bool_hasIdentityStep
plain-language theorem explainer
The Boolean realization of the Law of Logic satisfies the identity-step condition, asserting existence of a carrier element whose comparison cost to zero is nonzero. Researchers testing discrete carriers for Universal Forcing cite this to confirm the propositional instance passes the initial structural filter before arithmetic extraction. The proof is a direct one-line application of the general nontriviality lemma for LogicRealizations.
Claim. Let $R$ be the Boolean realization with carrier type Bool, cost type Nat, comparison function boolCost, and zero element false. Then $R$ satisfies the identity-step property: there exists $x$ in the carrier such that the comparison cost of $x$ to the zero element is nonzero.
background
The module DiscreteLogicRealization introduces the second Law-of-Logic realization: a discrete Boolean/propositional carrier serving as the first non-continuous test case for Universal Forcing. A LogicRealization is a structure supplying a carrier type, a cost type equipped with zero, a comparison map from carrier pairs to cost, and a distinguished zero element; each realization supplies its own topology or order through the propositions it carries. The identity-step property of a realization R is the proposition that there exists an element x in R.Carrier such that R.compare x R.zero is not equal to zero; this datum is the source from which ArithmeticOf extracts arithmetic.
proof idea
This is a one-line wrapper that applies the theorem hasIdentityStep_of_nontrivial to boolRealization.
why it matters
This result verifies that the discrete propositional carrier meets the identity-step requirement needed to extract arithmetic in the Recognition framework. It supplies a concrete non-continuous instance for the Universal Forcing program and supports downstream invariants such as arithmetic preservation and Peano-surface properties within the same module. The declaration sits inside the forcing chain where realizations must satisfy structural laws before constants or dimensions are derived.
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