TFPT scientific source excerpt
Original repository path: tfpt_explorer/marker_selection.py
Original file SHA-256: f6d9907f0e9fa242c4f50d67681c499d7bff046f0032abed37f66f332965842a
Inclusive original lines: 58-102
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000058 | def build_marker_data() -> dict[str, Any]:
000059 |     """Return the exact polar-transport and matching-selection calculation."""
000060 | 
000061 |     sqrt6 = sp.sqrt(6)
000062 |     identity5 = sp.eye(5)
000063 |     raw_metric = sp.diag(4, 12, 12, 12, 24)
000064 |     metric_sqrt = sp.diag(2, 2 * sp.sqrt(3), 2 * sp.sqrt(3), 2 * sp.sqrt(3), 2 * sp.sqrt(6))
000065 | 
000066 |     # The six native simplex columns and canonical sigma action are the ones
000067 |     # used by the existing P2 polar-transport calculation.  Sigma fixes
000068 |     # columns 0, 1, 5 and cycles 2 -> 3 -> 4 -> 2; the source convention takes
000069 |     # q*=fixed[-1]=5 rather than choosing a new basis or marker here.
000070 |     simplex = sp.Matrix(
000071 |         [
000072 |             [2, 2, -1, -1, -1, -1],
000073 |             [0, 0, -1, -1, 1, 1],
000074 |             [0, 0, -1, 1, -1, 1],
000075 |             [0, 0, 1, -1, -1, 1],
000076 |             [1, -1, 0, 0, 0, 0],
000077 |         ]
000078 |     )
000079 |     sigma_permutation = (0, 1, 3, 4, 2, 5)
000080 |     fixed_points = tuple(index for index, image in enumerate(sigma_permutation) if index == image)
000081 |     marked_point = fixed_points[-1]
000082 |     other_points = tuple(index for index in range(6) if index != marked_point)
000083 |     differences = simplex[:, other_points] - simplex[:, marked_point] * sp.ones(1, 5)
000084 |     positive_factor = identity5 + (1 / sqrt6 - 1) * sp.ones(5) / 5
000085 |     polar_transport = sp.simplify(metric_sqrt * differences * positive_factor / sp.sqrt(48))
000086 |     slot_charges = tuple(
000087 |         sp.Rational(-1, 3) if sigma_permutation[index] != index else sp.Rational(1, 2)
000088 |         for index in other_points
000089 |     )
000090 |     native_charge = sp.simplify(
000091 |         polar_transport * sp.diag(*slot_charges) * polar_transport.T
000092 |     )
000093 |     polar_isometry = sp.simplify(polar_transport.T * polar_transport) == identity5
000094 |     native_spectrum_polynomial = sp.factor(native_charge.charpoly().as_expr())
000095 | 
000096 |     # N and C^T are two orthonormal frames of the same sum-zero hyperplane.
000097 |     # The polar-transported charge above lives in the N frame.  The existing
000098 |     # process transpositions live in the C frame.  Their nontrivial orthogonal
000099 |     # relation N=O C^T must therefore be applied before taking overlaps or
000100 |     # commutators.
000101 |     simplex_frame = sp.simplify(metric_sqrt * simplex / sp.sqrt(48))
000102 |     f3 = sp.Matrix(