"""v97 -- Sheet-Conjugation Bridge (next.txt P1 follow-up): the branch-kernel Z2 of v96, the cusp-class conjugation of the Q-geometry (v69/v50) and the v92 chirality swap are ONE deck involution where the cover lives. [I]/[L] arithmetic; ONE remaining [P] identification. v92 proved: the two Lagrangian glues <(1,1)>, <(1,3)> (= the two chiralities) are swapped by EITHER single spinor conjugation and fixed by the simultaneous one. On the family side, conjugation acts on the cusp classes {0, 1/3, 2/3} (the Q_+ grading, v69) by negation: 0 fixed, 1/3 <-> 2/3. This script asks: does that conjugation have a generation- space realisation, and is it the same Z2 as the v96 branch-kernel transposition? RESULTS (all exact): [I] 1. ANCHOR FORCES THE CONJUGATION -- UNIQUELY. Q_+ has integer eigenvectors e1 = (0,0,1) [cusp 0], e2 = (0,1,2) [cusp 1/3], e3 = (1,0,0) [cusp 2/3]. Of the three possible "fix one, swap two" involutions, ONLY the cusp negation (fix e1, swap e2<->e3) admits an anchor-compatible scale: T.1 = 1 and T.a = a BOTH force the same scale s = 1, giving the integer involution T_A = [[0,1,0],[1,0,0],[2,-2,1]], T_A^2 = I, det T_A = -1. The two alternative pairings are REJECTED (T.a = a unsolvable). [I] 2. THE ANCHOR IS THE CONJUGATION-SYMMETRIC VECTOR: a = (1,1,2) = e2 + e3, 1 = a - e1, and the odd line of T_A is e2 - e3 = (-1,1,2). [I] 3. ONE DECK ACTION WHERE THE COVER LIVES: T_A acts as -1 on the quotient R^3/span{1,a} -- exactly like the v96 transposition sigma_12; the branch collapse direction (-1,1,0) is odd for both. (As matrices they differ -- T_A does not fix 1,a as covectors -- but the anchor block only sees the plane and its quotient.) [I] 4. PARITY MATCHES v92: det T_A = -1 = det sigma_12 (single conjugation = glue swap parity), det Sigma = +1 (simultaneous = glue-fixing parity). [L] 5. D4 CLOSURE: has order 8 with G = T_A*Sigma of order 4 and char(G) = (t+1)(t^2+1) -- the SAME D4 = Z4 |x Z2 as the established Q-geometry (v69) and the SAME char poly as the v70 mu4 homology matrix R_rot. The glue-swap reflection generates the D4 together with the sheet Sigma. [I] 6. SHEET-INDEX LEMMA: G and R_rot are rationally conjugate but NOT GL(3,Z)-conjugate: the intertwiner space S.G = R_rot.S has det S = 2*s8*(s6^2 + (s7+2*s8)^2) -- always EVEN, minimal |det S| = 2 = |Z2|. The generation-space mu4 and the puncture-homology mu4 differ by exactly one sheet doubling (and Q itself, det = N_fam, does NOT intertwine them: this is a new bridge, independent of v70's det Q = 3). HONEST TYPING: 1-6 are exact. The remaining [P] step of P1 is the lattice dictionary "Q_+ eigenspace grading of the generations = A3 discriminant grading", which would make T_A literally the v92 A3-side conjugation. Supporting evidence: the anchor rejects both alternative pairings (1), the parity matches (4), and the D4 closes (5) -- but the identification itself is not claimed. """ import itertools import sympy as sp from tfpt_constants import check, summary, reset Q = sp.Matrix([[3, 1, 0], [3, 2, 0], [3, 2, 1]]) SIG = sp.diag(1, -1, -1) QP = (Q + SIG * Q * SIG) / 2 ONE = sp.Matrix([1, 1, 1]) A = sp.Matrix([1, 1, 2]) RROT = sp.Matrix([[0, 0, -1], [1, 0, -1], [0, 1, -1]]) # v70 mu4 (homology) def integer_eigenvector(M, val): v = (M - val * sp.eye(3)).nullspace()[0] v = v / sp.gcd(tuple(v)) if sum(v) < 0: v = -v return v def swap_involution(fix, p, q_, scale): """Involution fixing `fix` and mapping p -> scale*q_, q_ -> p/scale.""" M = sp.Matrix.hstack(fix, p, q_) T = sp.Matrix.hstack(fix, scale * q_, p / scale) return T * M.inv() def run(): reset() print("v97 sheet-conjugation bridge (P1: one deck involution, one [P] left)") t, s = sp.symbols('t s') # 0. Q_+ integer eigenvectors and cusp dictionary lambda = 3w+1 e1 = integer_eigenvector(QP, 1) e2 = integer_eigenvector(QP, 2) e3 = integer_eigenvector(QP, 3) check("Q_+ integer eigenvectors: e1 = (0,0,1) [cusp 0], e2 = (0,1,2) " "[cusp 1/3], e3 = (1,0,0) [cusp 2/3]", e1 == sp.Matrix([0, 0, 1]) and e2 == sp.Matrix([0, 1, 2]) and e3 == sp.Matrix([1, 0, 0])) # 1. anchor forces the cusp negation, uniquely TA_s = swap_involution(e1, e2, e3, s) sol_one = sp.solve(TA_s * ONE - ONE, s, dict=True) sol_a = sp.solve(TA_s * A - A, s, dict=True) check("cusp negation (fix e1, swap e2<->e3): T.1 = 1 and T.a = a BOTH " "force the same scale s = 1", sol_one == [{s: 1}] and sol_a == [{s: 1}]) TA = sp.simplify(TA_s.subs(s, 1)) check("T_A = [[0,1,0],[1,0,0],[2,-2,1]] integer, T_A^2 = I, det = -1, " "fixes 1 and a", TA == sp.Matrix([[0, 1, 0], [1, 0, 0], [2, -2, 1]]) and TA * TA == sp.eye(3) and TA.det() == -1 and TA * ONE == ONE and TA * A == A) rejected = True for fix, p, q_ in ((e2, e1, e3), (e3, e1, e2)): T = swap_involution(fix, p, q_, s) rejected = rejected and sp.solve(T * A - A, s, dict=True) == [] check("the two alternative pairings (fix e2 or fix e3) admit NO " "anchor-compatible scale: T.a = a unsolvable -- the anchor " "selects the cusp negation uniquely", rejected) # 2. the anchor is the conjugation-symmetric vector odd = (TA + sp.eye(3)).nullspace() check("a = e2 + e3 (conjugation-symmetrisation of the swapped pair), " "1 = a - e1, odd line of T_A = e2 - e3 = (-1,1,2)", A == e2 + e3 and ONE == A - e1 and len(odd) == 1 and odd[0].cross(e2 - e3) == sp.zeros(3, 1) and TA * (e2 - e3) == -(e2 - e3)) # 3. one deck action on the quotient by the anchor plane d = sp.Matrix([-1, 1, 0]) sig12 = sp.Matrix([[0, 1, 0], [1, 0, 0], [0, 0, 1]]) co = sp.Matrix.hstack(ONE, A, d).solve(TA * d) check("T_A acts as -1 on R^3/span{1,a} (T_A.d = -d mod plane), exactly " "like sigma_12 (sigma_12.d = -d): ONE deck action where the " "cover lives", co[2] == -1 and sig12 * d == -d) # 4. parity matches v92 check("parity: det T_A = det sigma_12 = -1 (single conjugation = glue " "swap), det Sigma = +1 (simultaneous = glue-fixing)", TA.det() == -1 and sig12.det() == -1 and SIG.det() == 1) # 5. D4 closure G = TA * SIG els = {sp.ImmutableMatrix(sp.eye(3))} gens = [sp.ImmutableMatrix(TA), sp.ImmutableMatrix(SIG)] changed = True while changed: changed = False for e in list(els): for g in gens: for prod in (e * g, g * e): if prod not in els: els.add(prod) changed = True check("D4 CLOSURE: has order 8, G = T_A*Sigma has order 4, " "char(G) = (t+1)(t^2+1) = char(v70 mu4 matrix R_rot)", len(els) == 8 and G**4 == sp.eye(3) and G**2 != sp.eye(3) and sp.factor(G.charpoly(t).as_expr()) == (t + 1) * (t**2 + 1) and sp.factor(RROT.charpoly(t).as_expr()) == (t + 1) * (t**2 + 1)) # 6. sheet-index lemma: intertwiners all have even det, minimum |Z2| Svar = sp.Matrix(sp.symbols('s0:9')).reshape(3, 3) sols = sp.solve(sp.Matrix(Svar * G - RROT * Svar).vec(), list(Svar), dict=True)[0] Sgen = Svar.subs(sols) free = sorted(Sgen.free_symbols, key=str) detS = sp.factor(sp.expand(Sgen.det())) s6, s7, s8 = free check("intertwiner space S.G = R_rot.S has det S = 2*s8*(s6^2 + " "(s7+2*s8)^2): ALWAYS even -- G and R_rot are rationally but " "not GL(3,Z)-conjugate", sp.expand(detS - 2 * s8 * (s6**2 + (s7 + 2 * s8)**2)) == 0) dets = set() for vals in itertools.product(range(-2, 3), repeat=3): dv = detS.subs(dict(zip(free, vals))) if dv != 0: dets.add(abs(dv)) check("SHEET-INDEX LEMMA: minimal |det S| = 2 = |Z2| -- the generation " "mu4 and the homology mu4 differ by exactly one sheet doubling", min(dets) == 2) check("(and Q itself, det = N_fam, does NOT intertwine them: this " "det-2 bridge is independent of v70's det Q = 3)", sp.simplify(Q * G - RROT * Q) != sp.zeros(3, 3)) check("STATUS: branch-kernel Z2 (v96) = cusp conjugation = glue-swap " "parity, all [I]; the remaining [P] of P1 is the single lattice " "dictionary 'Q_+ grading = A3 discriminant grading'", True) return summary("v97 sheet-conjugation bridge") if __name__ == "__main__": raise SystemExit(1 if run() else 0)