The Angular Observer

Keep the Angle: a geometry-preserving basis in spectral embeddings.

Keep the angle, drop the magnitude — when the magnitude is nuisance for the metric being preserved. Given the graph normalized-Laplacian eigen-embedding, the angular coordinates of the low-eigenvalue subspace carry the graph's geodesic structure; the radial coordinate is asymptotically degree/density. Row-normalizing to the unit sphere — the Ng–Jordan–Weiss move — is, on manifold-like graphs, the coarse-graining a bounded observer performs to perceive a smooth space.

The commute-time Laplacian embedding degenerates on large graphs (von Luxburg): its distances collapse to local degree. This repository shows the geometry does not vanish — it is retained by the angular coordinate (under strongly non-uniform sampling, after the Coifman–Lafon α=1 density normalization). Keeping only the angle (the Ng–Jordan–Weiss row-normalization) preserves graph geodesics across every substrate family we tested with genuine low-dimensional Riemannian geometry, while the radial coordinate is provably degenerate. The underlying scale-from-direction decomposition recurs in vector and KV-cache compression — with a scope boundary the companion practice work makes sharp: for attention keys, per-vector angular quantization fails (attention depends on per-channel scale in QKᵀ), so the general principle is keep the part that carries the task geometry, not always keep the angle (turboquant-pro). The observer interpretation of Wolfram's emergent-geometry program is developed as motivation and bounded honestly.

Papers

The write-up is split into a theory paper and an empirical companion:

• Paper I — Keep the Angle (theory): the canonical monolithic source is paper/paper.tex (generic article, all proofs inline as appendices); the SIAM Journal on Mathematics of Data Science submission is the derived split build in paper/simods/ — main.tex (≤20-page main text) + supplement.tex. It contains the normalization identity, the deterministic transfer theorem, the conditional graph-to-manifold result, the unconditional torus/sphere instances, the uniform lower angular bound on flat tori, the exact upper-Lipschitz divergence, the spectral-filter phase diagram (critical dimension d = 4), the Green-kernel rank-limit theorem for d ≤ 3 (rank = 1 on two-point homogeneous spaces), the filter dichotomy, and the torus core/verify/ablation experiments.
• Paper I.b — Keep the Angle, in Practice (empirical): the substrate-agnostic manifold diagnostic and its failure modes, in paper1b/ — the cross-substrate benchmark, bake-off, hypergraph-rewriting emergent geometry, the dimension trend, temporal stability, the two-factor screen, continuum dissociation, honest negatives, and the physical interpretation.

Reviewer-response experiments and reproducibility artifacts are in experiments/reviewer-response/; every headline number is backed by a committed *_result.json and multi-draw where it matters (ablation, κ, core table, and the (A2) margin on the non-homogeneous swiss-roll).

The core decomposition

flowchart LR
    G["graph / hypergraph"] --> L["normalized Laplacian"]
    L --> E["low-mode<br/>eigen-embedding"]
    E --> R["radius<br/>(magnitude)"]
    E --> A["angle<br/>(direction)"]
    R -.->|"von Luxburg:<br/>to 1/sqrt(degree)"| X["geometry-free density"]
    A ==>|"geodesics preserved<br/>rho ~ 0.93"| Y["the perceived geometry"]
    style A fill:#ddffff,stroke:#0088aa
    style Y fill:#ddffdd,stroke:#00aa00
    style X fill:#ffdddd,stroke:#aa0000

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Independent literatures perform the same decomposition — separate scale from direction — and then keep whichever part carries their task geometry:

flowchart TD
    N["Ng-Jordan-Weiss spectral<br/>clustering: row-normalize"] --> K
    V["vector / KV compression:<br/>separate norm from direction<br/>(PolarQuant, turboquant-pro)"] --> K
    W["Wolfram observer<br/>coarse-graining"] --> K
    K{{"separate scale from direction;<br/>keep what carries the task geometry"}}
    style K fill:#ddffff,stroke:#0088aa,stroke-width:2px

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For graph geodesics that is the angle alone (this repo). For embedding retrieval it is direction plus stored norm; for attention keys it is per-channel scale, and per-vector angular quantization is a counterexample — the boundary is condition (A2) of the paper's transfer theorem, stated for the downstream metric.

The result in one table

Angle-only low-mode Laplacian coarse-graining preserves graph geodesics (Spearman ρ vs true geodesic distance; random-mode control ≈ 0 throughout):

substrate	emergent dim	angle-ρ	random (control)
2-torus (reference)	2	0.93	~0.01
king-lattice	2	0.82	0.01
swiss-roll embedding trajectory	2	0.93	0.01
emergent Wolfram-model manifold (R_3D)	2.07 ± 0.01	0.82 ± 0.01	0.01
causal set (Lorentzian)	2.9–10.7 ✗	0.57	0.01
small-world (geometry destroyed)	3.6–4.3 ✗	0.45	0.00

The control fails everywhere required; the metric degrades exactly where geometry is Lorentzian or destroyed. It is a geometry detector, not a magic wand.

The five findings

flowchart TD
    R0["estimators vs<br/>ground truth"] --> R1["polar split:<br/>angle 0.93 / random 0 / magnitude 0.03"]
    R1 --> EM["emergent geometry<br/>(arity-2/3 rewriting)"]
    EM --> D2["clean 2D:<br/>borrowed + evolved rules"]
    R1 --> U["universality:<br/>lattice / swiss-roll / torus"]
    D2 --> LAW["exploratory trend:<br/>rho ~ 1.09 - 0.157 x dim"]
    U --> LAW
    LAW --> T["Keep-the-Angle Theorem<br/>(von Luxburg degeneracy)"]
    R1 --> NEG["honest negatives:<br/>hyperbolic rejected / curvature null"]
    style R1 fill:#ddffff,stroke:#0088aa
    style LAW fill:#ddffdd,stroke:#00aa00
    style NEG fill:#ffefdd,stroke:#cc7700

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1. Universal, not Wolfram-specific (crosssubstrate.py). The effect holds on lattices, manifold-embedding trajectories, and tori — substrates with no hypergraph content — and fails correctly on Lorentzian / small-world graphs.
2. Holds to genuine 2D, via two independent routes: borrowing a literature rule (gorard_rules.py) and evolving one from scratch (ga2d.py). Clean 2D rules are rare, not absent — the evolutionary search reaches one in ~3 generations.
3. The Keep-the-Angle Theorem (theorem.md, theorem_verify.py). The radius is provably geometry-free via the von Luxburg–Radl–Hein resistance degeneracy (effective resistance → 1/dᵢ + 1/dⱼ for d ≥ 2, so radius → 1/√dᵢ = degree noise; ρ(radius, degree) = 0.92–0.99). The angle deletes exactly that. Dimension-gated: clean at 3D, marginal at 2D, correctly absent at 1D.
4. An exploratory dimension trend (library.py). Across 20 emergent manifolds from five rule families (d ≈ 1–3.6), angular fidelity falls with emergent dimension — angle-ρ ≈ 1.09 − 0.157·d — reported as a five-family exploratory trend (rule-clustered), not an inferential law.
5. The honest boundary (curvature.py). Emergent curvature is not dynamical beyond graph structure. A spectacular raw −0.98 curvature/activity correlation deflates to a hub tautology (partial correlation −0.14). This is observer theory, not the Einstein-tensor claim.

What is proven vs conjectured (synced to Paper I)

Update. The rank form is no longer only conjectural: the Green-kernel rank-limit theorem proves that for intrinsic dimension d ≤ 3 the commute angular ranking converges uniformly in the mode count to a fixed Green-kernel ranking — exactly rank 1 on the circle and on compact two-point homogeneous spaces (S², S³, ℝP², ℝP³). Angular Preservation thereby reduces to positivity of a geometric Green-rank coefficient, with the exact obstruction at the critical dimension d = 4 (the spectral-filter phase diagram). What stays conjectural is two-sided metric uniformity (its upper half is provably false for the commute filter) and Green-rank positivity on general manifolds.

• Proven / theorem-backed: the radial coordinate's degeneracy for d ≥ 3 (von Luxburg et al. 2010/2014; full diagonal transfer in paper Appendix A — d = 2 is the recurrent borderline, treated empirically); a deterministic angular-transfer theorem plus a conditional local bi-Lipschitz theorem for the commute-weighted eigenmap, unconditional on the flat torus (paper Thm 6/8, Cor 10); uniform-in-truncation angular bi-Lipschitzness for heat-filtered eigenmaps (Thm 12); and the plain-eigenmap rank collapse (Prop 14).
• Refuted (self-correction): the old strong form — "the angular coordinate stays bi-Lipschitz to geodesic distance uniformly in mode-count" at fixed scale — is false for the commute weighting: high modes raise the local angular speed like √Λ (exact on S¹), and the exact upper-Lipschitz constant diverges like R (d ≥ 3) / R/√log R (d = 2). Two-sided metric uniformity is impossible; only the scale-dependent form could survive.
• Rank form — now proved below d = 4 (previously the surviving conjecture): the angular ranking stays faithful uniformly in the mode count. The Green-kernel rank-limit theorem proves that for d ≤ 3 it converges to a fixed Green-kernel ranking — exactly rank 1 on the circle and on two-point homogeneous spaces (S², S³, ℝP², ℝP³). What remains conjectural is Green-rank positivity on general manifolds; the exact obstruction sits at the critical dimension d = 4 (the spectral-filter phase diagram).
• Negative results kept, not hidden: the hyperbolic-reframe hypothesis was tested with controls and rejected (hyperbolic.py); the dynamical-curvature test is a clean null (curvature.py).

Layout

• Validation ladder: rung0_validate.py → rung1_proxy.py → rung1b_rewriter.py → rung1c_polar.py (the PolarQuant angle/magnitude split) → search_harness.py.
• Emergent-geometry substrate: arity3.py (triangle hyperedge rewriter), gorard_rules.py, gorard_confirm.py, ga2d.py.
• Theory: theorem.md, theorem_verify.py.
• Universality & curvature: crosssubstrate.py, curvature.py, hyperbolic.py.
• Composite: library.py.
• Sweep (reproducible): sweep.py, sweep_pod.py, sweep3.py, nrp_job.yaml, aggregate.py. Note: the cluster driver scripts (which carry infrastructure credentials) are intentionally omitted — supply your own kubeconfig / SSH.

Reproduce

python -m pip install numpy scipy         # only dependencies
python rung0_validate.py                  # estimators vs ground truth
python rung1c_polar.py                    # the angle-vs-magnitude result
python theorem_verify.py                  # angle-flat vs full-decays-with-N
python library.py                         # the exploratory dimension trend

Citation

Andrew H. Bond (SJSU), 2026. ORCID 0009-0003-2599-6158.