decoherence.py — src/22-quantum

1## T6: the decoherence floor -- no spontaneous collapse, and the dilation floor.2## FRC is unitary below registration: the drive and every form-preserving3## propagator conserve ledger norms exactly, so an ISOLATED superposition keeps4## full fringe visibility at any mass, forever -- the framework forbids5## CSL/GRW-type spontaneous collapse outright (no parameter to tune). The only6## intrinsic floor is gravitational: mass is winding rate, so a composite's7## internal clocks read branch proper time, and a proper-time differential8## between interferometer arms writes which-path information into the internal9## state at exactly the rate of gravitational time-dilation decoherence10## (Pikovski et al. 2015) -- derived here from the FRC readings, with one11## FRC-specific discriminator: the spectrum is integer windings, so visibility12## REVIVES exactly at the cycle recurrence: dephasing, never collapse.13## Claims verified by exact arithmetic (Q(zeta_n), polys mod Phi_n):14##   (Q1) isolated doublet: fringe contrast = 1 exactly for every drive time and15##        every analyser setting sweep (unitarity: no intrinsic visibility loss);16##   (Q2) dilation dephasing: a cluster with internal winding distribution p_E17##        has branch visibility V(tau) = |sum_E p_E zeta^{E tau}| -- the18##        characteristic function of the internal energy distribution -- with19##        (a) the exact ring form verified, (b) Gaussian-envelope decay for a20##        binomial (thermal) distribution, matching the time-dilation law;21##   (Q3) exact recurrence: V(n) = 1 identically -- visibility revives at the22##        cycle period; decoherence in FRC is reversible dephasing, not collapse.23import numpy as np24from math import comb, gcd25from sympy import Poly, symbols, cyclotomic_poly, Rational2627def report(label, ok):28    print(('PASS ' if ok else 'FAIL ') + label)29    assert ok, label3031x = symbols('x')32n = 40                                                # internal winding cycle33PHI = Poly(cyclotomic_poly(n, x), x)34def zpow(k):  return Poly(x, x)**(k % n) % PHI35def zmul(a, b): return (a*b) % PHI36def zconj(a):37    out = Poly(0, x)38    for m, c in zip(a.monoms(), a.coeffs()):39        out = (out + c*zpow(-m[0])) % PHI40    return out4142# ---------------- Q1: isolated superposition keeps contrast 1 ----------------43# doublet (1, lambda) with lambda = zeta^j; drive tau multiplies the relative44# phase by zeta^{8 tau} (gap 8, as in the composite gate); analyser sweep j':45# w_+(tau, j') = |1 + zeta^{j + 8 tau + j'}|^2 in {0..4}: contrast (max-min)/46# (max+min) over the sweep = 1 exactly iff max = 4 and min = 0 both occur.47ok1 = True48for j in (0, 3, 11):49    for tau in (0, 1, 5, 17, 39):50        vals = []51        for jp in range(n):52            k = (j + 8*tau + jp) % n53            w = (Poly(2, x) + zpow(k) + zpow(-k)) % PHI      # |1+zeta^k|^254            vals.append(w)55        has4 = any(v.as_expr().equals(4) for v in vals)56        has0 = any(v.is_zero for v in vals)57        if not (has4 and has0): ok1 = False58report('Q1: isolated doublet contrast = 1 exactly, all drive times and phases '59       '(no spontaneous visibility loss at any mass: CSL forbidden)', ok1)6061# ---------------- Q2: dilation dephasing law ----------------62# internal energy distribution p_E: binomial on E = 0..M (thermal-like), shifted;63# branch visibility V(tau)^2 = |sum_E p_E zeta^{E tau}|^2, exact in the ring.64M = 865pE = {E: Rational(comb(M, E), 2**M) for E in range(M+1)}66def V2_exact(tau):67    s = Poly(0, x)68    for E, p in pE.items():69        s = (s + p*zpow(E*tau)) % PHI70    return zmul(s, zconj(s))71# (a) V(tau)^2 is a real, nonnegative ring element (a squared modulus), equal to 172#     at tau = 0 (full coherence, sum_E p_E = 1) -- exact ring identities.73ok2 = True74for tau in range(n):75    v2 = V2_exact(tau)76    if not ((v2 - zconj(v2)) % PHI).is_zero: ok2 = False         # V^2 is real (= its conjugate)77if not ((V2_exact(0) - Poly(1, x)) % PHI).is_zero: ok2 = False    # V(0)^2 = 1, full coherence78report('Q2a: V(tau)^2 = |characteristic function of internal windings|^2 is real for '79       'every drive time and equals 1 at tau = 0 (full coherence): exact ring identities', ok2)80# (b) Gaussian envelope: binomial variance M/4: V ~ exp(-sigma^2 theta^2/2)81ok2b = True82sig2 = M/483for tau in (1, 2, 3, 4):84    th = 2*np.pi*tau/n85    chi = abs(sum(float(p)*np.exp(1j*E*th) for E, p in pE.items()))86    gauss = np.exp(-sig2*th*th/2)87    if abs(chi - gauss) > 0.02: ok2b = False                 # envelope agreement88report('Q2b: Gaussian-envelope decay V = exp(-sigma^2 (omega tau)^2/2) -- the '89       'large-M numeric approximation of the exact ring characteristic function', ok2b)9091# ---------------- Q3: exact recurrence ----------------92v2 = V2_exact(n)93report('Q3: exact recurrence V(n)^2 = 1: visibility revives at the cycle '94       'period -- dephasing, never collapse', v2.as_expr().equals(1))9596# ---------------- Q4: distinct-eigenvalue superposition dephases to an integer count ----------------97# In 421/61/29, Object O_29 (cycle C_28, core Q4, quotient C_7, zeta7 = g_O^4), the98# same-channel superposition psi = psi_0 + psi_4 mixes windings of DISTINCT drive99# eigenvalues (1 and g_O^-4). The snapshot pair-count w_0(a) = |4(1+zeta7^a)|^2 =100# 16(2 + zeta7^a + zeta7^-a) is a real algebraic integer, NOT in Z. The REGISTERED101# weight is the drive-frequency count W_0(a) = sum_t |Psi_t^0(a)|^2 over the joint102# recurrence T = lcm(n_S, n_O) = lcm(60, 28) = 420; the cross term sums to zero,103# leaving an ordinary integer, uniform over the seven outcomes, so P = 1/7.104PHI7 = Poly(cyclotomic_poly(7, x), x)105def z7p(k):  return Poly(x, x)**(k % 7) % PHI7106def conj7(P):107    out = Poly(0, x)108    for m, c in zip(P.monoms(), P.coeffs()):109        out = (out + c*z7p(-m[0])) % PHI7110    return out111def isZ7(P):  return (P % PHI7).degree() <= 0112nO7 = 28113T7 = 60*nO7 // gcd(60, nO7)               # joint recurrence lcm(n_S, n_O) = lcm(60,28) = 420114snap, tot7 = [], Poly(0, x)115for a in range(7):116    Psi = (Poly(4, x)*(Poly(1, x) + z7p(a))) % PHI7117    w = (conj7(Psi)*Psi) % PHI7118    snap.append(w); tot7 = (tot7 + w) % PHI7119report('Q4a: snapshot pair-count w_0(1) = 16(2+zeta7+zeta7^-1) is a real algebraic '120       'integer, NOT in Z (the transient); sum over outcomes = 224 in Z',121       (not isZ7(snap[1])) and isZ7(tot7) and tot7.as_expr().equals(224))122W = []123for a in range(7):124    acc = Poly(0, x)125    for t in range(T7):126        Psi_t = (Poly(4, x)*(Poly(1, x) + z7p(a - t))) % PHI7127        acc = (acc + conj7(Psi_t)*Psi_t) % PHI7128    W.append(acc)129Wv = [int(w.as_expr()) for w in W]130report('Q4b: registered drive-frequency count W_0(alpha) = 13440 = 420*32 for every '131       'outcome (ordinary integer, uniform over T = lcm(60,28) = 420): the snapshot '132       'interference dephases over the recurrence',133       all(isZ7(w) for w in W) and len(set(Wv)) == 1 and Wv[0] == 13440)134report('Q4c: registered probability P_0(alpha) = 1/7 (rational); drive-average per '135       'step = 32 = incoherent diagonal sum |c0|^2+|c4|^2',136       Rational(Wv[0], sum(Wv)) == Rational(1, 7) and Wv[0] // T7 == 16 + 16)137138print('decoherence: all checks passed')139