references/run_sabrina_principal_series_plancherel_factorization.json

{
  "kernel": "run_sabrina_principal_series_plancherel_factorization",
  "role": "post-computation assertion only",
  "schema": "ouroboros_scientific_reference_assertion_v1",
  "scientific_payload": {
    "claim_boundary": {
      "capabilities_removed": [],
      "direct_gamma_substitution_at_nu_zero_used": false,
      "mass_independence_rederived": false,
      "off_principal_series_gauge_zero_modes_resolved": false,
      "source_completeness_extended_beyond_assumptions": false,
      "source_error_claimed": false,
      "strict_positivity_at_nu_zero_claimed": false
    },
    "classification": "principal_series_plancherel_measure_all_integer_d_factorization_exact",
    "dimension_rows": [
      {
        "closed_form": "nu*tanh(pi*nu)/(4*pi**2)",
        "d": 1,
        "measure_at_origin_by_limit": "0",
        "parity": "odd",
        "quadratic_zero_coefficient": "1/(4*pi)",
        "quadratic_zero_coefficient_positive": true,
        "shadow_even": true,
        "strict_positivity_domain": "real nu != 0"
      },
      {
        "closed_form": "nu**2/(4*pi**3)",
        "d": 2,
        "measure_at_origin_by_limit": "0",
        "parity": "even",
        "quadratic_zero_coefficient": "1/(4*pi**3)",
        "quadratic_zero_coefficient_positive": true,
        "shadow_even": true,
        "strict_positivity_domain": "real nu != 0"
      },
      {
        "closed_form": "nu*(4*nu**2 + 1)*tanh(pi*nu)/(16*pi**4)",
        "d": 3,
        "measure_at_origin_by_limit": "0",
        "parity": "odd",
        "quadratic_zero_coefficient": "1/(16*pi**3)",
        "quadratic_zero_coefficient_positive": true,
        "shadow_even": true,
        "strict_positivity_domain": "real nu != 0"
      },
      {
        "closed_form": "nu**2*(nu**2 + 1)/(4*pi**5)",
        "d": 4,
        "measure_at_origin_by_limit": "0",
        "parity": "even",
        "quadratic_zero_coefficient": "1/(4*pi**5)",
        "quadratic_zero_coefficient_positive": true,
        "shadow_even": true,
        "strict_positivity_domain": "real nu != 0"
      },
      {
        "closed_form": "nu*(4*nu**2 + 1)*(4*nu**2 + 9)*tanh(pi*nu)/(64*pi**6)",
        "d": 5,
        "measure_at_origin_by_limit": "0",
        "parity": "odd",
        "quadratic_zero_coefficient": "9/(64*pi**5)",
        "quadratic_zero_coefficient_positive": true,
        "shadow_even": true,
        "strict_positivity_domain": "real nu != 0"
      },
      {
        "closed_form": "nu**2*(nu**2 + 1)*(nu**2 + 4)/(4*pi**7)",
        "d": 6,
        "measure_at_origin_by_limit": "0",
        "parity": "even",
        "quadratic_zero_coefficient": "pi**(-7)",
        "quadratic_zero_coefficient_positive": true,
        "shadow_even": true,
        "strict_positivity_domain": "real nu != 0"
      },
      {
        "closed_form": "nu*(4*nu**2 + 1)*(4*nu**2 + 9)*(4*nu**2 + 25)*tanh(pi*nu)/(256*pi**8)",
        "d": 7,
        "measure_at_origin_by_limit": "0",
        "parity": "odd",
        "quadratic_zero_coefficient": "225/(256*pi**7)",
        "quadratic_zero_coefficient_positive": true,
        "shadow_even": true,
        "strict_positivity_domain": "real nu != 0"
      },
      {
        "closed_form": "nu**2*(nu**2 + 1)*(nu**2 + 4)*(nu**2 + 9)/(4*pi**9)",
        "d": 8,
        "measure_at_origin_by_limit": "0",
        "parity": "even",
        "quadratic_zero_coefficient": "9/pi**9",
        "quadratic_zero_coefficient_positive": true,
        "shadow_even": true,
        "strict_positivity_domain": "real nu != 0"
      },
      {
        "closed_form": "nu*(4*nu**2 + 1)*(4*nu**2 + 9)*(4*nu**2 + 25)*(4*nu**2 + 49)*tanh(pi*nu)/(1024*pi**10)",
        "d": 9,
        "measure_at_origin_by_limit": "0",
        "parity": "odd",
        "quadratic_zero_coefficient": "11025/(1024*pi**9)",
        "quadratic_zero_coefficient_positive": true,
        "shadow_even": true,
        "strict_positivity_domain": "real nu != 0"
      },
      {
        "closed_form": "nu**2*(nu**2 + 1)*(nu**2 + 4)*(nu**2 + 9)*(nu**2 + 16)/(4*pi**11)",
        "d": 10,
        "measure_at_origin_by_limit": "0",
        "parity": "even",
        "quadratic_zero_coefficient": "144/pi**11",
        "quadratic_zero_coefficient_positive": true,
        "shadow_even": true,
        "strict_positivity_domain": "real nu != 0"
      },
      {
        "closed_form": "nu*(4*nu**2 + 1)*(4*nu**2 + 9)*(4*nu**2 + 25)*(4*nu**2 + 49)*(4*nu**2 + 81)*tanh(pi*nu)/(4096*pi**12)",
        "d": 11,
        "measure_at_origin_by_limit": "0",
        "parity": "odd",
        "quadratic_zero_coefficient": "893025/(4096*pi**11)",
        "quadratic_zero_coefficient_positive": true,
        "shadow_even": true,
        "strict_positivity_domain": "real nu != 0"
      },
      {
        "closed_form": "nu**2*(nu**2 + 1)*(nu**2 + 4)*(nu**2 + 9)*(nu**2 + 16)*(nu**2 + 25)/(4*pi**13)",
        "d": 12,
        "measure_at_origin_by_limit": "0",
        "parity": "even",
        "quadratic_zero_coefficient": "3600/pi**13",
        "quadratic_zero_coefficient_positive": true,
        "shadow_even": true,
        "strict_positivity_domain": "real nu != 0"
      }
    ],
    "exact_checks": {
      "all_d1_through_d12_are_shadow_even": true,
      "all_d1_through_d12_vanish_at_origin": true,
      "all_quadratic_zero_coefficients_positive": true,
      "all_ten_dimension_recurrences_exact": true,
      "archive_hash_matches": true,
      "d1_base_is_nu_tanh_over_4pi2": true,
      "d2_base_is_nu2_over_4pi3": true,
      "member_hash_matches": true,
      "principal_shadow_maps_nu_to_minus_nu": true,
      "shadow_context_hash_matches": true,
      "shadow_context_unique": true,
      "source_equation_blocks_match": true,
      "source_equation_markers_unique": true,
      "source_states_mass_independence": true
    },
    "paper": {
      "arxiv_id": "1905.10052",
      "title": "Implications of Superrotations"
    },
    "recurrence_rows": [
      {
        "d_to_d_plus_2": "1->3",
        "exact": true,
        "residual": "0"
      },
      {
        "d_to_d_plus_2": "2->4",
        "exact": true,
        "residual": "0"
      },
      {
        "d_to_d_plus_2": "3->5",
        "exact": true,
        "residual": "0"
      },
      {
        "d_to_d_plus_2": "4->6",
        "exact": true,
        "residual": "0"
      },
      {
        "d_to_d_plus_2": "5->7",
        "exact": true,
        "residual": "0"
      },
      {
        "d_to_d_plus_2": "6->8",
        "exact": true,
        "residual": "0"
      },
      {
        "d_to_d_plus_2": "7->9",
        "exact": true,
        "residual": "0"
      },
      {
        "d_to_d_plus_2": "8->10",
        "exact": true,
        "residual": "0"
      },
      {
        "d_to_d_plus_2": "9->11",
        "exact": true,
        "residual": "0"
      },
      {
        "d_to_d_plus_2": "10->12",
        "exact": true,
        "residual": "0"
      }
    ],
    "result_version": "sabrina_principal_series_plancherel_factorization_v1",
    "source_evidence": {
      "archive_sha256": "f6dbbd7baa6b7eb93cb06b6460877b26f09ba30bacd6b447641ccf6843828df6",
      "block_hashes": {
        "orthonormality": "33ab352d1fe06108a33c8a2a706e2a52b1a11bdfdad1b1a807ce72b75fadd14f",
        "plancherel_measure": "5ba93a88abf2a4b0c64cfd76d2cf8fdad928059bf2a32876b19fbfaf768ee495",
        "principal_series": "2e1210ae9acd758068aa25342ee9c67505de433b6d850e0528cd6f50c3ea2c04"
      },
      "checks": {
        "archive_hash_matches": true,
        "member_hash_matches": true,
        "shadow_context_hash_matches": true,
        "shadow_context_unique": true,
        "source_equation_blocks_match": true,
        "source_equation_markers_unique": true,
        "source_states_mass_independence": true
      },
      "member_sha256": "0e505e56e7acd99f9a6190e4fe4a3625ea620a5067c3ea8d270ee947dffdab8c",
      "shadow_context_sha256": "1fbbaa27d16b0df0791fb61f386b558f7de3540e468b1ee2ab3c2dc65e4f3e55"
    },
    "status": "complete",
    "theorem": {
      "dimension_recurrence": "mu_(d+2)=((d/2)^2+nu^2) mu_d/pi^2",
      "even_d_2m": "nu^2 product_(k=1)^(m-1)(nu^2+k^2)/(4 pi^(2m+1))",
      "odd_d_2m_plus_1": "nu tanh(pi nu) product_(k=0)^(m-1)(nu^2+(k+1/2)^2)/(4 pi^(2m+2))",
      "origin": "quadratic zero obtained by limit, never direct gamma substitution",
      "positivity": "strict for real nu != 0",
      "shadow_action": "nu -> -nu",
      "source_measure": "Gamma(d/2+i nu)Gamma(d/2-i nu)/(4 pi^(d+1) Gamma(i nu)Gamma(-i nu))"
    }
  },
  "scientific_payload_sha256": "4047efe441ad0800b95a0d64b4354de371b4f6fdf6be4f1cbe59763680be8178"
}