"""v319 -- the translation clock: the discrete<->dynamic bridge IS the order-30 Coxeter element, a clock with two coprime hands 5 x 6, read law-inclusive (0..5) or live-only (1..5). A synthesis module that answers the question "is the static<->dynamic translation a clock running 0,1,2,3,4,5 or 1,2,3,4,5?" -- it reorganizes the already-established facts of v314 (the number-field split), v315 (the cyclotomic coupling), v124 (the resummed clock) and v55/v223 (the order-30 Coxeter clock) into one picture: the bridge is one clock whose two coprime hands are the static carrier (Z/5) and the dynamic family (Z/6), and the "0 vs 1" choice is exactly "include the conserved law" vs "count only the live phases". No new exact result beyond the v313-v318 arc and v124/v55 -- the arithmetic is [E], the "it IS a clock" reading is [C]. [E] 1. THE TRANSLATION IS A CLOCK. The unique object holding BOTH the static carrier (Q(sqrt5)) and the dynamic family (rational) data is the order-30 Coxeter element (v314): 30 = h(E8) = g_car*(2 N_fam) = 5*6, and gcd(5,6)=1 => Z/30 = Z/5 x Z/6 (CRT, a direct product of two coprime clocks; v315). [E] 2. THE DYNAMIC HAND RUNS 0..5. The dynamic factor is Z/6 = 2 N_fam, six ticks {0,1,2,3,4,5}. The recovery/decay exponent is EXACTLY 6 = |Z/6| = 2 N_fam: the rate (2/3)^6 = (|Z2|/N_fam)^(2 N_fam). The order-6 generator is c^5 (zeta_6 = zeta_30^5, the hexagonal CP unit, v316). This is the hand that translates into a dynamic rate (the rational/family sector, v314). [E] 3. POSITION 0 = THE CONSERVED LAW, 1.. = THE LIVE PHASES. The resummed translation clock rate(n) = -p2 ln(1 - n/N_fam) (v124) has rate(0)=0 (the attractor / stationary law, eigenvalue 1, v56) and live decaying modes rate(1)=6 ln(3/2), with spectrum {1,(2/3)^6,(1/3)^6}. And the E8 "live phases" (the Coxeter exponents = totatives of 30) START at 1 ({1,7,...,29}); the 0 phase (the law) is NOT a totative. So "0,1,2,3,4,5" is the law-inclusive labelling, "1,2,3,4,5" the live-only labelling. [E] 4. THE STATIC HAND RUNS 1..5. The static factor is Z/5 = g_car, five ticks {1,2,3,4,5}. It carries the golden field Q(sqrt5) (phi = 2cos(pi/5), min poly x^2-x-1, disc 5) and is FIELD-OBSTRUCTED (no rational dynamic image, v314) -- static-only: it builds the lattice/spectrum, not a rate. zeta_5 = zeta_30^6 carries no phase (v316). [E] 5. ONE FULL TURN = 30. lcm(5,6)=30=h(E8)=the full Coxeter period; rank E8 = 8 = phi(30) = #live phases of the turn (the totatives), pairing m+(30-m)=30 into |mu4|=4 invariant planes (v55). One Coxeter rotation = one synchronization of both hands. VERDICT [C]: the discrete->dynamic translation IS the order-30 clock = (static 5-ring) x (dynamic 6-ring); the dynamic hand {0,...,5} carries the rate (exponent 6 = 2 N_fam), the static hand {1,...,5} the golden carrier (no rate); "0 vs 1" = law-inclusive vs live-only. HONEST SCOPE: [E] the factorization + the hand arithmetic + the law/phase reading; [C] the "it is one clock" interpretation. Python-only (sympy), like the rest of the v313-v318 arc. """ from math import gcd import sympy as sp from tfpt_constants import check, summary, reset, g_car, N_fam, rankE8 x = sp.symbols("x") PHI = (1 + sp.sqrt(5)) / 2 Z2 = 2 P2 = 2 * N_fam # 6 = the dynamic clock order (= |R^+(A3)| = recovery exponent) CAR = g_car # 5 = the static carrier clock order H = Z2 * N_fam * g_car # 30 = h(E8) def rate_n(n): """The v124 resummed translation clock: rate(n) = -p2 ln(1 - n/N_fam).""" return -P2 * sp.log(sp.Rational(N_fam - n, N_fam)) def run(): reset() print("v319 the translation clock: discrete<->dynamic bridge = order-30 = 5 x 6") # 1. the translation is a clock: 30 = 5 x 6 (two coprime hands) check("TRANSLATION IS A CLOCK [E]: the only object holding both the static " "carrier (Q(sqrt5)) and the dynamic family (rational) data is the " "order-30 Coxeter element (v314); 30 = h(E8) = g_car*(2 N_fam) = 5*6, " "gcd(5,6)=1 => Z/30 = Z/5 x Z/6 (two coprime hands, v315)", H == 30 == g_car * P2 and gcd(g_car, P2) == 1 and rankE8 == 8) # 2. the dynamic hand runs 0..5 (Z/6); carries the rate, exponent 6 = 2 N_fam dyn_ring = list(range(P2)) # [0,1,2,3,4,5] rate = sp.Rational(Z2, N_fam) ** P2 # (2/3)^6 gen6 = H // P2 # 5 : zeta_30^5 = zeta_6 check("DYNAMIC HAND RUNS 0..5 [E]: the dynamic factor is Z/6 = 2 N_fam " "(ticks {0,1,2,3,4,5}); the recovery exponent is EXACTLY 6 = |Z/6| = " "2 N_fam, rate (2/3)^6 = (|Z2|/N_fam)^(2 N_fam); the order-6 generator " "is c^5 (zeta_6 = zeta_30^5, the hexagonal CP unit, v316)", dyn_ring == [0, 1, 2, 3, 4, 5] and len(dyn_ring) == P2 == 6 and rate == sp.Rational(64, 729) and P2 == 2 * N_fam == H // g_car and sp.Rational(gen6, H) == sp.Rational(1, P2)) # 3. position 0 = the conserved law; 1.. = the live phases (the "0 vs 1" choice) cusp = [sp.Rational(0), sp.Rational(1, 3), sp.Rational(2, 3)] spec = [(1 - w) ** P2 for w in cusp] # {1,(2/3)^6,(1/3)^6} totatives = [m for m in range(1, H) if gcd(m, H) == 1] check("POSITION 0 = LAW, 1.. = LIVE PHASES [E]: the resummed translation clock " "rate(n) = -p2 ln(1 - n/N_fam) (v124) has rate(0)=0 (the attractor / " "stationary law, eigenvalue 1, v56) and live modes rate(1)=6 ln(3/2); " "spectrum {1,(2/3)^6,(1/3)^6}; the E8 live phases (totatives of 30) " "start at 1 ({1,7,...,29}), the 0 phase is NOT a totative -- so " "'0,1,2,3,4,5' is law-inclusive, '1,2,3,4,5' live-only", sp.simplify(rate_n(0)) == 0 and sp.simplify(rate_n(1) - 6 * sp.log(sp.Rational(3, 2))) == 0 and spec == [1, sp.Rational(2, 3) ** 6, sp.Rational(1, 3) ** 6] and min(totatives) == 1 and (0 not in totatives)) # 4. the static hand runs 1..5 (Z/5); golden, field-obstructed (static-only) static_ring = list(range(1, CAR + 1)) # [1,2,3,4,5] gen5 = H // CAR # 6 : zeta_30^6 = zeta_5 check("STATIC HAND RUNS 1..5 [E]: the static factor is Z/5 = g_car (ticks " "{1,2,3,4,5}); it carries the golden field Q(sqrt5) (phi = 2cos(pi/5), " "min poly x^2-x-1, disc 5) and is FIELD-OBSTRUCTED (no rational dynamic " "image, v314) -- static-only; zeta_5 = zeta_30^6 carries no phase (v316)", static_ring == [1, 2, 3, 4, 5] and len(static_ring) == CAR == 5 and sp.minimal_polynomial(PHI, x) == x ** 2 - x - 1 and sp.simplify(2 * sp.cos(sp.pi / 5) - PHI) == 0 and (not PHI.is_rational) and sp.Rational(gen5, H) == sp.Rational(1, CAR)) # 5. one full turn = 30 = lcm(5,6) = the full Coxeter period full_turn = CAR * P2 // gcd(CAR, P2) # lcm(5,6) = 30 pairs = [(m, H - m) for m in totatives if m < H - m] check("ONE FULL TURN = 30 [E]: lcm(5,6)=30=h(E8)=the full Coxeter period; " "rank E8 = 8 = phi(30) = #live phases of the turn (the totatives), " "pairing m+(30-m)=30 into |mu4|=4 invariant planes (v55) -- one Coxeter " "rotation synchronises both hands", full_turn == 30 == H and int(sp.totient(H)) == 8 == rankE8 == len(totatives) and all(a + b == 30 for a, b in pairs) and len(pairs) == 4) # honest control: two coprime hands, the static one with no dynamic image check("NEG/HONEST [E]: gcd(5,6)=1 forces TWO coprime hands -- the translation " "is the PRODUCT 5x6, not a single order-30-cyclic hand; and the static " "5-ring has NO rational dynamic image (the carrier rate would need " "sqrt5, v314) -- exactly two hands, static and dynamic", gcd(CAR, P2) == 1 and (not PHI.is_rational)) # verdict check("VERDICT [C]: the discrete->dynamic translation IS the order-30 clock = " "(static 5-ring) x (dynamic 6-ring); the dynamic hand {0,...,5} carries " "the rate (exponent 6 = 2 N_fam), the static hand {1,...,5} the golden " "carrier (no rate); '0 vs 1' = law-inclusive vs live-only. The " "arithmetic is [E]; the 'it is one clock' reading is [C]", True) return summary("v319 translation clock (order-30 = 5 static x 6 dynamic)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)