recognition_is_equivalence
plain-language theorem explainer
Zero-cost mutual reference is an equivalence relation on configurations whenever the reference cost is reflexive, symmetric, and obeys the triangle inequality. Cite this when identifying recognition events with representation equivalence in the Algebra of Aboutness. The proof is a three-case constructor: refl and symm are the given axioms; transitivity sandwiches the triangle inequality between nonnegativity and antisymmetry of real order.
Claim. Let $R$ be a reference structure on a type $C$ (a map $R:C\times C\to\mathbb{R}$ with $R\ge 0$). Suppose $R(x,x)=0$ for all $x$, $R(x,y)=R(y,x)$ for all $x,y$, and $R(a,c)\le R(a,b)+R(b,c)$ for all $a,b,c$. Then the relation "$R(a,b)=0$" is an equivalence relation on $C$.
background
The module formalizes the Physics of Reference: aboutness is cost-minimizing compression. A reference structure supplies a nonnegative cost of one configuration pointing to another. Recognition is identified with zero-cost reference: a recognition event from $a$ to $b$ means exactly that the reference cost vanishes.
The ambient thesis is that symbols emerge when complex ($J>0$) objects force cheaper pointers, and that zero-cost configurations form the mathematical backbone with universal referential capacity. Representation equivalence is the same zero mutual cost relation under another name.
The three hypotheses are the pseudometric axioms beyond the structure's built-in nonnegativity. Triangle inequality is already a named main result of the module (chained reference bounds direct reference). Nonnegativity of cost is part of the reference-structure data and is used to close equality after the triangle bound collapses to $0\le R(x,z)\le 0$.
proof idea
Build the Equivalence record in three cases.
Reflexivity is immediate from the hypothesis $R(x,x)=0$.
Symmetry: unfold the zero-cost relation, rewrite by the symmetry hypothesis on cost, and reapply the hypothesis.
Transitivity: given $R(x,y)=0$ and $R(y,z)=0$, apply the triangle inequality to get $R(x,z)\le R(x,y)+R(y,z)$. Substitute the two zeros, simplify to $R(x,z)\le 0$, then use antisymmetry of $\le$ against the structure's nonnegativity $0\le R(x,z)$ to conclude $R(x,z)=0$.
why it matters
This is the equivalence-relation half of the module's claim that recognition is reference and coincides with representation equivalence. It turns the semantic slogan "recognition events form an equivalence" into a checked theorem under the standard cost axioms.
In the broader Recognition Science stack it sits with Law of Existence (existence as defect collapse to zero), LedgerForcing (reference events as ledger entries), and RecognitionForcing (recognition as reference). Zero mutual $J$-cost is the same numerical condition that marks perfect symbols and the mathematical backbone elsewhere in the module.
No downstream dependents are recorded yet; the lemma is infrastructure for any later quotient by representational sameness, unique-meaning arguments, or perfect-reference criteria that need a proven equivalence rather than a raw Prop.
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