{
  "schema_version": 1,
  "artifact_id": "qft.artifact.holography-quantum-gravity.quantum-cosmology-and-singularity-resolution-programs.uv-soft-power-and-flattened-kernel-control",
  "title": "UV-Soft Power Envelope and Finite Flattened Kernels",
  "source_revision": 1,
  "created_on": "2026-08-30",
  "creator": "OpenAI Codex, for QFT.org",
  "original_work": true,
  "reader_question": "How do the declared UV-soft Gaussian state and a finite initial surface regulate the power response and two operator-dependent flattened kernels?",
  "takeaway": "The phase-extremized power response is bounded and collapses exponentially toward the Bunch-Davies value. Both normalized finite-start kernels are finite at the flattened boundary, but F_0 has a zero at y = 2 pi while F_2 remains about 0.50107, so the side-lobe and zero structure is operator dependent rather than a universal pole.",
  "scientific_status": "quantitative analytic fixture and deterministic control figure",
  "claim_ceiling": "The artifact displays responses of one declared Gaussian state and abrupt finite-start kernels. It does not identify a trans-Planckian origin, supply a likelihood fit, prove matching-surface invariance, include every cubic operator or boundary kernel, or replace a full renormalized stress-tensor and perturbativity analysis.",
  "conventions": {
    "metric_signature": "(+---)",
    "physical_momentum": "p = k/a_0 on the initial slice; x = p/sigma is dimensionless.",
    "phase_band": "R_minus and R_plus are extrema over cos(Phi_k) in [-1,1]. The hatching is a phase-extremized band, not a statistical uncertainty band.",
    "infrared_endpoint": "x -> 0 is a formal analytic envelope endpoint. The declared subhorizon control scale is H/sigma = 0.01.",
    "uv_completion": "The Gaussian is assumed to provide a global UV-soft completion, but the EFT licenses predictions only below Lambda_EFT/sigma = 10. The cutoff lies outside the displayed x <= 4 signal window.",
    "flattened_coordinate": "q_j = k_l + k_m - k_j >= 0 on the physical momentum-triangle domain, y = q_j |eta_0|, and eta_0 < 0.",
    "kernel_normalization": "F_0 = |I_0|/|eta_0| and F_2 = 3|J_2|/|eta_0|^3 are normalized to one at y = 0.",
    "switching": "Both curves use an abrupt finite start at eta_0. Smoother switching or a rematched boundary action changes endpoint-sensitive side lobes."
  },
  "equations": {
    "state": "beta_k = b exp[-(p/sigma)^2] exp[i Phi_k], alpha_k = sqrt(1+|beta_k|^2)",
    "normalized_amplitude": "u(x) = b exp(-x^2)",
    "exact_power_ratio": "R_k = 1 + 2u^2 + 2u sqrt(1+u^2) cos(Phi_k)",
    "phase_extrema": "R_plus/minus(x) = [sqrt(1+u(x)^2) plus/minus u(x)]^2",
    "envelope_identity": "R_plus(x) R_minus(x) = 1",
    "occupation_energy": "rho_occ = b^2 sigma^4/(16 pi^2)",
    "infrared_energy_fraction": "f_rho_IR = 1 - [1 + 2(H/sigma)^2] exp[-2(H/sigma)^2]",
    "eft_tail_fraction": "f_rho_tail = [1 + 2(Lambda_EFT/sigma)^2] exp[-2(Lambda_EFT/sigma)^2]",
    "i0": "I_0(q) = integral_eta0^0 exp(i q eta) d eta = -eta_0 exp(i q eta_0/2) sinc(q eta_0/2)",
    "f0": "F_0(y) = |I_0(q)|/|eta_0| = |sinc(y/2)|",
    "j2": "J_2(q) = integral_eta0^0 (-eta)^2 exp(i q eta) d eta = |eta_0|^3 integral_0^1 t^2 exp(-i y t) dt",
    "f2": "F_2(y) = 3|J_2(q)|/|eta_0|^3 = 3|integral_0^1 t^2 exp(-i y t) dt|",
    "finite_limits": "F_0(0) = F_2(0) = 1; equivalently I_0(0) = |eta_0| and J_2(0) = |eta_0|^3/3."
  },
  "fixture": {
    "b": 0.01,
    "epsilon_H": 0.01,
    "H_over_M_Pl": 0.00004,
    "sigma_over_H": 100,
    "sigma_over_M_Pl": 0.004,
    "H_over_sigma": 0.01,
    "Lambda_EFT_over_sigma": 10,
    "displayed_x_range": [
      0,
      4
    ],
    "displayed_y_range": [
      0,
      12.566370614359172
    ],
    "regular_sample_intervals_per_panel": 400,
    "csv_rows_excluding_header": 402,
    "benchmark_y": 0.2
  },
  "anchors": {
    "x_zero": {
      "R_plus": 1.020200999975001,
      "R_minus": 0.9801990000249985,
      "product": 0.9999999999999996
    },
    "y_zero": {
      "F_0": 1,
      "F_2": 1
    },
    "y_0_2": {
      "F_0": 0.9983341664682815,
      "F_2": 0.9992502901316602
    },
    "y_2pi": {
      "y": 6.283185307179586,
      "F_0": 0,
      "F_2": 0.5010699784144922
    },
    "energy_controls": {
      "rho_occ_over_M_Pl4": 1.621138938277405e-16,
      "rho_occ_over_epsilon_H_M_Pl2_H2": 0.00001013211836423378,
      "f_rho_tail": 2.7816320187408423e-85,
      "f_rho_IR": 1.9997333533332794e-8
    }
  },
  "data": {
    "file": "uv-soft-power-and-flattened-kernel-control.csv",
    "row_model": "Each row carries an independent uniformly sampled panel-A coordinate x and a monotonically sampled panel-B coordinate y. The y grid is the 400-interval 0 to 4 pi grid plus the exact y = 0.2 benchmark.",
    "columns": {
      "x": "p/sigma",
      "u": "b exp(-x^2)",
      "alpha": "sqrt(1+u^2)",
      "r_minus": "lower phase extremum",
      "r_plus": "upper phase extremum",
      "r_product": "reciprocal-envelope control",
      "y": "q |eta_0|",
      "f0": "normalized absolute I_0 kernel",
      "f2": "normalized absolute J_2 kernel"
    }
  },
  "verification": {
    "analytic_algorithm": "F_0 uses a stable sinc expansion at the origin. The complex J_2 integral uses its convergent moment series for y <= 1 and a two-step integration-by-parts recurrence for larger y.",
    "independent_kernel_quadrature": {
      "method": "Composite Simpson quadrature of the real and imaginary t^2 exp(-i y t) integrands with 4000 intervals.",
      "values": {
        "y_0_2": 0.9992502901316612,
        "y_2pi": 0.5010699784145027
      }
    },
    "independent_energy_quadrature": {
      "method": "Composite Simpson quadrature of z^3 exp(-2 z^2): 40000 intervals on [0,12], 40000 on [10,14], and 4000 on [0,0.01].",
      "exact_total_dimensionless_integral": 0.125,
      "quadrature_total_dimensionless_integral": 0.12499999999999872,
      "exact_tail_dimensionless_integral": 3.477040023426053e-86,
      "quadrature_tail_dimensionless_integral": 3.477040023430794e-86,
      "exact_infrared_dimensionless_integral": 2.4996666916665992e-9,
      "quadrature_infrared_dimensionless_integral": 2.4996666916653332e-9
    },
    "acceptance_thresholds": {
      "audited_anchor_absolute_residual": 5e-15,
      "alpha_normalization_absolute_residual": 7e-16,
      "reciprocal_power_envelope_absolute_residual": 2e-15,
      "total_energy_quadrature_absolute_residual": 2e-14,
      "tail_and_infrared_quadrature_relative_residual": 1e-10,
      "independent_f2_quadrature_absolute_residual": 5e-13
    },
    "maxima": {
      "alpha_normalization_absolute_residual": 3.3306690738754696e-16,
      "reciprocal_power_envelope_absolute_residual": 9.992007221626409e-16,
      "sampled_f0_excess_above_one": 0,
      "sampled_f2_excess_above_one": 0,
      "total_energy_quadrature_absolute_residual": 1.27675647831893e-15,
      "tail_energy_quadrature_relative_residual": 1.3635759188446173e-12,
      "infrared_energy_quadrature_relative_residual": 5.064837438339964e-13,
      "f2_anchor_quadrature_absolute_residual": 1.0547118733938987e-14
    },
    "audited_reference_values": {
      "r_plus_at_x_zero": 1.020200999975001,
      "r_minus_at_x_zero": 0.9801990000249985,
      "f0_at_y_0_2": 0.9983341664682815,
      "f2_at_y_0_2": 0.9992502901316602,
      "f0_at_y_2pi": 0,
      "f2_at_y_2pi": 0.5010699784144923
    }
  },
  "uncertainty_and_failure_cases": [
    "The x = 0 endpoint is formal; using the subhorizon energy estimate below p = H requires the recorded infrared control rather than an unqualified extrapolation.",
    "The Gaussian tail is not compact support. The quoted tail controls only the assumed analytic completion, not an arbitrary continuation above Lambda_EFT.",
    "The diagonal occupation energy omits phase-sensitive squeezed terms, curvature terms, initial-surface counterterms, and the rest of the instantaneous renormalized stress tensor.",
    "The phase-extremized band is not an observed oscillation. A physical trace also requires eta_0, a_0 sigma, transfer functions, projection, and nuisance parameters.",
    "The finite kernels demonstrate regulated operator-dependent profiles. Their heights and phases in a bispectrum also depend on the cubic coupling, external legs, boundary kernels, and switching prescription.",
    "Neither panel licenses an inference about the ultraviolet origin of the state."
  ],
  "scientific_sources": [
    {
      "citation": "Holman, R., and A. J. Tolley. Enhanced Non-Gaussianity from Excited Initial States. Journal of Cosmology and Astroparticle Physics 2008, 5 (2008): 001.",
      "identifier": "DOI:10.1088/1475-7516/2008/05/001; arXiv:0710.1302",
      "locators": "sections 3.1-4 and 6",
      "use": "finite initial-time kernels, operator-dependent flattened enhancements, projection caveats, and claim ceiling"
    },
    {
      "citation": "Agarwal, N., R. Holman, A. J. Tolley, and J. Lin. Effective Field Theory and Non-Gaussianity from General Inflationary States. Journal of High Energy Physics 2013, 5 (2013): 085.",
      "identifier": "DOI:10.1007/JHEP05(2013)085; arXiv:1212.1172",
      "locators": "sections 2 and 5",
      "use": "independence of Gaussian power data and cubic initial kernels"
    },
    {
      "citation": "Collins, H., and R. Holman. The Renormalization of the Energy-Momentum Tensor for an Effective Initial State. Physical Review D 74 (2006): 045009.",
      "identifier": "DOI:10.1103/PhysRevD.74.045009; arXiv:hep-th/0605107",
      "locators": "sections III-IV",
      "use": "stress-tensor and initial-surface counterterm limitations on the diagonal occupation-energy control"
    }
  ],
  "accessibility": {
    "palette": "Explicit white canvas with black and gray strokes; hatching, line styles, direct labels, and marker shapes redundantly encode every distinction.",
    "reading_order": "Title and scope, panel A power band, panel B kernel comparison, deterministic controls, then claim ceiling.",
    "page_integration": "Embed in a keyboard-focusable figure-pan-region with visible focus, concise pan instructions, a full-size SVG link, and CSV/JSON downloads.",
    "structured_equivalent": "The semantic JSON preserves definitions, equations, domains, anchors, checks, caveats, caption, and alt text; the CSV preserves every plotted sample."
  },
  "caption": "Bounded and finite responses for the declared UV-soft benchmark. Panel A shows the exact phase extrema for b = 10^-2: the hatched band collapses exponentially toward the Bunch-Davies value, while the formal x -> 0 endpoint and Lambda_EFT/sigma = 10 cutoff are explicitly distinguished from the displayed response window. Panel B compares two normalized abrupt finite-start kernels. Both equal one at the flattened boundary, but F_0 vanishes at y = 2 pi while F_2 remains 0.501069978414, demonstrating operator-dependent zero and side-lobe structure. These are responses of a chosen state and matching surface, not evidence for a trans-Planckian origin.",
  "alt_text": "Two stacked plots show a hatched power-spectrum phase band narrowing from 0.980199 to 1.020201 at p over sigma equal to zero toward the Bunch-Davies ratio one, and two finite flattened kernels that both start at one but differ strongly: F0 reaches zero at q times absolute eta0 equal to two pi while F2 remains about 0.50107."
}