  [PASS] R0 registered reversal: S(1 3) D_φ = −D_φ  
  [PASS] R0' chamber reversal descends: C_* S₅ = Z C_*  
  [PASS] E1a g³ = −I (order 3 in PU(2)), ZgZ = g⁻¹, so ⟨g,Z⟩ ≅ S₃ ⊂ O(2)_readout  
  [PASS] E1b p ↦ ρ(p) ∈ {I,g,g²,Z,Zg,Zg²} is a projective homomorphism S₃ → PU(2) (Ad-action well defined)  
  [PASS] E1c (fact used) every S₃ ⊂ Aut(M₂)=SO(3) is conjugate to ⟨R(2π/3),Z⟩: one class of D₃ in SO(3)  det g = 1; the class statement is textbook SO(3) — labelled ASSUMED-STANDARD, not computed
  [PASS] E2a M₂ under Ad(S₃): ⟨χ,1⟩=1, ⟨χ,sgn⟩=1, ⟨χ,2⟩=1  (M₂ = 1 ⊕ sgn ⊕ 2)  (1, 1, 1)
  [PASS] E2b Q spans the sgn line: Ad(g)Q = Q, Ad(Z)Q = −Q  
  [PASS] E2c {σ_x, σ_z} is the 2: Ad(g) rotates the plane by ±120°, Ad(Z): σ_x ↦ −σ_x, σ_z ↦ σ_z  
  [PASS] E3a ℂ³ permutation rep: ⟨χ,1⟩=1, ⟨χ,sgn⟩=0, ⟨χ,2⟩=1  (ℂ³ = 1 ⊕ 2, NO sgn)  
  [PASS] E3b D_φ has zero sgn-projection and is the transposition-odd vector of the 2  
  [PASS] E4a dim Hom_{S₃}(ℂ³, M₂) = 2  free parameters [x11, x2]
  [PASS] E4b every equivariant Ξ has ⟨Ξ(D_φ), Q⟩ = 0 EXACTLY (the odd datum cannot reach the pseudoscalar)  Ξ(D_φ) coords (I,σx,Q,σz) = [0, 2*sqrt(3)*x11, 0, 0]
  [PASS] E4c and Ξ(D_φ) ∝ σ_x is realized (plane component nonzero for generic Ξ), odd under Z  
  [PASS] E4d the trivial component maps (1,1,1) ↦ c·I only (unitality is one linear condition on the 2 parameters)  
  [PASS] E5a Hom_{S₃}(2, sgn) = 0 and Hom_{S₃}(1, sgn) = 0  ⇒ Hom_{S₃}(ℂ³, sgn) = 0  
  [PASS] E5b explicit: Tr(σ_x Ad(g)A) ≠ Tr(σ_x A) while H_sgn is g-fixed ⇒ a plane-sourced channel into H_sgn is NOT 3-cycle covariant  
  [PASS] E6 sign-twisted marking σ·x = sgn(σ)P_σ x makes D_φ EVEN under the (1 3) reversal — contradicts SD_φS = −D_φ  
  [PASS] E7a M₃ under the frame S₃: ⟨χ,sgn⟩ = 1 (exactly one sign line = H_sgn) — and Ad(Z-type transposition)H_sgn = −H_sgn  
  [PASS] E7b dim Hom_{S₃}(ℂ³, M₃) = 5 [header expectation 4 FALSIFIED — arithmetic: M₃ = 2·1 ⊕ sgn ⊕ 3·2 ⇒ 2+3=5] and ⟨Ξ₃(D_φ), H_sgn⟩ = 0 for EVERY equivariant Ξ₃: the kill survives on the G₆ side  dim=5, projection=0
s1149: 19/19 checks passed; verdict PASS_S3_MARKING_ORBIT_DESCENT_CLOSED_BY_CHARACTER_COUNT
