_narrative_ok
The definition supplies an equivalence showing that the Peano carrier extracted from the arithmetic of the strict narrative realization matches LogicNat exactly. Domain auditors and forcing-chain verifiers cite it to confirm that narrative beat-succession data produces the same forced naturals as the core law-of-logic construction. The proof is a one-line wrapper that invokes the orbit equivalence already established for the corresponding lightweight realization.
claimLet $R$ be the strict narrative realization whose carrier is the event-state type and whose cost is the natural numbers. Then the carrier of the Peano structure inside the arithmetic derived from $R$ is equivalent to the inductive type LogicNat whose constructors are the zero-cost identity and the successor step.
background
StrictLogicRealization is the structure that packages a carrier type, a cost type with zero, a comparison map, and a composition operation, all required to be native to the law of logic with no external orbit supplied. The sibling definition strictNarrativeRealization instantiates this structure using beat succession as the generator, taking EventState as carrier and Nat as cost. LogicNat is the inductive type whose identity constructor represents the zero-cost multiplicative identity and whose step constructor iterates the generator, thereby reproducing the orbit {1, γ, γ², …} as the smallest subset of positive reals closed under multiplication by γ and containing 1. The arith operation extracts the forced arithmetic from the lightweight realization obtained by toLightweight, and the module AxiomAudit supplies surface-level checks that each domain realization recovers the same LogicNat.
proof idea
One-line wrapper that applies the orbitEquivLogicNat field of the lightweight realization produced by toLightweight on strictNarrativeRealization. No additional tactics or lemmas are invoked; the equivalence is obtained directly from the canonical construction that every strict realization yields arithmetic canonically equivalent to LogicNat.
why it matters in Recognition Science
The definition closes the narrative audit inside the strict Universal Forcing completion pass, confirming that beat-succession data produces the identical forced arithmetic as the core law-of-logic construction. It directly supports the upstream claim that every strict realization has forced arithmetic canonically equivalent to LogicNat. In the Recognition Science framework this step aligns with the T0–T8 forcing chain in which arithmetic emerges from the law of logic before spatial dimensions or physical constants are derived.
scope and limits
- Does not establish the equivalence for realizations other than the narrative one.
- Does not supply computational or efficiency properties of the equivalence.
- Does not address whether the equivalence preserves additional structure beyond the carrier.
- Does not discharge any open hypothesis about the full set of domain audits.
formal statement (Lean)
73def _narrative_ok :
74 (StrictLogicRealization.arith strictNarrativeRealization).peano.carrier
75 ≃ ArithmeticFromLogic.LogicNat :=
proof body
Definition body.
76 (StrictLogicRealization.toLightweight strictNarrativeRealization).orbitEquivLogicNat
77