s1258 — the missing functor: the braid 56 is the punctured-torus homology twelve times; the end flip is its orientation reversal
  representation       {"braid relation": "<1e-12", "(B1B2)^3 − (I − 2P_act)": "<1e-12", "B1 symplectic (s1250 Ω)": "<1e-12", "B2 symplectic": "<1e-12", "rank P_act": 24}
  natural_iso          {"dim Hom(H1, V56)": 12, "rank of the twelve images": 24, "images outside the active sector": "<1e-12", "dim Hom(trivial, V56)": 32, "dim Hom(Sym² H1, V56)": 0, "B1 in η-coords": [[[1.0, 1.0], [0.0, 1.0]], "<1e-12"], "B2 in η-coords": [[[1.0, 0.0], [-1.0, 1.0]], "<1e-12"]}
  figure8              {"W_h − B1²B2⁻¹B1⁻¹ (on 56)": "<1e-12", "W_h in η-coords": [[[3.0, -1.0], [1.0, 0.0]], "<1e-12"], "ρ_std(σ1σ2⁻¹)": [[2.0, 1.0], [1.0, 1.0]], "SL(2,ℤ) conjugator to [[2,1],[1,1]]": [[-2, 1], [-1, 0]], "rel |det(t − W_h)|act − (t²−3t+1)¹²| at 4 points": "<1e-09"}
  orientation_reversal {"O": {"σ1 ↦": "<1e-12", "σ2 ↦": "<1e-12", "preserves active": "<1e-12", "tensor rank-1 (2nd singular value)": "<1e-09", "H1 factor (normalised)": [[0.0, 1.0], [1.0, 0.0]], "det of H1 factor sign": -1, "U² = ": "+I", "U² residual": "<1e-10", "U eigen counts (real parts)": [[-1.0, 6], [1.0, 6]]}, "C56": {"σ1 ↦": "<1e-12", "σ2 ↦": "<1e-12", "preserves active": "<1e-12", "tensor rank-1 (2nd singular value)": "<1e-09", "H1 factor (normalised)": [[1.0, 0.0], [0.0, -1.0]], "det of H1 factor sign": -1, "U² = ": "+I", "U² residual": "<1e-10", "U eigen counts (real parts)": [[-1.0, 6], [1.0, 6]]}, "CP": {"σ1 ↦": "<1e-12", "σ2 ↦": "<1e-12", "preserves active": "<1e-12", "tensor rank-1 (2nd singular value)": "<1e-09", "H1 factor (normalised)": [[0.0, 1.0], [-1.0, 0.0]], "det of H1 factor sign": 1, "U² = ": "−I", "U² residual": "<1e-10", "U eigen counts (real parts)": [[0.0, 12]]}}
  gieseking            {"G² − B1B2⁻¹": "<1e-12", "G anti-symplectic": "<1e-12", "H1 factor of G (normalised to [0,0] = 1)": [[1.0, 1.0], [1.0, 0.0]], "tensor rank-1": "<1e-09", "eigenvalues on the active sector": [[-1.618034, 6], [-0.618034, 6], [0.618034, 6], [1.618034, 6]]}
  two_roots            {"H_φ² − W_h": "<1e-12", "H_φ symplectic": "<1e-12", "(B1⁻¹H_φB1)² − B1B2⁻¹": "<1e-12", "golden-bridge root eigenvalues on active": [[0.618034, 12], [1.618034, 12]], "Gieseking root eigenvalues on active": [[-1.618034, 6], [-0.618034, 6], [0.618034, 6], [1.618034, 6]]}
  ladder               {"1": {"type": "anti-symplectic", "dilatation": 1.618033989, "φⁿ": 1.618033989, "s1257 future/past at rung n": 1.618034}, "2": {"type": "symplectic", "dilatation": 2.618033989, "φⁿ": 2.618033989, "s1257 future/past at rung n": 2.618034}, "3": {"type": "anti-symplectic", "dilatation": 4.236067977, "φⁿ": 4.236067977, "s1257 future/past at rung n": 4.236067}, "4": {"type": "symplectic", "dilatation": 6.854101966, "φⁿ": 6.854101966, "s1257 future/past at rung n": 6.854099}}
  receipt   PASS filed_s1026_LANE_D1192_LENS_ARTIFACTS_npz
  receipt   PASS filed_d1410_solder_npz
  receipt   PASS app_T_is_the_printed_file_of_the_braid_representation_and_its_fence
  receipt   PASS s1257_results_hold_the_rung_ratios
  computed  PASS F2_the_registered_B3_representation_on_V56_braid_relation_central_element_and_the_s1250_symplectic_form
  computed  PASS F3_V56_is_twelve_copies_of_the_punctured_torus_homology_plus_32_trivials_with_B1_B2_acting_as_the_Dehn_twist_matrices_tensor_identity
  computed  PASS F4_the_hyperbolic_word_is_B1_squared_B2_inverse_B1_inverse_and_acts_as_the_figure8_monodromy_class_tensor_identity_with_charpoly_alexander_to_the_12
  computed  PASS F5_O_and_C56_extend_the_functor_to_orientation_reversing_classes_mirror_and_swap_mirror_with_det_minus1_on_homology_CP_is_orientation_preserving
  computed  PASS F6_B1_times_O_is_an_anti_symplectic_square_root_of_the_figure8_word_acting_as_the_Gieseking_monodromy_11_10_tensor_the_end_flip_multiplicity_involution
  computed  PASS F7_the_golden_bridge_and_the_Gieseking_element_are_two_square_roots_of_the_same_figure8_word_one_symplectic_positive_one_anti_symplectic
  computed  PASS F8_the_n_th_power_of_the_Gieseking_element_has_dilatation_phi_n_equal_to_s1257s_future_past_ratio_odd_powers_anti_symplectic_even_symplectic
  argument  ARG  topology
  argument  ARG  what_the_fence_becomes
  argument  ARG  the_ladder_and_the_end_flip
VERDICT the registered B3 representation is, by an exhibited natural isomorphism, 12 × H1(once-bordered torus) ⊕ 32 trivials; the hyperbolic word is the figure-8 monodromy class (charpoly = Alexander¹²); O and C56 extend it to the orientation-reversing mapping classes; B1·O is the Gieseking square root of the figure-8 word and its powers carry the golden ladder (odd rungs orientation-reversing).
RESULT_HASH 05168f6e4e114c9c
s1258: 7 computed, 4 receipt, 3 argument, 0 failed
