{
  "artifact_id": "qft.artifact.holography-quantum-gravity.black-hole-microstates-and-stringy-entropy.d1-d5-absorption-benchmark",
  "title": "D1–D5 Absorption Benchmark and Extrapolation Failure",
  "source_revision": 1,
  "created_on": "2026-08-29",
  "creator": "OpenAI Codex, for QFT.org",
  "original_work": true,
  "reader_question": "What exactly agrees in the D1–D5 absorption test, and how does the leading formula reveal an invalid all-frequency extrapolation?",
  "takeaway": "Bulk flux matching and CFT detailed balance give the same response in the declared dilute-gas, low-energy window. The cross section approaches the horizon area while the dimensionless s-wave probability vanishes; a deliberately invalid continuation eventually exceeds probability one and therefore advertises the approximation failure.",
  "fixture": {
    "units": "R_Q = 1 and hbar = c = k_B = 1",
    "channel": "one canonically normalized neutral massless minimally coupled T4 metric scalar, zero KK momentum, S3 angular momentum ell = 0",
    "r1_over_RQ": 1,
    "r5_over_RQ": 1,
    "TL_times_RQ": 0.01,
    "TR_times_RQ": 0.005,
    "TH_times_RQ": 0.006666666666666667,
    "delta_p": 0.34657359027997264,
    "r0_over_RQ": 0.044428829381583664,
    "rn_over_RQ": 0.015707963267948963,
    "string_length_over_RQ": 0.01,
    "area_over_RQ_cubed": 0.9301883004089945
  },
  "observable_definitions": {
    "q": "omega R_Q",
    "response": "sigma_abs / A_H",
    "transmission_probability": "P_0 = omega^3 sigma_abs / (4 pi)",
    "left_bose_factor": "b_L = 1 / expm1(q/(2 t_L))",
    "right_bose_factor": "b_R = 1 / expm1(q/(2 t_R))",
    "hawking_bose_factor": "b_H = 1 / expm1(q/t_H)",
    "bulk_cross_section": "sigma_hat = pi^3 q expm1(q/t_H) / [expm1(q/(2 t_L)) expm1(q/(2 t_R))]",
    "cft_cross_section": "sigma_hat = pi^3 q [(1+b_L)(1+b_R)-b_L b_R], with b_i the declared Bose factors",
    "emission_kernel": "sigma_hat b_H = pi^3 q b_L b_R"
  },
  "regime": {
    "source_domain": "dilute gas r0,rn much smaller than r1,r5; low frequency omega r1 and omega r5 much smaller than one; r1/r5 of order one; selected minimal scalar channel",
    "state_and_resolution": "left/right thermal ensemble in the thermodynamic long-string and continuum limit; not an individual finite-volume microstate",
    "semiclassical_control": "large-charge and weak-loop control is assumed separately and is not fixed by this dimensionless fixture",
    "declared_display_window": "0.0001 <= q <= 0.05",
    "display_boundary_status": "conservative visualization contract, not a sharp theorem boundary and not an uncertainty estimate",
    "display_classification_scope": "CSV display_status distinguishes the solid plotted window from the deliberately invalid dashed continuation; it is not a complete validity certificate",
    "maximum_omega_r1_or_r5": 0.05,
    "maximum_alpha_prime_omega_squared": 2.5000000000000004e-7,
    "extrapolation_control": "the same leading formula is continued with a dashed line to q=1 solely to expose failure; it is not a physical prediction there"
  },
  "analytic_checks": [
    "sigma_abs/A_H = 1 + q^2/(48 t_L t_R) + O(q^4)",
    "P_0 = (A_H/R_Q^3) q^3/(4 pi) + O(q^5)",
    "for t_L=t_R=t, sigma_abs/A_H = [q/(4t)] coth[q/(4t)]",
    "(1+b_L)(1+b_R)-b_L b_R = b_L b_R/b_H",
    "sigma_abs b_H = pi^3 r1^2 r5^2 omega b_L b_R",
    "sigma_abs has length cubed while P_0 is dimensionless"
  ],
  "numerical_checks": {
    "maximum_bulk_cft_relative_residual": 2.1341192248266464e-14,
    "declared_window_endpoint": {
      "q": 0.05,
      "response": 1.8270152412335938,
      "transmission_probability": 0.00001690492281162789
    },
    "probability_one_crossing_in_invalid_extension": 0.7978845608028668,
    "invalid_endpoint": {
      "q": 1,
      "transmission_probability": 2.4674011002723395,
      "status": "outside derivation; diagnostic only"
    }
  },
  "csv_columns": {
    "q": "dimensionless frequency omega R_Q",
    "response_bulk": "bulk exponential representation divided by A_H",
    "response_cft": "CFT gross-minus-reverse representation divided by A_H",
    "response_bulk_declared_window": "bulk response only in the declared display window",
    "response_cft_marker": "independently evaluated CFT values at selected marker frequencies",
    "relative_residual": "absolute bulk-CFT difference divided by their maximum magnitude",
    "transmission_probability": "dimensionless P_0 computed from the leading formula",
    "transmission_declared_window": "P_0 only in the declared display window",
    "transmission_extrapolated": "P_0 only in the dashed, not-licensed continuation",
    "display_status": "declared-low-energy-window or extrapolated-not-licensed; plot segmentation only, not a complete validity classification"
  },
  "scientific_sources": [
    {
      "citation": "Maldacena, Juan M., and Andrew Strominger. Black Hole Greybody Factors and D-Brane Spectroscopy. Physical Review D 55 (1997): 861-870.",
      "identifier": "arXiv:hep-th/9609026; DOI:10.1103/PhysRevD.55.861",
      "locators": "sections 2-4 and 6; equations 2.7, 2.9, 4.30-4.35",
      "use": "background parameters, validity regime, flux-to-cross-section normalization, response formula, and detailed balance"
    },
    {
      "citation": "David, Justin R., Gautam Mandal, and Spenta R. Wadia. Absorption and Hawking Radiation of Minimal and Fixed Scalars, and AdS/CFT Correspondence. Nuclear Physics B 544 (1999): 590-611.",
      "identifier": "arXiv:hep-th/9808168; DOI:10.1016/S0550-3213(99)00068-1",
      "locators": "sections 2.3-4",
      "use": "operator identification and independent normalization of the bulk-CFT coupling"
    },
    {
      "citation": "Das, Sumit R., Gary W. Gibbons, and Samir D. Mathur. Universality of Low Energy Absorption Cross Sections for Black Holes. Physical Review Letters 78 (1997): 417-419.",
      "identifier": "arXiv:hep-th/9609052; DOI:10.1103/PhysRevLett.78.417",
      "locators": "equation 16, page 418",
      "use": "independent low-frequency horizon-area check"
    },
    {
      "citation": "Das, Sumit R., and Gautam Mandal. Microstate Dependence of Scattering from the D1-D5 System. Journal of High Energy Physics 04 (2009): 036.",
      "identifier": "arXiv:0812.1358; DOI:10.1088/1126-6708/2009/04/036",
      "locators": "sections 1 and 4",
      "use": "finite-resolution microstate dependence in the two-charge orbifold laboratory; context for, not a proof of, the three-charge thermal formula"
    }
  ],
  "caption": "Computed benchmark for a neutral minimal s-wave with r1=r5=R_Q, T_L R_Q=0.01, T_R R_Q=0.005, and string length/R_Q=0.01. In the declared dilute-gas, low-energy window, the independently evaluated bulk curve and CFT markers coincide and approach sigma_abs/A_H=1, while P_0 vanishes as q^3. Large-charge and weak-loop control are separate assumptions. The solid curve stops at q=0.05. The dashed continuation is deliberately outside the derivation; its crossing of P_0=1 diagnoses approximation failure rather than physical nonunitarity.",
  "alt_text": "Bulk flux matching and CFT detailed balance give the same normalized low-energy D1-D5 response, which approaches the horizon area while the dimensionless transmission vanishes; a dashed extrapolation outside the derivation eventually violates the probability bound."
}