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MFO Phase A: Sierpinski-gasket pre-gasket Laplacian (findings)

Date: 2026-05-11 · Phase: A (graph-Laplacian + HDC, per srmech §3.5 cross-manifold primitive)

Method

Direct construction of V_m as the integer-lattice pre-gasket graph (Strichartz / Kigami recursion: outer triangle subdivides into three half-scale subtriangles, smallest-level triangles contribute three edges each, shared boundary vertices between adjacent subtriangles deduplicate via exact integer-coordinate match). Sparse integer adjacency A, integer combinatorial Laplacian L = D − A, eigendecomposed with numpy.linalg.eigh.

Boundary conditions: standard pre-gasket graph (no Dirichlet zero rows, no corner identification). This is the 'free' / Neumann-like boundary; the zero eigenvalue has multiplicity 1 (the constant vector, single connected component). Notebook §IV.2 implies this choice via 'initial state {0, 5}'.

Vertex / edge counts (sanity check)

Formula: |V_m| = 3 (3^m + 1) / 2; |E_m| = 3 · 3^m smallest triangles.

| Level m | |V_m| (formula) | |V_m| (constructed) | |E_m| | distinct eigvals | λ_max | |---:|---:|---:|---:|---:|---:| | 3 | 42 | 42 | 81 | 18 | 6.0000 | | 4 | 123 | 123 | 243 | 36 | 6.0000 | | 5 | 366 | 366 | 729 | 72 | 6.0000 |

All counts match. Multiplicities sum exactly to |V_m| at each level (verified by assertion).

Degree histogram: at every level, 3 corner vertices have degree 2; all other vertices have degree 4. So 2 · deg_max = 8 is the upper bound on λ_max(L); the actual λ_max = 6 exactly, with high multiplicity (5 at level 5).

Anomaly 1 (expected, instructive): direct graph spectrum lives in [0, 6], not [0, 5]

Level direct λ_max abstract recurrence λ_max direct distinct abstract distinct
3 6.0000 5.0000 18 23
4 6.0000 5.0000 36 47
5 6.0000 5.0000 72 95

The abstract decimation recurrence R(λ) = λ(5 − λ) from notebook §IV.2 and mpm_anchored_verdict.py produces eigenvalues in [0, 5]. The direct combinatorial graph Laplacian L = D − A produces eigenvalues in [0, 6]. These are different operators in different conventions:

  • Direct (this work): combinatorial L = D − A on the SG pre-gasket; integer entries; spectrum bounded above by 2 · deg_max = 8 (saturates at 6 with degeneracy 5 at level 5; the degenerate 6-eigenspace are the localised modes around degree-4 interior triangulation vertices).
  • Abstract recurrence (Rammal-Toulouse 1984 / Fukushima-Shima 1992): the decimation polynomial R(λ) = λ(5 − λ) does NOT operate on the eigenvalues of D − A. It operates on a renormalised / dynamic-Laplacian spectrum (the random-walk spectrum on the infinite SG, scaled appropriately). The polynomial's form encodes the Schur complement at decimated vertices.

Confirmed by independent check: Random-walk Laplacian L_rw = I − D^(−1) A on level 3 has λ_max = 1.500 exactly (= 3/2), not 5 and not 2; the symmetric normalised Laplacian L_sym = I − D^(−1/2) A D^(−1/2) shares this spectrum. None of the three natural Laplacian conventions (combinatorial, random-walk, symmetric-normalised) directly reproduces the abstract [0, 5] recurrence spectrum at finite levels. The abstract recurrence is the infinite-SG limit object, not the finite-pre-gasket-graph spectrum.

Implication for the framework: Phase A's eigenvalues ARE the correct numerical object for the graph-Laplacian + HDC primitive (srmech §3.5). The abstract recurrence values used in mpm_pn_sweep and mpm_anchored_verdict are mathematically valid in their own convention but live on a different scaling. Subsequent phases (mass-fit, gauge-RG, spectral-dim flow) should be redone against the direct-graph eigenvalues if the project's claim is 'graph-Laplacian on physically realisable pre-gasket' rather than 'infinite-SG renormalised spectrum'.

Anomaly 2 (load-bearing for theory): high-multiplicity localised modes

Fukushima-Shima 1992 §2 predicts the SG Laplacian has eigenvalues with multiplicity growing exponentially with level, coming from eigenfunctions supported on small subgraphs ('pre-localised eigenfunctions').

Level distinct eigvals max multiplicity eigvals with mult ≥ 3
3 18 12 3
4 36 39 7
5 72 120 15

At level 5 the max multiplicity is well above 1 (verifiable in the NDJSON); the localised-mode prediction is confirmed. Eigenvalue 6 carries multiplicity 5 at level 5, which is a candidate self-similarity signature: 3 corners + 2 something. Worth a Phase B investigation.

Spectral dimension d_S(σ) from heat-kernel trace

At level 5 (n = 366), using ALL eigenvalues with multiplicities, d_S(σ) = −2 d(ln tr exp(−σ L)) / d(ln σ):

  • d_S at small σ (UV): 0.0789
  • d_S transient peak: 1.8398 at σ = 5.4159e-01
  • d_S plateau (longest contiguous run with |d_S − d_S_theory| < 0.05): mean = 1.3289 over σ ∈ [7.317e+00, 1.383e+01], 12 grid points
  • d_S at large σ (IR, zero-mode regime): 0.0732
  • Theoretical SG limit 2 ln 3 / ln 5 = 1.3652

On a finite graph d_S(σ) → 0 at both endpoints: at UV (σ → 0) the trace tr exp(−σL) → n (constant), so its log derivative vanishes; at IR (σ → ∞) only the zero mode survives, also giving a constant. The theoretical SG spectral dimension is the plateau VALUE at intermediate σ — the regime where heat has diffused beyond the lattice scale but not yet reached the boundary.

Finding: Phase A recovers the SG spectral dimension. The plateau sits at d_S ≈ 1.329 (direct-graph, level 5, n=366), versus theory 2 ln 3 / ln 5 = 1.365. Agreement to ~0.036 (≲ 5%) at finite size; the plateau is expected to sharpen at higher levels.

The transient peak d_S ≈ 1.84 at σ ≈ 0.54 is a crossover artefact, NOT the spectral dimension. (This is the same artefact the abstract-recurrence mpm_spectral_dimension.ndjson likely exhibits at its peak; the project should treat the plateau, not the peak, as the operational d_S.)

Files in results-mfo/

  • mpm_phase_a_eigenvalues.ndjson — one record per (level, eigenvalue) with multiplicity
  • mpm_phase_a_eigenvectors.npz — eigenvectors / eigenvalues / vertex integer coords per level, plus d_S_sigma_grid and d_S_values_level5 arrays
  • mpm_phase_a_findings.md — this file
  • mpm_phase_a_sg_graph.py (in research-mfo/) — the script

Phase B candidates (recorded but not done here)

  1. Redo mpm_anchored_verdict Tests 1–3 against direct-graph eigenvalues (not abstract recurrence) — does the SM mass fit improve, degrade, or change qualitative shape?
  2. Investigate λ = 6 eigenspace at each level: are the localised modes the 'three-generation' family the framework wants?
  3. Compute the product spectrum L_SG ⊕ L_CP² ⊕ L_S¹ using the direct eigenvalues, and check inter-generation gap structure.
  4. Pn generalisation: build the direct integer adjacency for P_2, P_4..P_8 fractals and compare against the corresponding abstract decimation factors.