CompLex: A Cross-Disciplinary Structural Framework

• CompLex is a cross-disciplinary framework that treats complex variables, distributions, and structures as primary, emphasizing interaction, multiscale organization, and non-trivial invariants.
• It employs methodologies from non-Hermitian physics, complex-valued machine learning, and geometric topology to uncover critical phenomena and system dynamics.
• By unifying abstract theoretical models with practical optimization and synthesis challenges, CompLex offers actionable insights in areas like beamforming, RTL generation, and logical retrieval.

“CompLex” (Editor’s term) denotes a cross-disciplinary configuration of research in which complex variables, complex-valued distributions, complex logical structure, or the organization of strongly interacting systems are treated as primary mathematical structure rather than as a secondary reformulation. This synthesis is suggested by work spanning complexity science, statistical physics, non-Hermitian criticality, complex-valued machine learning, distributionally robust optimization, chemical reaction network theory, and several branches of geometry and topology (Gershenson, 2010, Gershenson, 2011).

1. Conceptual scope

In the foundational complexity-science sense, there is no single context-free definition of complexity. Carlos Gershenson explicitly treats “complexity” as a family of meanings: the amount of information required to describe a phenomenon, the length of the shortest program that computes that description, the time required to compute it, the minimal statistical model needed to describe it, and the interwoven organization of systems whose components are difficult to separate because interactions are constitutive rather than negligible (Gershenson, 2010). In the related essayistic account centered on the etymology of plexus, complexity is tied to interwovenness, scale dependence, emergence, computational irreducibility, and the limits of reductionist prediction (Gershenson, 2011).

This pluralism is important for any use of “CompLex.” It implies that the term cannot be reduced to “complex numbers” alone, nor to “complicatedness.” A plausible implication is that the unifying feature is structural: systems or models are “CompLex” when their correct description depends on interaction, composition, multiscale organization, or intrinsically complex-valued formalisms. The same body of work also cautions against a common misconception: complexity is not a single scalar observable that transfers unchanged across domains. It is relative to scale, encoding, and modeling frame (Gershenson, 2010).

A second recurrent intuition is that complexity often appears between order and disorder, sometimes heuristically associated with criticality. That theme reappears in later work on complex liquids, complex conformal field theory, and logically compositional retrieval, where broad temporal scales, exclusion semantics, or near-critical organization become empirically central (Gershenson, 2011).

2. Complexification in physics and critical phenomena

A mathematically explicit instance of complexification appears in the spherical $p$-spin model after the spin variables and random couplings are made complex. The stationary-point equations become a system of $N$ random polynomial equations of degree $p-1$, and the disorder-averaged number of solutions satisfies $\log \overline{\mathcal N}\sim N\log(p-1)$, saturating the algebraic Bézout bound in the large-$N$ limit (Kent-Dobias et al., 2020). The Hessian at a saddle has the form $C^\dagger C$, with $C$ a complex symmetric Gaussian matrix with a diagonal shift; its singular-value spectrum develops a gap that generalizes the threshold level familiar from the real $p$-spin problem. The real problem is recovered in the appropriate limit, but only the square root of the total number of complex solutions are real (Kent-Dobias et al., 2020).

A different physical realization occurs in non-Hermitian many-body systems governed by complex conformal field theory. There the relevant reduced density matrix is not the ordinary Hermitian one, but the biorthogonal object $$\hat{\rho}^{RL}_{\rm A}=\operatorname{tr}_{\rm B}(\ket{\psi}\langle\!\langle\psi|/\langle\!\langle\psi\ket{\psi})$$, built from right and left ground states. The associated von Neumann functional $$S=-\operatorname{tr}(\hat{\rho}^{RL}_{\rm A}\log \hat{\rho}^{RL}_{\rm A})$$ is generally complex, and in the non-Hermitian five-state Potts model its periodic-boundary scaling matches the complex-CFT prediction with central charge $N$0 (Shimizu et al., 4 Feb 2025). The same work stresses that this quantity is not simply the ordinary entanglement entropy continued to complex values: it is tied to a non-Hermitian reduced density matrix and does not preserve all standard entropy properties, including strong subadditivity (Shimizu et al., 4 Feb 2025).

Taken together, these results suggest a characteristic CompLex move in physics: instead of treating complexification as a formal trick, one promotes it to the natural arena in which state counting, critical scaling, or saddle geometry become structurally transparent.

3. Complex-valued learning and representation

In machine learning, complex-valued representation is treated in several distinct ways. A comparative study of CNN, parameter-matched CNNx2, and complex-valued CNNs on brain-tumour classification and segmentation in BraTS 2020 shows that CV-CNN models outperform both the baseline CNN and the widened CNNx2 across seven architectures and two tasks, even though the inputs are magnitude MR images with no phase information (Chatterjee et al., 2023). This directly addresses a frequent objection in the literature: the observed gains are not explained solely by doubled trainable parameter count, because CNNx2 was designed precisely to control for that factor (Chatterjee et al., 2023).

Gaudet and Maida argue for a more structural interpretation of those gains. In their account, the practical advantages attributed to complex and hypercomplex convolutions arise chiefly from weight sharing across dimensions and forced linear coupling of multicomponent inputs, not necessarily from strict adherence to the algebra of $N$1 or $N$2. Their vector map convolution generalizes these coupling patterns to arbitrary dimension through a circulant weight-sharing matrix modulated by a learnable coefficient matrix, and on CIFAR classification, colorization, and satellite segmentation it often matches or exceeds real and quaternion baselines while using fewer parameters (Gaudet et al., 2020).

On graphs, the same theme appears as complex diffusion. Complex-Weighted Convolutional Networks define a Hermitian complex edge-weight matrix $N$3, normalize via $N$4 and $N$5, and propagate through a complex random-walk operator. The central theorem is existential: for any node partition, there exists a complex-weighted graph such that the steady state of the diffusion assigns a distinct phase to each partition block, so class information survives the infinite-layer limit rather than being destroyed by oversmoothing (Amado et al., 17 Nov 2025). This does not imply that training always finds such a weighting, but it gives a precise expressiveness claim for complex-valued message passing that is stronger than merely noting that “complex numbers can be used in a GNN” (Amado et al., 17 Nov 2025).

4. Optimization and synthesis of complex artifacts

Optimization with genuinely complex variables introduces additional second-order structure through Hermitian covariance and pseudo-covariance. The framework for the Complex Chance-Constrained Problem (3CP) considers objectives of the form $N$6 with $N$7 and probabilistic constraints $N$8. Under Complex Elliptically Symmetric distributions and several distributionally robust ambiguity sets—known moments, bounded covariance, uncertain mean, support bounds, and empirical moments—the individual chance constraints admit deterministic convex reformulations, principally as SOCPs; joint constraints are treated by a Gumbel–Hougaard copula and bounded above and below by tractable conic approximations (Madani et al., 22 May 2026). The signal-processing application is minimum-variance distortionless-response beamforming, where the empirical out-of-sample behavior closely matches the prescribed probability levels (Madani et al., 22 May 2026).

A different but related sense of “complex” appears in automatic RTL generation. ComplexVCoder addresses the failure of prompt-to-Verilog systems on hierarchical, multi-module, real-world designs by introducing a two-stage generation pipeline: natural language is first mapped to a structured General Intermediate Representation, then GIR is converted into Verilog using rule-based alignment and domain-specific retrieval-augmented generation (Zuo et al., 29 Apr 2025). On the 55-design ComplexVDB benchmark, the framework exceeds prior Verilog-specific baselines such as CodeV and RTLCoder in functional correctness, while a Qwen2.5-32B instantiation reaches performance comparable to GPT-3.5 on the same benchmark (Zuo et al., 29 Apr 2025). Here the “complex” object is not complex-valued arithmetic but architectural depth, multi-level instantiation, and interface consistency across large designs.

These two lines of work show that CompLex optimization can mean either probabilistic reasoning over complex numbers or structured synthesis of artifacts whose combinatorial and hierarchical dependencies defeat direct one-shot generation.

5. Complex systems, logic, and kinetics

In liquid-state physics, a “complex liquid” is operationalized through a nearest-neighbor bond network whose memory decays as a stretched exponential,

$N$9

and whose waiting-time distribution satisfies $p-1$0 over a substantial range (Patashinski et al., 2012). In the two-dimensional Lennard-Jones mosaic regime, the stretching exponent reaches $p-1$1 near $p-1$2, while the waiting-time cutoff extends to roughly $p-1$3, consistent with a dynamically heterogeneous mosaic of long-living crystallites and less-ordered regions (Patashinski et al., 2012). The authors treat these signatures as evidence for a near-critical dynamic percolation regime rather than merely slow relaxation.

In information retrieval, compositionality itself becomes the source of complexity. ComLQ is a benchmark of 2,909 queries and 11,251 candidate passages designed around first-order logical operators such as conjunction, disjunction, and negation, with 14 query types ranging from $p-1$4 to $p-1$5 and a dedicated metric, Log-Scaled Negation Consistency, for exclusion handling (Xu et al., 15 Nov 2025). Existing sparse, dense, and LLM-augmented retrievers all show markedly weaker performance on negation-bearing and mixed logical forms than on simple projection-like queries, indicating that standard relevance-based retrieval does not robustly model logical exclusion (Xu et al., 15 Nov 2025). A common misconception is that multi-hop retrieval already covers this setting; ComLQ explicitly distinguishes logical composition over unstructured text from ordinary single- or multi-hop semantic lookup.

Chemical reaction network theory provides yet another specialized notion. A kinetic system is absolutely complex balanced when every positive equilibrium is complex balanced. The classical Horn–Jackson theorem establishes this for complex balanced mass-action systems, but the extension fails for all complex balanced PL-RDK systems: the paper constructs a weakly reversible, complex balanced PL-RDK example with $p-1$6 (Jose et al., 2021). Within the broader class of CLP systems—systems whose complex-balanced equilibria admit log parametrization—the exact criterion is bi-LP: absolute complex balancing holds if and only if the positive equilibrium set and the complex balanced equilibrium set are both of LP type and have the same flux space (Jose et al., 2021). This is a sharp instance of CompLex as equilibrium-set geometry in logarithmic coordinates.

6. Geometric, topological, and cohomological constructions

Several mathematical works develop “complex” objects in a more literal geometric sense. The variety of complete complexes generalizes complete collineations and complete quadrics to the Buchsbaum–Eisenbud variety of complexes. Its points are equivalence classes of reduced spectral sequences; the resulting projective variety is smooth, has normal-crossings boundary, and desingularizes the union of maximal strata of the original variety of complexes (Kapranov et al., 2017). The conceptual principle is that degeneration of a complex is controlled not by a single limiting differential but by an iterated sequence of differentials on successive cohomologies, hence by a spectral sequence (Kapranov et al., 2017).

The coupled alpha complex solves a more computational problem. Ordinary alpha complexes are not monotone under adding points: $p-1$7 in general when $p-1$8. The coupled construction replaces the Voronoi tessellation of $p-1$9 by the union of the separate Voronoi-ball families of $\log \overline{\mathcal N}\sim N\log(p-1)$0 and $\log \overline{\mathcal N}\sim N\log(p-1)$1, producing a simplicial complex $\log \overline{\mathcal N}\sim N\log(p-1)$2 such that $\log \overline{\mathcal N}\sim N\log(p-1)$3 and $\log \overline{\mathcal N}\sim N\log(p-1)$4 (Reani et al., 2021). The result is a monotone surrogate for alpha complexes that remains asymptotically sparse.

Complex intersection bodies extend Lutwak’s real theory to $\log \overline{\mathcal N}\sim N\log(p-1)$5. For a complex star body $\log \overline{\mathcal N}\sim N\log(p-1)$6, the complex intersection body $\log \overline{\mathcal N}\sim N\log(p-1)$7 is defined through complex hyperplane sections, and the paper proves the analogue of Busemann’s theorem: the complex intersection body of a symmetric complex convex body is convex (Koldobsky et al., 2012). Harmonic-analytic characterization is equally explicit: a complex star body is a complex intersection body if and only if $\log \overline{\mathcal N}\sim N\log(p-1)$8 is a positive definite distribution, equivalently if it is a $\log \overline{\mathcal N}\sim N\log(p-1)$9-intersection body compatible with the complex structure (Koldobsky et al., 2012).

Piovani’s treatment of the Bigolin, or Schweitzer, complex develops Hodge theory for an elliptic complex that interpolates between de Rham, Aeppli, and Bott–Chern cohomology. On compact Hermitian manifolds it yields orthogonal decompositions and harmonic representatives; on compact Kähler manifolds the corresponding harmonic spaces are exactly the $N$0-harmonic forms lying in the relevant region of the double complex (Piovani, 2024). On small deformations of the Iwasawa manifold, the dimensions $N$1 distinguish deformation classes as finely as Bott–Chern and Aeppli numbers, which suggests that Bigolin cohomology captures zigzag information in the double-complex decomposition that Hodge numbers alone do not (Piovani, 2024).

Across these examples, CompLex appears less as a single doctrine than as a recurrent methodological stance: one keeps the full complex, compositional, or interaction-generated structure of the object under study, and then seeks the right invariants—Bézout-saturating solution counts, complex entropies, phase-coded diffusions, log-parametrized equilibrium sets, spectral-sequence compactifications, or logical-consistency metrics—to make that structure analytically tractable.