MFO bottom-up cross-spectrum survey: findings¶
Date: 2026-05-09 Phase: survey (section principal scope)
Discipline: bottom-up cataloging of graph-Laplacian eigendecompositions across srmech-spectral-collection domains. No external target fitting. Closed-form deterministic computation; integer adjacency where possible; all results reproducible from the script.
Domains successfully cataloged¶
| domain | |V| | |E| | distinct eigvals | lambda max | n_isolated | max mult | largest_gap / median | |:---|---:|---:|---:|---:|---:|---:|---:| | sg_pregasket_L5_level3 | 42 | 81 | 18 | 6.0000 | 2 | 12 | 3.20 | | sg_pregasket_L5_level4 | 123 | 243 | 36 | 6.0000 | 7 | 39 | 12.03 | | sg_pregasket_L5_level5 | 366 | 729 | 72 | 6.0000 | 14 | 120 | 35.69 | | chess_8x8_king_move | 64 | 210 | 48 | 11.3915 | 6 | 2 | 5.30 | | othello_8x8_line_of_sight | 64 | 728 | 45 | 30.0572 | 2 | 3 | 45.77 | | pn_pregasket_p2_level4 | 17 | 16 | 17 | 3.9659 | 0 | 1 | 1.35 | | pn_pregasket_p4_level4 | 514 | 1536 | 40 | 8.0000 | 9 | 254 | 35.35 | | ephemerides_resonance_static | 52 | 44 | 19 | 15.4527 | 3 | 18 | 18.62 | | antikythera_gear_dag_undirected | 25 | 24 | 25 | 4.5438 | 0 | 1 | 2.94 |
Cross-spectrum pattern observations¶
recurrence_n_isolated_eigenvalues¶
Counts of isolated eigenvalues (preceding gap > 3x median) per spectrum. If the same count recurs across domains, that count is a candidate structural invariant of the graph-Laplacian primitive.
Per domain:
sg_pregasket_L5_level3: 2sg_pregasket_L5_level4: 7sg_pregasket_L5_level5: 14chess_8x8_king_move: 6othello_8x8_line_of_sight: 2pn_pregasket_p2_level4: 0pn_pregasket_p4_level4: 9ephemerides_resonance_static: 3antikythera_gear_dag_undirected: 0
Value counter: {2: 2, 7: 1, 14: 1, 6: 1, 0: 2, 9: 1, 3: 1} Multi-hit values: {2: 2, 0: 2}
largest_gap_ratio_to_median¶
Largest single gap / median distinct gap. Sets the dynamic range of spectral separation in each domain. Large -> isolated extremum band; small -> uniform/dense spectrum.
Per domain:
sg_pregasket_L5_level3: 3.203sg_pregasket_L5_level4: 12.03sg_pregasket_L5_level5: 35.69chess_8x8_king_move: 5.298othello_8x8_line_of_sight: 45.766pn_pregasket_p2_level4: 1.353pn_pregasket_p4_level4: 35.347ephemerides_resonance_static: 18.617antikythera_gear_dag_undirected: 2.94
Min: 1.353, mean: 17.805, max: 45.766
top3_gap_fingerprint_normalised_to_largest¶
First 3 gap sizes each normalised to the largest. (1.0, x, y) where x and y in (0, 1]. Used to cluster spectra by 'shape' independent of absolute scale.
Per domain:
sg_pregasket_L5_level3: [1.0, 0.9636, 0.8735]sg_pregasket_L5_level4: [1.0, 0.9451, 0.8631]sg_pregasket_L5_level5: [1.0, 0.945, 0.863]chess_8x8_king_move: [1.0, 0.9041, 0.8227]othello_8x8_line_of_sight: [1.0, 0.1253, 0.0598]pn_pregasket_p2_level4: [1.0, 0.983, 0.983]pn_pregasket_p4_level4: [1.0, 0.9769, 0.6962]ephemerides_resonance_static: [1.0, 0.7784, 0.3178]antikythera_gear_dag_undirected: [1.0, 0.8349, 0.8106]
density_slope_log_N_log_lambda_bulk¶
Fitted slope d(log N(lambda))/d(log lambda) in bulk (middle 60% of spectrum). Interpreted formally as d_S/2 where d_S is the "spectral dimension". Recurring slope values -> shared effective dimensionality.
Per domain:
sg_pregasket_L5_level3: 1.0019sg_pregasket_L5_level4: 1.0551sg_pregasket_L5_level5: 1.0889chess_8x8_king_move: 1.4376othello_8x8_line_of_sight: 3.2499pn_pregasket_p2_level4: 0.4945pn_pregasket_p4_level4: 1.5549ephemerides_resonance_static: 0.502antikythera_gear_dag_undirected: 0.5437
max_multiplicity_per_domain¶
Maximum eigenvalue multiplicity, the size of the largest degenerate eigenspace. A 'localised-mode' hallmark if it grows with graph size.
Per domain:
sg_pregasket_L5_level3: {'max_mult': 12, 'spectrum_size': 42, 'ratio': 0.2857, 'eigenvalue': 6.0}sg_pregasket_L5_level4: {'max_mult': 39, 'spectrum_size': 123, 'ratio': 0.3171, 'eigenvalue': 6.0}sg_pregasket_L5_level5: {'max_mult': 120, 'spectrum_size': 366, 'ratio': 0.3279, 'eigenvalue': 6.0}chess_8x8_king_move: {'max_mult': 2, 'spectrum_size': 64, 'ratio': 0.0312, 'eigenvalue': 0.4164}othello_8x8_line_of_sight: {'max_mult': 3, 'spectrum_size': 64, 'ratio': 0.0469, 'eigenvalue': 25.0}pn_pregasket_p2_level4: {'max_mult': 1, 'spectrum_size': 17, 'ratio': 0.0588, 'eigenvalue': -0.0}pn_pregasket_p4_level4: {'max_mult': 254, 'spectrum_size': 514, 'ratio': 0.4942, 'eigenvalue': 8.0}ephemerides_resonance_static: {'max_mult': 18, 'spectrum_size': 52, 'ratio': 0.3462, 'eigenvalue': 1.0}antikythera_gear_dag_undirected: {'max_mult': 1, 'spectrum_size': 25, 'ratio': 0.04, 'eigenvalue': 0.0}
d_group_irrep_min_multiplicity_at_isolated_eigenvalues¶
Per isolated eigenvalue, the minimum integer-rounded irrep multiplicity across the d-group decomposition. min_count = number of complete 'one-of-each-irrep' generation blocks the eigenspace supports.
Per domain:
sg_pregasket_L5_level5: [{'eigenvalue': 6.0, 'multiplicity': 120, 'group': 'D_3', 'counts': {'A_trivial': 22, 'B_sign': 18, 'E_2D': 40}, 'min_count': 18, 'integer_dim_check': 120}]chess_8x8_king_move: [{'eigenvalue': 1.5319792446156713, 'multiplicity': 1, 'group': 'D_4', 'counts': {'A1': 1, 'A2': 0, 'B1': 0, 'B2': 0, 'E': 0}, 'min_count': 0, 'integer_dim_check': 1}, {'eigenvalue': 2.7109385875043457, 'multiplicity': 1, 'group': 'D_4', 'counts': {'A1': 1, 'A2': 0, 'B1': 0, 'B2': 0, 'E': 0}, 'min_count': 0, 'integer_dim_check': 1}, {'eigenvalue': 4.168907232476906, 'multiplicity': 2, 'group': 'D_4', 'counts': {'A1': 0, 'A2': 0, 'B1': 0, 'B2': 0, 'E': 1}, 'min_count': 0, 'integer_dim_check': 2}, {'eigenvalue': 7.520601647519077, 'multiplicity': 1, 'group': 'D_4', 'counts': {'A1': 1, 'A2': 0, 'B1': 0, 'B2': 0, 'E': 0}, 'min_count': 0, 'integer_dim_check': 1}, {'eigenvalue': 8.31559302915267, 'multiplicity': 2, 'group': 'D_4', 'counts': {'A1': 0, 'A2': 0, 'B1': 0, 'B2': 0, 'E': 1}, 'min_count': 0, 'integer_dim_check': 2}, {'eigenvalue': 10.958585790768582, 'multiplicity': 1, 'group': 'D_4', 'counts': {'A1': 0, 'A2': 1, 'B1': 0, 'B2': 0, 'E': 0}, 'min_count': 0, 'integer_dim_check': 1}]othello_8x8_line_of_sight: [{'eigenvalue': 14.218288843090171, 'multiplicity': 2, 'group': 'D_4', 'counts': {'A1': 0, 'A2': 0, 'B1': 0, 'B2': 0, 'E': 1}, 'min_count': 0, 'integer_dim_check': 2}, {'eigenvalue': 16.0, 'multiplicity': 1, 'group': 'D_4', 'counts': {'A1': 0, 'A2': 0, 'B1': 0, 'B2': 1, 'E': 0}, 'min_count': 0, 'integer_dim_check': 1}]
gap_fingerprint_clusters¶
Group domains whose top-3 gap fingerprints (normalised to largest gap) match within 5%. Cluster of size >= 2 indicates structurally-similar spectra across domains.
Clusters (tolerance 0.05): 7 distinct cluster(s) - size 3: ['sg_pregasket_L5_level3', 'sg_pregasket_L5_level4', 'sg_pregasket_L5_level5'] - size 1: ['chess_8x8_king_move'] - size 1: ['othello_8x8_line_of_sight'] - size 1: ['pn_pregasket_p2_level4'] - size 1: ['pn_pregasket_p4_level4'] - size 1: ['ephemerides_resonance_static'] - size 1: ['antikythera_gear_dag_undirected']
Recurring patterns surfaced (what the data shows)¶
1. Density slope tracks topological dimensionality (strongest cross-domain regularity)¶
The bulk-fitted slope d(log N(lambda)) / d(log lambda) sorts the domains into three clean topological tiers, with two same-tier pairs matching to 4-9% of each other:
| tier | slope (= d_S / 2) | d_S estimate | domains |
|---|---|---|---|
| chain / tree | 0.49 - 0.54 | ~1.0 - 1.1 | P_2 (0.495), ephemerides static (0.502), antikythera gear DAG (0.544) |
| SG fractal family | 1.00 - 1.09 | ~2.0 - 2.2 | SG L3 (1.002), SG L4 (1.055), SG L5 (1.089) |
| 2-3D lattice | 1.44 - 1.55 | ~2.9 - 3.1 | chess king (1.438), P_4 (1.555) |
| near-complete | 3.25 | ~6.5 | othello line-of-sight (3.250) |
The three tree/chain domains cluster at d_S ≈ 1; the three SG levels cluster at d_S ≈ 2 (and increase monotonically with recursion level, hinting the finite-size truncation is pulling the slope up from the theoretical infinite-SG 2 ln 3 / ln 5 ≈ 1.365); chess king-move and P_4 are within 8% of each other despite very different graph constructions; othello is a clear outlier (near-complete-graph regime).
This is a genuine cross-domain structural fingerprint produced by the graph-Laplacian primitive. The slope reads the underlying graph's effective dimension off the eigenvalue distribution.
2. Self-similar gap fingerprint across SG recursion levels¶
The top-3 gap fingerprints (largest 3 gaps each normalised to the largest gap) for SG levels 3, 4, 5 are: - L3: (1.00, 0.964, 0.874) - L4: (1.00, 0.945, 0.863) - L5: (1.00, 0.945, 0.863)
L4 and L5 are identical to 4 decimal places; L3 matches to within 2%. This is the only multi-domain cluster the 5%-tolerance gap-fingerprint clustering surfaces (cluster of 3; all other domains are singletons). The fractal's top-3 gap structure converges as recursion depth grows -- a quantitative spectral signature of self-similarity.
3. Integer D-group irrep decomposition wherever the graph has global discrete symmetry¶
| domain | group | isolated/boundary eigval decomp | integer-clean? |
|---|---|---|---|
| SG L5, lambda=6 (mult 120) | D_3 | 22A + 18B + 40E | yes (dim=22+18+80=120) |
| chess king-move | D_4 | each isolated eigval = single irrep (1A1, 1A2, 1E, ...) | yes (dim_check matches multiplicity exactly) |
| othello LOS | D_4 | each isolated eigval = single irrep (1B2, 1E) | yes |
Every isolated chess eigenvalue is hosted by one D_4 irrep; no irrep mixing at gap-isolated points. Same for othello. This is structurally different from the SG lambda=6 case, where the 120-dim eigenspace mixes all three D_3 irreps richly. Both are "clean" in the sense the irrep counts are integers to machine precision, but the mode of cleanness differs (single-irrep vs equivariant-mix).
4. Max-multiplicity ratio recurrence (modulated by topology)¶
The ratio max_multiplicity / spectrum_size groups domains:
- High-multiplicity / fractal: P_4 (0.494), ephemerides (0.346), SG L5 (0.328), SG L4 (0.317), SG L3 (0.286)
- Low-multiplicity / sparse: P_2 (0.059), othello (0.047), antikythera (0.040), chess (0.031)
The five high-multiplicity domains all have some form of repeated-substructure (SG self-similar triangles; P_4 self-similar tetrahedra; ephemerides has 11 Jovian + 11 Saturnian moons with same-degree-1 attachment to their parent hub, producing high-mult eigenspaces at lambda=1). The low-multiplicity domains are either path-like (P_2, antikythera, mostly-chain) or have only local D_4 orbit structure (chess, othello). This is the Fukushima-Shima pre-localised-mode phenomenon manifesting whenever the graph has many similar substructures glued together.
5. Multi-hit n_isolated values¶
n_isolated = 0appears for both P_2 (path) and antikythera (gear chain) — both are essentially 1D chain-like graphs.n_isolated = 2appears for both SG-L3 (small SG) and othello (near-complete) — coincidence in this case (the underlying graphs are completely different; the count matches because both have a few extremum eigenvalues separated from the bulk).
The path/chain alignment at n_isolated = 0 is the more interesting hit -- it correlates with the density-slope ≈ 0.5 finding above.
Honest verdict¶
The graph-Laplacian primitive DOES produce a recognisable structural fingerprint across srmech's instantiations -- specifically via:
- Density slope (d_S/2) sorts domains into topological tiers and groups same-dimensional graphs together to 4-9%
- Top-3 gap fingerprint detects exact self-similarity across SG recursion levels (the only multi-domain cluster surfaced)
- Max-multiplicity ratio separates "many-repeated-substructures" graphs from chain-like graphs
This is stronger than "method-only" universality (the formulation in §3.5 'same architectural slot, parameterised by manifold'). The numbers produced by the primitive are individually domain-specific, but they cluster by graph topology in a quantitatively reproducible way. The §3.5 claim is supported at the level of method and at the level of topological-tier structural numbers, but not at the level of universal structural constants (no single number recurs across all nine spectra).
What's domain-specific (no recurrence): - Eigenvalue range / lambda_max (scales with degree structure) - Specific isolated-eigenvalue values (depend on graph particulars) - Number of isolated eigenvalues (correlates loosely with size + symmetry but no universal count) - D-group decomposition pattern (single-irrep at chess isolated vs. equivariant-mix at SG lambda=6)
Anomalies surfaced¶
- SG density slope rises monotonically with level (1.00 → 1.06 → 1.09). Either (a) the bulk-fit window is shifting position in the spectrum and reading different regimes, or (b) the true infinite-SG slope is approached from below as level grows. Worth a level-6 / level-7 check if compute allows.
- P_4 max-multiplicity ratio (0.494) is nearly double any other domain. lambda=8 hosts 254/514 eigenvalues. The (3-simplex) tetrahedral pre-gasket at level 4 has 4^4 = 256 smallest tetrahedra, and the localised-mode count tracks that closely. Phase-B-style D-group irrep decomposition on this eigenspace would be worth a follow-up (under the tetrahedral group S_4 or A_4).
- Ephemerides max-mult = 18 at lambda = 1.0 with ratio 0.35 (same range as SG levels). The 18 comes from the moons-attached-to-Jupiter-or-Saturn hub structure (each degree-1 leaf attached to the same hub contributes to the same eigenvalue). This is the tree-leaf-degeneracy mechanism, structurally distinct from but quantitatively similar to the SG localised-mode mechanism. Two different graph topologies producing similar max-mult ratios.
- Othello density slope 3.25 is far above any other domain. The line-of-sight graph has 728 edges on 64 vertices (average degree 22.75) -- approaching the regime where the Laplacian's spectrum starts to look like a complete graph's (which has spectrum {0, n, n, ..., n} for K_n). A linear-density-vs-edge-count study would be informative.
- D-group decomposition orthogonality: chess and othello use the same D_4 group on the same 64-vertex grid, yet their isolated eigenvalues lie in different irreps (chess emphasises A1; othello has a B2 isolated mode at lambda=16). The graph structure (king-move vs. line-of-sight) selects which D_4 irreps host isolated modes.
Future-survey domains (not in this run)¶
The srmech notebook §3.5 lists protein RIN GNM, audio mic-array, telecom ISL, and power Y-bus as additional graph-Laplacian instantiations. None of these have static-graph adjacency matrices computed in the project at present. Bringing them into a future cross-spectrum survey would test whether the density-slope-tier-classification continues to hold for biological / engineering domains.