obs_equiv_refl
plain-language theorem explainer
Any ledger configuration is observationally equivalent to itself for a fixed observer subsystem: the observer sees the same entries on both sides. This is the reflexivity half of the equivalence relation on configurations induced by a partial view. Anyone building measurement, outcome uniqueness, or apparent-randomness arguments in the RS ledger cites it. The proof is a one-line reflexivity of equality on observer indices.
Claim. Fix $N\in\mathbb{N}$, a subsystem partition $S$ of the $N$ ledger entries (observer block of size $K$ with $0<K<N$), and a configuration $c$ (positive real ratios on $\mathrm{Fin}\,N$). Then $c$ is observationally equivalent to itself relative to $S$: for every observer index $i$, the entry $c_i$ equals $c_i$.
background
Module F-009 formalizes measurement in Recognition Science: observers are internal subsystems of the ledger, not external agents. Determinism claimed that quantum randomness is projection through finite resolution; this module supplies the mechanism. An observer holds only $K$ of $N$ entries and never sees the complementary system block.
A configuration is an $N$-tuple of positive real ratios (ledger entries). A subsystem is a partition: observer indices of cardinality $K$ with $0<K<N$, and system indices the complement. Two configurations are observationally equivalent relative to that partition when they agree on every observer index. The observer cannot distinguish such pairs; that is a structural fact about partial access, not an approximation.
The module's key claim is that the full state fixes the measurement outcome while the partial view does not, so many full states share one observer view. Reflexivity is the first algebraic property of that indistinguishability relation.
proof idea
One-line term proof. Observational equivalence unfolds to pointwise equality of entries on the observer index set. Instantiating both configurations by the same $c$ yields $c_i=c_i$ at each observer index, discharged by rfl. No lemmas are required beyond the definition.
why it matters
Together with the sibling symmetry and transitivity facts, this makes observational equivalence an equivalence relation on configurations for each fixed observer partition. That relation is the formal carrier of "what the observer can see" in F-009: measurement outcomes, same-state-same-outcome, and the origin of apparent randomness all quantify over or quotient by this relation.
In the broader RS picture it supports the story that determinism plus finite internal resolution produces Born-like statistics without external collapse: the full variational trajectory (global update) fixes the outcome, while the observer's ignorance of complementary entries is the randomness. No downstream theorems currently depend on this declaration in the graph; it is infrastructure for the equivalence-relation package and for later outcome-determination lemmas in the same module.
It does not itself invoke J-cost, the forcing chain T0–T8, or the eight-tick structure; those enter when weighting equivalence classes or coupling via a measurement event.
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