In physical coordinates, the collisionless distribution function satisfies

where

Let Z = ( X , V ) and let ℱ t , t 0 denote the phase-space characteristic flow generated by a sufficiently regular background field Ψ 0 :

The Vlasov density is constant along characteristics,

Equivalently,

so deformation of the tangent map controls phase-space gradient amplification and filamentation. Thus, D ℱ t , t 0 is part of the kinetic transport itself. For a spatial deviation ξ and velocity deviation ν = ξ ˙ ,

whose configuration block is

Definition 1 (Reduced kinetic stability).On a finite interval,reduced kinetic stability means the invariant splitting,Lyapunov boundedness,exponential separation,and norm-dependent finite-time gain properties of the tangent map D ℱ t , t 0 associated with the Vlasov characteristic transport in a prescribed background.The term includes stable,unstable,and neutral local sectors in the standard dynamical-systems sense [32].It does not mean spectral,nonlinear,or energy-Casimir stability of the self-consistent distribution function.

For comparison, the self-consistent linearization about f 0 contains

The characteristic variation then contains the additional force − ∇ δ   Φ ( X 0 , t ) . Equations (8) and (9) describe collective feedback, Jeans-type modes, and phase mixing. They are not solved in the present model. The adjective “kinetic” therefore identifies the phase-space Vlasov transport from which the characteristic map is obtained, while the qualifier “reduced” records the omission of self-consistent perturbation feedback.

Write x = a ( t ) r , with comoving position r and H = a ˙ / a . In the usual peculiar-potential formulation, a reference trajectory and a neighboring comoving separation η obey

where

The physical separation is ξ = a η . Direct differentiation gives

For pressureless matter plus Λ,

so

Using Equation (12) gives tr   K phys = 4 π G ρ − Λ c 2 , as in the total physical-potential formulation.

The coordinate interpretation is exposed by a de Sitter check. If ϕ = 0 , a = e H t and H 2 = Λ c 2 / 3 , then

remains bounded, whereas

Thus the physical exponent H is background expansion, not a cosmic-web instability. The representation used for a finite-time estimate must be fixed in advance. Comoving variables are appropriate for peculiar deformation relative to the Hubble flow; physical variables measure actual physical separation over a stated finite interval. Asymptotic Lyapunov exponents need not be invariant under the unbounded transformation ξ = a η .

A numerical propagator norm is meaningless until configuration and velocity components have been placed on one declared physical scale. For a cosmological analysis window starting at t 0 , the primary convention of this paper is

The dimensionless full states are

The default reporting metric is the Euclidean norm of these dimensionless states, i.e., W ∗ = I 6 . In dimensional variables Y = ( q , q ˙ ) T , the same convention is

The common factor ℓ ∗ − 2 is a unit conversion rather than an independent regulator of induced gain. For every c > 0 ,

Consequently, changing ℓ ∗ alone does not change an operator gain or a normalized selected-state ratio. The smoothing length R s remains physically important because it determines the smoothed classifier tensor and Hessian, and hence the dynamics and morphology intervals. The relative weighting of the position and velocity blocks in the full-state metric is set by τ ∗ (and by any additional nonscalar choice of W ).

The scales and metric are fixed for the complete itinerary; they are not reset at a web transition. Cross-redshift or cross-cosmology comparisons must report R s as a dynamical/smoothing input and must report t 0 , τ ∗ and W as the phase-space metric convention. A sensitivity analysis in τ ∗ is mandatory whenever an alternative dynamical time is used. In a dimensionless algebraic benchmark with H = 0 , τ ∗ = 1 is simply part of the stated test units and carries no claim of cross-redshift comparability.

The local Newtonian description requires, in addition to sub-horizon scale,

where L is the analysed scale and u pec = a r ˙ . For a frozen background over one cell, one also requires

The peculiar Poisson equation is understood with the boundary convention of the simulation or reconstruction, typically a periodic box or a local patch with prescribed boundary data. The formally divergent potential of an infinite homogeneous Newtonian medium is not used. The physical Hessian is assembled from the background-subtracted peculiar Hessian and the Friedmann background as in Equation (15); this also prevents double counting of the Λ contribution.

At a physical point, the matter Hessian can be decomposed as

so

Density fixes only the trace. Individual principal-curvature signs depend on the traceless tidal field and on the boundary data. A density-only isotropic formula is therefore conditional on ℰ 0 ≃ 0 .

Morphology is assigned independently of the subsequent stability calculation. Smooth the comoving density contrast with a Gaussian kernel,

using the fiducial scale

Define

Let τ 1 ≥ τ 2 ≥ τ 3 be the eigenvalues of T R . With threshold τ th = 0 , the number

assigns void, wall, filament and node for N + = 0 , 1 , 2 , 3 , respectively [17][18]. The positive eigenvector is the wall normal; the two positive eigenvectors span the filament transverse plane. Numerical classification also requires a declared eigenvalue tolerance. Define the classifier margin

If d class < δ class , the label is flagged as unresolved rather than silently assigned. Such transition intervals remain in the propagator composition but are not used as evidence for a morphology-specific effect.

For consistent smoothing,

and the physical effective curvatures are

The morphology label retains only the count in Equation (28); it does not determine the values or signs in Equation (31). All local growth or confinement statements, therefore, use the complete Hessian.

Choose one representation σ ∈ { phys , com } for the entire itinerary. With a fixed time scale τ * > 0 , define

Let U ( t , t 0 ) be the exact propagator of z ˙ = L ( t ) z . In a cell [ t 0 , t 0 + Δ ] , freeze L 0 = L ( t 0 ) and use U 0 ( Δ ) = e L 0 Δ . For any submultiplicative matrix norm, set

Variation of constants gives the explicit estimate

Thus, the frozen-cell approximation is controlled directly by the variation of the full first-order generator, not merely by instantaneous eigenvalue ratios.

If E ( t ) is the orthogonal matrix of eigenvectors of K ( t ) , D = E T K E = diag ( κ i ) and Ω = E T E ˙ , then ξ = E q gives

The terms 2 Ω q ˙ , Ω ˙ q and Ω 2 q show why slow principal-frame rotation, its derivative, and spectral gaps must be checked. Quantities such as | κ ˙ i | / | κ i | 3 / 2 and ‖ Ω ‖ / | κ i | are useful diagnostics, but without an error estimate such as Equation (35), they are not sufficient theorems. Near eigenvalue crossings, the invariant subspace, rather than an individual eigenvector, must be propagated. More quantitatively, suppose a spectral cluster of K ( t 0 ) is separated from the complementary spectrum by a gap δ gap > 0 and ‖ K ( t ) − K ( t 0 ) ‖ 2 < δ gap / 2 . A Davis-Kahan projector estimate gives

for the corresponding invariant-subspace projectors [33]. This supplies a basis-independent localization check in addition to the full propagator error bound.

The stability information carried by a frozen symmetric Hessian is not tied to a particular eigenvector convention. Let P − , P 0 and P + be the spectral projectors of K onto its negative, zero, and positive spectral subspaces, and define

The triple ℑ ( K ) is the inertia of the quadratic form and n − is its Morse index. It is invariant under orthogonal frame changes and, more generally, under nonsingular congruences by Sylvester’s law of inertia [34]. In numerical work, the zero subspace is replaced by a declared tolerance | κ i | ≤ κ tol . The T-web count N + and the effective-Hessian inertia ℑ ( K ) are intentionally kept separate: the former is a morphology label, while the latter is the local stability signature. This inertia classification refers specifically to the frozen conservative physical-coordinate equation q ¨ = − K phys q . In the comoving representation, the Hubble block in L com changes the spectral type; consequently, comoving stability is classified by the full first-order generator or its finite-time propagator, not by the sign of K pec alone.

For the conservative physical frozen cell, write q = ξ , p = ξ ˙ and q α = P α q , p α = P α p . On the negative subspace set K − : = K | Ran   P − < 0 and M = ( − K − ) 1 / 2 . The hyperbolic modal vectors

obey

Their bilinear product

is an exact, generally indefinite invariant. On the positive subspace, K + : = K | Ran   P + > 0 and

is exact and positive definite. On the neutral subspace, p 0 is constant and q 0 may drift linearly. These quantities combine into the frozen-cell Hamiltonian

Table 1 lists the morphology-local specializations. The curvature condition, rather than the morphology label alone, is decisive.

There is also a structural invariant that survives arbitrary time dependence of a symmetric physical Hessian. For the unscaled canonical state Y = ( q , p ) T , set

Because A T J + J A = 0 , the physical tangent propagator satisfies

Thus, the symplectic form and phase volume are exact invariants even when the cell energy is not conserved [35][36].

The comoving equation has an analogous canonical statement. For any two solutions η 1 , η 2 of Equation (11),

is constant. Equivalently, with canonical momentum π = a 2 η ˙ ,

which is symplectic. The contraction of phase volume in the noncanonical variables ( η , η ˙ ) , therefore, reflects the time-dependent coordinate scaling, not the destruction of the canonical invariant.

The quadratic energy is only a local invariant when K is frozen. For a smoothly varying physical Hessian,

At an idealized cell boundary t j , where q and p are continuous but the frozen Hessian changes from K j to K j + 1 ,

This exact interface identity prevents local wall, filament, or void invariants from being misused as a single global conserved energy.

Table 1. Exact invariants for a frozen conservative cell in the physical-coordinate representation. Each row applies to an invariant spectral subspace of the full K phys ; comoving cells must be classified by L com or its propagator.

For a scalar principal curvature κ = − μ 2 < 0 , the frozen physical separation mode satisfies

The Chetaev function C ( q , p ) = q p has

in the forward-invariant cone q > 0 , p > 0 , establishing instability of the origin [37]. The phase-space modal coordinates

obey

Therefore, μ = − κ is the exact exponent of the growing phase-space mode a + , or of data restricted to p = μ q . In addition,

is conserved. In the scalar case, its zero level contains the stable and unstable separatrices p = ∓ μ q , while nonzero levels are hyperbola.

It is not a universal pointwise upper bound for the coordinate projection | q | . For q ( 0 ) = 0 , p ( 0 ) = 1 ,

Arbitrary projections must therefore be treated by the propagator of the full state.

In the isotropic physical void approximation ℰ 0 ≃ 0 , Equation (24) gives

This is a physical-separation modal exponent. It includes the background term and is not, by itself, a peculiar growth rate.

Hence | q ( t ) | ≤ 2 E ( 0 ) / ω . This is Lyapunov confinement [35][36][38]. It is not a monotonic contraction of a fixed Euclidean scaling. For

we have

Unless τ ∗ ω = 1 for that particular cell, the sign can be positive or negative. A scaling matched to one frequency is not generally matched to the next wall or filament cell.

For the unscaled state ( q , p ) T , the exact scalar cell propagator is

These matrices retain the phase information that a signed scalar “growth minus stabilization” budget discards.

For a constant symmetric 3 × 3 Hessian, an orthogonal transformation reduces the frozen physical problem to three scalar blocks of the form (60). Exact localization requires an invariant subspace of the complete Hessian, not merely positive directed curvature. If n is a fixed T-web wall normal, q n = n T q , p n = n T p and κ n = n T K n > 0 , then

Hence, the wall-normal energy is exact for all states if and only if K n ∥ n . For a filament transverse projector P ⊥ , the absence of coupling to the axial complement requires

When the classifier tensor and K are constructed with the same smoothing and background convention in Equations (30) and (31), this alignment is automatic because K is an affine combination of T R and the identity. If morphology and the stability Hessian are reconstructed independently, the commutator must be checked, and the exact invariants must be formulated with the spectral projectors of K . Under this condition, positive wall-normal or filament-transverse curvatures bound the associated configuration coordinates, while the remaining directions may be neutral or hyperbolic. Node intervals are ordinary dynamical intervals and cannot be deleted from an itinerary.

All primary gains are computed in the fixed dimensionless state of Equations (18) - (20). The default metric is W ∗ = I 6 ; the formulas below allow a different fixed W = W T > 0 only to make metric sensitivity explicit:

For an interval of duration Δ t > 0 , the associated metric Cauchy-Green tensor and the largest finite-time Lyapunov exponent are

This singular-value construction is standard [13]-[15]. The claimed contribution is not a new FTLE or Cauchy-Green measure, but the fixed-metric, maximal-interval, and background-referenced reporting protocol developed below.

Let m ( t ) be the T-web label after applying the declared threshold and classifier tolerance. A report interval I k = [ s k , s k + 1 ] is a maximal connected interval on which m ( t ) is constant and resolved. Ambiguous transition intervals are assigned the auxiliary label u and are retained in the propagation. Thus

The numerical integrator may use an arbitrarily fine mesh s k = t k , 0 < ⋯ < t k , M k = s k + 1 . Its substep propagators are first composed,

Only the norm of the composed propagator over the full maximal interval is reported as an interval gain. Integration substeps are not counted as separate morphology contributions.

Proposition 1(Finite-time report bound). For either the physical system(32) or the comoving system(33),provided one representation and one fixed matrix W are used throughout,

Proof. State continuity gives z ( T ) = U K − 1 ⋯ U 0 z ( 0 ) . The first inequality is the definition of the induced norm; the second is submultiplicativity. □

The proof contains no T-web property and would remain true after arbitrary relabeling. This label-blindness is important. For a fixed classifier and the maximal-interval rule, one may define the reproducible bookkeeping sums

but they are residence-time-labeled log-bound components, not invariant or causal effects of the environments. Scientific meaning requires an external demonstration that the conditional gain distributions differ from shuffled-label controls.

For one fixed report interval I = [ s 0 , s 1 ] , let U I = U ( s 1 , s 0 ) . If the same interval is represented by m exact substeps, then

The exact composed gain G I is independent of the bookkeeping partition, apart from numerical integration error. The upper product B m is not: refinement can increase it systematically. Therefore, B m is an integration error-control bound only; it is never inserted into Equation (65). A convergence report must show separately that the composed propagator U I converges as the integration mesh is refined and how the auxiliary product B m behaves.

Raw comoving propagation includes the common Hubble block and can have an induced norm larger than one even when the peculiar Hessian vanishes. To separate this background exactly, write

Let U bg ( t , t 0 ) and U ( t , t 0 ) be the background and full comoving propagators, respectively, with the same a ( t ) and H ( t ) . Define the interaction-picture propagator

It obeys

This is an exact change of variables and retains all noncommutativity between the background and peculiar generators.

For each maximal report interval I k = [ s k , s k + 1 ] , two exact background-normalized operators are useful. With one common anchor t 0 , define the globally composable factor; with only the interval endpoints, define the local two-point factor:

Let B k = U bg ( s k , t 0 ) . Exact propagator composition gives the similarity identity

Thus, R k ca and R ^ k have the same eigenvalues, but their induced norms in one fixed metric generally differ because B k is not orthogonal. In particular, G k ca ( W ) depends on the common anchor t 0 (equivalently, on the reported start redshift and preceding background transport) and is not a purely local property of I k .

The dependence can be written exactly as a transported-metric identity. With

one has

The anchor-free quantity G ^ k ( W ) instead uses the same fixed metric at the two endpoints of the segment. It is suitable for local interval comparisons, but the factors R ^ k do not telescope as a simple product without the omitted background conjugations.

The common-anchor factors do telescope:

When K pec = 0 , both R k ca and R ^ k are the identity, even though ‖ U bg ‖ W may exceed one because coordinate and velocity components shear in the chosen full-state norm. Background normalization, therefore, removes the homogeneous dynamics, but it does not make a common-anchor interval norm intrinsic or a morphology label causal. Every report must state t 0 ; local morphology comparisons should also report G ^ k ( W ) or, equivalently, the transported metric W k used to interpret G k ca ( W ) .

For the conservative physical system, Equation (45) shows that the canonical propagator is symplectic. Liouville’s formula gives det U k = 1 , and the singular values of W 1 / 2 U k W − 1 / 2 have product one. Hence

for every fixed positive-definite full-state metric. A conservative physical interval cannot be a strict operator contraction in all phase-space directions, although a particular state or coordinate projection can decrease. For noncanonical comoving variables,

The weighted Wronskian and the canonical system (47) show that this determinant reflects the time-dependent momentum scaling.

For a continuously varying generator, define the logarithmic norm in the fixed metric W by

Then

The same formula applies to the interaction generator B ( t ) for background-normalized evolution. The logarithmic norm is a standard metric-dependent tool for differential error and growth bounds [34][39][40]. Its integral is independent of a numerical output partition, although it may be looser than the norm of the exact composed propagator.

Let P ∈ ℝ k × 6 be a fixed observation matrix and A P ( t ) = ‖ P z ( t ) ‖ 2 . If P has a nontrivial kernel, A P ( T ) / A P ( 0 ) need not be finite because an initially unobserved component can rotate into the observed subspace. The generally valid estimate is

For a fixed propagator U , a uniform finite ratio relative to A P ( 0 ) requires ker P ⊆ ker ( P U ) . Sufficient interval-wise alternatives are invariance of ker P , restriction of initial data to an invariant subspace S on which P | S is injective, or an explicit observability inequality. The same statements hold with U replaced by the background-normalized propagator R .

A reduced non-Hamiltonian closure can produce a negative operator log-gain only after it is specified for the same state and metric. If a common W > 0 satisfies

throughout an interval, then

If different intervals use different Lyapunov matrices, transition factors between the metrics must be included. The principal results of this paper do not assume phenomenological wall or filament attenuation.

Use dimensionless comoving variables with τ ∗ = 1 , W = I 6 and constant H = 0.12 . For angles in degrees, define

where R x , R y , R z are the standard right-handed Cartesian rotations. On report interval I j , let

The dynamical data are listed in Table 2.

Table 2. Dynamical Hessians for the three-dimensional benchmark. The angles specify Q j = Q ( α j , β j , γ j ) in degrees.

The classifier tensors are constructed independently:

Their ordered eigenvalues, rotations, and zero-threshold labels are given in Table 3. The last column is the confidence margin from Equation (29).

Table 3. Independent classifier tensors. The environment follows only from the number N + of eigenvalues above the zero threshold.

The initial state has no zero component:

For the interaction-picture calculation, U bg is generated by Equation (67); the common-anchor interval propagators R j ca are defined by Equation (70) with t 0 = 0 . Let y j = U bg ( s j , 0 ) − 1 z ( s j ) .

Table 4 reports the raw comoving gains and the background-normalized common-anchor gains. The selected-state columns are distinct from operator norms. For wall and filament intervals, the final column gives

where P m , j is the classifier wall-normal or filament-transverse projector. The large values show that the corresponding morphology-local quadratic energies are not exact in this route; the full propagator must be used.

Table 4. Three-dimensional benchmark results. “Raw state” is ‖ z j + 1 ‖ 2 / ‖ z j ‖ 2 ; “CA state” is the common-anchor interaction-state ratio ‖ y j + 1 ‖ 2 / ‖ y j ‖ 2 .

For the complete route,

Hence, both verified chains have the required ordering:

Table 5 quantifies the distinction between the common-anchor gain in the fixed metric and the local two-point gain for the same segment. The difference grows along the route because the common-anchor factor is conjugated by a longer background prefix.

Table 5. Common-anchor versus local two-point background-normalized gains for the same benchmark segments. The last column is 100 ( G j ca / G ^ j − 1 ) .

The raw and common-anchor results differ substantially because the former includes the Hubble background. For a pure-background interval of length 0.30 with K pec = 0 , the raw operator gain in this metric is 1.12489, whereas both background-normalized propagators are exactly the identity and have a gain of one.

The benchmark is intentionally not aligned: the classifier wall normal and the positive spectral line of K 1 have a principal angle of about 64.0˚; the maximum principal angle between the classifier filament plane and the positive spectral plane of K 2 is about 85.3˚. This checks that the method does not infer exact local energies from the morphology label when the required commutator condition fails.

The partition issue can be isolated with the frozen physical wall segment

Let U ( Δ t ) = e L ( K p ) Δ t and split the same interval into m equal exact substeps. Table 6 distinguishes the norm of the composed propagator from the product of substep norms.

Table 6. Partition study for the same wall interval. The exact composed gain is unchanged; only the auxiliary product bound depends on substepping.

The product bound approaches exp [ μ 2 ( L ) Δ t ] = exp ( 0.30 ) = 1.34985881 , whereas the exact interval gain remains 1.33384168. This is why only the composed maximal-interval gain is used in Equation (65).

The same wall propagator has different Euclidean gains after changing only the configuration-velocity scale: