symmetry_status
plain-language theorem explainer
symmetry_status defines a string constant that reports completion of the symmetry constructions in the Recognition Geometry module. Auditors and developers tracking gauge-like redundancies in recognition-preserving maps cite it to confirm that RecognitionAutomorphism, quotient actions, and GaugeEquivalent are in place. The body is a direct string concatenation of status markers with no computation or lemmas applied.
Claim. symmetry_status is the constant string ``✓ RecognitionPreservingMap defined ✓ symmetry_preserves_indistinguishable proved ... SYMMETRY COMPLETE'' that records the algebraic closure of recognition-preserving transformations and gauge equivalence.
background
Recognition geometry treats symmetries as maps T : C → C that preserve the recognizer output R(T(c)) = R(c). Such maps descend to well-defined actions on the recognition quotient C_R and correspond to gauge redundancies invisible to the recognizer. GaugeEquivalent(r, c1, c2) holds when there exists a RecognitionAutomorphism T with T.toFun c1 = c2, making gauge equivalence an equivalence relation on configurations.
proof idea
This is a definition that directly constructs the status string by concatenating literal completion messages for RecognitionPreservingMap, symmetry preservation lemmas, idSymmetry, compSymmetry, RecognitionAutomorphism, and GaugeEquivalent.
why it matters
The definition serves as the module-level completion marker for the symmetry section of Recognition Geometry. It closes the algebraic structure (identity, composition, inverses) needed before gauge equivalence can be used in downstream recognition quotients and physical interpretations of gauge symmetries.
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