A Noether-Type Theorem for Admissible Gated Gradient Flows: Symmetry, Conservation, and Leakage

Joshua K. Cliff, 2026 · 32 pages · 11 sections · 3 appendices · CC BY 4. 0 Overview This article develops a Noether-type defect identity and conservation/leakage calculus for admissible gated gradient flows on a real Hilbert space, where the admissibility constraint is enforced through the mobility (scalar gates, constrained mobilities), through the free energy (soft barriers), or both. The paper extends the symmetry-conservation principle of classical Noether theory from variational/Hamiltonian dynamics to the dissipative gated regime — the analytic class that the Principal Dynamics (PD) program identifies as the right setting for constrained dissipative systems. It is positioned as a framework theorem paper: the admissible gated gradient flow class is imported from the PD foundation as a black box, and a symmetry, conservation, and leakage calculus is added on top of it. The article fixes four hypothesis packages — (S1) solution-class regularity, (F1) – (F2) free-energy regularity, (M1) – (M3) mobility regularity (accretive, with optional self-adjoint scalar-gate strengthening), and (Q1) charge regularity — together with an optional symmetry-group package (G1) used only when the free-energy symmetry is obtained from a one-parameter action. The defect identity and its corollaries are proved on the resulting class of admissible gated gradient flows. What the Paper Proves Gated Noether identity GNI. For a Noether pair (X, Q) with mobility-compatibility coefficient α and residual R fixed by a structural, baseline, or metric-projection convention, the charge obeys a single defect identity along the trajectory. Four-step Hilbert-space chain-rule proof under (S1), (F1), (M1), (Q1) ; (G1) needed only when the symmetry is obtained from a one-parameter action. Three-convention Noether-pair definition. Structural, baseline, and metric-projection conventions for α stated explicitly with their respective compatibility residuals; the metric-projection convention forces R perpendicular to X on the open set where X (C) ≠ 0 and uniquely determines α as the orthogonal-projection coefficient. Scalar-Gated Conservation Theorem SGC. Under the canonical scalar-gate mobility M (C) = G (C) M₀ with structural Noether pair M₀∇Q = X and free-energy symmetry DFX = 0, exact conservation of Q along the gated gradient flow, including hard scalar gates with γ (χcrit) = 0. Four-mode leakage taxonomy. Symmetry-breaking, mobility/gate-compatibility, admissibility/barrier, and projection/readout modes derived as named functionals from the same defect identity, each independently diagnosable. Soft-barrier leakage law SBL. Under Fgate = λg (χ − χcrit) ²₊, an explicit leakage law proportional to λg (χ − χcrit) ₊ DχX, with explicit C¹ regularity treatment of the soft barrier (locally Lipschitz; squared barrier C¹ but not C²) and a β_ε smoothing variant. Conditional boundary-multiplier statement. In the strong-coupling limit λg → ∞, a multiplier limit produces boundary-supported leakage provided trajectory compactness, weak multiplier convergence, and explicit boundary support hold. Sharply distinguished from the bridging theorem of the Foundation Paper, which is at the level of the overshoot functional rather than the pointwise multiplier; the multiplier-limit hypotheses are external inputs not supplied by the bridge. Projected leakage identity. Kinematic identity for the time-derivative of a projected observable charge Q = Q̄ ∘ Π under a C¹ readout Π: 𝒟 → 𝒴; explicitly distinguished from lumpability — kinematic conservation of the lifted charge does not imply autonomous evolution on the observable space. Baseline-charge integrability obstruction. The strict baseline relation M₀∇Q = X may fail on a simply connected open set even when the free energy is X-symmetric; rotational and U (1) phase generators supply the standard examples. The broader Noether-pair relation may admit local solutions with a variable integrating factor on punctured or multiply-connected subdomains (canonical example: angular coordinate Q = θ with integrating factor α (C) = ‖C‖⁻² on R² \ 0). Infinite-dimensional caveat correctly uses M (C) ⁻*, not M (C) ⁻¹, since the paper allows non-self-adjoint accretive mobility. Approximate-conservation bound. Under integrated bounds on the symmetry defect and mobility-compatibility defect, an explicit linear-in-time bound |Q (t) − Q (0) | ≤ t (AεF + BεM) — quantitative near-conservation rather than the binary exact-or-nothing of classical Noether. Gelfand-triple PDE specialization. The abstract Hilbert-space identity specializes to PDE applications via the standard triple Xₘ ↪ ℋ ↪ X'ₘ. Gated heat flow, Allen–Cahn, and complex Ginzburg–Landau use this specialization directly; fourth-order or H⁻¹-mobility branches such as Cahn–Hilliard require separate analytic packages noted in §10. Worked examples. Translation charge with scalar gate, soft radial barrier, anisotropic mobility, rotational symmetry with baseline obstruction and local angular first integral, gated mean-conservation on the torus, amplitude-budget leakage, dissipative U (1) gradient flow, and gated complex Allen–Cahn. Each example exhibits a distinct leakage mode or the obstruction in concrete form. Comparison with classical Noether theory (§9). Side-by-side comparison of dynamics, generator, symmetry object, conserved object, failure mode, and gate/barrier slot. Classical Noether's theorem is recovered as the structural special case where mobility-compatibility holds, no admissibility constraint is active, and no projection is involved. Non-theorem register (§10. 2). Eight bulleted non-claims demarcating what the paper does not establish: free-energy symmetry alone does not yield a conserved charge; Hamiltonian Noether theory does not transfer mechanically; the boundary-multiplier proposition is conditional; projection-leakage is kinematic only; no physics-regime closure; no benchmark or empirical standing; no claim across realization layers; no automatic transfer to unbounded-mobility, Wasserstein, sweeping, stochastic, H⁻¹, Cahn–Hilliard, or local-current branches. Position in the corpus The PD monograph (Cliff 2026, 10. 5281/zenodo. 19334802) is the authoritative Layer 1 corpus document — it defines the framework's axioms, theorem identifiers, scope taxonomy, hypothesis labels, and notation. The Foundation Paper (Cliff 2026, 10. 5281/zenodo. 19432514) presents the analytic/geometric realization layer with the proof-bearing spine of admissible gated gradient flows. The present article is a framework theorem paper drawing from the Foundation Paper as the source of well-posedness and admissibility infrastructure (S1) — the well-posedness package is cited but not re-proved. The theorems of §4–§7 are stated, proved, and used without further reference to the PD framework, so a reader approaching the paper purely as a contribution to mathematical analysis of dissipative gradient flows can read it end-to-end without prior PD knowledge; Appendix C records the PD-trace for readers who want the framework context. Related volumes: Foundation Monograph (authoritative Layer 1 corpus document): 10. 5281/zenodo. 19334802 Foundation Paper: 10. 5281/zenodo. 19432514 Physics Regime Extension I — Monograph-to-Physics Bridge: 10. 5281/zenodo. 19334815 Intelligence Regime Extension I — Monograph-to-Intelligence Bridge: 10. 5281/zenodo. 19334838 Keywords: Principal Dynamics; framework theorem paper; Noether-type theorem; admissible gated gradient flows; defect identity; conservation laws; symmetry; mobility compatibility; soft barrier; hard gate; baseline-charge integrability obstruction; integrating factor; projection / readout; lumpability; Gelfand triple; Hilbert-space chain rule; gradient-flow PDEs; dissipative dynamics MSC 2020: 35K58 (primary) ; 35B41, 37L30, 49J40, 58D25 (secondary)