Graph Wiring: a (surprising) ArrowSpace plus String Theory hypothesis

A recurring theme in my work on arrowspace is that vector spaces are not just geometry: they also hide a graph, and that graph exposes structure you can search, partition, and audit. In the last few weeks a new angle emerged from a preliminary assessment: the Dirichlet energy machinery already implemented in arrowspace may be more than a spectral trick: it may approximate a class of minimal surface computation in String Theory as recently demonstrated in the Meng at al. paper.

This post is a progress update. It’s exploratory, its proof is not fully written down yet but it’s concrete enough to outline what would need to be shown and why it could matter for both sciences (anything using minimising surfaces) and data engineering.

UPDATE: These ideas are now be put into testable code with surfface-rs, a library that will try to demonstrate the principles defined in this post.

The trigger: Meng et al. (2026) and surface optimization

Meng et al. (2026) study physical networks (connectomes, vasculature, corals, plants) and argue that classical “wiring minimization” (length/Steiner-tree style) misses key observed motifs. They propose that the right cost must account for surface constraints, and they report an exact mapping of surface minimization to higher-dimensional Feynman diagrams in string theory, with predictions like the emergence of trifurcations and stable branching morphologies.

What caught my attention is thge computation strategy: an intractable 3D geometric optimization becomes tractable by moving to a 2D surface (a “worldsheet” in String Theory) formulation. This is similar to what arrowspace does, trying to approximate high-dimensional spaces to spectral surfaces that are 2D-ish in terms of computational costs.

What ArrowSpace already computes: Dirichlet energy on kNN Laplacians

arrowspace builds Laplacians from neighbourhood graphs (kNN / \(\epsilon\)-graphs, typically after clustering/coarsening), then uses Rayleigh-quotient-style energies and $\tau$-bounded transforms to create compact spectral indices such as \(\lambda\tau\).

In the “energy maps” track (beside the base “eigen maps” track with similar characteristics), ArrowSpace also uses diffusion updates that are explicitly a discretised heat-flow step on the Laplacian:

\[x \leftarrow x - \eta L x\]

which is exactly how you numerically smooth signals on a graph manifold.

So the library already treats a dataset as: vectors → neighbourhood graph → Laplacian operator → energies/indices for search and analysis.

The hypothesis: “worldsheet” track as a Dirichlet-energy minimizer

The conjecture I’m exploring is (this has been worked using agentic AI):

1. The kNN Laplacian you build on semantic metadata (feature space / transposed space) can be interpreted as a discrete approximation of a Laplace–Beltrami operator on an underlying manifold (the classic Laplacian-eigenmaps story, under standard sampling assumptions).
2. ArrowSpace’s core energy primitive (Rayleigh/Dirichlet energy) is, in the continuum limit, the Dirichlet functional:

\[E(f) \approx f^\top L f\]

1. In differential geometry / string theory language, Dirichlet energy minimization corresponds to harmonic maps; and under specific gauge/metric conditions (e.g., conformal gauge in the Polyakov formulation), this is equivalent to minimal area (minimal surface / worldsheet) solutions.

If that chain holds under the right assumptions (sampling regime, weighting, normalization/density correction, etc.), then ArrowSpace wouldn’t just be “inspired by” minimal surfaces — it would be an efficient, graph-algorithmic route to approximate minimal-surface-like solutions in a discrete setting.

Why this could matter: scaling physical-network simulation

If confirmed, this becomes interesting for two reasons:

• Physics / biological networks: Meng et al.’s framework is powerful but the computational toolkit for minimal surfaces in complex branching systems is still non-trivial. ArrowSpace is engineered to run fast by coarsening first and then working on sparse Laplacians; the implementation aims to flatten practical costs toward \(O(n \log n \, d)\) behaviour by avoiding all-pairs work and building sparse neighbourhood structure over reduced representations (kNN is computed on top-3 neighbours to mirror the structure of a trifurcation of strings).
• Data engineering: regardless of the physics connection, “graph wiring” (vectors → Laplacian worldsheet → operations) is already a productive abstraction for dataset operations: spectral search, partitioning, motif spotting, drift detection, and cross-snapshot comparison. This further “missing link” to physical networks could only bring more capabilities to the table by providing an additional track or by inspiring a new library based on arrowspace.

What would “confirmation” look like?

A serious validation is not a vibe check — it’s a set of proofs and experiments and it will take at least few months:

• Show the graph Laplacian (built the ArrowSpace way) converges to the right continuous operator under stated assumptions (data sampled from a smooth manifold, k-scaling, kernel bandwidth, density correction).
• Show ArrowSpace’s energy objective corresponds to a Dirichlet functional in that limit.
• Show when (and only when) the harmonic-map ↔ minimal-surface equivalence applies, and what gauge/constraints are required.
• Empirically test on synthetic manifolds (where ground truth is known), then on real physical-network motifs.

That’s the core of the preprint I’m drafting.

Why I’m calling it “Graph Wiring”

I’m dubbing the broader technique in arrowspace Graph Wiring: taking any vector space and building a “worldsheet-like” Laplacian substrate that you can compute on search, partition, motif spotting, and more without being trapped in purely geometric similarity.

This also fits the arc of the project: I started in 2022 looking at fungal growth patterns under a microscope, built arrowspace in 2025 to improve graph computation for spectral search and I definitely did not expect that graph theory and String Theory may be so related. That’s a nice ad for Open Source software I guess.

Try arrowspace (and follow along)

If you want to play with this today:

• Rust: cargo add arrowspace (repo link in the docs)
• Python: pip install arrowspace (bindings repo linked from the Rust README)
• Start with the energy-informed search proof-of-concept and the “vectors + features + graphs” overview posts to see the operational framing.[^5][^3]
• If you’re interested in AI safety / RAG drift and “open search”, the memory-layer post gives the motivation for topology-aware retrieval.[^4]

Sponsorship / support

This work has been possible thanks to sponsors. If you want to follow progress live (code, preprints, benchmarks) and help me keep building, please consider sponsoring on GitHub.

Engineering in progress.