metaForcedArithmeticInvariance_self
plain-language theorem explainer
Any LogicRealization compared to itself induces the identity equivalence on its natural-number orbit under the universal forcing meta-theorem. Workers on the reflexive closure of the Recognition Science framework cite this result to establish that the meta-theorem itself satisfies the required structural shape. The proof applies the uniqueness property of the natural number object recursor to equate the canonical map with the identity.
Claim. For any meta-carrier $R$, the forced-arithmetic-invariance comparison of $R$ with itself equals the reflexive equivalence on the orbit of $R$.
background
The meta-carrier is the type of LogicRealization.{0,0} instances, living in Type 1. The module shows that the universal forcing meta-theorem fits the Law-of-Logic structural shape via a meta-cost that is zero on propositional equality, plus the three Aristotelian conditions and the forced-arithmetic-invariance condition. The orbit of any such realization carries a natural number object structure whose recursor uniqueness is the key prior result.
proof idea
Apply Equiv.ext to reduce to pointwise equality on the orbit. Unfold metaForcedArithmeticInvariance and the NNO equivalence, then build the identity map's preservation of zero and successor. Invoke recursor_unique from the orbit's NNO property and conclude by symmetry that the canonical equivalence equals the identity.
why it matters
This supplies the forced-arithmetic-invariance condition inside MetaRealizationCert, which records that the meta-theorem completes the Law-of-Logic structure for the reflexive case. It directly advances the reflexive closure described in the module documentation, confirming the framework is closed under its own comparison law without instantiating the full heavy LogicRealization axioms at the meta level.
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