Fluxamba: Topology in Segmentation & Quantum Anyons

• Fluxamba is a dual framework combining topology-aware segmentation for geological lineaments and non-Abelian flux braiding in quantum systems.
• It employs specialized components like ASG and PMF that enhance feature alignment and computational efficiency in challenging remote sensing conditions.
• The quantum branch realizes non-Abelian braiding of magnetic fluxons, establishing topological robustness and connecting to braid group representations.

Fluxamba encompasses two distinct but conceptually related frameworks: one as a topology-aware anisotropic state space model for geological lineament segmentation, and the other as a non-Abelian anyonic system derived from the braiding of magnetic fluxons in a two-dimensional Pauli Hamiltonian. Both share an underlying architecture of directional flux steering and topological robustness, manifest in either neural network design or quantum holonomy, respectively.

1. Topology-Aware Lineament Segmentation in Remote Sensing

Fluxamba is designed for high-fidelity segmentation of geological lineaments—fractures, ridges, rilles—characterized by highly anisotropic morphologies, tortuosity, and low contrast in multi-source remote sensing imagery (Bai et al., 24 Jan 2026). Traditional segmentation approaches, notably 2D CNNs and vision transformers, are constrained either by limited receptive field size (CNNs) or computational complexity scaling at $\mathcal{O}(H^2W^2)$ (transformers). State Space Models (SSMs) like Vision Mamba offer linear complexity $\mathcal{O}(HW)$ via fixed, axis-aligned scan paths, but introduce a topological mismatch: curvilinear targets spanning rows and columns result in spatial fragmentation, context loss, and feature erosion.

Fluxamba resolves this mismatch with a topology-aware feature rectification workflow that dynamically steers information flow along a feature’s intrinsic geometry, retaining SSM computational efficiency while overcoming serialization-induced artifacts.

2. Architectural Components: Structural Flux Block (SFB)

At the core of Fluxamba is the Structural Flux Block (SFB), comprising four specialized modules:

• Anisotropic Structural Gate (ASG): ASG computes a geometry-aware gating map $\mathbf{G}_{\mathrm{ASG}}$ from base features $\mathbf{X}_{\mathrm{base}}$. A coordinate-aware branch (axis-separated pooling) captures spatial indices; a strip-pooling branch (multi-scale, narrow kernels) aggregates elongated context. The concatenated result, projected and sigmoid-activated, selectively amplifies features likely aligned with lineaments’ primary orientation:

$$\mathbf{G}_{\mathrm{ASG}} = \sigma\bigl(\mathcal{C}_{1\times 1}[\mathrm{cat}(\mathbf{F}_{\mathrm{coord}}, \mathbf{F}_{\mathrm{strip}})]\bigr)$$

$$\mathbf{X}_{\mathrm{ASG}} = \mathbf{X}_{\mathrm{base}} + \mathbf{X}_{\mathrm{base}} \odot \mathbf{G}_{\mathrm{ASG}}$$

• Prior-Modulated Flow (PMF): PMF extends the four-directional Selective 2D Scan (FS2D) of SSMs ($0^\circ$, $45^\circ$, $90^\circ$, $135^\circ$). It produces state sequences $\mathcal{O}(HW)$0 and modulates their aggregation via softmax-normalized, locally and globally computed weight maps (conditioned on $\mathcal{O}(HW)$1):

$\mathcal{O}(HW)$2

$\mathcal{O}(HW)$3

PMF synthesizes a geometry-adaptive aggregation across scan directions, yielding robust, continuous flux propagation along the true path of the lineament.

• Hierarchical Spatial Regulator (HSR): HSR aligns $\mathcal{O}(HW)$4 to the backbone $\mathcal{O}(HW)$5 either via Lightweight Modulation Refinement (LMR, shallow stages, multi-dilated depthwise convolutions and gating) or via Global Transformer Reorganizer (GTR, deep stages, multi-head self-attention and feed-forward normalization). This guarantees multi-scale semantic integrity essential for both boundary accuracy and contextual coherence.
• High-Fidelity Focus Unit (HFFU): HFFU suppresses background variance by polarization across channels and spatial positions:

$\mathcal{O}(HW)$6

ensuring explicit enhancement of signal-to-noise ratio critical for detecting faint curvilinear features.

3. Network Structure and Optimization

Fluxamba’s encoder consists of four stages, each stacking depth-specific SFBs. Features at each scale are downsampled, fused in the decoder via boundary-modulated fusion, yielding the output segmentation map. The loss integrates weighted binary cross-entropy (WBCE), Dice, and boundary terms:

• $\mathcal{O}(HW)$7 for imbalanced positives/negatives
• $\mathcal{O}(HW)$8 for overlap-based quality
• $\mathcal{O}(HW)$9 for edge sensitivity

Fluxamba-Tiny configuration: depths $\mathbf{G}_{\mathrm{ASG}}$0, 3.39M parameters, 6.25 GFLOPs, 12.9MB model size, executes at 24.12 FPS (RTX 3090).

4. Empirical Evaluation and Ablation Analysis

Benchmarked against eight SOTA methods across LROC-Lineament, LineaMapper, and GeoCrack datasets, Fluxamba-Tiny yields:

• F1-score 89.22% and mIoU 89.87% on LROC-Lineament
• Superior ODS, OIS, F1, mIoU across all datasets
• Two orders of magnitude less compute than Swin-UMamba (616 GFLOPs)
• Maintained segmentation under severe Gaussian perturbations: mIoU drop only 16.7% vs. >40% for texture-dependent models

Module-wise ablation:

• PMF lifts mIoU +8.77% (restoring topological continuity)
• ASG (+1.12%) and HSR (+0.75%) sharpen delineation
• HFFU (+1.05%) purifies noise for optimal fidelity

A plausible implication is that content-adaptive flux gating via PMF and ASG yields SSM architectures genuinely competitive with transformer-based methods while remaining highly deployable on edge hardware.

5. Fluxamba in Pauli Hamiltonian: Non-Abelian Anyonic Braiding

In a distinct but etymologically related context, “Fluxamba” denotes the non-Abelian holonomy system realized by adiabatically braiding N subcritical magnetic fluxons ($\mathbf{G}_{\mathrm{ASG}}$1) in two-dimensional Pauli Hamiltonians (Kenneth et al., 2014). The system manifests D = N–1 gapless zero modes, with wavefunctions

$\mathbf{G}_{\mathrm{ASG}}$2

Adiabatic motion of fluxon positions $\mathbf{G}_{\mathrm{ASG}}$3 induces a Wilczek–Zee connection

$\mathbf{G}_{\mathrm{ASG}}$4

which governs parallel transport in the zero-mode bundle. The associated holonomy

$\mathbf{G}_{\mathrm{ASG}}$5

is topological (path-independent) whenever all fluxons are subcritical and $\mathbf{G}_{\mathrm{ASG}}$6, and non-Abelian for $\mathbf{G}_{\mathrm{ASG}}$7. For identical fluxes, the holonomy matches the Burau representation of the braid group $\mathbf{G}_{\mathrm{ASG}}$8.

6. Topological Robustness and Braid Group Representations

Braiding fluxons in the subcritical regime induces purely topological (path-independent) non-Abelian holonomies:

• For $\mathbf{G}_{\mathrm{ASG}}$9, the $\mathbf{X}_{\mathrm{base}}$0 free zero modes form a non-Abelian representation of pure braid group $\mathbf{X}_{\mathrm{base}}$1
• Monodromy matrix for braiding fluxon $\mathbf{X}_{\mathrm{base}}$2 around $\mathbf{X}_{\mathrm{base}}$3:

$\mathbf{X}_{\mathrm{base}}$4

where $\mathbf{X}_{\mathrm{base}}$5

• For equal fluxes, elementary braid exchanges $\mathbf{X}_{\mathrm{base}}$6 are realized through the Burau representation:

$\mathbf{X}_{\mathrm{base}}$7

This framework leverages classical fluxons to engineer anyonic systems with non-trivial quantum statistics.

7. Synthesis and Applicability

Fluxamba, in both machine learning and quantum mechanical contexts, is characterized by dynamic, topology-aware information flux. In segmentation, this yields state-of-the-art detection of geological lineaments under extreme anisotropy and noise, with computational tractability fit for edge deployment (Bai et al., 24 Jan 2026). In the quantum regime, it realizes unitary, topological, non-Abelian transformations of zero modes in two-dimensional Pauli Hamiltonians, connecting the physics of flux braiding to representations of braid groups (Kenneth et al., 2014). A plausible implication is that cross-fertilization between these domains—both driven by topological steering of information or phase—may offer new paradigms in robust feature modeling, either for geometric pattern recognition or quantum information processing.