TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/mixed-current-origin-20261014/check_continuous_time_sector_test.py
Original file SHA-256: bfab38f6ed057f368a28a037c068b26cfde3aadc856583ee79363de7f2244679
Inclusive original lines: 44-125
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
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000044 |     norm_dist2 = 2 - 2*s.sqrt(1-mass)
000045 |     need(expectation == n, 'first moment diverges')
000046 |     need(norm_dist2 <= 2*mass, 'state still norm-converges to vacuum')
000047 |     first_moment_controls.append({'N': n, 'mean_energy': str(expectation),
000048 |                                   'positive_time_norm_defect_bound': str(mass)})
000049 | 
000050 | # Off-grade weight identity, no assumption of a homogeneous microscopic field.
000051 | Z = s.diag(1, s.I, -1, -s.I)
000052 | v = s.Matrix([s.Rational(1, 3), s.I/s.Integer(4),
000053 |               -s.Rational(1, 5), (1+s.I)/7])
000054 | grade_cases = []
000055 | for r0 in range(4):
000056 |     defect = (Z - s.I**r0*s.eye(4))*v
000057 |     lhs = s.expand((defect.H*defect)[0])
000058 |     rhs = sum(s.Abs(s.I**r-s.I**r0)**2*s.Abs(v[r])**2 for r in range(4))
000059 |     off_norm = sum(s.Abs(v[r])**2 for r in range(4) if r != r0)
000060 |     need(s.simplify(lhs-rhs) == 0, 'exact register grade identity')
000061 |     need(s.simplify(lhs-2*off_norm) >= 0, 'off-grade norm bound')
000062 |     grade_cases.append({'initial_register': r0, 'commutator_norm2': str(lhs)})
000063 | 
000064 | # Closed heat-filtered pair formulas and the noncommuting sharp/time limits.
000065 | r, u = s.symbols('r u', positive=True)
000066 | F = ((1+r*u)*(1-r)/((1-r*u)*(1+r)))**4
000067 | need(s.cancel(F.subs(u, 1)-1) == 0, 'finite heat time has norm1')
000068 | need(s.limit(F.subs(u, s.Rational(1, 2)), r, 1, dir='-') == 0,
000069 |      'sharp limit loses all positive-time norm')
000070 | need(s.limit(F.subs(u, s.Rational(1, 2))/(1-r)**4, r, 1, dir='-')
000071 |      == s.Rational(81, 16), 'quantitative sharp-limit rate')
000072 | need(s.cancel((-u*s.diff(F,u)).subs(u,1) + 8*r/(1-r*r)) == 0,
000073 |      'time derivative gives the energy')
000074 | heat_controls = []
000075 | for rv in (s.Rational(1, 2), s.Rational(3, 4), s.Rational(7, 8), s.Rational(15, 16)):
000076 |     exact_energy = 8*rv/(1-rv*rv)
000077 |     exact_kernel = s.cancel(F.subs({r:rv,u:s.Rational(1,2)}))
000078 |     heat_controls.append({'exp_minus_2s': str(rv), 'mean_K': str(exact_energy),
000079 |                           'F_s_log2': str(exact_kernel)})
000080 | for stime, d, t in ((0.3, math.pi, 0.4), (0.7, 0.9, 0.2)):
000081 |     n = np.arange(1, 401, dtype=float)
000082 |     q = math.exp(-2*stime)
000083 |     amplitude2 = 4*np.exp(-2*stime*n)*(1-np.cos(n*d))/n
000084 |     e_sum = float(np.sum(n*amplitude2))
000085 |     e_closed = 4*(q/(1-q)-(q*np.exp(1j*d)/(1-q*np.exp(1j*d))).real)
000086 |     b = lambda x: 4*math.log(abs(1-x*np.exp(1j*d))/(1-x))
000087 |     f_sum = math.exp(-float(np.sum(amplitude2*(1-np.exp(-t*n)))))
000088 |     f_closed = math.exp(b(q*math.exp(-t))-b(q))
000089 |     need(abs(e_sum-e_closed) < 1e-11, 'heat energy exact geometric series')
000090 |     need(abs(f_sum-f_closed) < 1e-11, 'heat time kernel exact logarithmic sum')
000091 | 
000092 | # Actual source energies in all four original register backgrounds.
000093 | sys.path.insert(0, str(REPO/'verification'))
000094 | src = REPO/'verification/v1033_charged_disorder.py'
000095 | spec = importlib.util.spec_from_file_location('continuous_sector_v1033', src)
000096 | mod = importlib.util.module_from_spec(spec)
000097 | sys.modules[spec.name] = mod
000098 | spec.loader.exec_module(mod)
000099 | offset_controls = []
000100 | for nx in (8, 16, 32, 64, 128):
000101 |     energies = [mod.momentum_ground_energy(nx, k, ny=8) for k in range(4)]
000102 |     shifts = [(e-energies[1])*nx/(2*math.pi) for e in energies]
000103 |     need(abs(shifts[3]) < 1e-7, 'opposite quarter common vacuum energy')
000104 |     # Diagnostic only: the UNIFORM all-N bound is proved via BV Fourier tails.
000105 |     need(max(abs(x) for x in shifts) < 0.3, 'actual common-reference control')
000106 |     if nx == 8:
000107 |         for k in range(4):
000108 |             h = mod.qwz_cylinder(nx,8,1.,k)
000109 |             vals = np.linalg.eigvalsh(h)
000110 |             need(abs(float(np.sum(vals[vals<0]))-energies[k]) < 1e-10,
000111 |                  'Bloch and complete builder vacuum energies agree')
000112 |     offset_controls.append({'Nx':nx, 'scaled_offsets_relative_r1':shifts})
000113 | cusp = []
000114 | e0 = -np.sum(np.abs(np.linalg.eigvalsh(mod.strip_at_momentum(0.,8))))/2
000115 | for p in (1e-3, 1e-4, 1e-5):
000116 |     hp = mod.strip_at_momentum(p,8)
000117 |     ep = -np.sum(np.abs(np.linalg.eigvalsh(hp)))/2
000118 |     need(abs(np.trace(hp)) < 1e-12, 'original Bloch trace vanishes')
000119 |     cusp.append({'p':p,'one_sided_slope':float((ep-e0)/p)})
000120 | need(abs(cusp[-1]['one_sided_slope']+1) < 1e-3,
000121 |      'the two zero branches produce the stated -abs(p) cusp')
000122 | 
000123 | # ACTUAL nonunitary attempt: normalized exp(-s H_N) P_E Omega_N.
000124 | # Imaginary-time quadratic evolution preserves a Slater state. The Thouless
000125 | # matrix avoids cancellation of huge occupied-sea exponential factors.