Phase 4k: richer geometry (full Hessian spectrum + mode connectivity) - #7
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Tests whether the Phase-4j "graph tie = redundancy" verdict is real or an artifact of thin curvature features (only lambda_max + trace). Adds two opt-in feature groups behind [eval] richer_geometry and re-runs the 4j complementarity test on all four (task, reasoner) cells. - geometry/spectrum.py: full d×d Hessian spectrum (exact eigh) -> lmin, negative-eigenvalue fraction, effective rank, log-det (mean + at best restart). - geometry/connectivity.py: energy barriers along the path from the best restart to the others (mode connectivity) -> barrier mean/max, connected fraction. - curvature.py: batched_spectrum + batched_path_energy (all new JAX stays in the core, invariant 1; the two feature modules are pure NumPy). - evaluate/config/features: richer appended AFTER the base 14 (order-preserving), metrics["feature_set"], ablation guards empty groups. - aggregate: feature_set_of + canonical tables/figures default feature_set="base" so richer never pollutes T1/T2/T4 or the pooled figures (e.g. S1's K=12 point). - make_tables T5 (base vs richer per cell); make_figures excludes richer rows. Verdict: richer geometry helps NOWHERE. Crux graph-IRED conditional signal over softmax stays -0.002 (0/5); a wash where geometry already wins (arith-IRED +0.007, 4/5, unchanged) and slightly worse on the fixed-bowl basin-center cells. So the graph redundancy is a task property, not thin features. Base rows reproduce the 4j numbers exactly (purely additive). 109 tests green, ruff clean. Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
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What & why
The Phase-4j headline caveat — on graph, geometry is redundant with softmax — rested on a thin curvature description (only
λmax+tr(H)). This phase asks whether that redundancy is a task property or an artifact of feature poverty, by adding the two richer signals the paper names as future work and re-running the existing 4j complementarity test on all four (task, reasoner) cells.geometry/spectrum.py— fulld×dHessian spectrum (exacteigh, sincedis small):lmin, negative-eigenvalue fraction, effective rank (participation ratio), log-det — each mean-over-restarts and at the best restart.geometry/connectivity.py— mode connectivity: energy barriers along the straight-line path from the best restart to the others (barrier mean/max, connected fraction).batched_spectrum,batched_path_energy) lives incurvature.pyto keep invariant 1; the two feature modules are pure NumPy.[eval] richer_geometry; features appended after the base 14 so every prior result reproduces byte-identically. Aggregation is feature-set-aware (feature_set_of; canonical tables/figures defaultfeature_set="base"), so richer never pollutes T1/T2/T4 or the pooled figures.Verdict — redundancy confirmed as a task property
Richer geometry helps nowhere: the crux graph-IRED conditional signal stays −0.002 (0/5), it's a wash where geometry already wins (arith-IRED, unchanged), and slightly worse on the fixed-bowl basin-center cells (extra features add noise). So a well-calibrated softmax genuinely captures what even the richest descent geometry would add on graph. A decisive negative result that cements the headline caveat and closes the paper's stated "richer geometry" open question.
Tests
109 tests green (offline, CPU-only), ruff clean. New
test_spectrum.py/test_connectivity.pyverify exact spectra on a known quadratic and barriers on a double well; extended feature/evaluate/aggregate suites cover the richer path + feature-set aggregation. Paper rebuilds viatectonic.🤖 Generated with Claude Code