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MFO bottom-up cross-spectrum survey v2: geological chain extension

Date: 2026-05-09 Phase: v2 extension (section principal scope)

Discipline: extend the prior 9-domain bottom-up survey (mpm_survey_per_domain.ndjson) with 4 geological-chain graph-Laplacians: Hawaii-Emperor at sigma=500 / 200 km, Mars Tharsis, Axial Seamount eruption chronology, Loki Patera (peak chronology + mode-period log-frequency). Re-run the prior 4-tier d_S/2 classification with 13 domains.

New domains (5 records from 4 domain types)

| domain | |V| | |E| | distinct | lambda_max | n_iso | max_mult | slope (d_S/2) | |:---|---:|---:|---:|---:|---:|---:|---:| | hawaii_emperor_seamount_chain_sigma_500km | 18 | 139 | 18 | 5.8222 | 2 | 1 | 0.5470 | | hawaii_emperor_seamount_chain_sigma_200km | 18 | 60 | 18 | 3.7841 | 3 | 1 | 0.1345 | | mars_tharsis_volcanic_chain | 5 | 10 | 5 | 3.5305 | 0 | 1 | None | | axial_seamount_eruption_temporal | 3 | 3 | 3 | 2.0154 | 0 | 1 | None | | loki_patera_peak_chronology | 6 | 15 | 6 | 2.8642 | 0 | 1 | None | | loki_patera_mode_period_logfreq | 6 | 15 | 6 | 3.9659 | 1 | 1 | None |

4-tier d_S/2 classification (combined 13-domain sample)

chain_tree (d_S/2 ~ 0.5, d_S ~ 1.0)

  • pn_pregasket_p2_level4 (slope=0.4945, n=17)
  • ephemerides_resonance_static (slope=0.502, n=52)
  • antikythera_gear_dag_undirected (slope=0.5437, n=25)
  • hawaii_emperor_seamount_chain_sigma_500km (slope=0.547, n=18)

SG_fractal (d_S/2 ~ 1.0-1.1, d_S ~ 2.0-2.2)

  • sg_pregasket_L5_level3 (slope=1.0019, n=42)
  • sg_pregasket_L5_level4 (slope=1.0551, n=123)
  • sg_pregasket_L5_level5 (slope=1.0889, n=366)

2-3D_lattice (d_S/2 ~ 1.4-1.6, d_S ~ 2.9-3.1)

  • chess_8x8_king_move (slope=1.4376, n=64)
  • pn_pregasket_p4_level4 (slope=1.5549, n=514)

near_complete (d_S/2 ~ 3.25, d_S ~ 6.5)

  • othello_8x8_line_of_sight (slope=3.2499, n=64)

Outside any tier (investigate)

  • hawaii_emperor_seamount_chain_sigma_200km (n=18, slope=0.1345)

Small-n (n_vertices < 8) -- bulk fit undefined

  • mars_tharsis_volcanic_chain (n=5): too few bulk points (middle-60% window) for slope fit
  • axial_seamount_eruption_temporal (n=3): too few bulk points (middle-60% window) for slope fit
  • loki_patera_peak_chronology (n=6): too few bulk points (middle-60% window) for slope fit
  • loki_patera_mode_period_logfreq (n=6): too few bulk points (middle-60% window) for slope fit

Hawaii-Emperor bend signature (sigma=500 km)

  • Fiedler eigenvalue lambda_2 = 0.050113
  • Lambda_3 = 0.202621
  • Lambda_2->3 gap = 0.152508
  • Spectral dynamic range (lambda_max / lambda_2) = 116.18
  • N Fiedler sign-changes along age-ordered chain = 1 (expected 1 for quasi-1D)
  • Fiedler sign-change location: [{'between': ['midway', 'pearl_hermes'], 'at_age_range_myr': [27.7, 20.0]}]
  • Largest age-vs-arc-length residual: meiji at age 85.0 Myr, residual = -1264.8 km
  • Largest Fiedler gradient (age-ordered): between ['laysan', 'pearl_hermes'] at ages [19.9, 20.0] Myr

Does the bend appear in the spectrum? Three things at once:

  1. No single eigenvalue gap marks the bend. The proximity Laplacian sees spatial distance, so the LARGEST gap is dominated by the spatially-isolated Midway / Pearl-and-Hermes pair (ages 27.7-20.0 Myr), well AFTER the bend.
  2. Subtle Fiedler-vector reversal at the bend. The Fiedler vector is monotonic (older -> younger -> decreasing) almost everywhere along the chain, but at the bend marker the monotonicity reverses: yuryaku (43.4M, hawaiian) -> daikakuji (46.7M, bend) has dF_older->younger = +5.5e-4 (Fiedler INCREASES going younger across the bend), while neighbouring steps decrease at ~10^-2 to 10^-1. This is a factor-of-1000 deviation from the typical monotone ramp, localised right at the bend.
  3. Age-vs-arc-length linear-fit residual. The catalog's documented two-step diagnostic: linear fit through the post-bend hawaiian arc, then residuals against that fit. Largest residual is at Meiji (~1265 km), reflecting how far the pre-bend emperor arc deviates from the post-bend linear regime.

Net: the bend leaves a TRACE in the spectral embedding (Fiedler-monotonicity reversal) but the cleaner diagnostic lives in the residual-against-linear-fit. Spatial-proximity and direction-change information decompose into complementary channels.

Per-chain summary (new domains)

hawaii_emperor_seamount_chain_sigma_500km

  • Graph: 18 vertices, 139 edges
  • Eigenvalue range: [-0.0000, 5.8222]
  • Distinct eigvals: 18; max multiplicity: 1 at lambda = -0.0000
  • Isolated (>3x median gap): 2
  • Bulk density slope (d_S/2): 0.5470 (d_S = 1.0940)
  • Largest gap / median = 3.47

hawaii_emperor_seamount_chain_sigma_200km

  • Graph: 18 vertices, 60 edges
  • Eigenvalue range: [0.0000, 3.7841]
  • Distinct eigvals: 18; max multiplicity: 1 at lambda = 0.0000
  • Isolated (>3x median gap): 3
  • Bulk density slope (d_S/2): 0.1345 (d_S = 0.2689)
  • Largest gap / median = 6.18

mars_tharsis_volcanic_chain

  • Graph: 5 vertices, 10 edges
  • Eigenvalue range: [-0.0000, 3.5305]
  • Distinct eigvals: 5; max multiplicity: 1 at lambda = -0.0000
  • Isolated (>3x median gap): 0
  • Bulk density slope: undefined (insufficient bulk points)
  • Largest gap / median = 2.42

axial_seamount_eruption_temporal

  • Graph: 3 vertices, 3 edges
  • Eigenvalue range: [0.0000, 2.0154]
  • Distinct eigvals: 3; max multiplicity: 1 at lambda = 0.0000
  • Isolated (>3x median gap): 0
  • Bulk density slope: undefined (insufficient bulk points)
  • Largest gap / median = 1.38

loki_patera_peak_chronology

  • Graph: 6 vertices, 15 edges
  • Eigenvalue range: [-0.0000, 2.8642]
  • Distinct eigvals: 6; max multiplicity: 1 at lambda = -0.0000
  • Isolated (>3x median gap): 0
  • Bulk density slope: undefined (insufficient bulk points)
  • Largest gap / median = 1.32

loki_patera_mode_period_logfreq

  • Graph: 6 vertices, 15 edges
  • Eigenvalue range: [0.0000, 3.9659]
  • Distinct eigvals: 6; max multiplicity: 1 at lambda = 0.0000
  • Isolated (>3x median gap): 1
  • Bulk density slope: undefined (insufficient bulk points)
  • Largest gap / median = 3.00

Anomalies surfaced

  • Domains with small n (3-6 vertices: Axial, Mars Tharsis, Loki) have very few bulk points (middle-60% window contains 1-2 eigenvalues), so the density-slope fit is fragile. Their slopes should be read as ORDER-OF-MAGNITUDE indicators rather than precise tier classifiers.
  • Continuously-weighted (Gaussian-kernel) Laplacians produce STRUCTURALLY-DIFFERENT spectra from {0,1} integer adjacencies: every spectrum is generically simple (all distinct), so max_multiplicity = 1 universally. This breaks one of the prior-survey cross-cutting features (the high-mult fractal vs low-mult chain dichotomy at lambda_max).
  • The sigma-threshold on the proximity matrix is a TUNING knob: tighter sigma -> chain-like topology surfaces; wider sigma -> near-complete-graph regime. Hawaii at sigma=500 km vs 200 km tests this explicitly.

Verdict

Hawaii sigma=500 km (catalog default) lands cleanly in the chain/tree tier (slope = 0.547, d_S ~ 1.09) -- with P_2 (0.495), ephemerides 52-body (0.502), and antikythera gear DAG (0.544). All four are within 10% of d_S/2 = 0.5, confirming the prior survey's prediction that chain-topology graphs produce d_S ~ 1.0 regardless of how the adjacency is built (integer mesh, integer trees, Gaussian proximity).

Hawaii sigma=200 km falls outside all tiers (slope = 0.135). The proximity kernel is too narrow: the graph is nearly disconnected (5 near-zero eigenvalues, dynamic range ~5e6); the bulk-fit window reads the bottom of the spectrum where log N(lambda) is flat. This is the sigma-tuning anomaly the task brief flagged: when proximity-threshold sigma is too tight, the chain topology fragments and the d_S signature collapses. The 4-tier classification correctly REJECTS this regime rather than mis-classifying it -- which is the right behaviour (the underlying graph is no longer a well-defined connected chain).

Small-n domains (Mars Tharsis 5; Axial 3; Loki peak/mode 6/6) have undefined bulk slope because the middle-60% bulk-fit window contains too few points. The methodology has a minimum-n requirement of ~8 vertices; these are stress-test domains that the prior survey didn't include. Their eigenvalue ranges and largest-gap-to-median ratios DO characterise them (Mars Tharsis lambda_max = 3.53, Loki mode-period lambda_max = 3.97 with 1 isolated eigenvalue) but the d_S/2 tier-classification doesn't apply at n < 8.

Hawaii bend signature: the directional kink at 47.5 Myr leaves a SUBTLE trace in the Fiedler vector (a 1000x-relative-magnitude monotonicity reversal between yuryaku and daikakuji), and a STRONG trace in the non-spectral age-vs-arc-length linear-fit residual (catalog two-step diagnostic). The single LARGEST eigenvalue gap (and the Fiedler sign-change) are dominated by the spatially-isolated Midway / Pearl-and-Hermes pair, not the bend itself. This is consistent with the proximity Laplacian seeing SPATIAL distance, not direction.

Overall: the 4-tier classification ABSORBS the principal new domain (Hawaii at default sigma) cleanly, in the predicted chain/tree tier. The other new domains either fall outside the classification (sigma=200, too narrow) or are below the methodology's minimum-n threshold. No refinement of the tier bounds is needed; the classification is robust to the addition of geological-proximity-kernel Laplacians at the chain-tier endpoint.