## Exact arithmetic validation for "Quantum Observation in Finite Ring Continuum" ## Self-contained; integers only. Reproduces every numbered claim of the paper. from math import gcd, lcm def order(a, p): x, k = 1, 0 while True: x, k = (x * a) % p, k + 1 if x == 1: return k def report(label, ok): print(('PASS ' if ok else 'FAIL ') + label) assert ok, label # ---------- Z[i] ledger arithmetic (exact Gaussian integers) ---------- def gm(z, w): return (z[0]*w[0] - z[1]*w[1], z[0]*w[1] + z[1]*w[0]) def ga(z, w): return (z[0]+w[0], z[1]+w[1]) def gc(z): return (z[0], -z[1]) def gnorm(z): return z[0]*z[0] + z[1]*z[1] I4 = [(1,0), (0,1), (-1,0), (0,-1)] # i^l in Z[i] # ---------- Z[zeta_12] ledger (polys mod x^4 - x^2 + 1) ---------- def zmul(a, b): c = [0]*16 for i, ai in enumerate(a): if ai: for j, bj in enumerate(b): c[i+j] += ai*bj for k in range(15, 3, -1): if c[k]: c[k-2] += c[k]; c[k-4] -= c[k]; c[k] = 0 return c[:4] Z12 = {0: [1,0,0,0]} for k in range(1, 12): Z12[k] = zmul(Z12[k-1], [0,1,0,0]) def shell(p, e, nS, nO): N = p - 1 gS, gO = pow(e, N//nS, p), pow(e, N//nO, p) d = gcd(nS, nO) HS = {pow(gS, j, p) for j in range(nS)} HO = [pow(gO, j, p) for j in range(nO)] core = sorted(HS.intersection(HO)) return N, gS, gO, d, HS, HO, core # ===================== Carrier F_157, Subject 53, Object 13 ===================== p, e, nS, nO = 157, 5, 52, 12 N, gS, gO, d, HS, HO, core = shell(p, e, nS, nO) ip = pow(e, N//4, p) # oriented core element report('157: g=5 primitive', order(e, p) == N) report('157: core = {1,28,129,156}, |core| = 4 = gcd(52,12)', core == [1, 28, 129, 156] and d == 4) report('157: orientation I = g^{-39} = 28, I^2 = -1', pow(e, N - 39, p) == 28 and pow(28, 2, p) == p - 1) report('157: joint span lcm(52,12) = 156', lcm(nS, nO) == N) # Fibres: cosets of Q4 in Phi_O; h_alpha = gO^alpha, alpha = 0,1,2 fib = {a: [pow(gO, a + 3*l, p) for l in range(4)] for a in range(3)} report('157: three fibres partition Phi_O', sorted(sum(fib.values(), [])) == sorted(HO)) # Selection rule A(r,s) = 4*[r=s mod 4] over all 52x12 channel pairs ok = True for r in range(nS): for s in range(nO): A = sum(pow(e, ((N//4) * (s - r) * l) % N, p) for l in range(4)) % p if A != (4 if (r - s) % 4 == 0 else 0): ok = False report('157: selection rule A(r,s) over all 52x12 channels', ok) # Proposition (stationary states): psi_s(u) = u^s, drive u -> gO*u report('157: drive eigenstates U psi_s = gO^s psi_s, all s', all(all(pow((gO*u) % p, s, p) == (pow(gO, s, p)*pow(u, s, p)) % p for u in HO) for s in range(nO))) # Measurement vectors v_{r,a} in the Z[i] ledger; random Gaussian-integer state psi = {u: ((i % 13) - 6, (i % 11) - 5) for i, u in enumerate(HO)} # fixed state, no RNG red = lambda z: (z[0] + ip * z[1]) % p # Z[i] -> F_p, i -> 28 psip = {u: red(z) for u, z in psi.items()} def v(r, a): w = {u: (0, 0) for u in HO} for l in range(4): w[pow(gO, a + 3*l, p)] = I4[(r*l) % 4] return w def pair(x, y): s = (0, 0) for u in HO: s = ga(s, gm(gc(x[u]), y[u])) return s def amp(r, a): s = (0, 0) for l in range(4): s = ga(s, gm(gc(I4[(r*l) % 4]), psi[pow(gO, a + 3*l, p)])) return s def amp_fp(r, a): s = 0 for l in range(4): k = pow(ip, l, p) s = (s + pow(k, (4 - r) % 4, p) * psip[pow(gO, a + 3*l, p)]) % p return s report('157: basis orthogonality = 4*delta*delta', all(pair(v(r, a), v(r2, a2)) == ((4, 0) if (r, a) == (r2, a2) else (0, 0)) for r in range(4) for a in range(3) for r2 in range(4) for a2 in range(3))) lhs = sum(gnorm(amp(r, a)) for r in range(4) for a in range(3)) rhs = 4 * sum(gnorm(z) for z in psi.values()) report('157: Parseval sum |Psi|^2 = 4*||psi||^2 (%d = %d)' % (lhs, rhs), lhs == rhs) report('157: ledger reduction commutes, Z[i] -> F_157 on all channels', all(red(amp(r, a)) == amp_fp(r, a) for r in range(4) for a in range(3))) def proj(r, a, phi): A = (0, 0); vv = v(r, a) for l in range(4): u = pow(gO, a + 3*l, p) A = ga(A, gm(gc(vv[u]), phi[u])) return {u: gm(vv[u], A) for u in HO} phi1 = proj(2, 1, psi); phi2 = proj(2, 1, phi1) report('157: Lueders idempotence Pi^2 = 4*Pi', all(phi2[u] == gm((4, 0), phi1[u]) for u in HO)) # Z[zeta_12] ledger selection rule and its reduction (zeta -> gO = 22) okL, okR = True, True for s in range(12): for r in range(4): for a in range(3): S = [0, 0, 0, 0] for l in range(4): t = zmul(Z12[(s*(a + 3*l)) % 12], Z12[(-3*r*l) % 12]) S = [S[i] + t[i] for i in range(4)] tgt = [4*x for x in Z12[(s*a) % 12]] if (s - r) % 4 == 0 else [0]*4 if S != tgt: okL = False Sp = sum(c * pow(gO, k, p) for k, c in enumerate(S)) % p Tp = (4 * pow(gO, s*a, p)) % p if (s - r) % 4 == 0 else 0 if Sp != Tp: okR = False report('157: ledger selection rule Psi = 4*zeta^{s a} iff r = s mod 4', okL) report('157: ledger reduction matches carrier amplitudes (zeta -> 22)', okR) # ===================== Carrier F_421: two regimes ===================== p2, e2 = 421, 2 N2 = p2 - 1 report('421: g=2 primitive', order(e2, p2) == N2) # quantum pair: Subject 61 (C60) reads Object 29 (C28) _, _, gO2, d2, HS2, HO2, core2 = shell(p2, e2, 60, 28) report('421: quantum pair core = {1,29,392,420}, gcd = 4, lcm = 420', core2 == [1, 29, 392, 420] and d2 == 4 and lcm(60, 28) == 420) report('421: orientation I = g^{-105} = 2^{-105} = 392, I^2 = -1', pow(e2, N2 - 105, p2) == 392 and pow(392, 2, p2) == p2 - 1) ok = True for r in range(60): for s in range(28): A = sum(pow(e2, (105 * (s - r) * l) % N2, p2) for l in range(4)) % p2 if A != (4 if (r - s) % 4 == 0 else 0): ok = False report('421: selection rule over all 60x28 channels', ok) report('421: seven outcomes C28/Q4 = C7', 28 // d2 == 7) # classical pair: Subject 61 (C60) absorbs Object 13 (C12) _, _, _, d3, HS3, HO3, core3 = shell(p2, e2, 60, 12) report('421: classical pair gcd(60,12) = 12: total absorption, C1', d3 == 12 and set(HO3) <= HS3 and 12 // d3 == 1) print('all checks passed')