{
  "schema_version": 1,
  "artifact_id": "qft.artifact.holography-quantum-gravity.quantum-cosmology-and-singularity-resolution-programs.de-sitter-bo-finite-start-control",
  "title": "De Sitter BO/WKB Overlap and Finite-Start Control",
  "source_revision": 2,
  "created_on": "2026-08-30",
  "creator": "OpenAI Codex, for QFT.org",
  "original_work": true,
  "reader_question": "Why is the formal Bunch-Davies initialization outside this Born-Oppenheimer/WKB domain, and can finite starts inside a declared overlap recover the projected BKK coefficient?",
  "takeaway": "For H0/mP = 1e-5, k/k0 = 1, x_i = |k eta_i| >= 10, and epsilon_WKB <= 1e-2, the finite initialization band is 10 <= x_i <= 321.8297948685. Four starts inside that band approach C_BKK = 0.9875896344 nonmonotonically, reaching a relative discrepancy of -3.65e-5 at x_i = 300; the formal x_i to infinity prescription is beyond the declared WKB threshold.",
  "scientific_status": "quantitative internal-control benchmark",
  "claim_ceiling": "The artifact checks the linearized, real-frequency, projected-Gaussian one-mode prescription and its finite-start sensitivity. It does not establish Gaussian closure, mixed-mode control, renormalized backreaction, an anomaly-free continuum constraint algebra, or an observable quantum-gravity signal.",
  "conventions": {
    "metric_signature": "(+---)",
    "planck_mass": "m_P^2 = 3/(4 pi G) = 6 M_Pl^2",
    "de_sitter_branch": "a(eta) = -1/(H0 eta), eta < 0",
    "finite_box": "eta, k, v_k, and H_k are dimensionless after the declared reference-length rescaling; k0 restores the reference comoving scale.",
    "horizontal_coordinate": "x_i = |k eta_i| is a finite initialization choice, not cosmological time evolving from left to right.",
    "integration_constant_scaling": "After writing deltaOmega = (H0^2/m_P^2) deltaOmega_bar for k/k0 = 1, the order-suppressed BKK constant and the stored dimensionless constant obey c_1 = (H0^2/m_P^2) cbar_1. Every generated value named cbar_1 is dimensionless.",
    "gamma_branch": "Gamma(0,z) uses the principal branch. As eta -> 0-, z = -2 i eta - 2 approaches the negative real axis from Im z > 0, so Gamma(0,-2+i0) = -Ei(2) - i pi; only its real part enters C_BKK."
  },
  "equations": {
    "uncorrected_width": "Omega_0 = k^3 eta^2/(1+k^2 eta^2) + i/[eta(1+k^2 eta^2)]",
    "real_corrected_frequency": "Re(omega_tilde^2) = k^2 - 2/eta^2 - [H0^2 eta^4/(2 m_P^2)] k^3(11-k^2 eta^2)/(1+k^2 eta^2)^3",
    "linearized_riccati": "i deltaOmega_prime = 2 Omega_0 deltaOmega - (Re(omega_tilde^2)-omega^2)",
    "exact_solution_k1": "deltaOmega_bar(eta) = A(eta)[cbar_1+B(eta)] after factoring H0^2/m_P^2 and setting k/k0 = 1.",
    "amplitude_A_k1": "A(eta) = -eta^2 exp(-2 i eta)/(eta+i)^2",
    "bracket_B_k1": "B(eta) = (1/4){9 e^-2 Gamma(0,-2 i eta-2) + 3 e^2 Gamma(0,2-2 i eta) - e^(2 i eta)[1+eta(eta+6 i)]/(eta-i)^2}",
    "finite_start_rule": "deltaOmega(eta_i) = H0^2/(4 k^2 m_P^2); after writing deltaOmega = (H0^2/m_P^2) deltaOmega_bar and setting k=1, cbar_1 = (1/4)/A(eta_i) - B(eta_i).",
    "dimensional_constant_relation_k1": "c_1 = (H0^2/m_P^2) cbar_1",
    "asymptotic_coefficient": "C_BKK = -(1/4)[1 + 3 e^2 Gamma(0,2) + 9 e^-2 Re Gamma(0,-2+i0)]",
    "finite_start_coefficient": "C_i = C_BKK - Re(cbar_1)",
    "wkb_diagnostic": "epsilon_WKB = 3(H0/m_P)^2 x_i^3/k^3"
  },
  "fixture": {
    "H0_over_mP": 0.00001,
    "k_over_k0": 1,
    "subhorizon_minimum_x_i": 10,
    "epsilon_WKB_tolerance": 0.01,
    "finite_window": [
      10,
      321.82979486854316
    ],
    "formal_asymptotic_state_inside_window": false,
    "final_x_for_independent_ode_integration": 0.0001,
    "production_integrator_limits": {
      "maximum_delta_x": 0.01,
      "maximum_delta_log_x": 0.002
    },
    "refined_integrator_limits": {
      "maximum_delta_x": 0.005,
      "maximum_delta_log_x": 0.001
    }
  },
  "analytic_constants": {
    "gamma_0_2": {
      "re": 0.048900510708060896,
      "im": 0
    },
    "gamma_0_negative_2_upper": {
      "re": -4.954234356001891,
      "im": -3.141592653589793
    },
    "coefficient_bracket": -3.9503585374459984,
    "C_BKK": 0.9875896343614996
  },
  "finite_starts": [
    {
      "x_i": 10,
      "eta_i": -10,
      "epsilon_WKB": 3.0000000000000004e-7,
      "cbar_1_real": -0.009782229696624956,
      "cbar_1_imag": 0.030370870256320348,
      "C_i": 0.9973718640581246,
      "relative_discrepancy": 0.00990515630811517,
      "relative_discrepancy_percent": 0.9905156308115171,
      "inside_declared_window": true
    },
    {
      "x_i": 30,
      "eta_i": -30,
      "epsilon_WKB": 0.000008100000000000002,
      "cbar_1_real": 0.0033904202105607684,
      "cbar_1_imag": -0.0012210256758117682,
      "C_i": 0.9841992141509388,
      "relative_discrepancy": -0.0034330253099029706,
      "relative_discrepancy_percent": -0.34330253099029706,
      "inside_declared_window": true
    },
    {
      "x_i": 100,
      "eta_i": -100,
      "epsilon_WKB": 0.00030000000000000003,
      "cbar_1_real": -0.00016135326379526638,
      "cbar_1_imag": -0.0002820467890257139,
      "C_i": 0.9877509876252949,
      "relative_discrepancy": 0.00016338088025769082,
      "relative_discrepancy_percent": 0.016338088025769082,
      "inside_declared_window": true
    },
    {
      "x_i": 300,
      "eta_i": -300,
      "epsilon_WKB": 0.008100000000000001,
      "cbar_1_real": 0.000036080588475095876,
      "cbar_1_imag": 0.0000014659320605667214,
      "C_i": 0.9875535537730245,
      "relative_discrepancy": -0.000036533988632347913,
      "relative_discrepancy_percent": -0.0036533988632347913,
      "inside_declared_window": true
    }
  ],
  "verification": {
    "gamma_algorithm": "Convergent principal-log series for |z| < 12 and the upper-incomplete-gamma continued fraction elsewhere; frozen 50-digit audit references are checked at generation time.",
    "exact_solution_ode_residual_samples": [
      {
        "eta": -100,
        "scaledResidual": 9.165465559164375e-17
      },
      {
        "eta": -10,
        "scaledResidual": 7.044344449510712e-17
      },
      {
        "eta": -1,
        "scaledResidual": 4.847302891456678e-16
      },
      {
        "eta": -0.1,
        "scaledResidual": 2.7755575615628914e-17
      },
      {
        "eta": -0.01,
        "scaledResidual": 4.9065389333867974e-18
      },
      {
        "eta": -0.001,
        "scaledResidual": 0
      }
    ],
    "independent_ode_integration": "Classical fourth-order Runge-Kutta in log x evolves the normalized linearized complex Riccati equation from each finite start to x = 1e-4. Production and half-step runs are compared with the analytic cbar_1 coefficient.",
    "acceptance_thresholds": {
      "special_function_and_overlap_reference_absolute_residual": 2e-13,
      "finite_start_reference_absolute_residual": 5e-13,
      "exact_solution_ode_scaled_residual": 2e-12,
      "independent_ode_coefficient_absolute_residual": 1e-9,
      "ode_step_halving_absolute_shift": 1e-9
    },
    "maxima": {
      "specialFunctionReferenceResidual": 1.1368683772161603e-13,
      "exactSolutionOdeScaledResidual": 4.847302891456678e-16,
      "finiteStartCoefficientReferenceResidual": 1.7763568394002505e-15,
      "numericalIntegrationResidual": 2.8989477485197312e-11,
      "integrationStepHalvingShift": 4.315487966977116e-10
    },
    "audited_reference_values": {
      "gammaPositiveTwo": 0.04890051070806112,
      "gammaNegativeTwoUpperReal": -4.95423435600189,
      "coefficientBracket": -3.9503585374459917,
      "cBkk": 0.9875896343614979,
      "wkbUpperBoundary": 321.8297948685433,
      "starts": [
        [
          10,
          -0.009782229696624904,
          0.030370870256320393,
          0.9973718640581228
        ],
        [
          30,
          0.003390420210560767,
          -0.0012210256758117537,
          0.9841992141509373
        ],
        [
          100,
          -0.00016135326379529912,
          -0.0002820467890257485,
          0.9877509876252932
        ],
        [
          300,
          0.00003608058847506597,
          0.0000014659320605700107,
          0.9875535537730229
        ]
      ]
    }
  },
  "uncertainty": {
    "interpretation": "The coefficient differences are deterministic sensitivity to truncating the asymptotic initial condition at finite x_i. The ODE residual and step-halving values are implementation controls, not statistical error bars or uncertainties for the physical theory.",
    "finite_start_warning": "The finite-start boundary condition is imposed on the correction deltaOmega only. The imaginary part of the uncorrected Gaussian width Omega_0 is retained; resetting the full corrected width to a purely real beta at finite eta_i would introduce an order-1/x_i contamination unrelated to the quantum-gravity expansion.",
    "nonuniform_limit_warning": "The Planck-mass remainder is stated at fixed nonzero k/k0; it is not a derived O(lambda_k^2) remainder uniform as k/k0 -> 0."
  },
  "scientific_sources": [
    {
      "citation": "Brizuela, David, Claus Kiefer, and Manuel Kraemer. Quantum-Gravitational Effects on Gauge-Invariant Scalar and Tensor Perturbations during Inflation: The de Sitter Case. Physical Review D 93 (2016): 104035.",
      "identifier": "DOI:10.1103/PhysRevD.93.104035; arXiv:1511.05545v2",
      "locators": "sections VII-VIII, equations (121)-(147)",
      "use": "de Sitter background, Gaussian Riccati equation, real corrected frequency, exact incomplete-gamma solution, natural asymptotic state, and late coefficient"
    },
    {
      "citation": "Chataignier, Leonardo, and Manuel Kraemer. Unitarity of Quantum-Gravitational Corrections to Primordial Fluctuations in the Born-Oppenheimer Approach. Physical Review D 103 (2021): 066005.",
      "identifier": "DOI:10.1103/PhysRevD.103.066005; arXiv:2011.06426",
      "locators": "section IV, especially equations (115)-(160)",
      "use": "clock-conditioned inner product, induced non-Gaussian component, mixed-mode terms, and prescription dependence that bound the claim ceiling"
    }
  ],
  "data_file": "de-sitter-bo-finite-start-control.csv",
  "caption": "Finite-start control for the projected de Sitter one-mode benchmark. For H0/mP = 1e-5, k/k0 = 1, x_i = |k eta_i| >= 10, and epsilon_WKB <= 1e-2, the declared initialization window is 10 <= x_i <= 321.8298; the formal x_i -> infinity state lies outside it. Starts at x_i = 10, 30, 100, and 300 approach C_BKK = 0.9875896344 nonmonotonically, with relative discrepancies +0.9905%, -0.3433%, +0.01634%, and -0.003653%. This tests the linearized real-frequency projected-Gaussian prescription, not Gaussian closure, mixed modes, renormalized backreaction, a continuum constraint algebra, or observability.",
  "alt_text": "Two panels show a finite subhorizon Born-Oppenheimer/WKB initialization band from x_i = 10 to 321.8 and four finite-start correction coefficients converging nonmonotonically toward 0.98759; an arrow places the formal x_i to infinity state beyond WKB control."
}