{ "id": "T1-EMPTY-THRONE", "title": "T1 · The Empty Throne — Parameter-manifold geometry at ML fit", "semantic_name": "T1", "record_class": "SIMULATION", "version": "1.2.0", "status": "INCONCLUSIVE (architecture-dependent; PROVISIONAL SUPPORT for LTI/R1a; R2 qualitatively upholds; R6 baseline-failed)", "evidence_level": "computational_simulation", "replication_status": "internal_followups_completed_T1R1_T1R1a_T1R2_T1R6", "program": "structural_claims_of_the_valleys_t_series", "non_drift_question": "structural_arrival_vs_parametric_arrival_at_ml_fit", "date": "2026-07-30", "source_record": "OA-BAHA-VALLEYS", "related_records": [ "O0-CRP-035", "O0-CRP-037", "O0-CRP-038" ], "design_summary": { "source_classes": ["LTI-Gaussian (C fixed)", "small tanh MLP (Gaussian output)", "RBF GP"], "adversarial_controls": ["well-specified regression (null)", "colinear regression (positive control, known k_dup manifold)"], "capacities_tested": { "LTI": [4, 6], "MLP": [6, 8], "GP": [3] }, "N_train_range": {"LTI": [512, 2048, 8192], "MLP": [2048], "GP": [200]}, "estimators": ["E1 Fisher rank", "E2 orthogonal flat-walk verification", "E3 multi-init dispersion", "E4 trivial-vs-residual split"], "seed_range_lti_matched": [20260730, 20260800, 20260830], "preregistration_version": "1.1", "deviations": ["001 estimator redesign (v1.0->v1.1)", "002 LTI C fixed at truth", "003 R6 uses (A,Q,R)+EKF under nonlinear observation"] }, "primary_verdict": "INCONCLUSIVE overall; PROVISIONAL SUPPORT for LTI at deep-fit basins; R1a mechanism qualitatively upheld by R2; R6 α-trend uninformative (baseline failed); UNSUPPORTED for MLP and GP", "per_class_results": { "regression_baseline": {"d_E1": 0, "d_E2": 0, "d_E3": 0, "d_E4_raw": 0, "d_E4_res": 0, "verdict": "null_holds"}, "colinear_positive_control": {"d_E1_matches_k_dup": true, "d_E4_res": 0, "verdict": "controls_pass"}, "LTI_matched_bimodal": { "seed_20260730_random_init": {"d_E4_res": 0, "nll_train_fit": 4475.25}, "seed_20260800_random_init": {"d_E4_res": 2, "nll_train_fit": 5496.32}, "seed_20260830_random_init": {"d_E4_res": 11, "nll_train_fit": 6387.07}, "seed_20260730_truth_init": {"d_E4_res": 11, "nll_train_fit": 4473.77} }, "LTI_N_stability_truth_init": { "N_512": {"manifold_dim": 13, "cliff": 34.7}, "N_2048": {"manifold_dim": 12, "cliff": 54.2}, "N_8192": {"manifold_dim": 13, "cliff": 17.5} }, "LTI_over_n_M_6": {"d_E1": 7, "d_E2_verified": 5, "d_E4_res": 5, "d_trivial_predicted": 7}, "MLP_matched": {"d_E1": 0, "d_E2": 0, "d_E4_res": 0}, "MLP_over": {"d_E1": 0, "d_E2": 0, "d_E4_res": 0, "note": "predicted dead-unit d_trivial=10 not visited by ML fit"}, "GP": {"d_E1": 0, "d_E2": 0, "d_E4_res": 0} }, "supported_claims": [ "Adversarial controls pass: regression baseline d = 0 for p in {4,8,16}; colinear positive control d = k_dup with residual 0 for k_dup in {1,2,3}.", "LTI with C fixed at matched capacity shows a robust manifold at the deepest fit basin: d = 10-13, N-stable across a 16x range in training sample size.", "The LTI manifold is at the ML argmin (lower NLL_train than source parameters) and does not appear at source parameters, which are at a sharp higher-NLL minimum.", "LTI over-capacity partially recovers the excess-mode manifold: E1 exact (7 of 7); E2 verifies 5 of 7 under non-local walks." ], "not_established_claims": [ "Any generalization to physical, biological, or empirical systems.", "Any metaphysical claim about consciousness, unity, non-duality, or identity.", "That the LTI result would replicate under a different C or A distribution.", "That random-init L-BFGS reliably finds the deep basin in LTI (fit outcome is initialization-sensitive).", "That the linear-detection technique is the RIGHT operational translation of the source claim." ], "resolved_in_v1_1_0_addendum": [ "T1-R1 (innovations-form parametrization): manifold does NOT vanish. d shrinks from ~10-13 to ~8 across seeds. Spectral-factorization non-uniqueness alone does not account for the manifold; it accounts for 3-5 dim of it.", "T1-R1a (mechanism check): 8/8 flat-direction eigenvectors put dK columns entirely in ker(C) (fraction = 1.000), dL_S = 0, and preserve the observed innovation sequence to ~1e-7 relative. The residual manifold is mechanistically identified as the classical unobservable-subspace freedom under fixed C." ], "per_class_results_v1_1_0": { "LTI_R1_innovations_form_matched_n_M_4_N_2048": { "seed_20260730_random_init": {"d_E1": 8, "nll_train_fit": 4473.49, "improvement_over_v1_0_0": "yes (deeper basin from random init)"}, "seed_20260800_random_init": {"d_E1": 1, "nll_train_fit": 5496.29}, "seed_20260830_random_init": {"d_E1": 8, "nll_train_fit": 6386.63}, "seed_20260730_truth_init": {"d_E1": 8, "nll_train_fit": 4473.49} }, "LTI_R1a_mechanism_seed_20260730": { "n_flat_directions": 8, "n_dK_columns_in_ker_C": 8, "dL_S_norm_max": 0.000, "innovation_invariance_relative_at_step_0p01": 8.9e-8, "conclusion": "residual manifold is exactly the ker(C) hidden-state freedom" }, "LTI_R1a_mechanism_seed_20260830": { "n_flat_directions": 8, "n_dK_columns_in_ker_C": 8, "dL_S_norm_max": 0.000, "innovation_invariance_relative_at_step_0p01": 9.0e-6, "conclusion": "residual manifold is exactly the ker(C) hidden-state freedom" } }, "adversarial_notes": [ "Local-minimum sensitivity in LTI: our headline positive result requires the fit to reach the global-minimum basin. 3 restarts of random-init L-BFGS sometimes achieves this and sometimes does not.", "Parametrization dependence: the LTI finding is in a specific parametrization (C fixed, Cholesky-factored Q and R). Minimum-phase canonical form would likely give a very different answer.", "MLP over-capacity behavior: the closed-form d_trivial = 10 at H_M = 8 is a valid tangent-space prediction at points on the dead-unit submanifold. The ML fit is not on that submanifold, so the prediction is inapplicable to the specific fit measured.", "N-scaling: three N values is a limited scan. Larger N (up to 65k) would more cleanly distinguish structural manifold from slow finite-sample drift.", "GP fit is far from truth at N = 200 (log_l = -0.66 vs 0). Overfitting is expected but limits sharp-fit conclusion.", "The interpretation is not the finding: even if the LTI result survives all follow-up scrutiny, the reading that it 'supports the mystical claim' depends on the operational translation used." ], "replication_queue": [ {"id": "T1-R1", "kind": "innovations_form_parametrization", "target": "LTI in innovations form (A_pred, K, S_ss); does the manifold vanish under spectral-factorization quotient?", "status": "COMPLETED_2026_07_30", "verdict": "Manifold PERSISTS; shrinks from ~11 to ~8 dim. Spectral-factorization alone does not account for it."}, {"id": "T1-R1a", "kind": "mechanism_check", "target": "Decompose flat directions into (dA_pred, dK, dL_S); check dK cols in ker(C).", "status": "COMPLETED_2026_07_30", "verdict": "8/8 flat directions have dK columns entirely in ker(C); dL_S=0; innovation sequence preserved to numerical precision. Residual manifold is exactly the ker(C) hidden-state freedom."}, {"id": "T1-R2", "kind": "square_C_falsification", "target": "LTI with C square full-rank; ker(C) empty; predict d=0.", "status": "COMPLETED_2026_07_30", "verdict": "Strict prereg INCONCLUSIVE (d in {4,0,0}, mean 1.33). Post-hoc: seed-20260900 d=4 is gap-detector artifact (eigvals O(1) not O(1e-4)). Exploratory corroboration on original T1 seeds: d=0 on all 4 fits. Mechanism qualitatively upheld."}, {"id": "T1-R2b", "kind": "innovation_invariance_check", "target": "Apply R1a-style innovation-sequence invariance to seed-20260900 candidate flat directions.", "status": "REGISTERED"}, {"id": "T1-R3", "kind": "large_N_scan", "target": "LTI at N in {32768, 65536}; does manifold dim stabilize?", "status": "REGISTERED"}, {"id": "T1-R4", "kind": "many_restart_stat", "target": "LTI matched capacity with 100 random starts; distribution of basins", "status": "REGISTERED"}, {"id": "T1-R5", "kind": "mlp_active_dead_unit_init", "target": "MLP over-capacity initialized on dead-unit submanifold; does the fit stay?", "status": "REGISTERED"}, {"id": "T1-R6", "kind": "nonlinear_observation_channel", "target": "α-sweep of quadratic ker(C)-leaking observation; does manifold shrink as α grows?", "status": "COMPLETED_2026_07_30", "verdict": "Strict prereg REFUTES_R1A_MECHANISM driven by baseline failure (d(0)=0 < 5). α-trend uninformative. EKF pathology at α=1 seed-20260910 (d=27, ΔNLL=-3463). T1-R6b registered."}, {"id": "T1-R6b", "kind": "restored_baseline_alpha_sweep", "target": "Restore α=0 baseline to recover T1-R1 (N=2048, innovations or SS-Kalman, multi-restart); then re-run α-sweep.", "status": "REGISTERED"}, {"id": "T1-R7", "kind": "particle_filter_alpha1", "target": "UKF/particle-filter likelihood at α=1.0 to rule out EKF pathology.", "status": "REGISTERED"} ], "per_class_results_v1_2_0": { "T1_R2_square_C_prereg_seeds": { "seeds": [20260900, 20260901, 20260902], "d_E1": [4.0, 0.0, 0.0], "d_E1_mean": 1.33, "verdict_prereg": "INCONCLUSIVE", "posthoc": "seed-20260900 d=4 is gap-detector artifact; mechanism qualitatively upheld" }, "T1_R2_exploratory_original_seeds": { "seeds": [20260730, 20260800, 20260830], "d_E1": [0, 0, 0, 0], "note": "includes truth-init on 20260730; all d=0 across E1/E2/E4" }, "T1_R6_alpha_sweep": { "alphas": [0.0, 0.3, 1.0], "d_E1_mean_by_alpha": {"0.0": 0.0, "0.3": 0.0, "1.0": 13.5}, "verdict_prereg": "REFUTES_R1A_MECHANISM", "driver": "baseline d(0)=0 < 5; α-trend uninformative; α=1 seed-20260910 degenerate" } }, "figures": [], "files": { "preregistration": "preregistration.md", "deviations": "deviations.md", "scientific_record": "scientific_record.md", "code": [ "src/model_interface.py", "src/common.py", "src/lti.py", "src/lti_innov.py", "src/lti_square_c.py", "src/lti_nonlin_obs.py", "src/nl_ssm.py", "src/mlp.py", "src/gp.py", "src/regression_baseline.py", "src/estimators.py", "src/pilot_regression.py", "src/pilot_positive_control.py", "src/pilot_lti.py", "src/pilot_lti_innov.py", "src/pilot_lti_r2.py", "src/pilot_lti_square_c.py", "src/pilot_lti_r6.py", "src/pilot_nl_ssm.py", "src/pilot_mlp.py", "src/pilot_gp.py", "src/full_run_lti.py", "src/debug_lti_innov_mechanism.py", "src/debug_r2_seed900.py" ], "results": [ "runs/REG/pilot/pilot_results.json", "runs/REG/positive_control/positive_control_results.json", "runs/LTI/pilot/pilot_results.json", "runs/LTI/full_run/results.json", "runs/LTI_INNOV/pilot/results.json", "runs/LTI_R2/pilot_results.json", "runs/LTI_SQUARE_C/pilot/results.json", "runs/LTI_R6/pilot_results.json", "runs/NL_SSM/pilot/results.json", "runs/MLP/pilot/pilot_results.json", "runs/GP/pilot/pilot_results.json", "logs/r1a_mechanism.log", "logs/r2_pilot.log", "logs/r2_square_c.log", "logs/r2_seed900_diagnostic.log", "logs/r6_pilot.log", "logs/r6_nl_ssm.log" ] }, "provenance": { "philosophical_source": "Baha'u'llah, The Seven Valleys / The Four Valleys; Sussmann 1992 for MLP theory; Kailath et al. 2000 for LTI identifiability", "conceptual_relationship": "structural_end_state_claim", "disclaimer": "This study tests only the loss-surface geometry of specific model classes. No metaphysical claim is evaluated." }, "runtime_seconds": 2069 }