{ "schema_version": 1, "artifact_id": "qft.artifact.holography-quantum-gravity.quantum-cosmology-and-singularity-resolution-programs.bkl-billiard-kasner-map", "title": "Bianchi IX Wall Encounters and the Kasner Map", "source_revision": 2, "created_on": "2026-08-30", "creator": "OpenAI Codex, for QFT.org", "original_work": true, "reader_question": "What does an actual Bianchi IX integration retain that an isotropic minisuperspace truncation discards, and do its resolved Kasner plateaus follow both branches of the BKL map?", "takeaway": "For the declared diagonal homogeneous vacuum fixture, the initial momenta have exact zero-wall kinetic-limit u = 7/5, and two resolved curvature-wall encounters divide three measured kinetic windows whose ordered Kasner parameters follow approximately 1.4 to 2.5 to 1.5. The first transition uses the invert branch and the second uses the subtract branch; the recorded residual and step-halving controls are numerical implementation checks, not evidence for the generic inhomogeneous BKL conjecture.", "scientific_status": "quantitative homogeneous-vacuum benchmark", "claim_ceiling": "This trajectory checks one implementation of diagonal homogeneous vacuum Bianchi IX and two asymptotic Kasner-map transitions. It does not establish generic chaos, inhomogeneous BKL locality, quantum suppression, or singularity resolution.", "conventions": { "metric_signature": "(+---)", "invariant_forms": "d sigma_i = (1/2) epsilon_ijk sigma_j wedge sigma_k", "alpha": "ln[(a_1 a_2 a_3)^(1/3)]; the singular direction is decreasing alpha", "scale_factors": [ "a_1 = exp(alpha + beta_plus + sqrt(3) beta_minus)", "a_2 = exp(alpha + beta_plus - sqrt(3) beta_minus)", "a_3 = exp(alpha - 2 beta_plus)" ], "constraint": "C = (1/2) exp(-3 alpha) (-p_alpha^2 + p_plus^2 + p_minus^2) + exp(alpha) V = 0", "potential": "V = [exp(-8 beta_plus) + 2 exp(4 beta_plus)(cosh(4 sqrt(3) beta_minus)-1) - 4 exp(-2 beta_plus) cosh(2 sqrt(3) beta_minus)]/6", "reduced_branch": "p_alpha = -H, H = sqrt(p_plus^2 + p_minus^2 + 2 exp(4 alpha) V)", "epoch_extraction": "In kinetic windows, compute the three axis exponents from d beta_plus/d alpha and d beta_minus/d alpha, sort them, and set u = p_ordered_max / p_ordered_mid." }, "initial_conditions": { "alpha": -5, "beta_plus": 0, "beta_minus": 0, "p_plus": -143, "p_minus": "-95 sqrt(3)", "zero_wall_kinetic_limit": { "H": "218", "ordered_exponents": [ "-35/109", "60/109", "84/109" ], "ordered_u": "7/5" }, "finite_alpha_qualification": "At alpha = -5, V(0,0) = -1/2 gives H^2 = 218^2 - exp(-20), so the full finite-alpha extraction differs infinitesimally from the exact zero-wall kinetic-limit value." }, "solver": { "method": "fixed-step classical fourth-order Runge-Kutta in alpha", "interval": [ -5, -50 ], "production_step": -0.0005, "refinement_step": -0.00025, "trace_sampling_delta_alpha": -0.05, "implementation": "figures-src/holography-quantum-gravity/bkl-billiard-kasner-map.mjs" }, "selection_rules": { "wall_encounters": { "observable": "R_V = abs(2 exp(4 alpha) V) / (p_plus^2 + p_minus^2)", "threshold": 0.05, "grid": "Evaluate every point of each full integration grid: Delta alpha = -0.0005 for production and -0.00025 for refinement; do not detect peaks on the Delta alpha = -0.05 sampled trace.", "peak_rule": "Accept a grid point when R_V exceeds the threshold, is strictly greater than the preceding point, and is greater than or equal to the following point along decreasing alpha.", "pairing": "Order accepted peaks by encounter along decreasing-alpha integration and pair production and refined peaks by ordinal position; both runs must contain exactly two accepted peaks.", "scope": "This total-absolute-potential rule is a declared classifier for this fixed fixture, not a universal decomposition into individual Bianchi IX walls." }, "kinetic_windows": { "method": "Use the nearest point on each full integration grid to each hard-coded fixture-specific alpha value; the windows are not selected by an automatic threshold.", "fixed_alphas": [ -8, -20, -48 ], "production_grid_step": -0.0005, "refinement_grid_step": -0.00025, "diagnostics": [ "wall_ratio R_V", "force_ratio R_F", "Kasner-square residual" ] } }, "kinetic_windows": [ { "epoch": 0, "alpha": -8, "ordered_exponents": [ -0.32110091689739756, 0.5504587152428645, 0.770642201654533 ], "u_measured": 1.4000000005713829, "u_predicted": null, "map_branch": "initial", "map_residual": null, "wall_ratio": 1.5148192703515408e-9, "force_ratio": 6.059278505185703e-9, "step_halving_u_shift": 4.218847493575595e-15 }, { "epoch": 1, "alpha": -20, "ordered_exponents": [ -0.256410256410227, 0.358974358974296, 0.897435897435931 ], "u_measured": 2.500000000000532, "u_predicted": 2.499999996428857, "map_branch": "invert", "map_residual": 3.571674955793469e-9, "wall_ratio": 1.0596917795045942e-28, "force_ratio": 4.720193922104505e-28, "step_halving_u_shift": 9.103828801926284e-13 }, { "epoch": 2, "alpha": -48, "ordered_exponents": [ -0.3157894736842009, 0.5263157894736437, 0.7894736842105572 ], "u_measured": 1.5000000000001743, "u_predicted": 1.500000000000532, "map_branch": "subtract", "map_residual": 3.5771385853422544e-13, "wall_ratio": 6.349582889933904e-20, "force_ratio": 2.5398331559735615e-19, "step_halving_u_shift": 7.820410985459603e-13 } ], "wall_encounters": [ { "collision": 1, "alpha_at_peak_wall_ratio": -13.370000000000001, "beta_plus": 5.413977831385356, "beta_minus": 6.100497101032741, "peak_wall_ratio": 6.380199880754509, "refined_alpha_at_peak": -13.370000000000001 }, { "collision": 2, "alpha_at_peak_wall_ratio": -40.8585, "beta_plus": 6.3415178124463765, "beta_minus": -21.092091484992313, "peak_wall_ratio": 0.6802719340947873, "refined_alpha_at_peak": -40.85875 } ], "verification": { "maxima": { "mapResidual": 3.571674955793469e-9, "stepHalvingUShift": 9.103828801926284e-13, "collisionAlphaShift": 0.0002500000000011937, "c3PotentialScaledResidual": 8.700484118278642e-16, "c3GradientCovarianceScaledResidual": 8.076773998132063e-16, "c3RotationClosureScaledResidual": 5.551115123125783e-17, "kineticWindowWallRatio": 1.5148192703515408e-9, "kineticWindowKasnerSquareResidual": 1.0098796243696029e-9, "algebraicConstraintReconstructionResidual": 8.612588571177799e-17 }, "algebraic_constraint_diagnostic": { "identity": "-H^2 + p_plus^2 + p_minus^2 + 2 exp(4 alpha) V = 0", "normalization": "abs(identity) / max(H^2 + p_plus^2 + p_minus^2 + abs(2 exp(4 alpha) V), smallest positive number)", "independent_conservation_test": false, "legacy_controls_csv_alias": "constraint_reconstruction_residual_maximum", "interpretation": "This floating-point residual reconstructs the same algebraic square-root identity used to define H; it is not an independent Hamiltonian-constraint conservation test.", "maximum": 8.612588571177799e-17 }, "isotropic_control": { "V_at_origin": -0.5, "gradient_at_origin": [ 0, 0 ], "H_squared_with_zero_anisotropy_momenta": "-exp(4 alpha)", "real_vacuum_constraint_trajectory": false, "interpretation": "beta_plus = beta_minus = p_plus = p_minus = 0 removes anisotropy motion and wall scattering but leaves V(0,0) = -1/2, the closed-FLRW spatial-curvature term. In this vacuum model H^2 = -exp(4 alpha), so this formal isotropic restriction is not a real constrained vacuum trajectory; matter or a cosmological term is required for a physical isotropic closed-FLRW comparison." }, "c3_symmetry_control": { "transformation": "(beta_plus, beta_minus) -> (-beta_plus/2 - sqrt(3) beta_minus/2, sqrt(3) beta_plus/2 - beta_minus/2)", "residual_scaling": "Each scalar or Euclidean-vector residual is divided by max(1, magnitudes of the compared quantities).", "test_points": [ [ 0.2, -0.1 ], [ -0.7, 0.3 ], [ 1.1, -0.5 ] ], "potential_invariance_maximum": 8.700484118278642e-16, "gradient_covariance_maximum": 8.076773998132063e-16, "rotation_cubed_identity_maximum": 5.551115123125783e-17 } }, "uncertainty": { "interpretation": "The map residual, half-step u shift, and collision-alpha shift are deterministic discretization diagnostics for this fixed fixture. They are not statistical error bars, theorem bounds, or uncertainty estimates for other initial data.", "map_residual_maximum": 3.571674955793469e-9, "step_halving_u_shift_maximum": 9.103828801926284e-13, "collision_alpha_step_halving_shift_maximum": 0.0002500000000011937 }, "scientific_sources": [ { "citation": "Belinskii, Vladimir A., Isaak M. Khalatnikov, and Evgeny M. Lifshitz. Oscillatory Approach to a Singular Point in the Relativistic Cosmology. Advances in Physics 19 (1970): 525-573.", "identifier": "DOI:10.1080/00018737000101171", "locators": "sections 2-7, especially section 3 for transitions and section 6 for the statistical map", "use": "Kasner epochs, curvature-driven transitions, era map, and the inhomogeneous asymptotic proposal" }, { "citation": "Misner, Charles W. Mixmaster Universe. Physical Review Letters 22 (1969): 1071-1074.", "identifier": "DOI:10.1103/PhysRevLett.22.1071", "locators": "pages 1071-1073", "use": "anisotropy-plane and moving-wall formulation of homogeneous Bianchi IX" }, { "citation": "Bojowald, Martin, David Brizuela, Paula Calizaya Cabrera, and Sara F. Uria. Chaotic Behavior of the Bianchi IX Model under the Influence of Quantum Effects. Physical Review D 109 (2024): 044038.", "identifier": "DOI:10.1103/PhysRevD.109.044038; arXiv:2307.00063", "locators": "section II, equations (1)-(11), pages 2-4", "use": "invariant one-forms, Misner variables, complete potential and constraint normalization, and alpha-reduced Hamilton equations after the declared overall signature and canonical-convention translation" }, { "citation": "Heinzle, J. Mark, and Claes Uggla. Mixmaster: Fact and Belief. Classical and Quantum Gravity 26 (2009): 075016.", "identifier": "arXiv:0901.0776; DOI:10.1088/0264-9381/26/7/075016", "locators": "section 3, equations (16)-(17); section 5, equations (23)-(28); section 6, theorem 6.1 and equations (32)-(33)", "use": "ordered u parameter, epoch and era maps, homogeneous attractor result, and claim limits" } ], "data_files": { "trace": "bkl-billiard-kasner-map.csv", "epoch_summary": "bkl-billiard-kasner-map-epochs.csv", "collision_summary": "bkl-billiard-kasner-map-collisions.csv", "numerical_controls": "bkl-billiard-kasner-map-controls.csv" }, "caption": "Computed diagonal vacuum Bianchi IX benchmark for alpha decreasing from -5 to -50. The anisotropy-plane path has two resolved curvature-wall encounters and three kinetic windows. Their measured ordered parameters follow u = 1.4 to 2.5 by the invert branch and 2.5 to 1.5 by the subtract branch, with maximum map residual 3.6e-9 and maximum half-step u shift 9.2e-13. The walls move with alpha and are therefore not drawn as static contours. This one homogeneous trajectory checks an implementation; it does not establish generic chaos, inhomogeneous BKL locality, quantum suppression, or singularity resolution.", "alt_text": "A computed homogeneous vacuum Bianchi IX anisotropy trajectory bends at two numbered curvature-wall encounters; its three measured Kasner plateaus follow the invert map from 1.4 to 2.5 and the subtract map from 2.5 to 1.5 within the recorded numerical controls." }