composition_status
plain-language theorem explainer
The composition_status definition supplies a status string that confirms completion of the composite recognizer construction and its refinement properties in Recognition Geometry RG6. Workers on multi-recognizer measurement models cite it to confirm that indistinguishability under the product equals the conjunction of component relations and that the composite quotient refines both inputs. The definition is a direct string literal concatenation of verification bullets followed by an #eval directive.
Claim. The status string for the composition module asserts that the composite recognizer $R_1 ⊗ R_2$ is defined, that $c_1 ∼_{12} c_2$ if and only if both $c_1 ∼_1 c_2$ and $c_1 ∼_2 c_2$, that the composite refines each component, that resolution cells are intersections, and that the quotient maps are surjective, with the overall refinement theorem holding.
background
In Recognition Geometry, a Recognizer on configuration space C with event space E maps configurations to events and induces an indistinguishability relation ∼ on C. The CompositeRecognizer is the product construction $R_1 ⊗ R_2$ that returns the pair of events; its induced relation is the conjunction of the two component relations. The module develops the Refinement Theorem showing that the composite quotient is finer than either factor quotient. This setting rests on the Recognition Composition Law and the earlier forcing chain T0–T8 that fixes the functional form of J and the emergence of three spatial dimensions.
proof idea
The declaration is a definition that builds the status string by successive string concatenation of fixed verification messages. Each bullet names a sibling result (CompositeRecognizer, composite_indistinguishable_iff, composite_refines_left, quotientMapLeft_surjective, etc.) whose proofs appear earlier in the same file. The final line invokes #eval to force evaluation and display.
why it matters
This definition closes the RG6 module by recording that the composite recognizer supplies strictly more distinguishing power than either factor, directly supporting the information-monotonicity theorems composite_info_monotone_left and composite_info_monotone_right. It therefore supplies the concrete mechanism by which richer geometry arises from multiple measurements, feeding any later work that composes recognizers to recover spatial structure or the alpha band. No open scaffolding remains inside the listed items.
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