Artian's A1-Faraday Verdet Reference Theorem: Material Rotation, the Conditional pi/8 Spinor Character, and Finite Terminal-Reference Closure

Can Faraday rotation separate a material response from a terminal-reference clock factor? For a homogeneous, local, optically nonmagnetic medium in linear response, the Faraday angle is fixed by the difference between the two circular propagation indices: \[ \Phi_F(\omega,B,L) =\frac{\omega L}{2c}\,[n_L(\omega,B)-n_R(\omega,B)]. \] The corresponding Verdet response functional is \[ V_{\mathcal M}(\omega,T) =\frac{\omega}{2c}\operatorname{Re} \left[ \frac{1}{n_0} \left.\frac{\partial g_{\mathcal M}}{\partial B}\right|_{B=0} \right]. \] Here \(g_{\mathcal M}\) is the material's gyrotropic off-diagonal response. Its Kubo representation depends on the material Hamiltonian, populations, transition matrix elements, and causal line structure. The Verdet coefficient is therefore a material response packet, not a universal pure number. The paper then isolates a different conditional object. If A1 assigns the source-clock/laboratory-clock plane the quarter-turn \(\Theta_A=\pi/4\), A4's spinor lift gives \[ U_A =\exp\!\left(-J\frac{\pi}{8}Z\right), \qquad c_A =\frac12\operatorname{Tr}U_A =\cos\frac{\pi}{8}. \] Version 1.1 closes the four-history exponent with a finite terminal-reference complex rather than an assigned lookup table: \[ 0\longrightarrow C_2^F \xrightarrow{D_F}C_1^F \xrightarrow{\partial_F}C_0^F \longrightarrow0, \qquad \partial_FD_F=0, \] \[ \kappa_F=\partial_F^{\mathsf T}s_F, \qquad n_{ab}=\kappa_F^{\mathsf T}p_{ab}, \qquad \chi_{ab}=n_{ab}^2, \] \[ (\chi_{LL},\chi_{LA},\chi_{AL},\chi_{AA})=(0,1,1,0). \] The completed square makes the count independent of whether the same reference conversion is represented at the input or output terminal. In the declared minimal integer-cochain class, the primitive clock cochain is unique up to global orientation. For a detector certified to be homogeneous of degree \(d\) in the affected amplitude, the material-corrected same-sample cross-ratio becomes \[ \widehat R_F^{(d)} =\exp[-(\Delta_K+\Delta_N)] \sqrt{\left|\frac{S_{LA}S_{AL}}{S_{LL}S_{AA}}\right|} =c_A^d. \] This yields the frozen paired target \[ H_{\rm std}:(1,1,1), \qquad H_{A1}:\left(c_A,c_A^2,1\right) \] for coherent, unnormalized-power, and normalized readouts. The material functional, conditional A1+A4 character, and finite path exponent are closed at their stated levels. Physical realization of the autonomous Faraday boundary and laboratory execution remain open. Stable anchors Quantum Traction Theory: Main Book v10.01 A1-CHSH Spinor-Character Theorem Absolute-Clock Two-Test Preregistration Absolute-Clock Laboratory Discriminator Atlas QTT Lexicon Derivation Atlas