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v520_celestial_wp5e_measure_decision.py
verification/v520_celestial_wp5e_measure_decision.py
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"""v520 -- CELEST.WP5E.MEASURE.01: the WP5e BCOV measure question DECIDED
at probe level -- "single-valuedness is derived, the completion reading
wins" (delta-1e + delta-1f consolidated, v518-consolidation style).
Question (the named [O] of v516/v518): which of the two exact, mutually
contradictory measures on PT/Z4 is the true BCOV measure -- (A) the
DECLARED completion reading (v516: unphased sector trace, w_m = 4 h_m,
supplies psi = 64) or (B) the DERIVED chiral measure (v518: multiplier
character chi_4 on Gamma_1(4); all three testers fail)? Answer:
ERFOLG-A on the decision layer -- single-valuedness of the physical
one-loop lattice factor is DERIVED from two independent constructive
sources (F-independence of the moduli integral + the Quillen/BCOV
conjugate pairing), NOT postulated; with single-valuedness forced, the
exact invariant subspace of the 16-dim Weil system forces the unphased
E4 numerator, and the consistency circle to v516 closes levelwise.
[E] 1. THE INVARIANT SUBSPACE (delta-1e D2): dim_Q {v : rho(T) v = v,
rho(S) v = v} = 8 = 4 x 2 in the dressed 16-component Weil system
(exact rational nullspace); EVERY kernel vector lies in the
Q(zeta_8)-span of {e_H, e_H'} -- the two Lagrangian gluings, both
theta = E4, level-1 quartic = Q^{(0)} = the unphased trace; strict
trivial-character solutions at physical weight (m, k) = (0, 0):
ZERO for every dressing (11 combinations, both orientations).
[E] 2. SOURCE 1 -- THE F-LEMMA (delta-1f A1): the only strict chiral
families are chi_4 (dims (3,3)) and chi_10 (dims (1,1)), both with
nontrivial T-character EXACTLY (chi_4(T) = e(1/3), chi_10(T) =
e(5/6)); the canonical column-matched chi_4 member's total block
sum obeys G(gamma tau) = chi_4(gamma) G(tau) pointwise (certificate
3.9e-16 over two fresh tau, nonvacuous |G| >= 59.6); EXACT change
of variables: int_{gamma F} G dmu = chi_4(gamma) int_F G dmu --
fundamental-domain independence forces int = 0 for EVERY chiral
integrand: under TP-1 + TP-2 the chiral route can never source
psi = 64 -- the v518 route was never a well-defined nonvanishing
moduli integral (the kill is SHARPENED, its unfolded values are
convention bookkeeping).
[E] 3. SOURCE 2 -- THE QUILLEN PAIRING (delta-1f A2): the doubled
hol x antihol transports satisfy |t conj(t) - 1| < 8.3e-40 on all
15 pairs and both generators (measured unitarity, not constructed);
|chi_4|^2 = 1 exactly vs one-sided chi_4^2 = e(2/3) != 1 (T-fixed:
640 != 0 mod 960, squared defects on 15/15 pairs); the doubled
256-dim Weil system closes STRICTLY at trivial character and
physical weight (nullspaces (28, 17) >= 11), contains the unphased
diagonal (~2e-15) AND the physical columns w_b (x) conj(w_b)
(sigma 4.9e-16 / 8.3e-16); negative control hol (x) hol: EMPTY
nullspace (0, 0) -- only the conjugate pairing implements
single-valuedness.
[E] 4. THE CONSISTENCY CLOSURE (delta-1f A3): the forced unphased
numerator inside the v518 scaffold reproduces sum_m V prop
(1, 2, 1, 0, 0) = <x,x>^2 EXACTLY on every level n <= 8 (levelwise
Okubo / 4-design); J-weighted spreads ~5e-41, T5/T3 fractions
~7e-42, zero negative-exponent cells; psi(skeleton) = +0.2302 =
-psi(contact) -- the J-weighted mirror image of psi(A_fix) = +64 /
psi(contact) = -64; linearity skeleton + contact = unphased at
4.6e-41: the circle to v516 is CLOSED.
[E] 5. D3 -- THE Z2/EH ANCHOR (delta-1e): only the declared reading
hits the anchor (9<x,x>^2 = (1/4) x 36<x,x>^2, psi(A_2) = -8,
scale tooth c = 1/2 = |Z2| h^{A1} unique); ALL THREE derived Z2
instantiations fail the anchor testers (spreads 1.12 / 0.895 /
0.516, psi sign wrong).
[E] 6. D1 -- COLUMN CANONICALISATION (delta-1e): the physical AB
column sum_m i^{bm} e_{(m,m)} is contained EXACTLY in the chi_4
family (sigma 1.8e-16 / 1.7e-16, one scalar per orbit --
canonicalising the (N1, N2) freedom); the canonical member STILL
fails every tester (sharpened kill, counted per the delta-1e
preregistration).
[E] NEGATIVE CONTROLS: SO(16)/D8 (T5 = 20 / T3 = -40 exact, mu_2-only
supply), wrong form (|Gauss|^2 = 64 != 16, S^2 = C breaks), wrong
weight (transport dev 0.78 at exponent 2), shuffled weights
(T5 = -14 / T3 = 36), planted family detected (1.5e-16), hol (x)
hol empty.
[C] TYPED PREMISES (the mandatory fence): TP-1 (the twisted one-loop
contribution is an F-independent moduli-space integral), TP-2
(it is nonzero -- it must source psi = 64), TP-3 (the v518
kernel-family weight convention), TP-4 (BCOV F1 is Quillen/
Ray-Singer: the one-loop factor pairs hol x conj(hol)).
[O] the constructive BCOV derivation of the w_m normalisation itself
(the residual open item); SEAM.EQUIV.TWISTOR.01 stays Open.
VERDICT: ERFOLG-A per the preregistered delta-1f criteria (both
constructive sources fire independently, the consistency closure
carries, all negative controls behave); the [O] question "which is
the true BCOV measure" is DECIDED at probe level in favour of (A)
under the typed premises TP-1..TP-4. NO marker moves anywhere.
Status: [E] exact Weil/Q(zeta_8) algebra, exact invariant subspace
(rational nullspace), characters + 1/960 grid arithmetic, ledgers,
completion/contact/psi identities, levelwise 4-design, Z2 anchor
arithmetic, change-of-variables corollary, controls; numeric
(documented) = block transports (28+ digits, grid 25+), SVD solves
and memberships (double, thresholds 1e-8/1e-3), function certificates
at fresh tau (< 1e-8), HM kernels (30+ digits, cut-6 vs cut-8 scan,
negative-cell audit). Python; Wolfram-mirrored (Weil relations,
invariant-subspace forcing, characters |chi_4|^2 = 1 vs chi_4^2 =
e(2/3), completion/contact/psi arithmetic, Z2 anchor + scale teeth,
levelwise Okubo, so16/wrong-form/shuffle controls); the Harvey-Moore
integrals and SVD memberships are numerical and stay Python-only,
flagged in the .wl mirror. Discovery provenance:
experiments/tfpt-discovery/
celestial_seam_wp5e_delta1e_bcov_measure_decision_probe.py (28/28),
..._delta1f_single_valuedness_probe.py (23/23) (2026-07-22)."""
import math
from fractions import Fraction as F
from itertools import combinations, product
from math import comb, gcd
import mpmath as mp
import numpy as np
import sympy as sp
from tfpt_constants import check, summary, reset, N_fam, rankE8
MU4 = N_fam + 1 # 4 = |mu4|, the seam clock order
NCLS = MU4 * MU4 # 16 discriminant classes of D5 (+) A3
L_PHASE = 960
GRID_CUT = F(8)
GRID_CUT_LOW = F(6)
MAXN_LED = 16
MAXN_CERT = 20
BASES4 = {0: F(-1, 2), 1: F(-5, 16), 2: F(-1, 4), 3: F(-5, 16)}
SIGMA3 = [1, 9, 28, 73, 126, 252, 344, 585]
NPROD = 120
TAU_A = mp.mpc(mp.mpf(13) / 100, mp.mpf(51) / 50)
TAU_B = mp.mpc(mp.mpf(-31) / 100, mp.mpf(99) / 100)
TAU_C = mp.mpc(mp.mpf(7) / 100, mp.mpf(101) / 100)
TAU_D = mp.mpc(mp.mpf(-19) / 100, mp.mpf(103) / 100)
def fmt(xs):
return "(" + ", ".join(str(x) for x in xs) + ")"
def fmtn(xs, nd=6):
return "(" + ", ".join(mp.nstr(mp.mpf(x) if not isinstance(x, mp.mpc)
else x, nd) for x in xs) + ")"
# ---------------------------------------------------------------------------
# exact Q(zeta_8) arithmetic (delta-1c/1d/1e machinery, verbatim)
# ---------------------------------------------------------------------------
Z8_ZERO = (F(0), F(0), F(0), F(0))
Z8_ONE = (F(1), F(0), F(0), F(0))
def z8_add(a, b):
return tuple(x + y for x, y in zip(a, b))
def z8_mul(a, b):
out = [F(0)] * 4
for i in range(4):
if a[i] == 0:
continue
for j in range(4):
if b[j] == 0:
continue
k = i + j
if k < 4:
out[k] += a[i] * b[j]
else:
out[k - 4] -= a[i] * b[j]
return tuple(out)
def z8_scal(f, a):
return tuple(f * x for x in a)
def z8_root(k):
k %= 8
s = F(1)
if k >= 4:
k -= 4
s = F(-1)
v = [F(0)] * 4
v[k] = s
return tuple(v)
def z8_conj(a):
return (a[0], -a[3], -a[2], -a[1])
def z8_to_mpc(a):
z = mp.exp(mp.mpc(0, mp.pi / 4))
return (mp.mpf(a[0].numerator) / a[0].denominator
+ z * mp.mpf(a[1].numerator) / a[1].denominator
+ z ** 2 * mp.mpf(a[2].numerator) / a[2].denominator
+ z ** 3 * mp.mpf(a[3].numerator) / a[3].denominator)
def sum_z8(xs):
out = Z8_ZERO
for x in xs:
out = z8_add(out, x)
return out
def mat_mul(A, B):
n = len(A)
return [[sum_z8([z8_mul(A[i][k], B[k][j]) for k in range(n)])
for j in range(n)] for i in range(n)]
def mat_eq(A, B):
return all(A[i][j] == B[i][j] for i in range(len(A))
for j in range(len(A)))
def mat_id(n):
return [[Z8_ONE if i == j else Z8_ZERO for j in range(n)]
for i in range(n)]
def mat_conjT(A):
n = len(A)
return [[z8_conj(A[j][i]) for j in range(n)] for i in range(n)]
# ---------------------------------------------------------------------------
# discriminant module D = Z4 x Z4, q = (5x^2 + 3y^2)/8 mod 1
# ---------------------------------------------------------------------------
MU = [(x, y) for x in range(4) for y in range(4)]
IDX = {m: i for i, m in enumerate(MU)}
DIAG_H = [(a, a) for a in range(4)]
ANTI_H = [(0, 0), (1, 3), (2, 2), (3, 1)]
def q8(mu):
x, y = mu
return (5 * x * x + 3 * y * y) % 8
def bil8(mu, nu):
return (2 * (5 * mu[0] * nu[0] + 3 * mu[1] * nu[1])) % 8
def neg(mu):
return ((-mu[0]) % 4, (-mu[1]) % 4)
def build_weil(qfun, bfun, group, sigma_conj):
n = len(group)
rt = F(1)
m = n
while m % 4 == 0:
m //= 4
rt *= 2
assert m == 1
T = [[z8_root(qfun(group[i])) if i == j else Z8_ZERO
for j in range(n)] for i in range(n)]
S = [[z8_scal(F(1, rt), z8_mul(sigma_conj,
z8_root(-bfun(group[i], group[j]))))
for j in range(n)] for i in range(n)]
return T, S
def weil_matrices():
return build_weil(q8, bil8, MU, Z8_ONE)
def rho_np(exact):
return np.array([[complex(z8_to_mpc(exact[i][j])) for j in range(16)]
for i in range(16)])
# ---------------------------------------------------------------------------
# lattice ledgers (delta-1c/1d/1e machinery, verbatim)
# ---------------------------------------------------------------------------
def build_d5_vecs(maxnorm):
out = {x: [] for x in range(4)}
lim = 16 * maxnorm
r = int(maxnorm ** 0.5) + 1
for v in product(range(-r, r + 1), repeat=5):
n16 = 16 * sum(t * t for t in v)
if n16 > lim:
continue
x = 0 if sum(v) % 2 == 0 else 2
out[x].append((n16, tuple(2 * t for t in v)))
ru = int((4 * maxnorm) ** 0.5) + 1
odd = [k for k in range(-ru, ru + 1) if k % 2 != 0]
for u in product(odd, repeat=5):
n16 = 4 * sum(t * t for t in u)
if n16 > lim:
continue
x = 1 if (sum(u) - 5) % 4 == 0 else 3
out[x].append((n16, u))
return out
def build_a3_vecs(maxnorm):
out = {y: [] for y in range(4)}
lim = 16 * maxnorm
r = int(maxnorm ** 0.5) + 2
for y in range(4):
rng = [4 * m - y for m in range(-r, r + 2)]
for w3 in product(rng, repeat=3):
w4 = -(w3[0] + w3[1] + w3[2])
if (w4 + y) % 4 != 0:
continue
n16 = w3[0] ** 2 + w3[1] ** 2 + w3[2] ** 2 + w4 ** 2
if n16 <= lim:
out[y].append((n16, w3 + (w4,)))
return out
def factor_moments(vecs, form_re, form_im, kmax, maxn16):
out = {}
for cls, lst in vecs.items():
acc = {}
for n16, u in lst:
if n16 > maxn16:
continue
dre = sum(a * b for a, b in zip(u, form_re))
dim_ = (sum(a * b for a, b in zip(u, form_im))
if form_im else 0)
ms = acc.get(n16)
if ms is None:
ms = acc[n16] = [[0, 0] for _ in range(kmax + 1)]
pr, pi = 1, 0
for k in range(kmax + 1):
ms[k][0] += pr
ms[k][1] += pi
pr, pi = pr * dre - pi * dim_, pr * dim_ + pi * dre
out[cls] = acc
return out
def combine_series(m5x, m3y, k_ins, maxn16):
out = {}
for n5, a in m5x.items():
for n3, b in m3y.items():
n = n5 + n3
if n > maxn16:
continue
cre = cim = 0
for k in range(k_ins + 1):
c = comb(k_ins, k)
ar, ai = a[k]
br, bi = b[k_ins - k]
cre += c * (ar * br - ai * bi)
cim += c * (ar * bi + ai * br)
if cre or cim:
o = out.get(n, (0, 0))
out[n] = (o[0] + cre, o[1] + cim)
return out
def eval_series16(ser, tau, maxn16):
tot = mp.mpc(0)
for n16, (cre, cim) in ser.items():
if n16 > maxn16:
continue
tot += mp.mpc(cre, cim) * mp.exp(1j * mp.pi * tau * n16 / 16)
return tot
SAMPLES = [
(1, 0, 0, 0, 0, 0, 0, 0),
(0, 0, 0, 0, 0, 1, 0, 0),
(0, 0, 0, 0, 0, 1, 1, 1),
(1, 2, 0, 0, 0, 0, 0, 0),
(1, 0, 0, 0, 0, 1, 0, 0),
(1, 1, 1, 0, 0, 1, 1, 1),
]
def basis_at(pt):
xs, ys = pt[:5], list(pt[5:])
ys4 = ys + [-sum(ys)]
S5 = sum(x * x for x in xs)
S3 = sum(y * y for y in ys4)
T5 = sum(x ** 4 for x in xs)
T3 = sum(y ** 4 for y in ys4)
return (S5 * S5, S5 * S3, S3 * S3, T5, T3)
BMAT = [basis_at(p) for p in SAMPLES]
def solve_cell(bvals):
A = sp.Matrix([[BMAT[k][i] for i in range(5)] for k in range(6)])
b = sp.Matrix([sp.Rational(x.numerator, x.denominator) for x in bvals])
sol, params = A.gauss_jordan_solve(b)
assert len(params) == 0 and A * sol == b
return [F(sp.Rational(x).p, sp.Rational(x).q) for x in sol]
def build_ledgers16(d5vecs, a3vecs, maxnorm):
lim16 = 16 * maxnorm
md5 = {}
for x in range(4):
acc = {}
for n16, u in d5vecs[x]:
if n16 > lim16:
continue
ms = acc.get(n16)
if ms is None:
ms = acc[n16] = [[0] * 6 for _ in range(5)]
for si, s in enumerate(SAMPLES):
d = 2 * (u[0] * s[0] + u[1] * s[1] + u[2] * s[2]
+ u[3] * s[3] + u[4] * s[4])
p = 1
for k in range(5):
ms[k][si] += p
p *= d
md5[x] = acc
ma3 = {}
for y in range(4):
acc = {}
for n16, w in a3vecs[y]:
if n16 > lim16:
continue
ms = acc.get(n16)
if ms is None:
ms = acc[n16] = [[0] * 6 for _ in range(5)]
for si, s in enumerate(SAMPLES):
d = ((w[0] - w[3]) * s[5] + (w[1] - w[3]) * s[6]
+ (w[2] - w[3]) * s[7])
p = 1
for k in range(5):
ms[k][si] += p
p *= d
ma3[y] = acc
out4 = {m: {} for m in MU}
out2 = {m: {} for m in MU}
CNT = {m: {} for m in MU}
for (x, y) in MU:
acc4, acc2 = {}, {}
for n5, m5 in md5[x].items():
for n3, m3 in ma3[y].items():
tot = n5 + n3
if tot == 0 or tot > lim16:
continue
lev = F(tot, 32)
cell4 = acc4.get(lev)
if cell4 is None:
cell4 = acc4[lev] = [0] * 6
acc2[lev] = [0, 0]
for si in range(6):
cell4[si] += sum(comb(4, k) * m5[k][si] * m3[4 - k][si]
for k in range(5))
for si in range(2):
acc2[lev][si] += (m5[2][si] * m3[0][si]
+ 2 * m5[1][si] * m3[1][si]
+ m5[0][si] * m3[2][si])
CNT[(x, y)][lev] = (CNT[(x, y)].get(lev, 0)
+ m5[0][0] * m3[0][0])
for lev, cell in acc4.items():
out4[(x, y)][lev] = solve_cell([F(c, 256) for c in cell])
b0, b1 = acc2[lev]
out2[(x, y)][lev] = (F(b0, 16), F(b1, 32))
return out4, out2, CNT
def build_theta_ser0():
d5v = build_d5_vecs(MAXN_CERT)
a3v = build_a3_vecs(MAXN_CERT)
lim = 16 * MAXN_CERT
zero5 = (0, 0, 0, 0, 0)
zero4 = (0, 0, 0, 0)
m5_0 = factor_moments(d5v, zero5, None, 0, lim)
m3_0 = factor_moments(a3v, zero4, None, 0, lim)
return {mu: combine_series(m5_0[mu[0]], m3_0[mu[1]], 0, lim)
for mu in MU}, lim
# ---------------------------------------------------------------------------
# E8 roots in glue coordinates (replication anchor)
# ---------------------------------------------------------------------------
HALF = F(1, 2)
def build_glue_roots():
d5_roots, d5_v = [], []
for i, j in combinations(range(5), 2):
for si in (1, -1):
for sj in (1, -1):
v = [F(0)] * 5
v[i], v[j] = F(si), F(sj)
d5_roots.append(tuple(v))
for i in range(5):
for s in (1, -1):
v = [F(0)] * 5
v[i] = F(s)
d5_v.append(tuple(v))
d5_s, d5_c = [], []
for signs in product((1, -1), repeat=5):
v = tuple(HALF * s for s in signs)
(d5_s if signs.count(-1) % 2 == 0 else d5_c).append(v)
a3_roots = []
for i in range(4):
for j in range(4):
if i != j:
v = [F(0)] * 4
v[i], v[j] = F(1), F(-1)
a3_roots.append(tuple(v))
def wclass(k):
out = []
for sub in combinations(range(4), k):
v = [F(-k, 4)] * 4
for i in sub:
v[i] += 1
out.append(tuple(v))
return out
z5, z4 = tuple([F(0)] * 5), tuple([F(0)] * 4)
roots = {}
for r in d5_roots:
roots[r + z4] = 0
for r in a3_roots:
roots[z5 + r] = 0
for d in d5_s:
for w in wclass(1):
roots[d + w] = 1
for d in d5_v:
for w in wclass(2):
roots[d + w] = 2
for d in d5_c:
for w in wclass(3):
roots[d + w] = 3
return roots
# ---------------------------------------------------------------------------
# block numerics: gauge M = G/eta^8 and the fibre blocks f_m
# ---------------------------------------------------------------------------
def qp(tau, e):
return mp.exp(2j * mp.pi * tau * e)
def geo_val(N, a, b, tau):
zb = mp.exp(2j * mp.pi * b / N)
zbb = mp.exp(-2j * mp.pi * b / N)
ep = mp.mpf(a) / N if a else mp.mpf(1)
em = 1 - mp.mpf(a) / N if a else mp.mpf(1)
Eb = -mp.mpf(1) / 24 + (mp.mpf(a) / N) * (1 - mp.mpf(a) / N) / 4 \
if a else -mp.mpf(1) / 24
val = qp(tau, 4 * Eb)
for n in range(NPROD):
val /= (1 - zb * qp(tau, ep + n)) ** 2
val /= (1 - zbb * qp(tau, em + n)) ** 2
if a == 0 and b % N != 0:
val /= (1 - zb) * (1 - zbb)
return val
def eta_val(tau):
val = qp(tau, mp.mpf(1) / 24)
for n in range(1, NPROD + 1):
val *= (1 - qp(tau, n))
return val
def M_val(N, a, b, tau):
return geo_val(N, a, b, tau) / eta_val(tau) ** 8
def f_val(N, m, a, b, tau):
aa = (m * a) % N
bb = (m * b) % N
if aa == 0 and bb == 0:
return mp.mpc(1)
z = mp.exp(2j * mp.pi * bb / N)
zc = mp.exp(-2j * mp.pi * bb / N)
ep = mp.mpf(aa) / N if aa else mp.mpf(1)
em = 1 - mp.mpf(aa) / N if aa else mp.mpf(1)
E = -mp.mpf(1) / 24 + (mp.mpf(aa) / N) * (1 - mp.mpf(aa) / N) / 4 \
if aa else -mp.mpf(1) / 24
val = qp(tau, 2 * E)
for n in range(NPROD):
val /= (1 - z * qp(tau, ep + n))
val /= (1 - zc * qp(tau, em + n))
if aa == 0 and bb % N != 0:
val /= (1 - z)
return val
def ax_val(N, ws, a, b, tau):
v = mp.mpc(1)
for m in ws:
v *= f_val(N, m, a, b, tau)
return v
def measure_transport_w(N, pairs, fM, wexp):
t, s = {}, {}
dev = mp.mpf(0)
for (a, b) in pairs:
Tp = (a, (a + b) % N)
Sp = (b % N, (-a) % N)
vals_t, vals_s = [], []
for tau in (TAU_A, TAU_B):
vals_t.append(fM(a, b, tau + 1) / fM(*Tp, tau))
vals_s.append(fM(a, b, -1 / tau) * (-1j * tau) ** wexp
/ fM(*Sp, tau))
dev = max(dev, abs(vals_t[0] - vals_t[1]),
abs(vals_s[0] - vals_s[1]))
t[(a, b)] = vals_t[0]
s[(a, b)] = vals_s[0]
return t, s, dev
def recog(z):
lm = mp.log(abs(z))
ph = mp.arg(z) / (2 * mp.pi)
k = int(mp.nint(ph * L_PHASE))
dev = abs(ph - mp.mpf(k) / L_PHASE)
return lm, k % L_PHASE, dev
# ---------------------------------------------------------------------------
# contraction solve machinery (generalised to n dims for the doubled system)
# ---------------------------------------------------------------------------
def sl2z_char(k):
return (np.exp(2j * np.pi * k / 12), np.exp(-2j * np.pi * k / 4))
def solve_orbit(pairs, N, tconst, sconst, rhoT_np, rhoS_np, base, k=0):
n = rhoT_np.shape[0]
chT, chS = sl2z_char(k)
TT = rhoT_np.T
ST = rhoS_np.T
Lam = {base: np.eye(n, dtype=complex)}
queue = [base]
rows = []
seen_edges = set()
while queue:
p = queue.pop(0)
a, b = p
for tag, p2, Mx in (
('T', (a, (a + b) % N),
complex(tconst[p]) / chT * TT),
('S', ((b % N), (-a) % N),
complex(sconst[p]) / chS * ST)):
if (p, tag) in seen_edges:
continue
seen_edges.add((p, tag))
cand = Mx @ Lam[p]
if p2 not in Lam:
Lam[p2] = cand
queue.append(p2)
else:
rows.append(cand - Lam[p2])
A = np.vstack(rows) if rows else np.zeros((1, n), dtype=complex)
_, sv, Vh = np.linalg.svd(A)
sv = list(sv) + [0.0] * (n - len(sv))
tol = max(sv) * 1e-8 if max(sv) > 0 else 1e-8
null = [Vh[i].conj() for i in range(n) if sv[i] < tol]
return sorted(Lam.keys()), Lam, null, sv
def member_test(Lam, null, targets):
"""stacked homogeneous system [Lam[p] V | -t_p] u = 0 with ONE
shared scalar; returns (sigma_min/sigma_max, |c|/||u||, u, V)."""
if not null:
return None
V = np.array(null).T
rows = []
for p, t in targets.items():
A = Lam[p] @ V
rows.append(np.hstack([A, -t.reshape(len(t), 1)]))
Mst = np.vstack(rows)
_, svs, Vh = np.linalg.svd(Mst)
ratio = svs[-1] / svs[0]
u = Vh[-1].conj()
c_rel = abs(u[-1]) / np.linalg.norm(u)
return ratio, c_rel, u, V
def ray_residual(Lam, null, p, t):
"""per-pair least-squares containment residual of target t in the
family column space at pair p."""
if not null:
return None
V = np.array(null).T
A = Lam[p] @ V
x, *_ = np.linalg.lstsq(A, t, rcond=None)
return float(np.linalg.norm(A @ x - t) / np.linalg.norm(t))
def project_seed(Lam, null, seed):
if not null:
return None, 0.0
V = np.array(null).T
G = sum(Lam[p].conj().T @ Lam[p] for p in Lam)
Avec = sum(Lam[p].conj().T @ seed[p] for p in Lam)
Gv = V.conj().T @ G @ V
c = np.linalg.solve(Gv, V.conj().T @ Avec)
b0 = V @ c
beta = {p: Lam[p] @ b0 for p in Lam}
n_pr = np.sqrt(sum(np.vdot(beta[p], beta[p]).real for p in beta))
n_sd = np.sqrt(sum(np.vdot(seed[p], seed[p]).real for p in seed))
ov = sum(np.vdot(seed[p], beta[p]).real for p in beta)
cos = ov / (n_pr * n_sd) if n_pr > 0 else 0.0
return beta, cos
def dressed_consts(tg, sg, d, orient, pairs):
if orient == 'hol':
tt = {p: complex(tg[p]) * complex(d['t'][p]) for p in pairs}
ss = {p: complex(sg[p]) * complex(d['s'][p]) for p in pairs}
else:
tt = {p: complex(tg[p]) * np.conj(complex(d['t'][p]))
for p in pairs}
ss = {p: complex(sg[p]) * np.conj(complex(d['s'][p]))
for p in pairs}
return tt, ss
# ---------------------------------------------------------------------------
# exact invariant subspace of the 16-dim Weil representation (delta-1e D2)
# ---------------------------------------------------------------------------
def mulmat_q8(u):
M = [[F(0)] * 4 for _ in range(4)]
for i in range(4):
if u[i] == 0:
continue
for j in range(4):
k = i + j
if k < 4:
M[k][j] += u[i]
else:
M[k - 4][j] -= u[i]
return M
def invariant_subspace(Tx, Sx):
rows = []
for Op in (Tx, Sx):
for i in range(16):
block = [[sp.Rational(0)] * 64 for _ in range(4)]
for j in range(16):
Mm = mulmat_q8(Op[i][j])
for r in range(4):
for cc in range(4):
if Mm[r][cc]:
block[r][4 * j + cc] += sp.Rational(
Mm[r][cc].numerator, Mm[r][cc].denominator)
for r in range(4):
block[r][4 * i + r] -= 1
rows.extend(block)
A = sp.Matrix(rows)
return A.nullspace()
def in_lagrangian_span(vec64):
comp = [tuple(F(sp.Rational(vec64[4 * j + c]).p,
sp.Rational(vec64[4 * j + c]).q) for c in range(4))
for j in range(16)]
alpha = comp[IDX[(1, 1)]]
beta = comp[IDX[(1, 3)]]
for mu in MU:
j = IDX[mu]
in_h = mu in DIAG_H
in_hp = mu in ANTI_H
if mu == (0, 0):
want = z8_add(alpha, beta)
elif in_h and in_hp:
want = z8_add(alpha, beta)
elif in_h:
want = alpha
elif in_hp:
want = beta
else:
want = Z8_ZERO
if comp[j] != want:
return False
return True
# ---------------------------------------------------------------------------
# exact dressing series + fundamental-domain kernel J (delta-1b/1d/1e)
# ---------------------------------------------------------------------------
def gadd(z1, z2):
return (z1[0] + z2[0], z1[1] + z2[1])
def gmul(z1, z2):
return (z1[0] * z2[0] - z1[1] * z2[1], z1[0] * z2[1] + z1[1] * z2[0])
def ipow(b):
return [(F(1), F(0)), (F(0), F(1)), (F(-1), F(0)), (F(0), F(-1))][b % 4]
def uroot(N, k):
if N == 2:
return [(F(1), F(0)), (F(-1), F(0))][k % 2]
return ipow(k)
def ser_mul(A, B, cut):
out = {}
for e1, v1 in A.items():
for e2, v2 in B.items():
e = e1 + e2
if e > cut:
continue
out[e] = gadd(out.get(e, (F(0), F(0))), gmul(v1, v2))
return {e: v for e, v in out.items() if v != (F(0), F(0))}
def tower(zeta, e0, cut):
out = {F(0): (F(1), F(0))}
n = 0
while e0 + n <= cut:
e = e0 + n
n += 1
if e <= 0:
continue
fac = {F(0): (F(1), F(0))}
k = 1
zk = zeta
while k * e <= cut:
fac[k * e] = (F(k + 1) * zk[0], F(k + 1) * zk[1])
zk = gmul(zk, zeta)
k += 1
out = ser_mul(out, fac, cut)
return out
def tower1(zeta, e0, cut):
out = {F(0): (F(1), F(0))}
n = 0
while e0 + n <= cut:
e = e0 + n
n += 1
if e <= 0:
continue
fac = {F(0): (F(1), F(0))}
k = 1
zk = zeta
while k * e <= cut:
fac[k * e] = zk
zk = gmul(zk, zeta)
k += 1
out = ser_mul(out, fac, cut)
return out
def geo_block(N, a, b, cut):
ab = F(a, N)
e_plus = ab if a else F(1)
e_minus = (1 - ab) if a else F(1)
block = ser_mul(tower(uroot(N, b), e_plus, cut),
tower(uroot(N, -b), e_minus, cut), cut)
zm = None
if a == 0 and b % N != 0:
wb, wmb = uroot(N, b), uroot(N, -b)
zm = gmul((F(1) - wb[0], -wb[1]), (F(1) - wmb[0], -wmb[1]))
return block, zm
def p8_series(nmax):
out = [F(1)] + [F(0)] * nmax
for d in range(1, nmax + 1):
geo = [F(0)] * (nmax + 1)
k = 0
while d * k <= nmax:
geo[d * k] = F(int(sp.binomial(8 + k - 1, k)))
k += 1
new = [F(0)] * (nmax + 1)
for i, av in enumerate(out):
if av == 0:
continue
for j, bv in enumerate(geo[:nmax + 1 - i]):
new[i + j] += av * bv
out = new
return out
def gauge_dress_series(N, a, b, cut):
geo, zm = geo_block(N, a, b, cut + 1)
w = (F(1), F(0))
if zm is not None:
nrm = zm[0] * zm[0] + zm[1] * zm[1]
w = (zm[0] / nrm, -zm[1] / nrm)
p8 = p8_series(int(cut) + 1)
dress = {}
for e, cv in geo.items():
for m2, pv in enumerate(p8):
if pv == 0:
continue
ex = e + m2
if ex > cut:
continue
dress[ex] = gadd(dress.get(ex, (F(0), F(0))),
gmul(w, (cv[0] * pv, cv[1] * pv)))
return dress
def f_series(N, m, a, b, cut):
aa = (m * a) % N
bb = (m * b) % N
if aa == 0 and bb == 0:
return {F(0): (F(1), F(0))}, F(0)
ep = F(aa, N) if aa else F(1)
em = (1 - F(aa, N)) if aa else F(1)
ser = ser_mul(tower1(uroot(N, bb), ep, cut),
tower1(uroot(N, -bb), em, cut), cut)
if aa == 0 and bb % N != 0:
w = uroot(N, bb)
d = (F(1) - w[0], -w[1])
nrm = d[0] * d[0] + d[1] * d[1]
winv = (d[0] / nrm, -d[1] / nrm)
ser = {e: gmul(winv, v) for e, v in ser.items()}
shift = 2 * (F(-1, 24) + F(aa, N) * (1 - F(aa, N)) / 4)
return ser, shift
_GLX, _GLW = np.polynomial.legendre.leggauss(160)
_TH = [(math.pi / 12) * (x + 1) for x in _GLX]
_WTH = [(math.pi / 12) * w for w in _GLW]
_J_CACHE = {}
def Jval(delta, s):
key = (delta, s)
if key in _J_CACHE:
return _J_CACHE[key]
d = mp.mpf(delta.numerator) / mp.mpf(delta.denominator)
cusp = (mp.sin(mp.pi * d) / (mp.pi * d)
* (2 * mp.pi * d) ** (s - 1) * mp.gammainc(1 - s, 2 * mp.pi * d))
df = float(d)
sinpd = math.sin(math.pi * df)
arc = 0.0
for th, wt in zip(_TH, _WTH):
ct, st = math.cos(th), math.sin(th)
arc += wt * (st * ct ** (-s) * math.exp(-2 * math.pi * df * ct)
* (sinpd - math.sin(2 * math.pi * df * st))
/ (math.pi * df))
val = cusp + mp.mpf(arc)
_J_CACHE[key] = val
return val
def hm_integral_pp(beta, dressmaps, basemap, V4, V2, CNT, cut, weight):
I_main = [mp.mpc(0)] * 5
I_comp = [mp.mpc(0)] * 5
W = {}
n_neg = 0
wt_neg = mp.mpf(0)
for p, dress in dressmaps.items():
a = p[0]
base = basemap[p]
bp = beta[p]
for i_mu, mu in enumerate(MU):
bc = complex(bp[i_mu])
if abs(bc) < 1e-14:
continue
bz = mp.mpc(bc.real, bc.imag)
for ex, cv in dress.items():
cz = bz * mp.mpc(mp.mpf(cv[0].numerator) / cv[0].denominator,
mp.mpf(cv[1].numerator) / cv[1].denominator)
for lev, vec4 in V4[mu].items():
D = base + ex + lev
if D > cut:
continue
if D <= 0:
n_neg += 1
wt_neg = max(wt_neg, abs(cz))
continue
z0 = cz * Jval(D, 0)
for i in range(5):
z = z0 * (mp.mpf(vec4[i].numerator)
/ vec4[i].denominator)
I_main[i] += weight * z
if mu[0] == mu[1]:
key = (a, lev)
W[key] = W.get(key, mp.mpc(0)) + weight * z0
u, v = V2[mu][lev]
z1 = cz * Jval(D, 1) * mp.mpf(-3) / (2 * mp.pi)
uu = mp.mpf(u.numerator) / u.denominator
vv = mp.mpf(v.numerator) / v.denominator
I_comp[0] += weight * z1 * uu
I_comp[1] += weight * z1 * (uu + vv)
I_comp[2] += weight * z1 * vv
z2 = (cz * Jval(D, 2) * mp.mpf(3)
/ (16 * mp.pi ** 2) * CNT[mu][lev])
I_comp[0] += weight * z2
I_comp[1] += weight * z2 * 2
I_comp[2] += weight * z2
I_hat = [I_main[i] + I_comp[i] for i in range(5)]
return I_main, I_hat, W, n_neg, wt_neg
def build_dress(beta_keys, ws):
dm, basemap = {}, {}
for p in beta_keys:
a, b = p
d = gauge_dress_series(4, a, b, GRID_CUT)
shift = F(0)
for m in ws:
fs, sh = f_series(4, m, a, b, GRID_CUT)
d = ser_mul(d, fs, GRID_CUT)
shift += sh
dm[p] = d
basemap[p] = BASES4[a] + shift
return dm, basemap
def run_integral(beta, ws, V4, V2, CNT, weight):
dm, basemap = build_dress(list(beta.keys()), ws)
I_main, I_hat, W, n_neg, wt_neg = hm_integral_pp(
beta, dm, basemap, V4, V2, CNT, GRID_CUT, weight)
dml = {p: {e: v for e, v in dm[p].items() if e <= GRID_CUT_LOW}
for p in dm}
I_lo, _, _, _, _ = hm_integral_pp(
beta, dml, basemap, V4, V2, CNT, GRID_CUT_LOW, weight)
scale = max(abs(x) for x in I_main)
im_rel = max(abs(x.imag) for x in I_main) / scale if scale > 0 else 0
eps_rel = (max(abs(I_main[i] - I_lo[i]) for i in range(5)) / scale
if scale > 0 else 0)
return I_main, W, n_neg, wt_neg, im_rel, eps_rel
def spread_vs(I_main, tgt):
scale = max(abs(x) for x in I_main)
rats = [I_main[i] / tgt[i] for i in range(5) if tgt[i] != 0]
spread = (max(abs(rats[i] - rats[j]) for i in range(len(rats))
for j in range(len(rats)))
/ max(abs(r) for r in rats))
zrel = max([abs(I_main[i]) / scale for i in range(5) if tgt[i] == 0]
+ [mp.mpf(0)])
return spread, zrel
def function_certificate(beta, ws, k_chi, th):
chT = mp.exp(2j * mp.pi * k_chi / 12)
chS = mp.exp(-2j * mp.pi * k_chi / 4)
def blockval(p, tau):
v = (M_val(4, p[0], p[1], tau)
* ax_val(4, ws, p[0], p[1], tau))
return v * sum(complex(beta[p][i]) * th[tau][i]
for i in range(16))
max_dev = mp.mpf(0)
for p in beta:
a, b = p
for tau_from, q2, ch in (
(TAU_C + 1, (a, (a + b) % 4), chT),
(-1 / TAU_C, ((b % 4), (-a) % 4), chS)):
if q2 not in beta:
continue
lhs = blockval(p, tau_from)
rhs = ch * blockval(q2, TAU_C)
if abs(rhs) < mp.mpf(10) ** (-25):
continue
max_dev = max(max_dev, abs(lhs / rhs - 1))
return max_dev
PSIV = [F(3), F(-1), F(-1), F(0), F(-1, 4)]
def psi_of(v):
return sum(float(PSIV[i]) * v[i] for i in range(5))
# ---------------------------------------------------------------------------
# pair sets, candidates, columns, targets
# ---------------------------------------------------------------------------
PAIRS4 = [(a, b) for a in range(4) for b in range(4) if (a, b) != (0, 0)]
ORB1 = [p for p in PAIRS4 if gcd(gcd(p[0], p[1]), 4) == 1]
ORB2 = [p for p in PAIRS4 if gcd(gcd(p[0], p[1]), 4) == 2]
CAND_WS = {
'f2': (2,),
'f1f3': (1, 3),
'f1f2f3': (1, 2, 3),
'wrong112': (1, 1, 2),
'wrong222': (2, 2, 2),
}
QTAB = {0: [12, 0, 6, 4, 8], 1: [12, 24, 0, -8, 16],
2: [0, 24, 30, 12, -40], 3: [12, 24, 0, -8, 16]}
DETS = {1: 2, 2: 4, 3: 2}
AFIX = [9, -30, -15, 0, 32]
A2VEC = [-3, -6, 9, 8, -16]
Q1VEC = QTAB[1]
CONTACT_A = [36, 120, 60, 0, -32]
OKUBO = [F(1), F(2), F(1), F(0), F(0)]
ZERO5 = [F(0)] * 5
CONTAIN_TOL = 1e-8
ROBUST_TOL = 1e-3
E00 = np.zeros(16, dtype=complex)
E00[IDX[(0, 0)]] = 1.0
def w_col(b, shift=0):
v = np.zeros(16, dtype=complex)
for m in range(4):
v[IDX[(m, m)]] = 1j ** (((b * m) + shift * m) % 4)
return v
def diag_unph():
v = np.zeros(16, dtype=complex)
for m in range(4):
v[IDX[(m, m)]] = 1.0
return v
def col_contact(j):
v = np.zeros(16, dtype=complex)
for m in range(4):
v[IDX[(m, m)]] = 1.0 - 1j ** ((j * m) % 4)
return v
COLUMN_MODELS = {
'diag': {1: E00, 3: E00, 2: E00},
'spread': {1: w_col(1), 3: w_col(3), 2: w_col(2)},
}
WRONG_MODEL = {1: w_col(1, shift=1), 3: w_col(3, shift=1),
2: w_col(2, shift=1)}
def make_seed2():
seed = {}
for p in ORB2:
v = np.zeros(16, dtype=complex)
v[IDX[(p[0] % 4, p[0] % 4)]] = 1j ** ((p[0] * p[1]) % 4)
seed[p] = v
return seed
# ---------------------------------------------------------------------------
# S0 -- exact foundations: roots, ledgers, targets, Z2 anchor, Weil, chars
# ---------------------------------------------------------------------------
def section0(V4, CNT):
print(" -- S0: exact foundations (roots, ledgers, targets, Weil)")
roots = build_glue_roots()
counts = [sum(1 for c in roots.values() if c == m) for m in range(4)]
check("S0.1 [ROOTS] 240 roots, all norm 2, class split %s = "
"(52, 64, 60, 64); 16 = |mu4|^2 discriminant classes "
"(v492/v505 replication)" % fmt(counts),
len(roots) == 240 and counts == [52, 64, 60, 64]
and all(sum(x * x for x in r) == 2 for r in roots)
and NCLS == 16 and MU4 == 4 and rankE8 == 8)
ok_q = all(V4[(a, a)].get(F(1)) == QTAB[a] for a in range(4))
tots = [sum(CNT[(a, a)].get(F(n), 0) for a in range(4))
for n in range(1, 9)]
tots_anti = [sum(CNT[m].get(F(n), 0) for m in ANTI_H)
for n in range(1, 9)]
check("S0.2 [16-CLASS LEDGER] diagonal slice = quartic ledger "
"(V_{(a,a),1} = Q_a exactly); diagonal AND anti-diagonal "
"counts = 240 sigma_3(n) for n <= 8 (both Lagrangian "
"gluings are E4; delta-1c/v518 replication)",
ok_q and tots == [240 * s for s in SIGMA3]
and tots_anti == [240 * s for s in SIGMA3])
I = sp.I
R = sp.Rational
Qv = {m: [R(x) for x in QTAB[m]] for m in range(4)}
Qtw = {j: [sp.nsimplify(sp.expand(
sum(I ** (j * m) * Qv[m][i] for m in range(4))))
for i in range(5)] for j in range(4)}
h = [R(m * (4 - m), 8) for m in range(4)]
w = [sp.nsimplify(sp.expand(
sum((1 - I ** (j * m)) / DETS[j] for j in (1, 2, 3))))
for m in range(4)]
Afix = [sp.nsimplify(sum(Qtw[j][i] / DETS[j] for j in (1, 2, 3)))
for i in range(5)]
contact = {j: [sp.nsimplify((Qtw[0][i] - Qtw[j][i]) / DETS[j])
for i in range(5)] for j in (1, 2, 3)}
total = [sum(contact[j][i] for j in (1, 2, 3)) for i in range(5)]
sums = {j: [sp.nsimplify(Qtw[j][i] / DETS[j] + contact[j][i])
for i in range(5)] for j in (1, 2, 3)}
psiS = lambda v: sp.nsimplify(3 * v[0] - v[1] - v[2] - R(1, 4) * v[4])
check("S0.3 [A-CORE, EXACT] Q^{(0)} = %s = (36,72,36,0,0); A_fix = "
"%s = (9,-30,-15,0,32); completion weights w_m = %s = "
"(0, 3/2, 2, 3/2) = 4 h_m, h_2 : h_1 = %s = 4:3; contact "
"total = %s = (36,120,60,0,-32); per-sector perfect squares "
"(18/9/18)<x,x>^2; psi(A_fix) = %s = +64, psi(contact) = %s "
"= -64 (v516 replication)"
% (fmt(Qtw[0]), fmt(Afix), fmt(w), h[2] / h[1], fmt(total),
psiS(Afix), psiS(total)),
Qtw[0] == [36, 72, 36, 0, 0] and Afix == AFIX
and w == [0, R(3, 2), 2, R(3, 2)]
and all(w[m] == 4 * h[m] for m in range(4))
and h[2] / h[1] == R(4, 3)
and total == CONTACT_A
and sums[1] == [18, 36, 18, 0, 0]
and sums[2] == [9, 18, 9, 0, 0]
and sums[3] == [18, 36, 18, 0, 0]
and psiS(Afix) == 64 and psiS(total) == -64)
A2 = [sp.nsimplify(Qtw[2][i] / 4) for i in range(5)]
contact2 = [sp.nsimplify(R(1, 2) * (Qv[1][i] + Qv[3][i]))
for i in range(5)]
tot2 = [A2[i] + contact2[i] for i in range(5)]
psiA2 = psiS(A2)
leftovers = {}
for c in (F(1, 4), F(1, 2), F(1), F(2)):
cc = sp.Rational(c.numerator, c.denominator)
t5l = sp.nsimplify(A2[3] + cc * (Qv[1][3] + Qv[3][3]))
t3l = sp.nsimplify(A2[4] + cc * (Qv[1][4] + Qv[3][4]))
leftovers[c] = (t5l, t3l)
ok_scale = (leftovers[F(1, 2)] == (0, 0)
and all(leftovers[c] != (0, 0)
for c in (F(1, 4), F(1), F(2))))
check("S0.4 [Z2 ANCHOR TARGETS, EXACT] A_2 = Q^{(g^2)}/4 = %s = "
"(-3,-6,9,8,-16); contact_2 = (1/2)(Q_1 + Q_3) = Q_1; "
"A_2 + contact_2 = %s = 9<x,x>^2 = (1/4) x 36<x,x>^2, "
"coupling scale 2 = |Z2| (weight 1/2 = 2 x h^{A1}); "
"psi(A_2) = %s = -8; SCALE TEETH: only c = 1/2 clears both "
"leftovers %s"
% (fmt(A2), fmt(tot2), psiA2,
str({str(k): fmt(v) for k, v in leftovers.items()})),
A2 == A2VEC and contact2 == Q1VEC
and tot2 == [9, 18, 9, 0, 0] and psiA2 == -8 and ok_scale)
Tx, Sx = weil_matrices()
n = 16
C = [[Z8_ONE if MU[j] == neg(MU[i]) else Z8_ZERO for j in range(n)]
for i in range(n)]
S2 = mat_mul(Sx, Sx)
ST = mat_mul(Sx, Tx)
ST3 = mat_mul(mat_mul(ST, ST), ST)
T8 = mat_id(n)
for _ in range(8):
T8 = mat_mul(T8, Tx)
ok_sym = all(Sx[i][j] == Sx[j][i] for i in range(n) for j in range(n))
eH = [Z8_ONE if MU[i] in DIAG_H else Z8_ZERO for i in range(n)]
eHp = [Z8_ONE if MU[i] in ANTI_H else Z8_ZERO for i in range(n)]
SeH = [sum_z8([z8_mul(Sx[i][j], eH[j]) for j in range(n)])
for i in range(n)]
TeH = [z8_mul(Tx[i][i], eH[i]) for i in range(n)]
SeHp = [sum_z8([z8_mul(Sx[i][j], eHp[j]) for j in range(n)])
for i in range(n)]
TeHp = [z8_mul(Tx[i][i], eHp[i]) for i in range(n)]
iso_diag = all(q8(m) == 0 for m in DIAG_H + ANTI_H)
check("S0.5 [WEIL EXACT] S symmetric + unitary, S^2 = C, (ST)^3 = "
"S^2, S^4 = 1, T^8 = 1 in Q(zeta_8); e_H (glue diagonal) and "
"e_H' (anti-diagonal) exactly T- and S-fixed, all classes "
"isotropic (v518/delta-1e replication)",
ok_sym and mat_eq(mat_mul(Sx, mat_conjT(Sx)), mat_id(n))
and mat_eq(S2, C) and mat_eq(ST3, S2)
and mat_eq(mat_mul(S2, S2), mat_id(n))
and mat_eq(T8, mat_id(n))
and SeH == eH and TeH == eH and SeHp == eHp and TeHp == eHp
and iso_diag)
e = lambda x: sp.exp(2 * sp.pi * sp.I * x)
chi4T = e(R(1, 3))
chi10T = e(R(5, 6))
ok_chars = (sp.simplify(chi4T - 1) != 0
and sp.simplify(chi10T - 1) != 0
and sp.simplify(chi4T * sp.conjugate(chi4T) - 1) == 0
and sp.simplify(chi4T ** 2 - e(R(2, 3))) == 0
and sp.simplify(chi4T ** 2 - 1) != 0
and sp.simplify((-1) * e(R(-1, 6)) - chi4T) == 0)
check("S0.6 [CHARACTERS, EXACT] chi_4(T) = e(1/3) and chi_10(T) = "
"e(5/6) are NONTRIVIAL; |chi_4(T)|^2 = 1 exactly (conjugate "
"pairing cancels) while chi_4(T)^2 = e(2/3) != 1 (one-sided "
"squaring does NOT); T-fix mechanism (-1) x e(-1/6) = "
"chi_4(T) replicated; real tau_2 powers carry no phase",
ok_chars)
return Qv, Qtw, Tx, Sx
# ---------------------------------------------------------------------------
# S1 -- transports, derived families, the k = 0 slice
# ---------------------------------------------------------------------------
def section1(Tx, Sx):
print(" -- S1: block transports + derived families")
tg, sg, dev = measure_transport_w(
4, PAIRS4, lambda a, b, tau: M_val(4, a, b, tau), 4)
tgx = {}
devmax = mp.mpf(0)
for p in PAIRS4:
lt, kt, dt = recog(tg[p])
_, _, ds = recog(sg[p])
devmax = max(devmax, dt, ds)
tgx[p] = (lt, kt)
axd = {}
for name, ws in CAND_WS.items():
tA, sA, devA = measure_transport_w(
4, PAIRS4, lambda a, b, tau: ax_val(4, ws, a, b, tau), 0)
tAx = {}
dmax = mp.mpf(0)
for p in PAIRS4:
lt, kt, dt = recog(tA[p])
_, _, ds = recog(sA[p])
dmax = max(dmax, dt, ds)
tAx[p] = (lt, kt)
axd[name] = dict(ws=ws, t=tA, s=sA, tx=tAx, dev=devA, rec=dmax)
ok_fix = all(tgx[(0, b)][1] == L_PHASE // 2
and abs(tgx[(0, b)][0]) < mp.mpf(10) ** (-30)
and axd['f1f3']['tx'][(0, b)][1]
== (-L_PHASE // 6) % L_PHASE for b in (1, 2, 3))
check("S1.1 [TRANSPORTS] gauge + fibre transport constants constant "
"over two tau (max dev %s < 1e-28) and recognised on the "
"1/960 grid (max %s < 1e-25); T-fixed nodes: gauge k = 480 "
"(= -1), fibre f1f3 k = 800 (= e(-1/6)) exactly (v518 "
"replication)"
% (mp.nstr(max(dev, max(d['dev'] for d in axd.values())), 3),
mp.nstr(max(devmax, max(d['rec'] for d in axd.values())),
3)),
dev < mp.mpf(10) ** (-28) and devmax < mp.mpf(10) ** (-25)
and all(d['dev'] < mp.mpf(10) ** (-28) for d in axd.values())
and all(d['rec'] < mp.mpf(10) ** (-25) for d in axd.values())
and ok_fix)
rhoT = rho_np(Tx)
rhoS = rho_np(Sx)
fams = {}
for orient in ('hol', 'anti'):
tt, ss = dressed_consts(tg, sg, axd['f1f3'], orient, PAIRS4)
for k in range(12):
_, L1, n1, _ = solve_orbit(ORB1, 4, tt, ss, rhoT, rhoS,
(0, 1), k)
if not n1:
continue
_, L2, n2, _ = solve_orbit(ORB2, 4, tt, ss, rhoT, rhoS,
(0, 2), k)
if not n2:
continue
fams[(orient, k)] = dict(L1=L1, n1=n1, L2=L2, n2=n2)
for (orient, k), d in sorted(fams.items()):
print(" family %-4s chi_%-2d dims (%d, %d)"
% (orient, k, len(d['n1']), len(d['n2'])))
hol_ks = sorted(k for (o, k) in fams if o == 'hol')
dims_ok = (('hol', 4) in fams and ('hol', 10) in fams
and len(fams[('hol', 4)]['n1']) == 3
and len(fams[('hol', 4)]['n2']) == 3
and len(fams[('hol', 10)]['n1']) == 1
and len(fams[('hol', 10)]['n2']) == 1)
check("S1.2 [THE ONLY STRICT CHIRAL FAMILIES] hol f1f3 closes "
"exactly at k = %s with chi_4 dims (3,3) and chi_10 dims "
"(1,1), both characters nontrivial (S0.6); anti k = %s "
"(membership only) -- every strict chiral closure at "
"physical weight carries a NONTRIVIAL multiplier (v518 S4 "
"replication)"
% (str(hol_ks), str(sorted(k for (o, k) in fams
if o == 'anti'))),
hol_ks == [4, 10] and dims_ok)
k0 = {}
scans = [('none', 'hol')] + [(nm, o) for nm in CAND_WS
for o in ('hol', 'anti')]
for nm, orient in scans:
if nm == 'none':
tt = {p: complex(tg[p]) for p in PAIRS4}
ss = {p: complex(sg[p]) for p in PAIRS4}
else:
tt, ss = dressed_consts(tg, sg, axd[nm], orient, PAIRS4)
_, _, n1, _ = solve_orbit(ORB1, 4, tt, ss, rhoT, rhoS, (0, 1), 0)
_, _, n2, _ = solve_orbit(ORB2, 4, tt, ss, rhoT, rhoS, (0, 2), 0)
k0[(nm, orient)] = (len(n1), len(n2))
n_k0 = sum(1 for v in k0.values() if v[0] > 0 and v[1] > 0)
check("S1.3 [NO SINGLE-VALUED CHIRAL OPTION, THE CORE FACT] strict "
"trivial-character two-orbit solutions at physical weight "
"(m, k) = (0, 0): count %d = 0 across the bare gauge blocks "
"and ALL five dressings in BOTH orientations (the eta^{+-8} "
"hits of v518 are weight-shifted bookkeeping, NOT the "
"physical weight): the one-sided chiral combination KEEPS "
"its multiplier, always" % n_k0, n_k0 == 0)
return tg, sg, tgx, axd, rhoT, rhoS, fams, n_k0
# ---------------------------------------------------------------------------
# S2 -- D1: the untwisted-column boundary criterion (canonicalisation)
# ---------------------------------------------------------------------------
def section2(fams, Qv, Qtw, V4, V2, CNT, ser0, lim):
print(" -- S2: D1 -- column canonicalisation + sharpened kill")
I = sp.I
ok_id = True
for b in (1, 2, 3):
lhs = [sp.nsimplify(sp.expand(
sum(I ** (b * m) * Qv[m][i] for m in range(4))))
for i in range(5)]
if lhs != Qtw[b]:
ok_id = False
check("S2.1 [A PASSES THE COLUMN CRITERION, EXACT] the spread "
"column sum_m i^{bm} Q_m = Q^{(b)} for b = 1, 2, 3 -- the "
"physical (0,b) contraction reproduces the AB skeleton "
"numerators exactly; completing each channel with the "
"unphased loop Q^{(0)}/det_j is reading (A) by construction",
ok_id)
rng = np.random.default_rng(23)
b0_star = E00.copy()
null_fake = [b0_star,
rng.normal(size=16) + 1j * rng.normal(size=16),
rng.normal(size=16) + 1j * rng.normal(size=16)]
null_fake = [v / np.linalg.norm(v) for v in null_fake]
Lam1 = fams[('hol', 4)]['L1']
targets_fake = {p: Lam1[p] @ b0_star for p in [(0, 1), (0, 3)]}
r_fake, c_fake, _, _ = member_test(Lam1, null_fake, targets_fake)
check("S2.2 [NC: PLANTED FAMILY] a synthetic family containing its "
"own column data is detected: sigma ratio %.2e < 1e-10 with "
"|c| = %.3f > 0.1 (membership machinery teeth)"
% (r_fake, c_fake), r_fake < 1e-10 and c_fake > 0.1)
print(" membership table (strong condition, one scalar per "
"orbit): sigma_min/sigma_max [|c| rel]")
results = {}
for (orient, k), fam in sorted(fams.items()):
for model, cols in list(COLUMN_MODELS.items()) + [
('wrongph', WRONG_MODEL)]:
t1 = {(0, 1): cols[1], (0, 3): cols[3]}
r1 = member_test(fam['L1'], fam['n1'], t1)
t2 = {(0, 2): cols[2]}
r2 = member_test(fam['L2'], fam['n2'], t2)
results[(orient, k, model)] = (r1, r2)
print(" %-4s chi_%-2d %-7s orb1 %.2e [%.3f] "
"orb2 %.2e [%.3f]"
% (orient, k, model, r1[0], r1[1], r2[0], r2[1]))
contained = []
grey = []
for (orient, k, model), (r1, r2) in results.items():
if model == 'wrongph':
continue
ok1 = r1[0] < CONTAIN_TOL and r1[1] > 1e-6
ok2 = r2[0] < CONTAIN_TOL and r2[1] > 1e-6
if ok1 and ok2:
contained.append((orient, k, model))
elif (r1[0] < ROBUST_TOL or r2[0] < ROBUST_TOL) \
and not (ok1 and ok2):
grey.append((orient, k, model, r1[0], r2[0]))
check("S2.3 [D1 MEMBERSHIP RESULT] the physical AB spread column "
"sum_m i^{bm} e_{(m,m)} is contained in exactly ONE derived "
"family (both orbits, one scalar each -- CANONICALISING the "
"(N1, N2) freedom): %s = [('hol', 4, 'spread')]; grey-zone "
"entries: %s"
% (contained, [(g[0], g[1], g[2]) for g in grey]),
contained == [('hol', 4, 'spread')]
and [(g[0], g[1], g[2]) for g in grey]
== [('anti', 8, 'spread')])
wrong_contained = []
for (orient, k, model), (r1, r2) in results.items():
if model != 'wrongph':
continue
if (r1[0] < CONTAIN_TOL and r1[1] > 1e-6
and r2[0] < CONTAIN_TOL and r2[1] > 1e-6):
wrong_contained.append((orient, k))
check("S2.4 [NC: WRONG-PHASE COLUMNS] the shifted-phase control "
"columns i^{(b+1)m} are contained in NO family: %s"
% (wrong_contained if wrong_contained else "NONE"),
not wrong_contained)
orient, k, model = contained[0]
fam = fams[(orient, k)]
cols = COLUMN_MODELS[model]
_, _, u1, V1 = member_test(
fam['L1'], fam['n1'], {(0, 1): cols[1], (0, 3): cols[3]})
_, _, u2, V2m = member_test(
fam['L2'], fam['n2'], {(0, 2): cols[2]})
b01 = V1 @ (u1[:-1] / u1[-1])
b02 = V2m @ (u2[:-1] / u2[-1])
beta = {}
for p in ORB1:
beta[p] = fam['L1'][p] @ b01
for p in ORB2:
beta[p] = fam['L2'][p] @ b02
th = {}
for tau in (TAU_C, TAU_C + 1, -1 / TAU_C):
th[tau] = [eval_series16(ser0[mu], tau, lim) for mu in MU]
cert = function_certificate(beta, (1, 3), k, th)
I_main, W, n_neg, wt_neg, im_rel, eps_rel = run_integral(
beta, (1, 3), V4, V2, CNT, mp.mpf(1) / 4)
tol = max(eps_rel * 100, mp.mpf(10) ** (-7))
scale = max(abs(x) for x in I_main)
t5rel = abs(I_main[3]) / scale
sprA, _ = spread_vs(I_main, [-x for x in AFIX])
sprC, _ = spread_vs(I_main, CONTACT_A)
psiI = psi_of(I_main)
passes = (t5rel < tol
and (sprA < tol * 10 or sprC < tol * 10
or (psiI.real < 0
and abs(psiI.imag) < abs(psiI.real) * 1e-6)))
print(" [D1 canonical member %s chi_%d %s] cert %s; I_main = "
"%s; neg-cells %d; T5 frac %s; spread(-A_fix) %s; "
"spread(contact_A) %s; psi = %s"
% (orient, k, model, mp.nstr(cert, 3), fmtn(I_main), n_neg,
mp.nstr(t5rel, 3), mp.nstr(sprA, 3), mp.nstr(sprC, 3),
mp.nstr(psiI, 6)))
check("S2.5 [D1 SHARPENED KILL] the column-matched canonical chi_4 "
"member (function certificate %s < 1e-8) STILL FAILS all "
"testers (T5 frac %s > 0.1, spreads %s / %s > 0.5, psi > 0): "
"the (N1, N2)-canonicalised derived route is boundary-"
"consistent yet tester-dead -- the v518 kill is sharpened"
% (mp.nstr(cert, 3), mp.nstr(t5rel, 3), mp.nstr(sprA, 3),
mp.nstr(sprC, 3)),
cert < mp.mpf(10) ** (-8) and not passes
and t5rel > mp.mpf('0.1'))
return contained, grey
# ---------------------------------------------------------------------------
# S3 -- D2: the exact invariant subspace / single-valuedness forcing
# ---------------------------------------------------------------------------
def section3(Tx, Sx, V4, CNT, n_k0, tg, sg, axd, rhoT, rhoS):
print(" -- S3: D2 -- invariant subspace / single-valuedness")
ns = invariant_subspace(Tx, Sx)
dim_q = len(ns)
all_in_span = all(in_lagrangian_span(v) for v in ns)
check("S3.1 [THE FULL INVARIANT SUBSPACE, EXACT RATIONAL] "
"dim_Q {v : rho(T) v = v, rho(S) v = v} = %d = 4 x 2 "
"([Q(zeta_8):Q] = 4 x dim 2 over the field); EVERY exact "
"nullspace vector lies in the Q(zeta_8)-span of {e_H, e_H'}: "
"%s -- the invariants are EXACTLY span{e_H, e_H'}, every "
"single-valued lattice factor is a combination of the two "
"Lagrangian thetas" % (dim_q, all_in_span),
dim_q == 8 and all_in_span)
q_unph = [sum(V4[(a, a)][F(1)][i] for a in range(4))
for i in range(5)]
check("S3.2 [BOTH INVARIANTS CARRY THE UNPHASED TRACE] Theta_H and "
"Theta_H' = E4 (S0.2); the level-1 quartic of the "
"H-contraction = %s = (36, 72, 36, 0, 0) = Q^{(0)}: the "
"single-valued lattice factor is FORCED to the unphased "
"sector trace" % fmt(q_unph),
q_unph == [36, 72, 36, 0, 0])
check("S3.3 [DERIVED SIDE CANNOT BE SINGLE-VALUED AT PHYSICAL "
"WEIGHT] chi_4(T), chi_10(T) nontrivial (S0.6); strict "
"(m, k) = (0, 0) solutions: %d = 0 for every dressing "
"(S1.3); real tau_2 powers cannot compensate a phase; the "
"eta^{+-8} absorption shifts the block weight by -+4 and "
"breaks the fixed 'block -4 x theta +4' bookkeeping (TP-3)"
% n_k0, n_k0 == 0)
z2tab = {}
for orient in ('hol', 'anti'):
tt, ss = dressed_consts(tg, sg, axd['f1f3'], orient, ORB2)
for k in range(12):
_, L2, n2, _ = solve_orbit(ORB2, 4, tt, ss, rhoT, rhoS,
(0, 2), k)
if n2:
z2tab[(orient, k)] = dict(L2=L2, n2=n2)
ks_hol = sorted(k for (o, k) in z2tab if o == 'hol')
check("S3.4 [Z2 STANDALONE CHARACTER TABLE] the Z2 sub-orbifold "
"system (dressed) closes strictly at hol k = %s ({4, 10} "
"expected as a subset); trivial character k = 0 is NOT "
"admitted at Z2 either" % ks_hol,
set([4, 10]).issubset(set(ks_hol)) and 0 not in ks_hol)
return dim_q, z2tab, ks_hol
# ---------------------------------------------------------------------------
# S4 -- D3: the Z2/EH anchor at the integral level
# ---------------------------------------------------------------------------
def section4(z2tab, V4, V2, CNT, ser0, lim):
print(" -- S4: D3 -- the Z2/EH anchor at the integral level")
th = {}
for tau in (TAU_C, TAU_C + 1, -1 / TAU_C):
th[tau] = [eval_series16(ser0[mu], tau, lim) for mu in MU]
seed2 = make_seed2()
hol_ks = sorted(k for (o, k) in z2tab if o == 'hol')
outcomes = []
for k in hol_ks:
dat = z2tab[('hol', k)]
evals = [('seed-proj', project_seed(dat['L2'], dat['n2'],
seed2)[0])]
for model, cols in COLUMN_MODELS.items():
r2 = member_test(dat['L2'], dat['n2'], {(0, 2): cols[2]})
if r2 is not None and r2[0] < CONTAIN_TOL and r2[1] > 1e-6:
b0 = np.array(dat['n2']).T @ (r2[2][:-1] / r2[2][-1])
beta_m = {p: dat['L2'][p] @ b0 for p in ORB2}
evals.append(('column-' + model, beta_m))
for tag, beta in evals:
if beta is None:
continue
nrm = max(np.linalg.norm(beta[p]) for p in beta)
if nrm < 1e-12:
continue
cert = function_certificate(beta, (1, 3), k, th)
I_main, W, n_neg, wt_neg, im_rel, eps_rel = run_integral(
beta, (1, 3), V4, V2, CNT, mp.mpf(1) / 2)
tol = max(eps_rel * 100, mp.mpf(10) ** (-7))
sprQ1, zQ1 = spread_vs(I_main, Q1VEC)
psiI = psi_of(I_main)
ratio_t5t3 = (I_main[3] / I_main[4]
if abs(I_main[4]) > 0 else mp.inf)
okQ1 = sprQ1 < tol * 10 and zQ1 < tol * 10
ok_ratio = abs(ratio_t5t3 + mp.mpf(1) / 2) < mp.mpf('0.05')
ok_psi = (psiI.real > 0
and abs(psiI.imag) < abs(psiI.real) * 1e-6)
anchor_pass = okQ1 or (ok_ratio and ok_psi)
print(" [Z2 chi_%d %s] cert %s; I = %s; spread(Q1) %s; "
"psi = %s (Soll +8N)"
% (k, tag, mp.nstr(cert, 3), fmtn(I_main),
mp.nstr(sprQ1, 3), mp.nstr(psiI, 6)))
outcomes.append(dict(k=k, tag=tag, cert=cert,
anchor=anchor_pass))
check("S4.%d [Z2 DERIVED INTEGRAL, chi_%d, %s] function "
"certificate %s < 1e-8; the Z2 anchor testers "
"(V ~ Q_1 / T5:T3 = -1:2 with psi > 0) FAIL "
"(spread(Q1) %s, psi < 0)"
% (len(outcomes), k, tag, mp.nstr(cert, 3),
mp.nstr(sprQ1, 3)),
cert < mp.mpf(10) ** (-8) and not anchor_pass)
all_fail = bool(outcomes) and not any(o['anchor'] for o in outcomes)
check("S4.X [D3 RESULT] declared reading hits the Z2 anchor "
"exactly (S0.4, arithmetic identity); derived Z2 "
"evaluations: %d run, anchor passes: %d = 0 -- D3 fires "
"for A (every derived instantiation fails at the anchor)"
% (len(outcomes), sum(1 for o in outcomes if o['anchor'])),
len(outcomes) == 3 and all_fail)
return outcomes, all_fail
# ---------------------------------------------------------------------------
# S5 -- SOURCE 1: the F-lemma (fundamental-domain independence)
# ---------------------------------------------------------------------------
def section5(fams, ser0, lim):
print(" -- S5: source-1 -- the fundamental-domain lemma")
r1 = member_test(fams[('hol', 4)]['L1'], fams[('hol', 4)]['n1'],
{(0, 1): w_col(1), (0, 3): w_col(3)})
r2 = member_test(fams[('hol', 4)]['L2'], fams[('hol', 4)]['n2'],
{(0, 2): w_col(2)})
contained = (r1[0] < CONTAIN_TOL and r1[1] > 1e-6
and r2[0] < CONTAIN_TOL and r2[1] > 1e-6)
b01 = np.array(fams[('hol', 4)]['n1']).T @ (r1[2][:-1] / r1[2][-1])
b02 = np.array(fams[('hol', 4)]['n2']).T @ (r2[2][:-1] / r2[2][-1])
beta = {}
for p in ORB1:
beta[p] = fams[('hol', 4)]['L1'][p] @ b01
for p in ORB2:
beta[p] = fams[('hol', 4)]['L2'][p] @ b02
taus = []
for t0 in (TAU_C, TAU_D):
taus += [t0, t0 + 1, -1 / t0]
th = {tau: [eval_series16(ser0[mu], tau, lim) for mu in MU]
for tau in taus}
def G_total(tau):
tot = mp.mpc(0)
for p, bp in beta.items():
v = (M_val(4, p[0], p[1], tau)
* ax_val(4, (1, 3), p[0], p[1], tau))
tot += v * sum(complex(bp[i]) * th[tau][i] for i in range(16))
return tot
chT = mp.exp(2j * mp.pi * mp.mpf(4) / 12)
chS = mp.exp(-2j * mp.pi * mp.mpf(4) / 4)
cert = mp.mpf(0)
gmin = mp.inf
for t0 in (TAU_C, TAU_D):
g0 = G_total(t0)
gmin = min(gmin, abs(g0))
cert = max(cert, abs(G_total(t0 + 1) / (chT * g0) - 1))
cert = max(cert, abs(G_total(-1 / t0) / (chS * g0) - 1))
print(" canonical chi_4 column member: sigma ratios orb1 "
"%.2e / orb2 %.2e; |G(tau)| >= %s; total-sum certificate "
"max dev %s over two fresh tau"
% (r1[0], r2[0], mp.nstr(gmin, 3), mp.nstr(cert, 3)))
ok = bool(contained and cert < mp.mpf(10) ** (-8)
and gmin > mp.mpf(10) ** (-10))
check("S5.1 [F-LEMMA CERTIFICATE + COROLLARY] the canonical "
"member's TOTAL block sum obeys G(tau+1) = chi_4(T) G(tau), "
"G(-1/tau) = chi_4(S) G(tau) pointwise (cert %s < 1e-8, "
"nonvacuous |G| > 1e-10). EXACT CHANGE OF VARIABLES: "
"int_{gamma F} G dmu = chi_4(gamma) int_F G dmu with "
"chi_4(T) = e(1/3) != 1 -- an F-independent value forces "
"int = 0: under TP-1 + TP-2 NO chiral integrand at physical "
"weight can source psi = 64; single-valuedness is FORCED "
"for any nonzero one-loop lattice contribution"
% mp.nstr(cert, 3), ok)
return ok
# ---------------------------------------------------------------------------
# S6 -- SOURCE 2: the Quillen pairing
# ---------------------------------------------------------------------------
def section6(tg, sg, axd, rhoT, rhoS):
print(" -- S6: source-2 -- the Quillen pairing")
ttH, ssH = dressed_consts(tg, sg, axd['f1f3'], 'hol', PAIRS4)
ktot = {}
dbl_dev = mp.mpf(0)
for p in PAIRS4:
_, kt, _ = recog(mp.mpc(ttH[p]))
ktot[p] = kt
tf = mp.mpc(tg[p]) * mp.mpc(axd['f1f3']['t'][p])
sf = mp.mpc(sg[p]) * mp.mpc(axd['f1f3']['s'][p])
dbl_dev = max(dbl_dev, abs(tf * mp.conj(tf) - 1),
abs(sf * mp.conj(sf) - 1))
sq_def_T = {p: (2 * ktot[p]) % L_PHASE for p in PAIRS4}
fib_sq = (2 * ((-L_PHASE // 6) % L_PHASE)) % L_PHASE
check("S6.1 [CONJUGATE PAIRING CANCELS -- MEASURED UNITARITY] the "
"doubled hol x antihol transports t conj(t) = |t|^2 equal 1 "
"to %s < 1e-30 on all 15 pairs and both generators; the "
"one-sided SQUARED combination keeps defects: T-fixed fibre "
"2 x 800 = %d = e(2/3) != 0 on the exact grid, squared-"
"defect table nonzero on %d of 15 pairs -- the phase dies "
"ONLY in the conjugate pairing (|chi_4|^2 = 1 vs chi_4^2 = "
"e(2/3), S0.6)"
% (mp.nstr(dbl_dev, 3), fib_sq,
sum(1 for p in PAIRS4 if sq_def_T[p] != 0)),
dbl_dev < mp.mpf(10) ** (-30) and fib_sq == 640
and any(v != 0 for v in sq_def_T.values()))
rhoT2 = np.kron(rhoT, rhoT.conj())
rhoS2 = np.kron(rhoS, rhoS.conj())
tt2 = {p: ttH[p] * np.conj(ttH[p]) for p in PAIRS4}
ss2 = {p: ssH[p] * np.conj(ssH[p]) for p in PAIRS4}
_, L1d, n1d, _ = solve_orbit(ORB1, 4, tt2, ss2, rhoT2, rhoS2,
(0, 1), 0)
_, L2d, n2d, _ = solve_orbit(ORB2, 4, tt2, ss2, rhoT2, rhoS2,
(0, 2), 0)
d1, d2 = len(n1d), len(n2d)
vecI = np.zeros(256, dtype=complex)
for i in range(16):
vecI[16 * i + i] = 1.0
vecI /= np.linalg.norm(vecI)
resI = []
for nd in (n1d, n2d):
V = np.array(nd).T
proj = V @ np.linalg.solve(V.conj().T @ V, V.conj().T @ vecI)
resI.append(float(np.linalg.norm(vecI - proj)))
check("S6.2 [DOUBLED SYSTEM CLOSES SINGLE-VALUEDLY] the 256-dim "
"hol x antihol Weil system (rho (x) conj(rho), transports "
"t conj(t)) closes STRICTLY at trivial character and "
"physical weight: nullspace dims (%d, %d) >= 11 = 3^2 + 1^2 "
"+ 1; the unphased diagonal sum_mu e_mu (x) conj(e_mu) lies "
"inside (residuals %.2e / %.2e < 1e-8): single-valuedness "
"is IMPLEMENTED by the pairing, not postulated"
% (d1, d2, resI[0], resI[1]),
d1 >= 11 and d2 >= 11 and max(resI) < 1e-8)
W1 = np.kron(w_col(1), w_col(1).conj())
W3 = np.kron(w_col(3), w_col(3).conj())
W2 = np.kron(w_col(2), w_col(2).conj())
rd1 = member_test(L1d, n1d, {(0, 1): W1, (0, 3): W3})
rd2 = member_test(L2d, n2d, {(0, 2): W2})
check("S6.3 [PHYSICAL DOUBLED COLUMNS CONTAINED] the physical "
"untwisted-column data w_b (x) conj(w_b) is contained in "
"the doubled single-valued family with ONE scalar per orbit "
"(sigma %.2e / %.2e < 1e-8, coefficient nonvanishing): the "
"physical |.|^2 object lives entirely inside the single-"
"valued surface" % (rd1[0], rd2[0]),
rd1[0] < CONTAIN_TOL and rd1[1] > 1e-6
and rd2[0] < CONTAIN_TOL and rd2[1] > 1e-6)
rhoT2w = np.kron(rhoT, rhoT)
rhoS2w = np.kron(rhoS, rhoS)
tt2w = {p: ttH[p] * ttH[p] for p in PAIRS4}
ss2w = {p: ssH[p] * ssH[p] for p in PAIRS4}
_, _, n1w, sv1w = solve_orbit(ORB1, 4, tt2w, ss2w, rhoT2w, rhoS2w,
(0, 1), 0)
_, _, n2w, sv2w = solve_orbit(ORB2, 4, tt2w, ss2w, rhoT2w, rhoS2w,
(0, 2), 0)
check("S6.4 [NC: WRONG PAIRING EMPTY] the hol (x) hol doubled "
"system (squared transports, characters chi^2 != 1) has NO "
"trivial-character closure: nullspace dims (%d, %d) = "
"(0, 0), min sv ratios %.2e / %.2e -- only the CONJUGATE "
"pairing implements single-valuedness"
% (len(n1w), len(n2w),
min(sv1w) / max(sv1w), min(sv2w) / max(sv2w)),
len(n1w) == 0 and len(n2w) == 0)
_, _, devw = measure_transport_w(
4, [(0, 1), (1, 0), (1, 1), (2, 1)],
lambda a, b, tau: M_val(4, a, b, tau), 2)
check("S6.5 [NC: WRONG WEIGHT] the gauge transport measured at "
"weight exponent 2 instead of 4 is NOT constant over tau "
"(dev %s > 1e-3): the physical weight bookkeeping (TP-3) "
"has teeth" % mp.nstr(devw, 3),
devw > mp.mpf(10) ** (-3))
rng = np.random.default_rng(23)
bstar = n1d[0]
fake_null = [bstar,
rng.normal(size=256) + 1j * rng.normal(size=256),
rng.normal(size=256) + 1j * rng.normal(size=256)]
fake_null = [v / np.linalg.norm(v) for v in fake_null]
targets_fake = {p: L1d[p] @ bstar for p in [(0, 1), (0, 3)]}
rf = member_test(L1d, fake_null, targets_fake)
check("S6.6 [NC: PLANTED DOUBLED FAMILY] a synthetic doubled "
"family containing its own column data is detected: sigma "
"%.2e < 1e-10 with |c| = %.3f > 0.1 (machinery teeth at "
"256 dims)" % (rf[0], rf[1]),
rf[0] < 1e-10 and rf[1] > 0.1)
A2ok = bool(dbl_dev < mp.mpf(10) ** (-30) and d1 >= 11 and d2 >= 11
and max(resI) < 1e-8
and rd1[0] < CONTAIN_TOL and rd2[0] < CONTAIN_TOL
and len(n1w) == 0 and len(n2w) == 0)
return A2ok
# ---------------------------------------------------------------------------
# S7 -- the point-4 consistency closure
# ---------------------------------------------------------------------------
def section7(V4, V2, CNT):
print(" -- S7: the point-4 consistency closure")
levels_H = sorted(set().union(*[set(V4[(m, m)].keys())
for m in range(4)]))
levels_Hp = sorted(set().union(*[set(V4[m].keys()) for m in ANTI_H]))
ok_H = True
ok_Hp = True
c_samples = []
for lev in levels_H:
s = [sum(V4[(m, m)].get(lev, ZERO5)[i] for m in range(4))
for i in range(5)]
c = s[0]
if s != [c, 2 * c, c, 0, 0]:
ok_H = False
if lev <= 3:
c_samples.append((lev, c))
for lev in levels_Hp:
s = [sum(V4[m].get(lev, ZERO5)[i] for m in ANTI_H)
for i in range(5)]
c = s[0]
if s != [c, 2 * c, c, 0, 0]:
ok_Hp = False
check("S7.1 [EXACT LEVELWISE OKUBO / 4-DESIGN] the unphased "
"diagonal ledger sum_m V_{(m,m),lev} AND the anti-diagonal "
"(H') sum are EXACTLY proportional to <x,x>^2 = (1, 2, 1, "
"0, 0) at EVERY level (H: %d levels, H': %d levels, n <= 8; "
"samples (lev, c): %s): the forced E4 numerator is a "
"perfect Okubo square LEVEL BY LEVEL"
% (len(levels_H), len(levels_Hp), str(c_samples[:4])),
ok_H and ok_Hp)
pairs_col = [(0, 1), (0, 2), (0, 3)]
dets = {1: 2.0, 2: 4.0, 3: 2.0}
dm, basemap = build_dress(pairs_col, (1, 3))
beta_unph = {p: diag_unph() / dets[p[1]] for p in pairs_col}
beta_ph = {p: w_col(p[1]) / dets[p[1]] for p in pairs_col}
beta_ct = {p: col_contact(p[1]) / dets[p[1]] for p in pairs_col}
def run_cols(beta, cut):
dml = {p: {e: v for e, v in dm[p].items() if e <= cut}
for p in beta}
return hm_integral_pp(beta, dml, basemap, V4, V2, CNT, cut,
mp.mpf(1) / 4)
I_tot, _, _, nneg_t, _ = run_cols(beta_unph, GRID_CUT)
I_lo, _, _, _, _ = run_cols(beta_unph, GRID_CUT_LOW)
scale = max(abs(x) for x in I_tot)
eps_rel = max(abs(I_tot[i] - I_lo[i]) for i in range(5)) / scale
im_rel = max(abs(x.imag) for x in I_tot) / scale
spr_tot, z_tot = spread_vs(I_tot, OKUBO)
ch_dat = []
for p in pairs_col:
I_ch, _, _, _, _ = run_cols({p: beta_unph[p]}, GRID_CUT)
spr, zz = spread_vs(I_ch, OKUBO)
ch_dat.append((p[1], spr, zz))
print(" I(unphased columns) = %s; cut-6 vs cut-8 %s; |Im| %s"
% (fmtn(I_tot), mp.nstr(eps_rel, 3), mp.nstr(im_rel, 3)))
check("S7.2 [UNPHASED NUMERATOR x v518 SCAFFOLD = OKUBO] the "
"J-weighted unphased column integral is proportional to "
"<x,x>^2 PER CHANNEL and in TOTAL (total spread %s, T5/T3 "
"fraction %s, channel spreads all < 1e-8), with ZERO "
"negative-exponent cells (%d): the v516 per-sector perfect "
"squares are REPRODUCED inside the v518 integral machinery"
% (mp.nstr(spr_tot, 3), mp.nstr(z_tot, 3), nneg_t),
spr_tot < mp.mpf(10) ** (-8) and z_tot < mp.mpf(10) ** (-8)
and all(spr < mp.mpf(10) ** (-8)
and zz < mp.mpf(10) ** (-8) for _, spr, zz in ch_dat)
and nneg_t == 0 and im_rel < mp.mpf(10) ** (-8))
I_ct, _, _, _, _ = run_cols(beta_ct, GRID_CUT)
psi_ct = psi_of(I_ct)
V4_1 = {mu: ({F(1): V4[mu][F(1)]} if F(1) in V4[mu] else {})
for mu in MU}
I_ct1, _, _, _, _ = hm_integral_pp(
beta_ct, dm, basemap, V4_1, V2, CNT, GRID_CUT, mp.mpf(1) / 4)
psi_ct1 = psi_of(I_ct1)
print(" I(contact columns) = %s; psi = %s; level-1 slice psi "
"= %s [arithmetic contact psi = -64]"
% (fmtn(I_ct), mp.nstr(psi_ct, 6), mp.nstr(psi_ct1, 6)))
ok_psi = (psi_ct.real < 0
and abs(psi_ct.imag) < abs(psi_ct.real) * 1e-6
and psi_ct1.real < 0)
check("S7.3 [CONTACT SUPPLIES THE psi SLICE, RIGHT SIGN] the "
"J-weighted contact (unphased minus phased columns) carries "
"psi = %s < 0 (full) and %s < 0 (level-1 slice) -- the "
"-64-slice sign of v511/v516 is reproduced by the FORCED "
"numerator inside the scaffold"
% (mp.nstr(psi_ct, 6), mp.nstr(psi_ct1, 6)), ok_psi)
I_ph, _, _, _, _ = run_cols(beta_ph, GRID_CUT)
psi_ph = psi_of(I_ph)
print(" I(skeleton/phased columns) = %s; psi = %s "
"[arithmetic A_fix psi = +64]"
% (fmtn(I_ph), mp.nstr(psi_ph, 6)))
lin = max(abs(I_ph[i] + I_ct[i] - I_tot[i]) for i in range(5)) / scale
check("S7.4 [THE CIRCLE CLOSES LINEARLY] skeleton (phased, psi = "
"%s > 0) + contact (completion, psi < 0) = unphased columns "
"inside the integral machinery (rel residual %s < 1e-12): "
"the v516 identity skeleton_j + contact_j = Q^(0)/det_j "
"holds cellwise in the v518 scaffold -- the circle to v516 "
"is closed" % (mp.nstr(psi_ph, 6), mp.nstr(lin, 3)),
lin < mp.mpf(10) ** (-12) and psi_ph.real > 0)
A3ok = bool(ok_H and ok_Hp and spr_tot < mp.mpf(10) ** (-8)
and z_tot < mp.mpf(10) ** (-8) and nneg_t == 0
and ok_psi and lin < mp.mpf(10) ** (-12))
return A3ok
# ---------------------------------------------------------------------------
# S8 -- remaining negative controls (exact)
# ---------------------------------------------------------------------------
def section8(Qv):
print(" -- S8: negative controls (exact)")
R = sp.Rational
I = sp.I
so_tot = [sp.nsimplify(R(5, 4) * (Qv[0][i] + Qv[2][i]))
for i in range(5)]
okmu2 = all(((m * p[1]) % 4) % 2 == 0 for p in ORB2
for m in (1, 2, 3))
ok_so = (so_tot == [15, 30, 45, 20, -40])
check("S8.1 [NC: SO(16)/D8] even with single-valuedness granted "
"the so16 unphased trace is NO Okubo square: completion "
"total = %s leaves T5 = 20, T3 = -40 (exact) -- the KILL "
"fires for so16, E8 stays doubly special; the order-4 "
"supply is structurally absent on the derived side (all "
"ORB2 fibre b-characters in mu_2: %s)"
% (fmt(so_tot), okmu2), ok_so and okmu2)
dets_f = {1: 2 * I, 2: 4, 3: -2 * I}
Qtw = {j: [sp.nsimplify(sp.expand(
sum(I ** (j * m) * Qv[m][i] for m in range(4))))
for i in range(5)] for j in range(4)}
wf = [sp.nsimplify(sp.expand(
sum((1 - I ** (j * m)) / dets_f[j] for j in (1, 2, 3))))
for m in range(4)]
Af = [sp.nsimplify(sp.expand(
sum(Qtw[j][i] / dets_f[j] for j in (1, 2, 3))))
for i in range(5)]
h = [R(m * (4 - m), 8) for m in range(4)]
check("S8.2 [NC: diag(i, i)] fake zero modes (2i, 4, -2i): "
"skeleton degenerates to the Z2 target %s = A_2; fake "
"weights %s break the conjugation pairing (w_1 != w_3) and "
"the index bridge (w != 4h) (v516 S4.4 replication)"
% (fmt(Af), fmt(wf)),
Af == A2VEC and wf == [0, R(-1, 2), 0, R(3, 2)]
and wf[1] != wf[3]
and any(wf[m] != 4 * h[m] for m in range(4)))
gw = sum_z8([z8_root((2 * x * x + 2 * y * y) % 8)
for x in range(4) for y in range(4)])
nrm = z8_mul(gw, z8_conj(gw))
Tw, Sw = build_weil(lambda m: (2 * m[0] ** 2 + 2 * m[1] ** 2) % 8,
lambda m, n2: (4 * (m[0] * n2[0]
+ m[1] * n2[1])) % 8,
MU, Z8_ONE)
Cm = [[Z8_ONE if MU[j] == neg(MU[i]) else Z8_ZERO
for j in range(16)] for i in range(16)]
s2w = mat_mul(Sw, Sw)
ok_wf = (nrm == z8_scal(F(64), Z8_ONE) and not mat_eq(s2w, Cm))
check("S8.3 [NC: WRONG FORM] q = (x^2+y^2)/4: |Gauss|^2 = %s = "
"64 != 16 and S^2 = C breaks EXACTLY -- no invariant-"
"subspace forcing is available for the wrong discriminant "
"form (the E4 forcing is form-specific)" % str(nrm[0]),
ok_wf)
wsh = [0, R(1, 2), R(3, 8), R(1, 2)]
T5sh = sp.nsimplify(sum(4 * wsh[m] * Qv[m][3] for m in range(4)))
T3sh = sp.nsimplify(32 + sum(4 * wsh[m] * Qv[m][4]
for m in range(4)))
check("S8.4 [NC: SHUFFLED WEIGHTS] h_1 <-> h_2 swap leaves "
"T5 = %s != 0 AND T3 leftover %s != 0 (exact): the ch2 "
"pattern carries the levelwise cancellation (v516 S4.2 "
"replication)" % (T5sh, T3sh), T5sh == -14 and T3sh == 36)
return bool(ok_so and okmu2 and ok_wf and T5sh == -14)
# ---------------------------------------------------------------------------
# S9 -- verdict per the preregistered branches
# ---------------------------------------------------------------------------
def section9(d1_sharpened, dim_q, n_k0, z2_all_fail, A1, A2, A3, NC_ok):
print(" -- S9: verdict per the preregistered branches")
d2_fires = (dim_q == 8 and n_k0 == 0)
erfolg_a = bool(A1 and A2 and A3 and NC_ok and d2_fires
and z2_all_fail and d1_sharpened)
check("S9.1 [VERDICT: ERFOLG-A (decision layer)] single-valuedness "
"of the physical one-loop lattice factor on PT/Z4 is DERIVED "
"from two independent constructive sources, not postulated "
"-- (source-1) F-independence: every strict chiral option at "
"physical weight carries a nontrivial character, and a "
"character-covariant integrand has NO F-independent nonzero "
"moduli integral (exact change of variables); (source-2) the "
"Quillen/BCOV pairing hol x conj(hol) cancels the multiplier "
"identically and the doubled system closes single-valuedly "
"with the physical columns contained. With single-"
"valuedness forced, the invariant subspace (exact: "
"span{e_H, e_H'} = E4 = the unphased trace) FORCES the v516 "
"numerator, and the point-4 circle closes levelwise. "
"Consequences: the WP5e measure question is DECIDED at probe "
"level in favour of reading (A) (v516) under the typed "
"premises TP-1..TP-4; the v518 kill is SHARPENED (the chi_4 "
"route was never a well-defined nonzero moduli integral: its "
"unfolded values are convention bookkeeping; the column-"
"canonicalised member is boundary-consistent yet tester-"
"dead); psi = 64 is sourced by the now-forced unphased "
"completion; D3 confirms at the Z2/EH anchor. NO marker "
"moves -- any status upgrade goes through the regular "
"promotion workflow", erfolg_a)
check("S9.2 [HONEST BOOKKEEPING + TYPED PREMISES] exact = Weil "
"relations, invariant subspace (rational nullspace), "
"characters and 1/960 grid arithmetic, completion/contact/"
"psi identities, levelwise 4-design ledgers, Z2 anchor "
"arithmetic, controls, change-of-variables corollary; "
"numeric (documented) = block transports (28+ digits, grid "
"25+), SVD solves and memberships (double, thresholds "
"1e-8/1e-3), total-sum certificates at two fresh tau "
"(< 1e-8), HM kernels (30+ digits, cut-6 vs cut-8, "
"negative-cell audit); typed premises TP-1 (F-independent "
"moduli integral), TP-2 (nonvanishing -- must source "
"psi = 64), TP-3 (v518 kernel convention), TP-4 (Quillen "
"structure of BCOV F1) named in the verdict; the residual "
"[O] is the constructive BCOV derivation of the w_m "
"normalisation itself", True)
# ---------------------------------------------------------------------------
def run():
reset()
# the transport/recognition certificates need 40 digits; restore the
# working precision in case an earlier suite module lowered it
mp.mp.dps = 40
_J_CACHE.clear()
print("v520 CELEST.WP5E.MEASURE.01: the WP5e BCOV measure question "
"decided at probe level -- single-valuedness derived from two "
"constructive sources, the completion reading (A) wins "
"(delta-1e + delta-1f consolidated)")
print(" [building 16-class exact ledgers to level 8 ...]")
d5led = build_d5_vecs(MAXN_LED)
a3led = build_a3_vecs(MAXN_LED)
V4, V2, CNT = build_ledgers16(d5led, a3led, MAXN_LED)
print(" [building coset theta series for certificates ...]")
ser0, lim = build_theta_ser0()
Qv, Qtw, Tx, Sx = section0(V4, CNT)
tg, sg, tgx, axd, rhoT, rhoS, fams, n_k0 = section1(Tx, Sx)
contained, grey = section2(fams, Qv, Qtw, V4, V2, CNT, ser0, lim)
dim_q, z2tab, ks_hol = section3(Tx, Sx, V4, CNT, n_k0, tg, sg, axd,
rhoT, rhoS)
z2_outcomes, z2_all_fail = section4(z2tab, V4, V2, CNT, ser0, lim)
A1 = section5(fams, ser0, lim)
A2 = section6(tg, sg, axd, rhoT, rhoS)
A3 = section7(V4, V2, CNT)
NC_ok = section8(Qv)
d1_sharpened = (contained == [('hol', 4, 'spread')])
section9(d1_sharpened, dim_q, n_k0, z2_all_fail, A1, A2, A3, NC_ok)
return summary("v520 CELEST.WP5E.MEASURE.01: the WP5e measure "
"question is DECIDED at probe level -- the invariant "
"subspace of the 16-dim Weil system is exactly "
"span{e_H, e_H'} (dim_Q = 8 = 4 x 2, both = E4 = "
"the unphased trace); single-valuedness is DERIVED "
"from two independent constructive sources "
"(F-independence of the moduli integral kills every "
"chiral option: exact change of variables with "
"chi_4(T) = e(1/3) != 1; the Quillen pairing "
"cancels the multiplier: |t conj(t) - 1| < 8.3e-40 "
"on all 15 pairs, the doubled 256-dim system closes "
"at trivial character with the physical columns "
"contained, hol (x) hol empty); the forced unphased "
"numerator reproduces the v516 Okubo squares "
"levelwise inside the v518 scaffold and supplies "
"the psi < 0 contact slice (circle closed); the "
"declared reading also wins D1 (column "
"canonicalisation: the chi_4 member is boundary-"
"consistent yet fails every tester -- kill "
"sharpened) and D3 (Z2/EH anchor: all three derived "
"instantiations fail); ERFOLG-A under the typed "
"premises TP-1..TP-4; residual [O] = the "
"constructive BCOV derivation of w_m; no marker "
"moves")
if __name__ == "__main__":
raise SystemExit(1 if run() else 0)
Recorded source fingerprint
62ab1dcda9330ced112ec8c5cf0fdd25ffe5c73cfa46c4b94cc8ab2116904131
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