Dynamic Laplacian via fiber-driven edge weights — FALSIFIED (2026-05-12)¶

Origin: user question following PR #334 merge: "Now that we can draw our hypervector as voxels, does that mean we could also use fiber content to 'reshape' voxels for dynamic graph-laplacian things?"

Pre-flighted as: "might yield nothing." Falsification spike.

Headline verdict: FALSIFIED at natural voxelization parameters. The dynamic Laplacian via fiber phase coherence is vacuous for the protein-bipartition use case because the voxelization step itself pre-encodes the bipartition into the voxel-grid topology — positive and negative Fiedler regions form geometrically-separated voxel blobs that don't share neighbors, so the dynamic Laplacian has nothing to discover. The "reshape" the user proposed has already happened in the encoding step.

Reproduce: python -X utf8 docs/srmech/notes/dynamic_laplacian_fiber_reshape_script.py. Runtime ~10s. Deterministic seed 20260512.

Setup¶

Voxel grid as base manifold; HDC phase per voxel as U(1) fiber. Dynamic edge weights:

\[w_{ij}(\psi) = \cos^2\left(\frac{\theta_i - \theta_j}{2}\right) = \frac{1 + \cos(\theta_i - \theta_j)}{2}\]

Same phase → strong edge (w=1); opposite phase → weak edge (w=0).

Iterative fixed-point loop: 1. Build dynamic weights from current phases 2. Compute Fiedler eigenvector of L(ψ) 3. Update phases from new Fiedler via tanh-smooth map → [0, π] 4. Sign-align Fiedler against previous Fiedler (not previous phases — avoids spurious sign-flip cycles) 5. Check convergence

Test protein: 1UBQ (ubiquitin, 76 Cα). Voxel grid 32³, σ=3.5Å, active threshold |field|>0.05.

Diagnostic finding — the active voxel region is disconnected¶

Connectivity check on the 6-connected lattice over active voxels:

Statistic	Value
Total voxels (32³)	32,768
Active voxels (	field
Edges in 6-connected active subgraph	12,442
BFS reachable from voxel 0	2,532
Unreachable	2,089
Phase positive (>π/2) count	2,532
Phase negative (<π/2) count	2,089

The numbers match exactly: 2532 + 2089 = 4621 active voxels split into two BFS components matching the positive/negative phase bipartition counts. The Fiedler partition is baked into the voxel topology by the Gaussian-smear + threshold step itself.

Why this happens: with σ=3.5Å smear, the positive Fiedler region (β-strand cluster) and negative Fiedler region (α-helix cluster) form separated 3D blobs. The zero contour of the Fiedler eigenvector in real space passes through "low-density" voxels which fall below the |field|>0.05 threshold and are excluded. Result: two non-touching blobs in the active subgraph.

Iteration results — trivial convergence in 2 steps¶

With sign-aligned Fiedler against previous iteration's Fiedler:

| Variant | Weight floor ε | Steps to convergence | Final ||Δphase|| | λ_Fiedler | |---|---:|---:|---:|---:| | A (naive) | 0.0 | 2 | 0.0006 | 0.0 | | B (small floor) | 0.1 | 2 | 0.0001 | 0.0 | | C (large floor) | 0.3 | 2 | 0.0001 | 0.0 |

λ_Fiedler = 0 across all variants because the active subgraph is disconnected → multiple zero eigenvalues in the kernel of L(ψ). The "Fiedler" eigenvector eigsh returns is just the indicator function of one component vs the other — exactly the initial bipartition. Trivially self-consistent. Nothing learned.

All three variants behave identically because edge weights don't matter when the graph topology is already disconnected: zero weights and floor=0.3 weights give the same disconnected-component structure.

What was actually tested vs what we thought we were testing¶

We thought	Reality
Dynamic Laplacian discovers self-consistent partition through iteration	Initial voxelization already separates the bipartition into geometric components
Fiber phase coherence reshapes the graph	Graph was already "reshaped" by Gaussian-smear + threshold
Need weight floor to keep graph connected	Active subgraph itself is disconnected; weights are irrelevant
Period-2 cycle from first run was dynamical	Period-2 cycle was sign-flip artifact in Fiedler eigsh on degenerate spectrum (fixed with sign-align to previous Fiedler)

The alignment-fix moved the outcome from "limit cycle" to "trivial fixed point in 2 steps." Both diagnoses describe the same underlying issue — the active voxel region is geometrically partitioned before the iteration starts.

What this falsifies¶

The naive form of "use fiber content to reshape voxels for dynamic graph-Laplacian" — i.e., compute fiber-derived edge weights and recompute Fiedler iteratively — is vacuous for any voxelization where the Fiedler positive/negative regions don't share voxels. This includes:

Protein bipartitions at natural σ and threshold (PR #333 setup)
Any sparsely-supported scalar field where positive and negative regions are geometrically separated
Any HDC encoding where the encoded data IS the partition (sign-only encoding from PR #333)

The user's intuition was sound at the architectural level (fiber feedback to base is a real category) but the specific instantiation with naive phase-coherence weights on the protein bipartition is a degenerate case.

What the falsification rules OUT for the catalog¶

Sign-only encoded bipartitions on naturally-thresholded voxel grids will never support non-trivial dynamic-Laplacian iteration. The geometry has done the work the iteration was supposed to do.
Don't add a "dynamic Fiedler-driven Laplacian" primitive to srmech §3.5.4 catalog. It's a vacuous operator at the bipartition level.

What might still work — flagged but not tested here¶

To actually test the user's deeper intuition, one would need:

Connected substrate: lower threshold or full 32³ grid. The transition zone between positive and negative regions would then be in the graph, giving the iteration something to do. Predicted outcome: either a softened bipartition fixed point (with smooth zero contour) or a non-trivial sub-partition emerging within each region.
Smoother phase function: don't use tanh saturation. Linear phase = π * (Fiedler − Fiedler_min) / (Fiedler_max − Fiedler_min) keeps phase continuous and ambiguous in low-Fiedler regions where the iteration could find structure.
Different weight functional: w_{ij} = α + β·cos((θ_i − θ_j)/2) with α > 0 keeps the graph connected regardless of phase pattern. The α/β ratio is a "coupling strength" knob — α=1, β=0 gives static Laplacian; α=0, β=1 gives naive case; intermediate gives a real test.
Non-bipartition test: try the dynamic Laplacian on a non-bipartition field (e.g., the L₁ edge Fiedler from PR #333 Hodge spike, or a random scalar field) where the geometry doesn't pre-partition.
Gauge-theoretic connection (most ambitious): instead of edge weights, use the phase as a gauge connection — (ψ_i, ψ_j) → ⟨ψ_i, U_{ij} ψ_j⟩ for some link variable U_{ij}. This is the discrete Wilson-line setup from lattice gauge theory; gives a complex Hermitian Laplacian whose spectrum captures holonomy not just coherence. Would be a genuinely different primitive.

None of these are this PR's scope. Falsifying the naive form is the load-bearing content here.

Connection to architectural framework¶

§4.2 substrate-vs-config split (srmech notebook): state-dependent operators are substrate primitives (don't close under pure spectral form). The dynamic Laplacian was the project's first attempt at this category for a graph operator (graphics-domain Perona-Malik is the existing example). This spike showed: even substrate primitives need to be non-degenerate to be interesting. Adding "state-dependence" to an operator doesn't automatically buy you anything if the state already determines the geometry the operator computes against.

PR #333 encoding lesson: sign-only encoding matches sign-valued bipartition data → preserves partition cleanly. This spike showed the corollary: when encoding does its job too well (perfect bipartition preservation in geometry), there's nothing left for downstream operators to discover. Encoding is information-preserving; downstream processing requires information to be non-redundant at the substrate level for any new finding to emerge.

MFO §VII.4.1.1 (PR #332 Hopf bundle): connection (= fiber feedback to base curvature) is the continuum analog of what the user proposed. The Hopf bundle's curvature is non-zero (Chern class 1) — that's what gives the Δ_{S³} = Δ_{S²} + fiber-harmonics decomposition real content. For our voxel substrate at natural parameters, the "curvature" is degenerate (graph disconnects) → no content. This spike instantiates the boundary case where the framework gives nothing.

Honest summary¶

User asked an architecturally sound question; the specific instantiation we tested falsifies for the natural protein-bipartition setup at PR #333 voxelization parameters. Three weight-floor variants confirm: the issue is geometric (active-voxel-region disconnection), not numerical. The architectural slot remains live but needs a non-degenerate substrate.

Falsified specifically: naive cos²(Δθ/2) edge weights on Fiedler-bipartition-derived phases over standard PR #333 voxelization.

Not falsified: gauge-theoretic dynamic Laplacian, connected substrate variants, non-bipartition initial conditions. These are flagged as separable future spikes.

Files¶

dynamic_laplacian_fiber_reshape_script.py — three-variant spike (~10s)
dynamic-laplacian-fiber-reshape-per-step-2026-05-12.ndjson — per-step + final outcomes
dynamic-laplacian-trajectory-2026-05-12.png — convergence plot + phase-field snapshots

Citations¶

Perona-Malik 1990: anisotropic diffusion with state-dependent weights — the canonical substrate-primitive precedent
Imrich-Klavžar: Cartesian product graph spectra (factor-sum identity used in PR #331)
Bohigas-Giannoni-Schmit 1984 + Wigner-Dyson universality (PR #334 connection)