s1315 — the zeta lens, typed
  Q1_artin_mazur                     {"N_n = |det(Aⁿ − I)|, n = 1..12": [1, 5, 16, 45, 121, 320, 841, 2205, 5776, 15125, 39601, 103680], "N_n − (L₂ₙ − 2)": [0], "exp(Σ N_n zⁿ/n) − (1 − z)²/(1 − 3z + z²) to z¹²": "0", "Lefschetz numbers L(fⁿ) = 2 − tr Aⁿ, n = 1..5": [-1, -5, -16, -45, -121], "poles of the Artin–Mazur zeta": ["3/2 - sqrt(5)/2", "sqrt(5)/2 + 3/2"], "tr W_hⁿ − (12 L₂ₙ + 32), n = 1..8 (max)": "<1e-09", "charpoly(W_h) − (t² − 3t + 1)¹²(t − 1)³², max relative coefficient deviation": "<1e-09"}
  Q2_golden_residue                  {"L(1, χ₅) (digamma)": "0.4304089409640040388894332", "L(1, χ₅) (finite Gauss-sum formula) − digamma": "5.74e-42", "τ = 2 ln φ": "0.9624236501192068949955178", "√5·L(1, χ₅) − τ": "-1.15e-41", "L(2, χ₅) − 4π²/(25√5)": "0.0", "ζ(2)·L(2, χ₅) − 2π⁴/(75√5)  [ζ_ℚ(√5)(2)]": "0.0"}
  Q3_k0_fib_is_the_golden_integers   {"N_τ": "[[0, 1], [1, 1]]", "charpoly N_τ": "t**2 - t - 1", "det N_τ (= N(φ))": -1, "N_τ²": "[[1, 1], [1, 2]]", "P ∈ SL(2, ℤ) with P N_τ² P⁻¹ = A": "[[-1, -2], [0, -1]]", "Zagier-reduced forms of discriminant 5 (one cycle ⇔ one class)": [[1, 3, 1]], "reduced positive forms of discriminant −3": [[1, 1, 1]]}
  Q4_ramification                    {"disc(t² − 3t + 1)": 5, "Δ(−1) for Δ = t² − 3t + 1": 5, "t² − 3t + 1 mod 5": "(t + 1)**2", "A⁵ mod 5": "[[4, 0], [0, 4]]", "primes dividing the discriminant": [5], "χ₅ = Legendre (·/5)": true}
  Q5_norm_minus_one                  {"Gᵀ Ω G + Ω (anti-symplectic)": "<1e-10", "G² − B₁B₂⁻¹": "<1e-10", "eigenvalues φ and −1/φ present (product −1 = N(φ))": [true, true], "tr Gⁿ, n = 1..6": [0.0, 68.0, 0.0, 116.0, 0.0, 248.0]}
  Q6_volume_dedekind_zeta            {"Vol = 2 Im Li₂(e^{iπ/3})": "2.02988321281930725", "(3√3/2)·L(χ₋₃, 2) − Vol": "-7.89e-31", "9√3·ζ_ℚ(√−3)(2)/π² − Vol": "-3.94e-31", "Kashaev estimate (2π/N)(ln⟨4₁⟩_N − (3/2) ln N), N = 200, 800, 3200": ["2.02134237", "2.02773154", "2.02934427"], "⟨4₁⟩₅ − (46 + 2√5)": "1.34e-29"}
  Q7_fibonacci_arithmetic            {"Σ dᵢ²θᵢ/D − e^{2πic/8}": "3.94e-31", "T phases h − c/24": ["-7/60", "17/60"], "order of T": 60, "Rogers–Ramanujan: |χ(−1/τ) − S_LY χ(τ)| (two τ)": "3.96e-31", "S_F: S₀₁/S₀₀": "1.61803398875", "S_LY: S₀₁/S₀₀ (the Galois image φ ↦ −1/φ)": "-0.61803398875"}
  Q8_prime_geodesic_fenced           {"ℓ(W_h) = 2 arccosh(tr/2)": "1.92484730023841379", "ℓ − 4 ln φ": "0.0", "tr A (smallest hyperbolic |trace| in SL(2, ℤ) is 3)": 3}
  receipt   PASS filed_d1414_house_frames_npz
  receipt   PASS s1302_results_lucas_ladder_and_fibonacci_data
  receipt   PASS s1307_results_A1785_checks_class_number_one
  computed  PASS Q1_the_lucas_ladder_is_the_artin_mazur_zeta_of_the_figure8_monodromy_and_W_h_on_the_56_has_lefschetz_zeta_one_over_alexander_to_the_12_times_1_minus_z_to_the_32
  computed  PASS Q2_the_banked_tau_2_ln_phi_is_root5_times_L1_chi5_the_residue_of_the_dedekind_zeta_of_q_root5
  computed  PASS Q3_k0_fib_is_z_phi_and_tau_squared_is_sl2z_conjugate_to_the_figure8_monodromy_one_reduced_cycle_of_discriminant_5_class_number_one
  computed  PASS Q4_every_five_in_the_knot_and_fibonacci_record_is_the_conductor_of_chi5_the_ramified_prime_of_q_root5
  computed  PASS Q5_the_orientation_reversing_square_root_G_carries_the_norm_minus_one_pair_phi_and_minus_inverse_phi_and_its_square_is_the_golden_braid_word
  computed  PASS Q6_the_figure8_volume_is_9root3_zeta_q_root_minus3_of_2_over_pi_squared_and_the_kashaev_invariant_grows_towards_it
  computed  PASS Q7_fibonacci_modular_data_have_gauss_sum_e_2pi_i_c_over_8_conductor_60_and_their_galois_conjugate_is_realised_by_the_rogers_ramanujan_functions
  computed  PASS Q8_W_h_is_the_shortest_prime_geodesic_class_of_the_modular_group_length_4_ln_phi
  argument  ARG  reading
  argument  ARG  fence
  argument  ARG  typing_correction_TC1
  argument  ARG  scope
VERDICT: the figure-8 monodromy carries an exact zeta web — Artin–Mazur (Lucas ladder), ζ_ℚ(√5) (τ = √5·L(1, χ₅); K₀(Fib) = ℤ[φ]; the 5s; the norm −1 pair), ζ_ℚ(√−3) (volume), Rogers–Ramanujan (Fibonacci conjugate). The Riemann ζ enters only through a fenced prime geodesic.
RESULT_HASH 7b79e3ddd1f04740
s1315: 8 computed, 3 receipt, 4 argument, 0 failed
