A Critical Monotonicity Principle A modular identity–rigidity package for constrained gradient dynamics

A Critical Monotonicity Principle A modular identity–rigidity package for constrained gradient dynamics Mateus R. de Maria∗ Universidade Federal do Ceará (UFC) Fortaleza–CE, Brazil Abstract We formalize a Critical Monotonicity Principle (CMP) for normalized/constrained gradient dynamics in a Hilbert/Dirichlet framework. The core mechanism is a structural defect identity whose induced criticality observable is monotone and whose equality case enforces rigidity (selection of a critical state / eigenstate). We present a localized version with cutoffs and leakage, and a quantitative threshold criterion turning monotonicity into a usable selection mechanism. As a model application, we connect CMP to Rayleigh-type normalized flows generated by a self-adjoint dissipation operator and a constraint functional, clarifying how energy decay, rigidity, and localization emerge from the same identity. Keywords. monotonicity; gradient-flow; constraint; rigidity; localization; Rayleigh-flow. MSC (2020). Primary 49J40, 47J35; Secondary 35B40, 37N20. ∗Corresponding author. Email: [email protected]. 1 1 Introduction 1.1 Motivation: normalized gradient dynamics and critical selection Many optimization and evolution mechanisms in analysis and geometry can be phrased as gradient dynamics driven by a Lyapunov functional. In numerous settings, however, the dynamics is constrained or normalized: the trajectory is required to remain on a constraint manifold (e.g. a unit sphere, a mass constraint, or a moment constraint). This introduces a nontrivial interaction between dissipation and normalization, and it often produces a rigid equality case: if a natural monotonicity quantity becomes constant, the flow must collapse onto a critical state (eigenstate/steady state). The purpose of this paper is to formalize a modular mechanism behind such phenomena. We identify a structural defect identity in a broad Hilbert/Dirichlet-form framework, from which one reads off: •acriticality observable that is monotone along normalized flows; •an equality case implying rigidity (selection of a critical state); •alocalized version with cutoffs and leakage terms; •athreshold criterion turning monotonicity into a quantitative selection tool. We call this package the Critical Monotonicity Principle (CMP). 1.2 Main results Framework. Let Hbe a real Hilbert space with inner product ⟨·,·⟩ and norm ∥·∥. Let B:H → H be a bounded, strictly positive, self-adjoint operator inducing the constraint manifold SBdef ={u∈ H :⟨Bu, u⟩= 1}. Let E:H → Rbe a C2functional with (ambient) gradient ∇E. We consider normalized gradient dynamics of the form ∂tu=−∇E(u) + λ(t)Bu, u(t)∈ SB,(1.1) where λ(t)is chosen so that the constraint is preserved. Theorem A (global defect identity and rigidity). Under standing assumptions recorded in Chapter 2, there exists a canonical defect identity d dt C(u(t)) = −D(u(t)) ≤0 for a natural criticality observable Cand a nonnegative dissipation D, such that D(u(t)) ≡ 0forces u(t)to be stationary (hence a critical state of the constrained problem). A representative abstract formulation is given below. Theorem 1.1 (CMP on the B–sphere).Assume (H1)–(Hk) from Chapter 2. Along any sufficiently regular solution of (1.1) one has an identity of the form d dt C(u(t)) + D(u(t)) = 0, where D ≥ 0. Moreover, if Cis constant on a time interval then uis stationary there. 2 Theorem B (localized CMP with leakage and threshold silence). We develop a localized version allowing cutoffs and commutator/leakage terms. In particular, if the leakage is integrable and a quantitative barrier holds, then the criticality defect must vanish asymptotically (an asymptotic silence principle), yielding convergence to a critical state in a large class of situations. Theorem 1.2 (Localized CMP and asymptotic silence).Assume the localized framework of Chapters 5–6. Then one has a localized defect identity d dt Cϕ(u(t)) + Dϕ(u(t)) = Leakϕ(u(t)), together with a quantitative criterion: if Leakϕ∈L1 tand a barrier condition holds, then Dϕ(u(t)) →0along a sequence tj→ ∞ (and in particular the flow selects a localized critical state). Theorem C (Rayleigh-type application and PDE avatar). As a model application, we connect CMP to Rayleigh-type normalized flows generated by a self-adjoint dissipation operator and a constraint functional, clarifying how energy decay and selection are encoded by the defect identity. We also present a PDE avatar illustrating the same mechanism. Theorem 1.3 (Rayleigh-type normalized flow and selection).In the Rayleigh model setting of Chapters 7–8, the CMP identity reduces to a monotonicity formula for a Rayleightype quotient. The equality case forces convergence to an eigenstate/critical state selected by the constraint. 1.3 Related work and conceptual positioning CMP is inspired by recurring patterns in: the Hilbert/metric-space perspective and Lyapunov methods (e.g. [1, 4]), and the use of second-variation information for asymptotics and rigidity (e.g. [6] and the Łojasiewicz–Simon inequality; see also [3]). Related mechanisms also appear in geometric flows (e.g. [5]), and in diffusion/Dirichlet-form frameworks (e.g. [2, 7]). Our emphasis is different: CMP is packaged as a modular identity–rigidity interface, with a localized version and a quantitative threshold mechanism designed to be reusable. 1.4 Organization The paper is organized as follows. Chapter 2 fixes the standing assumptions and the abstract Hilbert/Dirichlet-form setup. Chapter 3 records the normalized dynamics and the role of the Lagrange multiplier. Chapter 3 proves the global structural identity behind Theorem 1.1. Chapter 5 develops the localized framework and leakage structure. Chapter 6 proves the integrable Grönwall mechanism supporting Theorem 1.2. Chapters 7–8 present the Rayleigh-type model and the PDE avatar, proving Theorem 1.3. Appendices collect technical estimates and auxiliary material. 3 2 Standing assumptions and abstract framework 2.1 Hilbert space, constraint and operators Let (H,⟨·,·⟩)be a real Hilbert space. Fix a bounded, strictly positive, self-adjoint operator B:H → H and define the B–sphere SBdef ={u∈ H :⟨Bu, u⟩= 1}. We also fix a densely-defined self-adjoint nonnegative operator A(or a closed quadratic form) that will play the role of a dissipation generator, and a C2energy functional E: H → R. 2.2 Gradient structure and constrained criticality Let G:H → H denote the (ambient) gradient map of E, so G(u) = ∇E(u). Critical points on SBare u∈ SBsuch that there exists λ∈Rwith G(u) = λBu. We emphasize that λis not prescribed: it is selected by the constraint. 2.3 Regularity and admissible solutions Throughout we assume solutions are sufficiently regular so that all pairings and time derivatives below are justified. (When needed, the framework can be made fully rigorous by approximation and density arguments; see Appendix A.) 2.4 A functional-analytic viewpoint The CMP identity is designed to be compatible with operator and Dirichlet-form viewpoints. In particular, the dissipation term may be expressed using a closed form associated to A, while the constraint uses the B–inner product. 2.5 Model case preview: Rayleigh-type flows In Chapters 7–8 we discuss the model case where Eis quadratic and the normalized flow reduces to a Rayleigh-type quotient evolution, serving as a clean avatar for the general identity. 4 3 Lyapunov structure and the canonical defect identity 3.1 Energy decay (first-order Lyapunov) ∂tut=−ΠB ut∇BE(ut), u0∈ SB.(3.1) Along the projected flow (3.1) on SB, the energy decreases: d dtE[ut] = DE(ut)[∂tut] = ⟨∇BE(ut), ∂tut⟩B=− ΠB ut(∇BE(ut)) 2 B=−D[ut],(3.2) where D[u]def =∥g(u)∥2 Bis the squared speed from (3.5). Thus Eis a Lyapunov functional on SB. If Eis bounded below on SB, then Z∞ 0D[ut] dt≤ E[u0]−inf SBE<∞. Remark 3.1 (Rigor and standing hypotheses).Identity (3.2) is rigorous under (H1)–(H2) from Appendix A; see Lemma A.4 and Lemma A.3. 3.2 The canonical defect and the criticality observable The constraint geometry induces a canonical tangential defect: g(u)def = ΠB u(∇BE(u)) ∈TuSB,C[u]def =∥g(u)∥2 B.(3.3) g(u) = 0 ⇐⇒ ∇BE(u) = λu for some λ∈R.(3.4) D[u]def =∥g(u)∥2 B.(3.5) By construction C[u]≥0, and C[u] = 0 if and only if uis constrained-critical on SB in the sense of (3.4). Moreover, since ∂tut=−g(ut)along (3.1), one has C[ut] = ∥∂tut∥2 B=D[ut]. In the fixed-sphere setting, the criticality observable coincides with the squared B–speed. After localization (Chapter 5), the corresponding windowed observables no longer coincide with the speed, and commutator/leakage terms appear. 3.3 An exact second-order identity on the SB(structural CMP identity) The key point is that Citself satisfies an exact differential identity with no hidden remainder terms. Introduce the canonical multiplier λ(u) = ⟨∇BE(u), u⟩B, and the tangential Jacobian (constrained linearization) J(u)def =HB(u)−λ(u) Id .(3.6) Proposition 3.2 (Exact identity for d dtCon SB).Let u(t)∈ SBsolve (3.1). Set g(t)def = g(u(t)) and C(t)def =∥g(t)∥2 B. Then 1 2 d dtC(t) = − ⟨J(u(t)) g(t), g(t)⟩B=− ⟨(HB(u(t)) −λ(u(t)) Id) g(t), g(t)⟩B.(3.7) Dependencies: (H1)–(H2) (Appendix A, Lemma A.7). 5 Structural proof (rigorous under Appendix A). On SBwe have the explicit formula g(u) = ∇BE(u)−λ(u)u, ∂tu=−g(u). Differentiate g(u(t)): ∂tg=HB(u)∂tu−(∂tλ)u−λ ∂tu. Pairing with geliminates the (∂tλ)uterm because ⟨u, g⟩B= 0 (tangency). Using ∂tu=−g yields (3.7). Remark 3.3 (Why no remainder appears (fixed constraint)).The only potentially problematic contribution is (∂tλ)u, and it cancels exactly because g(u)∈TuSB. Once one localizes or introduces varying windows (Chapter 5), genuine commutator and windowvariation terms reappear; Appendix A records the rigorous differentiated forms used later. 3.4 Positivity notions: trajectory-wise vs. tangent-wise Identity (3.7) shows that the sign of d dtC[ut]is controlled by the quadratic form induced by J(u)evaluated on the direction selected by the dynamics, namely g(ut). It is convenient to distinguish two positivity notions. Definition 3.4 (Trajectory-wise positivity and tangent-wise positivity).We say that E satisfies: 1. Trajectory-wise positivity along utif ⟨J(ut)g(ut), g(ut)⟩B≥0for all t. (3.8) 2. Tangent-wise positivity on SBif for every u∈ SB, ⟨J(u)v, v⟩B≥0for all v∈TuSB.(3.9) Remark 3.5 (Why the distinction is structural).Tangent-wise positivity implies trajectorywise positivity for every trajectory. Conversely, monotonicity of C[ut]only controls J(ut) on the single direction g(ut)and does not yield (3.9) in general. This is a key issue in localized/PDE settings, where one typically proves coercivity only on selected windows and up to commutator leakage. Corollary 3.6 (Monotone criticality under tangent-wise positivity).If (3.9) holds, then C[ut]is nonincreasing along (3.1).Dependencies: (H1)–(H2). Corollary 3.7 (Strict tangential coercivity implies exponential selection).Assume there exists κ > 0such that ⟨J(u)v, v⟩B≥κ∥v∥2 Bfor all u∈ SB, v ∈TuSB.(3.10) Then along (3.1), d dtC[ut]≤ −2κC[ut], hence C[ut]≤e−2κtC[u0]and in particular C[ut]→0as t→ ∞.Dependencies: (H1)– (H2). 3.5 Preview: Rayleigh quotients and perfect squares In quadratic/spectral regimes one often upgrades the CMP identity to a perfect-square dissipation law for a Rayleigh-type quotient. This yields monotonicity without assuming (3.9); see Chapter 7. 6 4 Monotonicity identity and rigidity: the CMP package 4.1 CMP on the fixed sphere: an exact dissipation identity In the fixed Hilbert-sphere framework of Chapter 2, CMP is not an additional assumption: it begins as an identity. Combining Proposition 3.2 with C[u] = ∥g(u)∥2 Byields d dtC[ut] = −2⟨J(ut)g(ut), g(ut)⟩B | {z } def =Dcmp[ut] ,J(u) = HB(u)−λ(u) Id .(4.1) Thus the evolution of the criticality defect is governed by the constrained linearization J(u)evaluated on a natural tangential direction g(u). Remark 4.1 (Two “dissipations” in this paper).Along the projected flow, C[ut] = ∥∂tut∥2 B, so in the fixed-sphere setting Ccoincides with the squared speed (energy dissipation) D from (3.2). In contrast, Dcmp in (4.1) is a quadratic form induced by J(u). It is not automatically nonnegative; monotonicity of Crequires a positivity mechanism such as (3.8) or (3.9). 4.2 Rigidity: equality case ⇒constrained criticality under coercivity The equality case of (4.1) reads ⟨J(ut)g(ut), g(ut)⟩B≡0. To upgrade this to g≡0, one needs quantitative positivity on tangent directions. Proposition 4.2 (Rigidity under strict tangential coercivity).Assume the strict tangential coercivity (3.10) holds: there exists κ > 0such that ⟨J(u)v, v⟩B≥κ∥v∥2 Bfor all u∈ SB, v ∈TuSB. Then C[ut]is strictly decreasing unless g(ut)≡0. In particular, if C[ut]is constant on a time interval, then utis constrained-critical there. Dependencies: (H1)–(H2). Proof (rigorous under Appendix A). If Cis constant on an interval, then (4.1) gives Dcmp[ut]≡0there, hence ⟨J(ut)g(ut), g(ut)⟩B= 0. Coercivity forces g(ut) = 0, and (3.4) implies constrained criticality on that interval. 4.3 An abstract CMP theorem (monotonicity + compactness ⇒criticality) CMP identities are often used through a common functional-analytic template. We record it explicitly, since it will be invoked verbatim in localized PDE contexts. Theorem 4.3 (Abstract CMP package (no leakage)).Assume there exist functionals C≥0and D ≥ 0such that along a flow ut: d dtC[ut]≤ −D[ut](in the a.e. / distributional sense).(4.2) Assume moreover that t7→ C[ut]is absolutely continuous and that D[u]=0 ⇐⇒ uis constrained-critical. Then: 7 1. C[ut]is nonincreasing and converges as t→ ∞. 2. R∞ 0D[ut]dt≤C[u0]. 3. If {ut}t≥0is precompact in H, then every accumulation point is constrained-critical. 4. (Rigidity) If C[ut]is constant on an interval, then D[ut] = 0 a.e. there; under continuity of t7→ ut(or closedness of the critical set), this implies utis constrained-critical on that interval. Dependencies: absolute continuity of t7→ C[ut]; closedness of the critical set (or continuity of ut) for the rigidity upgrade. Proof sketch. Integrate (4.2) to obtain monotonicity and the dissipation bound. If Cis constant on an interval, the integral of Don that interval vanishes, hence D= 0 a.e. Precompactness yields convergent subsequences; the equivalence D= 0 ⇐⇒ critical gives that limits are critical. 4.4 A general “square + leakage” template (bridge to localization) The fixed-sphere identity (4.1) has no remainder because both the constraint and the metric are fixed. Once one introduces cutoffs, varying windows, or nonlocal generators, genuine commutator/boundary terms appear. The point of the next template is to isolate them cleanly. Proposition 4.4 (Square plus leakage template (structural)).Assume a normalized flow has the form ∂tu=−G(u) + λ(u)u, ⟨u, u⟩B= 1, where G(u)is a dissipative field and λ(u)enforces the constraint. Let the tangential defect be ∆(u)def = ΠB u(G(u)) = G(u)−λ(u)u∈TuSB. Then for any bounded B–self–adjoint nonnegative window K(possibly depending on tor u), the quantity CK[u]def =⟨K∆(u),∆(u)⟩B admits a decomposition d dtCK[ut] = −2Dcmp,K[ut] + LeakK[ut], where Dcmp,K is the windowed quadratic form induced by the linearization of G(and, in the sphere-gradient case, by J(u)), while LeakKcollects explicitly: 1. commutators between the linearized operator and K(localization mismatch), 2. multiplier/constraint terms (e.g. λ′(t)effects), 3. window-variation terms when Kdepends on tor on u. Dependencies: differentiability of t7→ CK[ut](Appendix A). Remark 4.5 (Sphere-gradient specialization and rigorous localized form).If G(u) = ∇BE(u)and the constraint is the fixed sphere SB, choosing K≡Id yields (4.1) with LeakK≡0: everything is absorbed into J(u) = HB(u)−λ(u) Id. For general bounded B–self–adjoint windows K, the rigorous commutator + multiplier form of LeakKis given in Appendix A, Proposition A.9. 8 4.5 Link to Rayleigh-type perfect squares In spectral/quadratic regimes, one often obtains perfect-square identities for quotient observables (Rayleigh), yielding monotonicity without assuming tangent-wise positivity. This is developed in Chapter 7. 9 7 Model application: Rayleigh quotients and perfect-square selection 7.1 Quadratic energies on the B–sphere Let Hbe a real Hilbert space. Let A, B :H → H be bounded self–adjoint operators and assume that Bis coercive (uniformly positive) as in (A.1), so that B−1exists and is bounded. For simplicity assume A≥0. Consider the quadratic energy E(u) = 1 2⟨Au, u⟩on SB={u∈ H :⟨Bu, u⟩= 1}. As recalled in Chapter 2, constrained critical points on SBare precisely generalized eigenvectors: Au =λ Bu. (7.1) 7.2 Rayleigh quotient and canonical multiplier For u= 0 define the Rayleigh quotient R[u]def =⟨Au, u⟩ ⟨Bu, u⟩.(7.2) On the B–sphere SBone has R[u] = ⟨Au, u⟩. The projected B–gradient flow for Eon SBreads ∂tu=−ΠB uB−1Au=−B−1Au +λ(u)u, ⟨Bu, u⟩ ≡ 1,(7.3) with the canonical multiplier λ(u) = B−1Au, uB=⟨Au, u⟩=R[u]on SB.(7.4) 7.3 Perfect-square CMP identity (Theorem C) This is the flagship regime where the CMP dissipation collapses to a closed perfect square. Theorem 7.1 (Rayleigh perfect-square monotonicity and rigidity).Along the Rayleigh flow (7.3) one has d dtR[ut] = −2∥Aut−R[ut]But∥2 B−1≤0.(7.5) Moreover, d dtR[ut] = 0 at some time if and only if Aut=R[ut]But, i.e. utis a generalized eigenvector and the dynamics is rigid. Dependencies: bounded self–adjoint A, B with coercive B; classical differentiability for ut. Proof. On SB,R[u] = ⟨Au, u⟩, hence in a real Hilbert space d dtR[ut] = 2 ⟨Aut, ∂tut⟩. 16 Using (7.3) and (7.4), ⟨Au, ∂tu⟩=Au, −B−1Au +R[u]u =−Au, B−1Au+R[u]⟨Au, u⟩ =−Au, B−1Au+R[u]2. On the other hand, since B−1Bu =uand ⟨Bu, u⟩= 1 on SB, ∥Au −R[u]Bu∥2 B−1=B−1(Au −R[u]Bu), Au −R[u]Bu =Au, B−1Au−2R[u]⟨u, Au⟩+R[u]2⟨u, Bu⟩ =Au, B−1Au−R[u]2. Combining the last two displays yields (7.5). Rigidity follows since the square vanishes iff Au =R[u]Bu. Remark 7.2 (CMP viewpoint).Define the Rayleigh defect ∆(u)def =Au −R[u]Bu. Then ∥∆(u)∥2 B−1= B−1∆(u) 2 B, so the perfect square (7.5) is exactly “monotonicity ⇒eigenstate selection”. This is an illustrative operator-theoretic example for the abstract CMP package. Remark 7.3 (Complex Hilbert spaces).If His complex, replace 2⟨Au, ∂tu⟩by 2ℜ⟨Au, ∂tu⟩. 7.4 Unbounded generators (form interpretation) In PDE settings Ais typically unbounded self–adjoint and the same computation is read in the associated form/domain framework: the algebra is unchanged, while rigor is supplied by density and chain rules (Appendix A). 17 8 Worked PDE example: a normalized confining heat flow with an explicit cutoff window 8.1 Model and Hilbert setting Let H=L2(Rd)with the standard real inner product ⟨f, g⟩=RRdfg dx. We take B= Id, so SB={u∈L2:∥u∥2= 1}. Fix a confining Schrödinger generator Adef =−∆ + V(x),(8.1) where V∈C∞(Rd)is bounded below and confining (e.g. V(x)≥c|x|2−C). Consider the quadratic energy E(u) = 1 2⟨Au, u⟩on SB. The projected gradient flow on SBreads ∂tu=−Au +λ(t)u, ∥u(t)∥2≡1,(8.2) where the canonical multiplier is the Rayleigh quotient λ(t) = λ(u(t)) = ⟨Au(t), u(t)⟩.(8.3) Define the tangent defect g(t)def =Au(t)−λ(t)u(t),⟨g(t), u(t)⟩= 0 on SB,(8.4) so that (8.2) is simply ∂tu=−g. 8.2 Global perfect-square dissipation A direct computation (cf. Chapter 7) yields λ′(t) = −2∥g(t)∥2 2≤0,(8.5) with equality iff g(t)=0(i.e. Au =λu, an eigenstate). Since λis monotone and bounded below by inf σ(A), it converges and Z∞ 0|λ′(t)|dt < ∞.(8.6) This is precisely the L1input that makes multiplier leakage harmless below. 8.3 Cutoff window and windowed criticality Fix χ∈C∞ c(Rd)radial with 0≤χ≤1,χ≡1on B1(0), and supp(χ)⊂B2(0). For R≥1, set χR(x)def =χ(x/R)and let KRfdef =χRf. Define the windowed criticality CR(t)def =CKR[u(t)] = ⟨χRg(t), g(t)⟩=ZRd χR(x)|g(t, x)|2dx. (8.7) 18 8.4 Promoted statement: explicit “square + leakage” with a referee-safe bound This proposition instantiates the abstract localization taxonomy (Chapter 5) in a concrete unbounded setting, and points to Appendix C for the commutator algebra. Proposition 8.1 (Explicit cutoff localization: identity and leakage bound).Let u(t)solve (8.2) and define g(t),λ(t)by (8.3)–(8.4). Fix R≥1and define CR(t)by (8.7). Assume there exists T∗>0such that for a.e. t≥T∗, g(t)∈H1(Rd),Z∞ T∗∥∇g(t)∥2 2dt < ∞.(8.8) Then for a.e. t≥T∗one has the localized identity d dtCR(t) = −2 CMPR(t) + LeakR(t),(8.9) where CMPR(t)def =⟨χR(A−λ(t)Id) g(t), g(t)⟩, LeakR(t)def =−2⟨[A, χR]g(t), g(t)⟩−2λ′(t)⟨χRg(t), u(t)⟩. Moreover, letting AnnRdef ={R≤ |x| ≤ 2R}, there exists Cχ>0(depending only on χ) such that for a.e. t≥T∗, |LeakR(t)|≤|λ′(t)|CR(t) + |λ′(t)|+Cχ R∥g(t)∥2 L2(AnnR)+∥∇g(t)∥2 L2(AnnR).(8.10) Dependencies: Appendix A (chain rules in the unbounded/form setting) and Appendix C (commutator computation and the L1-coefficient Grönwall closing). 8.5 How the threshold closure works (one line) Assume a windowed coercivity regime holds: CMPR(t)≥κcoer CR(t)for a.e. t≥T∗,(8.11) for some κcoer >0. Then (8.9) and (8.10) yield a differential inequality for CRwith an L1coefficient a(t) = |λ′(t)|(using (8.6)) plus an integrable remainder. The closing step is exactly Lemma 6.2 from Chapter 6, hence CR(t)→0. 8.6 Interpretation This avatar shows the role of localized CMP bookkeeping: after fixing a window, the only obstruction to monotone selection is leakage. Here leakage is completely explicit: multiplier leakage contributes an L1coefficient (automatic), and commutator leakage is an annulus effect suppressed by R−1and controlled by local H1information on g. 19 9 Conclusion and outlook 9.1 Summary of contributions This paper isolates a portable mechanism by which normalized (constrained) gradient flows produce a canonical defect field whose evolution admits a sharp identity. The resulting “monotonicity ⇒rigidity” logic is designed to be reusable across Hilbert/Dirichlet settings and compatible with localization by windows/cutoffs. •Canonical defect and exact identity on a fixed constraint. On the B–sphere (and more generally on constraint manifolds), normalization induces a canonical tangent defect g(u)=ΠB u(∇BE(u)) and a tangential Jacobian J(u). The squared defect C[u] = ∥g(u)∥2 Bsatisfies an exact identity whose sign is governed by a single quadratic form. Tangential positivity yields monotonicity, while strict tangential coercivity yields rigidity and quantitative decay. •Localization as “square + leakage” (net gap principle). Introducing a window operator Kyields d dtCK[ut] = −2 CMPK[ut] + LeakK[ut], where leakage splits into explicit commutator/multiplier (and, if applicable, windowvariation) terms. A coercivity regime plus L1bookkeeping closes the estimate via a single net-gap condition (Chapter 6). •Quadratic flagship: Rayleigh perfect square (Theorem 1.3). In the quadratic regime the dissipation becomes an exact perfect square and the equality case enforces generalized eigenstates. This provides a clean operator-theoretic showcase for the abstract package, and motivates the localized bookkeeping in unbounded/PDE settings. •PDE avatar with explicit cutoff leakage (Chapter 8). A confining normalized heat flow illustrates how localization reduces asymptotic selection to one coercivity regime plus explicit commutator/multiplier bounds. 9.2 A short reuse guide (what to check in a new model) Given a normalized gradient dynamics in a Hilbert/Dirichlet framework, the CMP workflow is: 1. Identify the defect. Compute the canonical tangent defect g(u)induced by the constraint (sphere/multi-constraint) in the chosen metric. 2. Identify the Jacobian. Compute the tangential Jacobian (or its form-domain analogue) governing the sign of d dt∥g(u)∥2. 3. Localize if needed. Choose a window family Kand write the localized identity, isolating commutator and multiplier terms as leakage. 4. Verify a regime. Prove a coercivity/positivity regime for the induced dissipation (often triggered by a barrier functional or a spectral gap). 5. Close by L1bookkeeping. Control leakage by an L1coefficient and/or integrable remainders, then apply the Grönwall-type closing step (Chapter 6). 20 9.3 Analytic scope and limitations The abstract identities provide monotonicity and rigidity at the level of defect observables. They do not, by themselves, guarantee convergence of utin the ambient space. Convergence typically requires additional compactness (or a Łojasiewicz–Simon inequality near a constrained-critical set) and is model dependent. For unbounded generators (typical in PDE), the identities should be interpreted in a quadratic-form/domain framework: the algebraic structure is unchanged, while analytic justification uses standard chain rules and density arguments, recorded in Appendix A and instantiated explicitly in the worked PDE avatar. 9.4 Immediate extensions The framework admits natural upgrades with minimal additional overhead in concrete models: 1. Multi-constraint manifolds: codimension-kconstraints fit the same projection/defect formalism (Appendix B). 2. Commutator-free localization: spectral windows commuting with the generator eliminate commutator leakage (Appendix B). 3. Intrinsic geometric notation: rewriting the package on constraint manifolds with connections clarifies which terms are structural versus coordinate artifacts. 9.5 Closing remark The main point is that the normalization constraint canonically produces a defect whose evolution is governed by a single sharp identity. Once this object is identified, monotonicity, equality cases, and localized threshold statements follow from explicit bookkeeping. 21 A Appendix A: Functional-analytic toolbox A.1 Dependency map (what is used where) For the reader who wants a quick “referee map”, we indicate the minimal analytic inputs behind each block of computations: •Chapters 2–4: (H1)–(H2) suffice for differentiations leading to (3.2) and the exact defect identity (3.7). •Chapter 5: in addition, (H4) (window regularity) is used to differentiate windowed observables and isolate commutator/multiplier leakage. •Unbounded generators / quadratic forms: Proposition A.10 provides a clean template that justifies the same identities in the self–adjoint (possibly unbounded) setting used in the PDE avatar. •Long-time conclusions (compactness/convergence): one adds a model-dependent compactness input (C1)–(C3) below, and optionally an Łojasiewicz–Simon inequality (LS) to obtain finite length. These are standard in Hilbert/Dirichlet settings; we collect them here so the main text can remain structural. A.2 Standing hypotheses (minimal package) Throughout, we work on the constraint sphere SB={u∈ H :∥u∥B= 1}, with Bbounded, B–self–adjoint and coercive on H. mB∥v∥2≤ ⟨Bv, v⟩ ≤ MB∥v∥2for all v∈ H,(A.1) for some constants 0< mB≤MB<∞. In particular, B−1exists and is bounded. To justify the computations in Chapters 2–6 (and the worked example in Chapter 8), it suffices to assume: (H1) Regularity near the sphere. E ∈ C2(U;R)on a B–open neighborhood Uof SB. Equivalently, ∇BEis C1on U, with Fréchet derivative HB(u) continuous in u. (H2) Local Lipschitz vector field. The projected vector field F(u)def =−ΠB u∇BE(u) is locally Lipschitz on U ∩{u:∥u∥B≥1/2}. (H3) Global existence on the sphere (template assumption). For every u0∈ SBthe solution of ∂tu=F(u)exists for all t≥0and stays in SB. (H4) Windows and their regularity. Whenever a window Kis used, assume K∈ L(H)is bounded and B–self–adjoint. If a model requires K≥0, assume nonnegativity in the B–sense: ⟨Kv, v⟩B≥0. If a model uses K=K(t) 22 or K=K(u), assume the corresponding map is C1on the regime where differentiation is performed. (H5) Operator/Dirichlet-form compatibility (intended reading). When Eis generated by a closed symmetric quadratic form (or by a self– adjoint operator), identities are interpreted in the corresponding weak/duality sense, with commutators and window terms defined on the relevant form domain. Proposition A.10 records a minimal self-contained template in the unbounded setting. Remark A.1 (How to ensure (H3) in practice).(H3) is automatic if the vector field Fis globally Lipschitz on a B–neighborhood of SB, or more generally if Fsatisfies a linear growth bound on SBthat precludes finite-time blow-up. In PDE settings, (H3) typically follows from semigroup theory and a priori bounds. A.3 Well-posedness and invariance of the projected flow on the SB We record standard ODE statements in Banach/Hilbert spaces. Lemma A.2 (Local well-posedness).Assume (H1)–(H2). For every u0∈ SBthere exists T > 0and a unique solution u∈C1([0, T); H)of ∂tu=−ΠB u(∇BE(u)), u(0) = u0, defined on a maximal interval [0, Tmax). Lemma A.3 (Sphere invariance).Assume u∈C1([0, T); H)solves the projected flow with u(0) ∈ SB. Then u(t)∈ SBfor all t∈[0, T). Proof. Since ΠB u(·)∈TuSB, we have ⟨∂tu, u⟩B= 0. Hence d dt⟨u, u⟩B= 2 ⟨∂tu, u⟩B= 0, so ⟨u(t), u(t)⟩Bis constant and equals 1. A.4 Chain rules used in Chapters 2–4 All computations below can be justified under u∈ACloc([0,∞); H)and (H1), with u(t)∈ Ua.e. Lemma A.4 (Energy chain rule).Assume (H1) and u∈ACloc([0,∞); H)with u(t)∈ U. Then t7→ E(u(t)) is absolutely continuous and for a.e. t, d dtE(u(t)) = DE(u(t))[u′(t)] = ∇BE(u(t)), u′(t)B. Lemma A.5 (Derivative of the multiplier λ).Assume (H1) and u∈C1([0, T); H)with u(t)∈ U. Let λ(u) = ⟨∇BE(u), u⟩B. Then λ(u(t)) is C1and d dtλ(u(t)) = HB(u(t)) u′(t), u(t)B+∇BE(u(t)), u′(t)B. 23 A.5 Differentiating the tangential defect and criticality (Chapter 3 identity) On SBone has the convenient explicit formula g(u)=ΠB u(∇BE(u)) = ∇BE(u)−λ(u)u. Lemma A.6 (Differentiability of g(u(t))).Assume (H1) and u∈C1([0, T); H)with u(t)∈ SB. Then g(u(t)) is C1and d dtg(u(t)) = HB(u(t))u′(t)−λ′(t)u(t)−λ(t)u′(t), λ(t)def =λ(u(t)). Lemma A.7 (Criticality chain rule and the exact identity).Assume (H1) and let u∈ C1([0, T); H)solve the projected flow on SB. Define C(t)def =∥g(u(t))∥2 B. Then Cis C1 and 1 2C′(t) = − ⟨(HB(u(t)) −λ(t) Id) g(u(t)), g(u(t))⟩B, which is the rigorous version of the identity proved in Chapter 3. Sketch. Combine Lemma A.6 with C(t) = ⟨g, g⟩Band the flow identity u′(t) = −g(u(t)). The λ′(t)u(t)term cancels after pairing with gbecause ⟨u, g⟩B= 0 on SB. A.6 Windowed observables and localized identities (Chapter 5) We isolate the analytic steps needed to differentiate CK(t) = ⟨Kg, g⟩Band to recover the localized CMP identity with commutator leakage. Lemma A.8 (Windowed chain rule).Assume (H1) and let u∈C1([0, T); H)with u(t)∈ SB. Let K∈ L(H)be bounded and B–self–adjoint. Then t7→ CK[u(t)] def = ⟨K g(u(t)), g(u(t))⟩Bis C1and d dtCK[u(t)] = 2 K g′(u(t)), g(u(t))B. If K=K(t)is C1, add the term D˙ K(t)g, gEB; if K=K(u)is C1, add ⟨K′(u)[u′]g, g⟩B. Proposition A.9 (Localized identity with explicit commutator + multiplier leakage). Assume (H1) and let u∈C1([0, T); H)solve the projected flow on SB. Assume moreover that for each tthe operator J(u(t)) def =HB(u(t))−λ(t) Id is B–self–adjoint. Let K∈ L(H) be bounded and B–self–adjoint, time-independent. Then t7→ CK[u(t)] is C1and d dtCK[u(t)] = −2⟨KJ(u(t)) g(u(t)), g(u(t))⟩B −2⟨[J(u(t)), K]g(u(t)), g(u(t))⟩B −2λ′(t)⟨Ku(t), g(u(t))⟩B. (A.2) with [J, K]def =JK−KJ. In particular, defining Dcmp,K[u]def =⟨KJ(u)g(u), g(u)⟩B, one recovers d dtCK[ut] = −2Dcmp,K[ut] + LeakK[ut], where LeakKis exactly the sum of the last two terms in (A.2). 24 Sketch. By Lemma A.8, 1 2 d dtCK[u(t)] = Kg, g′, with all pairings in ⟨·,·⟩B. Insert Lemma A.6 and u′=−g: g′=−HB(u)g+λg −λ′u=−J(u)g−λ′u. Hence 1 2C′ K=−⟨Kg, Jg⟩−λ′⟨Kg, u⟩. Use B–self–adjointness of Jto rewrite ⟨Kg, Jg⟩=⟨JKg, g⟩=⟨KJg, g⟩+⟨[J, K]g, g⟩. Multiply by 2and rearrange to obtain (A.2). A.7 A referee-safe unbounded template (self–adjoint generators / closed forms) Proposition A.10 (CMP identities for quadratic energies generated by a self–adjoint operator).Let (H,⟨·,·⟩)be a real Hilbert space and let Bbe bounded, B–self–adjoint and coercive. Let Abe a (possibly unbounded) self–adjoint operator on Hbounded below, with domain D(A). Consider the quadratic energy E(u) = 1 2⟨Au, u⟩, u ∈ D(A), and the B–sphere SB={u∈ H :⟨u, u⟩B= 1}. Assume u∈C1([0, T]; H)∩C([0, T]; D(A)) satisfies the normalized flow ∂tu=−B−1Au +λ(t)u, λ(t)def =⟨Au(t), u(t)⟩.(A.3) Define the defect g(t)def =B−1Au(t)−λ(t)u(t)∈ H,so that ut=−g(t). Then: 1. (Constraint preservation.)∥u(t)∥B≡1and ⟨g(t), u(t)⟩B= 0 on [0, T]. 2. (Perfect-square identity.) The Rayleigh multiplier is absolutely continuous and λ′(t) = −2∥g(t)∥2 B≤0for a.e. t∈(0, T).(A.4) 3. (Windowed identity with commutator & multiplier leakage.) Let K∈ L(H) be bounded and B–self–adjoint, and assume in addition that K(D(A)) ⊂ D(A)so that [B−1A, K]is meaningful on D(A). Then the windowed criticality CK(t)def =⟨K g(t), g(t)⟩B is absolutely continuous and for a.e. tone has d dtCK(t) = −2K(B−1A−λ(t) Id) g(t), g(t)B −2[B−1A, K]g(t), g(t)B −2λ′(t)⟨Ku(t), g(t)⟩B. (A.5) 25 Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Data availability No data were generated or analyzed in this study. 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