"""v422 -- The Galois<->Net bridge: the seam mu4 = Gal(Q(zeta5)) is the SAME cyclic Z/4 as the (E8)_1 simple-current glue -- not a mere order-4 coincidence. This connects the Galois "gearbox" of v419 to G_net / SEAM.EQUIV.01 (the (D5)_1 (x) (A3)_1 -> (E8)_1 extension). The discipline (v100): the match is to the CYCLIC Z/4 specifically, and cyclic-vs-Klein is LOAD-BEARING -- cyclic Z/4 gives E8 (det 1), the Klein Z2xZ2 gives D8/SO(16) (det 4). [E] lattice/Galois, [C] the canonical identification. [E] 1. THE CARRIER GLUE IS CYCLIC Z/4 (and D5 is cyclic because rank 5 is ODD). disc(A3) = coker(Cartan A3) = Z/4 and disc(D5) = Z/4 (one invariant factor 4 in the Smith normal form -- CYCLIC, not Klein), det = 4 each. D_n discriminant is Z/4 for n ODD and Z2xZ2 for n EVEN; the carrier is D5 (rank 5 = g_car, odd) -> cyclic Z/4; the control D8 (rank 8, even) -> Z2xZ2 (Klein). [E] 2. THE CARRIER-GALOIS IS THE SAME CYCLIC Z/4 (and 5 is what makes it so). Gal(Q(zeta5)) = (Z/5)^x = <2> is cyclic of order 4 (max multiplicative order 4, NOT Klein); among the atoms only the carrier prime 5 gives order 4: |(Z/2)^x|=1, |(Z/3)^x|=2, |(Z/5)^x|=4. [E] 3. THE E8 GLUE IS THE CYCLIC Z/4 <(1,1)>. In disc(D5 (+) A3) = Z4 x Z4 (order 16 = dim S^+) the simple current (1,1) has order 4 (cyclic <(1,1)>); the Lagrangian quotient is 16/|glue|^2 = 16/4^2 = 1, so the extension is UNIMODULAR = (E8)_1 (one primary, holomorphic, v89/v154). [E] 4. CYCLIC vs KLEIN IS LOAD-BEARING (negative control, v100). The order-4 Klein group Z2xZ2 has max element order 2 (NOT cyclic); the even-rank D8 carries it, and the order-2 halfway glue <(2,2)> gives 16/2^2 = 4 = |disc(D8)| = SO(16) (det Cartan = 4, four primaries), NOT E8 (det 1). So cyclic Z/4 -> E8, Klein/order-2 -> D8: the carrier-Galois being cyclic Z/4 (5 prime) matches the E8 glue, never the D8 Klein. [C] 5. THE BRIDGE = THE FOUR-FOLD mu4 IDENTITY. ONE cyclic Z/4 threads the FOUR mu4 readings: the seam DECK / divisor mu4 (z->iz on the 4-mark P^1, v419/ v177), the carrier-GALOIS Gal(Q(zeta5))=(Z/5)^x (v419), the DISCRIMINANT group disc(D5)=disc(A3)=Z/4 (bullet 1), and the SIMPLE-CURRENT glue <(1,1)> -> (E8)_1 (v1/v89/v154). So the Galois gearbox (Z/30)^x of v419 IS the simple-current glue of the holomorphic net -- a direct bridge from the Galois side to G_net (SEAM.EQUIV.01). All four are computably cyclic-Z/4 [E]; the canonical identification (one functorial Z/4) is the [C] bridge, cyclicity forced (5 odd/prime), the Klein alternative excluded. TYPING (v428): this four-fold identity is a COMPRESSION -- ONE object read four ways, a unification that makes the carrier less axiomatic -- NOT four independent witnesses that multiply over-determination. Mirrored in wolfram/tfpt_readouts_extension.wl (exact lattice/Galois facts). """ import sympy as sp from sympy.matrices.normalforms import smith_normal_form from tfpt_constants import check, summary, reset, g_car, dim_Splus # Cartan matrices A3 = sp.Matrix([[2, -1, 0], [-1, 2, -1], [0, -1, 2]]) D5 = sp.Matrix([[2, -1, 0, 0, 0], [-1, 2, -1, 0, 0], [0, -1, 2, -1, -1], [0, 0, -1, 2, 0], [0, 0, -1, 0, 2]]) D8 = sp.Matrix([[2, -1, 0, 0, 0, 0, 0, 0], [-1, 2, -1, 0, 0, 0, 0, 0], [0, -1, 2, -1, 0, 0, 0, 0], [0, 0, -1, 2, -1, 0, 0, 0], [0, 0, 0, -1, 2, -1, 0, 0], [0, 0, 0, 0, -1, 2, -1, -1], [0, 0, 0, 0, 0, -1, 2, 0], [0, 0, 0, 0, 0, -1, 0, 2]]) def invariant_factors(C): """The Smith-normal-form diagonal entries > 1 = the cyclic factors of coker(C).""" snf = smith_normal_form(C) return [snf[i, i] for i in range(C.shape[0]) if abs(snf[i, i]) > 1] def mult_order(u, n): o, x = 1, u % n while x != 1: x = (x * u) % n o += 1 return o def run(): reset() print("v422 Galois<->Net bridge: mu4 = Gal(Q(zeta5)) is the (E8)_1 glue Z/4") # ---- 1. the carrier glue is cyclic Z/4 (D5 cyclic because rank 5 is odd) ---- f_a3, f_d5, f_d8 = invariant_factors(A3), invariant_factors(D5), invariant_factors(D8) check("CARRIER GLUE CYCLIC Z/4 [E]: disc(A3)=Z/4 and disc(D5)=Z/4 (ONE " "invariant factor 4 = cyclic, not Klein), det=4 each; D_n disc is Z/4 " "for n ODD, Z2xZ2 for n EVEN -- carrier D5 (rank 5=g_car, odd) is " "cyclic, control D8 (rank 8, even) is Klein (factors %s)" % (f_d8,), f_a3 == [4] and f_d5 == [4] and abs(A3.det()) == 4 and abs(D5.det()) == 4 and f_d8 == [2, 2] and len(f_d5) == 1 and len(f_d8) == 2 and g_car == 5 and g_car % 2 == 1) # ---- 2. the carrier-Galois is the SAME cyclic Z/4 (5 makes it so) ---- ord5 = max(mult_order(u, 5) for u in (1, 2, 3, 4)) check("CARRIER-GALOIS CYCLIC Z/4 [E]: Gal(Q(zeta5))=(Z/5)^x=<2> is cyclic of " "order 4 (max mult order 4, not Klein); among the atoms only 5 gives " "order 4 (|(Z/2)^x|=1, |(Z/3)^x|=2, |(Z/5)^x|=4)", ord5 == 4 and sp.totient(5) == 4 and sp.is_primitive_root(2, 5) and sp.totient(2) == 1 and sp.totient(3) == 2) # ---- 3. the E8 glue is the cyclic Z/4 <(1,1)> ---- def order_in_Z4xZ4(a, b): return sp.ilcm(4 // sp.gcd(a, 4), 4 // sp.gcd(b, 4)) glue_order = order_in_Z4xZ4(1, 1) disc_carrier = 4 * 4 e8_disc = disc_carrier // glue_order**2 check("E8 GLUE CYCLIC Z/4 [E]: in disc(D5(+)A3)=Z4xZ4 (order 16=dim S^+) the " "simple current (1,1) has order 4 (cyclic <(1,1)>); the Lagrangian " "quotient 16/4^2 = 1 => the extension is unimodular = (E8)_1 (one " "primary, holomorphic, v89/v154)", glue_order == 4 and disc_carrier == 16 == dim_Splus and e8_disc == 1) # ---- 4. cyclic vs Klein is load-bearing (negative control) ---- halfway_order = order_in_Z4xZ4(2, 2) # <(2,2)> = order 2 d8_disc = disc_carrier // halfway_order**2 # 16/2^2 = 4 = |disc(D8)| check("CYCLIC vs KLEIN LOAD-BEARING [E] (neg control): the Klein Z2xZ2 has " "max element order 2 (not cyclic); the order-2 halfway glue <(2,2)> " "gives 16/2^2 = 4 = |disc(D8)| = SO(16) (det Cartan 4, four primaries), " "NOT E8 (det 1). So cyclic Z/4 -> E8, Klein/order-2 -> D8 -- the " "carrier-Galois (cyclic Z/4, 5 prime) matches E8, never the D8 Klein", halfway_order == 2 and d8_disc == 4 == abs(D8.det()) and e8_disc != d8_disc and f_d8 == [2, 2]) # ---- 5. the bridge = the four-fold mu4 identity (typed [C]; compression per v428) ---- check("FOUR-FOLD mu4 IDENTITY [C]: ONE cyclic Z/4 threads the FOUR mu4 readings " "-- the seam deck/divisor mu4 (z->iz, v419/v177), the carrier-Galois " "Gal(Q(zeta5)) (v419), the discriminant disc(D5)=disc(A3)=Z/4, and the " "(E8)_1 simple-current glue <(1,1)> (v1/v89/v154); the Galois gearbox " "(Z/30)^x of v419 IS the holomorphic-net glue (a direct Galois->G_net " "bridge). All four computably cyclic-Z/4 [E], the canonical identification " "[C] (cyclicity forced by 5, Klein excluded). Per v428 this is a " "COMPRESSION (one object read four ways), NOT four independent multipliers", f_d5 == f_a3 == [4] and ord5 == 4 and glue_order == 4) return summary("v422 Galois<->Net bridge / four-fold mu4 identity (deck=Galois=" "discriminant=simple-current = one cyclic Z/4 -> (E8)_1; Klein->D8 " "excluded; a compression per v428, not four witnesses)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)