Trimmed Structure-Preserving Linearization

• Trimmed structure-preserving linearization is a design principle that converts complex nonlinear or higher-order systems into linear forms by eliminating redundant or non-essential components.
• It is applied across diverse settings such as Carleman lifting in polynomial dynamics, internal rewriting in the structural resource λ-calculus, and spectral deflation in quadratic eigenvalue problems for damped systems.
• Common mechanisms include symmetry reduction, exponential reduction, and projection techniques that preserve essential dynamical, derivational, and physical invariants without compromising solution accuracy.

Searching arXiv for the cited papers and closely related work to ground the article. Trimmed structure-preserving linearization denotes a family of linearization strategies in which a nonlinear, higher-order, or polynomially parameterized object is converted into a linear representation after redundant, duplicated, or spectrally inessential components have been removed, while the defining structure of the original object is retained. In the supplied literature, the term appears in three technically distinct settings: polynomial dynamical systems treated by duplicate-aware Carleman lifting, strongly normalizing $\lambda$-terms treated by internal rewriting in the structural resource $\lambda$-calculus, and quadratic eigenvalue problems for damped mass-spring systems treated by deflation of critical eigenvalues. Across these settings, “trimmed” refers to elimination of redundancy, latent structural rules, or critical eigenvalue blocks, whereas “structure-preserving” refers to preservation of exact truncated affine dynamics, derivational and $\beta$-structure, or Hermitian and dissipative Hamiltonian structure, respectively (Akiba et al., 7 May 2026).

1. Conceptual scope and defining features

The common conceptual core is not a single algorithm but a design principle: linearization is performed in a way that reduces superfluous representation while preserving the mathematically relevant invariants of the source problem. In polynomial dynamics, the redundant objects are duplicated monomial contributions generated during Carleman assembly; in the structural resource $\lambda$-calculus, they are nested structural morphisms corresponding to contraction, weakening, and permutations; in quadratic eigenvalue problems, they are critical eigenvalues at $0$, $\infty$, and on the imaginary axis that are projected out before solving the remaining problem.

The supplied literature treats “structure preservation” in a strong sense. For duplicate-aware shift-and-lift Carleman linearization, key-resolved triplet assembly is stated to yield the same truncated affine pair $(A_{ZZ}, b_Z)$ as term-by-term expansion in exact arithmetic, differing only by floating-point summation order (Akiba et al., 7 May 2026). For linearization via rewriting, exponential reduction preserves coherence and does not alter the $\beta$-structure of the term, while the overall structural reduction is strongly normalizing and confluent (Lago et al., 6 Mar 2025). For damped mass-spring quadratic eigenvalue problems, trimmed linearizations preserve nondeflated eigenvalues and their partial multiplicities while retaining Hermitian or $J$-Hermitian structure (Alam et al., 21 Jul 2025).

A plausible implication is that the phrase “trimmed structure-preserving linearization” is best understood as an umbrella characterization for methods that reduce linearization overhead without changing the target semantics of the retained model.

2. Shared mechanisms of trimming and preservation

Although the three settings differ sharply in formalism, the mechanisms used to achieve trimming are structurally analogous. Each starts from an exact or canonical lifting, expansion, or linearization, and then introduces a reduction step that removes representation-level redundancy without changing the retained linear object.

In duplicate-aware Carleman linearization, trimming operates at the level of basis design and sparse operator assembly. The method uses symmetry-reduced monomial bases, packed exponent-key indexing, and sparse triplet coalescing during construction rather than post hoc correction. The basis is already trimmed because monomials differing only by permutations of variables are not duplicated, and the assembly is trimmed because repeated targets $\gamma = \alpha - e_i + \beta$ are coalesced immediately rather than emitted as separate triplets (Akiba et al., 7 May 2026).

In the structural resource $\lambda$0-calculus, trimming is accomplished by exponential reduction. Structural resource terms use labeled abstractions $\lambda$1, where the binder morphism $\lambda$2 specifies allowed structural operations. Exponential reduction pushes these structural operations into the body by contravariant action, removes nested structural morphisms from binders, and reaches a normal form that the paper identifies as a special kind of linear resource term, the planar ones (Lago et al., 6 Mar 2025).

In the quadratic eigenvalue setting, trimming is spectral. A trimmed linearization is defined as an $\lambda$3 pencil $\lambda$4 with $\lambda$5 that preserves all nonzero eigenvalues and their partial multiplicities while deflating a known critical eigenvalue. The key mechanisms are congruence transformations, projection to lower-dimensional subspaces, Schur complement reduction, and $\lambda$6-orthogonal modal decompositions (Alam et al., 21 Jul 2025).

This suggests a unifying pattern: trimming is not an approximation step in itself, but a representation-theoretic or spectral consolidation step applied before or during linearization.

3. Duplicate-aware shift-and-lift Carleman linearization

For polynomial ODEs

$\lambda$7

Carleman linearization lifts the dynamics into monomials

$\lambda$8

producing an infinite linear system that is truncated at degree $\lambda$9 to obtain a finite-dimensional affine system

$\beta$0

The duplicate-aware shift-and-lift architecture separates a shift stage from a lift stage. The vector field is first re-expressed around a shift point $\beta$1 using the multinomial expansion

$\beta$2

yielding a constant term $\beta$3 and a shifted polynomial coefficient matrix $\beta$4. Lifting then applies the Carleman derivative rule

$\beta$5

and retains only contributions whose target exponent lies in the truncated lifted basis $\beta$6 (Akiba et al., 7 May 2026).

The central trimming issue is duplication of exponent targets. Distinct tuples $\beta$7 may map to the same target exponent

$\beta$8

To resolve this, the method encodes multi-indices by packed exponent keys

$\beta$9

with the constraint $\lambda$0. Lookup by $\lambda$1 allows immediate accumulation of duplicate contributions into the corresponding sparse entry. The paper states a proposition of exactness under fixed truncation: for fixed $\lambda$2 and fixed shift $\lambda$3, key-resolved triplet assembly yields the same truncated affine pair $\lambda$4 as term-by-term expansion in exact arithmetic, differing only by floating-point summation order (Akiba et al., 7 May 2026).

The method is also explicitly local. Because high-order truncation intensifies local convergence and closure sensitivity for higher-order nonlinearities, the shift-and-lift architecture is paired with a moving-center expansion so that shift and lift are updated jointly around evolving local centers. The resulting workflow combines symmetry-reduced monomial bases, packed exponent-key indexing, and sparse triplet coalescing to preserve truncated affine dynamics while reducing index-resolution overhead and write-path irregularity. Comparative evaluation against Jacobian linearization is organized by fixed-step error, admissible step size, and cost-at-target-accuracy criteria. On the bilinear driver and logistic interaction benchmarks, both approaches converge under refinement, and the proposed method shows regime-dependent accuracy gains rather than universal superiority (Akiba et al., 7 May 2026).

A plausible implication is that, in this setting, “trimmed structure-preserving linearization” refers to an algebraically exact reduction of assembly redundancy rather than a weakening of the truncated model.

4. Internal linearization in the structural resource $\lambda$5-calculus

In the structural resource $\lambda$6-calculus, trimmed structure-preserving linearization is realized as an internal rewriting process on typed terms and type derivations. Resource types are built from atomic types and list-based intersections,

$\lambda$7

and structural resource terms have the form

$\lambda$8

where the abstraction carries a morphism $\lambda$9 describing structural permissions on the bound variable. Because derivations are unique, a typed term essentially is its derivation, which makes rewriting of terms simultaneously a rewriting of derivations (Lago et al., 6 Mar 2025).

The distinction between linear and exponential reduction is fundamental. Linear reduction is $0$0-like and performs linear substitution. Exponential reduction is the linearization mechanism proper: it modifies binder labels by eliminating nested structural morphisms and propagating their action into the body. The ground exponential step rewrites an abstraction with non-identity structural annotation into one with reduced annotation and a transformed body obtained through the contravariant action of type morphisms. This internalization is what the paper identifies as the first example of a system where the linearization process takes place internally, while remaining purely finitary and rewrite-based (Lago et al., 6 Mar 2025).

The sense in which the process is structure-preserving is precise. Exponential steps do not perform substitution, so the $0$1-structure of terms is untouched. The shape of derivations, including context splitting and application trees, is preserved, and each step is labeled with the corresponding context and type morphisms. The paper further states that exponential reduction preserves coherence, so linearization by rewriting does not change the qualitative $0$2-content of a term (Lago et al., 6 Mar 2025).

The sense in which it is trimmed is equally precise. Exponential reduction is strongly normalizing, linear reduction is strongly normalizing, and the union is confluent. Normal forms for exponential reduction are a special kind of linear resource terms, the planar ones, and the final normal forms of the whole structural reduction are exactly the planar terms in $0$3-normal form. In this setting, the trimmed linearization is therefore the exponential normal form: no latent structural rules remain in binders, and all duplication and weakening information has been made explicit in the resource structure of the term (Lago et al., 6 Mar 2025).

This suggests that the $0$4-calculus instance supplies a proof-theoretic analogue of trimming: redundancy is not combinatorial duplication of coefficients but deferred structural bureaucracy in the typing and term syntax.

5. Deflation-based trimmed linearizations for quadratic eigenvalue problems

For damped mass-spring systems, the governing quadratic eigenvalue problem is

$0$5

with Hermitian coefficients $0$6, $0$7, and $0$8. The associated critical eigenvalues are $0$9, $\infty$0, and the nonzero imaginary eigenvalues $\infty$1. The paper develops structure-preserving deflation strategies that remove these eigenvalues through trimmed structure-preserving linearizations before computing the remaining eigenpairs (Alam et al., 21 Jul 2025).

The central Hermitian pencil is

$\infty$2

with $\infty$3. Lemma 2.1 establishes that

$\infty$4

and that partial multiplicities are preserved for every nonzero eigenvalue. This motivates the formal definition of a trimmed linearization: an $\infty$5 pencil $\infty$6 with $\infty$7 nonsingular is a trimmed linearization of $\infty$8 if $\infty$9, if $(A_{ZZ}, b_Z)$0, and if $(A_{ZZ}, b_Z)$1 for all $(A_{ZZ}, b_Z)$2 (Alam et al., 21 Jul 2025).

Zero eigenvalues are treated by projecting away the common null directions of $(A_{ZZ}, b_Z)$3 and $(A_{ZZ}, b_Z)$4. Under $(A_{ZZ}, b_Z)$5, Theorem 2.5 constructs a reduced Hermitian pencil

$(A_{ZZ}, b_Z)$6

together with a smaller quadratic polynomial

$(A_{ZZ}, b_Z)$7

such that

$(A_{ZZ}, b_Z)$8

If $(A_{ZZ}, b_Z)$9 is semidefinite then $\beta$0, so all zero eigenvalues are deflated (Alam et al., 21 Jul 2025).

Deflation at $\beta$1 is obtained by applying the same construction to the reversed polynomial $\beta$2. Purely imaginary eigenvalues are isolated by an $\beta$3-unitary basis $\beta$4 satisfying

$\beta$5

so that the reduced polynomial $\beta$6 has spectrum $\beta$7. Repeated application separates an entire set of purely imaginary eigenvalues into a block-diagonal factor (Alam et al., 21 Jul 2025).

The structure-preserving aspect is tied to physical and algebraic invariants. The reduced pencils remain Hermitian, and for hyperbolic problems the paper passes to a standard $\beta$8-Hermitian trimmed linearization

$\beta$9

where $J$0 is $J$1-Hermitian. Theorem 3.3 states that $J$2 is a definite pencil if and only if $J$3 is hyperbolic, and Theorem 3.4 states that in the hyperbolic case $J$4 has definite spectrum with exactly $J$5 $J$6-negative and $J$7 $J$8-positive eigenvalues (Alam et al., 21 Jul 2025).

6. Comparative perspective, limitations, and significance

The three settings instantiate different meanings of linearization and different retained structures. In Carleman lifting, the retained object is a truncated affine dynamical system whose coefficients and sparsity pattern match exact truncation. In the structural resource $J$9-calculus, the retained object is a canonical linear resource term that preserves derivational content and $\gamma = \alpha - e_i + \beta$0-structure. In damped mass-spring QEPs, the retained object is a smaller Hermitian or $\gamma = \alpha - e_i + \beta$1-Hermitian pencil with the noncritical spectrum intact.

Their limitations are also domain-specific. The duplicate-aware shift-and-lift approach does not claim universal superiority over Jacobian linearization; the paper emphasizes regime dependence, closure sensitivity, and conditioning effects at high truncation order (Akiba et al., 7 May 2026). The structural resource $\gamma = \alpha - e_i + \beta$2-calculus applies to strongly normalizing $\gamma = \alpha - e_i + \beta$3-terms represented by type derivations; the setting is proof-theoretic rather than numerical, and its finitary character depends on strong normalization of exponential reduction (Lago et al., 6 Mar 2025). The QEP framework is tailored to Hermitian mass-spring pencils and uses assumptions such as $\gamma = \alpha - e_i + \beta$4, $\gamma = \alpha - e_i + \beta$5, and $\gamma = \alpha - e_i + \beta$6 in key theorems, with special treatment for hyperbolic problems and parametric damping (Alam et al., 21 Jul 2025).

Even with these differences, the supplied literature supports a coherent encyclopedic interpretation. Trimmed structure-preserving linearization is a methodology for converting a structured problem into a linear one after removing representational or spectral excess, while preserving the invariants that define the intended semantics of the reduced model. In one case the excess is duplicated monomial assembly, in another it is latent structural morphism content, and in another it is critical eigenvalue blocks. This suggests that the term identifies a cross-domain research pattern rather than a single canonical construction.