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v653_bond_defect.py
verification/v653_bond_defect.py
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#!/usr/bin/env python3
"""v653 -- QGEO.BONDDEF.01: HARDENING THE BOND-DEFECT PREMISE -- the last [C]
of the orbifold front (v652_orbifold_arf.py A4.1): "the premise that the
seam's twist sectors are the bond-defect realization on the one cover
lattice (grounded in the v622/v623 mark/bond geometry, but a reading,
not a theorem)". This probe upgrades the reading to a LATTICE THEOREM
along three routes:
P1 (DECK CANONICITY -- the theorem):
On the mu3-cover (v623: sheet shift +1 per mark-bond crossing, one
3M-site NS circle) the Z3 twist IS the deck shift by definition. We
construct the deck lift EXACTLY (D_geo = sign-gauge x shift-by-M,
[D_geo, H_cover] = 0, D_geo^3 = +1) and its canonical section lift
D_can = -D_geo with D_can^3 = -1 (the NS-borne Arf sign of arf A2.2,
now an operator identity: the quarter clock L fixes the lift,
L^4 = D_can, see P2.4). The explicit unitary equivalence
A_r = sqrt(3) R P_r : (deck-character-r block of the cover chain)
-> (M-site base chain, ONE twisted bond)
(R = restriction to the fundamental domain = M consecutive walk
sites, P_r = deck character projector) is verified as EXACT matrix
identities over the cyclotomic ring Q(zeta_6):
3 B_r B_r^+ = 1_M, 3 B_r^+ B_r = P_r, sum_r P_r = 1,
3 B_r H_cover B_r^+ = h_def(lambda_r) with lambda_r = conj(chi_r)
i.e. THE DEFECT PHASE IS THE DECK CHARACTER (one global orientation
+ one global lift sign fix all three sectors simultaneously), and
h_def(lambda_r) IS the Arf-probe bond-defect Hamiltonian
h[N-1,0] = -exp(i pi (gt+3)/3)/2 verbatim, gt = 2r - 2 mod 6 (all
even: the Z3 family). All 3M cover modes match the M defect modes
per sector as exact integer sets ({2m+1 : m = r mod 3} =
{6n + 2r + 1}); executed at M = 16 (the seam) AND M = 48 (the Arf
lattice: 144-cover -> eps6(48, gt) exactly). NS-vs-R obstruction:
on the NS cover every deck-block BC satisfies lambda^3 = -1 => only
the EVEN gt (Z3) sectors arise geometrically; the odd (half-twist)
gt would need lambda^3 = +1 = the R cover, which v622 D2 / arf A2.1
exclude as the seam cover -- the mu6 half-twist sectors are ladder
decorations (fermionic Z6 lift), exactly as the Arf bookkeeping has
them.
P2 (POSITION INDEPENDENCE + THE GEOMETRIC PIN):
The single-defect position is a GAUGE choice: the telescoping
diagonal gauge maps ANY unit-phase bond configuration to the
canonical one with the same holonomy (symbolic sympy theorem on all
16 bond phases; exact gauge identities + isospectrality for all 16
single-bond positions). The geometry nevertheless pins the gauge
class: (i) SHEET-GAUGE REDUCTION: restricting the deck-character
block to ONE SHEET (the geometric fundamental domain) produces
phases supported EXACTLY on the four mark bonds, with a UNIFORM
twist unit zeta^gt per mark relative to the untwisted block (the
canonical cocycle support = the mark set; total holonomy = the
sector BC; 4 = 1 mod 3 is the v623 connectivity magic); (ii) WINDOW
CENSUS: the walk-window reduction puts the collapsed defect at base
bond i0 mod 16; it lands ON A MARK iff the window boundary is one
of the 12 crossings (i0 = 0 mod 4); the canonical v623 walk (i0 = 0)
puts it at base bond 0 = the mu4 = +1 mark (v622 D3 anchor
re-verified); (iii) THE CLOCK: L = sign-gauge x shift-by-M/4
commutes with H_cover exactly and L^4 = D_can EXACTLY (v623 V3
"L^4 = deck" as a one-particle identity -- and it FIXES the section
lift sign); per block L16 = A_r L A_r^+ is unitary, commutes with
h_def, and L16^4 = chi_r x 1 (the zeta_12 clock per twist sector);
the mark-uniform twist field is 4-periodic (clock-covariant), the
single-bond twist field is not -- the clock + mu4 anchoring fix the
gauge class to "one twist unit per mark".
P3 (EXCLUSION CENSUS):
(i) PURE-PHASE census: every unit-phase bond configuration with the
sector holonomy is EXACTLY gauge-equivalent to the bond defect
(structured families exact incl. 4-mark, site-centered split,
spread; 200 random configs numeric). (ii) SYMMETRY census: the
order-3 lattice-local symmetries of the NS base chain (magnetic
translations x U(1) x reflections) are EXACTLY the two global Z3
charge rotations omega^{+-1} (no order-3 translation: 3 does not
divide 16, T^s scalar iff 16 | s with T^16 = -1; no order-3 in the
reflection coset: parity; no antiunitary order 3): the ONLY Z3
available for twisting is the deck/charge one, whose twist IS the
bond defect. (iii) Alternatives kill table: on-site Z3 rotation
("site-centered twist") is pure gauge (holonomy unchanged -- it
cannot create a twist, exact); site-potential defects are killed by
the EXACT trace obstructions (Tr H = 0, Tr H^2 = 8 for the twisted
multiset) + scan; amplitude defects by Tr H^2; the sign defect is
exactly the R sector (killed by the v628 Casimir table: nu = 0 gives
+1/12 vs the twisted 1/72, and by the A1.4 charge minima); the
alternating mark weights (1,2,1,2) trivialize the holonomy (the
spectral avatar of the v623 V1 disconnection kill). (iv) HONEST
isospectral hunt: random-restart search over ALL real NN chains
(free hoppings + potentials, 32 parameters) for the exact twisted
multiset; if an isospectral real chain exists, the SPECTRAL data
alone cannot discriminate and the discrimination must come from the
measured string/correlator data (v639/A1 exponents ARE dressed
charge-string two-points): a correlator + string certificate is
computed for the best candidate. (v) The charge-string bridge:
conjugation by the mu6 string exp(i lam N_arc) creates EXACTLY a
bond-defect PAIR at the arc ends (exact identity): the measured mu6
ladder (arf A1) is bond-defect spectroscopy -- the v639/A1 data
measure THIS realization.
Checks:
(S0.1) [exact] the v623 walk skeleton at M = 16: one 48-site orbit,
12 crossings at walk steps i = 3 mod 4, deck = walk shift by
16, fundamental-domain boundary bond crosses base bond 0
(a mark).
(S0.2) [exact] the same skeleton at M = 48 (marks {0,12,24,36}):
one 144-orbit, 12 crossings at i = 11 mod 12, deck = shift 48.
(S0.3) [machine] Q(zeta_6) ring self-test + H_cover(48) spectrum =
NS mode sums (1e-13).
(P1.1) [exact] the deck lift: [D_geo, H_cover] = 0, D_geo^3 = +1,
D_can = -D_geo, D_can^3 = -1 (NS-borne sign as an operator
identity), D_can P_r = chi_r P_r with chi_r = zeta^{-(2r+1)},
tr P_r = M, P_r P_s = delta_rs P_r, sum_r P_r = 1.
(P1.2) [exact, THE THEOREM, M = 16] the unitary equivalence:
3 B_r B_r^+ = 1_16, 3 B_r^+ B_r = P_r, and
3 B_r H48 B_r^+ = h_def(16, lambda_r), lambda_r =
conj(chi_r) = zeta^{2r+1} -- and h_def(16, lambda_r) equals
the Arf bond-defect operator (phase pi(gt+3)/3, gt = 2r-2
mod 6) VERBATIM: twist sector = deck boundary condition =
bond defect, canonically.
(P1.3) [exact] all modes: the integer set identity
{2m+1 : m = r mod 3} = {6n + 2r+1} per sector (M = 16 and
48), the symbolic bulk-row eigen identity, the boundary-row
congruence e^{iMq} = lambda_r, and the numeric union of the
three block spectra = all 3M NS cover modes (1e-13).
(P1.4) [exact] the Arf lattice (M = 48): 3 B_r H144 B_r^+ = the
verbatim arf A2.3 Hamiltonian for gt in {4, 0, 2}; spectra =
eps6(48, gt) (1e-13); mode fractions exact.
(P1.5) [exact] NS-vs-R: NS-cover deck blocks have lambda^3 = -1
(even gt only -- exact parity obstruction against odd gt);
the R-cover control delivers exactly gt in {3, 5, 1}; with NS
forced (v622 D1/D2, arf A2.1) the geometric quotient sectors
are exactly the Z3 family.
(P2.1) [exact+E] defect position = gauge: the symbolic telescoping
theorem (16 free bond phases, sympy) + the 16-position table
(exact gauge identity, numeric max spectral dev < 1e-12).
(P2.2) [exact] sheet-gauge reduction: support = the four mark bonds
EXACTLY, twist unit UNIFORM = zeta^gt per mark, holonomy =
lambda_r, explicit exact gauge back to the single defect.
(P2.3) [exact] window census: defect base bond = i0 mod 16; ON a
mark iff the window boundary is a crossing (i0 = 0 mod 4;
12 of 48 windows = the 12 lifted marks of v623 V6); the
canonical walk lands on base bond 0 = the mu4 = +1 mark
(v622 D3 midpoint identity re-verified symbolically).
(P2.4) [exact] the clock: [L, H_cover] = 0, L^4 = D_can (the clock
FIXES the section lift), L^12 = -1, L^24 = 1; per block:
L16 unitary, [L16, h_def] = 0, L16^4 = chi_r; the mark twist
field is 4-periodic (S4-covariant, exact commutator 0), the
single-bond twist field is not (exact nonzero): geometry
fixes the gauge class.
(P3.1) [exact+machine] pure-phase census: structured configs
(4-mark, site-centered split, 3-bond spread, random single
bonds) exactly gauge-equivalent; 200 random unit-phase
configs isospectral at 1e-12.
(P3.2) [exact] symmetry census: order-3 lattice-local symmetries of
the NS base chain = exactly {omega x 1, omega-bar x 1}; no
order-3 translation (T^s scalar iff 16 | s, T^16 = -1,
gcd(3,16) = 1); reflection coset excluded by parity;
antiunitary excluded (an odd power of an antiunitary is
antiunitary).
(P3.3) [exact+E] alternatives kill table: (a) on-site Z3 rotation =
gauge, holonomy unchanged (exact); (b) site potentials:
Tr H = mu != 0 resp. Tr H^2 = 8 + sum mu^2 > 8 (exact) +
scan floor; (c) amplitude defect: Tr H^2 obstruction (exact)
+ scan floor; (d) sign defect = R sector exactly (v628 kill:
nu = 0 coefficient 1/12 vs twisted 1/72); (e) alternating
mark weights: holonomy trivializes to the NS class (exact) --
the v623 disconnection kill, spectrally.
(P3.4) [E, honest] the isospectral hunt over real NN chains: report
the best loss; if an isospectral real chain exists, verify
that the gauge-invariant correlator + mu6-string data
(the measured v639/A1 observables) discriminate it from the
bond-defect realization.
(P3.5) [exact] the charge-string bridge: exp(i lam N_arc)
conjugation = bond-defect PAIR at the arc ends (exact matrix
identity for lam = pi gt/3, gt = 1..5): the measured mu6
ladder is bond-defect spectroscopy.
(P4.1) [C -> lattice theorem] typing.
Verdict enums (frozen): BOND-DEFECT-CANONICAL (P1 theorem + P2 pin +
P3 census all pass: the premise is a lattice theorem, geometrically
pinned, alternative-free within realizations and discriminated
outside), BOND-DEFECT-GAUGE-ONLY (P1/P2 pass, P3 finds an
undiscriminated alternative), BOND-DEFECT-UNDECIDED (P1 fails),
MIXED (machinery breaks).
FIREWALL: GATE.QGEO does not move; no marker changes; the continuum
CFT statement and the seam identification stay typed open.
Python-only, counted per GATE.WOLFRAM.02.
PROVENANCE: discovery probe bond_defect_premise_probe.py (2026-08-02,
18/18, verdict BOND-DEFECT-CANONICAL).
Conventions inherited read-only from v652_orbifold_arf.py,
v651_orbifold_assembly.py, v622/v623/v628/v639. Bond b = the pair
(b, b+1 mod n); the v622 mark bonds {0,4,8,12} (between sites b-1, b)
are probe bond indices {15, 3, 7, 11}; holonomy = prod_b phases[b]
with phases[b] = -2 H[b, b+1].
"""
from fractions import Fraction
import numpy as np
import sympy as sp
CHECKS = []
def check(name, ok, detail=""):
CHECKS.append((name, bool(ok)))
print("[%s] %s%s" % ("PASS" if ok else "FAIL", name,
(" -- " + detail) if detail else ""))
# ================================================================ Q(zeta_6)
# exact cyclotomic ring: x = a + b zeta, zeta = exp(i pi/3), zeta^2 = zeta - 1
class Cyc(object):
__slots__ = ("a", "b")
def __init__(self, a=0, b=0):
self.a = Fraction(a)
self.b = Fraction(b)
def __add__(self, o):
return Cyc(self.a + o.a, self.b + o.b)
def __sub__(self, o):
return Cyc(self.a - o.a, self.b - o.b)
def __neg__(self):
return Cyc(-self.a, -self.b)
def __mul__(self, o):
# (a1 + b1 z)(a2 + b2 z), z^2 = z - 1
return Cyc(self.a * o.a - self.b * o.b,
self.a * o.b + self.b * o.a + self.b * o.b)
def conj(self):
# conj(z) = 1 - z
return Cyc(self.a + self.b, -self.b)
def is_zero(self):
return self.a == 0 and self.b == 0
def __eq__(self, o):
return self.a == o.a and self.b == o.b
def to_c(self):
return complex(self.a) + complex(self.b) * np.exp(1j * np.pi / 3)
def __repr__(self):
return "(%s%+s z)" % (self.a, self.b)
CZERO, CONE, CMONE = Cyc(0), Cyc(1), Cyc(-1)
MHALF = Cyc(Fraction(-1, 2))
THIRD = Cyc(Fraction(1, 3))
_ZP = {0: Cyc(1, 0), 1: Cyc(0, 1), 2: Cyc(-1, 1),
3: Cyc(-1, 0), 4: Cyc(0, -1), 5: Cyc(1, -1)}
def zeta(k):
"""zeta^k exactly (zeta = e^{i pi/3})."""
return _ZP[k % 6]
# sparse exact matrices: dict {(i, j): Cyc}
def mmul(A, B):
rows = {}
for (i, j), v in B.items():
rows.setdefault(i, []).append((j, v))
C = {}
for (i, k), v in A.items():
for (j, w) in rows.get(k, ()):
key = (i, j)
s = C.get(key)
C[key] = v * w if s is None else s + v * w
return {k: v for k, v in C.items() if not v.is_zero()}
def mdag(A):
return {(j, i): v.conj() for (i, j), v in A.items()}
def madd(A, B):
C = dict(A)
for k, v in B.items():
s = C.get(k)
C[k] = v if s is None else s + v
return {k: v for k, v in C.items() if not v.is_zero()}
def msub(A, B):
return madd(A, {k: -v for k, v in B.items()})
def mscale(c, A):
return {k: c * v for k, v in A.items() if not (c * v).is_zero()}
def meq(A, B):
return len(msub(A, B)) == 0
def mident(n, c=None):
c = CONE if c is None else c
return {(i, i): c for i in range(n)} if not c.is_zero() else {}
def mpow(A, p):
R = None
for _ in range(p):
R = dict(A) if R is None else mmul(R, A)
return R
def to_np(A, nr, nc=None):
M = np.zeros((nr, nc or nr), dtype=complex)
for (i, j), v in A.items():
M[i, j] = v.to_c()
return M
def mtrace(A, n):
t = CZERO
for i in range(n):
t = t + A.get((i, i), CZERO)
return t
# ================================================================ builders
def marks_of(M):
return (0, M // 4, M // 2, 3 * M // 4)
def build_walk(M, weights=(1, 1, 1, 1)):
"""v623 successor walk on the mu3-cover of the M-ring."""
marks = marks_of(M)
walk, crossings = [], []
a, s = 0, 0
for i in range(3 * M):
walk.append((a, s))
a2, s2 = (a + 1) % M, s
if a2 in marks:
s2 = (s + weights[marks.index(a2)]) % 3
if s2 != s:
crossings.append(i)
a, s = a2, s2
return walk, crossings, (a, s) == (0, 0)
def h_def(n, lam):
"""n-ring, hopping -1/2, BC psi(x+n) = lam psi(x) realized as ONE
twisted bond: H[n-1, 0] = -lam/2 (the verbatim Arf-probe form with
lam = e^{i pi (gt+3)/3}). Modes: e^{iqn} = lam, eps = -cos q."""
H = {}
for a2 in range(n - 1):
H[(a2, a2 + 1)] = MHALF
H[(a2 + 1, a2)] = MHALF
H[(n - 1, 0)] = MHALF * lam
H[(0, n - 1)] = MHALF * lam.conj()
return H
def h_cfg(n, phases):
"""n-ring with unit phases per bond b = (b, b+1): H[b, b+1] =
-phases[b]/2. Holonomy = prod phases. h_def(lam) = phases with
lam at bond n-1."""
H = {}
for b in range(n):
H[(b, (b + 1) % n)] = MHALF * phases[b]
H[((b + 1) % n, b)] = MHALF * phases[b].conj()
return H
def shift(n, step):
"""(S psi)(i) = psi(i - step)."""
return {(i, (i - step) % n): CONE for i in range(n)}
def telescope_gauge(phases):
"""diagonal d with d_0 = 1, d_{b+1} = d_b phases_b: conjugation
moves every phase onto the last bond (holonomy)."""
d = [CONE]
for b in range(len(phases) - 1):
d.append(d[-1] * phases[b])
return d
def apply_gauge(H, d):
return {(i, j): d[i] * v * d[j].conj() for (i, j), v in H.items()}
def phases_of(H, n):
"""extract phases[b] = -2 H[b, b+1]."""
m2 = Cyc(-2)
return [m2 * H.get((b, (b + 1) % n), CZERO) for b in range(n)]
def holonomy(phases):
h = CONE
for p in phases:
h = h * p
return h
def np_spec(A, n):
return np.sort(np.linalg.eigvalsh(to_np(A, n)))
def modes6_np(N, gt):
return np.sort(-np.cos(2.0 * np.pi * (np.arange(N) + 0.5 + gt / 6.0) / N))
def deck_machinery(M):
"""cover chain + deck + clock + projectors + reductions for base M."""
N3 = 3 * M
walk, crossings, closed = build_walk(M)
Hc = h_def(N3, CMONE) # NS cover = lam = -1
S_M = shift(N3, M)
G = {(i, i): (CONE if i < M else CMONE) for i in range(N3)}
Dgeo = mmul(G, S_M)
Dcan = mscale(CMONE, Dgeo)
q = M // 4
GL = {(i, i): (CONE if i < q else CMONE) for i in range(N3)}
L = mmul(GL, shift(N3, q))
# deck-character projectors: chi_r = zeta^{-(2r+1)}, P_r =
# (1/3) sum_t chi_r^{-t} D_can^t
P = {}
D2 = mmul(Dcan, Dcan)
for r in range(3):
acc = mident(N3, THIRD)
acc = madd(acc, mscale(THIRD * zeta(2 * r + 1), Dcan))
acc = madd(acc, mscale(THIRD * zeta(2 * (2 * r + 1)), D2))
P[r] = acc
Rsel = {(a2, a2): CONE for a2 in range(M)} # walk fundamental domain
B = {r: mmul(Rsel, P[r]) for r in range(3)}
return dict(N3=N3, walk=walk, crossings=crossings, closed=closed,
Hc=Hc, Dgeo=Dgeo, Dcan=Dcan, L=L, P=P, B=B)
# ================================================================== S0
print("=" * 72)
print("S0: machinery (walk skeleton + exact ring)")
print("=" * 72)
MK = {}
for M in (16, 48):
MK[M] = deck_machinery(M)
w16 = MK[16]
deck_ok = all(w16["walk"][(i + 16) % 48]
== (w16["walk"][i][0], (w16["walk"][i][1] + 1) % 3)
for i in range(48))
cross_ok = (len(w16["crossings"]) == 12
and all(i % 4 == 3 for i in w16["crossings"]))
bdry_cross_base = (0 + 16) % 16 # boundary crosses bond 0
check("S0.1 [exact] the v623 walk at M = 16: one closed 48-orbit, "
"12 crossings at walk steps i = 3 mod 4, deck = walk shift by 16 "
"((a, s) -> (a, s+1)), and the fundamental-domain boundary bond "
"(walk 47 -> 0) crosses base bond 0 -- a MARK",
w16["closed"] and len(set(w16["walk"])) == 48 and deck_ok
and cross_ok and bdry_cross_base in marks_of(16))
w48 = MK[48]
deck_ok48 = all(w48["walk"][(i + 48) % 144]
== (w48["walk"][i][0], (w48["walk"][i][1] + 1) % 3)
for i in range(144))
check("S0.2 [exact] the same skeleton at M = 48 (marks {0,12,24,36}): "
"one closed 144-orbit, 12 crossings at i = 11 mod 12, deck = "
"walk shift by 48",
w48["closed"] and len(set(w48["walk"])) == 144 and deck_ok48
and len(w48["crossings"]) == 12
and all(i % 12 == 11 for i in w48["crossings"]))
ring_ok = (zeta(1) * zeta(5) == CONE and zeta(3) == CMONE
and zeta(2) == zeta(1) * zeta(1)
and zeta(1).conj() == zeta(5)
and abs(zeta(1).to_c() - np.exp(1j * np.pi / 3)) < 1e-15)
dev_ns = np.abs(np_spec(w16["Hc"], 48)
- np.sort(-np.cos(2 * np.pi * (np.arange(48) + 0.5) / 48))
).max()
check("S0.3 [machine] Q(zeta_6) ring self-test (zeta^2 = zeta - 1, "
"zeta^3 = -1, conj = inverse) and H_cover(48) spectrum = the NS "
"mode sums", ring_ok and dev_ns < 1e-13,
"max |spec dev| = %.2e" % dev_ns)
# ================================================================== P1
print("=" * 72)
print("P1: deck canonicity -- the lattice theorem")
print("=" * 72)
# ---- P1.1 the deck lift
ok11 = True
det11 = []
for M in (16, 48):
mk = MK[M]
N3, Hc, Dg, Dc, P = mk["N3"], mk["Hc"], mk["Dgeo"], mk["Dcan"], mk["P"]
comm = meq(mmul(Dg, Hc), mmul(Hc, Dg))
cube_geo = meq(mpow(Dg, 3), mident(N3))
cube_can = meq(mpow(Dc, 3), mident(N3, CMONE))
proj = all(meq(mmul(P[r], P[s]),
P[r] if r == s else {}) for r in range(3)
for s in range(3))
part = meq(madd(madd(P[0], P[1]), P[2]), mident(N3))
chi_eig = all(meq(mmul(Dc, P[r]), mscale(zeta(-(2 * r + 1)), P[r]))
for r in range(3))
tr = all(mtrace(P[r], N3) == Cyc(M) for r in range(3))
ok11 &= (comm and cube_geo and cube_can and proj and part
and chi_eig and tr)
det11.append("M=%d: comm %s, D_geo^3=+1 %s, D_can^3=-1 %s, "
"chi-eig %s, tr P_r = %d %s"
% (M, comm, cube_geo, cube_can, chi_eig, M, tr))
for d in det11:
print(" " + d)
check("P1.1 [exact] the deck lift: [D_geo, H_cover] = 0, D_geo^3 = +1 "
"(function picture), the canonical section lift D_can = -D_geo "
"has D_can^3 = -1 (the NS-borne Arf sign, arf A2.2, as an exact "
"operator identity), D_can P_r = zeta^{-(2r+1)} P_r, projectors "
"orthogonal/complete with tr P_r = M -- at M = 16 AND 48", ok11)
# ---- P1.2 the theorem at M = 16
GT_OF_R = {r: (2 * r - 2) % 6 for r in range(3)}
ok12 = True
lam_dev = 0.0
for r in range(3):
B = MK[16]["B"][r]
Bd = mdag(B)
iso = meq(mscale(Cyc(3), mmul(B, Bd)), mident(16))
proj_back = meq(mscale(Cyc(3), mmul(Bd, B)), MK[16]["P"][r])
Hred = mscale(Cyc(3), mmul(mmul(B, MK[16]["Hc"]), Bd))
lam = zeta(2 * r + 1)
equiv = meq(Hred, h_def(16, lam))
# verbatim Arf constructor (float) vs the exact reduction
gt = GT_OF_R[r]
h_arf = np.zeros((16, 16), dtype=complex)
for j in range(15):
h_arf[j, j + 1] = -0.5
h_arf[j + 1, j] = -0.5
phi = np.pi * (gt + 3) / 3.0
h_arf[15, 0] = -0.5 * np.exp(1j * phi)
h_arf[0, 15] = -0.5 * np.exp(-1j * phi)
dev = np.abs(to_np(Hred, 16) - h_arf).max()
lam_dev = max(lam_dev, dev)
# defect phase = deck character (conj = orientation convention)
canon = (lam == zeta(-(2 * r + 1)).conj())
ok12 &= iso and proj_back and equiv and canon and dev < 1e-15
print(" r=%d: 3BB^+=1 %s, 3B^+B=P %s, 3BHB^+ = h_def(zeta^%d) %s, "
"= Arf h(16, gt=%d) dev %.1e" % (r, iso, proj_back,
2 * r + 1, equiv, gt, dev))
check("P1.2 [exact, THE THEOREM] the explicit unitary equivalence at "
"M = 16: A_r = sqrt(3) R P_r maps the deck-character-r block of "
"the 48-cover chain EXACTLY onto the 16-site chain with ONE "
"twisted bond, phase lambda_r = zeta^{2r+1} = conj(chi_r): the "
"defect phase IS the deck character (one global orientation), "
"and h_def(16, lambda_r) is the VERBATIM Arf bond-defect "
"operator with gt = 2r-2 mod 6 in {4, 0, 2} -- twist sector = "
"deck boundary condition = bond defect, canonically",
ok12, "max |exact - Arf float| = %.1e" % lam_dev)
# ---- P1.3 all modes, exactly
q = sp.symbols("q", real=True)
bulk = sp.simplify(-(sp.exp(sp.I * q) + sp.exp(-sp.I * q)) / 2
+ sp.cos(q)) == 0
ok13 = bulk
for M in (16, 48):
for r in range(3):
cover = sorted((2 * m + 1) % (6 * M)
for m in range(3 * M) if m % 3 == r)
block = sorted((6 * n + 2 * r + 1) % (6 * M) for n in range(M))
ok13 &= (cover == block)
# boundary congruence: e^{iMq} = zeta^{2r+1} for every block mode
ok13 &= all((6 * n + 2 * r + 1) % 6 == (2 * r + 1) % 6
for n in range(M))
# numeric union at M = 16
spec_union = np.sort(np.concatenate(
[np_spec(h_def(16, zeta(2 * r + 1)), 16) for r in range(3)]))
dev_union = np.abs(spec_union
- np.sort(-np.cos(2 * np.pi
* (np.arange(48) + 0.5) / 48))).max()
ok13 &= dev_union < 1e-13
check("P1.3 [exact] all 3M/M modes: {2m+1 : m = r mod 3} = "
"{6n + 2r + 1} as exact integer sets (M = 16 and 48), the "
"symbolic bulk-row eigen identity, the boundary congruence "
"e^{iMq} = zeta^{2r+1}, and the numeric union of the three "
"block spectra = ALL 48 NS cover modes", ok13,
"union dev = %.2e" % dev_union)
# ---- P1.4 the Arf lattice (M = 48)
ok14 = True
for r in range(3):
B = MK[48]["B"][r]
Hred = mscale(Cyc(3), mmul(mmul(B, MK[48]["Hc"]), mdag(B)))
gt = GT_OF_R[r]
equiv = meq(Hred, h_def(48, zeta(gt + 3)))
dev = np.abs(np_spec(Hred, 48) - modes6_np(48, gt)).max()
ok14 &= equiv and dev < 1e-13
print(" r=%d -> gt=%d: 3BH144B^+ = arf h(48, gt) %s, "
"spec vs eps6(48, gt) dev %.1e" % (r, gt, equiv, dev))
check("P1.4 [exact] the Arf lattice: the SAME theorem at M = 48 "
"(144-cover, marks {0,12,24,36}): the deck blocks are VERBATIM "
"the arf A2.3 bond-defect Hamiltonians for gt in {4, 0, 2}, "
"spectra = eps6(48, gt) exactly: the Arf probe's Z3 twist "
"sectors ARE the deck-character blocks of the covered seam",
ok14)
# ---- P1.5 NS vs R
ns_parity = all((3 * (2 * r + 1)) % 6 == 3 for r in range(3))
# every NS block: lambda^3 = zeta^{3(2r+1)} = zeta^3 = -1
odd_gt_obstruct = all((3 * (gt + 3)) % 6 == 0 for gt in (1, 3, 5))
# odd gt: lambda^3 = zeta^{3(gt+3)} = +1 != -1 -- excluded on NS
# R-cover control: no NS sign, plain shift deck
HR = h_def(48, CONE)
SR = shift(48, 16)
ok_r = meq(mmul(SR, HR), mmul(HR, SR)) and meq(mpow(SR, 3), mident(48))
PR = {}
S2 = mmul(SR, SR)
for c in range(3):
acc = mident(48, THIRD)
acc = madd(acc, mscale(THIRD * zeta(2 * c), SR))
acc = madd(acc, mscale(THIRD * zeta(4 * c), S2))
PR[c] = acc
gts_r = []
for c in range(3):
BR = mmul({(a2, a2): CONE for a2 in range(16)}, PR[c])
HRred = mscale(Cyc(3), mmul(mmul(BR, HR), mdag(BR)))
lamr = phases_of(HRred, 16)[15]
gt_found = next(g for g in range(6) if zeta(g + 3) == lamr)
gts_r.append(gt_found)
ok_r &= meq(HRred, h_def(16, lamr))
check("P1.5 [exact] NS-vs-R bookkeeping: every NS-cover deck block "
"has lambda^3 = zeta^{3(2r+1)} = -1 => ONLY the even-gt (Z3) "
"sectors arise as geometric deck blocks; odd gt (lambda^3 = +1) "
"is EXACTLY obstructed and appears instead as the deck blocks "
"of the R cover (control: gt found = %s); since the cover circle "
"is NS, measured and forced (v622 D1/D2, arf A2.1), the "
"geometric quotient sectors are exactly the Z3 family -- the "
"odd (half-twist) sectors are the fermionic-Z6 ladder "
"decorations, as the Arf bookkeeping has them"
% sorted(gts_r),
ns_parity and odd_gt_obstruct and ok_r
and sorted(gts_r) == [1, 3, 5])
# ================================================================== P2
print("=" * 72)
print("P2: position independence + the geometric pin")
print("=" * 72)
# ---- P2.1 the symbolic telescoping theorem + 16-position table
th = sp.symbols("th0:16", real=True)
u_sym = [sp.exp(sp.I * th[b]) for b in range(16)]
Hsym = sp.zeros(16, 16)
for b in range(16):
Hsym[b, (b + 1) % 16] += -u_sym[b] / 2
Hsym[(b + 1) % 16, b] += -sp.conjugate(u_sym[b]) / 2
dsym = [sp.Integer(1)]
for b in range(15):
dsym.append(dsym[-1] * u_sym[b])
Gs = sp.diag(*dsym)
Hp = Gs * Hsym * Gs.conjugate().T
Ht = sp.zeros(16, 16)
for b in range(15):
Ht[b, b + 1] = -sp.Rational(1, 2)
Ht[b + 1, b] = -sp.Rational(1, 2)
tot = sp.exp(sp.I * sum(th))
Ht[15, 0] = -tot / 2
Ht[0, 15] = -sp.conjugate(tot) / 2
thm = all(sp.simplify(sp.expand(Hp[i, j] - Ht[i, j])) == 0
for i in range(16) for j in range(16))
lam_ref = zeta(1) # gt = 4 sector as reference
spec_ref = np_spec(h_def(16, lam_ref), 16)
print(" 16-position gauge table (defect phase zeta^1, gt = 4):")
tab_ok = True
for b in range(16):
ph = [CONE] * 16
ph[b] = lam_ref
Hb = h_cfg(16, ph)
d = telescope_gauge(ph)
exact = meq(apply_gauge(Hb, d), h_def(16, lam_ref))
dev = np.abs(np_spec(Hb, 16) - spec_ref).max()
tab_ok &= exact and dev < 1e-12
print(" bond %2d: exact gauge identity %s, max |spec dev| = %.2e"
% (b, exact, dev))
check("P2.1 [exact+E] defect position = gauge: the telescoping "
"diagonal gauge maps ANY unit-phase bond configuration to the "
"canonical single defect carrying the total holonomy (symbolic "
"theorem in all 16 free bond phases: %s) and all 16 single-bond "
"positions are exactly gauge-equivalent and isospectral"
% thm, thm and tab_ok)
# ---- P2.2 sheet-gauge reduction
first_visit = {}
for i, pt in enumerate(w16["walk"]):
first_visit[pt] = i
ok22 = True
print(" sheet-0 reduction (geometric fundamental domain):")
MARK_BONDS_PROBE = (3, 7, 11, 15) # pairs (3,4),(7,8),(11,12),(15,0)
u_by_r = {}
for r in range(3):
R0 = {(a2, first_visit[(a2, 0)]): CONE for a2 in range(16)}
B0 = mmul(R0, MK[16]["P"][r])
iso0 = meq(mscale(Cyc(3), mmul(B0, mdag(B0))), mident(16))
H0 = mscale(Cyc(3), mmul(mmul(B0, MK[16]["Hc"]), mdag(B0)))
ph0 = phases_of(H0, 16)
supp = [b for b in range(16) if not (ph0[b] == CONE)]
unit = all((ph0[b] * ph0[b].conj()) == CONE for b in range(16))
hol = holonomy(ph0)
d = telescope_gauge(ph0)
back = meq(apply_gauge(H0, d), h_def(16, zeta(2 * r + 1)))
u_by_r[r] = ph0
supp_ok = (set(supp) <= set(MARK_BONDS_PROBE)
if r == 1 else supp == list(MARK_BONDS_PROBE))
ok22 &= (iso0 and unit and supp_ok
and hol == zeta(2 * r + 1) and back)
print(" r=%d: support %s v622 mark bonds %s (found %s), "
"holonomy = zeta^%d %s, exact gauge back to the single "
"defect %s"
% (r, "<=" if r == 1 else "==", supp_ok, supp, 2 * r + 1,
hol == zeta(2 * r + 1), back))
# twist part relative to the untwisted block (r = 1, lam = -1)
uni_ok = True
for r in (0, 2):
gt = GT_OF_R[r]
ratios = [u_by_r[r][b] * u_by_r[1][b].conj() for b in range(16)]
uni_ok &= all(ratios[b] == CONE for b in range(16)
if b not in MARK_BONDS_PROBE)
uni_ok &= all(ratios[b] == zeta(gt) for b in MARK_BONDS_PROBE)
check("P2.2 [exact] the sheet-gauge (geometric) reduction: restricting "
"each deck block to ONE SHEET puts the phases ONLY on the four "
"v622 mark bonds (off-mark phases exactly 1; both twisted "
"blocks carry ALL four marks, the untwisted block's NS sign "
"cancels one mark phase to +1), with a UNIFORM twist unit "
"zeta^gt per mark relative to the untwisted block (4 marks x "
"one unit, total = zeta^{4 gt} = zeta^gt: the 4 = 1 mod 3 "
"connectivity magic), holonomy = the sector BC, and an explicit "
"exact gauge back to the single-bond defect: the canonical "
"cocycle support IS the mark set", ok22 and uni_ok)
# ---- P2.3 window census
cross_set = set(w16["crossings"])
census_ok = True
n_mark_windows = 0
for i0 in range(48):
bond_base = i0 % 16
boundary_is_crossing = ((i0 + 15) % 48) in cross_set
at_mark = bond_base in marks_of(16)
census_ok &= (at_mark == boundary_is_crossing == (i0 % 4 == 0))
n_mark_windows += at_mark
mid_ok = all((sp.exp(2 * sp.pi * sp.I * sp.Rational(b, 16)) ** 4 == 1)
== (b % 4 == 0) for b in range(16))
check("P2.3 [exact] the window census: the walk-window reduction puts "
"the collapsed defect at base bond i0 mod 16; it lands ON A "
"MARK iff the window boundary is one of the 12 crossings "
"(i0 = 0 mod 4; %d of 48 windows -- the 12 lifted marks of "
"v623 V6); the canonical v623 walk (i0 = 0) lands on base bond "
"0 = the mu4 = +1 mark (midpoint e^{2 pi i b/16} in mu4 iff "
"4 | b, v622 D3 re-verified)"
% n_mark_windows, census_ok and n_mark_windows == 12 and mid_ok)
# ---- P2.4 the clock
ok24 = True
for M in (16, 48):
mk = MK[M]
N3, L, Dc, Hc = mk["N3"], mk["L"], mk["Dcan"], mk["Hc"]
commL = meq(mmul(L, Hc), mmul(Hc, L))
L4 = mpow(L, 4)
fix = meq(L4, Dc)
L12 = meq(mpow(L4, 3), mident(N3, CMONE))
L24 = meq(mpow(mpow(L4, 3), 2), mident(N3))
ok24 &= commL and fix and L12 and L24
print(" M=%d: [L, H] = 0 %s, L^4 = D_can %s, L^12 = -1 %s, "
"L^24 = 1 %s" % (M, commL, fix, L12, L24))
for r in range(3):
B = MK[16]["B"][r]
L16 = mscale(Cyc(3), mmul(mmul(B, MK[16]["L"]), mdag(B)))
unit = meq(mmul(L16, mdag(L16)), mident(16))
commd = meq(mmul(L16, h_def(16, zeta(2 * r + 1))),
mmul(h_def(16, zeta(2 * r + 1)), L16))
quart = meq(mpow(L16, 4), mident(16, zeta(-(2 * r + 1))))
ok24 &= unit and commd and quart
print(" r=%d: L16 unitary %s, [L16, h_def] = 0 %s, L16^4 = chi_r "
"%s" % (r, unit, commd, quart))
# covariance contrast of the twist fields (S4 = plain quarter shift)
S4b = shift(16, 4)
tw_mark = [zeta(GT_OF_R[0]) if b in MARK_BONDS_PROBE else CONE
for b in range(16)]
tw_single = [zeta(GT_OF_R[0]) if b == 15 else CONE for b in range(16)]
Hm, Hs = h_cfg(16, tw_mark), h_cfg(16, tw_single)
cov_mark = meq(mmul(S4b, Hm), mmul(Hm, S4b))
cov_single = meq(mmul(S4b, Hs), mmul(Hs, S4b))
ok24 &= cov_mark and (not cov_single)
check("P2.4 [exact] the clock pins the gauge: [L, H_cover] = 0 and "
"L^4 = D_can EXACTLY (v623 V3 'L^4 = deck' as a one-particle "
"identity -- the quarter clock FIXES the canonical section "
"lift), L^12 = -1, L^24 = 1; per block L16 is unitary, commutes "
"with h_def and L16^4 = chi_r (the zeta_12 clock per twist "
"sector); the mark-uniform twist field is 4-periodic "
"(S4-covariant, exact) while the single-bond twist field is not "
"(covariant %s): clock covariance + the mu4 mark anchoring fix "
"the gauge class to 'one twist unit per mark'"
% cov_single, ok24)
# ================================================================== P3
print("=" * 72)
print("P3: exclusion census")
print("=" * 72)
RNG = np.random.default_rng(61)
lam_tw = zeta(1) # reference twisted sector gt = 4
spec_tw = np_spec(h_def(16, lam_tw), 16)
spec_ns = np_spec(h_def(16, CMONE), 16)
spec_r0 = np_spec(h_def(16, CONE), 16)
# ---- P3.1 pure-phase census
fam_ok = True
# (a) 4-mark realization, (b) site-centered split around site 8,
# (c) 3-bond zeta spread, (d) random single bonds with extra coboundary
fams = {}
fams["4-mark (sheet gauge)"] = u_by_r[0]
split = [CONE] * 16
split[7] = zeta(4) # bonds (7,8) and (8,9): around
split[8] = zeta(3) # site 8: zeta^4 * zeta^3 = zeta^1
fams["site-centered split"] = split
spread = [CONE] * 16
spread[2], spread[9], spread[13] = zeta(5), zeta(1), zeta(1)
fams["3-bond spread"] = spread
for name, ph in fams.items():
okf = (holonomy(ph) == lam_tw
and meq(apply_gauge(h_cfg(16, ph), telescope_gauge(ph)),
h_def(16, lam_tw)))
fam_ok &= okf
print(" family %-22s holonomy zeta^1: exact gauge equivalence %s"
% (name, okf))
dev_rand = 0.0
for _ in range(200):
thr = RNG.uniform(0, 2 * np.pi, 16)
thr[15] += (np.pi / 3 - thr.sum()) % (2 * np.pi)
Hn = np.zeros((16, 16), dtype=complex)
for b in range(16):
Hn[b, (b + 1) % 16] = -0.5 * np.exp(1j * thr[b])
Hn[(b + 1) % 16, b] = np.conj(Hn[b, (b + 1) % 16])
dev_rand = max(dev_rand,
np.abs(np.sort(np.linalg.eigvalsh(Hn)) - spec_tw).max())
check("P3.1 [exact+machine] the pure-phase census: every unit-phase "
"bond configuration with the sector holonomy is gauge-"
"equivalent to the bond defect -- structured families (4-mark, "
"site-centered split, 3-bond spread) EXACT, 200 random configs "
"isospectral", fam_ok and dev_rand < 1e-12,
"random max |spec dev| = %.2e" % dev_rand)
# ---- P3.2 symmetry census
GT16 = {(i, i): (CONE if i == 0 else CMONE) for i in range(16)}
Tb = mmul(GT16, shift(16, 1))
Hns = h_def(16, CMONE)
t_comm = meq(mmul(Tb, Hns), mmul(Hns, Tb))
t16 = meq(mpow(Tb, 16), mident(16, CMONE))
scalar_only_16 = all(any(i != j for (i, j) in mpow(Tb, s))
for s in range(1, 16))
no_t3 = all((3 * t) % 16 != 0 for t in range(1, 16))
# reflection coset: g = phase T^t R has permutation part
# a -> (t + 15 - a) mod 16 (orientation-reversing); its odd powers are
# again flips, never the identity permutation => no order 3
refl_no3 = all(any(((t + 15 - a2) % 16) != a2 for a2 in range(16))
for t in range(16))
refl_parity = refl_no3
check("P3.2 [exact] the symmetry census: the order-3 lattice-local "
"symmetries of the NS base chain are EXACTLY the two global Z3 "
"charge rotations omega^{+-1}: the NS translation T satisfies "
"[T, H] = 0, T^16 = -1, T^s is never scalar for 0 < s < 16, and "
"3t = 0 mod 16 forces t = 0 (3 does not divide 16 -- no spatial "
"Z3 on the base; the spatial Z3 lives ONLY on the cover, as the "
"deck); the reflection coset has no order-3 (odd powers reverse "
"orientation); antiunitaries have no order 3 (odd powers are "
"antiunitary). The unique Z3 twist of the seam is the "
"deck/charge twist -- the bond defect",
t_comm and t16 and scalar_only_16 and no_t3 and refl_parity
and refl_no3)
# ---- P3.3 alternatives kill table
print(" alternatives kill table (target: twisted class gt = 4, "
"offset 1/6 pair):")
# (a) on-site Z3 rotation: gauge, holonomy unchanged
ons = [CONE] * 16
ons[7], ons[8] = zeta(2), zeta(4) # e^{2pi i/3} on site 8: gauge
Ha = h_cfg(16, ons)
kill_a = (holonomy(ons) == CONE
and np.abs(np_spec(Ha, 16) - spec_r0).max() < 1e-13)
# careful: base must be NS; on-site rotation applied to the NS chain:
ons_ns = list(ons)
ons_ns[15] = CMONE
Ha_ns = h_cfg(16, ons_ns)
kill_a &= (np.abs(np_spec(Ha_ns, 16) - spec_ns).max() < 1e-13
and np.abs(spec_ns - spec_tw).max() > 1e-2)
print(" (a) on-site Z3 rotation: holonomy unchanged (exact), "
"spectrum = NS class, NOT twisted (class gap %.3f): no twist "
"created" % np.abs(spec_ns - spec_tw).max())
# (b) site potential
tr_tw = sp.simplify(sum(-sp.cos(2 * sp.pi * (sp.Rational(m2) +
sp.Rational(1, 6)) / 16)
for m2 in range(16))) == 0
tr2_tw = sp.simplify(sum(sp.cos(2 * sp.pi * (sp.Rational(m2) +
sp.Rational(1, 6)) / 16) ** 2
for m2 in range(16)) - 8) == 0
mus = np.linspace(-2, 2, 161)
dmin_b, mu_best = np.inf, None
Hns_np = to_np(Hns, 16)
for mu in mus:
if abs(mu) < 1e-9:
continue
Hb = Hns_np.copy()
Hb[5, 5] += mu
dv = np.abs(np.sort(np.linalg.eigvalsh(Hb)) - spec_tw).max()
if dv < dmin_b:
dmin_b, mu_best = dv, mu
kill_b = tr_tw and tr2_tw and dmin_b > 1e-2
print(" (b) site potential: Tr H = mu != 0 = Tr(twisted) (exact); "
"balanced pairs: Tr H^2 = 8 + sum mu^2 > 8 (exact); scan floor "
"min_mu max|spec dev| = %.4f at mu = %.3f" % (dmin_b, mu_best))
# (c) amplitude defect
dmin_c = np.inf
for de in np.linspace(-1.5, 1.5, 121):
if abs(de) < 1e-9 or abs(de + 2) < 1e-9:
continue
Hc2 = Hns_np.copy()
Hc2[2, 3] *= (1 + de)
Hc2[3, 2] *= (1 + de)
dmin_c = min(dmin_c,
np.abs(np.sort(np.linalg.eigvalsh(Hc2)) - spec_tw).max())
kill_c = dmin_c > 1e-2
print(" (c) amplitude defect: Tr H^2 = 8 + (2 delta + delta^2)/2 "
"!= 8 (exact, delta not in {0, -2}); scan floor = %.4f" % dmin_c)
# (d) sign defect = R sector
sgn = [CONE] * 16
sgn[15] = CMONE
sgn[6] = CMONE
Hd = h_cfg(16, sgn)
kill_d = (holonomy(sgn) == CONE
and np.abs(np_spec(Hd, 16) - spec_r0).max() < 1e-13)
cas = {nu: sp.Rational(6, 1) * nu ** 2 - 6 * nu + 1
for nu in (sp.Rational(0), sp.Rational(1, 6), sp.Rational(1, 2))}
kill_d &= (cas[sp.Rational(0)] / 12 == sp.Rational(1, 12)
and cas[sp.Rational(1, 6)] / 12 == sp.Rational(1, 72))
print(" (d) sign defect: holonomy +1 => EXACTLY the R sector; "
"killed by the v628 Casimir table (nu = 0: +1/12 vs twisted "
"1/72) and the A1.4 charge minima")
# (e) alternating mark weights
alt_w = [CONE] * 16
for b, w in zip(MARK_BONDS_PROBE, (1, 2, 1, 2)):
alt_w[b] = zeta(GT_OF_R[0] * w)
alt_w[15] = alt_w[15] * CMONE # NS background
Halt = h_cfg(16, alt_w)
kill_e = (holonomy(alt_w) == CMONE
and np.abs(np_spec(Halt, 16) - spec_ns).max() < 1e-13)
print(" (e) alternating mark weights (1,2,1,2): total holonomy "
"trivializes (zeta^{6 gt} = 1) => NS class, no twist: the "
"spectral avatar of the v623 V1 disconnection kill")
check("P3.3 [exact+E] the alternatives kill table: the site-centered "
"rotation is gauge (no twist), site potentials and amplitude "
"defects are killed by EXACT trace obstructions (Tr H = 0, "
"Tr H^2 = 8 for the twisted multiset: %s/%s) with scan floors "
"%.3f / %.3f, the sign defect is exactly the R sector (v628 "
"kill), and the alternating weights trivialize (v623 kill)"
% (tr_tw, tr2_tw, dmin_b, dmin_c),
kill_a and kill_b and kill_c and kill_d and kill_e)
# ---- P3.4 the honest isospectral hunt
from scipy.optimize import minimize
def loss_and_grad(x, target):
t, mu = x[:16], x[16:]
H = np.zeros((16, 16))
for j in range(16):
H[j, (j + 1) % 16] = t[j]
H[(j + 1) % 16, j] = t[j]
H[j, j] = mu[j]
ev, V = np.linalg.eigh(H)
d = ev - target
Gm = V @ np.diag(2 * d) @ V.T
gt_ = np.array([2 * Gm[(j + 1) % 16, j] for j in range(16)])
gmu = np.diag(Gm).copy()
return float(d @ d), np.concatenate([gt_, gmu])
best = (np.inf, None)
for _ in range(150):
x0 = np.concatenate([-0.5 + 0.2 * RNG.standard_normal(16),
0.1 * RNG.standard_normal(16)])
res = minimize(loss_and_grad, x0, args=(spec_tw,), jac=True,
method="L-BFGS-B", options=dict(maxiter=600))
if res.fun < best[0]:
best = (res.fun, res.x)
def build_real(x):
t, mu = x[:16], x[16:]
H = np.zeros((16, 16))
for j in range(16):
H[j, (j + 1) % 16] = t[j]
H[(j + 1) % 16, j] = t[j]
H[j, j] = mu[j]
return H
# Gauss-Newton polish of the best candidate (eigenvalue Jacobian)
x = best[1].copy()
for _ in range(60):
ev, V = np.linalg.eigh(build_real(x))
F = ev - spec_tw
J = np.zeros((16, 32))
for i in range(16):
for j in range(16):
J[i, j] = 2 * V[j, i] * V[(j + 1) % 16, i]
J[i, 16 + j] = V[j, i] ** 2
dx, *_ = np.linalg.lstsq(J, -F, rcond=None)
x = x + dx
if np.abs(F).max() < 1e-14:
break
ev_fin = np.linalg.eigvalsh(build_real(x))
resid = np.abs(ev_fin - spec_tw).max()
print(" isospectral hunt over real NN chains (32 params, 150 "
"restarts + Gauss-Newton polish): best loss = %.3e, final max "
"|eig dev| = %.3e" % (best[0], resid))
found = resid < 1e-12
best = (float(np.sum((ev_fin - spec_tw) ** 2)), x)
def corr_half(Hnp):
ev, V = np.linalg.eigh(Hnp)
return V[:, :8] @ V[:, :8].conj().T
def string_vals(C):
out = []
for x in range(2, 9):
Cs = C[:x, :x]
Mx = np.eye(x, dtype=complex) \
+ (np.exp(1j * np.pi / 3) - 1.0) * Cs
out.append(abs(np.linalg.det(Mx)))
return np.array(out)
if found:
Hcand = build_real(best[1])
C_true = corr_half(to_np(h_def(16, lam_tw), 16))
C_cand = corr_half(Hcand)
nn_true = np.array([abs(C_true[j, (j + 1) % 16]) for j in range(16)])
nn_cand = np.array([abs(C_cand[j, (j + 1) % 16]) for j in range(16)])
dev_nn = np.abs(np.sort(nn_true) - np.sort(nn_cand)).max()
sv_t, sv_c = string_vals(C_true), string_vals(C_cand)
dev_str = np.abs(sv_t - sv_c).max()
disc = dev_nn > 1e-3 or dev_str > 1e-3
print(" candidate found: gauge-invariant discrimination: "
"max |dev| |C_nn| = %.4f, mu6-string values = %.4f"
% (dev_nn, dev_str))
check("P3.4 [E, honest] an isospectral real NN chain EXISTS (max "
"|eig dev| %.1e): purely spectral data (incl. the v628 "
"Casimir, a spectral functional) cannot exclude it -- but "
"it is NOT a Z3-twist realization (real holonomy +-1, fails "
"the P3.2 census) and the gauge-invariant correlator + "
"mu6-string data (the measured v639/A1 observables are "
"dressed string two-points) discriminate it (max devs "
"|C_nn| = %.4f, string = %.4f, > 1e-3): the measured data "
"decide, and they pick the bond defect"
% (resid, dev_nn, dev_str), disc)
else:
check("P3.4 [E, honest] the isospectral hunt over ALL real NN "
"chains (hoppings + potentials, 150 restarts + polish) does "
"not reach the twisted multiset (final max |eig dev| "
"%.3e): no real isospectral alternative found -- the census "
"stands on the spectra alone" % resid, resid > 1e-6)
# ---- P3.5 the charge-string bridge
ok35 = True
for gt in range(1, 6):
arc = list(range(4, 11)) # arc A = sites 4..10
Ustr = {(i, i): (zeta(gt) if i in arc else CONE) for i in range(16)}
Hconj = mmul(mmul(Ustr, Hns), mdag(Ustr))
pair = [CONE] * 16
pair[15] = CMONE
pair[3] = zeta(-gt) # entering bond (3,4)
pair[10] = zeta(gt) # leaving bond (10,11)
ok35 &= meq(Hconj, h_cfg(16, pair))
check("P3.5 [exact] the charge-string bridge: conjugating the seam "
"chain by the mu6 string exp(i pi gt N_arc/3) creates EXACTLY a "
"bond-defect PAIR (zeta^{-gt}, zeta^{+gt}) at the arc ends, for "
"every gt = 1..5: the measured defect ladder (arf A1, v639) is "
"bond-defect spectroscopy -- the measured exponents 1/18, 2/9, "
"1/2 are measurements OF this realization", ok35)
# ================================================================== P4
print("=" * 72)
print("P4: typing")
print("=" * 72)
check("P4.1 [C -> lattice theorem] TYPING: (i) P1 is a THEOREM at the "
"lattice level -- the deck-character blocks of the covered seam "
"ARE the one-bond-defect chains, unitarily and exactly, at the "
"seam size (16/48) and at the Arf lattice size (48/144), with "
"defect phase = deck character (one global orientation) and the "
"NS-borne D_can^3 = -1 fixed by the clock (L^4 = D_can); "
"(ii) P2: the single-bond position is pure gauge, and the "
"geometry pins the gauge class: sheet reduction = one uniform "
"twist unit per mark (cocycle support = mark set), window "
"boundary at a crossing <=> defect at a mark, clock covariance "
"selects the mark-uniform field; (iii) P3: within twist "
"realizations the bond defect is alternativlos (pure-phase "
"census + the symmetry census: the base chain has NO other Z3), "
"and everything outside is either exactly killed (traces, "
"holonomy, v628 Casimir table, v623 connectivity) or "
"discriminated by the measured string/correlator data. WHAT "
"REMAINS [C]: the continuum/CFT-level orbifold statement (the "
"lattice theorem does not by itself construct the interacting "
"twist fields), and the identification of THIS lattice with the "
"physical seam rests on v622/v623 as before. GATE.QGEO does "
"not move.", True)
# ================================================================== summary
print("=" * 72)
n_pass = sum(1 for _, ok in CHECKS if ok)
print("%d/%d checks passed" % (n_pass, len(CHECKS)))
p1_ok = all(ok for nm, ok in CHECKS if nm.startswith("P1"))
p2_ok = all(ok for nm, ok in CHECKS if nm.startswith("P2"))
p3_ok = all(ok for nm, ok in CHECKS if nm.startswith("P3"))
s0_ok = all(ok for nm, ok in CHECKS if nm.startswith("S0"))
if n_pass == len(CHECKS):
print("ALL CHECKS PASSED")
print("VERDICT: BOND-DEFECT-CANONICAL -- the bond-defect premise is a")
print("lattice THEOREM: twist sector = deck boundary condition = bond")
print("defect (exact unitary equivalence on all modes, at 16/48 and")
print("48/144), the defect position is gauge with the gauge class")
print("pinned by the mark geometry and the clock, and the realization")
print("is alternative-free within twist realizations and discriminated")
print("outside by the measured v628/v639/A1 data.")
elif s0_ok and p1_ok and p2_ok and not p3_ok:
print("SOME CHECKS FAILED")
print("VERDICT: BOND-DEFECT-GAUGE-ONLY -- the theorem and the")
print("geometric pin stand, but the exclusion census found an")
print("undiscriminated alternative.")
elif s0_ok and not p1_ok:
print("SOME CHECKS FAILED")
print("VERDICT: BOND-DEFECT-UNDECIDED -- the canonical equivalence")
print("failed; the premise stays a [C] reading.")
else:
print("SOME CHECKS FAILED")
print("VERDICT: MIXED")
def run():
"""run_all entry point: the checks above already ran at import."""
return len([1 for _, ok in CHECKS if not ok])
if __name__ == "__main__":
raise SystemExit(1 if run() else 0)
Recorded source fingerprint
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