strict_metaphysical_ground_unique
This definition equates the Peano carrier of the arithmetic derived from any strict logic realization R to the logic-forced natural numbers LogicNat. Researchers tracing the strict universal forcing chain to its metaphysical ground would cite it to confirm the identification. The definition is a one-line wrapper that applies the orbit equivalence after converting R to the lightweight realization interface.
claimFor any strict logic realization $R$, the Peano carrier of the arithmetic extracted from $R$ is canonically equivalent to the logic-forced natural numbers LogicNat.
background
A StrictLogicRealization supplies a carrier type, cost type with zero element, comparison, and composition, all native to the law of logic with no supplied orbit. The arith operation extracts the forced arithmetic structure from the lightweight conversion of R. LogicNat is the inductive type whose identity constructor is the zero-cost multiplicative identity and whose step constructor iterates the generator, forming the smallest orbit closed under multiplication by γ and containing 1.
proof idea
The definition is a one-line wrapper that applies the orbitEquivLogicNat field of the lightweight realization produced by toLightweight R.
why it matters in Recognition Science
The definition supplies the structural identification required by the axiom audit, directly feeding the _ethics_ok declaration that aligns ethical realizations with forced arithmetic. It completes the metaphysical grounding step in the strict universal forcing package, linking the realization carrier to LogicNat without extra hypotheses. It touches no open questions and closes the identification for downstream audits.
scope and limits
- Does not prove existence or uniqueness of any strict logic realization.
- Does not apply to non-strict or supplied-orbit realizations.
- Does not incorporate physical constants, forcing chain steps, or J-cost.
- Does not address computational or constructive content of the equivalence.
formal statement (Lean)
43noncomputable def strict_metaphysical_ground_unique (R : StrictLogicRealization) :
44 (StrictLogicRealization.arith R).peano.carrier ≃ ArithmeticFromLogic.LogicNat :=
proof body
Definition body.
45 (StrictLogicRealization.toLightweight R).orbitEquivLogicNat
46
47end Metaphysical
48end Strict
49end UniversalForcing
50end Foundation
51end IndisputableMonolith