sorkin.py — src/22-quantum

1## T1: Sorkin hierarchy in the FRC ledger.2## Claims verified by exact integer arithmetic (no floats):3##   (S1) second-order interference is generically nonzero (the theory interferes);4##   (S2) third- and fourth-order interference vanish identically on ledger states,5##        for every channel, outcome, and random state -- the Sorkin nullity;6##   (S3) sub-horizon (d*||psi||^2 < p) the carrier shadow determines the exact7##        counts, so the observed Sorkin combination is exactly zero;8##   (S4) super-window, the observed (lifted-shadow) Sorkin combination is always9##        an integer multiple of p: wrap deviations are quantised at the horizon,10##        and small nonzero kappa is forbidden at every scale.11from math import gcd1213def gm(z, w): return (z[0]*w[0] - z[1]*w[1], z[0]*w[1] + z[1]*w[0])14def ga(z, w): return (z[0]+w[0], z[1]+w[1])15def gc(z):    return (z[0], -z[1])16def gnorm(z): return z[0]*z[0] + z[1]*z[1]17I4 = [(1,0), (0,1), (-1,0), (0,-1)]1819def report(label, ok):20    print(('PASS ' if ok else 'FAIL ') + label)21    assert ok, label2223def subsets(slits):24    out = []25    for mask in range(1, 1 << len(slits)):26        out.append(tuple(i for i in range(len(slits)) if mask >> i & 1))27    return out2829def sorkin_run(p, e, nO, d, nslits, coeff):30    """Random nslits Z[i] components on the object cycle; full Sorkin hierarchy.31    coeff: callable returning one random Gaussian-integer coefficient."""32    N = p - 133    gO = pow(e, N//nO, p)34    HO = [pow(gO, j, p) for j in range(nO)]35    q  = nO // d                                    # number of outcomes36    coreg = nO // d                                 # fibres indexed mod nO/d37    # pointer characters on the core ~ Z_d via i-powers only valid for d=4:38    assert d == 439    slits = []40    for _ in range(nslits):41        slits.append({u: coeff() for u in HO})42    def amp(S, r, a):43        s = (0, 0)44        for l in range(4):45            u = pow(gO, a + coreg*l, p)46            comp = (0, 0)47            for i in S:48                comp = ga(comp, slits[i][u])49            s = ga(s, gm(gc(I4[(r*l) % 4]), comp))50        return s51    W = {}52    for S in subsets(range(nslits)):53        for r in range(4):54            for a in range(q):55                W[(S, r, a)] = gnorm(amp(S, r, a))56    return W, q5758def hier(W, q, k, nslits):59    """k-th order Sorkin combination for the first k slits, all channels/outcomes."""60    from itertools import combinations61    vals = []62    base = tuple(range(k))63    for r in range(4):64        for a in range(q):65            tot = 066            for m in range(1, k+1):67                sgn = (-1)**(k-m)68                for c in combinations(base, m):69                    tot += sgn * W[(tuple(sorted(c)), r, a)]70            vals.append(tot)71    return vals7273_c = [0]                                            # deterministic counter, no RNG74def _nxt():75    _c[0] += 1; return _c[0]76wide   = lambda: (((_nxt()*5) % 19) - 9, ((_nxt()*13) % 19) - 9)77binary = lambda: (_nxt() % 2, 0)78deep   = lambda: (7*(((_nxt()*5) % 19) - 9), 7*(((_nxt()*13) % 19) - 9))7980for (p, e, nO, tag) in [(157, 5, 12, '157/53/13'), (421, 2, 28, '421/61/29')]:81    W, q = sorkin_run(p, e, nO, 4, 4, coeff=wide)82    I2 = hier(W, q, 2, 4); I3 = hier(W, q, 3, 4); I4v = hier(W, q, 4, 4)83    report('%s: I2 generically nonzero (max |I2| = %d)' % (tag, max(map(abs, I2))),84           any(v != 0 for v in I2))85    report('%s: I3 = 0 exactly, all channels and outcomes' % tag,86           all(v == 0 for v in I3))87    report('%s: I4 = 0 exactly, all channels and outcomes' % tag,88           all(v == 0 for v in I4v))8990# (S3)/(S4): shadow readout. Lift L(w) = (w mod p); observed Sorkin combination91# from lifted shadows. Sub-horizon: zero exactly. Super-window: multiple of p.92p, e, nO = 157, 5, 1293W, q = sorkin_run(p, e, nO, 4, 3, coeff=binary)    # 0/1 coefficients: w <= 144 < p94sub = all(w < p for w in W.values())95I3hat = []96from itertools import combinations97for r in range(4):98    for a in range(q):99        tot = 0100        for m in range(1, 4):101            for c in combinations((0, 1, 2), m):102                tot += (-1)**(3-m) * (W[(tuple(sorted(c)), r, a)] % p)103        I3hat.append(tot)104report('157: sub-horizon tallies (all w < p): observed I3-hat = 0 exactly',105       sub and all(v == 0 for v in I3hat))106107W, q = sorkin_run(p, e, nO, 4, 3, coeff=deep)       # forced super-window tallies108big = any(w >= p for w in W.values())109I3hat, nz = [], 0110for r in range(4):111    for a in range(q):112        tot = 0113        for m in range(1, 4):114            for c in combinations((0, 1, 2), m):115                tot += (-1)**(3-m) * (W[(tuple(sorted(c)), r, a)] % p)116        I3hat.append(tot)117        if tot != 0: nz += 1118report('157: super-window tallies present and wrap quantisation I3-hat in pZ '119       '(nonzero cases: %d, all multiples of %d)' % (nz, p),120       big and all(v % p == 0 for v in I3hat) and nz > 0)121122print('sorkin: all checks passed')123