Reproducible source and supporting text for the associated calculation.
View complete source excerpt
"""Minimal balanced carriers and their induced local spatial rotation forms."""
from datetime import datetime, timezone
from hashlib import sha256
from importlib.util import spec_from_file_location, module_from_spec
from itertools import combinations
from math import comb
from pathlib import Path
import json
import mpmath as mp
import numpy as np
import sympy as sp
ROOT = Path(__file__).resolve().parent
spec = spec_from_file_location('forward', ROOT/'forward-count-action.py')
forward = module_from_spec(spec)
spec.loader.exec_module(forward)
COUNTS, ERRORS = {}, {}
def check(value, label):
if not bool(value):
raise AssertionError(label)
COUNTS[label] = COUNTS.get(label, 0)+1
def close(actual, expected, label, tolerance=3e-8):
error = float(np.linalg.norm(np.asarray(actual)-np.asarray(expected))/max(1., np.linalg.norm(expected)))
ERRORS[label] = max(ERRORS.get(label, 0.), error)
check(error < tolerance, label)
def preparation(depth):
generators = forward.radial.gammas(depth+1)
alpha, beta = generators[:depth], generators[depth]
rank, eye = len(beta), np.eye(len(beta))
ambient = [beta]+[-1j*a@beta for a in alpha]
low = ambient+[1j*ambient[i]@ambient[j] for i, j in combinations(range(depth+1), 2)]
if depth in [2, 4]:
raw = eye.copy()
elif depth == 3:
chirality = np.eye(rank, dtype=complex)
for gamma in ambient:
chirality = chirality@gamma
check(np.array_equal(chirality, chirality.conj().T), 'the depth-three ambient chirality is Hermitian')
check(np.array_equal(chirality@chirality, eye), 'the depth-three ambient chirality squares to identity')
columns = sp.Matrix(eye+chirality).columnspace()
raw = np.array(sp.Matrix.hstack(*columns), complex)
complement = [chirality]+[1j*chirality@gamma for gamma in ambient]
full = [eye]+low+complement
close(np.array([[np.trace(a@b) for b in full] for a in full]), rank*np.eye(rank*rank),
'depth-three low moments and five complementary Clifford matrices exhaust the Hermitian density space')
for i, a in enumerate(complement):
for j, b in enumerate(complement):
close(a@b+b@a, 2*eye if i == j else np.zeros_like(eye),
'the depth-three density complement is a five-generator Clifford system with doubly degenerate eigenvalues')
else:
raw = np.zeros((rank, 1), complex)
raw[0, 0] = raw[-1, 0] = 1
norm = np.vdot(raw, raw).real
for gamma in low:
check(np.vdot(raw, gamma@raw) == 0,
'exact Gaussian-integer seed arithmetic annihilates every ambient vector and bivector moment')
seed = raw/np.sqrt(norm)
minimal = {2: 2, 3: 2, 4: 4, 5: 1, 6: 1, 10: 1}[depth]
check(np.linalg.matrix_rank(raw) == minimal and raw.shape[1] == minimal,
'the explicit carrier attains its proved minimal column rank')
if depth in [2, 4]:
full = [eye]+ambient if depth == 2 else [eye]+low
close(np.array([[np.trace(a@b) for b in full] for a in full]), rank*np.eye(rank*rank),
'low Clifford moments exhaust the Hermitian density space at depths two and four')
phi = np.zeros((depth+1,)*3)
for i, j, k in combinations(range(depth+1), 3):
value = np.vdot(raw, 1j*ambient[i]@ambient[j]@ambient[k]@raw).real/norm
for order in [(i, j, k), (j, k, i), (k, i, j)]:
phi[order] = value
for order in [(j, i, k), (i, k, j), (k, j, i)]:
phi[order] = -value
return alpha, beta, ambient, seed, phi
def rotation_form(field, alpha, scale):
result = np.zeros((len(alpha), len(alpha)))
for i, j in combinations(range(len(alpha)), 2):
result[i, j] = np.vdot(field, 1j*alpha[i]@alpha[j]@field).real/scale
result[j, i] = -result[i, j]
return result
def main():
mp.mp.dps = 90
rng = np.random.default_rng(202610120843)
prepared = {depth: preparation(depth) for depth in forward.radial.DEPTHS}
sectors, cases = [], []
for depth, (alpha, beta, ambient, seed, phi) in prepared.items():
sectors.append(dict(depth=depth, spinor_rank=len(beta), minimal_column_rank=seed.shape[1],
nonzero_three_form_entries=[dict(indices=list(ix), value=float(phi[ix]))
for ix in combinations(range(depth+1), 3) if phi[ix] != 0]))
for name, depth, index in forward.radial.ASSIGNMENTS:
alpha, beta, ambient, seed, phi = prepared[depth]
rank, eye = len(beta), np.eye(len(beta))
level, h = index-1, index-1+depth/2
count, first, _ = forward.count_data(depth, level)
eps = mp.mpf(3)/10/max(forward.radial.polynomial(depth, n-1)[1]
for _, d, n in forward.radial.ASSIGNMENTS if d == depth)
c = float(eps*mp.mpf(first.numerator)/first.denominator)
x = np.sqrt(1-c*c)
a, b = np.sqrt((1+c)/2), np.sqrt((1-c)/2)
mass, lam, omega = h*x/c, h/c, float(eps*count.numerator/count.denominator)
eta = omega-lam
local_forms = []
for sample in range(5):
position = rng.normal(size=depth+1); position /= np.linalg.norm(position)
frame, _ = np.linalg.qr(np.column_stack([position, rng.normal(size=(depth+1, depth))]))
t, v = position[0], frame[0, 1:]
av = sum(vi*al for vi, al in zip(v, alpha))
aa, bb, ap, bp = forward.orientation.polynomials(depth, level, np.array(t))
matrix = aa*eye-1j*bb*av
# Any actual Killing-frame value is an ambient Spin rotor. Test several values.
killing = eye.astype(complex)
for turn in range(3):
i, j = rng.choice(depth+1, 2, replace=False)
angle = rng.normal()
killing = (np.cos(angle/2)*eye+np.sin(angle/2)*ambient[i]@ambient[j])@killing
normal = np.sqrt(comb(level+depth-1, depth-1))
bare = matrix@killing@seed/normal
bare_density = np.vdot(bare, bare).real
close(np.vdot(bare, beta@bare).real, 0., 'the actual reduced spatial kernel has zero bare scalar current')
close([np.vdot(bare, al@bare).real for al in alpha], np.zeros(depth),
'the actual reduced spatial kernel has zero bare spatial vector current')
jets = [(ap*v[i]*eye-1j*(bp*v[i]*av-t*bb*al)+.5j*matrix@al)@killing@seed/normal
for i, al in enumerate(alpha)]
dbare = -1j*sum(al@jet for al, jet in zip(alpha, jets))
close(dbare, h*bare, 'actual full Clifford spatial jets preserve the positive shell after carrier reduction')
field = (a*eye+b*beta)@bare
dfield = (a*eye-b*beta)@dbare
rho, scalar, current, proper, support = forward.bilinears(field, alpha, beta)
close(rho, bare_density, 'the reduced massive field retains the complete bare density')
close(scalar, x*rho, 'the reduced massive field retains the same local constitutive ratio')
close(current, np.zeros(depth), 'the reduced massive field retains exactly zero summed spatial current')
close((dfield+forward.source(field, h, float(eps), alpha, beta, depth))/lam,
omega*field/lam, 'the complete proper-current field equation preserves every count clock on the minimal carrier', 3e-6)
form = rotation_form(field, alpha, x*rho)
squares = np.linalg.eigvalsh(-form@form)
if depth in [2, 6]:
close(squares, np.ones(depth), 'the normalized induced rotation form has equal strength in every spatial plane at depth two or six')
elif depth == 5:
close(squares, [0, 1, 1, 1, 1], 'the normalized depth-five rotation form has two equal planes and one kernel direction')
elif depth in [3, 4]:
close(squares, np.zeros(depth), 'the canonical lower-depth balanced matrix seed has vanishing local spin form')
else:
rotor = matrix@killing/np.sqrt(aa*aa+bb*bb*np.dot(v, v))
effective_normal = rotor.conj().T@beta@rotor
axial = np.trace(beta@effective_normal).real/rank
close(squares, sorted([1., 1.]+[axial*axial]*8),
'the depth-ten induced rotation form has one unit plane and four axis-dependent planes')
local_forms.append(dict(squared_singular_values=squares.tolist()))
cases.append(dict(particle=name, depth=depth, inferred_index=index,
minimal_columns=seed.shape[1], local_rotation_form_samples=local_forms,
classification='exact reduced-carrier stationary mass state and local spin-density form, not observed particle spin'))
print(json.dumps(dict(completed=name, depth=depth, minimal_columns=seed.shape[1])), flush=True)
stamp = datetime.now(timezone.utc).strftime('%Y%m%dT%H%M%S.%fZ')
report = dict(experiment_key='minimal-balanced-count-carrier', created_utc=datetime.now(timezone.utc).isoformat(),
sources={'experiments/'+p.name: sha256(p.read_bytes()).hexdigest() for p in
[Path(__file__), ROOT/'forward-count-action.py', ROOT/'inverse-current-radial.py',
ROOT/'excited-shell-orientation-current.py']},
evaluations=sum(COUNTS.values()), conditions=COUNTS, maximum_errors=ERRORS,
sectors=sectors, cases=cases,
limits=['the rank theorem assumes vanishing ambient Clifford vector and bivector seed moments',
'the local spin-density form omits orbital angular momentum and does not derive observed total spin',
'the same exact stationary equations survive; the full-matrix background persistence theorem is not transferred',
'a local kernel direction is not an established fixed encounter axis',
'depth-ten rotor-dependent rank is not a demonstrated decay or instability',
'ordinary sphere geometry, count constitutive law and sector scales remain conditional'])
output = ROOT/('minimal-count-carrier-results-'+stamp+'.json')
output.write_text(json.dumps(report, indent=2)+'\n')
print(json.dumps(dict(report=str(output), evaluations=report['evaluations'], maximum_errors=ERRORS)), flush=True)
if __name__ == '__main__':
main()