# Quantum Reference Frames and Single-Outcome Dynamics — A Compatibility Framework for Generative Presentism

Part 35 of 40 (1,323 words)



# ---- 15. equal ensembles, different realized-history measures (two unravellings)

plus = np.array([1, 1])/np.sqrt(2); minus = np.array([1, -1])/np.sqrt(2)

I1 = [np.diag([1, 0]), np.diag([0, 1])]

I2 = [np.eye(2)/np.sqrt(2), np.diag([1, -1])/np.sqrt(2)]

Xm = [np.outer(plus, plus), np.outer(minus, minus)]

rho = rq.normal(size=(2, 2)) + 1j*rq.normal(size=(2, 2))

rho = rho @ rho.conj().T; rho /= np.trace(rho)

ch = lambda I, r: sum(K @ r @ K.conj().T for K in I)

same_channel = np.allclose(ch(I1, rho), ch(I2, rho))

def jd(I):

    return np.array([[np.linalg.norm(M @ K @ plus)**2 for M in Xm] for K in I])

J1, J2 = jd(I1), jd(I2)

def mi(Jm):

    pa, pb = Jm.sum(1), Jm.sum(0)

    return sum(Jm[i, j]*np.log2(Jm[i, j]/(pa[i]*pb[j])) for i in range(2) for j in range(2)

               if Jm[i, j] > 1e-15)

print(f"15. same channel: {same_channel}; joint law of (first event, later X outcome):")

print(f"    {J1.round(3).tolist()} vs {J2.round(3).tolist()};")

print(f"    mutual information {mi(J1):.3f} vs {mi(J2):.3f} bits")



# ---- 16. imperfect quantum clocks: identity of events vs clock-relative timing

Mc, th = 32, 0.9

Sc = np.roll(np.eye(Mc), 1, axis=0)

Uc = Sc @ expm(-1j*th*(Sc + Sc.T))

step = np.abs(Uc[:, 0])**2                               # law of one recorded clock step

signed = [(m if m <= Mc//2 else m - Mc) for m in range(Mc)]

p_inv = sum(step[m] for m in range(Mc) if signed[m] < 0)

p_stall = step[0]

from scipy.special import jv

bes_inv = (1 - jv(0, 2*th)**2 - 2*jv(1, 2*th)**2)/2; bes_stall = jv(1, 2*th)**2

far = sum(step[m] for m in range(Mc) if abs(signed[m]) >= 8)

print(f"16. drifting quantum clock, exact on the 32-site ring (signed displacement in -15..16):")

print(f"    P(backward) = {p_inv:.9f}, P(no advance) = {p_stall:.9f};")

print(f"    infinite-lattice Bessel values {bes_inv:.9f}, {bes_stall:.9f}")

print(f"    max difference = {max(abs(p_inv-bes_inv), abs(p_stall-bes_stall)):.1e}; "

      f"P(|displacement| >= 8) = {far:.1e}")

per = [n % 2 for n in range(5)]

print(f"    period-2 counter readings at stages 0-4: {per};")

print("    superposed counter: reading at stage n is n or n+4, each with probability 0.5")





# ---- 17. gapless detectors on a free field: three-region (Sorkin-type) signaling test

# two-mode realization of the commutation pattern E_AB = 0, E_AC = E_CB = 1:

# phi_A = q1, phi_C = p1 + q2, phi_B = p2; vacuum state; Fock cutoff nc per mode

def sorkin(nc, lamA=0.9, lamC=0.8, lamB=0.7, kick=1.5):

    am = np.diag(np.sqrt(np.arange(1, nc)), 1); I_ = np.eye(nc)

    qm = (am + am.T)/np.sqrt(2); pm = 1j*(am.T - am)/np.sqrt(2)

    phA, phC, phB = np.kron(qm, I_), np.kron(pm, I_) + np.kron(I_, qm), np.kron(I_, pm)

    v0 = np.zeros(nc*nc, complex); v0[0] = 1

    def fn(Hm, f):

        w, V = np.linalg.eigh(Hm); return (V * f(w)) @ V.conj().T

    det = lambda ph, l: [fn(ph, lambda x: np.cos(l*x)), 1j*fn(ph, lambda x: np.sin(l*x))]

    gg = lambda x: 0.5*(1 + np.tanh(2*x))

    lud = lambda ph: [fn(ph, lambda x: np.sqrt(gg(x))), fn(ph, lambda x: np.sqrt(1 - gg(x)))]

    def bob(KA, KC, KB):

        pb = 0.0

        for Ka in KA:

            va = Ka @ v0

            for Kc in KC: pb += np.linalg.norm(KB[1] @ (Kc @ va))**2

        return pb

    KB = det(phB, lamB); Id = [np.eye(nc*nc)]; kA = [fn(phA, lambda x: np.exp(1j*kick*x))]

    out = {}

    for nm, KC in (('gapless', det(phC, lamC)), ('luders', lud(phC))):

        base = bob(Id, KC, KB)

        out[nm] = (base, bob(det(phA, lamA), KC, KB) - base, bob(kA, KC, KB) - base)

    return out

s40, s50 = sorkin(40), sorkin(50)

ana = 0.5 - 0.5*np.cos(2*0.7*0.8)*np.exp(-0.7**2)

print(f"17. three-region test, Bob's P(b=1) and its shift under Alice's detector / kick:")

print(f"    gapless Charlie: P = {s50['gapless'][0]:.10f} (analytic {ana:.10f}); shifts "

      f"{s50['gapless'][1]:.1e}, {s50['gapless'][2]:.1e}")

print(f"    unsharp Lueders Charlie: P = {s50['luders'][0]:.6f}; shifts "

      f"{s50['luders'][1]:.6f}, {s50['luders'][2]:.6f}")

print(f"    cutoff 40 vs 50: max change = "

      f"{max(abs(s40[k][i] - s50[k][i]) for k in s40 for i in range(3)):.1e}")



# ---- 18. Gaussian collapse of a smeared field at C (Weyl-diagonal): same three-region test

def gauss_test(nc=40, lamA=0.9, lamB=0.7, kick=1.5, sig=0.6, p_hit=1.0, dy=0.02):

    am = np.diag(np.sqrt(np.arange(1, nc)), 1); I_ = np.eye(nc)

    qm = (am + am.T)/np.sqrt(2); pm = 1j*(am.T - am)/np.sqrt(2)

    phA, phC, phB = np.kron(qm, I_), np.kron(pm, I_) + np.kron(I_, qm), np.kron(I_, pm)

    v0 = np.zeros(nc*nc, complex); v0[0] = 1

    def fn(Hm, f):

        w, V = np.linalg.eigh(Hm); return (V * f(w)) @ V.conj().T

    wC, VC = np.linalg.eigh(phC); ys = np.arange(-10, 10 + dy/2, dy)

    KB1 = 1j*fn(phB, lambda x: np.sin(lamB*x))

    def bob(KA):

        pb = 0.0

        for Ka in KA:

            w = VC.conj().T @ (Ka @ v0)

            if p_hit < 1: pb += (1 - p_hit)*np.linalg.norm(KB1 @ (VC @ w))**2

            for y in ys:

                k = (2*np.pi*sig**2)**-0.25*np.exp(-(wC - y)**2/(4*sig**2))

                pb += p_hit*dy*np.linalg.norm(KB1 @ (VC @ (k*w)))**2

        return pb

    detA = [fn(phA, lambda x: np.cos(lamA*x)), 1j*fn(phA, lambda x: np.sin(lamA*x))]

    base = bob([np.eye(nc*nc)])

    ana = 0.5 - 0.5*np.exp(-lamB**2)*((1 - p_hit) + p_hit*np.exp(-lamB**2/(2*sig**2)))

    shifts = (bob(detA) - base, bob([fn(phA, lambda x: np.exp(1j*kick*x))]) - base)

    return base, ana, shifts

for ph in (1.0, 0.4):

    b_, a_, sh = gauss_test(p_hit=ph)

    print(f"18. Gaussian collapse at C (hit probability {ph}): P = {b_:.10f} "

          f"(analytic {a_:.10f});")

    print(f"    shifts under Alice's detector / kick: {sh[0]:.1e}, {sh[1]:.1e}")

import sys, scipy

print(f"Environment: Python {sys.version.split()[0]}, NumPy {np.__version__}, "

      f"SciPy {scipy.__version__}")

print("Seeds: 7 (initial state; present state of item 11); 1000-1199 (item 9); "

      "2026 (items 14-15)")



Output:



1. reductions unitary; V_AB = explicit permutation: True

   groupoid law V_BC V_AB = V_AC: True

2. instrument complete: True | all generators G-invariant: True

3. intertwining V K^A V^dag = K^B for W and every Kraus operator: True

4. 4096 histories of 3 stages; max |p_A-p_B| = 2.78e-17, max |p_A-p_neutral| = 1.39e-17

   max(1-|<V phi_A|phi_B>|) = 6.66e-16, total prob = 1.000000000000

5. naive same-form frame-B rule: total-variation distance from correct record measure = 0.180

6. records [(0, 1, 1), (0, 2, 3)] vs [(0, 2, 3), (0, 1, 1)] -> same present ray: True

7. entanglement entropy: frame A 0.000 bits, frame B 1.000 bits

8. absolute collapse: ||(1-Pi) P_abs Psi|| / ||P_abs Psi|| = 0.894

9. continuous time: 200 trajectories, 1158 events; identical event records in A and B: True

   max(1-|<V psi_A|psi_B>|) at final time = 6.7e-16

10. generic parameters, 4096 three-stage histories: exact coincidences = 0; largest overlap = 0.9852

    minimum error probability for telling that pair apart (Helstrom) = 0.414

11. two different 3-stage histories, same present ray: |<end|Phi>| = 1.000000000000, 1.000000000000

    history probabilities 4.15e-05, 7.62e-05; overlap of the two initial states = 0.552

12a. finite width: both branches end at the same ray (|<end|Phi>| = 1.000000000000, 1.000000000000),

     probabilities 5.04e-05, 1.62e-03; next-event total variation = 0.1000 (lambda/6 = 0.1000)

12b. sharp, fixed initial state: same present ray: True; probabilities 8.254e-04, 1.238e-03;

     next-event total variation = 0.1000

13. conditioned on (1, 2, 0) at stage 3: Doob forward kernel vs conditioned law, max diff = 8.3e-17

14. causally local instruments: all 136 foliations (linear extensions), 256 outcome histories;

    max |p - p'| = 6.9e-18, max state deviation = 4.4e-16

    instruments changed outside the predecessor set of slot 3: max change = 7.8e-16

    order ideal [0, 1, 3]: law from its own instruments vs marginal of full law, max diff = 1.1e-16

    non-commuting 'spacelike' pair: total variation between orders = 0.165

15. same channel: True; joint law of (first event, later X outcome):

    [[0.25, 0.25], [0.25, 0.25]] vs [[0.5, 0.0], [0.0, 0.5]];

    mutual information 0.000 vs 1.000 bits

16. drifting quantum clock, exact on the 32-site ring (signed displacement in -15..16):

    P(backward) = 0.104042655, P(no advance) = 0.338161965;

    infinite-lattice Bessel values 0.104042655, 0.338161965

    max difference = 1.1e-16; P(|displacement| >= 8) = 7.4e-09

    period-2 counter readings at stages 0-4: [0, 1, 0, 1, 0];

    superposed counter: reading at stage n is n or n+4, each with probability 0.5

17. three-region test, Bob's P(b=1) and its shift under Alice's detector / kick:

    gapless Charlie: P = 0.3665447170 (analytic 0.3665447170); shifts -2.2e-16, -6.1e-16

    unsharp Lueders Charlie: P = 0.275038; shifts -0.021492, -0.046628

    cutoff 40 vs 50: max change = 8.8e-08

18. Gaussian collapse at C (hit probability 1.0): P = 0.3449027185 (analytic 0.3449027185);

    shifts under Alice's detector / kick: -3.3e-16, -5.0e-14

18. Gaussian collapse at C (hit probability 0.4): P = 0.2541731691 (analytic 0.2541731691);

    shifts under Alice's detector / kick: -8.3e-16, -2.0e-14

Environment: Python 3.13.16, NumPy 2.5.3, SciPy 1.18.1

Seeds: 7 (initial state; present state of item 11); 1000-1199 (item 9); 2026 (items 14-15)



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