{
  "schema_version": "qft.relative-entropy-interval-benchmark.v1",
  "generated_on": "2026-09-01",
  "generated_by": "scripts/generate-relative-entropy-interval-benchmark.mjs",
  "deterministic_generation": {
    "random_seed": null,
    "stochastic_steps": false,
    "arithmetic": "IEEE-754 binary64 JavaScript Number arithmetic",
    "eigensolver": "deterministic cyclic Jacobi rotations for real symmetric matrices",
    "output_order": "fixed parameter-loop order encoded in the generator",
    "generator_source_sha256": "43ccf9d229bb530a5f24114f44b6756620a4da534f0052fe36b2760e533d6653"
  },
  "question": "Does a regulated vacuum-versus-coherent-state interval calculation approach the exact continuum relative entropy, and does one fixed regulated state obey inclusion monotonicity on nested interval algebras?",
  "orientation_and_support": {
    "reported_quantity": "D(rho_coherent restricted to A || rho_vacuum restricted to A)",
    "reference_state": "the vacuum restriction is the second argument",
    "finite_dimensional_formula": "D(rho||sigma)=Tr[rho(log rho-log sigma)] when supp(rho) is a subspace of supp(sigma), and +infinity otherwise",
    "continuum_formula": "S(omega_coherent||omega_vacuum)=-<xi_coherent, log Delta_(vacuum,coherent) xi_coherent>",
    "support_statement": "The finite-chain interval vacuum is faithful before numerical spectral conditioning. The smooth coherent displacement is locally normal to it. Faithfulness does not by itself replace the modular form-domain condition in the continuum."
  },
  "model": {
    "continuum": "1+1-dimensional free massless real scalar at t=0, with the infrared zero mode controlled by a finite Dirichlet box before B tends to infinity",
    "interval": "A=(-R,R)",
    "classical_coherent_data": {
      "field_mean_F": "F(x)=A exp[-x^2/(r^2-x^2)] for |x|<r, and F(x)=0 for |x|>=r",
      "momentum_mean_G": "G(x)=0",
      "derivative_inside_support": "F'(x)=-2 A r^2 x exp[-x^2/(r^2-x^2)]/(r^2-x^2)^2 for |x|<r",
      "peak_normalization": "F(0)=A",
      "parameters": {
        "interval_half_length_R": 1,
        "bump_support_half_length_r": 0.5,
        "peak_amplitude_A": 1
      }
    },
    "continuum_target_formula": "D=2 pi integral_(-R)^R [(R^2-x^2)/(2R)] [F'(x)^2+G(x)^2]/2 dx",
    "lattice_regulator": {
      "box": "Dirichlet endpoints at x=-B and x=B; only the 2B/h-1 interior oscillators are retained",
      "canonical_variables": "q_j=sqrt(h) Phi(x_j), p_j=sqrt(h) Pi(x_j), with [q_j,p_k]=i delta_jk",
      "stiffness": "K_jk=h^-2(2 delta_jk-delta_(j,k+1)-delta_(j,k-1))",
      "vacuum_covariance": "X=K^(-1/2)/2 and P=K^(1/2)/2",
      "coherent_displacement": "d_q,j=sqrt(h) F(x_j-center), d_p,j=0",
      "interval_site_rule": "retain oscillator centers satisfying |x_j-center|<L; endpoints are excluded",
      "relative_entropy_evaluation": "Williamson-diagonalize diag(X_A,P_A); because both states have the same covariance, D=1/2 d_A^T G_A d_A and Delta S_A=0 exactly",
      "grid_phase_convention": "The chain and Dirichlet box remain fixed; center=(grid phase)h shifts the interval and bump relative to the lattice. This also makes the two distances to the finite-box walls differ by O(h)."
    }
  },
  "sources_and_exact_locators": [
    {
      "citation": "H. Araki, Relative Entropy of States of von Neumann Algebras, Publ. RIMS Kyoto Univ. 11 (1976) 809-833",
      "doi": "https://doi.org/10.2977/prims/1195191148",
      "locators": "Eqs. (1.1)-(1.2) for orientation and the finite-dimensional reduction; Section 2 for positivity and lower semicontinuity in the faithful case"
    },
    {
      "citation": "H. Araki, Relative Entropy of States of von Neumann Algebras II, Publ. RIMS Kyoto Univ. 13 (1977) 173-192",
      "doi": "https://doi.org/10.2977/PRIMS/1195190105",
      "locators": "Definition 3.1 for the support condition and +infinity branch; Remarks 3.4-3.5 for the relative-modular expression and faithful compression"
    },
    {
      "citation": "D. Bostelmann, D. Cadamuro, and S. Del Vecchio, Relative entropy of coherent states on general CCR algebras, Commun. Math. Phys. 389 (2022) 661-691",
      "doi": "https://doi.org/10.1007/s00220-021-04249-x",
      "locators": "Theorem 2.13 and Eq. (2.32) for coherent-state relative entropy and its form domain; Appendix Eq. (A.1) for support orientation"
    },
    {
      "citation": "A. Garbarz and G. Palau, Relative Entropy of an Interval for a Massless Boson at Finite Temperature, Phys. Rev. D 107 (2023) 125016",
      "doi": "https://doi.org/10.1103/PhysRevD.107.125016",
      "locators": "Eqs. (72)-(73) for the coherent-data setup; Eq. (91), p. 125016-9, for T_00=(f_prime^2+g^2)/2; and Section IV.B.2, Eq. (95), p. 125016-10, for the vacuum bounded-interval relative entropy"
    },
    {
      "citation": "M. M. Wilde, M. Tomamichel, S. Lloyd, and M. Berta, Gaussian Hypothesis Testing and Quantum Illumination, Phys. Rev. Lett. 119 (2017) 120501",
      "doi": "https://doi.org/10.1103/PhysRevLett.119.120501",
      "locators": "Eqs. (3)-(6) for quadratures, covariance, and the Gibbs matrix; Eq. (9) for Gaussian relative entropy"
    }
  ],
  "analytic_benchmark": {
    "canonical_target_decimal_90_digit_working_precision": "8.4159644887460075759108223396229092656000419590061841955637452824658996437093613607277615587825700171650928951722986",
    "canonical_target_nats_binary64": 8.415964488746008,
    "canonical_target_method": "Independent 90-digit Wolfram Language NIntegrate with WorkingPrecision=90 and AccuracyGoal=PrecisionGoal=75; the generator checks it with two native-JavaScript quadratures.",
    "canonical_target_reproduction_command": "wolframscript -code 'NIntegrate[2 Pi ((1-x^2)/2) (1/2) ((-2 x (1/2)^2/(((1/2)^2-x^2)^2)) Exp[-x^2/((1/2)^2-x^2)])^2,{x,-1/2,1/2},WorkingPrecision->90,AccuracyGoal->75,PrecisionGoal->75,Method->{\"GlobalAdaptive\",\"SymbolicProcessing\"->0}]'",
    "adaptive_simpson_nats": 8.415964488746006,
    "composite_simpson_32768_panels_nats": 8.415964488746054,
    "composite_simpson_65536_panels_nats": 8.415964488745846,
    "adaptive_difference_from_canonical_nats": -1.7763568394002505e-15,
    "fine_composite_difference_from_canonical_nats": -1.616484723854228e-13,
    "composite_refinement_difference_nats": -2.078337502098293e-13
  },
  "controls": {
    "varied_lattice_spacings_h": [
      0.125,
      0.08333333333333333,
      0.0625
    ],
    "varied_lattice_spacings_exact": [
      "R/8",
      "R/12",
      "R/16"
    ],
    "sites_per_R": [
      8,
      12,
      16
    ],
    "varied_box_half_lengths_B": [
      4,
      6,
      8
    ],
    "varied_grid_phases_in_units_of_h": [
      0,
      0.5
    ],
    "symplectic_gap_floors": [
      1e-8,
      1e-10,
      1e-12
    ],
    "preferred_symplectic_gap_floor": 1e-10,
    "preferred_finite_regulator_point": {
      "box_half_length_B": 8,
      "lattice_spacing_h": 0.0625,
      "grid_phase_in_units_of_h": 0
    },
    "precision_and_conditioning": "Raw covariance sums and Jacobi diagonalizations use binary64. Only arccoth(2 nu) is evaluated at max(nu,1/2+epsilon); epsilon is a numerical conditioner, not a physical regulator. Every row reports its clamped count and epsilon sweep."
  },
  "empirical_cross_regulator_convergence_not_DPI": {
    "interpretation": "Rows with different h, B, or grid phase belong to different finite systems. Their approach to the continuum target is empirical convergence, not data-processing monotonicity.",
    "rows": [
      {
        "box_half_length_B": 4,
        "box_half_length_in_units_of_R": 4,
        "lattice_spacing_h": 0.125,
        "lattice_spacing_exact": "R/8",
        "sites_per_R": 8,
        "grid_phase_in_units_of_h": 0,
        "interval_center": 0,
        "global_chain_site_count": 63,
        "interval_site_count": 15,
        "relative_entropy_nats": 7.889235754390109,
        "signed_difference_from_analytic_target_nats": -0.5267287343558991,
        "absolute_difference_from_analytic_target_nats": 0.5267287343558991,
        "numerical_conditioner_sweep": [
          {
            "relative_entropy_nats": 7.634205518002998,
            "symplectic_gap_floor": 1e-8,
            "clamped_mode_count": 8
          },
          {
            "relative_entropy_nats": 7.889235754390109,
            "symplectic_gap_floor": 1e-10,
            "clamped_mode_count": 7
          },
          {
            "relative_entropy_nats": 7.957635917561133,
            "symplectic_gap_floor": 1e-12,
            "clamped_mode_count": 5
          }
        ],
        "numerical_conditioner_spread_nats": 0.32343039955813424,
        "diagnostics": {
          "interval_site_count": 15,
          "first_global_site_index_zero_based": 24,
          "last_global_site_index_zero_based": 38,
          "raw_minimum_symplectic_eigenvalue": 0.49999999999996886,
          "raw_uncertainty_violation_below_one_half": 3.114175584073564e-14,
          "position_covariance_minimum_eigenvalue": 0.03141492273200165,
          "position_eigensolver_sweeps": 6,
          "xp_eigensolver_sweeps": 7,
          "maximum_eigensolver_reconstruction_residual": 2.7144952952085077e-14,
          "maximum_eigenvector_orthogonality_residual": 3.552713678800501e-15,
          "momentum_williamson_reconstruction_residual_max_abs": 6.483702463810914e-13
        }
      },
      {
        "box_half_length_B": 4,
        "box_half_length_in_units_of_R": 4,
        "lattice_spacing_h": 0.08333333333333333,
        "lattice_spacing_exact": "R/12",
        "sites_per_R": 12,
        "grid_phase_in_units_of_h": 0,
        "interval_center": 0,
        "global_chain_site_count": 95,
        "interval_site_count": 23,
        "relative_entropy_nats": 8.553542517662697,
        "signed_difference_from_analytic_target_nats": 0.13757802891668902,
        "absolute_difference_from_analytic_target_nats": 0.13757802891668902,
        "numerical_conditioner_sweep": [
          {
            "relative_entropy_nats": 8.15275693989563,
            "symplectic_gap_floor": 1e-8,
            "clamped_mode_count": 15
          },
          {
            "relative_entropy_nats": 8.553542517662697,
            "symplectic_gap_floor": 1e-10,
            "clamped_mode_count": 13
          },
          {
            "relative_entropy_nats": 8.715045160367385,
            "symplectic_gap_floor": 1e-12,
            "clamped_mode_count": 12
          }
        ],
        "numerical_conditioner_spread_nats": 0.5622882204717552,
        "diagnostics": {
          "interval_site_count": 23,
          "first_global_site_index_zero_based": 36,
          "last_global_site_index_zero_based": 58,
          "raw_minimum_symplectic_eigenvalue": 0.49999999999964284,
          "raw_uncertainty_violation_below_one_half": 3.5715874702191286e-13,
          "position_covariance_minimum_eigenvalue": 0.020880679135359282,
          "position_eigensolver_sweeps": 7,
          "xp_eigensolver_sweeps": 7,
          "maximum_eigensolver_reconstruction_residual": 3.38149647172159e-14,
          "maximum_eigenvector_orthogonality_residual": 6.439293542825908e-15,
          "momentum_williamson_reconstruction_residual_max_abs": 3.750777466393629e-12
        }
      },
      {
        "box_half_length_B": 4,
        "box_half_length_in_units_of_R": 4,
        "lattice_spacing_h": 0.0625,
        "lattice_spacing_exact": "R/16",
        "sites_per_R": 16,
        "grid_phase_in_units_of_h": 0,
        "interval_center": 0,
        "global_chain_site_count": 127,
        "interval_site_count": 31,
        "relative_entropy_nats": 8.77547554141513,
        "signed_difference_from_analytic_target_nats": 0.35951105266912187,
        "absolute_difference_from_analytic_target_nats": 0.35951105266912187,
        "numerical_conditioner_sweep": [
          {
            "relative_entropy_nats": 8.4056613042549,
            "symplectic_gap_floor": 1e-8,
            "clamped_mode_count": 23
          },
          {
            "relative_entropy_nats": 8.77547554141513,
            "symplectic_gap_floor": 1e-10,
            "clamped_mode_count": 21
          },
          {
            "relative_entropy_nats": 8.954827865483606,
            "symplectic_gap_floor": 1e-12,
            "clamped_mode_count": 19
          }
        ],
        "numerical_conditioner_spread_nats": 0.5491665612287058,
        "diagnostics": {
          "interval_site_count": 31,
          "first_global_site_index_zero_based": 48,
          "last_global_site_index_zero_based": 78,
          "raw_minimum_symplectic_eigenvalue": 0.4999999999994499,
          "raw_uncertainty_violation_below_one_half": 5.501155087017651e-13,
          "position_covariance_minimum_eigenvalue": 0.015644670928486294,
          "position_eigensolver_sweeps": 8,
          "xp_eigensolver_sweeps": 8,
          "maximum_eigensolver_reconstruction_residual": 4.193694003173931e-14,
          "maximum_eigenvector_orthogonality_residual": 8.881784197001252e-15,
          "momentum_williamson_reconstruction_residual_max_abs": 1.0414780149403668e-11
        }
      },
      {
        "box_half_length_B": 4,
        "box_half_length_in_units_of_R": 4,
        "lattice_spacing_h": 0.125,
        "lattice_spacing_exact": "R/8",
        "sites_per_R": 8,
        "grid_phase_in_units_of_h": 0.5,
        "interval_center": 0.0625,
        "global_chain_site_count": 63,
        "interval_site_count": 16,
        "relative_entropy_nats": 9.20224532603609,
        "signed_difference_from_analytic_target_nats": 0.7862808372900822,
        "absolute_difference_from_analytic_target_nats": 0.7862808372900822,
        "numerical_conditioner_sweep": [
          {
            "relative_entropy_nats": 8.63329820218096,
            "symplectic_gap_floor": 1e-8,
            "clamped_mode_count": 9
          },
          {
            "relative_entropy_nats": 9.20224532603609,
            "symplectic_gap_floor": 1e-10,
            "clamped_mode_count": 8
          },
          {
            "relative_entropy_nats": 9.348628750859008,
            "symplectic_gap_floor": 1e-12,
            "clamped_mode_count": 6
          }
        ],
        "numerical_conditioner_spread_nats": 0.7153305486780486,
        "diagnostics": {
          "interval_site_count": 16,
          "first_global_site_index_zero_based": 24,
          "last_global_site_index_zero_based": 39,
          "raw_minimum_symplectic_eigenvalue": 0.4999999999998051,
          "raw_uncertainty_violation_below_one_half": 1.9489965197294623e-13,
          "position_covariance_minimum_eigenvalue": 0.031395147246969665,
          "position_eigensolver_sweeps": 6,
          "xp_eigensolver_sweeps": 7,
          "maximum_eigensolver_reconstruction_residual": 2.609024107869118e-14,
          "maximum_eigenvector_orthogonality_residual": 4.440892098500626e-15,
          "momentum_williamson_reconstruction_residual_max_abs": 2.5552893134772603e-12
        }
      },
      {
        "box_half_length_B": 4,
        "box_half_length_in_units_of_R": 4,
        "lattice_spacing_h": 0.08333333333333333,
        "lattice_spacing_exact": "R/12",
        "sites_per_R": 12,
        "grid_phase_in_units_of_h": 0.5,
        "interval_center": 0.041666666666666664,
        "global_chain_site_count": 95,
        "interval_site_count": 24,
        "relative_entropy_nats": 9.011888718974483,
        "signed_difference_from_analytic_target_nats": 0.5959242302284746,
        "absolute_difference_from_analytic_target_nats": 0.5959242302284746,
        "numerical_conditioner_sweep": [
          {
            "relative_entropy_nats": 8.518995996758779,
            "symplectic_gap_floor": 1e-8,
            "clamped_mode_count": 16
          },
          {
            "relative_entropy_nats": 9.011888718974483,
            "symplectic_gap_floor": 1e-10,
            "clamped_mode_count": 14
          },
          {
            "relative_entropy_nats": 9.22726764195431,
            "symplectic_gap_floor": 1e-12,
            "clamped_mode_count": 13
          }
        ],
        "numerical_conditioner_spread_nats": 0.7082716451955307,
        "diagnostics": {
          "interval_site_count": 24,
          "first_global_site_index_zero_based": 36,
          "last_global_site_index_zero_based": 59,
          "raw_minimum_symplectic_eigenvalue": 0.49999999999966976,
          "raw_uncertainty_violation_below_one_half": 3.302358386747528e-13,
          "position_covariance_minimum_eigenvalue": 0.020876844052317628,
          "position_eigensolver_sweeps": 7,
          "xp_eigensolver_sweeps": 7,
          "maximum_eigensolver_reconstruction_residual": 3.727226860483768e-14,
          "maximum_eigenvector_orthogonality_residual": 6.8833827526759706e-15,
          "momentum_williamson_reconstruction_residual_max_abs": 4.842348744205083e-12
        }
      },
      {
        "box_half_length_B": 4,
        "box_half_length_in_units_of_R": 4,
        "lattice_spacing_h": 0.0625,
        "lattice_spacing_exact": "R/16",
        "sites_per_R": 16,
        "grid_phase_in_units_of_h": 0.5,
        "interval_center": 0.03125,
        "global_chain_site_count": 127,
        "interval_site_count": 32,
        "relative_entropy_nats": 9.033558808812995,
        "signed_difference_from_analytic_target_nats": 0.6175943200669867,
        "absolute_difference_from_analytic_target_nats": 0.6175943200669867,
        "numerical_conditioner_sweep": [
          {
            "relative_entropy_nats": 8.605739446252496,
            "symplectic_gap_floor": 1e-8,
            "clamped_mode_count": 24
          },
          {
            "relative_entropy_nats": 9.033558808812995,
            "symplectic_gap_floor": 1e-10,
            "clamped_mode_count": 22
          },
          {
            "relative_entropy_nats": 9.232152391170796,
            "symplectic_gap_floor": 1e-12,
            "clamped_mode_count": 20
          }
        ],
        "numerical_conditioner_spread_nats": 0.6264129449183002,
        "diagnostics": {
          "interval_site_count": 32,
          "first_global_site_index_zero_based": 48,
          "last_global_site_index_zero_based": 79,
          "raw_minimum_symplectic_eigenvalue": 0.4999999999994738,
          "raw_uncertainty_violation_below_one_half": 5.261902025210929e-13,
          "position_covariance_minimum_eigenvalue": 0.015643467676252076,
          "position_eigensolver_sweeps": 8,
          "xp_eigensolver_sweeps": 9,
          "maximum_eigensolver_reconstruction_residual": 5.062616992290714e-14,
          "maximum_eigenvector_orthogonality_residual": 1.021405182655144e-14,
          "momentum_williamson_reconstruction_residual_max_abs": 8.494538406011998e-12
        }
      },
      {
        "box_half_length_B": 6,
        "box_half_length_in_units_of_R": 6,
        "lattice_spacing_h": 0.125,
        "lattice_spacing_exact": "R/8",
        "sites_per_R": 8,
        "grid_phase_in_units_of_h": 0,
        "interval_center": 0,
        "global_chain_site_count": 95,
        "interval_site_count": 15,
        "relative_entropy_nats": 7.4784390804323495,
        "signed_difference_from_analytic_target_nats": -0.9375254083136584,
        "absolute_difference_from_analytic_target_nats": 0.9375254083136584,
        "numerical_conditioner_sweep": [
          {
            "relative_entropy_nats": 7.24902531256066,
            "symplectic_gap_floor": 1e-8,
            "clamped_mode_count": 8
          },
          {
            "relative_entropy_nats": 7.4784390804323495,
            "symplectic_gap_floor": 1e-10,
            "clamped_mode_count": 6
          },
          {
            "relative_entropy_nats": 7.524178540717946,
            "symplectic_gap_floor": 1e-12,
            "clamped_mode_count": 5
          }
        ],
        "numerical_conditioner_spread_nats": 0.2751532281572864,
        "diagnostics": {
          "interval_site_count": 15,
          "first_global_site_index_zero_based": 40,
          "last_global_site_index_zero_based": 54,
          "raw_minimum_symplectic_eigenvalue": 0.49999999999996675,
          "raw_uncertainty_violation_below_one_half": 3.325117958752344e-14,
          "position_covariance_minimum_eigenvalue": 0.03141502380340096,
          "position_eigensolver_sweeps": 6,
          "xp_eigensolver_sweeps": 7,
          "maximum_eigensolver_reconstruction_residual": 3.758104938356155e-14,
          "maximum_eigenvector_orthogonality_residual": 3.1086244689504383e-15,
          "momentum_williamson_reconstruction_residual_max_abs": 1.0897949209720537e-12
        }
      },
      {
        "box_half_length_B": 6,
        "box_half_length_in_units_of_R": 6,
        "lattice_spacing_h": 0.08333333333333333,
        "lattice_spacing_exact": "R/12",
        "sites_per_R": 12,
        "grid_phase_in_units_of_h": 0,
        "interval_center": 0,
        "global_chain_site_count": 143,
        "interval_site_count": 23,
        "relative_entropy_nats": 8.107781473047583,
        "signed_difference_from_analytic_target_nats": -0.3081830156984253,
        "absolute_difference_from_analytic_target_nats": 0.3081830156984253,
        "numerical_conditioner_sweep": [
          {
            "relative_entropy_nats": 7.794504477686766,
            "symplectic_gap_floor": 1e-8,
            "clamped_mode_count": 15
          },
          {
            "relative_entropy_nats": 8.107781473047583,
            "symplectic_gap_floor": 1e-10,
            "clamped_mode_count": 13
          },
          {
            "relative_entropy_nats": 8.24999491506796,
            "symplectic_gap_floor": 1e-12,
            "clamped_mode_count": 12
          }
        ],
        "numerical_conditioner_spread_nats": 0.45549043738119455,
        "diagnostics": {
          "interval_site_count": 23,
          "first_global_site_index_zero_based": 60,
          "last_global_site_index_zero_based": 82,
          "raw_minimum_symplectic_eigenvalue": 0.49999999999968575,
          "raw_uncertainty_violation_below_one_half": 3.1424862712015056e-13,
          "position_covariance_minimum_eigenvalue": 0.020880691748795498,
          "position_eigensolver_sweeps": 7,
          "xp_eigensolver_sweeps": 7,
          "maximum_eigensolver_reconstruction_residual": 4.172530376767014e-14,
          "maximum_eigenvector_orthogonality_residual": 6.217248937900877e-15,
          "momentum_williamson_reconstruction_residual_max_abs": 3.7765346405649325e-12
        }
      },
      {
        "box_half_length_B": 6,
        "box_half_length_in_units_of_R": 6,
        "lattice_spacing_h": 0.0625,
        "lattice_spacing_exact": "R/16",
        "sites_per_R": 16,
        "grid_phase_in_units_of_h": 0,
        "interval_center": 0,
        "global_chain_site_count": 191,
        "interval_site_count": 31,
        "relative_entropy_nats": 8.334137671932519,
        "signed_difference_from_analytic_target_nats": -0.08182681681348924,
        "absolute_difference_from_analytic_target_nats": 0.08182681681348924,
        "numerical_conditioner_sweep": [
          {
            "relative_entropy_nats": 8.032496883172538,
            "symplectic_gap_floor": 1e-8,
            "clamped_mode_count": 22
          },
          {
            "relative_entropy_nats": 8.334137671932519,
            "symplectic_gap_floor": 1e-10,
            "clamped_mode_count": 21
          },
          {
            "relative_entropy_nats": 8.479104019152137,
            "symplectic_gap_floor": 1e-12,
            "clamped_mode_count": 19
          }
        ],
        "numerical_conditioner_spread_nats": 0.44660713597959933,
        "diagnostics": {
          "interval_site_count": 31,
          "first_global_site_index_zero_based": 80,
          "last_global_site_index_zero_based": 110,
          "raw_minimum_symplectic_eigenvalue": 0.49999999999941047,
          "raw_uncertainty_violation_below_one_half": 5.895284260759581e-13,
          "position_covariance_minimum_eigenvalue": 0.01564467384586425,
          "position_eigensolver_sweeps": 8,
          "xp_eigensolver_sweeps": 11,
          "maximum_eigensolver_reconstruction_residual": 5.5566662382489085e-14,
          "maximum_eigenvector_orthogonality_residual": 8.659739592076221e-15,
          "momentum_williamson_reconstruction_residual_max_abs": 1.1707967928487051e-11
        }
      },
      {
        "box_half_length_B": 6,
        "box_half_length_in_units_of_R": 6,
        "lattice_spacing_h": 0.125,
        "lattice_spacing_exact": "R/8",
        "sites_per_R": 8,
        "grid_phase_in_units_of_h": 0.5,
        "interval_center": 0.0625,
        "global_chain_site_count": 95,
        "interval_site_count": 16,
        "relative_entropy_nats": 8.71046903772945,
        "signed_difference_from_analytic_target_nats": 0.2945045489834417,
        "absolute_difference_from_analytic_target_nats": 0.2945045489834417,
        "numerical_conditioner_sweep": [
          {
            "relative_entropy_nats": 8.230117394740857,
            "symplectic_gap_floor": 1e-8,
            "clamped_mode_count": 8
          },
          {
            "relative_entropy_nats": 8.71046903772945,
            "symplectic_gap_floor": 1e-10,
            "clamped_mode_count": 7
          },
          {
            "relative_entropy_nats": 8.84253311300766,
            "symplectic_gap_floor": 1e-12,
            "clamped_mode_count": 6
          }
        ],
        "numerical_conditioner_spread_nats": 0.6124157182668029,
        "diagnostics": {
          "interval_site_count": 16,
          "first_global_site_index_zero_based": 40,
          "last_global_site_index_zero_based": 55,
          "raw_minimum_symplectic_eigenvalue": 0.4999999999997799,
          "raw_uncertainty_violation_below_one_half": 2.2010171463193728e-13,
          "position_covariance_minimum_eigenvalue": 0.03139518696145502,
          "position_eigensolver_sweeps": 6,
          "xp_eigensolver_sweeps": 6,
          "maximum_eigensolver_reconstruction_residual": 4.0023540037736893e-14,
          "maximum_eigenvector_orthogonality_residual": 3.552713678800501e-15,
          "momentum_williamson_reconstruction_residual_max_abs": 2.4069635173873394e-12
        }
      },
      {
        "box_half_length_B": 6,
        "box_half_length_in_units_of_R": 6,
        "lattice_spacing_h": 0.08333333333333333,
        "lattice_spacing_exact": "R/12",
        "sites_per_R": 12,
        "grid_phase_in_units_of_h": 0.5,
        "interval_center": 0.041666666666666664,
        "global_chain_site_count": 143,
        "interval_site_count": 24,
        "relative_entropy_nats": 8.538072751456072,
        "signed_difference_from_analytic_target_nats": 0.12210826271006425,
        "absolute_difference_from_analytic_target_nats": 0.12210826271006425,
        "numerical_conditioner_sweep": [
          {
            "relative_entropy_nats": 8.154048524135785,
            "symplectic_gap_floor": 1e-8,
            "clamped_mode_count": 16
          },
          {
            "relative_entropy_nats": 8.538072751456072,
            "symplectic_gap_floor": 1e-10,
            "clamped_mode_count": 14
          },
          {
            "relative_entropy_nats": 8.718014069570225,
            "symplectic_gap_floor": 1e-12,
            "clamped_mode_count": 13
          }
        ],
        "numerical_conditioner_spread_nats": 0.5639655454344403,
        "diagnostics": {
          "interval_site_count": 24,
          "first_global_site_index_zero_based": 60,
          "last_global_site_index_zero_based": 83,
          "raw_minimum_symplectic_eigenvalue": 0.4999999999996797,
          "raw_uncertainty_violation_below_one_half": 3.2029934260435766e-13,
          "position_covariance_minimum_eigenvalue": 0.020876849162237577,
          "position_eigensolver_sweeps": 7,
          "xp_eigensolver_sweeps": 8,
          "maximum_eigensolver_reconstruction_residual": 5.934142066621462e-14,
          "maximum_eigenvector_orthogonality_residual": 6.439293542825908e-15,
          "momentum_williamson_reconstruction_residual_max_abs": 5.0679460628089146e-12
        }
      },
      {
        "box_half_length_B": 6,
        "box_half_length_in_units_of_R": 6,
        "lattice_spacing_h": 0.0625,
        "lattice_spacing_exact": "R/16",
        "sites_per_R": 16,
        "grid_phase_in_units_of_h": 0.5,
        "interval_center": 0.03125,
        "global_chain_site_count": 191,
        "interval_site_count": 32,
        "relative_entropy_nats": 8.575461141744524,
        "signed_difference_from_analytic_target_nats": 0.15949665299851645,
        "absolute_difference_from_analytic_target_nats": 0.15949665299851645,
        "numerical_conditioner_sweep": [
          {
            "relative_entropy_nats": 8.209619372793123,
            "symplectic_gap_floor": 1e-8,
            "clamped_mode_count": 23
          },
          {
            "relative_entropy_nats": 8.575461141744524,
            "symplectic_gap_floor": 1e-10,
            "clamped_mode_count": 22
          },
          {
            "relative_entropy_nats": 8.727375427666288,
            "symplectic_gap_floor": 1e-12,
            "clamped_mode_count": 20
          }
        ],
        "numerical_conditioner_spread_nats": 0.5177560548731659,
        "diagnostics": {
          "interval_site_count": 32,
          "first_global_site_index_zero_based": 80,
          "last_global_site_index_zero_based": 111,
          "raw_minimum_symplectic_eigenvalue": 0.4999999999994417,
          "raw_uncertainty_violation_below_one_half": 5.5827564793276e-13,
          "position_covariance_minimum_eigenvalue": 0.015643468874112425,
          "position_eigensolver_sweeps": 8,
          "xp_eigensolver_sweeps": 9,
          "maximum_eigensolver_reconstruction_residual": 4.8405723873656825e-14,
          "maximum_eigenvector_orthogonality_residual": 1.0658141036401503e-14,
          "momentum_williamson_reconstruction_residual_max_abs": 9.341860618405917e-12
        }
      },
      {
        "box_half_length_B": 8,
        "box_half_length_in_units_of_R": 8,
        "lattice_spacing_h": 0.125,
        "lattice_spacing_exact": "R/8",
        "sites_per_R": 8,
        "grid_phase_in_units_of_h": 0,
        "interval_center": 0,
        "global_chain_site_count": 127,
        "interval_site_count": 15,
        "relative_entropy_nats": 7.291650562500557,
        "signed_difference_from_analytic_target_nats": -1.124313926245451,
        "absolute_difference_from_analytic_target_nats": 1.124313926245451,
        "numerical_conditioner_sweep": [
          {
            "relative_entropy_nats": 7.080464856819262,
            "symplectic_gap_floor": 1e-8,
            "clamped_mode_count": 8
          },
          {
            "relative_entropy_nats": 7.291650562500557,
            "symplectic_gap_floor": 1e-10,
            "clamped_mode_count": 6
          },
          {
            "relative_entropy_nats": 7.338428951710041,
            "symplectic_gap_floor": 1e-12,
            "clamped_mode_count": 5
          }
        ],
        "numerical_conditioner_spread_nats": 0.2579640948907791,
        "diagnostics": {
          "interval_site_count": 15,
          "first_global_site_index_zero_based": 56,
          "last_global_site_index_zero_based": 70,
          "raw_minimum_symplectic_eigenvalue": 0.4999999999999888,
          "raw_uncertainty_violation_below_one_half": 1.1213252548714081e-14,
          "position_covariance_minimum_eigenvalue": 0.03141507488477233,
          "position_eigensolver_sweeps": 6,
          "xp_eigensolver_sweeps": 6,
          "maximum_eigensolver_reconstruction_residual": 4.163336342344337e-14,
          "maximum_eigenvector_orthogonality_residual": 3.9968028886505635e-15,
          "momentum_williamson_reconstruction_residual_max_abs": 9.805489753489383e-13
        }
      },
      {
        "box_half_length_B": 8,
        "box_half_length_in_units_of_R": 8,
        "lattice_spacing_h": 0.08333333333333333,
        "lattice_spacing_exact": "R/12",
        "sites_per_R": 12,
        "grid_phase_in_units_of_h": 0,
        "interval_center": 0,
        "global_chain_site_count": 191,
        "interval_site_count": 23,
        "relative_entropy_nats": 7.9163546089756505,
        "signed_difference_from_analytic_target_nats": -0.4996098797703574,
        "absolute_difference_from_analytic_target_nats": 0.4996098797703574,
        "numerical_conditioner_sweep": [
          {
            "relative_entropy_nats": 7.635559299284075,
            "symplectic_gap_floor": 1e-8,
            "clamped_mode_count": 15
          },
          {
            "relative_entropy_nats": 7.9163546089756505,
            "symplectic_gap_floor": 1e-10,
            "clamped_mode_count": 13
          },
          {
            "relative_entropy_nats": 8.052849777703605,
            "symplectic_gap_floor": 1e-12,
            "clamped_mode_count": 12
          }
        ],
        "numerical_conditioner_spread_nats": 0.41729047841953015,
        "diagnostics": {
          "interval_site_count": 23,
          "first_global_site_index_zero_based": 84,
          "last_global_site_index_zero_based": 106,
          "raw_minimum_symplectic_eigenvalue": 0.4999999999996494,
          "raw_uncertainty_violation_below_one_half": 3.5060843117662444e-13,
          "position_covariance_minimum_eigenvalue": 0.02088069813597766,
          "position_eigensolver_sweeps": 7,
          "xp_eigensolver_sweeps": 7,
          "maximum_eigensolver_reconstruction_residual": 4.651834473179406e-14,
          "maximum_eigenvector_orthogonality_residual": 5.551115123125783e-15,
          "momentum_williamson_reconstruction_residual_max_abs": 3.715250329605624e-12
        }
      },
      {
        "box_half_length_B": 8,
        "box_half_length_in_units_of_R": 8,
        "lattice_spacing_h": 0.0625,
        "lattice_spacing_exact": "R/16",
        "sites_per_R": 16,
        "grid_phase_in_units_of_h": 0,
        "interval_center": 0,
        "global_chain_site_count": 255,
        "interval_site_count": 31,
        "relative_entropy_nats": 8.143612173659628,
        "signed_difference_from_analytic_target_nats": -0.2723523150863798,
        "absolute_difference_from_analytic_target_nats": 0.2723523150863798,
        "numerical_conditioner_sweep": [
          {
            "relative_entropy_nats": 7.852674569401324,
            "symplectic_gap_floor": 1e-8,
            "clamped_mode_count": 22
          },
          {
            "relative_entropy_nats": 8.143612173659628,
            "symplectic_gap_floor": 1e-10,
            "clamped_mode_count": 21
          },
          {
            "relative_entropy_nats": 8.277414692406403,
            "symplectic_gap_floor": 1e-12,
            "clamped_mode_count": 19
          }
        ],
        "numerical_conditioner_spread_nats": 0.4247401230050789,
        "diagnostics": {
          "interval_site_count": 31,
          "first_global_site_index_zero_based": 112,
          "last_global_site_index_zero_based": 142,
          "raw_minimum_symplectic_eigenvalue": 0.49999999999945666,
          "raw_uncertainty_violation_below_one_half": 5.433431482515516e-13,
          "position_covariance_minimum_eigenvalue": 0.01564467532508509,
          "position_eigensolver_sweeps": 8,
          "xp_eigensolver_sweeps": 9,
          "maximum_eigensolver_reconstruction_residual": 4.614711390793502e-14,
          "maximum_eigenvector_orthogonality_residual": 8.881784197001252e-15,
          "momentum_williamson_reconstruction_residual_max_abs": 9.15001407975069e-12
        }
      },
      {
        "box_half_length_B": 8,
        "box_half_length_in_units_of_R": 8,
        "lattice_spacing_h": 0.125,
        "lattice_spacing_exact": "R/8",
        "sites_per_R": 8,
        "grid_phase_in_units_of_h": 0.5,
        "interval_center": 0.0625,
        "global_chain_site_count": 127,
        "interval_site_count": 16,
        "relative_entropy_nats": 8.498558015060823,
        "signed_difference_from_analytic_target_nats": 0.08259352631481498,
        "absolute_difference_from_analytic_target_nats": 0.08259352631481498,
        "numerical_conditioner_sweep": [
          {
            "relative_entropy_nats": 8.055173515405073,
            "symplectic_gap_floor": 1e-8,
            "clamped_mode_count": 8
          },
          {
            "relative_entropy_nats": 8.498558015060823,
            "symplectic_gap_floor": 1e-10,
            "clamped_mode_count": 7
          },
          {
            "relative_entropy_nats": 8.630219148225793,
            "symplectic_gap_floor": 1e-12,
            "clamped_mode_count": 6
          }
        ],
        "numerical_conditioner_spread_nats": 0.5750456328207196,
        "diagnostics": {
          "interval_site_count": 16,
          "first_global_site_index_zero_based": 56,
          "last_global_site_index_zero_based": 71,
          "raw_minimum_symplectic_eigenvalue": 0.49999999999976713,
          "raw_uncertainty_violation_below_one_half": 2.328692794151266e-13,
          "position_covariance_minimum_eigenvalue": 0.031395200152999236,
          "position_eigensolver_sweeps": 6,
          "xp_eigensolver_sweeps": 6,
          "maximum_eigensolver_reconstruction_residual": 3.6914915568786455e-14,
          "maximum_eigenvector_orthogonality_residual": 4.440892098500626e-15,
          "momentum_williamson_reconstruction_residual_max_abs": 2.850164548817702e-12
        }
      },
      {
        "box_half_length_B": 8,
        "box_half_length_in_units_of_R": 8,
        "lattice_spacing_h": 0.08333333333333333,
        "lattice_spacing_exact": "R/12",
        "sites_per_R": 12,
        "grid_phase_in_units_of_h": 0.5,
        "interval_center": 0.041666666666666664,
        "global_chain_site_count": 191,
        "interval_site_count": 24,
        "relative_entropy_nats": 8.336284178238884,
        "signed_difference_from_analytic_target_nats": -0.07968031050712376,
        "absolute_difference_from_analytic_target_nats": 0.07968031050712376,
        "numerical_conditioner_sweep": [
          {
            "relative_entropy_nats": 7.992857132443856,
            "symplectic_gap_floor": 1e-8,
            "clamped_mode_count": 16
          },
          {
            "relative_entropy_nats": 8.336284178238884,
            "symplectic_gap_floor": 1e-10,
            "clamped_mode_count": 14
          },
          {
            "relative_entropy_nats": 8.504958536371594,
            "symplectic_gap_floor": 1e-12,
            "clamped_mode_count": 13
          }
        ],
        "numerical_conditioner_spread_nats": 0.5121014039277387,
        "diagnostics": {
          "interval_site_count": 24,
          "first_global_site_index_zero_based": 84,
          "last_global_site_index_zero_based": 107,
          "raw_minimum_symplectic_eigenvalue": 0.4999999999994655,
          "raw_uncertainty_violation_below_one_half": 5.345168752057816e-13,
          "position_covariance_minimum_eigenvalue": 0.02087685086329351,
          "position_eigensolver_sweeps": 7,
          "xp_eigensolver_sweeps": 7,
          "maximum_eigensolver_reconstruction_residual": 5.206945985491984e-14,
          "maximum_eigenvector_orthogonality_residual": 7.327471962526033e-15,
          "momentum_williamson_reconstruction_residual_max_abs": 5.6132876125047915e-12
        }
      },
      {
        "box_half_length_B": 8,
        "box_half_length_in_units_of_R": 8,
        "lattice_spacing_h": 0.0625,
        "lattice_spacing_exact": "R/16",
        "sites_per_R": 16,
        "grid_phase_in_units_of_h": 0.5,
        "interval_center": 0.03125,
        "global_chain_site_count": 255,
        "interval_site_count": 32,
        "relative_entropy_nats": 8.378717911062218,
        "signed_difference_from_analytic_target_nats": -0.03724657768379025,
        "absolute_difference_from_analytic_target_nats": 0.03724657768379025,
        "numerical_conditioner_sweep": [
          {
            "relative_entropy_nats": 8.02646118950526,
            "symplectic_gap_floor": 1e-8,
            "clamped_mode_count": 23
          },
          {
            "relative_entropy_nats": 8.378717911062218,
            "symplectic_gap_floor": 1e-10,
            "clamped_mode_count": 22
          },
          {
            "relative_entropy_nats": 8.515287196185366,
            "symplectic_gap_floor": 1e-12,
            "clamped_mode_count": 20
          }
        ],
        "numerical_conditioner_spread_nats": 0.4888260066801049,
        "diagnostics": {
          "interval_site_count": 32,
          "first_global_site_index_zero_based": 112,
          "last_global_site_index_zero_based": 143,
          "raw_minimum_symplectic_eigenvalue": 0.4999999999994157,
          "raw_uncertainty_violation_below_one_half": 5.843103778602199e-13,
          "position_covariance_minimum_eigenvalue": 0.0156434692733454,
          "position_eigensolver_sweeps": 8,
          "xp_eigensolver_sweeps": 10,
          "maximum_eigensolver_reconstruction_residual": 5.084821452783217e-14,
          "maximum_eigenvector_orthogonality_residual": 9.325873406851315e-15,
          "momentum_williamson_reconstruction_residual_max_abs": 1.2422063377925951e-11
        }
      }
    ],
    "two_ansatz_extrapolation_summaries": [
      {
        "box_half_length_B": 4,
        "grid_phase_in_units_of_h": 0,
        "input_sites_per_R": [
          8,
          12,
          16
        ],
        "constant_plus_h_fit": {
          "ansatz": "D_h = D_0 + c h^1",
          "intercept_D0_nats": 9.70895262467609,
          "slope_c": -14.431768838992017,
          "signed_intercept_difference_from_analytic_target_nats": 1.2929881359300825,
          "maximum_absolute_fit_residual_nats": 0.04723729623594153
        },
        "constant_plus_h_squared_fit": {
          "ansatz": "D_h = D_0 + c h^2",
          "intercept_D0_nats": 9.075257377993855,
          "slope_c": -75.82495426989185,
          "signed_intercept_difference_from_analytic_target_nats": 0.6592928892478476,
          "maximum_absolute_fit_residual_nats": 0.004847322098646245
        },
        "extrapolation_model_spread_nats": 0.633695246682235,
        "interpretation": "Empirical cross-regulator extrapolation only; neither fit is a monotonicity theorem or a rigorous error bound."
      },
      {
        "box_half_length_B": 4,
        "grid_phase_in_units_of_h": 0.5,
        "input_sites_per_R": [
          8,
          12,
          16
        ],
        "constant_plus_h_fit": {
          "ansatz": "D_h = D_0 + c h^1",
          "intercept_D0_nats": 8.814794408717336,
          "slope_c": 2.9660663175565367,
          "signed_intercept_difference_from_analytic_target_nats": 0.39882991997132855,
          "maximum_absolute_fit_residual_nats": 0.05007788287256609
        },
        "constant_plus_h_squared_fit": {
          "ansatz": "D_h = D_0 + c h^2",
          "intercept_D0_nats": 8.94085273222513,
          "slope_c": 16.057545083104998,
          "signed_intercept_difference_from_analytic_target_nats": 0.524888243479122,
          "maximum_absolute_fit_residual_nats": 0.04047474299443188
        },
        "extrapolation_model_spread_nats": 0.12605832350779345,
        "interpretation": "Empirical cross-regulator extrapolation only; neither fit is a monotonicity theorem or a rigorous error bound."
      },
      {
        "box_half_length_B": 6,
        "grid_phase_in_units_of_h": 0,
        "input_sites_per_R": [
          8,
          12,
          16
        ],
        "constant_plus_h_fit": {
          "ansatz": "D_h = D_0 + c h^1",
          "intercept_D0_nats": 9.227685548042407,
          "slope_c": -13.893040315254545,
          "signed_intercept_difference_from_analytic_target_nats": 0.8117210592963993,
          "maximum_absolute_fit_residual_nats": 0.03784928460972026
        },
        "constant_plus_h_squared_fit": {
          "ansatz": "D_h = D_0 + c h^2",
          "intercept_D0_nats": 8.616856079697538,
          "slope_c": -72.904981500313,
          "signed_intercept_difference_from_analytic_target_nats": 0.20089159095152986,
          "maximum_absolute_fit_residual_nats": 0.0027900128977815797
        },
        "extrapolation_model_spread_nats": 0.6108294683448694,
        "interpretation": "Empirical cross-regulator extrapolation only; neither fit is a monotonicity theorem or a rigorous error bound."
      },
      {
        "box_half_length_B": 6,
        "grid_phase_in_units_of_h": 0.5,
        "input_sites_per_R": [
          8,
          12,
          16
        ],
        "constant_plus_h_fit": {
          "ansatz": "D_h = D_0 + c h^1",
          "intercept_D0_nats": 8.387487588577397,
          "slope_c": 2.442609840730554,
          "signed_intercept_difference_from_analytic_target_nats": -0.028476900168611152,
          "maximum_absolute_fit_residual_nats": 0.05296565718220414
        },
        "constant_plus_h_squared_fit": {
          "ansatz": "D_h = D_0 + c h^2",
          "intercept_D0_nats": 8.490084556125659,
          "slope_c": 13.361283621676831,
          "signed_intercept_difference_from_analytic_target_nats": 0.07412006737965093,
          "maximum_absolute_fit_residual_nats": 0.04479849648678602
        },
        "extrapolation_model_spread_nats": 0.10259696754826209,
        "interpretation": "Empirical cross-regulator extrapolation only; neither fit is a monotonicity theorem or a rigorous error bound."
      },
      {
        "box_half_length_B": 8,
        "grid_phase_in_units_of_h": 0,
        "input_sites_per_R": [
          8,
          12,
          16
        ],
        "constant_plus_h_fit": {
          "ansatz": "D_h = D_0 + c h^1",
          "intercept_D0_nats": 9.032042838484514,
          "slope_c": -13.82588739809615,
          "signed_intercept_difference_from_analytic_target_nats": 0.6160783497385065,
          "maximum_absolute_fit_residual_nats": 0.03646905366581521
        },
        "constant_plus_h_squared_fit": {
          "ansatz": "D_h = D_0 + c h^2",
          "intercept_D0_nats": 8.42404185709357,
          "slope_c": -72.5385402137343,
          "signed_intercept_difference_from_analytic_target_nats": 0.008077368347562341,
          "maximum_absolute_fit_residual_nats": 0.003947385522542923
        },
        "extrapolation_model_spread_nats": 0.6080009813909442,
        "interpretation": "Empirical cross-regulator extrapolation only; neither fit is a monotonicity theorem or a rigorous error bound."
      },
      {
        "box_half_length_B": 8,
        "grid_phase_in_units_of_h": 0.5,
        "input_sites_per_R": [
          8,
          12,
          16
        ],
        "constant_plus_h_fit": {
          "ansatz": "D_h = D_0 + c h^1",
          "intercept_D0_nats": 8.205918956534626,
          "slope_c": 2.1998888667989474,
          "signed_intercept_difference_from_analytic_target_nats": -0.21004553221138167,
          "maximum_absolute_fit_residual_nats": 0.052958850528987966
        },
        "constant_plus_h_squared_fit": {
          "ansatz": "D_h = D_0 + c h^2",
          "intercept_D0_nats": 8.297776534987392,
          "slope_c": 12.095263452738072,
          "signed_intercept_difference_from_analytic_target_nats": -0.1181879537586159,
          "maximum_absolute_fit_residual_nats": 0.045487241836966064
        },
        "extrapolation_model_spread_nats": 0.09185757845276576,
        "interpretation": "Empirical cross-regulator extrapolation only; neither fit is a monotonicity theorem or a rigorous error bound."
      }
    ],
    "preferred_row": {
      "box_half_length_B": 8,
      "box_half_length_in_units_of_R": 8,
      "lattice_spacing_h": 0.0625,
      "lattice_spacing_exact": "R/16",
      "sites_per_R": 16,
      "grid_phase_in_units_of_h": 0,
      "interval_center": 0,
      "global_chain_site_count": 255,
      "interval_site_count": 31,
      "relative_entropy_nats": 8.143612173659628,
      "signed_difference_from_analytic_target_nats": -0.2723523150863798,
      "absolute_difference_from_analytic_target_nats": 0.2723523150863798,
      "numerical_conditioner_sweep": [
        {
          "relative_entropy_nats": 7.852674569401324,
          "symplectic_gap_floor": 1e-8,
          "clamped_mode_count": 22
        },
        {
          "relative_entropy_nats": 8.143612173659628,
          "symplectic_gap_floor": 1e-10,
          "clamped_mode_count": 21
        },
        {
          "relative_entropy_nats": 8.277414692406403,
          "symplectic_gap_floor": 1e-12,
          "clamped_mode_count": 19
        }
      ],
      "numerical_conditioner_spread_nats": 0.4247401230050789,
      "diagnostics": {
        "interval_site_count": 31,
        "first_global_site_index_zero_based": 112,
        "last_global_site_index_zero_based": 142,
        "raw_minimum_symplectic_eigenvalue": 0.49999999999945666,
        "raw_uncertainty_violation_below_one_half": 5.433431482515516e-13,
        "position_covariance_minimum_eigenvalue": 0.01564467532508509,
        "position_eigensolver_sweeps": 8,
        "xp_eigensolver_sweeps": 9,
        "maximum_eigensolver_reconstruction_residual": 4.614711390793502e-14,
        "maximum_eigenvector_orthogonality_residual": 8.881784197001252e-15,
        "momentum_williamson_reconstruction_residual_max_abs": 9.15001407975069e-12
      }
    },
    "box_size_control_at_h_one_sixteenth_phase_zero": {
      "rows": [
        {
          "box_half_length_B": 4,
          "relative_entropy_nats": 8.77547554141513
        },
        {
          "box_half_length_B": 6,
          "relative_entropy_nats": 8.334137671932519
        },
        {
          "box_half_length_B": 8,
          "relative_entropy_nats": 8.143612173659628
        }
      ],
      "spread_nats": 0.6318633677555017
    },
    "grid_phase_control_at_B_eight_h_one_sixteenth": {
      "rows": [
        {
          "grid_phase_in_units_of_h": 0,
          "interval_site_count": 31,
          "relative_entropy_nats": 8.143612173659628
        },
        {
          "grid_phase_in_units_of_h": 0.5,
          "interval_site_count": 32,
          "relative_entropy_nats": 8.378717911062218
        }
      ],
      "spread_nats": 0.23510573740258955
    }
  },
  "fixed_master_lattice_inclusion_DPI_test": {
    "status": "This is the actual data-processing/inclusion-monotonicity test.",
    "theorem_direction": "For A_L1 contained in A_L2, restriction from the larger algebra to the smaller algebra gives D_L1 <= D_L2.",
    "common_master_state": {
      "master_id": "dirichlet-B8-h1over16-phase0-vacuum-plus-fixed-bump",
      "box_half_length_B": 8,
      "lattice_spacing_h": 0.0625,
      "grid_phase_in_units_of_h": 0,
      "global_chain_site_count": 255,
      "covariance": "One materialized 255-site global vacuum X and P, analytically assembled once from the Dirichlet normal modes; every row is a literal principal submatrix",
      "displacement": "One global q_j=sqrt(h)F(x_j), p_j=0 array; every row is a literal site-index slice",
      "nesting_rule": "The listed site-index ranges are increasing centered subsets of this single master chain."
    },
    "rows": [
      {
        "interval_half_length_L": 0.25,
        "interval_site_count": 7,
        "first_global_site_index_zero_based": 124,
        "last_global_site_index_zero_based": 130,
        "relative_entropy_nats": 0.7458923206079616,
        "clamped_mode_count": 0,
        "displacement_slice_identity_residual": 0,
        "covariance_slice_identity_residual": 0,
        "raw_minimum_symplectic_eigenvalue": 0.5000000004788626
      },
      {
        "interval_half_length_L": 0.375,
        "interval_site_count": 11,
        "first_global_site_index_zero_based": 122,
        "last_global_site_index_zero_based": 132,
        "relative_entropy_nats": 1.1797733784639635,
        "clamped_mode_count": 3,
        "displacement_slice_identity_residual": 0,
        "covariance_slice_identity_residual": 0,
        "raw_minimum_symplectic_eigenvalue": 0.499999999999896
      },
      {
        "interval_half_length_L": 0.5,
        "interval_site_count": 15,
        "first_global_site_index_zero_based": 120,
        "last_global_site_index_zero_based": 134,
        "relative_entropy_nats": 2.7145984662575637,
        "clamped_mode_count": 6,
        "displacement_slice_identity_residual": 0,
        "covariance_slice_identity_residual": 0,
        "raw_minimum_symplectic_eigenvalue": 0.4999999999999113
      },
      {
        "interval_half_length_L": 0.625,
        "interval_site_count": 19,
        "first_global_site_index_zero_based": 118,
        "last_global_site_index_zero_based": 136,
        "relative_entropy_nats": 4.358232534885054,
        "clamped_mode_count": 10,
        "displacement_slice_identity_residual": 0,
        "covariance_slice_identity_residual": 0,
        "raw_minimum_symplectic_eigenvalue": 0.49999999999963146
      },
      {
        "interval_half_length_L": 0.75,
        "interval_site_count": 23,
        "first_global_site_index_zero_based": 116,
        "last_global_site_index_zero_based": 138,
        "relative_entropy_nats": 5.759384396385076,
        "clamped_mode_count": 13,
        "displacement_slice_identity_residual": 0,
        "covariance_slice_identity_residual": 0,
        "raw_minimum_symplectic_eigenvalue": 0.4999999999995952
      },
      {
        "interval_half_length_L": 0.875,
        "interval_site_count": 27,
        "first_global_site_index_zero_based": 114,
        "last_global_site_index_zero_based": 140,
        "relative_entropy_nats": 6.991721328289371,
        "clamped_mode_count": 17,
        "displacement_slice_identity_residual": 0,
        "covariance_slice_identity_residual": 0,
        "raw_minimum_symplectic_eigenvalue": 0.49999999999956585
      },
      {
        "interval_half_length_L": 1,
        "interval_site_count": 31,
        "first_global_site_index_zero_based": 112,
        "last_global_site_index_zero_based": 142,
        "relative_entropy_nats": 8.143612173659628,
        "clamped_mode_count": 21,
        "displacement_slice_identity_residual": 0,
        "covariance_slice_identity_residual": 0,
        "raw_minimum_symplectic_eigenvalue": 0.49999999999945666
      }
    ],
    "increments": [
      {
        "from_interval_half_length_L": 0.25,
        "to_interval_half_length_L": 0.375,
        "relative_entropy_increment_nats": 0.43388105785600184
      },
      {
        "from_interval_half_length_L": 0.375,
        "to_interval_half_length_L": 0.5,
        "relative_entropy_increment_nats": 1.5348250877936003
      },
      {
        "from_interval_half_length_L": 0.5,
        "to_interval_half_length_L": 0.625,
        "relative_entropy_increment_nats": 1.64363406862749
      },
      {
        "from_interval_half_length_L": 0.625,
        "to_interval_half_length_L": 0.75,
        "relative_entropy_increment_nats": 1.401151861500022
      },
      {
        "from_interval_half_length_L": 0.75,
        "to_interval_half_length_L": 0.875,
        "relative_entropy_increment_nats": 1.2323369319042952
      },
      {
        "from_interval_half_length_L": 0.875,
        "to_interval_half_length_L": 1,
        "relative_entropy_increment_nats": 1.1518908453702572
      }
    ],
    "minimum_increment_nats": 0.43388105785600184,
    "numerical_conditioner_robustness": [
      {
        "symplectic_gap_floor": 1e-8,
        "rows": [
          {
            "interval_half_length_L": 0.25,
            "relative_entropy_nats": 0.7458921552700368
          },
          {
            "interval_half_length_L": 0.375,
            "relative_entropy_nats": 1.1797733610806298
          },
          {
            "interval_half_length_L": 0.5,
            "relative_entropy_nats": 2.7145916285625984
          },
          {
            "interval_half_length_L": 0.625,
            "relative_entropy_nats": 4.337496292230515
          },
          {
            "interval_half_length_L": 0.75,
            "relative_entropy_nats": 5.672704776253964
          },
          {
            "interval_half_length_L": 0.875,
            "relative_entropy_nats": 6.866861386921988
          },
          {
            "interval_half_length_L": 1,
            "relative_entropy_nats": 7.852674569401324
          }
        ],
        "minimum_increment_nats": 0.433881205810593,
        "passes_nondecreasing_test": true
      },
      {
        "symplectic_gap_floor": 1e-10,
        "rows": [
          {
            "interval_half_length_L": 0.25,
            "relative_entropy_nats": 0.7458923206079616
          },
          {
            "interval_half_length_L": 0.375,
            "relative_entropy_nats": 1.1797733784639635
          },
          {
            "interval_half_length_L": 0.5,
            "relative_entropy_nats": 2.7145984662575637
          },
          {
            "interval_half_length_L": 0.625,
            "relative_entropy_nats": 4.358232534885054
          },
          {
            "interval_half_length_L": 0.75,
            "relative_entropy_nats": 5.759384396385076
          },
          {
            "interval_half_length_L": 0.875,
            "relative_entropy_nats": 6.991721328289371
          },
          {
            "interval_half_length_L": 1,
            "relative_entropy_nats": 8.143612173659628
          }
        ],
        "minimum_increment_nats": 0.43388105785600184,
        "passes_nondecreasing_test": true
      },
      {
        "symplectic_gap_floor": 1e-12,
        "rows": [
          {
            "interval_half_length_L": 0.25,
            "relative_entropy_nats": 0.7458923206079616
          },
          {
            "interval_half_length_L": 0.375,
            "relative_entropy_nats": 1.1797733869597071
          },
          {
            "interval_half_length_L": 0.5,
            "relative_entropy_nats": 2.714598647733988
          },
          {
            "interval_half_length_L": 0.625,
            "relative_entropy_nats": 4.3585967144390825
          },
          {
            "interval_half_length_L": 0.75,
            "relative_entropy_nats": 5.8144078561040775
          },
          {
            "interval_half_length_L": 0.875,
            "relative_entropy_nats": 7.096434512270574
          },
          {
            "interval_half_length_L": 1,
            "relative_entropy_nats": 8.277414692406403
          }
        ],
        "minimum_increment_nats": 0.4338810663517455,
        "passes_nondecreasing_test": true
      }
    ],
    "minimum_increment_across_all_conditioners_nats": 0.43388105785600184,
    "literal_site_subset_test_passes": true,
    "numerical_tolerance_nats": 2e-8,
    "passes_nondecreasing_test": true
  },
  "adversarial_validity_tests": {
    "mismatched_support_orientation": {
      "subject_rho_diagonal": [
        0.9,
        0.1
      ],
      "reference_sigma_diagonal": [
        1,
        0
      ],
      "reported_orientation": "D(rho||sigma)",
      "support_condition": "supp(rho) must be a subspace of supp(sigma)",
      "support_condition_passes": false,
      "result": "positive_infinity",
      "reason": "rho assigns probability 0.1 to the second basis vector while the reference sigma assigns it probability zero",
      "reversed_orientation": {
        "reported_orientation": "D(sigma||rho)",
        "support_condition_passes": true,
        "result_nats": 0.10536051565782628,
        "exact_result": "log(10/9)"
      },
      "illicit_projection_control": {
        "projected_and_renormalized_subject_diagonal": [
          1,
          0
        ],
        "relative_entropy_after_projection_nats": 0,
        "interpretation": "The apparent zero is not a regularization of D(rho||sigma); projection deleted the unsupported event and changed the state-comparison problem."
      },
      "passes": true
    },
    "mismatched_algebras": {
      "left_state_domain": "a two-outcome algebra C^2",
      "right_state_domain": "a three-outcome algebra C^3",
      "common_embedding_restriction_or_channel_supplied": false,
      "result": "undefined_without_a_common_algebra_or_specified_channel",
      "distinction_from_support_failure": "Support failure is a statement about two states on one algebra and yields +infinity in this orientation. Different domains do not define a relative entropy until both states are transported to one declared algebra.",
      "passes": true
    }
  },
  "independent_checks": {
    "analytic_quadrature_agreement": {
      "tolerance_nats": 2e-11,
      "maximum_difference_nats": 1.616484723854228e-13,
      "passes": true
    },
    "fixed_master_DPI_nondecreasing": {
      "minimum_increment_nats": 0.43388105785600184,
      "minimum_increment_across_all_conditioners_nats": 0.43388105785600184,
      "tolerance_nats": 2e-8,
      "passes": true
    },
    "fixed_master_literal_restrictions": {
      "maximum_covariance_slice_identity_residual": 0,
      "maximum_displacement_slice_identity_residual": 0,
      "literal_site_subset_test_passes": true,
      "tolerance": 1e-14,
      "passes": true
    },
    "adversarial_support_orientation": {
      "forward_result": "positive_infinity",
      "reverse_result_nats": 0.10536051565782628,
      "projected_problem_result_nats": 0,
      "passes": true
    },
    "mismatched_algebra_is_not_a_support_test": {
      "result": "undefined_without_a_common_algebra_or_specified_channel",
      "passes": true
    },
    "positivity": {
      "minimum_reported_relative_entropy_nats": 0.7458923206079616,
      "tolerance_nats": 1e-10,
      "passes": true
    },
    "amplitude_quadratic_scaling": {
      "rows": [
        {
          "peak_amplitude_A": 0.5,
          "relative_entropy_nats": 2.035903043414907,
          "expected_A_squared_times_unit_result_nats": 2.035903043414907,
          "scaling_residual_nats": 0
        },
        {
          "peak_amplitude_A": 1,
          "relative_entropy_nats": 8.143612173659628,
          "expected_A_squared_times_unit_result_nats": 8.143612173659628,
          "scaling_residual_nats": 0
        },
        {
          "peak_amplitude_A": 2,
          "relative_entropy_nats": 32.57444869463851,
          "expected_A_squared_times_unit_result_nats": 32.57444869463851,
          "scaling_residual_nats": 0
        }
      ],
      "maximum_absolute_residual_nats": 0,
      "tolerance_nats": 2e-12,
      "passes": true
    },
    "coherent_displacement_entropy_difference": {
      "delta_S_A": 0,
      "covariance_identity_residual": 0,
      "reason": "A Weyl displacement changes first moments but not covariance or the symplectic spectrum, so the two reduced Gaussian states have exactly equal von Neumann entropy.",
      "passes": true
    },
    "covariance_uncertainty_before_conditioning": {
      "maximum_raw_violation_below_one_half": 5.895284260759581e-13,
      "binary64_tolerance": 2e-11,
      "passes": true
    },
    "matrix_factorization": {
      "maximum_reported_residual": 1.2422063377925951e-11,
      "tolerance": 2e-10,
      "passes": true
    }
  },
  "limitations": [
    "The h-refinement, finite-box, and grid-phase tables are empirical regulator controls. They do not establish theorem monotonicity and their two extrapolation ansatzes are not rigorous error bars.",
    "The fixed-master nesting table is a genuine finite-dimensional DPI check, but its numerical arccoth conditioner can perturb nearly pure entanglement modes. The reported floor sweep and clamped-mode counts expose that limitation.",
    "Binary64 cannot resolve symplectic gaps exponentially close to one half. The preferred epsilon=1e-10 is a numerical stabilization choice, not a physical cutoff; continuum claims must remain within the reported conditioner sensitivity.",
    "Dirichlet walls regulate the massless scalar zero mode. The B sweep tests residual wall dependence but does not prove the infinite-volume limit.",
    "The grid-phase control shifts the interval and bump on one fixed box, so it combines site-centering effects with an O(h/B) left-right wall-distance asymmetry.",
    "The interval is represented by oscillator centers strictly inside its endpoints. No boundary-cell improvement or rigorous lattice-to-continuum theorem is supplied.",
    "Only time-symmetric coherent data G=0 are benchmarked. The exact continuum formula also permits smooth compactly supported momentum data G.",
    "Delta S=0 is checked structurally through identical covariance, not by subtracting two separately evaluated large entanglement entropies.",
    "The benchmark concerns normalized states on a common interval algebra. It does not assign a relative entropy to states on mismatched algebras and does not turn the support condition into a numerical projection rule."
  ],
  "strongest_supported_claim": "For the stated smooth coherent profile, the exact continuum interval formula evaluates numerically to approximately 8.4159644887460076 nats at 90-digit working precision; its binary64 rendering is 8.415964488746008 nats. The regulated harmonic-chain values provide reproducible empirical h/B/grid-phase comparisons, while only the fixed-master nested-site table tests finite-dimensional data-processing monotonicity."
}