{
  "format": "tight39-native-data-v1",
  "scope": "Exact transcription of existing pinned finite coefficients and explicit evaluation recipes. No scientific identity was replayed by this extraction.",
  "provenance": {
    "source_repository": "https://github.com/macgregor-sudo/tight-39-code",
    "source_commit": "46f01c135aacea9003435e7f1dc99b4a69568e83",
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      {
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      {
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      {
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      {
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      {
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      {
        "path": "Proofs/HTPhaseCompleteHPhase.lean",
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      {
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      {
        "path": "Proofs/NativePhaseReducedQuadratic.lean",
        "sha256": "60187c246766d96703c3a10c8c71cafdbc7f676e43f35794da02d0c582ccb11d"
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      {
        "path": "Proofs/NativePhaseAlongSheetH.lean",
        "sha256": "1c1d83f8c8d57f1bcb08edcacd175e230cb643f2b516e8241682f3500b32f0c2"
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      {
        "path": "Proofs/NativeHAnchorSupport.lean",
        "sha256": "fdafd338668d09ce7917bc29618b58ac06dcd0f15c8671d096ead99eacf34095"
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      {
        "path": "Proofs/NativeHEquilibriumEndpoint.lean",
        "sha256": "2c7afa90a51813c518a8714e75158fcadbc6e6904349a861d3cbc9014a6b9468"
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      {
        "path": "Proofs/BookHTQuartic.lean",
        "sha256": "cfc780e582ababdf414a4534ea7e2fb860ed136dfea5d5b16a67c9ab1651ac5c"
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        "sha256": "40874782e2c9486b4091da6d57c5b41541058c4923c1de13e468421349a09d82"
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        "sha256": "7fd23a941f96514b1d28ec99d5c39a1bda2673d4f706caba08faf6dd8cec4530"
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    "license": {
      "identifier": "Apache-2.0",
      "license_sha256": "b40930bbcf80744c86c46a12bc9da056641d722716c378f5659b9e555ef833e1",
      "notices_sha256": "ad0559e2562428ee03a851b42ea4c323adf438f994cc87210b59b5da536aa173",
      "scope": "First-party mathematical coefficient data from the listed modules. No copied Mathlib implementation. No standalone manuscript license selected.",
      "modifications": "Lean typed literal lists and declaration metadata transcribed into JSON integer arrays, with exact tuple/order preservation and explicitly stated evaluation formulas.",
      "copyright": "No separate individual holder is asserted; source first-party/AI-generation disclosures retained by reference.",
      "acknowledgment": "Thanks to the source contributors. If I have seen further it is by standing on the sholders of Giants. \u2014 Isaac Newton, letter to Robert Hooke, 1676."
    }
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    ],
    "K_recipe": "For each eigen_pair [a,b] with R=real_difference and S=imaginary_sum: K_a=(R-i*S)/2 and K_b=-(R+i*S)/2. These are the fixed complex operators with eigen label j, before application to a state.",
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          [
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          [
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      }
    ],
    "seed_column_recipe": "Define D_b from seed_columns[b]. For eigen label j, E_j=D_eigen_index[j]. Then V_base(z)=sum_j y_j(z)*E_j, which is real by the native conjugation identities. The fixed K recipe satisfies K_j*P_base=E_j. No additional factor2; columns already include common denominator384540.",
    "norm": "Each z_i has its usual complex Euclidean norm. Native eigen-coordinate norm squared sum_j|y_j|^2=2*sum_i||z_i||^2. The criterion E norm squared is sum_i||z_i||^2, not the outer Book maximum norm.",
    "all_state_action_and_nonclaim": "Every fixed K_j is applied to each of the39 states: (opApply(M,X))[27*p+k]=sum_c M[k,c]*X[27*p+c]. There is no point-dependent mask. The matrices are partial-family operators; a global27-dimensional linear map need not be a derivation or a global Albert automorphism. No such derivation/global-automorphism assertion about K_j is made."
  },
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      [
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    "cartan_pairs": [
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        "k_exponents": [
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            -1
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          [
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        ]
      }
    ],
    "cartan_evaluation": "d_i(u) is Appendix A's determinant-one Tits pair action F_(g,k), where g,k are diagonal with entries u^exponent. E_i=d_i(u^-1)=d_i(u)^-1. On the selected twelve quartet states E_i P_xi=P_base.",
    "row_action": "rho_E applies E to each idempotency and frame-sum 27-vector and is identity on each trace/pair scalar. With row covector convention lambda_raw=lambda_base*rho_E, the new vector-block coefficient at m is sum_n lambda_base[n]*E[n,m].",
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    "normalization": "For H component(i,a), c_ia=u^character_exponents[i][a]; lambda_H(xi)=c_ia^-1*lambda_base_oriented*rho_E_i. Scalar rows also receive c_ia^-1. No coefficient conjugation occurs in the covector contraction itself.",
    "contraction": "L_H(y)_(i,a)=sum_r lambda_H(i,a,r)*y_r for real residual y, viewed in C.",
    "sheet_field": "For raw family R_i(p)=g_i(p)S_i(p), component field on the slice representative S_i(p) is L_H(xi)*rho_g_i(p). Its value at0 is L_H(xi).",
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    "inverseCartan_direct_evaluation": "Fix coordinates0,1,2. For each plane j=0..11, set x=X[3+2j],y=X[4+2j], d=native_plane_exponents[i][j], chi=u^(-d), chip=u^d. E_i sends (x,y) to ((chi*(x-i*y)+chip*(x+i*y))/2, i*(chi*(x-i*y)-chip*(x+i*y))/2). Equivalently its 2x2 block is [[(chi+chip)/2,i*(chip-chi)/2],[i*(chi-chip)/2,(chi+chip)/2]]. This is a complex-linear native-coordinate formula and yields real coordinates on real input when |u|=1. No reordering by cyclic octonion slots is made."
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        [
          "id",
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        [
          "sum",
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        [
          "sum",
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          10
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        [
          "sum",
          2,
          9
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        [
          "sum",
          2,
          10
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        [
          "sum",
          3,
          9
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        [
          "sum",
          3,
          10
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        [
          "sum",
          4,
          13
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        [
          "sum",
          4,
          14
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        [
          "sum",
          5,
          13
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        [
          "sum",
          5,
          14
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        [
          "sum",
          6,
          13
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        [
          "sum",
          6,
          14
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        [
          "sum",
          7,
          23
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        [
          "sum",
          7,
          24
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        [
          "sum",
          8,
          23
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        [
          "sum",
          8,
          24
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        [
          "sum",
          9,
          23
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        [
          "sum",
          9,
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        [
          "pair",
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          12
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        [
          "pair",
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          "pair",
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          15
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        [
          "pair",
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          17
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          "pair",
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          "pair",
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          "pair",
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          22
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          "pair",
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          "pair",
          3,
          25
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          "pair",
          3,
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          "pair",
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          28
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        [
          "pair",
          3,
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          "pair",
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          13
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        [
          "pair",
          4,
          14
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        [
          "pair",
          4,
          16
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        [
          "pair",
          4,
          17
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        [
          "pair",
          4,
          19
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        [
          "pair",
          4,
          20
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        [
          "pair",
          4,
          21
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        [
          "pair",
          4,
          23
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        [
          "pair",
          4,
          24
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        [
          "pair",
          4,
          26
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        [
          "pair",
          4,
          27
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        [
          "pair",
          4,
          29
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        [
          "pair",
          5,
          12
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        [
          "pair",
          5,
          13
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          "pair",
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          "pair",
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          "pair",
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          "pair",
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          "pair",
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          "pair",
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          "pair",
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        [
          "pair",
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          "pair",
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          "pair",
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          "pair",
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          "pair",
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        [
          "pair",
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          "pair",
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          17
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        [
          "pair",
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          "pair",
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          "pair",
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          "pair",
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          "pair",
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        [
          "pair",
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        [
          "pair",
          6,
          28
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        [
          "pair",
          6,
          29
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          "pair",
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          13
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        [
          "pair",
          7,
          14
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        [
          "pair",
          7,
          16
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        [
          "pair",
          7,
          17
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        [
          "pair",
          7,
          19
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        [
          "pair",
          7,
          20
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        [
          "pair",
          7,
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        [
          "pair",
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          23
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        [
          "pair",
          7,
          24
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        [
          "pair",
          7,
          26
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        [
          "pair",
          7,
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        [
          "pair",
          7,
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          "pair",
          8,
          12
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        [
          "pair",
          8,
          13
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        [
          "pair",
          8,
          15
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        [
          "pair",
          8,
          16
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        [
          "pair",
          8,
          18
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        [
          "pair",
          8,
          19
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        [
          "pair",
          8,
          21
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        [
          "pair",
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          22
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        [
          "pair",
          8,
          24
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        [
          "pair",
          8,
          25
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        [
          "pair",
          8,
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        [
          "pair",
          8,
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          "pair",
          9,
          12
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        [
          "pair",
          9,
          14
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        [
          "pair",
          9,
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          "pair",
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          "pair",
          9,
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          "pair",
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          "pair",
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          "pair",
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          "pair",
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        [
          "pair",
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        [
          "pair",
          9,
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        [
          "pair",
          9,
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          "pair",
          10,
          13
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        [
          "pair",
          10,
          14
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        [
          "pair",
          10,
          16
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        [
          "pair",
          10,
          17
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          "pair",
          10,
          19
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        [
          "pair",
          10,
          20
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        [
          "pair",
          10,
          21
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        [
          "pair",
          10,
          23
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        [
          "pair",
          10,
          24
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        [
          "pair",
          10,
          26
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        [
          "pair",
          10,
          27
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        [
          "pair",
          10,
          29
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        [
          "pair",
          11,
          12
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        [
          "pair",
          11,
          13
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          "pair",
          11,
          15
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        [
          "pair",
          11,
          16
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        [
          "pair",
          11,
          18
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        [
          "pair",
          11,
          19
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        [
          "pair",
          11,
          21
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        [
          "pair",
          11,
          22
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        [
          "pair",
          11,
          24
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        [
          "pair",
          11,
          25
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        [
          "pair",
          11,
          27
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        [
          "pair",
          11,
          28
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          "pair",
          12,
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          "pair",
          12,
          22
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          "pair",
          12,
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          "pair",
          12,
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          "pair",
          12,
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          "pair",
          12,
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          "pair",
          12,
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        [
          "pair",
          12,
          28
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        [
          "pair",
          12,
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        [
          "pair",
          13,
          21
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        [
          "pair",
          13,
          22
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          "pair",
          13,
          23
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          "pair",
          13,
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          "pair",
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          "pair",
          13,
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          "pair",
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          "pair",
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          "pair",
          13,
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          "pair",
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          "pair",
          14,
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          "pair",
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          "pair",
          17,
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          "pair",
          17,
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          "pair",
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          "pair",
          17,
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          "pair",
          17,
          27
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          "pair",
          17,
          28
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          "pair",
          18,
          21
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          "pair",
          18,
          22
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          "pair",
          18,
          23
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          "pair",
          18,
          24
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          "pair",
          18,
          26
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          "pair",
          18,
          27
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          "pair",
          18,
          29
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        [
          "pair",
          19,
          21
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        [
          "pair",
          19,
          22
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        [
          "pair",
          19,
          23
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        [
          "pair",
          19,
          25
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        [
          "pair",
          19,
          26
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        [
          "pair",
          19,
          28
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        [
          "pair",
          19,
          29
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        [
          "pair",
          20,
          21
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        [
          "pair",
          20,
          22
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        [
          "pair",
          20,
          23
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        [
          "pair",
          20,
          24
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        [
          "pair",
          20,
          25
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        [
          "pair",
          20,
          27
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        [
          "pair",
          20,
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      ]
    }
  ],
  "identities_in_pinned_source": {
    "status": "These are exact mathematical identities proved in the pinned source; not independently recomputed in this transcription run.",
    "H_equilibrium": "L_H(xi) DF_Pxi=0, all1053 columns, all unit xi; HTPhaseCompleteHPack.H_equilibrium.",
    "T_equilibrium": "L_T(xi) DF_Pxi=0, all1053 columns, all unit xi; HTPCovAll.T_equilibrium.",
    "H_reproduction": "L_H(xi) F(Pxi+Vxi z)=H(z), all real z and all unit xi; HTPhaseCompleteHPhase.H_reproduction_phase.",
    "T_reproduction": "L_T(xi) F(Pxi+Vxi z)=T(z), all real z and all unit xi; HTPhaseCompleteTValue.T_reproduction_phase.",
    "H_anchor_support": "The entire contracted ambient polynomial uses only coordinates on the indicated quartet; NativeHAnchorSupport.hasAnchorSupport, transported using rho_E.",
    "raw_family_annihilation": "L_H(xi) DF_Ri(p)=0 on all ambient directions; NativePhaseAlongSheetH.raw_H_annihilates.",
    "nonclaims": [
      "No claim that all covectors span the full cokernel.",
      "No supported T field along a sheet is asserted.",
      "No finite-data identity was executed by this extractor; only transcription, index, tuple, source hash and serialization checks."
    ]
  },
  "finite_linear_system": {
    "purpose": "Defines the exact obligations on a covector, with the supplied lists giving explicit candidates. The transcription run does not solve or verify this system.",
    "variables": "lambda_r in C, r=0..2183, original residual row order.",
    "equilibrium": "For each packed ambient coordinate k=0..1052 impose sum_r lambda_r*(DF_Pxi)[r,k]=0.",
    "quadratic_basis": "Let e_alpha (alpha=0..23) be the real coordinate basis of z=(C^3)^4, in order block0 Re0,Im0,Re1,Im1,Re2,Im2, then block1,etc. Put v_alpha=V_xi(e_alpha). For alpha=beta set A_r,alpha,alpha=Q_F,r(v_alpha). For alpha<beta set A_r,alpha,beta=Q_F,r(v_alpha+v_beta)-Q_F,r(v_alpha)-Q_F,r(v_beta). These are all300 real-coordinate monomial coefficients.",
    "reproduction": "For target H_i,a or T_i,a impose sum_r lambda_r*A_r,alpha,beta equal to its corresponding quadratic coefficient obtained by the same diagonal/polar recipe on the explicit H/T formulas. This is300 complex linear equations, in addition to equilibrium.",
    "H_support": "Set lambda_r=0 off the row incidence of anchor frames[0,1+3i,2+3i,3+3i]: an id/trace row uses an anchor point; a sum row uses an anchor frame; a pair row has both points in the anchor. This stronger row support implies full ambient quadratic support. No analogous T support constraint is asserted.",
    "explicit_solution": "Use H_base_covectors with the stated orientation, inverseCartan row action and character normalization; use T_phase_covectors with the stated Laurent evaluation. These are explicit solution candidates whose identities are proved in the pinned source, rather than a claim of consistency based on row counts.",
    "optional_canonical_solve": "If a solution independent of the supplied table values is desired, exact row-reduce this displayed linear system in ascending row/variable order and set free variables0. Consistency still requires proof; the supplied solutions are the witnesses. This extractor neither performs RREF nor claims its output equals the stored source solution."
  }
}