"""v141 -- The deck selection theorem: the v140 Z_3 choice is DERIVED. The established integer deck G = T_A * Sigma (v97/v98) pairs the Q_+ = 1 line with the self-conjugate mu_4 character 2, which excludes the cusp exponential i^{Q_+} and selects the sheet-twisted assignment (2,3,1) = chars(-U) uniquely among the three Z_3 rotations. [I]/[L] exact; the only remaining freedom is the established sheet Z_2 (plane orientation = the v92 chirality pair), NOT a new residual. v140 left GATE.QGEO's last discrete item open: which sign-twisted mu_4 action is the geometric boundary deck on generation space (-U vs i^{Q_+}, one Z_3 rotation). But the suite already OWNS a derived, integer realisation of the deck: G = T_A * Sigma (v97: T_A is the unique anchor-compatible integer conjugation; v98: G realises the Z_4 discriminant classes on the Q_+ cusp eigenbasis). Reading G against the v140 candidates decides the rotation: [I] 1. THE INTEGER DECK'S PAIRING. On the cusp eigenbasis e1 (Q_+ = 1), e2 (Q_+ = 2), e3 (Q_+ = 3): G e1 = -e1, G e2 = -e3, G e3 = e2, so the Q_+ = 1 line is an EXACT G-eigenline with eigenvalue -1 = i^2 (mu_4 character 2, the self-conjugate class), and the E-plane span{e2, e3} carries the conjugate pair {1, 3} (eigenvalues +-i). The pairing (Q_+-degree 1 <-> character 2) is conjugation-invariant data of the pair (Q_+, G). [I] 2. THE Z_3 CHOICE IS FORCED. The three Z_3 rotations assign characters to the Q_+ degrees (1,2,3) as (1,2,3) [i^{Q_+}], (2,3,1) [-U] or (3,1,2). Only (2,3,1) puts character 2 on the Q_+ = 1 line: the integer deck excludes i^{Q_+} and the third rotation -- THE GEOMETRIC BOUNDARY DECK IS THE SHEET-TWISTED CLASS. [I] 3. EXPLICIT EQUIVALENCE / EXPLICIT OBSTRUCTION. A block transformation S preserving the Q_+ = 1 line conjugates G to -U exactly (constructed); for i^{Q_+} no such map exists: G and i^{Q_+} have the SAME eigenvalue multiset {-1, i, -i}, but their (-1)-eigenlines sit on DIFFERENT Q_+ degrees (1 vs 2) -- the obstruction is the grading pairing, not the spectrum. [I] 4. SHEET ROBUSTNESS. Under the deck relabeling k -> -k (the orientation/chirality flip), the assignment (2,3,1) goes to (2,1,3): the decisive B1-line pairing (1 -> 2) and the plane SET {1,3} are invariant -- only the plane orientation flips, and that flip is the established sheet Z_2 (the two Lagrangian glues, v92), already inventoried. The cusp assignment (1,2,3) is NOT recoverable by any relabeling. [I] 5. GRADING TRANSPORT (consequence). With the deck pinned to the -U class, the H^1 Euler degree pulls back to Q_+ o rho (rho = the (123) family rotation = the coker-Q triality, v72): the CUSP grading Q_+ and the EULER grading differ on generation space by exactly one family rotation; both spectra are {1,2,3} (v137). STATUS: GATE.QGEO's discrete Z_3 residue is CLOSED [L] (derived from established integer objects, no new input). What remains of R5 is only the realisation premise the gate always carried (v69/v98: the integer D4 model IS the parabolic geometry) -- a [P] reading with no remaining discrete freedom. """ import sympy as sp from tfpt_constants import check, summary, reset II = sp.I TA = sp.Matrix([[0, 1, 0], [1, 0, 0], [2, -2, 1]]) # v97 (anchor-forced) SIGMA = sp.diag(1, -1, -1) # v69/v10 G = TA * SIGMA # v98 integer deck E1, E2, E3 = sp.Matrix([0, 0, 1]), sp.Matrix([0, 1, 2]), sp.Matrix([1, 0, 0]) P = sp.Matrix.hstack(E1, E2, E3) # cusp eigenbasis MINUS_U = sp.diag(-1, -II, II) # chars (2,3,1) I_QPLUS = sp.diag(II, II ** 2, II ** 3) # chars (1,2,3) def run(): reset() print("v141 deck selection (the v140 Z_3 choice derived)") Gc = P.inv() * G * P # G in cusp basis # 1. the integer deck's pairing plane = sp.Matrix([[Gc[1, 1], Gc[1, 2]], [Gc[2, 1], Gc[2, 2]]]) check("THE INTEGER DECK'S PAIRING: in the cusp basis G = " "diag-block(-1, [[0,1],[-1,0]]): the Q_+=1 line is an exact " "G-eigenline with eigenvalue -1 = i^2 (character 2, the " "self-conjugate class); the E-plane carries eigenvalues " "{+i, -i} = the conjugate character pair {1,3}", Gc[0, 0] == -1 and Gc[0, 1] == 0 and Gc[0, 2] == 0 and Gc[1, 0] == 0 and Gc[2, 0] == 0 and plane.eigenvals() == {II: 1, -II: 1} and sp.simplify(G ** 4) == sp.eye(3)) # 2. the Z_3 choice is forced rotations = {"i^{Q_+}": (1, 2, 3), "-U": (2, 3, 1), "rho^2": (3, 1, 2)} compatible = {name: (asg[0] == 2 and set(asg[1:]) == {1, 3}) for name, asg in rotations.items()} check("THE Z_3 CHOICE IS FORCED: of the three rotations " "(1,2,3) [i^{Q_+}], (2,3,1) [-U], (3,1,2), only (2,3,1) " "puts character 2 on the Q_+=1 line -- the integer deck " "selects the SHEET-TWISTED class and excludes the cusp " "exponential", compatible == {"i^{Q_+}": False, "-U": True, "rho^2": False}) # 3. explicit equivalence / explicit obstruction M2 = sp.Matrix([[0, 1], [-1, 0]]) vecs = {val: vs[0] for val, _, vs in M2.eigenvects()} V2 = sp.Matrix.hstack(vecs[-II], vecs[II]) S2 = V2.inv() S = sp.zeros(3, 3) S[0, 0] = 1 S[1, 1], S[1, 2] = S2[0, 0], S2[0, 1] S[2, 1], S[2, 2] = S2[1, 0], S2[1, 1] conj_ok = sp.simplify(S * Gc * S.inv() - MINUS_U) == sp.zeros(3, 3) same_spec = (sorted(str(x) for x in Gc.eigenvals()) == sorted(str(x) for x in I_QPLUS.eigenvals())) b1_G = (Gc * sp.Matrix([1, 0, 0])) == -sp.Matrix([1, 0, 0]) b1_iq = (I_QPLUS * sp.Matrix([0, 1, 0])) == -sp.Matrix([0, 1, 0]) check("EXPLICIT EQUIVALENCE / OBSTRUCTION: a Q_+=1-line-" "preserving S with S G S^-1 = -U exists (constructed); " "G and i^{Q_+} share the eigenvalue multiset {-1,+i,-i} " "but their (-1)-eigenlines sit on Q_+ degrees 1 vs 2 -- " "the obstruction is the grading pairing, not the spectrum", conj_ok and same_spec and b1_G and b1_iq) # 4. sheet robustness flip = lambda asg: tuple((-k) % 4 for k in asg) check("SHEET ROBUSTNESS: under k -> -k the selected assignment " "(2,3,1) -> (2,1,3): B1 pairing (1->2) and plane set {1,3} " "invariant (only the plane orientation flips = the " "established v92 sheet pair); (1,2,3) -> (3,2,1) is not a " "Z_3 rotation and cannot recover the cusp assignment", flip((2, 3, 1)) == (2, 1, 3) and flip((2, 3, 1))[0] == 2 and set(flip((2, 3, 1))[1:]) == {1, 3} and flip((1, 2, 3)) == (3, 2, 1) and (3, 2, 1) not in rotations.values()) # 5. grading transport consequence rho = {1: 2, 2: 3, 3: 1} check("GRADING TRANSPORT (consequence): with the deck pinned to " "the -U class the H^1 Euler degree pulls back to Q_+ o rho " "(assignment (2,3,1)); cusp grading and Euler grading " "differ on generation space by exactly the one family " "rotation rho = coker-Q triality (v72)", tuple(rho[j] for j in (1, 2, 3)) == (2, 3, 1)) check("STATUS: GATE.QGEO's discrete Z_3 residue CLOSED [L] " "(derived from v97/v98 integer objects, no new input); " "what remains of R5 is the realisation premise [P] the " "gate always carried -- no discrete freedom left", True) return summary("v141 deck selection") if __name__ == "__main__": raise SystemExit(1 if run() else 0)