"""v416 -- The atom trichotomy: {2,3,5} are ramified / inert / split in the two exceptional CM rings, and each atom is the unique ramified prime of one quadratic facet of the order-30 clock field (v403). This sharpens v222/v230 (which read the atoms only as CM NORMS) and v390/v394/v403 (the three facets) into one prime-splitting table. [E] number theory, [C] the seam/flavor/golden reading. The atoms are e3,p0,e2 of the anchor a=(1,1,2): |Z2|=2, N_fam=3, g_car=5, with product 2*3*5 = 30 = h(E8) (v394). [E] 1. THE Z[i] (SQUARE / SEAM, tau=i) TRICHOTOMY: 2 = |Z2| RAMIFIES (2 = -i(1+i)^2; 2 | disc Q(i) = -4), 3 = N_fam INERT (3 = 3 mod 4; norm 9 = N_fam^2), 5 = g_car SPLITS (5 = (2+i)(2-i); 5 = 1 mod 4). So on the square seam the carrier rank g_car=5 FACTORS. [E] 2. THE Z[omega] (HEX / FLAVOR, tau=omega) TRICHOTOMY: 3 = N_fam RAMIFIES (3 = -omega^2 (1-omega)^2; N(1-omega)=3), 2 = |Z2| INERT (2 = 2 mod 3; norm 4), 5 = g_car INERT (5 = 2 mod 3; norm 25 = g_car^2). [E] 3. RAMIFIED PRIME = THE 'OWN' ATOM. The discriminant prime of each ring is exactly the atom that belongs to it: Z[i] ramifies at 2 = |Z2| (sheet/seam), Z[omega] ramifies at 3 = N_fam (family/flavor). [E] 4. THE THREE FACETS (v403) RAMIFY OVER ONE ATOM EACH: Q(i) disc -4 -> ramified only over 2 (square/seam, mu4), Q(sqrt-3) disc -3 -> ramified only over 3 (hex/flavor, CP zeta6), Q(sqrt5) disc 5 -> ramified only over 5 (REAL/golden, g_car, v313), product of ramified primes 2*3*5 = 30 = h(E8); two imaginary (CM / dynamic) {2,3} + one real (RM / static) {5} = the v403 reading at the prime level. Mirrored in wolfram/tfpt_readouts_extension.wl (exact arithmetic only). """ import sympy as sp from tfpt_constants import check, summary, reset, g_car, N_fam i = sp.I H_E8 = 30 def N_gauss(a, b): """Norm in Z[i]: |a + b i|^2 = a^2 + b^2.""" return a * a + b * b def N_eisen(a, b): """Norm in Z[omega], omega = e^{2 pi i/3}: |a + b omega|^2 = a^2 - a b + b^2.""" return a * a - a * b + b * b def run(): reset() print("v416 atom trichotomy: {2,3,5} ramified/inert/split in Z[i] vs " "Z[omega]; each atom one facet") # the atoms are the anchor elementary-symmetric data (v394) check("ATOMS = anchor data [E]: (|Z2|, N_fam, g_car) = (2,3,5), product " "2*3*5 = 30 = h(E8)", (2, N_fam, g_car) == (2, 3, 5) and 2 * 3 * 5 == H_E8) # ---- 1. Z[i] trichotomy ---- check("Z[i] TRICHOTOMY [E] (square/seam, tau=i): 2 RAMIFIES " "(2 = -i(1+i)^2), 3 INERT (3 = 3 mod 4, norm 9 = N_fam^2), 5 SPLITS " "(5 = (2+i)(2-i), 5 = 1 mod 4) -- g_car=5 factors on the seam", sp.expand(-i * (1 + i)**2) == 2 and 3 % 4 == 3 and N_gauss(3, 0) == 9 == N_fam**2 and sp.expand((2 + i) * (2 - i)) == 5 and 5 % 4 == 1 and N_gauss(2, 1) == g_car) # ---- 2. Z[omega] trichotomy ---- check("Z[omega] TRICHOTOMY [E] (hex/flavor, tau=omega): 3 RAMIFIES " "(N(1-omega) = 3), 2 INERT (2 = 2 mod 3, norm 4), 5 INERT " "(5 = 2 mod 3, norm 25 = g_car^2)", N_eisen(1, -1) == 3 == N_fam and 2 % 3 == 2 and N_gauss(2, 0) == 4 and 5 % 3 == 2 and N_eisen(5, 0) == 25 == g_car**2) # ---- 3. ramified prime = the 'own' atom ---- check("RAMIFIED PRIME = OWN ATOM [E]: Z[i] ramifies at 2 = |Z2| " "(sheet/seam), Z[omega] ramifies at 3 = N_fam (family/flavor) -- " "each ring's discriminant prime is its atom", sp.factorint(4) == {2: 2} and sp.factorint(3) == {3: 1} and 2 == 2 and 3 == N_fam) # ---- 4. the three facets ramify over one atom each (v403) ---- facets = { "Q(i)": (-4, 2), "Q(sqrt-3)": (-3, 3), "Q(sqrt5)": (5, 5), } ram_ok = all(set(sp.factorint(abs(disc)).keys()) == {atom} for disc, atom in facets.values()) check("THE THREE FACETS (v403) [E]: Q(i) disc -4 -> ram 2 (square/seam), " "Q(sqrt-3) disc -3 -> ram 3 (hex/flavor), Q(sqrt5) disc 5 -> ram 5 " "(real/golden, v313); each ramifies over exactly ONE atom; product " "2*3*5 = 30 = h(E8); 2 imaginary (CM/dynamic) + 1 real (RM/static)", ram_ok and (-4) % 4 == 0 and (-3) % 4 == 1 and 5 % 4 == 1 # disc residues and 2 * 3 * 5 == H_E8) # ---- 5. negative control: the ring assignment is rigid ---- check("NEG CONTROL [E]: the rings are not interchangeable -- N_i(2,1)=5 " "but N_omega(2,1)=3; 5 splits in Z[i] yet is inert in Z[omega]; the " "square<->seam / hex<->flavor split (v222) is rigid", N_gauss(2, 1) == 5 and N_eisen(2, 1) == 3 and 5 % 4 == 1 and 5 % 3 == 2) return summary("v416 atom trichotomy ({2,3,5} ram/inert/split; " "each atom one facet of Q(i,sqrt-3,sqrt5))") if __name__ == "__main__": raise SystemExit(1 if run() else 0)