s1286 — chirality at the edges of F0
  Q1_discrete_elements_on_the_D56_end      {"P = diag(I, −I)": {"gᵀΩg": "−Ω", "g² = I": true, "X ↦": "neither", "D56 ↦": "+", "D_sol ↦": "+", "Z ↦": "+", "face(+D56) ↦ ±face(+D56)": "−", "face(+D56) ↦ ±face(−D56)": "neither"}, "O (end flip)": {"gᵀΩg": "−Ω", "g² = I": true, "X ↦": "+", "D56 ↦": "−", "D_sol ↦": "−", "Z ↦": "+", "face(+D56) ↦ ±face(+D56)": "neither", "face(+D56) ↦ ±face(−D56)": "+"}, "C56": {"gᵀΩg": "+Ω", "g² = I": true, "X ↦": "+", "D56 ↦": "+", "D_sol ↦": "neither", "Z ↦": "−", "face(+D56) ↦ ±face(+D56)": "+", "face(+D56) ↦ ±face(−D56)": "neither"}, "CP = C56·O": {"gᵀΩg": "−Ω", "g² = I": true, "X ↦": "+", "D56 ↦": "−", "D_sol ↦": "neither", "Z ↦": "−", "face(+D56) ↦ ±face(+D56)": "neither", "face(+D56) ↦ ±face(−D56)": "+"}, "exp(πK)": {"gᵀΩg": "+Ω", "g² = I": true, "X ↦": "−", "D56 ↦": "+", "D_sol ↦": "+", "Z ↦": "+", "face(+D56) ↦ ±face(+D56)": "−", "face(+D56) ↦ ±face(−D56)": "neither"}, "orbifold-type at the +D56 end (keeps D56, maps the end gauging to ±itself, inverts the twist Z)": ["C56"]}
  Q2_C56_projection_at_the_D56_end         {"compact gauge algebra commuting with D56 and C56: dim; ideals (dim, Killing)": [6, [[3, [-1.0, -1.0, -1.0]], [3, [-1.0, -1.0, -1.0]]]], "Casimir of each ideal on the 8 (normalised spectrum)": [{"1.0": 8}, {"1.0": 8}], "C56 on Λ²8: unitary, involution, (−1)-multiplicity": ["<1e-10", "<1e-10", 16], "[Λ²(C56 lift), Λ²(1 + k)] over the surviving algebra": "<1e-09", "ideals: max bracket leaving an ideal; max bracket between ideals": ["<1e-09", "<1e-09"], "each ideal: spectrum of one generator on the 8 (|imaginary| / max; doublets only ⇔ {1.0: 8})": [{"1.0": 8}, {"1.0": 8}]}
  Q3_twist_neutral_fermions                {"Z itself: neutral gravitini, neutral spin-½ (of 56)": [0, 0], "+D56 face": {"twists": 1, "twist in T0": "<1e-08", "neutral gravitini": 0, "neutral spin-½": 0, "u_of residual": "<1e-09"}, "+D_sol face": {"twists": 1, "twist in T0": "<1e-08", "neutral gravitini": 0, "neutral spin-½": 0, "u_of residual": "<1e-09"}, "four-form chamber (0.3, −0.2, 0.1)": {"twists": 1, "twist in T0": "<1e-08", "neutral gravitini": 6, "neutral spin-½": 26, "u_of residual": "<1e-09"}, "four-form chamber (1, 1, 1)": {"twists": 1, "twist in T0": "<1e-08", "neutral gravitini": 6, "neutral spin-½": 26, "u_of residual": "<1e-09"}, "four-form chamber (1, 1, 0)": {"twists": 1, "twist in T0": "<1e-08", "neutral gravitini": 6, "neutral spin-½": 26, "u_of residual": "<1e-09"}, "10 generic faces (s1284 Q3 seed): (neutral gravitini, neutral spin-½)": [[0, 0]]}
  Q4_four_form_chamber_orbifold_candidates {"P = diag(I, −I)": {"H ↦": "−", "face ↦": "neither", "twist ↦": "+"}, "O (end flip)": {"H ↦": "+", "face ↦": "+", "twist ↦": "+"}, "C56": {"H ↦": "neither", "face ↦": "neither", "twist ↦": "neither"}, "CP = C56·O": {"H ↦": "neither", "face ↦": "neither", "twist ↦": "neither"}, "exp(πK)": {"H ↦": "+", "face ↦": "−", "twist ↦": "+"}, "orbifold-type for the chamber face (keeps H, face ↦ ±face, twist inverted)": []}
  receipt   PASS filed_d1414_house_frames_npz
  receipt   PASS filed_chatgpt_d1414_maps_npz
  receipt   PASS filed_chatgpt_d1417_maps_npz
  receipt   PASS s1283_results_construction_and_machinery
  receipt   PASS s1284_results_face_and_twist_definitions
  computed  PASS Q1_among_the_registered_discrete_elements_only_C56_meets_the_orbifold_conditions_at_the_D56_end
  computed  PASS Q2_the_C56_projection_at_the_D56_end_leaves_su2_plus_su2_with_8_equal_two_copies_of_2_2_and_a_4_plus_4_lift_so_even_and_odd_gravitini_are_the_same_representation
  computed  PASS Q3_no_twist_neutral_fermion_at_the_D56_or_D_sol_edges_six_gravitini_and_26_spin_half_at_the_four_form_chambers_none_at_generic_edges
  computed  PASS Q4_no_registered_discrete_element_is_orbifold_type_for_the_four_form_chamber_face
  argument  ARG  forcing
  argument  ARG  reading
  argument  ARG  scope
VERDICT: at the edges where Z (or Q₁₂) is the KK twist every fermion is twist-charged — no zero modes, no chirality by any orbifold; C56, the only registered orbifold-type parity at the D56 end, leaves su(2) ⊕ su(2) and vectorlike content; the four-form chambers carry 6 + 26 twist-neutral fermions but no registered orbifold-type element. Open: that element.
RESULT_HASH 64dea880752383e7
s1286: 4 computed, 5 receipt, 3 argument, 0 failed
