Lnal Vm L Step Core Breath

The breathing step

A recognition cycle, in the Recognition Science framework, is a discrete record of events where each event costs something to account for. The framework's virtual machine tracks that cost in a counter it calls breath. The declaration lStep_core_breath is a theorem about that counter: it proves that one step of the machine's execution always leaves the breath value below its configured limit of 1024.

The theorem is a safety property. It says that no matter what program the machine runs, after a single step the breath counter is still within its bound. The proof is direct from an arithmetic lemma, and the machine-checked library records it as preservation_breath. A companion theorem, progress, proves the machine always has a next step to take. Together they say the machine never gets stuck and never exhausts its per-step budget.

The declaration does not claim that the breath counter never reaches its limit over many steps. It proves only that each step preserves the bound, not that the bound is never hit. It also does not claim anything about what the breath counter means physically, or that the limit 1024 is derived from the framework's forcing chain. Those are separate questions that the declaration leaves open.

/-- Preservation (breath bound): stepping keeps breath within period. -/
theorem preservation_breath (P : LProgram) (s : LState) :
    BreathBound (lStep P s) := by
  -- direct from arithmetic lemma
  simpa [BreathBound] using l_breath_lt_period P s

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-- Hardware test vectors and invariants
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/-- Progress: every state can take a small‑step (possibly self if halted). -/
theorem progress (P : LProgram) (s : LState) : ∃ s', LStepRel P s s' := by
  exact ⟨lStep P s, lStep_as_rel P s⟩

What this page does not claim

The breath counter never reaches its limit over many steps. The breath limit of 1024 is derived from the framework's forcing chain. The breath counter has a physical meaning beyond being a cost tracker.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/LNAL/VM.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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