LogicRealization
plain-language theorem explainer
LogicRealization supplies the minimal interface of a carrier set equipped with a comparison cost, zero element, generator step, and an orbit satisfying induction together with equivalence to LogicNat. Researchers extracting uniform arithmetic from disparate Law-of-Logic settings cite the structure when constructing ArithmeticOf objects. The declaration is a direct structure definition that assembles the listed fields and proposition carriers without further reduction steps.
Claim. A structure consists of a carrier type $C$, a cost type $K$ with zero, a comparison map $C×C→K$, a zero element $0∈C$, a successor $s:C→C$, an orbit type $O$ with its zero and successor, an interpretation $ι:O→C$ that preserves zero and successor, an equivalence $O≃LogicNat$ preserving zero and successor, the axioms compare$(x,x)=0$ and compare$(x,y)=$compare$(y,x)$, the existence of a nontrivial element with compare$(x,0)≠0$, and three proposition fields for excluded middle, composition, and action invariance.
background
LogicRealization provides the setting-independent interface for the Universal Forcing program. The module creates the common object into which different Law-of-Logic settings, continuous positive ratios, discrete propositions, or categorical structures, can be mapped. The invariant target is the arithmetic object extracted from the identity-step data rather than the ambient carrier.
proof idea
This declaration is a structure definition. It directly encodes the carrier, cost, zero, step, orbit, interpretation, equivalence to LogicNat, and the listed axioms and proposition fields without invoking lemmas or tactics.
why it matters
LogicRealization supplies the data from which ArithmeticOf extracts the Peano object and its initiality, including the canonical and extracted variants that produce Peano surfaces. It fills the interface step in the Universal Forcing program, allowing realizations from kinship graphs, nucleosynthesis tiers, and cellular automata to feed into uniform arithmetic extraction. The structure ensures the extracted arithmetic remains independent of the specific topology or order supplied by each realization.
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papers checked against this theorem (showing 1 of 1)
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TTD homomorphism yields low-memory split-beam dictionary
"Corollary 1. ∀Φ1, Φ2 ∈ V1, PΦ1+Φ2 = PΦ1 ⋆ PΦ2; <PT, ⋆> defines a group structure"