s1280 — the D1416 return checked in-house
  Q1_specializations         {"homology": {"(t/π-arg, c/π-arg) with Hom ≠ 0: (dim Hom, intertwiner invertible)": [[[1.0, 0.0], [1, true]]], "algebra dim, commutant dim": [4, 1]}, "Fibonacci": {"(t/π-arg, c/π-arg) with Hom ≠ 0: (dim Hom, intertwiner invertible)": [[[0.4, 1.2], [1, true]], [[1.6, 0.6], [1, true]]], "algebra dim, commutant dim": [4, 1]}, "lane intertwiners tested on the house matrices": {"homology": {"lane (t/π-arg, c/π-arg)": [1.0, 0.0], "residual s_i P − P cρ_t(σ_i)": "<1e-12", "det P ≠ 0": true}, "Fibonacci_tminus": {"lane (t/π-arg, c/π-arg)": [1.6, 0.6], "residual s_i P − P cρ_t(σ_i)": "<1e-12", "det P ≠ 0": true}, "Fibonacci_tplus": {"lane (t/π-arg, c/π-arg)": [0.4, 1.2], "residual s_i P − P cρ_t(σ_i)": "<1e-12", "det P ≠ 0": true}}, "J(t): ρ_t J = (−t) J ρ_{1/t} exact; det J": [true, "t**2 + t + 1"]}
  Q2_figure8_word            {"tr ρ_t(W_h) − (1 − t − 1/t), det − 1, (σ₁σ₂)³ − t³I": ["0", "0", "True"], "numerator of tr − 3": "-(t + 1)**2", "W_h homology": {"trace": 3.0, "imag(trace)": "<1e-12", "spectral radius": 2.618034, "class": "hyperbolic"}, "W_h Fibonacci": {"trace": 0.381966, "imag(trace)": "<1e-12", "spectral radius": 1.0, "class": "elliptic"}, "Fibonacci eigenvalue minimal polynomial; irreducible over Q; cyclotomic n; |poly(λ)| at the eigenvalues": ["lam**4 - 3*lam**3 + 3*lam**2 - 3*lam + 1", true, [], "<1e-12"], "Res(χ_hom, χ_Fib) − φ⁴ mod (φ² − φ − 1)": "0", "dim Hom between homology and c·Fibonacci, both directions, 5 characters": [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], "det(I − s W_h(−1))": "s**2 - 3*s + 1", "closure Δ: W_h, (σ₁σ₂⁻¹)²": ["1/t", "-(t**2 - 3*t + 1)/t**2"], "Fibonacci normalized closure: W_h, (σ₁σ₂⁻¹)², Jones(4₁) at e^{±2πi/5}": [1.0, -1.236068, -1.236068, -1.236068, "<1e-12"]}
  Q3_tl_jw_and_image_groups  {"3 strands: paths (total 1, total τ); TL residual δ = φ; ‖R − (A + A⁻¹E₁)‖, ‖FRF − (A + A⁻¹E₂)‖ (A = e^{3πi/5}); Eᵢ on the total-1 path": [[1, 2], "<1e-12", "<1e-12", "<1e-12", "<1e-12"], "4 strands: paths (total 1, total τ); TL residual; ‖f₄‖; ‖f₃‖, f₃² − f₃; [5]_φ; [2]_0, [5]_0": [[2, 3], "<1e-12", "<1e-12", 1.0, "<1e-12", "<1e-12", 0.0, 1.0], "lane f₄ and path U: ‖lane f₄‖; lane U spectrum = house E spectrum": ["<1e-12", true], "ν = e^{πi/10}: unitarity, |det − 1|, braid, a² − b³, a⁴ − I, b³ + I": ["<1e-12", "<1e-12", "<1e-12", "<1e-12", "<1e-12", "<1e-12"], "homology: a² + I, b³ + I (SL(2, ℤ) = ⟨a, b | a² = b³, a⁴ = 1⟩)": ["<1e-12", "<1e-12"], "ν¹² = e^{−4πi/5}: roots that descend; with det 1; e^{πi/10} among them": [12, 2, true], "lane twisted generators − (νR, νFRF)": "<1e-12", "[σ₁¹⁰, σ₂]: homology; Fibonacci − I (any character); lane reverse word": [[[91, 100], [10, 11]], "<1e-12", [[91, 100], [10, 11]]]}
  Q4_centres_and_mirrors     {"centre: homology + I; Fibonacci / e^{2πi/5} − I; ν-twisted + I": ["<1e-12", "<1e-12", "<1e-12"], "Fibonacci centre order": 5, "lane full56 S1, S2, centre − (M ⊗ I₁₂) ⊕ I₃₂; trace; multiplicities (−1, +1)": ["<1e-12", 8.0, [24, 32]], "family: C_t = diag(1, t)ι inverts σᵢ; O_t = swap·ι inverts and swaps (exact)": [true, true], "Fibonacci: conj(σᵢ) − σᵢ⁻¹; F conj(σ₁) F − σ₂⁻¹, F conj(σ₂) F − σ₁⁻¹": ["<1e-12", "<1e-12"], "homology at t = −1: C, O as real det −1 matrices — C S_i C⁻¹ − S_i⁻¹, O S₁ O⁻¹ − S₂⁻¹; det": ["<1e-12", "<1e-12", [-1.0, -1.0]]}
  receipt   PASS filed_chatgpt_d1416_intertwiners_npz
  receipt   PASS lane_results_json_d1416
  computed  PASS Q1_homology_is_burau_at_t_minus1_c1_fibonacci_at_exactly_two_twisted_points_each_hom_dim1_and_the_lane_intertwiners_hold_on_house_matrices
  computed  PASS Q2_trace_1_minus_t_minus_1_over_t_separates_3_from_2_minus_phi_for_every_character_resultant_phi4_closure_of_Wh_is_unknot_and_fig8_braid_gives_minus2_over_phi
  computed  PASS Q3_fibonacci_path_tl_reproduces_R_and_FRF_f4_vanishes_on_five_paths_nu_twist_gives_sl2z_to_su2_and_sigma1_10_commutator_obstructs_the_reverse
  computed  PASS Q4_centres_minus_I_and_e_2pii_5_the_lane_56_model_is_the_canonical_sum_and_the_semilinear_mirrors_invert_the_generators
  argument  ARG  reading
  argument  ARG  scope
VERDICT: both braid sides are character-twisted specializations of one Burau / TL parent (homology t = −1; Fibonacci t = e^{∓2πi/5}, Hom dim 1 each); W_h's character-free trace 1 − t − t⁻¹ separates them (3 vs 2 − φ; Res φ⁴); W_h closes to the unknot, the figure-8 braid (σ₁σ₂⁻¹)² gives −2/φ at Fibonacci; f₄ = 0; the ν-twisted Fibonacci image is a quotient of SL(2, ℤ).
RESULT_HASH 9c3704254495a138
s1280: 4 computed, 2 receipt, 2 argument, 0 failed
