Order-Constrained Spectral Non-Invariance

• Order-constrained spectral non-invariance defines a property where spectral structure and global ordering conflict, leading to non-standard permutation behaviors in matrices.
• It integrates spectral, nilpotent, and geometric constraints to classify spectrum- and commutativity-preserving maps, highlighting discontinuities in transformation processes.
• The framework has practical implications for optimization, dynamical systems, and physical predictions by challenging traditional unitary invariance and enabling novel causality analyses.

Order-constrained spectral non-invariance governs the interplay between spectral structure, operator topology, and permutation symmetry when linear operators or matrices are subject to both spectral constraints and explicit orderings. In this context, spectral invariance—usually associated with unitary or similarity transformations—is replaced by strictly weaker forms that may exhibit combinatorial freedom, discontinuity, or robust breaking under function application or geometric constraints. The phenomenon manifests across pure matrix theory, operator algebras, optimization, and dynamical systems, and has emerged as critical in recent generalizations of spectrum-commutativity preservers, optimization under linear eigenvalue constraints, and matrix orderings in complex domains. The essential signature of order-constrained spectral non-invariance is that allowable transformations preserving spectrum and commutativity can fail to commute with global orderings, with consequences for functional calculus, optimization, causality analysis, and physical predictions.

1. Foundations: Spectrum-Preserving Maps and Ordered Spectra

Order-constrained spectral non-invariance originates in the classification of spectrum- and @@@@1@@@@ $$\phi: \mathcal{N}_{n|\Lambda}^* \to \mathcal{M}_n(\mathbb{C})$$, where $\mathcal{N}_{n|\Lambda}^*$ is the set of normal matrices with spectrum in a prescribed subset $\Lambda \subseteq \mathbb{C}$, and $n \ge 3$ (Chirvasitu, 3 Jan 2026). Spectrum-preserving means $\sigma(\phi(A)) = \sigma(A)$, while commutativity-preserving ensures $[A,B]=0 \implies [\phi(A), \phi(B)]=0$. Imposing continuity (in the operator-norm or entrywise topology) restricts $\phi$ further, defining CS-preserving maps.

A principal result is that, when $\Lambda$ is the image of a continuous injection (a simple curve or interval), any continuous CS-preserving $\phi$ admits only three archetypes:

• Conjugations: $\phi(A) = S A S^{-1}$ for $S \in GL_n(\mathbb{C})$
• Transpose-conjugations: $\phi(A) = S A^T S^{-1}$
• Spectral reordering: $$\phi(A) = \sum_{j=1}^n \lambda_{\sigma(j)}(A) E_j(A)$$, where $\sigma$ is a permutation tracing eigenvalue order along $\Lambda$'s orientation and $E_j(A)$ are the corresponding eigenspaces.

This spectral reordering entwines the geometry of $\Lambda$ and the combinatorics of flag manifolds, controlled by the symmetric group $S_n$. Crucially, when $\Lambda$ is homeomorphic to an interval, such reorderings persist; when $\Lambda$ is a simple closed curve, global ordering breaks down and only conjugation and transpose-conjugation survive.

2. Spectral and Nilpotent Ordering (SNO): Generalized Order Structures

The SNO framework (Chang, 6 Sep 2025, Chang, 17 Jan 2025) extends classical matrix comparison beyond Hermitian matrices by introducing a total order on $\mathbb{C}$ (lexicographic: $z_1 \preceq_C z_2$ iff either $\Re z_1 < \Re z_2$, or $\Re z_1 = \Re z_2$ and $\Im z_1 \le \Im z_2$), and combining spectral and nilpotent (Jordan block size) structures into a unified partial order. Two matrices $A,B$ of the same size admit the following:

• Spectral ordering: $\Lambda(A) \prec_w \Lambda(B)$ in weak majorization (ordered sums in $\preceq_C$ lex order).
• Nilpotent ordering: If spectra coincide, dominance for each block size partition $M(A) \le_N M(B)$ under the classical dominance order on integer partitions. The composite SNO order $A \preceq_{SN} B$ holds if either spectral majorization or, on coincident spectra, nilpotent block-size dominance.

Order-constrained spectral non-invariance in SNO arises when function application $f$ is not monotone in $\preceq_C$; even direct sum, scalar, or non-monotone function transformations can disrupt the order, as demonstrated by the $f(z) = 1 - z$ counterexample (Chang, 17 Jan 2025).

3. Optimization under Order-Constrained Spectra and Symmetry Breaking

Spectrally constrained optimization (Garner et al., 2023) models smooth objectives with linear inequality constraints on ordered eigenvalues of real symmetric matrices: $$\min_{X \in S^n} F(X)\quad \text{s.t.} \quad A \lambda(X) \leq b$$ where $A$ encodes linear constraints on the spectrum. The ordering constraint $D_n \lambda(X) \leq 0$ (with $D_n$ encoding differences enforcing $\lambda_1 \geq \cdots \geq \lambda_n$) ensures order preservation.

Order constraints fracture traditional spectral (unitary) invariance. While projection and linear objectives can still decouple into eigenvalue-level problems via Fan's inequality, gradient steps and Frank-Wolfe directions become inextricably tied to both spectral and eigenvector structure. Algorithm trajectories inhabit the full matrix manifold $S^n$; pure spectrum iteration is no longer sufficient, and computational direction selection involves both eigenbasis and eigenvalues.

4. Geometric and Topological Constraints: Curve-Bound Spectra

The combinatorics of spectral permutation and invariance depend critically on $\Lambda$'s topology (Chirvasitu, 3 Jan 2026). When $\Lambda$ is an interval, the global ordering is well-defined and arbitrary spectral reordering is possible. For a simple closed curve ($S^1$), orderings confront a topological obstruction: traversing the loop forces a discontinuity, breaking continuity of any single cyclic eigenvalue permutation. This loss yields order-constrained spectral non-invariance—permutations valid locally cannot extend globally (case (c) is impossible).

The extension of CS-preservers to semisimple operators with $\Lambda$-bounded spectra introduces further subtlety. The involutive map $X \mapsto X^\rho$ (complex-conjugating eigenvalues, preserving eigenspaces) is spectrum-commutativity-preserving and continuous provided regularity conditions on $\Lambda$ (difference-quotients for $\lambda \mapsto \bar\lambda$).

5. Applications: Dynamical Systems, Causality, and Physical Systems

In linear dynamical systems, order-constrained spectral non-invariance is central to hierarchies of stability. As the ordering tightens—from spectral abscissa to spectral majorization to SNO—the set of comparable matrices shrinks. Small perturbations maintaining exponential decay rates may still violate majorization or nilpotent dominance, so finer transient control is lost (Chang, 6 Sep 2025).

Order-constrained spectral causality (Dominguez, 3 Jan 2026) formalizes directional influence in high-dimensional time series: causality is detected whenever spectral summaries of second-order dependence operators vary under admissible order-preserving temporal deformations of a source. Orthogonally invariant spectral functionals (e.g., largest eigenvalue, normalized trace, empirical measures) are used to construct sup–inf statistics for non-invariance. Randomization inference is exact under group invariance, and—under Gaussian assumptions—the method coincides with Granger causality. Extensions capture nonlinear and distributed effects missed by pairwise predictability.

6. Physical Interpretation: Gauge and Pseudo-Tensor Non-Invariance

Spectral non-invariance appears in general relativity at higher order, where the "spurious gauge-invariance" problem highlights that formally gauge-invariant constructs (scalar-induced tensor anisotropic stress at second order) yield parametrically different spectral predictions in different gauges, particularly outside the sound horizon (Giovannini, 2020). Distinct energy-momentum pseudo-tensors (Ford–Parker vs. Landau–Lifshitz) convolved with gauge-dependent sources produce inequivalent results for the spectral energy density of relic gravitons. Thus, physical predictions must specify both the gauge and the pseudo-tensor, as theoretical uncertainties escalate for scalar-induced corrections.

7. Implications and Commentary on Combinatorial Freedom

Order-constrained spectral non-invariance is combinatorially governed by $S_n$ acting on ordered flags of eigenspaces. Assigning an orientation to $\Lambda$ enables, in intervals, freedom to permute eigenvalues along the curve while fixing eigenspaces; in closed curves, topological barriers render this freedom non-continuous. Unlike classical conjugation or transpose-conjugation preservers, spectral reordering is combinatorially new and exists only where the geometry permits a global order.

A plausible implication is that in any framework imposing both continuity and commutativity preservation, spectral non-invariance is both a constraint and an analytical diagnostic: identification of functional or topological non-invariance marks boundaries on implementable spectral manipulations, optimization protocols, causality constructs, and even physical predictions. The regularity of $\Lambda$, function monotonicity in total orderings, and algebraic topology jointly determine whether spectral structure is robust or subject to discontinuous permutation and loss of invariance.