IdentityL
plain-language theorem explainer
Identity law for comparison operators on the recovered real line: self-comparison of any positive element returns zero. Anyone packaging the recovered-real Law of Logic cites this field. It is a pure Prop definition (no proof obligation), the native LogicReal mirror of the analytic identity axiom.
Claim. A binary comparison operator $C$ on the recovered real line satisfies the identity law when $C(x,x)=0$ for every recovered real $x>0$, where $0$ is the image of the ordinary real zero under the canonical embedding.
background
This module is the recovered-real mirror of the analytic Law-of-Logic development. Structural axioms (identity, reciprocal symmetry, scale invariance, non-triviality) are stated directly on LogicReal; analytic regularity is transported later via the real embedding.
LogicReal is the Cauchy completion of the recovered rationals, wrapped so Mathlib's completed line can be reused without polluting global instances. The map that sends an ordinary real into this line is the canonical transport used to name the zero on the right-hand side.
A comparison operator here is simply a binary map on recovered reals. The identity law says that comparing any positive element to itself yields that transported zero: self-recognition carries no cost.
proof idea
Definitional only. The declaration is a Prop abbreviation: the universal quantification over positive recovered reals with self-comparison equal to transported zero. No tactics, no lemmas, no proof term beyond the predicate itself.
why it matters
This is one of the four structural fields of the recovered-real Law of Logic package. The parent structure bundles identity with non-contradiction, scale invariance, and non-triviality, then attaches a transported analytic regularity surface.
The companion transport theorem lifts the native identity field to the already-verified real identity axiom by unfolding the comparison transport and rewriting positivity and zero along the real embedding. Together they keep the structural half of the Law of Logic native to recovered reals while reusing the analytic half proved on ordinary reals.
In the broader forcing chain this is bookkeeping for the cost/comparison side of recognition, not a new T5–T8 step; it ensures the J-cost identity $J(1)=0$ style constraint is available on the logic-recovered continuum.
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