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          "checks": [
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              "separation_over_radius": 0.1,
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              "feynman_relative_error": 5.667462453095635e-16,
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              "separation_over_radius": 0.1,
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      ],
      "exact_formula": "For analytic stereographic cores with rho=A^2(1+|x|^2/R^2)^(1-d), the prepared difference-current pair potential per eta*omega0*h_k*s^k is 4*r*A^4*R^d*h^2*J_d*F_d(h), h=delta/R, J_d=pi^(d/2)*Gamma(3d/2)/Gamma(2d), F_d=2F1(3d/2,d;d+1/2;-h^2/4). The positive-sum quartic prepared pair per 4*g*r*A^4*R^d*s^k is J_base*F_base(h)-h^2*J_d*F_d(h), with J_base=pi^(d/2)*Gamma(3d/2-2)/Gamma(2d-2) and F_base=2F1(3d/2-2,d-1;d-1/2;-h^2/4).",
      "scope": "Same-depth translated analytic upper cores only. d=2 has divergent canonical norm; odd-depth stereographic polarization lacks the maintained Clifford realization. This is an initial potential, not a solved encounter or bound composite."
    },
    "depth_two_polarization": {
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            8.881784197001252e-16,
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        },
        {
          "first": "elliptical_a",
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          "jones_overlap_squared": 0.27794285808655483,
          "quartic_pair_factor": 1.1117714323462184,
          "common_physical_rotation_errors": [
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      ],
      "relative_rotations": [
        {
          "relative_rotation": 0.0,
          "linear_pair_factor": 3.999999999999999,
          "circular_pair_factor": 3.999999999999999
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        {
          "relative_rotation": 0.3,
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        {
          "relative_rotation": 0.8,
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          "relative_rotation": 1.5707963267948966,
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      ],
      "identity": "For normalized transverse Jones vectors e,f, B_e=2ee^dagger-I obeys Tr B_e B_f=4|e^dagger f|^2-2. The positive-sum two-current prepared pair factor is 2+Tr B_e B_f=4|e^dagger f|^2.",
      "scope": "Explicit linear, circular and elliptical transverse Jones controls with common and relative physical-plane rotations. Identifying these Jones vectors with occupied IDWT photon states and their field equation remains a physical premise."
    },
    "shared_shell_grade_response": {
      "experiment": "two-current-shared-shell-pair-response",
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      ],
      "identity": "F_{M,d}(s)=sum_{k=0}^M binom(k+d-1,d-1)s^k, M=n-1. At s=1 this is D_M=binom(M+d,d); for 0<=s<1 the infinite sum is (1-s)^(-d).",
      "scope": "Exact retained-order weight comparison on prepared same-depth pairs. It does not determine the physical ray overlap, encounter rate, or cross-depth embedding."
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          "max_imaginary_frequency": 1.054017984003508e-14
        },
        {
          "depth": 4,
          "mixing": -0.5,
          "max_imaginary_frequency": 7.087580878215172e-15
        },
        {
          "depth": 4,
          "mixing": 0.0,
          "max_imaginary_frequency": 1.9984014443252818e-14
        },
        {
          "depth": 4,
          "mixing": 0.25,
          "max_imaginary_frequency": 0.6301917940761607
        },
        {
          "depth": 4,
          "mixing": 0.5,
          "max_imaginary_frequency": 1.2952388306473486
        },
        {
          "depth": 4,
          "mixing": 0.75,
          "max_imaginary_frequency": 2.6078916566919483
        },
        {
          "depth": 6,
          "mixing": -1.0,
          "max_imaginary_frequency": 2.185751579730777e-14
        },
        {
          "depth": 6,
          "mixing": -0.5,
          "max_imaginary_frequency": 3.552713678800501e-14
        },
        {
          "depth": 6,
          "mixing": 0.0,
          "max_imaginary_frequency": 2.0872192862952943e-14
        },
        {
          "depth": 6,
          "mixing": 0.25,
          "max_imaginary_frequency": 0.6301917940761638
        },
        {
          "depth": 6,
          "mixing": 0.5,
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        },
        {
          "depth": 6,
          "mixing": 0.75,
          "max_imaginary_frequency": 2.6078916566919563
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      ],
      "formula": "U=g sum_k h_k (Tr C_k^2-q Tr K_k^2); on a coherent constant-density upper field rho, omega_N=2g(1-q)rho D_N and the same law gives pair cross energy 2g rho sigma [r-q Tr(B_A B_B)] s^k.",
      "scope": "One quartic contact on an assumed constant-density, spatial zero-mode branch. No common physical compact geometry, self-consistent Q, spatial localization, cross-depth scale, or full stability has been derived."
    }
  },
  "component_sha256": {
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    "two-current-tower.py": "b197a0ce401d30fe83e11ba7524daced4bc6ef7bc94e7a7c8a4eaba2cfc40309",
    "two-current-spatial-overlap.py": "e620efe597e54c51ef76a9bf64807ef9c05a44340fec45f9b4396a487bc6b014",
    "two-current-photon-control.py": "090fab27b49472522491a718699f830168053ca0061cd1cc7bf7e0c5708c9d9a",
    "two-current-grade-response.py": "eb1c0b98743afe1685ad3c44636bc8d20e71a2c159a2d2ccf1c86391171c170b",
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  },
  "scope": "One comparison of smooth same-field quartic pair contact, full retained order-zero source, conditional displaced analytic cores, and a single-quartic constant-density self-mass branch. None derives the common physical carrier, cross-depth normalization, or complete stable mass operator.",
  "sources": {
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  }
}
