The flow-invariant criterion

A stabiliser is reconstructively legitimate only if it originates in the intrinsic structure of the admissible-state flow. This revision computes what that structure permits, classifies the legitimate invariants geometrically, and turns the former “third species” from an anomaly into a second legitimate family. Its base is the companion note Causal Succession by Segments: Tunnel Effect, Superposition, and Measurement in the Reconstructive Order (the “segment-order article”). Status markers. E established, granting the segment-order base · C conditional / to verify · O open — named and located, not claimed. Purpose. Why should a conserved quantity protect an object from decay? Rather than accept conservation laws at face value, the note asks whether the corresponding stability follows from the geometry and dynamics of the reconstructive flow itself. This revision does more than ask: it computes the legitimate symmetry group, classifies the invariants, and locates the flavour and extremality questions precisely. The flow and its symmetries — the symmetry group is now computed. On the manifold of admissible states with Fisher metric gF and confining reference μF, the closure potential is the relative entropy φ = H(·∥μF), a strict potential of the causal order (the segment-order article). The covering flow is the entropy-producing gradient flow ∂tρ = −grad φ, with de Bruijn dissipation φ̇ = −IF ≤ 0. A flow symmetry preserves both gF and μF (hence φ and the flow). The flow-symmetry group is Sym(Φ) = Isom(M, gF, μF), the isotropy of the reference — established for both signs of curvature by Wasserstein rigidity, the lone non-rigid base being one the programme already excludes. E Central reframe — two species, split geometrically. The flow-invariants split not by physical type but by their position relative to the critical set Crit(φ): Stratum invariants — tangent to Crit(φ): Noether charges (orbit coordinates within a stratum) and topological classes (separating strata between themselves). This is symmetry and topology. The electron lives here: protected by a charge, present in every snapshot. Germ invariants — transverse to Crit(φ): the Łojasiewicz exponent θ, relaxation rates λ, critical exponents — analytic data of the transverse germ. This is the world of universality and resonance (quasiparticles, decay widths, transport rates), the former “hidden modulus,” now a legitimate family rather than a defect. The two are exhaustive by the splitting lemma: every flow-invariant is stratum data plus germ data, with no third generator — conditional on a stratified-regularity hypothesis, not on a “no hidden modulus” premise. C A germ identity is carried by the exponent, not a conserved state: the identity of a relaxation pattern, read against gF and the de Bruijn clock. Where gauge lives — horizontal versus vertical. The computed group is horizontal (state-moving). Gauge symmetries are the vertical kernel fixing |ψ|², invisible to the push-forward; in the programme’s three dimensions the vertical Clifford commutant is U(1) (electromagnetism), with chirality not primitive but created by the flow (γ5 = γ0·vol). At the level of the measure, invariance of μF under a chiral gauge symmetry is — by Fujikawa — the vanishing of an index, so anomaly cancellation is the same legitimacy criterion read functionally: a necessary filter on a chiral gauge sector, not a generator of one. Legitimate versus illegitimate stabilisation. A stabiliser is legitimate iff it is a flow symmetry or a topological / germ class; a merely-conserved charge that is neither is reconstructively illegitimate (the “illegitimate WIMP” corollary — a statement about reconstructive content, not about our universe). The criterion is explicitly non-circular: membership in the flow-symmetry group is verified from the geometry, never assumed from the existence of a charge. E Global obstruction — inadmissibility as holonomy. Inadmissible reconstruction is a holonomy: the obstruction lives in H¹(R, Sym(Φ)), non-vacuous whenever π1(R) ≠ 0 and independently of the fibre. A concrete transverse class is exhibited — the Maslov ℤ/2 of the totally-geodesic density submanifold in ℂP(L²). E Flavour interstice — a chain of no-goes for Koide. The single datum that would close the internal sector is the projection of the causal order onto the fibre, πF(≺) = ≺F — which would define the internal generator and, if balanced, force the charged-lepton (Koide) null cone. The note develops, then closes, every internal route: A candidate oriented datum (excess-entropy cost CF, conditional potential φF) reduces Koide to a falsifiable computation — a fibre de Bruijn identity plus a spectral check. As an internal closure RG, Koide reduces to whether the closure sum is Kramers–Wannier self-dual. The ℤ₃ closure sum does carry a KW (Potts) self-duality — but its self-dual locus is the democratic point, not Koide; the Koide-fixing involution is the amplitude duality ∥s∥ ↔ ∥d∥, a non-Fourier, non-linearly-realised map. The natural self-duality is the wrong one. The Onsager–Machlup form makes this geometric: the Koide coordinate is KW-even but the hierarchy coordinate is not — the duality reaches Koide, never the hierarchy. The no-go. The amplitude self-duality does not lift to an admissible Fisher isometry of the fibre (the singlet stratum is a point, the doublet a circle). So Koide does not arise from any admissible internal self-duality of the present programme — only from an enriched fibre (new input) or an external origin. The escape. Lifted to the pre-observational fibre F̃ ≅ S³ (the singlet carrying the segment-order article’s closure phase, projected out by observation), the amplitude duality becomes a metric isometry fixing Koide — so Koide may be a pre-observational symmetry broken by the projection, conditional on that phase being physical, and still silent on the hierarchy. O The modular bridge and the Cauchy horizon. The inadmissibility wall may be realised distinctively as the time-reversal / Maslov ℤ/2 holonomy at the Cauchy horizon — at regular curvature, hence distinct from the semiclassical ⟨Tμν⟩ divergence. The transport carrying it would be the Tomita–Takesaki modular conjugation J of the de Bruijn arrow — a proposed modular bridge resting on two conjectural joints: that the de Bruijn covering is a half-sided modular inclusion with geometric modular flow (reduced, via Brunetti–Guido–Longo, to the conformal-net axioms for the chiral core), and that the outer-horizon conjugation governs the inner one. As an antiunitary holonomy obstruction it is independent of the density of states in the bosonic case J² = +1. O The germ and stratum families fuse here: at the extremal degeneracy (λ → 0, θ → 0) the anomalous Łojasiewicz exponent and the inadmissibility wall are the analytic and topological faces of one degeneracy. Markov / Petz rigidity. Metric admissibility reduces to a finite-order Markov condition, and an information-losing stabiliser is forced to be a measured isometry (Petz sufficiency, equality case of data processing) — discharging the no-hidden-modulus hypothesis for the symmetry species. C The topological and kinetic species are non-map modes of protection. Status triage. Established (granting the segment-order base): the gradient structure; the criterion’s well-posedness and non-circularity; the topological species; the symmetry group Sym(Φ) = Isom(M, gF, μF) for both curvature signs; the Markov reduction and the topological locus of the only irreducible non-local invariant; the Maslov class; the global holonomy obstruction. E To verify: the stratified-regularity hypothesis (Łojasiewicz–Simon, made plausible by W2-displacement convexity) on which stratum/germ exhaustiveness rests; the Markov reduction’s two joints; the pre-observational Koide escape. C Open — not claimed: the covering generator as an indexed operator (the chiral half of the transverse species); the existence of an irreducible germ invariant in the programme (the Łojasiewicz θ candidate); the modular bridge’s two conjectural joints; and the single missing datum πF(≺) = ≺F that would define the internal generator and decide Koide. O The one load-bearing chain. The entire AdS₂ / extremality sector hangs on a single chain: de Bruijn ⇒ a modular structure with geometric flow ⇒ λ = the modular gap ⇒ λ → 0 at extremality — reducing to one antecedent, that the chiral core is a conformal net. The sector is labelled a candidate realisation conditional on that antecedent. If the chain breaks, degradation is graceful: the germ species collapses (Bakry–Émery forces θ = ½ everywhere), the physical realisation (extremality, AdS₂, the Cauchy locus) becomes a candidate — but two things survive chain-independently: the wall as an abstract holonomy obstruction, and the regular-regime Reduction Cliff (the measurement-sector article), which remains the programme’s sole forced prediction, in the parameter-rigidity sense (the identity line R(p) ≡ 1 and a p-independent Fano factor with Îmut fixed from τD). Conclusion. The note shifts protection from the existence of a conserved quantity to the existence of a reconstructively legitimate flow invariant — and now computes the legitimate group, classifies the invariants into a tangent (symmetry / topology) and a transverse (universality / resonance) family, and locates the flavour and extremality questions precisely: Koide as a chain of internal no-goes with a single pre-observational escape, the inadmissibility wall as a candidate modular / Cauchy-horizon holonomy. Conservation alone remains insufficient; what protects is symmetry, topology, or a relaxation germ — and what the programme can force, it forces; what it cannot, it names and locates.