repr_equiv_refl
plain-language theorem explainer
If a same-type reference structure has vanishing self-cost, representation equivalence is reflexive: every configuration is perfectly equivalent to itself. Anyone building the Algebra of Aboutness cites this as the first equivalence-law check. The proof is a two-line constructor application of the self-cost hypothesis.
Claim. Let $R$ be a reference structure on a common configuration type $C$ (nonnegative cost of one configuration pointing to another). If $R.\mathrm{cost}(x,x)=0$ for every $x\in C$, then for every $x$, $x$ is representationally equivalent to itself: mutual reference costs vanish, $R.\mathrm{cost}(x,x)=R.\mathrm{cost}(x,x)=0$.
background
The module formalizes the Physics of Reference: aboutness is cost-minimizing compression. A reference structure supplies a nonnegative cost of symbol $s$ pointing to object $o$. When symbol and object share a type $C$, one can ask when two configurations are the same for referential purposes.
Representation equivalence is the induced semantic relation: $x$ and $y$ are equivalent when mutual reference costs both vanish ($R.\mathrm{cost}(x,y)=0$ and $R.\mathrm{cost}(y,x)=0$). Perfect self-reference is not automatic from nonnegativity alone; it is an extra hypothesis that self-loops cost zero, matching the RS picture that $J$-balanced or zero-defect self-identity is free.
This sits beside ratio-induced reference from the RS cost $J(x)=\frac12(x+1/x)-1$, the triangle inequality for chained reference, and the claim that zero-cost configurations form a mathematical backbone with universal referential capacity.
proof idea
Term-style tactic proof with no external lemmas. Introduce an arbitrary $x$. Representation equivalence at $(x,x)$ is the pair of equalities $R.\mathrm{cost}(x,x)=0$ and $R.\mathrm{cost}(x,x)=0$. Both conjuncts are exactly the self-cost hypothesis applied at $x$, so the constructor closes immediately.
why it matters
Module main results list representation equivalence as the semantic equivalence relation induced by reference (perfect mutual zero cost). Reflexivity is the first of the three equivalence laws; without it the relation cannot serve as identity of meaning under cost-minimizing aboutness.
It anchors the Algebra of Aboutness next to reference-from-asymmetry, the mathematical backbone of zero-cost configurations, ratio-induced reference from $J$, and the effectiveness principle for near-balanced symbols. In RS terms it aligns with existence as defect collapse to zero and with recognition as reference: free self-reference is the ledger identity case.
No downstream dependents are recorded yet; the natural next steps are symmetry and transitivity of the same relation, so the full equivalence can support quotienting configurations by perfect mutual reference.
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