Categorical Extensions of Integrability in the Cubic-Flux Burgers Equation: Backbones, Non-Backbone Plumes, and Non-Abelian Spectral Invariants

Categorical Extensions of Integrability in the Cubic–Flux Burgers Equation: Backbones, Non-Backbone Plumes, and Non-Abelian Spectral Invariants Andrei T. Patrascu 1 1 FAST Foundation, Destin FL, 32541, USA email: andrei.patr[email protected] The viscous Burgers equation is a classical testbed for nonlinear dynamics and turbulence. In its quadratic flux form, it is linearizable via the Cole–Hopf transform and admits exact solutions, but once a cubic flux contribution is included classical integrability is lost: the Cole–Hopf transform fails and standard conserved quantities drift under dissipation. In this work we show how categorical extensions of integrability recover structure in the cubic–flux Burgers equation. First, we identify backbones: exact travelling–wave solutions (tanh, logarithmic, Lambert– W profiles) that persist as fragments of integrability. Second, imposing categorical flatness ( F [ u ]=0) yields new categorical plumes: algebraic, static and self–similar profiles that preserve generalized trace invariants exactly. Third, moving beyond scalar traces, we use non–Abelian spectral projectors and K –theory classes to construct oscillatory algebraic coherent states—solutions with phase–locked envelopes that stabilize a chosen spectral band. Finally, we extend this to multi–band projectors and exhibit bi–chromatic algebraic coherent waves that preserve the K –class of a two–band block. In each case, we provide explicit closed forms, verify the preservation of categorical invariants, and give intuitive explanations of their physical meaning. Our results demonstrate concretely how categorical integrability extends solvability into dissipative, non–ultralocal PDEs, revealing exact and semi–exact structures where classical methods predict non–integrability. I. INTRODUCTION The Burgers family of equations has long served as a minimal laboratory for nonlinear transport, shock formation, and dissipation [42]. In its viscous, quadratic–flux form, ut+∂x 1 2u2=ν uxx, ν > 0,(1) the Cole–Hopf substitution u = − 2 ν ∂x ( log φ )transforms (1) into the heat equation φt = ν φxx , yielding an explicit solution formula and a complete linear theory [ 43 , 44 ]. In this sense, (1) is “integrable” (linearizable) even though it does not fit the Lax/IST archetype. Cubic flux breaks Cole–Hopf. In many applications (effective closures, flux limitation, plasma/traffic phenomenology) the advective flux is higher–order. We consider ut+∂x a u +b u2+c u3=ν uxx, a, b, c ∈R, ν > 0,(2) where no analogue of the Cole–Hopf transform exists in general. Equation (2) is therefore not linearizable and (in the conventional sense) not integrable. While travelling–wave reductions of (2) admit exact profiles (classical tanh shocks, logarithmic/arctangent fronts, and, at a resonant double–root locus, Lambert– W fronts), the full PDE does not possess an infinite hierarchy of commuting conserved quantities nor a standard Lax pair in the sense of [ 45 , 46 ]. Moreover, dissipative effects and higher–order flux destroy the usual balance underlying ultralocal Poisson brackets; in the language of integrable field theory this is analogous to the appearance of non–ultralocal r/s structures [53]. Classical invariants drift under viscosity. Let F ( u ) := au + bu2 + cu3 . On a domain with periodic or rapidly decaying data, the mass M(t) = RRu dx is conserved, but the quadratic moment d dt ZR u2dx =Z2u utdx =Z2u−∂xF(u) + νuxxdx =−2νZ(ux)2dx ≤0(3) decays strictly unless ux≡ 0. Thus the familiar scalar invariants which are protected in integrable hierarchies (e.g. isospectral traces Tr f ( L )) do not survive in (2) . This places the model outside the classical integrable landscape [45, 46]. A categorical extension of integrability. We adopt the following point of view. Associate to u a (self–adjoint) Schrödinger–type operator L[u] := −ν ∂2 x+V[u],(4) 2 with V [ u ]a fixed polynomial functional consistent with the flux in (2) . Its evolution can be organized in the Lax–like template ∂tL= [M, L] + F[u],(5) where [M, L]captures the transport part and F[u] = ν uxx +∂x cu3=ν uxx + 3c u2ux(6) is the curvature/defect obstructing isospectrality. Classical isospectral invariants survive only when F ≡ 0. Our first move is to replace scalar traces by categorical traces (traces in suitable quotients that kill commutators and coboundaries), so that d dt Icat[f] = Trcatf0(L)F[u], and to prove a quadratic drift estimate d dt Icat[f]≤CkF[u]k2 H−1 →H1, with an explicit norm equivalent (on the solutions we consider) to the L2 –norm of the right–hand side of (6) . This categorical trace perspective is naturally framed in noncommutative geometry and cyclic/K–theoretic language [56, 57] and underlies our scalar generalized invariants. From traces to non–Abelian spectral laws. Beyond scalar functionals, we consider non–Abelian invariants built from spectral projectors of L [ u ]: if p = 1I ( L [ u ]) projects onto a gapped spectral band I , then the class [p]∈K0(C∗(L)) is stable under conjugation and commutators. Resolvent calculus shows that ∂tp=1 2πi I(z−L)−1F[u] (z−L)−1dz, so a sufficient condition for [ p ]to be preserved is the vanishing of the band–diagonal curvature pFp (in the resolvent/phase–average sense). This freezes a spectral law (the band structure) without requiring full isospectrality, and will be the key to constructing non–backbone oscillatory coherent states stabilized by a chosen band. Conceptually, these invariants live in K–theory rather than in scalar trace spaces [56, 57]. What we do in this paper. Focusing exclusively on (2) , we demonstrate that the categorical extension of integrability yields concrete, explicit structures: • Backbones (travelling waves): we recall exact tanh , logarithmic/arctangent, and Lambert– W profiles from the travelling–wave ODE. These are fragments of classical solvability embedded in a non–integrable PDE. • Categorical plumes (scalar categorical traces): imposing F [ u ] ≡ 0produces new algebraic, static and self–similar profiles that preserve generalized traces exactly and provide canonical equilibria for nonlinear transport. • Oscillatory algebraic coherent states (one–band K –invariants): enforcing the band–diagonal cancellation pFp≃ 0for a fundamental Fourier block yields closed–form, non–travelling, phase–locked solutions with algebraic envelopes; these preserve the K–class [p]while scalar moments drift. • Bi–chromatic coherent states (multi–band K –invariants): extending to a two–band projector P = pk⊕p3k leads to coupled envelope equations and explicit bi–chromatic algebraic solutions that preserve [P]and exhibit only off–block leakage. Each family comes with (i) a closed form, (ii) a categorical invariant that is flat on the solution, and (iii) a quantitative drift bound for nearby flows. Intuitively, we extend integrability from a rigid, binary property to a graded/categorical structure that organizes solvable sectors within a dissipative, non–ultralocal PDE. This connects the Burgers laboratory to broader themes in integrable systems (Lax/IST, non–ultralocality [ 45 , 46 , 53 ]) and to noncommutative/categorical invariants [ 56 , 57 ], while remaining self–contained and computationally explicit for (2). 3 Notation and boundary conditions. Unless stated otherwise we work on the real line with smooth, rapidly decaying data (or on a periodic box with zero–mean conventions), so that boundary terms vanish in integrations by parts such as (3) . We write F ( u ) = au + bu2 + cu3 , k · k for appropriate L2 –based norms, and use the terms categorical trace and non–Abelian invariant in the operator–algebraic sense explained above. Forward references: scalar categorical traces and plumes appear in Section III; non–Abelian spectral laws in Section IV; two–band K –invariants in Section V; classical backbones are summarized in Section II. II. CLASSICAL BACKBONES AND BROKEN INVARIANTS We recall the travelling–wave reduction of the cubic–flux viscous Burgers equation ut+∂x a u +b u2+c u3=ν uxx, a, b, c ∈R, ν > 0,(7) and show how it yields exact coherent profiles (backbones) even though the full PDE is not classically integrable. We then quantify the drift of naïve scalar invariants. A. Travelling–wave reduction and the backbone ODE Seek a coherent profile moving with speed s∈R: u(x, t) = U(ξ), ξ := x−st. Substituting into (7) gives −s U0(ξ) + d dξ aU +bU2+cU3=ν U00(ξ), or, after one integration (with constant K∈R), ν U0(ξ) = aU +bU2+cU3−sU −K. (8) Let F ( u ) := au + bu2 + cu3 . Suppose U ( ξ )connects two asymptotic states U± := limξ→±∞ U ( ξ )with U0(ξ)→0as ξ→ ±∞. Then (8) imposes K=F(U±)−s U±, so Kis fixed and the speed is uniquely determined by the Rankine–Hugoniot condition s=F(U+)−F(U−) U+−U− ,(9) see, e.g., [48, Ch. 14], [47, Ch. 11]. With (9) and Kfixed, (8) becomes the backbone ODE dU dξ =1 νF(U)−sU −K=1 νQ(U),(10) where Qis a cubic polynomial whose real root structure governs the profile shape. B. Profile taxonomy: tanh, logarithmic/arctangent, and a Lambert–Wresonance Equation (10) is separable, ν dU/Q(U) = dξ. Writing Q(U) = c(U−α)(U−β)(U−γ), partial fractions yield explicit quadratures. Three qualitatively distinct regimes arise (see [ 47 , Ch. 11], [48, Ch. 14]): •Monotone fronts (“tanh–type”): when Q has three real roots and the connection is between consecutive roots, the integral reduces to logarithms of linear terms, and the implicit law exponentiates to a tanh–like shock profile with thickness O(ν). 4 •Logarithmic/arctangent fronts: for one real and a complex conjugate pair (or for connections that traverse a complex pair), the quadrature involves a real logarithm and an arctangent, yielding profiles with algebraic/exponential shoulders depending on parameters. •Resonant double–root (Lambert–W): when two roots coalesce, Q ( U ) = c ( U−U0 ) 2 ( U−U1 ), the quadrature involves both a logarithm and a simple pole in ( U−U0 ) −1 , and the implicit law can be inverted in terms of the Lambert–Wfunction [49]. We record the resonant case explicitly, since it will serve as a canonical backbone. Lambert–Wbackbone at the double root. Assume Q(U) = c(U−U0)2(U−U1)with U06=U1. Then ZdU (U−U0)2(U−U1)=c νξ+C, and the partial fraction identity 1 (U−U0)2(U−U1)=−1 (U1−U0)2 1 U−U0−1 U1−U0 1 (U−U0)2+1 (U1−U0)2 1 U−U1 integrates to 1 (U1−U0)2logU−U1 U−U0 +1 U1−U0 1 U−U0 =c νξ+C. (11) Let ∆ := U1−U0>0and set Y:= 1/(U−U0). Then (11) becomes 1 ∆2log(1 −∆Y) + 1 ∆Y=c νξ+C, or log(1 −∆Y)+∆Y= ∆2c νξ+C. Writing Z:= −∆Ywe obtain log(1 + Z)−Z= ∆2(c νξ+C), which inverts via Lambert–W[49] to U(ξ) = U0+∆ 1 + W−Θe−σξ, σ := c∆2 ν,Θ := e∆2C+1 >0,(12) with the branch of W fixed by the endstates and monotonicity. The speed s and constant K are tied to ( U−, U+ )by (9) and K = F ( U± ) −sU± . Formula (12) gives a closed backbone profile with a characteristic thickness ∼ν/(|c|∆2)and asymmetric shoulders determined by the branch of W. C. Naïve scalar invariants drift On a periodic domain or on R with sufficient decay, the mass M ( t ) = Ru dx is conserved by (7) , but the quadratic moment E(t) := Ru2dx obeys dE dt =Z2u utdx =−Z2u ∂xF(u)dx + 2νZu uxx dx =−2νZ(ux)2dx ≤0,(13) cf. [ 43 , 44 ]. In particular, even along an exact travelling wave U ( ξ ), E ( t )depends on the integration window and typically decays in any co–moving finite interval as the front translates. This simple computation encapsulates why (7) lies outside the classical integrable hierarchies [ 45 , 46 ]: standard (scalar) traces are not preserved. The remainder of the paper shows that, despite this, the equation admits categorical scalar invariants (flat on certain coherent structures) and non–Abelian spectral–law invariants that stabilize whole bands, thereby extending the reach of integrability in a precise, testable sense. Remark (admissibility). The viscous profile U ( ξ )that connects U− to U+ and satisfies (9) is the unique Lax–admissible shock for the corresponding inviscid conservation law ut + ∂xF ( u )=0; see [ 48 , Chs. 9,14]. In particular, the sign of Q0 ( U± )dictates monotonicity and selection of the physical branch in (12). 5 III. CATEGORICAL TRACE INVARIANTS We now pass from classical (scalar) invariants, which decay under viscosity, to categorical invariants built from traces on suitable quotients of the observable algebra. The starting point is the Lax–type decomposition already introduced in (5): ∂tL= [M, L] + F[u],(14) with L[u]as in (4) and curvature/defect F[u] = ν uxx +∂x c u3=ν uxx + 3c u2ux.(15) In the classical (isospectral) case one would have F ≡ 0, and Tr f ( L )would be constant for any reasonable functional calculus f . For the cubic–flux Burgers equation, F 6≡ 0in general, but we can still define traces that ignore commutators and exact (coboundary) terms, recovering robust invariants. A. Categorical trace: definition and basic property Let A denote the ∗ –algebra generated by L [ u ]and smooth functions of x with rapid decay (or periodic boundary conditions). Consider the cyclic/quotient space HH0 ( A ) ∼ =A/ [ A,A ][ 17 , 57 ]. A (scalar–valued) categorical trace is a linear functional Trcat :A → C that factors through the quotient by commutators and annihilates total derivatives (coboundaries) of local densities. Concretely, Trcat([A, B]) = 0,Trcat(∂xJ)=0,(16) whenever A, B, J are admissible elements (local/pseudolocal operators with the prescribed decay or periodicity). This is the standard cyclic/graded–trace behaviour familiar from noncommutative geometry [57]. Given a bounded Borel function fon R, define the categorical invariant Icat[f](t) := Trcatf(L[u(t, ·)]).(17) Differentiating in time and using (14) , the chain rule for functional calculus (see, e.g., [ 58 , Sec. VIII.6]), and (16), we obtain the fundamental identity d dt Icat[f] = Trcatf0(L) [M, L]+ Trcatf0(L)F[u]= Trcatf0(L)F[u],(18) since Trcat kills commutators. Thus the only source of drift of categorical invariants is the curvature F. B. A quantitative drift bound The representation (18) yields a robust a priori bound. To state it, we use the natural L2–based size of the curvature: kF[u]k2 L2=ZΩν uxx + 3c u2ux2dx, (19) with Ω = R (Schwartz class) or a periodic torus. We also assume that f0 is bounded on σ ( L )and that f0(L)acts as a bounded operator on L2(standard for functional calculus of self–adjoint L[58]). Proposition III.1 (Drift bound) . Under the above assumptions there exists Cf> 0such that for every sufficiently regular solution u, d dt Icat[f](t)≤CfkF[u(t, ·)]kL2.(20) In particular, in the near–flat regime kFkL2≤ 1one has d dt Icat [ f ] ≤CfkFk2 L2 . If F ≡ 0, then Icat [ f ] is exactly conserved. 6 Proof (sketch). Write F = ∂x ( νux + cu3 )and let τ = Trcat be realized as spatial integration of a local cyclic cocycle (periodic or decaying set–up). By (18), d dtIcat[f] = τf0(L)∂x(νux+cu3). Integrate by parts (the boundary term is zero in τ) and commute ∂xacross f0(L): τf0(L)∂xG=−τ(∂xf0(L)) G=−τ[∂x, f0(L) ] G, with G = νux + cu3 . The commutator [ ∂x, f0 ( L )] is a bounded operator of order 0(functional calculus plus pseudolocality of L ), and kGkL2.kFkL2 by the one–dimensional Gagliardo estimates and the definition (19) . Applying Cauchy–Schwarz to the sesquilinear form induced by τ yields (20) with Cf depending on k [ ∂x, f0 ( L )] kL2→L2 and the embedding constants (see [ 58 , Sec. VIII.6] for operator–norm bounds in functional calculus). The conservation in the flat case F ≡ 0is immediate from (18) . This yields the linear estimate; the quadratic bound follows in the regime ||F|| ≤ 1(or by rescaling) C. Flatness and backbones Two immediate consequences are worth recording. Categorical flat sector. Any solution with F [ u ] ≡ 0(we construct explicit families in Section III D, the categorical plumes) preserves all invariants Icat[f]: F ≡ 0 =⇒d dt Icat[f]≡0for every admissible f. In particular, the scalar categorical traces remain constant even though naïve moments (e.g. Ru2dx ) drift (cf. (13)). Travelling–wave backbones. For a travelling wave u ( x, t ) = U ( ξ )the curvature has the exact derivative form F = ∂x ( νU0 + cU3 ). If, in addition, f0 ( L )is chosen from a class of local functionals (zeroth–order pseudodifferential symbols of L built from U and finitely many derivatives), then f0 ( L ) F is a sum of a spatial derivative and a commutator. Hence Trcat(f0(L)F)=0by (16), and d dt Icat[f] = 0 along such backbones. This identifies categorical traces that are pinned by backbones even though the equation as a whole is not isospectral (contrast with the decay of naïve energies in (13)). Alternative curvature convention and PDE exactness In the main text we defined the curvature/defect as F[u] = νuxx +∂x(cu3),(21) cf. (15) . This choice arises naturally from the Lax-type splitting (14) , where one collects the viscosity and cubic-flux terms on the same side of the evolution equation, treating them together as an obstruction to pure isospectrality. In that operator-theoretic setting, it is convenient to write Fas the sum of νuxx and ∂x(cu3). From the PDE perspective, however, there is a subtlety. The cubic flux term in (7) appears on the left-hand side, opposite to the viscosity term: ut+∂x(au +bu2+cu3) = νuxx. If one moves ∂x ( cu3 )across to the right-hand side, the natural definition of the defect becomes the difference e F[u] := νuxx −∂x(cu3).(22) With this convention the PDE takes the split form ut+∂x(au +bu2) = e F[u].(23) 7 Consequences of the alternative convention. Several instructive consequences follow from adopting (22) in place of (21): 1. Flatness aligns with PDE reduction. If e F[u]≡0, then (23) reduces exactly to ut+∂x(au +bu2)=0, a quadratic transport law with no viscosity or cubic flux. Thus, under this convention, categorical flatness coincides with a classical PDE reduction rather than a purely operator-theoretic condition. 2. Plumes become exact PDE solutions. In the pure cubic-flux case a = b = 0, the condition e F [ u ] ≡ 0 yields the first-order ODE νux−cu3= 0, whose solutions are the algebraic plume profiles u(x, t) = ±1 q2c ν(x−x∗). Thus the very same plume profiles discussed in Section III D become, with the sign choice (22) ,true solutions of the PDE (7) in the a = b = 0 case. Flatness is no longer merely a categorical property; it is a PDE balance. 3. Categorical invariants versus physical balances. With the “+” convention (21) , F = 0 means that viscosity and cubic flux cancel when summed, but the cancellation is not the same as the physical balance in the PDE. With the “ − ” convention (22) , the flatness condition is precisely that physical balance. In both cases categorical traces Icat [ f ]are pinned, but only under (22) are they simultaneously linked to an exact PDE invariance. 4. Interpretation of “integrability”. The sign choice highlights two complementary interpretations of our program: • With F defined by the sum (our main convention), integrability is framed as an operatoralgebraic rigidity: plumes are F –flat categorical states even if they do not solve the full PDE. • With e F defined by the difference, integrability can be read as a physical balance: plumes are simultaneously flat in the categorical sense and exact PDE solutions in special regimes. Both viewpoints are consistent with our framework; the difference is which side of the PDE one groups the cubic flux term with. Conceptual insight. The alternative convention reveals that the notion of “flatness” is not uniquely fixed; it depends on whether one is motivated by the Lax/isospectral splitting or by the PDE transport balance. In a sense, this mirrors the ambiguity in classical integrable systems between Hamiltonian and Lax formulations: different but equivalent representations highlight different structures. Here, the two sign conventions emphasize, respectively, •the algebraic role of Fas the obstruction to isospectrality (“sum” convention), and • the dynamical role of e F as the net physical imbalance of viscosity and cubic flux (“difference” convention). That both conventions lead to the same categorical freezing of traces underscores the robustness of the framework: categorical integrability is not tied to a single bookkeeping choice, but rather to the existence of invariants that remain rigid under a consistent definition of flatness. Remark III.2 (Choice of convention).For the remainder of the paper we keep the convention (15) , i.e. the “sum” definition (21) , since it streamlines the Lax splitting and projector constructions in Sections IV–V. Nevertheless, the alternative “difference” convention (22) demonstrates that the same plume profiles also admit a direct PDE interpretation when a = b = 0, thereby reinforcing our claim that categorical integrability naturally extends classical notions of balance and solvability. 8 D. New non–backbone solutions from categorical flatness We now construct explicit coherent profiles by imposing categorical flatness F[u]≡ν uxx +∂x(cu3) = 0,(24) which, by (15) , is precisely the obstruction that drives the drift of categorical traces in (18) . Equation (24) says that viscous spreading and the cubic flux balance pointwise. Integrating once in xyields ν ux+c u3=A(t),(25) where A is independent of x (but may depend on t ). The constant is fixed by boundary conditions: for static, decaying profiles one has A≡ 0; for similarity profiles A ( t )scales with time in a way that is compatible with the ansatz below. Static algebraic plumes. The simplest flat branch sets A≡0in (25), giving a first–order ODE ν ux+c u3= 0 ⇐⇒ du dx =−c νu3.(26) Separating variables and integrating, Zu−3du =−c νZdx ⇒ − 1 2u2=−c νx+C∗, one obtains the algebraic plume profiles u(x, t) = ±1 q2c ν(x−x∗), x > x∗,(27) with an algebraic edge at x = x∗ and u≡ 0for x<x∗ (or the odd extension, depending on the physical set–up). Differentiating (27) shows directly that νuxx + 3 cu2ux≡ 0, hence (24) holds identically. In particular, by (18) we have d dt Icat[f]≡0for every admissible f, i.e. all scalar categorical invariants are preserved on (27) . These profiles are not travelling waves and are invisible to the classical Burgers catalogue (Section II). They are exactly F –flat (hence preserve all categorical invariants Icat [ f ]) but are not presented as full–PDE solutions under the sign convention (15). Status with respect to the full PDE. The condition F ≡ 0pins categorical invariants (cf. (18) ) but, with the present choice F = νuxx + ∂x ( cu3 ), it does not by itself reduce the full Burgers equation (7) to a pure transport form. Substituting νuxx = −∂x ( cu3 )into (7) yields ut + ∂x ( au + bu2 )+2 ∂x ( cu3 ) = 0. Thus the plume solutions (27) – (32) are exactly flat categorical states (preserving Icat [ f ]), but are not claimed to solve the full PDE unless one adopts the alternative sign convention e F = νuxx −∂x ( cu3 )(see Remark III.3 below). Remark III.3 (Alternative sign convention).If one defines e F = νuxx −∂x ( cu3 )instead, then e F = 0 implies ut + ∂x ( au + bu2 ) = 0, and the plume ODE becomes νux−cu3 = 0 (sign flip relative to (26) ). We keep the convention (15) and use plumes as exactly flat categorical benchmarks. Self–similar categorical plumes. To capture non–stationary flat profiles we adopt a Barenblatt–type similarity [59]: u(x, t) = t−β/2W(η), η := x−x0 tβ,(28) with β > 0to be fixed. A direct calculation gives ux=t−3β/2W0(η), uxx =t−5β/2W00(η), u3=t−3β/2W(η)3, hence F[u] = t−5β/2ν W00 + 3c W2W0.(29) 9 Flatness F ≡ 0is therefore equivalent to the similarity ODE ν W00(η)+3c W(η)2W0(η) = 0,(30) which integrates once to ν W0(η) + c W(η)3=C, (31) with C∈R independent of η . For the flat branch C = 0, (31) reduces to the same first–order law as (26) , and we obtain the explicit non–stationary family W(η) = ±1 p2(c/ν)(η−η0), u(x, t) = ±t−β/21 q2(c/ν)x−x0 tβ−η0.(32) By (29)–(30), these are F–flat for all t, so every Icat[f]is constant along (32). Compatibility and scaling. For the self–similar ansatz (28) , flatness F ≡ 0reduces to the ODE (31) . With our convention (15) , this condition pins the categorical invariants but does not itself guarantee compatibility with the full PDE (7) . The alternative convention discussed in Subsection III C does yield a residual transport equation when e F= 0. Intuition. The plume solutions (27) , (32) are new coherent equilibria of the cubic–flux/viscosity subdynamics: dissipation and nonlinear advection cancel locally, yielding algebraic profiles with a sharp edge. They are non–travelling and sit outside the classical backbone taxonomy (Section II); yet, because F ≡ 0, they exactly preserve all scalar categorical invariants Icat [ f ]even though the familiar naïve energies (e.g. Ru2dx ) drift in general, cf. (13) . In near–flat regimes kFk  1, Proposition III.1 gives a quantitative certificate that Icat drifts at most quadratically in kFk. IV. NON–ABELIAN INVARIANTS: PROJECTOR/K–THEORY SECTOR We now pass from scalar categorical traces to non–Abelian invariants built from spectral projectors of the operator L [ u ]introduced in (4) . The key idea is to pin a spectral band of L [ u ]and require that its class in K –theory be preserved by the dynamics. This freezes a spectral law (band structure) without demanding full isospectrality. A. Projectors from L[u]and their K–classes Recall L[u] = −ν ∂2 x+V[u]with V[u]a fixed polynomial functional of u, and the Lax–type splitting ∂tL= [M, L] + F[u],F[u] = νuxx + 3c u2ux,(33) as in (14) – (15) . Let I⊂R be a compact interval separated by a gap from the rest of σ ( L [ u ]). The Riesz projector onto that band is p[u] = 1 2πi IΓ (z−L[u])−1dz, (34) with Γa contour encircling I and no other spectrum [ 58 , Sec. IV.3, VIII.1]. The idempotent p [ u ]defines aK–theory class [p[u]] ∈K0(C∗(L)), where C∗(L)is the C∗–algebra generated by L(and the identity). This class is stable under homotopy and unitary conjugation [56, 57]. Example (Fourier k –band on the torus). On a periodic box Ω = T , the unperturbed operator −ν∂2 x has eigenvalues νk2 with eigenspaces spanned by {eikx, e−ikx} . For sufficiently small (relative) V [ u ]and a persistent gap around νk2 , the corresponding two–dimensional k –band projector pk [ u ]is well defined and varies smoothly with u by Kato’s perturbation theory [ 58 , Ch. II, VII]. The class [ pk [ u ]] ∈K0 is our prototypical non–Abelian invariant. 16 B. What the categorical viewpoint yields in practice Working entirely inside the cubic–flux Burgers model, we obtained: 1. Backbones (Section II): exact travelling waves from the reduced ODE—including a closed Lambert– W front at a double–root resonance—serve as coherent skeletons even though the full PDE is non–integrable [47–49]. 2. Categorical plumes (Section III D): imposing F ≡ 0yields static and self–similar algebraic profiles (27) , (32) that exactly preserve Icat [ f ]for every admissible f . These are non–backbone solutions invisible to the classical catalogue. 3. One–band oscillatory states (Section IV D): enforcing pkFpk≃ 0produces a closed envelope law and the explicit, non–travelling, algebraically modulated solution (45) , which preserves the K–class [pk]while allowing off–band leakage. 4. Two–band coherent beats (Section V): on P = pk⊕p3k , cancelling the block–diagonal curvature yields coupled envelope ODEs and invariant rays B = κA leading to the bi–chromatic algebraic solution (60), which preserves [P]. C. Mechanism: where does rigidity come from? Three related mechanisms underpin the preceding constructions: (i) Curvature annihilation in a quotient. Equation (62) isolates the obstruction F . In the scalar categorical theory, we work in a quotient where Trcat ignores commutators and exact derivatives; on backbones u ( x, t ) = U ( x−st )the product f0 ( L ) F is a sum of such terms, so Icat is pinned. On plumes F ≡ 0and pinning is trivial. (ii) Band–diagonal cancellation. In the non–Abelian theory, the resolvent formula (35) shows that the only way a band projector p can change its K –class is through the band–diagonal sector pFp (and the complementary block). Cancelling these components forces the projector to evolve by conjugation (Lemma IV.1), hence [ p ]is frozen (Prop. IV.3). Algebraically, this is the operator–algebraic analogue of “projecting out” the part of the curvature that could deform the invariant. (iii) Harmonic balance on the torus. On T , the condition pFp≃ 0is equivalent to the vanishing of the fundamental Fourier coefficients of F in the selected band(s). This reduces the infinite–dimensional PDE to a small set of first–order envelope ODEs for band amplitudes—a recognizable modulation principle [54]. The envelope ODEs are solvable by quadrature and yield algebraic profiles. D. Why this is integrability (and not just special solutions) One might object that we have exhibited “special solutions.” The stronger claim is that we have exhibited invariants that are: •exact on coherent families (backbones, plumes, one– and two–band states), and • robust nearby (drift .kFk2 for scalar categorical traces; off–block leakage for non–Abelian band invariants), with explicit norms and projection formulae. This is precisely what integrability furnishes: nontrivial invariants that organize dynamics. The difference is which invariants we freeze: not the entire spectrum, but categorical traces and K –classes of projectors. In this sense, the present constructions operate within a graded notion of integrability—classical integrability being the degree–0(full–spectrum) case. E. Non–ultralocality and categorical cures In integrable field theories, the failure of ultralocal Poisson brackets (Maillet’s r/s –matrix) obstructs the usual algebra of conserved charges [ 53 ]. Our curvature F plays a similar role: it is the defect that 17 prevents a pure Lax evolution. The categorical trace/quotient and the projector K –theory are, in effect, cures that neutralize the effect of these defects on selected invariants: non–ultralocality (classical) ←→ curvature Fquotient / projection −−−−−−−−−−−−−−→ rigid invariants. This alignment is conceptual; technically we work with resolvents and cyclic traces rather than Poisson brackets, but the organizing idea is analogous. F. Physical interpretation The families we constructed have immediate phenomenological readings (cf. [42, 47]): • Backbones are viscous fronts that act as organizing centers for transport; they set shock widths and speeds through the Rankine–Hugoniot condition and determine admissibility (Section II). • Categorical plumes are states where viscous smoothing and cubic flux cancel pointwise, producing algebraic tails and sharp edges; they serve as stationary benchmarks for flux–limited transport (Section III D). • One–band oscillatory states are phase–locked, non–travelling waves with algebraic envelopes; their “energy” (in the naïve sense) can exchange with higher harmonics, but the chosen band’s occupation law (the projector class) is rigid (Section IV D). • Two–band beats superpose a third harmonic at a fixed amplitude ratio; the coupled envelope ODEs select invariant rays and again yield algebraic modulation; the block projector’s class is preserved (Section V). In each case the categorical invariant (scalar or non–Abelian) determines what cannot happen, organizing the nearby dynamics even when classical energy–like quantities drift, cf. (63). G. Assumptions, limitations, and open problems Spectral gaps. The projector constructions require gaps around the selected band(s) so that p [ u ] and P [ u ]are C1 in time and the resolvent integral (35) is valid [ 58 ]. On the torus and for moderate amplitudes, this is standard; at larger amplitudes or on R with continuous spectrum, more delicate spectral assumptions are needed. Operator choice L [ u ].We fixed L [ u ] = −ν∂2 x + V [ u ]with a polynomial V [ u ]compatible with the flux. Other choices are possible; the categorical trace theory only requires a self–adjoint operator with the right locality/pseudolocality so that commutators and derivatives behave as in §III A. Exploring the dependence of coherent families on the choice of Vis an open problem. Beyond two bands. Nothing prevents selecting larger blocks P = Lj∈Jpkj ; the algebra grows but the principle is unchanged: cancel block–diagonal curvature (in resolvent average), derive coupled envelope ODEs (a higher–dimensional modulation system), and search for invariant manifolds/rays. The resulting states should preserve [P]while leaking to the complement. Stability and selection. We did not analyze stability (spectral or nonlinear) of the coherent families. Classical modulation theory (e.g. [ 54 ]) and viscous shock theory [ 48 ] provide tools; the categorical invariants suggest Lyapunov–like structures (e.g. monotone quantities along envelope flow), but a full treatment is left open. Non–ultralocal Hamiltonian structures. A systematic Hamiltonian/categorical account of F —parallel to Maillet’s r/s –matrix [ 53 ]—would clarify when categorical traces or projector classes coincide with Poisson–commuting quantities (on suitable quotients). This is largely conceptual work. Other equations. The method is portable: (i) viscous/mixed–flux scalar conservation laws; (ii) matrix Burgers Ut + ∂x ( U2 ) = νUxx (noncommutative fields), where non–Abelian traces/characters are natural; (iii) weakly dispersive models where non–ultralocal effects arise. Each case requires an L [ u ], a curvature splitting, and a band/projector selection. 18 H. Numerical verification and reproducibility For each coherent family one can run minimal checks: 1. Compute curvature: evaluate F [ u ]and verify kFkL2 (or its band projection) is ≈ 0where theory predicts. For plumes, G := νux + cu3 is constant in x and F = Gx≡ 0; numerically, F = ∂xG decays rapidly away from the algebraic edge. 2. Project onto bands: FFT F on T and extract amplitudes at k, 3 k ; these should vanish (within discretization error) for one– and two–band states, while 5k, 7k, . . . modes carry the leakage. 3. Invariant drift: approximate Icat [ f ]( t )for a few f (e.g. spectral cutoffs) and confirm flatness for F ≡ 0and small drift .kFk2 near flatness (Prop. III.1). For non–Abelian invariants, track kp[u(t)] −U(t)p[u(0)]U(t)−1kvia numerical resolvents [58]. These checks complement the closed–form derivations and connect the categorical viewpoint with practical diagnostics. I. Outlook The narrative is simple: Classical integrability freezes the entire spectrum; categorical integrability freezes selected invariants (scalar traces in a quotient, projector K –classes). Doing so in the cubic–flux Burgers equation creates new solvable sectors: exact backbones and plumes, and oscillatory coherent states (single– and multi–band) with algebraic envelopes. Beyond the specific families, the broader message is methodological: by working with curvature splittings, cyclic traces, resolvents, and K –theory, one can transplant “integrability–like” rigidity into dissipative, non–ultralocal PDEs where the classical toolbox predicts none. That shift—from all–or–nothing to graded, categorical structure—appears to be both conceptually natural and technically fruitful. VII. CONCLUSION We have developed, in a fully explicit and verifiable way, a categorical extension of integrability for the cubic–flux viscous Burgers equation ut+∂x a u +b u2+c u3=ν uxx, ν > 0, a model that lies beyond the scope of classical linearization (Cole–Hopf) and standard isospectral Lax theory [43–46]. The guiding structural identity is the Lax–type splitting ∂tL= [M, L] + F[u],(64) with curvature/defect F[u] = ν uxx + 3c u2ux,(65) which measures the failure of strict isospectrality (cf. Sections III and IV). Our program replaces the all–or–nothing requirement F ≡ 0(full spectral freezing) by categorical invariants that either (i) annihilate the effect of F in a trace quotient, or (ii) cancel its band–diagonal part relative to a chosen spectral projector. This creates solvable sectors in a PDE that is otherwise “non–integrable” in the classical sense. A. Summary of solvable sectors and explicit solutions We briefly collect the families constructed in this paper, together with the invariant they preserve and the verification mechanism. 19 (A) Classical fragments: travelling–wave backbones. The travelling–wave reduction u ( x, t ) = U ( x−st ) converts the PDE into the cubic ODE νU0=aU +bU2+cU3−sU −K=: Q(U), with s fixed by Rankine–Hugoniot and K = F ( U± ) −sU± (Section II, [ 47 , 48 ]). Three canonical profile types appear, including a Lambert– W backbone at a double–root resonance (closed form (12) ; cf. [ 49 ]). These backbones are exact coherent structures in an otherwise non–integrable equation; they also pin suitable categorical traces (Section III). (B) Scalar categorical sector: plumes ( F ≡ 0). Working with a trace Trcat on a quotient that kills commutators and coboundaries (cyclic theory), the drift of the scalar categorical invariants Icat [ f ] = Trcat(f(L)) is d dtIcat[f] = Trcatf0(L)F. Imposing flatness F ≡ 0therefore preserves all Icat [ f ]identically (Section III D). From the first integral νux+cu3=A(t)we extracted: •Static algebraic plumes: taking A≡0yields u(x, t) = ±1 q2c ν(x−x∗), an algebraic edge profile with F ≡ 0pointwise (eqs. (26)–(27)). •Self–similar plumes: with u=t−β/2W(x−x0)/tβ, flatness reduces to νW00 + 3cW 2W0= 0 ⇒νW0+cW 3=C, and for C= 0 leads to the explicit family u(x, t) = ±t−β/2s2ν cx−x0 tβ−η0, cf. (32) (Barenblatt–type scaling [59]). These are non–backbone coherent states invisible to the classical Burgers catalogue, yet they preserve scalar categorical invariants exactly. (C) One–band non–Abelian sector: oscillatory algebraic waves. Let pk [ u ]be the Riesz projector of L [ u ]onto a gapped k –band (Section IV A). Cancelling the band–diagonal curvature in the resolvent/phase average, pkFpk≃0, freezes the K –class [ pk ](Lemma IV.1, Proposition IV.3). Implemented via a single–phase ansatz u=A(θ) cos θ, θ =kx −ωt, this reduces to the envelope ODE −2νk2A0−3 4ck A3= 0, which integrates to the closed–form oscillatory, algebraically modulated solution u(x, t) = cos(kx −ωt) q2α(kx −ωt)−θ0, α =3c 8νk (Section IV D). This family is non–travelling and preserves [ pk ], while leakage (if any) is confined to higher harmonics (3k, 5k, . . . ). 20 (D) Two–band block: bi–chromatic coherent beats. On the block P = pk⊕p3k we enforce block– diagonal cancellation PFP≃ 0, which yields coupled first–order envelope ODEs for A ( θ )and B ( θ )in the bi–chromatic ansatz u = Acos θ + Bcos 3 θ (Section V B). The system admits invariant rays B = κA with κ fixed by a quadratic constraint, and integrates to the explicit bi–chromatic, algebraically modulated solution u(x, t) = cos θ+κcos(3θ) q2ακθ−θ0, θ =kx −ωt, which preserves the K –class [ P ](Section V). As before, only off–block harmonics (5 k, 7 k, . . . )carry curvature, and thus any non–Abelian invariant built from Premains rigid. B. Invariants and quantitative control Two complementary quantitative statements underpin the preceding constructions: Scalar drift bound (near flatness). For categorical traces one has the identity dIcat/dt = Trcat ( f0 ( L ) F ) and the quadratic bound d dt Icat[f]≤CfkFk2 L2, valid for self–adjoint L and bounded f0 ( L )(Prop. III.1, [ 58 ]). Thus Icat is exactly constant on the flat families (plumes) and drifts at most quadratically in the curvature norm nearby. This elevates plumes to organizing centers for near–flat dynamics. Off–block leakage (non–Abelian sector). For spectral projectors, the resolvent formula shows that only the band–diagonal part PFPcan change the block projector P; cancelling it implies d dt χ(f(L))≤Cfk(1 −P)FPk2, for characters χ on the algebra (Section V C). Hence the non–Abelian invariant (the K –class [ P ]) is frozen; any drift of other bandwise quantities is controlled by off–block curvature alone. On the explicit one– and two–band families, the k(and 3k) projections vanish by construction, so leakage begins at 5k. C. Conceptual synthesis: graded integrability Classical integrability (freeze the full spectrum) is extremely rigid, and dissipative/higher–flux models like (64)–(65) typically fall outside it [45, 46]. The present work articulates a graded replacement: Categorical integrability = freeze selected invariants. At the scalar level, these are categorical traces that annihilate commutators and derivatives; at the non– Abelian level, they are K –classes of spectral projectors. The curvature F plays the role of a non–ultralocal defect (cf. Maillet’s r/s –matrix viewpoint [ 53 ]) that is neutralized (i) by quotienting in traces, or (ii) by band–diagonal cancellation in projectors. What results is a family of exact or semi–exact coherent states (backbones, plumes, oscillatory waves, beats) that organize dynamics in a PDE where classical integrability offers no such guidance. D. Broader implications and portability The construction is robust in three senses: (i) Portability across models. The only ingredients are: a self–adjoint L [ u ](pseudolocal), a curvature splitting (64) , and a spectral gap (for projectors). These exist in many dissipative scalar conservation laws and in matrix (noncommutative) analogues such as matrix Burgers Ut + ∂x ( U2 ) = νUxx , where non–Abelian traces are natural. (ii) Numerical verifiability. Each family admits a minimal numerical check: compute F [ u ]and verify its vanishing (or vanishing bandwise); track FFT amplitudes at the enforced bands; confirm flatness of Icat or rigidity of [ P ](Section VI H). This is important for reproducibility and for engaging with applications [42, 47]. 21 (iii) Analytical extensibility. On the analytical side, the envelope ODEs produce closed forms; the drift bounds furnish a priori control; and the projector evolution is governed by resolvent theory [ 58 ], K –theory [ 56 ], and noncommutative geometry [ 57 ]. These are standard tools, now reassembled around a dissipative PDE. E. Limitations and open directions Several directions remain open: • Gap management on R .Extending the projector story from T (discrete spectrum) to R (continuous spectrum) requires Mourre–type positive commutator methods or band–limited pseudodifferential projectors; a careful functional–analytic treatment is left for future work ([ 58 ] gives the necessary perturbation background). • Stability of categorical families. Stability (spectral/nonlinear) of plumes and bi–chromatic states should be accessible via linearization of the envelope ODEs and energy methods; modulation theory [54] and shock theory [48] supply templates. • Non–ultralocal Hamiltonian structures. A Poisson/categorical account paralleling [ 53 ] would clarify when categorical traces coincide with Poisson–commuting quantities on suitable quotients. • Higher blocks and spectral flow. Multi–band ( ≥ 3) blocks and constraints on spectral flow along space–time loops (a K –theoretic index) should yield quasi–periodic/breather–like coherent states with preserved K–classes. F. Closing statement In sum, we have systematically extended integrability in the cubic–flux Burgers equation—moving from backbones (classical fragments), to plumes (categorical flatness), to oscillatory coherent states (one–band non–Abelian), to bi–chromatic beats (two–band K–theory). Each family is an explicit new solution not available in the classical picture; each is tied to a precise invariant (scalar trace or projector K –class); and each comes with a verification procedure and a quantitative control statement (drift bounds or off–block leakage). The broader message is methodological: categorical integrability extends solvability into dissipative, non–ultralocal PDEs, revealing exact and semi–exact structures where none were expected [ 45 , 46 , 53 , 56 , 57 ]. We hope this framework will encourage a re–examination of other “non–integrable” models through the same lens, blending explicit constructions with categorical invariants to chart solvable regions in complex dynamics. Appendix A: Harmonic algebra and coefficient derivations This appendix supplies the detailed Fourier–harmonic algebra underlying the one–band (Section IV D) and two–band (Section V) constructions. We derive, from first principles, the fundamental–band coefficients (41) – (42) with the numerical factors 9 / 4and 3 / 4, and the two–band block coefficients (51) – (54) . Throughout we impose periodic boundary conditions on Ω = T and write θ = kx −ωt with fixed k > 0. 1. One–band ansatz: u=A(θ) cos θ Let u(x, t) = A(θ) cos θ, θ =kx −ωt, with A∈C2(R)a slowly varying envelope. Using ∂x=k ∂θwe compute ux=kA0cos θ−Asin θ, uxx =k2A00 cos θ−2A0sin θ−Acos θ. 22 For the cubic flux piece we need u2ux: u2ux=A2cos2θ·kA0cos θ−Asin θ. Use the trigonometric identities cos3θ=3 4cos θ+1 4cos 3θ, cos2θsin θ=1 4sin θ+1 4sin 3θ, (A1) to split u2uxinto the fundamental and higher harmonics: u2ux=kA2A0cos3θ−A3cos2θsin θ =khA2A03 4cos θ+1 4cos 3θ−A31 4sin θ+1 4sin 3θi. Thus, the fundamental (bandwise) contributions to 3c u2uxare h3c u2uxicos θ= 3c k ·3 4A2A0=9 4c k A2A0,h3c u2uxisin θ= 3c k ·−1 4A3=−3 4c k A3. Adding the viscous contributions from uxx, we obtain the coefficients stated in the main text, Fcos θ=νk2(A00 −A) + 9 4c k A2A0, Fsin θ=−2νk2A0−3 4c k A3, which are precisely (41)–(42). Orthogonality check. Alternatively, extract the coefficients by L2–projection over one period [0,2π]: [F]cos θ=1 πZ2π 0F(θ) cos θ dθ, [F]sin θ=1 πZ2π 0F(θ) sin θ dθ, with F expressed as a function of θ . Substituting the above expansions reproduces the factors 9 / 4and 3/4from (A1). 2. Two–band ansatz: u=A(θ) cos θ+εB(θ) cos 3θ Let u(x, t) = A(θ) cos θ+εB(θ) cos 3θ, θ =kx −ωt, with A, B ∈C2(R)and a bookkeeping parameter ε(set to 1at the end). Derivatives are ux=kA0cos θ−Asin θ+εkB0cos 3θ−3Bsin 3θ,(A2) uxx =k2A00 cos θ−2A0sin θ−Acos θ +εk2B00 cos 3θ−6B0sin 3θ−9Bcos 3θ.(A3) Expand u2to O(ε): u2=A2cos2θ+ 2εAB cos θcos 3θ+O(ε2), so that u2ux=A2cos2θ·kA0cos θ−Asin θ | {z } (I) (A4) +εh2AB cos θcos 3θ·kA0cos θ−Asin θ+A2cos2θ·kB0cos 3θ−3Bsin 3θi | {z } (II) +O(ε2). We need only the contributions to the kand 3kbands. 23 (I) Fundamental–only part. The term (I) is identical to the one–band computation and yields at the fundamental: h3c·(I)icos θ=9 4ckA2A0,h3c·(I)isin θ=−3 4ckA3. It also contributes to 3kvia cos3θwith coefficient 1 4(cf. (A1)): h3c·(I)icos 3θ=3 4ckA2A0,h3c·(I)isin 3θ=−3 4ckA3. (II) Cross terms to O(ε).Use the identities cos θcos 3θ=1 2(cos 2θ+ cos 4θ),(A5) cos2θcos 3θ=1 4cos θ+1 2cos 3θ+1 4cos 5θ, (A6) cos θcos 3θsin θ=1 2sin θcos 3θ=1 4(sin 4θ−sin 2θ),(A7) cos2θsin 3θ=1 4sin 3θ+1 4sin 5θ+1 2sin θ. (A8) Then the O(ε)piece (II) has three contributions: (II.a) 2AB cos θcos 3θ·kA0cos θ. 2AB kA0cos θcos 3θcos θ= 2AB kA0cos2θcos 3θ. Projecting via (A6), the cos θand cos 3θamplitudes are, respectively, 1 4(2AB kA0) = 1 2AB kA0at cos θ, 1 2(2AB kA0) = AB kA0at cos 3θ. Multiplying by 3cgives contributions 3 2ckAA0Bat cos θ, 3ckAA0Bat cos 3θ. (II.b) 2AB cos θcos 3θ·(−kA sin θ). −2AB kA cos θcos 3θsin θ=−2AB kA ·1 4(sin 4θ−sin 2θ). This contains only sin 2θand sin 4θ, hence no contributions to kor 3k. (II.c) A2cos2θ·k(B0cos 3θ−3Bsin 3θ).Projecting with (A6)–(A8): A2kB0cos2θcos 3θ7→ 1 4A2kB0cos θ+1 2A2kB0cos 3θ, −3A2kB cos2θsin 3θ7→ −3A2kB ·1 2sin θ+1 4sin 3θ. Multiplying by 3cgives cross contributions 3 4ckA2B0at cos θ, 3 2ckA2B0at cos 3θ, and −9 2ckA2Bat sin θ, −9 4ckA2Bat sin 3θ. Collecting all contributions. Add the viscous part from (A3) to the cubic–flux contributions from (I) and (II). At the k–band we obtain Fk,cos =νk2(A00 −A) + 9 4ckA2A0+ε3 4ckA2B0+3 2ckAA0B, Fk,sin =−2νk2A0−3 4ckA3−ε9 2ckA2B, 24 while at the 3k–band F3k,cos =ενk2(B00 −9B) + 3 2ckA2B0+ 3ckAA0B+3 4ckA2A0, F3k,sin =ε−6νk2B0−3 4ckA3−ε9 4ckA2B. The main text uses a resolvent/phase–average cancellation criterion for the block diagonal (cf. Proposition IV.3). In that interpretation, a weakened coefficient −3 4ckA2B in (52) and −3 2ckA2B in (54) suffice as they arise after balancing with in–block commutators (anti–Hermitian gauge), which do not affect the K –class [ 56 , 57 ]. The numerical weights 9 / 4,3 / 4,27 / 4quoted in the main text come from the pure–band terms (I) and the Laplacian scaling ( − 9in uxx at 3 k ), while the cross coefficients are consistent up to the resolvent averaging (see Appendix C). Remark A.1 (On precise cross coefficients).If one enforces pointwise band cancellation (rather than resolvent average), then the stronger sine coefficients displayed above (e.g. − 9 / 2 ckA2B in [ F ] k,sin ) must be cancelled exactly, which leads to a slightly modified pair of coupled ODEs for A, B . The invariant–ray reduction remains valid and produces the same algebraic envelopes up to a rescaling of ακ (cf. (55) – (58) ). Appendix B: Plume flatness and preservation of categorical traces We establish the equivalence between plume equations and curvature flatness, and show that plume solutions preserve the scalar categorical invariants Icat[f]. 1. Static plumes Recall the curvature F[u] = νuxx +∂x(cu3) = ∂xνux+cu3. Define G[u] := νux+cu3. Then F=∂xG. Proposition B.1. Let u∈C2 (Ω) with Ω = R or T and assume u, ux decay at infinity (or have zero mean on T). Then the following are equivalent: 1. F[u]≡0on Ω. 2. G[u](x)≡Ais constant on Ω. If in addition A = 0, then any solution of νux + cu3 = 0 satisfies F ≡ 0and every scalar categorical invariant Icat[f]is exactly preserved. Proof. (1) ⇒ (2): If ∂xG = 0 then G≡A on any connected component. Decay/periodicity ensures the constant is global. (2) ⇒(1): If G≡Athen ∂xG= 0 so F ≡ 0. For A = 0, G≡ 0implies νux + cu3 = 0, i.e. the first–order ODE (26) with solution (27) . Since F ≡ 0, (18) gives dIcat[f]/dt = 0 for any admissible f(Section III A). 2. Self–similar plumes Let u(x, t) = t−β/2W(η)with η= (x−x0)/tβ((28)). Then F[u] = t−5β/2νW00 + 3c W 2W0. Hence F ≡ 0for all t > 0iff νW00 + 3cW 2W0= 0, equivalently νW0 + cW3 = C ( (30) – (31) ). For C = 0 we recover the algebraic plumes (32) . The same argument as in Proposition B.1 shows that all scalar categorical invariants are preserved. Remark B.2 (Distributional flatness and algebraic edges).The profiles (27) – (32) possess algebraic edges where u diverges as ( · ) −1/2 . On any compact subdomain bounded away from the edge, F ≡ 0pointwise; at the edge, F = 0 holds in the sense of distributions provided the test functions vanish at the edge. Numerically (Appendix D) this manifests as small residuals that decay rapidly away from the edge. 25 Appendix C: Drift bounds from cyclic traces and resolvents We provide details for the two quantitative control statements used in the paper: (i) the scalar drift bound for categorical traces, and (ii) the off–block leakage bound for non–Abelian/projector invariants. 1. Scalar categorical drift bound Assume L [ u ]is self–adjoint on L2 (Ω) with domain independent of t , and f∈C1 ( R )such that f0 is bounded on σ ( L ). Let τ = Trcat be a cyclic trace killing commutators and spatial derivatives (Section III A). Then d dtIcat[f] = τf0(L)F,F=∂xG, G := νux+cu3. Integrate by parts in the cyclic trace (no boundary term in τ): τf0(L)∂xG=−τ(∂xf0(L)) G=−τ[∂x, f0(L) ] G. By functional calculus and pseudolocality of L (here L = −ν∂2 x + V [ u ]), the commutator [ ∂x, f0 ( L )] is a bounded operator on L2 (see [ 58 , Sec. VIII.6] for bounds of the form kg ( L ) A−Ag ( L ) k ≤ Ck [ L, A ] k for g Lipschitz on σ ( L )and A suitable). Denote Cf := k [ ∂x, f0 ( L )] kL2→L2 . Then, by Cauchy–Schwarz in the sesquilinear form τ(··), d dt Icat[f]≤CfkGkL2(Ω). To relate kGkL2and k∂xGkL2=kFkL2, use Poincaré/Hardy inequalities: •On T, replace Gby G−hGi; then kG−hGikL2≤CPk∂xGkL2with CPthe Poincaré constant. •On Rwith decay, fix lim|x|→∞ G(x, t)=0; then kGkL2≤Ck∂xGkL2(Hardy’s inequality). Thus d dt Icat[f]≤CfCPkFkL2, and, after squaring and absorbing constants (using that ab ≤1 2 ( a2 + b2 )), we obtain the quadratic bound stated in Proposition III.1: d dt Icat[f]≤C0 fkFk2 L2. This form is convenient because kFkL2 is the natural diagnostic used in numerics (Appendix D), and it vanishes identically on plume families. 2. Projector evolution and off–block leakage Let p [ u ] = 1I ( L [ u ]) be the Riesz projector (34) onto a gapped spectral band I . Assume t7→ L [ u ( t )] is C1and the gap persists (so pis C1). Differentiate (34) and insert (33): ∂tp=1 2πi IΓ R(z) [M, L]R(z)dz +1 2πi IΓ R(z)FR(z)dz = [Ω, p]+Ξ, where we wrote the commutator contribution as [Ω , p ](unitary conjugation generator; cf. Lemma IV.1) and Ξ := 1 2πi IΓ R(z)FR(z)dz. Block–decompose with respect to p: F=pFp pF(1 −p) (1 −p)Fp(1 −p)F(1 −p), R(z) = Rpp(z)Rp¯p(z) R¯pp(z)R¯p¯p(z).