean 4, the no-go side excludes twelve base direct-measure classes (two unconditional, six scalar growth, four tracked vector / pair), with arctic / tropical matrix continuations, a WPO-facing polynomial-branch corollary, and a KBO obstruction. Four meta-theorems organize the stack: projected-primary dominance, scalar-projection lift, mixed-matrix scalarization, and the symbolic comparator barrier. The surface spans 72 schema-level dup-step impossibilities and 80 concrete-system global-step theorems, with a 76-row RDRS method-universe closeout and a semantic capstone proving every payload-erasing semantic direct measure is counter-dominated.
The successful side carries a transparency-essentiality witness, a dependency-pair projection escape, a generalized polynomial barrier under frozen-base failure, computable witness extractors, a coefficient-table decision procedure, and mutual-recursion / synchronized-SCC barriers. The witness calculus KO7 has a two-layer chain. Its guarded fragment is strongly normalizing, root-confluent, and normalizable, with single-exponential contextual derivation bounds and an exact $ω^ω$ ordinal calibration below $ω^ω\cdot 2$. The full unguarded system is root-terminating via a nonlinear polynomial witness and a specialized MPO, with context-closed strong normalization lifted through every constructor position. A checked TPDB export and a Lean-side replay of the FAST certificate connect the development to TTT2 / CeTA. To our knowledge, this is the first mechanized object-level barrier theorem on a fixed terminating system, proved without reductions or undecidability arguments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_00081 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Orientation Boundary for Step-Duplicating Recursors: Mechanized Impossibility, Escape, and Certification Rahnama, Moses Logic in Computer Science Logic 68Q42, 03F07, 03B40 F.4.1; F.3.1 We formalize the orientation boundary for first-order step-duplicating recursors, centered on the Right-Duplicating Recursor Schema (RDRS), $\mathrm{recur}(b,s,\mathrm{succ}(n))\to\mathrm{wrap}(s,\mathrm{recur}(b,s,n))$. In Lean 4, the no-go side excludes twelve base direct-measure classes (two unconditional, six scalar growth, four tracked vector / pair), with arctic / tropical matrix continuations, a WPO-facing polynomial-branch corollary, and a KBO obstruction. Four meta-theorems organize the stack: projected-primary dominance, scalar-projection lift, mixed-matrix scalarization, and the symbolic comparator barrier. The surface spans 72 schema-level dup-step impossibilities and 80 concrete-system global-step theorems, with a 76-row RDRS method-universe closeout and a semantic capstone proving every payload-erasing semantic direct measure is counter-dominated. The successful side carries a transparency-essentiality witness, a dependency-pair projection escape, a generalized polynomial barrier under frozen-base failure, computable witness extractors, a coefficient-table decision procedure, and mutual-recursion / synchronized-SCC barriers. The witness calculus KO7 has a two-layer chain. Its guarded fragment is strongly normalizing, root-confluent, and normalizable, with single-exponential contextual derivation bounds and an exact $ω^ω$ ordinal calibration below $ω^ω\cdot 2$. The full unguarded system is root-terminating via a nonlinear polynomial witness and a specialized MPO, with context-closed strong normalization lifted through every constructor position. A checked TPDB export and a Lean-side replay of the FAST certificate connect the development to TTT2 / CeTA. To our knowledge, this is the first mechanized object-level barrier theorem on a fixed terminating system, proved without reductions or undecidability arguments. |
| title | The Orientation Boundary for Step-Duplicating Recursors: Mechanized Impossibility, Escape, and Certification |
| topic | Logic in Computer Science Logic 68Q42, 03F07, 03B40 F.4.1; F.3.1 |
| url | https://arxiv.org/abs/2512.00081 |