bibkey: ramtm2026divider authors: Szymon Toruńczyk and Codex 5.6 year: 2026 title: “Computability and polynomial-time equivalence of Turing machines and word RAMs” doi: null url: https://github.com/szymtor/RAM-TM/tree/6fe5a3d94f6c2da46cc2c5ab98cd36997b130737 claim: “The fixed Boolean-stack restoring divider has exact positive-divisor quotient and remainder semantics with a quadratic source-step runtime.” strata_touched:

• D5/S0/Computability/PhysicalDivider/WordArithmetic
• D5/S0/Computability/PhysicalDivider/Machine
• D5/S0/Computability/PhysicalDivider/Inspection
• D5/S0/Computability/PhysicalDivider/Shift
• D5/S0/Computability/PhysicalDivider/SubtractLoops
• D5/S0/Computability/PhysicalDivider/Finalize
• D5/S0/Computability/PhysicalDivider/Rounds
• D5/S0/Computability/PhysicalDivider/Positive
• D5/S0/Computability/PhysicalDivider/StackGrowth
• D5/S0/Computability/PhysicalDivider/ArithmeticExecution
• D5/S0/Computability/PhysicalDivider/CallExecution license: Apache-2.0 triage: anchor

Boolean-stack restoring division

The source is RAM-TM commit 6fe5a3d94f6c2da46cc2c5ab98cd36997b130737. Its manifest.yaml credits Szymon Toruńczyk and Codex 5.6. The retained source files contain no additional copyright notice. The complete upstream license is reproduced below; the source archive contains no NOTICE file. No Lax13 source is included: it is absent from the selected declaration closure.

Upstream pins are Lean v4.30.0 and Mathlib c5ea00351c28e24afc9f0f84379aa41082b1188f. These files use this repository’s Lean v4.33.0 and Mathlib db584cd6d46c92f209a44c0f1c829460d327499d. The original namespace Lax51Proofs.RamToTM and substantive proof arguments are preserved. Import routing, explicit unfolding of generated defaults and Nat.ModEq, and equivalent congruence/simplifier syntax are the adaptations. Merely binding helper proofs are placed locally in their original consumers, with source locations retained at each insertion.

The arithmetic theorem reaches the source .done configuration. It is not a theorem about physical bit microsteps, reusable physical return frames, or all-prefix physical storage. The stack-growth theorem must be applied within short rounds when a linear width bound is needed.

Retirement condition: when this repository’s own adopted Mathlib pin provides equivalent declarations, consumers should use those declarations directly. Any retirement or migration remains subject to the repository’s frozen-content rules. Acceptance into a different or hypothetical Mathlib revision is not the retirement condition.

The physical call proof also retains the bitsValue_append_false argument from WordOperations.lean:37-41 locally to identify the executed padding. The inverse fixedBits_bitsValue supplies the fixed-word equality; no additional arithmetic declaration is introduced.

Exact source files

All source paths below are relative to the immutable repository root.

Declaration mapping

Names below are in Lax51Proofs.RamToTM. An inline name identifies original proof material retained locally, rather than an additional top-level declaration.

Full upstream license

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