The flow-invariant criterion

The Flow-Invariant Criterion This conceptual seed proposes a criterion for distinguishing physically meaningful stabilising structures from merely formal conserved quantities. The central idea is that a stability mechanism is reconstructively legitimate only if it originates from the intrinsic structure of the admissible-state flow. Purpose The note addresses a foundational question: why should a conserved quantity protect an object from decay? Rather than accepting conservation laws at face value, it asks whether the corresponding stability can be derived from the geometry and dynamics of the reconstructive flow itself. Flow Symmetries The admissible-state manifold is equipped with a Fisher geometry and a reference measure defining the relative-entropy potential. The covering flow is the entropy-producing gradient flow generated by this potential. A flow symmetry is defined as a transformation preserving both the Fisher metric and the reference measure. Such transformations automatically preserve the relative-entropy potential and commute with the reconstructive flow. Legitimate versus Illegitimate Stabilisation The note introduces a distinction between: Legitimate stabilisers, arising from intrinsic flow symmetries or from topological classes preserved by admissible deformations; Illegitimate stabilisers, corresponding to quantities that are formally conserved but are not realised as symmetries of the reconstructive flow and do not label a topological sector. The criterion is explicitly non-circular: membership in the flow-symmetry group must be verified from the underlying geometry rather than assumed from the existence of a conserved charge. Structure Theorem Under regularity and completeness assumptions, the note proposes a structural classification of all flow invariants. Two species emerge: continuous invariants associated with one-parameter symmetry groups (Noether-type structures); discrete invariants associated with homotopy or cohomology classes (topological structures). The resulting theorem states that every reconstructively legitimate stabiliser belongs to one of these two classes. Hidden-Modulus Problem A major open issue concerns the possibility of hidden invariant directions inside degenerate critical strata. If such directions exist, they would define a third species of reconstructive identity, neither symmetry-based nor topological. The note identifies the Łojasiewicz exponent of the entropy potential as a possible candidate for such a hidden modulus and discusses the conditions under which it could arise. Physical Interpretation The criterion provides a method for testing whether a proposed stabilising charge carries genuine reconstructive content. A quantity protects a state only if it corresponds to a symmetry of the admissible-state flow or to a topological sector preserved under admissible deformations. Conservation alone is insufficient. Status The geometric definition of the flow-symmetry group and the non-circularity of the criterion are established. The proposed classification theorem depends on explicit regularity and no-hidden-modulus hypotheses, which remain open mathematical obligations. Conclusion The note shifts the notion of protection from the existence of a conserved quantity to the existence of a reconstructively legitimate flow invariant. Its central claim is that genuine stability must arise either from symmetry, topology, or a yet-unresolved third species associated with hidden moduli.