s1288 — the D56 tower as a deformation problem
  Q1_grading_and_912_by_weight   {"D56 on 56 (weight: multiplicity)": {"-6": 1, "-2": 27, "2": 27, "6": 1}, "e7 by ad-D56 weight": {"-4": 27, "0": 79, "4": 27}, "912 by D56 weight (trace of the projector block)": {"10": 0, "6": 78, "2": 378, "-2": 378, "-6": 78, "-10": 0}, "blocks: max(‖P² − P‖, ‖P − Pᵀ‖)": "<1e-12", "P_912 Θ_X − Θ_X (all weights)": "<1e-12", "Θ_X entries outside weights ±6, ±2": "<1e-12"}
  Q2_E6_content                  {"e6: dim, max ‖[g, D56]‖": [78, "<1e-12"], "Casimir on the 27 of the 56": 1.444444, "X_6: Casimir / c27, eigen residual": [1.384615, "<1e-12"], "X_2: Casimir / c27, eigen residual": [1.923077, "<1e-12"], "X_-2: Casimir / c27, eigen residual": [1.923077, "<1e-12"], "X_-6: Casimir / c27, eigen residual": [1.384615, "<1e-12"]}
  Q3_order_one                   {"ker L₂ (first-order deformations of the face in 912₂)": 153, "e7₋₄·X₆ (trivial ones): rank": 27, "L₂·(e7₋₄·X₆)": "<1e-12", "L₂·X₂": "<1e-12", "|X₂ − best fit by e7₋₄·X₆| / |X₂|": 1.0, "E7 action on Θ vs on tensors (one generator, X)": "<1e-12"}
  Q4_completion_line             {"ker L₋₂": 21, "L₋₂·X₋₂": "<1e-12", "order-3 solvability: rank on ker L₋₂, |cos(null, X₋₂)|": [20, 1.0], "L₋₆: rank of 78 (injective), smallest singular value": [78, 2.618615], "QC of X by order 0..6 (Ω form)": ["<1e-12", "<1e-12", "<1e-12", "<1e-12", "<1e-12", "<1e-12", "<1e-12"], "qc(X₊), qc(X₋), qc(X) (closure, Ω)": [["<1e-12", "<1e-12"], ["<1e-12", "<1e-12"], ["<1e-12", "<1e-12"]], "Ω-form cross term Qs(X₊, X₋)": "<1e-12"}
  Q5_the_line                    {"λ = 1": {"algebra (dim, derived, Killing +,−,0)": [28, 28, [12, 16, 0]], "commutant of ad": 1, "V(o) + (3/2)(1−λ)²": "<1e-12", "|∇V(o)| − |1−λ²|/(2√2)": "<1e-12", "V(o)": 0.0}, "λ = 0.5": {"algebra (dim, derived, Killing +,−,0)": [28, 28, [12, 16, 0]], "commutant of ad": 1, "V(o) + (3/2)(1−λ)²": "<1e-12", "|∇V(o)| − |1−λ²|/(2√2)": "<1e-12", "V(o)": -0.375}, "λ = 2": {"algebra (dim, derived, Killing +,−,0)": [28, 28, [12, 16, 0]], "commutant of ad": 1, "V(o) + (3/2)(1−λ)²": "<1e-12", "|∇V(o)| − |1−λ²|/(2√2)": "<1e-12", "V(o)": -1.5}, "λ = 0": {"algebra (dim, derived, Killing +,−,0)": [28, 27, [0, 16, 12]], "commutant of ad": null, "V(o) + (3/2)(1−λ)²": "<1e-12", "|∇V(o)| − |1−λ²|/(2√2)": "<1e-12", "V(o)": -1.5}, "λ = -0.5": {"algebra (dim, derived, Killing +,−,0)": [28, 28, [0, 28, 0]], "commutant of ad": 1, "V(o) + (3/2)(1−λ)²": "<1e-12", "|∇V(o)| − |1−λ²|/(2√2)": "<1e-12", "V(o)": -3.375}, "λ = -1": {"algebra (dim, derived, Killing +,−,0)": [28, 28, [0, 28, 0]], "commutant of ad": 1, "V(o) + (3/2)(1−λ)²": "<1e-12", "|∇V(o)| − |1−λ²|/(2√2)": "<1e-12", "V(o)": -6.0}}
  Q6_dyonic_direction            {"X + T(η62, π/4)": "<1e-12", "orbit tangent dim (e7·X + scale)": 106, "|off-tangent part of ∂λX_λ| / |X₋| at λ = 1": 0.353553, "|cos| between off-tangent ∂λ and off-tangent ∂ω": 1.0}
  Q7_C2_search_SAMPLED           {"residual² at Λ = 0 (|X₋₂|² + |X₋₆|²)": 336.0, "starts": 6, "final residual² over starts (min, max)": [294.0, 294.0], "C₂ member on the line (λ = 0): weights of X₊": [2, 6]}
  Q8_mirror                      {"O·X − X": "<1e-12", "O·X₊ − X₋": "<1e-12", "O D56 O⁻¹ + D56": "<1e-12"}
  receipt   PASS filed_d1414_house_frames_npz
  receipt   PASS s1283_results_T_family_pot_and_gradient_formulas
  receipt   PASS s1287_results_tower_and_invariant_machinery
  computed  PASS Q1_912_has_D56_weights_6_2_minus2_minus6_only_dims_78_378_378_78_so_the_tower_stops_at_eps_cubed
  computed  PASS Q2_X6_and_Xminus6_are_pure_78_and_X2_and_Xminus2_pure_351_with_no_27_part
  computed  PASS Q3_the_face_has_126_nontrivial_first_order_classes_and_X2_is_orthogonal_to_the_trivial_ones_so_C1_fails_at_eps_one
  computed  PASS Q4_given_X6_and_X2_the_graded_completions_are_exactly_the_line_lambda_times_the_lower_half_and_each_half_is_consistent
  computed  PASS Q5_on_the_line_lambda_positive_is_so62_lambda_negative_compact_simple_so8_lambda_zero_a_contraction_and_o_is_critical_only_at_lambda_plus_minus_one
  computed  PASS Q6_the_lambda_direction_is_off_the_E7_orbit_and_parallel_there_to_the_dyonic_omega_direction
  computed  PASS Q7_SAMPLED_no_lower_unipotent_frame_found_that_removes_the_lower_half_of_X_every_start_stops_above_250
  computed  PASS Q8_the_house_map_O_fixes_X_and_exchanges_its_two_halves
  argument  ARG  reading
  argument  ARG  scope
VERDICT: the D56 face plus its first-order term fix the tower up to one real number λ: X₆ + X₂ + λ(X₋₂ + X₋₆), each half consistent alone; λ is the dyonic direction and the Minkowski condition at o fixes λ = 1. Twist-only parents fail at ε¹; twist-plus-351 parents reach the upper half and (sampled) not X.
RESULT_HASH 30367e0659197ec2
s1288: 8 computed, 3 receipt, 2 argument, 0 failed
