s1135 — AX6 parametrisation audit
  target value: sin²θ_W = φ⁻³ = 0.236067977500 (A820/s115a: 1/(2(Ω₁₂+Ω₂₃)), a constant identity)

PART A — statement (i): Λ = 4tan²ψ on the I₄ = φ³ circle (A262 :37); readout sin²θ_W = 3/(4+Λ), Λ = 3P/Q(Y) (D213ii, registered at D237 :85);
  AX6_pol Λ⋆ = 6φ−1 (App A :373 prints tan²ψ⋆ = (6φ−1)/4)
  Λ⋆ = 6φ−1 = 8.708203932499 · tan²ψ⋆ = 2.177050983125 (≈ 2.177 ✓) · ψ⋆(i) = 55.872778° · cos²ψ⋆(i) = 0.314757303333
  [PASS] 3 + 6φ = 3φ³ EXACTLY (φ³ = 2φ+1): the identity behind the value
  [PASS] cos²ψ⋆(i) = 4/(3φ³) EXACTLY ⇒ sin²θ_W = ¾cos²ψ⋆(i) = 3/(4+Λ⋆) = φ⁻³: statement (i) IS an f(ψ⋆) = φ⁻³ relation (f = ¾cos², rational in cos²)
  cos(6ψ⋆(i)) = 0.908045729282
  [PASS] cos(6ψ⋆(i)) ≠ ½ (|Δ| > 0.1): the AX6_pol angle is NOT a solution of the Det² condition

PART B — statement (ii): cos(6ψ⋆) = ½, ψ⋆ = 50° (p4 :343, App X :83, p7 :54; Det² selector D326/A351; A651)
  solutions of cos(6ψ) = ½ in [0°,180°): [10.0, 50.0, 70.0, 110.0, 130.0, 170.0]
  [PASS] ψ = 50° is a solution of cos(6ψ) = ½ and ψ⋆(i) ≈ 55.87° is NOT (nearest solution ≥ 4° away)
  sin²θ_W = ¾cos²ψ at each Det² solution (the (i) readout applied to the (ii) angle):
    ψ =   10.0° → ¾cos²ψ = 0.727385  (φ⁻³ = 0.236068; Δ = +0.491317)
    ψ =   50.0° → ¾cos²ψ = 0.309882  (φ⁻³ = 0.236068; Δ = +0.073814)
    ψ =   70.0° → ¾cos²ψ = 0.087733  (φ⁻³ = 0.236068; Δ = -0.148335)
    ψ =  110.0° → ¾cos²ψ = 0.087733  (φ⁻³ = 0.236068; Δ = -0.148335)
    ψ =  130.0° → ¾cos²ψ = 0.309882  (φ⁻³ = 0.236068; Δ = +0.073814)
    ψ =  170.0° → ¾cos²ψ = 0.727385  (φ⁻³ = 0.236068; Δ = +0.491317)
  [PASS] NO Det² solution gives φ⁻³ under the (i) readout (min |Δ| > 0.05): 50° and 55.87° are DIFFERENT objects, not one angle in two coordinates — HOUSE EXPECTATION FALSIFIED (kept)

PART C — which angle did the three CW no-gos (A617/A620/A621) evaluate? A621 prints R(ψ⋆) = 27.105 (60-digit)
  R(ψ⋆(i) = 55.8728°) = 27.105003 · R(50°) = 130.646096 · R(45°) = R_max → ∞ (the CW critical point)
  [PASS] R(ψ⋆(i)) = 27.105 to 3 decimals — the CW no-gos evaluated the AX6_pol angle (i), i.e. 55.87°, not 50°
  [PASS] R(50°) ≠ 27.105 (the printed A621 number is not the 50° angle)

PART D — A667's degree argument: 'cos²(50°) has degree 3 over ℚ while ℚ(φ) has degree 2 ⇒ no f(ψ⋆) = φ⁻³'
  [PASS] sin 10° satisfies 8x³ − 6x + 1 = 0 (|residual| ≤ 1e-12): cos²50° = (1 − sin 10°)/2 has degree 3 over ℚ — A667's premise holds for 50°
  cos²ψ⋆(i) = 4/(3φ³) = 4(√5−2)/3 = 0.314757303333; residual of 9y² + 48y − 16 = 0.0e+00
  [PASS] cos²ψ⋆(i) satisfies 9y² + 48y − 16 = 0 (|residual| ≤ 1e-12): DEGREE 2 over ℚ — A667's degree-3 argument does NOT apply to the AX6_pol angle (i)

PART E — print census (Rev32.1 tex, locators from the S295 grep; not modified here)
  object (i)  Λ⋆ = 6φ−1 / tan²ψ⋆ = (6φ−1)/4 / ψ⋆ ≈ 55.87°: appendix_a :373 (D237) · A621 R = 27.105 · A262 · A658 'AX6 condition 3P = (6φ−1)Q(Y)'
  object (ii) cos(6ψ⋆) = ½ / ψ⋆ = 50°: p4 :343, :346 · appendix_x :83 · p7 :54 · A651 'Det² selector' · A667 degree argument
  both under the ONE label 'AX6' / 'ψ⋆'. The Det² selector (ii) was proposed (D326/A351) as a SELECTOR for the polarisation angle;
  it selects 50°, which is not the AX6_pol angle 55.87° and does not give φ⁻³ under the registered readout ¾cos²ψ.

VERDICT (typing): 'ψ⋆' is a label collision (S294.P2). AX6 proper is the polarisation postulate Λ⋆ = 6φ−1 (ψ⋆ ≈ 55.87°,
  sin²θ_W = ¾cos²ψ⋆ = φ⁻³ exactly); the Det² condition cos(6ψ) = ½ selects a DIFFERENT angle (50°) at which the readout is 0.310,
  and A667's Galois decoupling is a theorem about the 50° object only. Consequences (PI-gated, nothing moved here): (1) the
  p4/App X/p7 sentence 'AX6, written as cos(6ψ⋆) = ½ with ψ⋆ = 50°' conflates two objects — REV33 queue candidate; (2) D1230 W2-A
  carried the same conflation in one sentence — a one-line correction rider is owed to the lane; (3) the Det² selector is NOT
  'consistent with AX6' (A651 :112) — it is falsified as a selector of the AX6_pol angle by this arithmetic; (4) W2-A's 'derive
  AX6' target is Λ⋆ = 6φ−1, whose angle lies in ℚ(φ) — the Galois obstruction does not stand in its way.

RESULT: ALL GATES PASS
