perfect_implies_representational_equivalence
plain-language theorem explainer
Mutual zero reference cost between two configurations yields representational equivalence under a self-referential reference structure. Anyone working the Algebra of Aboutness or the Fundamental Theorem of Reference cites this packaging step. The proof is a one-line term constructor: the two vanishing-cost hypotheses are exactly the defining conjunction.
Claim. Let $R$ be a reference structure on a common configuration type $C$ (nonnegative cost of one configuration pointing to another). If $R(x,y)=0$ and $R(y,x)=0$, then $x$ and $y$ are representationally equivalent under $R$: mutual perfect reference holds.
background
This module formalizes the Physics of Reference: aboutness is cost-minimizing compression. A reference structure $R$ on symbol and object types supplies a nonnegative cost $R(s,o)$ for $s$ pointing to $o$. When both types are the same configuration space $C$, one obtains a self-referential structure.
Representational equivalence is the semantic relation induced by perfect mutual reference: $x\sim_R y$ means $R(x,y)=0$ and $R(y,x)=0$. The module thesis is that zero-cost configurations form the mathematical backbone, while positive $J$-cost forces genuine referential asymmetry. Upstream cost notions (observer $J$-cost, ledger balance, multiplicative recognizer cost) supply the nonnegative real-valued costs that reference structures package.
The local setting is the Algebra of Aboutness: reference is forced by cost asymmetry, ratio-induced self-reference costs zero, and near-balanced configurations can refer broadly.
proof idea
Pure definitional packaging. Representational equivalence is defined as the conjunction of the two directed zero-cost statements. The term proof is the anonymous constructor $\langle h_{xy}, h_{yx}\rangle$ that assembles the two hypotheses into that pair. No lemmas, no rewriting, no cost calculus.
why it matters
Closes the elementary bridge from perfect mutual reference to the named equivalence relation used throughout the Algebra of Aboutness. The module's Fundamental Theorem of Reference lists representational equivalence among its five pillars: zero mutual cost is the semantic identity of configurations under a reference structure, while non-mathematical reference carries positive cost and self-reference (ratio-induced) costs zero.
Downstream use is not yet wired in this graph (no used_by edges), but the declaration sits immediately before the Fundamental Theorem block and supplies the clean introduction rule for the equivalence. It does not itself invoke the forcing chain (T5 $J$-uniqueness, T6 $\varphi$, eight-tick octave), yet it is the right interface for later theorems that treat zero-cost pairs as interchangeable symbols in ledger or recognition arguments.
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