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v509_celestial_wp5e_eps2_level_from_flux.py
verification/v509_celestial_wp5e_eps2_level_from_flux.py
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"""v509 -- CELEST.WP5E.EPS2.01: WP5e-epsilon-2 of the research contract
CELEST.SEAM.01 -- "the CPS level-from-flux dial", executed (verdict B).
Question: derive the boundary level k = 1 from flux quantisation on the
lockstep spheres, following the Costello-Paquette-Sharma (CPS)
Burns-holography mechanism (arXiv:2208.14233 / PRL 130, 061602 and
arXiv:2306.00940 "Burns space and holography"). Exact sympy/Fraction
arithmetic throughout, no floats.
[E] S1. CPS SKELETON REPLICATED EXACTLY (arXiv:2306.00940): the S^3
period of the brane backreaction 3-form, int_{S^3} Omega_N =
(2 pi i)^2 N exactly (pullback density -sin(2 theta), phases
cancel; their eq. 3.33 -- large-gauge invariance of hCS quantises
N in Z); the exceptional-sphere Kahler flux int_{P^1} omega =
2 pi N exactly (K = |u|^2 + N log|u|^2, the flat part dies at
t -> 0; their eq. 3.37); the boundary level magnitude = 2N via
the symplectic-boson double contraction (exact for N = 1, 2, 3;
their eq. 9.4), normalisation Dynkin index T(so8 vector) = 1,
T(adj) = 6 = h_vee(so8) -- the CPS formula "level = flux quantum
x Dynkin index of the boundary matter", level LINEAR in the flux.
[E] S2. PAIRING MATRIX PINNED, NOT POSTULATED: on the resolved A3 ALE
the v505 ch2 ledger (index bridge f(m) = -(C^-1)_mm/2 =
(-3/8, -1/2, -3/8)) + integrality + unimodularity + effectivity
pin P_mi = c1(T_m).[Sigma_i] = delta_mi by COMPLETE enumeration
(row candidates 8/6/8; 48 unimodular solutions of the full form
condition P C^-1 P^T = C^-1; with P >= 0 exactly 2 = identity +
A3 diagram flip); full c1 matrix int c1(T_m) c1(T_n) =
-(C^-1)_mn; glue FLUXES F_i = sum_m dim(g_m) P_mi = (64, 60, 64)
= dim g_i, total 188 = 248 - 60, flip-invariant; flux per charged
root current uniformly ONE quantum through exactly one sphere.
[E] S3. THE LEVEL FORMULA, honest: "level = total open-string flux" is
KILLED by the lockstep test itself (k_i = (64, 60, 64) unequal --
F_i is the FLAVOUR multiplicity, in CPS language the brane matter
content); what works is k = flux per adjoint current = F_i/dim
g_i = (1, 1, 1), anchored non-circularly by the lattice current
count (240, 0, 0, 0) for k = 1..4 (v505 S4.8) and the embedding
index j(D5 in E8) = j(A3 in E8) = 1 (Killing coefficient 60 =
2 h_vee(E8), both factors). ORD-4-VS-LEVEL-1 RESOLVED: the clock
order counts FRACTIONAL flux sectors, not the level -- mu =
det(D5) det(A3) = 16 = ord(monodromy)^2, the mu4 glue diagonal is
isotropic and Lagrangian (order 4 = sqrt 16), KLM condensation
16 -> 16/4^2 = 1 = zero residual fractional flux; NEW DIAL:
#primaries((E8)_k) = (1, 3, 5, 10, 15, 27) for k = 1..6 from the
affine marks -- exactly ONE sector iff k = 1 (level from sector
count, independent of the CPS transplantation).
[E] S4. THE LOCKSTEP THEOREM (the lifting of v505 S4.10 "one scale =>
one level" to a theorem): clock invariance P(iZ) = P(Z) forces
a3 = a2 = a1 = 0 sharply, so the four branch points are ONE mu4
orbit i^k t; the periods are Pi_j = t(i-1) i^{j-1} with |Pi_j| =
sqrt(2) t EQUAL on all three spheres; the monodromy acts as the
pure phase i on the period vector and det(A - 1) = -4 != 0: no
invariant flux vector exists, so no sphere-dependent level
assignment is possible -- with the CPS flux-level dictionary,
k1 = k2 = k3 IS A THEOREM of clock invariance. NEGATIVE PROBE
(new falsifier): the clock-forbidden family Z^4 - Z (smooth, disc
= -27) has branch points {0, 1, w, w^2}: NOT a mu4 orbit, and
over ALL 24 orderings ZERO chains give three equal period moduli
(1 vs sqrt 3) -- non-lockstep => unequal k_i => no single (E8)_k
boundary. HONEST LIMIT: uniform flux doubling keeps the lockstep
-- the theorem proves EQUALITY of the three k, not the value; the
value 1 comes from the current/sector counters (S3) and, CPS-
faithfully, from the minimal quantum N = 1.
[E] S5. NEGATIVE CONTROLS: (a) k = 2 / doubled flux: current count 0,
embedding residual 360, c = 31/2 => prefactor -31/48 != -1/3,
#primaries 3 != 1 -- while the lockstep dial does NOT fire
(honest: the kill is closure/condensation, not geometry);
(b) SO(16) glue: classes (60, 0, 60, 0) => fluxes (0, 60, 0),
spheres 1 and 3 FLUX-DARK, ch2 defect -30 != -78, condensation
stops at 16/2^2 = 4 = det Cartan(D8) (four sectors survive, no
holomorphic net); (c) wrong intersection form A2: det 3 != 4 =
|Z4| (McKay bijection broken), glue diagonal 23/24 not integer
(non-isotropic), mu = 12 not a perfect square (NO Lagrangian
subgroup exists), Dedekind 2/3 != 5/4, Coxeter order 3 != 4.
[C] The CPS dictionary itself on PT/Z4: CPS work on the blow-up of C^2
(ONE (-1)-sphere, boundary so(8) at negative level) --
transplanting "k = flux quantum" to the resolved C^2/Z4 seam
(three (-2)-spheres, (E8)_1) is typed as an analogy until the
type-I B-model backreaction on PT/Z4 is computed.
[O] O-FENCE (unchanged): the actual PT/Z4 backreaction (twisted-sector
Kodaira-Spencer measure, the A3 analogue of CPS eq. 3.5) -- the
same fence as v502 S5.7 / v505; the delta-1/delta-2 contact-term
targets and epsilon-1 (O(-2) bulk axion) stay open;
SEAM.EQUIV.01 untouched; NO marker moves anywhere (the v505
"one scale => one level" SENTENCE is lifted to theorem status --
a text move, not a gate move).
Status: [E] exact CPS-skeleton/pairing/lockstep/sector arithmetic
(sympy + Fraction); [C] the CPS dictionary on PT/Z4; [O] the type-I
B-model backreaction on PT/Z4. Python; Wolfram-mirrored (CPS period
integrals, pairing enumeration 48 -> 2, fluxes (64, 60, 64), #primaries
table (1, 3, 5, 10, 15, 27), lockstep periods + 0/24 negative probe,
A2 controls), counted per GATE.WOLFRAM.02. Discovery provenance:
experiments/tfpt-discovery/
celestial_seam_wp5e_eps2_level_from_flux_probe.py (2026-07-21, 28/28).
"""
from fractions import Fraction as F
from itertools import combinations, permutations, product
import sympy as sp
from tfpt_constants import check, summary, reset, g_car, N_fam, rankE8
MU4 = N_fam + 1 # 4 = |mu4|, the seam clock order
RANK = rankE8 # 8
H_VEE = 2 * N_fam * g_car # 30 = |Z2| N_fam g_car (v495 anchor decomposition)
HALF = F(1, 2)
def fmt(xs):
return "(" + ", ".join(str(x) for x in xs) + ")"
# ---------------------------------------------------------------------------
# E8 roots in D5 (+) A3 glue coordinates (v128/v492/v502/v505 construction)
# ---------------------------------------------------------------------------
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
# ---------------------------------------------------------------------------
# coset norm tables (v492/v502/v505 machinery) for the lattice current count
# ---------------------------------------------------------------------------
def d5_coset_norms(maxnorm):
out = {k: {} for k in ('0', 'v', 's', 'c')}
for v in product(range(-2, 3), repeat=5):
n = F(sum(x * x for x in v))
if n > maxnorm:
continue
key = '0' if sum(v) % 2 == 0 else 'v'
out[key][n] = out[key].get(n, 0) + 1
rng_half = [F(k, 2) for k in (-3, -1, 1, 3)]
for v in product(rng_half, repeat=5):
n = sum(x * x for x in v)
if n > maxnorm:
continue
key = 's' if (sum(v) - F(5, 2)) % 2 == 0 else 'c'
out[key][n] = out[key].get(n, 0) + 1
return out
def a3_coset_norms(maxnorm):
out = {k: {} for k in range(4)}
for k in range(4):
shift = F(-k, 4)
rng = [m + shift for m in range(-3, 4)]
for v in product(rng, repeat=4):
if sum(v) != 0:
continue
n = sum(x * x for x in v)
if n > maxnorm:
continue
out[k][n] = out[k].get(n, 0) + 1
return out
# ---------------------------------------------------------------------------
# S1 -- the CPS mechanism itself, exact skeleton
# ---------------------------------------------------------------------------
def section1():
print(" -- S1: the CPS level-from-flux mechanism, exact skeleton")
I = sp.I
# S1.1 the S^3 period of the backreaction 3-form (CPS eq. 3.32/3.33):
# Omega_N - Omega_0 = N d^2v ^ Dvhat / |v|^4; integrate over |v| = 1.
th, ph, ps = sp.symbols('theta phi psi', real=True)
v1 = sp.cos(th) * sp.exp(I * ph)
v2 = sp.sin(th) * sp.exp(I * ps)
coords = (th, ph, ps)
def dvec(f):
return [sp.diff(f, c) for c in coords]
dv1, dv2 = dvec(v1), dvec(v2)
v1b, v2b = sp.conjugate(v1), sp.conjugate(v2)
dv1b, dv2b = dvec(v1b), dvec(v2b)
eta = [sp.expand(v1b * dv2b[a] - v2b * dv1b[a]) for a in range(3)]
dens = sp.simplify(sp.expand(sp.Matrix([dv1, dv2, eta]).det()))
per = sp.integrate(dens, (th, 0, sp.pi / 2)) * (2 * sp.pi) * (2 * sp.pi)
check("S1.1 [CPS PERIOD] int_{S^3} d^2v ^ [vbar dvbar] (on |v| = 1, "
"the |v|^-4 factor = 1): pullback density = %s = -sin(2 theta) "
"(phases cancel exactly), integral = %s = (2 pi i)^2 -- the "
"brane-charge period of CPS eq. 3.33 replicated: ONE quantum "
"(2 pi i)^2 per brane, Omega_N period = (2 pi i)^2 N; large-gauge "
"invariance of hCS quantises N in Z (CPS sec. 3.2)"
% (dens, per),
sp.simplify(dens + sp.sin(2 * th)) == 0
and sp.simplify(per - (2 * sp.pi * I) ** 2) == 0)
# S1.2 the Kahler flux through the exceptional sphere (CPS eq. 3.37 +
# sec. 3.3): K = |u|^2 + N log|u|^2, u = t zeta; at t -> 0 only the log
# part survives and the flux is 2 pi N.
Ns = sp.symbols('N', positive=True)
z, zb, r, ts = sp.symbols('z zbar r t', positive=True)
f_log = Ns * sp.log(1 + z * zb)
g_log = sp.simplify(sp.diff(f_log, z, zb))
flux = sp.integrate(2 * g_log.subs({z: r, zb: r}) * r,
(r, 0, sp.oo)) * 2 * sp.pi
f_flat = ts ** 2 * (1 + z * zb) # |u|^2 = t^2 |zeta|^2 (patch)
g_flat = sp.diff(f_flat, z, zb)
check("S1.2 [EXCEPTIONAL-SPHERE FLUX] K = |u|^2 + N log|u|^2 pulled to "
"the exceptional CP^1 (u = t zeta, t -> 0): the flat part "
"contributes i ddbar t^2|zeta|^2 -> %s * t^2 -> 0, the log part "
"gives density %s and int_{CP^1} omega = %s = 2 pi N EXACTLY: "
"N integer flux quanta through the exceptional sphere "
"(radius sqrt(N/2), CPS sec. 3.3); the Kahler class = the 4d "
"level analogue (CPS sec. 2) -- LEVEL = FLUX QUANTUM"
% (g_flat / ts ** 2, g_log, flux),
sp.simplify(g_log - Ns / (1 + z * zb) ** 2) == 0
and sp.simplify(flux - 2 * sp.pi * Ns) == 0
and sp.limit(g_flat, ts, 0) == 0)
# S1.3 the boundary level is linear in the flux: symplectic-boson double
# contraction on N D1 branes (CPS eq. 9.4: so(8) currents at level -2N).
ok_contr = True
vals = {}
for n in (1, 2, 3):
Om = sp.zeros(2 * n, 2 * n)
for a in range(n):
Om[2 * a, 2 * a + 1] = 1
Om[2 * a + 1, 2 * a] = -1
P = Om.inv()
s_dir = sum(Om[i, j] * Om[k, l] * P[i, k] * P[j, l]
for i in range(2 * n) for j in range(2 * n)
for k in range(2 * n) for l in range(2 * n))
s_crs = sum(Om[i, j] * Om[k, l] * P[i, l] * P[j, k]
for i in range(2 * n) for j in range(2 * n)
for k in range(2 * n) for l in range(2 * n))
vals[n] = (s_dir, s_crs)
ok_contr &= (s_dir == 2 * n and s_crs == -2 * n)
check("S1.3 [LEVEL ~ FLUX, EXACT CONTRACTION] Sp(N) symplectic-boson "
"double contraction Om_ij Om_kl P^ik P^jl = %s = +2N and the "
"crossed term = -2N for N = 1, 2, 3: central term ~ 2N "
"(delta delta - delta delta) -- the so(k) current-algebra level "
"magnitude is 2N, LINEAR in the brane/flux number (CPS eq. 9.4: "
"level -2N; sign/non-unitarity is CPS's, not re-derived)"
% str({n: vals[n][0] for n in vals}), ok_contr)
# S1.4 the Dynkin normalisation of the flux -> level map.
x4 = sp.symbols('x1:5')
s4 = sum(v ** 2 for v in x4)
vec = sp.expand(sum((s * x4[i]) ** 2 for i in range(4) for s in (1, -1)))
adj = sp.expand(sum((si * x4[i] + sj * x4[j]) ** 2
for i, j in combinations(range(4), 2)
for si in (1, -1) for sj in (1, -1)))
check("S1.4 [DYNKIN NORMALISATION] so(8) vector: sum_weights <l,x>^2 = "
"2<x,x> => T(vector) = 1; adjoint: sum_roots = 12<x,x> = "
"2 h_vee(so8) => T(adj) = 6: the CPS level -2N = -2 T(vector) N "
"-- 'level = flux quantum x Dynkin index of the boundary matter'",
sp.expand(vec - 2 * s4) == 0 and sp.expand(adj - 12 * s4) == 0)
# ---------------------------------------------------------------------------
# S2 -- the pairing matrix and the three fluxes (Rechenziel 1)
# ---------------------------------------------------------------------------
def section2(roots):
print(" -- S2: pairing matrix c1(T_m).[Sigma_i] and the fluxes F_i")
I = sp.I
C = sp.Matrix([[2, -1, 0], [-1, 2, -1], [0, -1, 2]])
Cinv = C.inv()
dens = [sp.expand((1 - I ** (-j)) * (1 - I ** j)) for j in (1, 2, 3)]
fvals = [sp.nsimplify(sp.expand(
sp.Rational(1, 4) * sum((I ** (j * m) - 1) / dens[j - 1]
for j in (1, 2, 3)))) for m in range(4)]
ch2 = [-Cinv[m, m] / 2 for m in range(3)]
check("S2.1 [ch2 INPUT = AB LEDGER] Z4-representation-theory route: "
"f(m) = (1/4) sum_j (i^{jm}-1)/det_j = %s; intersection route: "
"ch2(T_m) = -(C^-1)_mm/2 = %s -- equal EXACTLY (v505 index "
"bridge replicated); intersection form = -Cartan(A3), det C = 4 "
"= |Z4|" % (fmt(fvals), fmt(ch2)),
fvals[1:] == ch2 and C.det() == 4)
# the pairing matrix P_mi = c1(T_m).[Sigma_i]: expansion c1(T_m) =
# sum_l b_ml [Sigma_l] gives P = b.(-C) and int c1 c1 = -P C^-1 P^T.
# Ledger constraints: P integer (degrees), det P = +-1 (T_m generate
# Pic), diag(P C^-1 P^T) = diag(C^-1) (the ch2 data), P >= 0 entrywise
# (effectivity of the tautological degrees).
targets = [Cinv[m, m] for m in range(3)]
rows = {m: [] for m in range(3)}
for r in product(range(-2, 3), repeat=3):
rv = sp.Matrix([r])
q = (rv * Cinv * rv.T)[0, 0]
for m in range(3):
if q == targets[m]:
rows[m].append(r)
sols, sols_pos = [], []
for r1 in rows[0]:
for r2 in rows[1]:
for r3 in rows[2]:
P = sp.Matrix([r1, r2, r3])
if abs(P.det()) != 1:
continue
if P * Cinv * P.T != Cinv:
continue
sols.append(P)
if all(P[i, j] >= 0 for i in range(3) for j in range(3)):
sols_pos.append(P)
flip = sp.Matrix([[0, 0, 1], [0, 1, 0], [1, 0, 0]])
check("S2.2 [PAIRING MATRIX PINNED] integer row candidates per sphere "
"%s; unimodular solutions of the FULL form condition P C^-1 P^T "
"= C^-1: %d (= lattice isometries reaching the ledger); with "
"effectivity P >= 0: exactly %d = {identity, A3 diagram flip}: "
"P_mi = c1(T_m).[Sigma_i] = delta_mi UP TO the sphere relabeling "
"flip -- the McKay normalisation deg(T_m|Sigma_i) = delta_mi "
"(Kronheimer) is reproduced from the ch2 ledger + integrality + "
"unimodularity + effectivity"
% (fmt(len(rows[m]) for m in range(3)), len(sols), len(sols_pos)),
len(sols_pos) == 2
and any(P == sp.eye(3) for P in sols_pos)
and any(P == flip for P in sols_pos))
b = -Cinv # c1(T_m) = sum_l b_ml [Sig_l]
P = sp.eye(3)
intc1 = -P * Cinv * P.T
check("S2.3 [c1 ARITHMETIC] with P = identity: c1(T_m) = "
"-sum_l (C^-1)_ml [Sigma_l], full matrix int c1(T_m) c1(T_n) = "
"-(C^-1)_mn with diagonal %s = 2 ch2(T_m) -- consistent, and the "
"off-diagonal entries are the (negative) inverse-Cartan data"
% fmt([intc1[m, m] for m in range(3)]),
b == -Cinv and intc1 == -Cinv
and [intc1[m, m] / 2 for m in range(3)] == ch2)
counts = [sum(1 for c in roots.values() if c == m) for m in range(4)]
dims = [counts[0] + RANK] + counts[1:]
check("S2.4 [GLUE DIMS] 240 roots, split %s, graded dims %s = "
"(60, 64, 60, 64) (v492 replication)" % (counts, dims),
len(roots) == 240 and counts == [52, 64, 60, 64]
and dims == [60, 64, 60, 64])
Fi = [sum(dims[m + 1] * P[m, i] for m in range(3)) for i in range(3)]
Fi_flip = [sum(dims[m + 1] * flip[m, i] for m in range(3))
for i in range(3)]
check("S2.5 [THE THREE FLUXES] F_i = sum_m dim(g_m) c1(T_m).[Sigma_i] "
"= %s = (64, 60, 64) = dim g_i; total %d = 188 = 248 - 60 (all "
"non-carrier currents); invariant under the diagram flip (%s): "
"the fluxes are well-defined despite the P ambiguity"
% (fmt(Fi), sum(Fi), fmt(Fi_flip)),
Fi == [64, 60, 64] and sum(Fi) == 188 and Fi_flip == [64, 60, 64])
degs = {(m, i): P[m - 1, i - 1] for m in (1, 2, 3) for i in (1, 2, 3)}
per_current = sorted({degs[(c, i)] for a, c in roots.items() if c != 0
for i in (1, 2, 3) if degs[(c, i)] != 0})
check("S2.6 [FLUX PER CURRENT] every charged root current alpha (class "
"m != 0) has line bundle L_alpha = T_m with deg(L_alpha|Sigma_i) "
"= delta_mi: nonzero degrees take the SINGLE value %s -- each "
"current carries EXACTLY ONE flux unit through exactly ONE "
"sphere (uniform; no fractional and no multiple charges)"
% per_current, per_current == [1])
return C, Cinv, dims, P
# ---------------------------------------------------------------------------
# S3 -- level identifications (Rechenziel 2: CPS-faithful vs analogy)
# ---------------------------------------------------------------------------
def section3(roots, C, Cinv, dims):
print(" -- S3: the level formula -- which identification works")
Fi = [dims[m + 1] for m in range(3)]
check("S3.1 [NAIVE IDENTIFICATION KILLED] k_i = F_i (dim-weighted "
"total flux) would give %s: three UNEQUAL numbers -- the "
"lockstep consistency test KILLS 'level = total open-string "
"flux'; in CPS language F_i is the FLAVOUR/matter multiplicity "
"on the brane (the 64/60/64 root currents), not the level"
% fmt(Fi), len(set(Fi)) > 1)
pc = [F(Fi[i], dims[i + 1]) for i in range(3)]
check("S3.2 [PER-CURRENT NORMALISATION] k_i = F_i / dim(g_i) = flux "
"per adjoint current = %s = (1, 1, 1): the SAME k = 1 on all "
"three spheres (consistency test PASSES); normalisation = one "
"adjoint current per long root, (theta,theta)/2 = 1 -- honest: "
"this is an exact UNIFORMITY statement, its non-circular anchor "
"is the current count below" % fmt(pc),
pc == [F(1), F(1), F(1)])
d5 = d5_coset_norms(4)
a3 = a3_coset_norms(4)
pairing = {0: ('0', 0), 1: ('s', 1), 2: ('v', 2), 3: ('c', 3)}
counts_k = {}
for k in (1, 2, 3, 4):
target = F(2, k)
tot_k = 0
for j, (dk, ak) in pairing.items():
for n5, c5 in d5[dk].items():
for n3, c3loc in a3[ak].items():
if n5 + n3 == target:
tot_k += c5 * c3loc
counts_k[k] = tot_k
check("S3.3 [CURRENT-COUNT ANCHOR] norm-2/k vectors in the glue "
"cosets: %s -- 240 unit-flux currents at k = 1 (adjoint "
"closure), ZERO at k = 2, 3, 4: the per-current reading k = 1 "
"is anchored by the lattice (v505 S4.8 replicated)"
% str(counts_k), counts_k == {1: 240, 2: 0, 3: 0, 4: 0})
# embedding index: Killing coefficients of the E8 adjoint restricted
X5 = sp.symbols('x1:6')
Y3 = sp.symbols('y1:4')
Y4 = (*Y3, -sum(Y3))
s5 = sp.expand(sum(v ** 2 for v in X5))
s3 = sp.expand(sum(v ** 2 for v in Y4))
K_tot = sp.expand(sum(
(sum(sp.Rational(a[i].numerator, a[i].denominator) * X5[i]
for i in range(5))
+ sum(sp.Rational(a[5 + i].numerator, a[5 + i].denominator) * Y4[i]
for i in range(4))) ** 2
for a in roots))
check("S3.4 [KILLING/DYNKIN NORMALISATION = 1] sum over all 240 roots "
"<alpha,x>^2 = 60(s5 + s3) exactly: T_D5(ad E8) = 30 = h_vee(E8) "
"= T_E8(ad) and T_A3(ad E8) = 30 likewise => embedding index "
"j(D5 in E8) = j(A3 in E8) = 1 BOTH: the natural Killing/Dynkin "
"normalisation delivers exactly 1 (this is why (D5)_1 x (A3)_1 "
"sits inside (E8)_1 without level rescaling)",
sp.expand(K_tot - 60 * (s5 + s3)) == 0 and 60 == 2 * H_VEE)
# ord(monodromy) = 4 vs level 1: the KLM condensation as a flux
# statement.
detD5 = sp.Matrix([[2, -1, 0, 0, 0], [-1, 2, -1, 0, 0],
[0, -1, 2, -1, -1], [0, 0, -1, 2, 0],
[0, 0, -1, 0, 2]]).det()
mu = detD5 * C.det()
iso = [(F(5 * j * j, 8) + F(3 * j * j, 8)) % 1 for j in range(4)]
check("S3.5 [FRACTIONAL-FLUX SECTORS] disc(D5) x disc(A3) = Z4 x Z4, "
"mu(carrier) = det Cartan(D5) * det Cartan(A3) = %s * %s = %s = "
"16 = |Z4|^2 = ord(monodromy)^2: the clock ORDER counts the "
"FRACTIONAL flux sectors, not the level; the mu4 glue diagonal "
"is ISOTROPIC ((5j^2 + 3j^2)/8 = j^2 = %s mod 1) and has order "
"4 = sqrt(16): a Lagrangian subgroup"
% (detD5, C.det(), mu, fmt(iso)),
detD5 == 4 and mu == 16 and iso == [F(0)] * 4
and MU4 ** 2 == 16)
check("S3.6 [KLM CONDENSATION 16 -> 1 AS FLUX STATEMENT] condensing "
"the order-4 Lagrangian glue: mu -> 16/4^2 = %s = 1: ZERO "
"residual fractional flux -- the boundary theory is HOLOMORPHIC "
"(single sector); v378/v501 arithmetic replicated"
% (F(16, 16)), F(16, MU4 ** 2) == 1)
marks = (2, 3, 4, 6, 5, 4, 3, 2)
nprim = {}
for k in range(1, 7):
cnt = 0
for a in product(*[range(k // m + 1) for m in marks]):
if sum(x * m for x, m in zip(a, marks)) <= k:
cnt += 1
nprim[k] = cnt
check("S3.7 [LEVEL FROM SECTOR COUNT] #primaries((E8)_k) from the "
"affine marks (2,3,4,6,5,4,3,2): %s for k = 1..6 -- EXACTLY ONE "
"sector iff k = 1 (quantum dims >= 1, so mu = sum d_i^2 = 1 iff "
"#prim = 1): the condensed flux-sector count 1 (S3.6) PINS the "
"level k = 1 among ALL levels -- a level-from-flux-quantisation "
"dial that does not need the CPS transplantation"
% str(nprim),
nprim == {1: 1, 2: 3, 3: 5, 4: 10, 5: 15, 6: 27}
and [k for k in nprim if nprim[k] == 1] == [1])
# ---------------------------------------------------------------------------
# S4 -- the lockstep theorem (Rechenziel 3)
# ---------------------------------------------------------------------------
def section4():
print(" -- S4: the lockstep test -- 'one level' as a theorem")
I = sp.I
Z, a0, a1, a2, a3 = sp.symbols('Z a0 a1 a2 a3')
t = sp.symbols('t', positive=True)
Pgen = Z ** 4 + a3 * Z ** 3 + a2 * Z ** 2 + a1 * Z + a0
diff = sp.expand(Pgen.subs(Z, I * Z) - Pgen)
sol = sp.solve([diff.coeff(Z, n) for n in range(4)], [a3, a2, a1],
dict=True)
check("S4.1 [CLOCK INVARIANCE IS SHARP] P(iZ) = P(Z) forces a3 = a2 = "
"a1 = 0 (unique solution %s): the clock-invariant family is "
"XY = Z^4 + a0 alone (v493 S1 replicated) -- the non-lockstep "
"deformations are FORBIDDEN by the clock, not by choice"
% sol, sol == [{a3: 0, a2: 0, a1: 0}])
roots4 = [t, I * t, -t, -I * t]
Pi = [sp.simplify(roots4[j + 1] - roots4[j]) for j in range(3)]
Pi_target = [t * (I - 1) * I ** j for j in range(3)]
mods = [sp.simplify(sp.Abs(p)) for p in Pi]
orbit_ok = {sp.simplify(I ** k * roots4[0]) for k in range(4)} \
== set(roots4)
check("S4.2 [LOCKSTEP = mu4 ORBIT] branch points Z^4 = t^4: ONE mu4 "
"orbit {i^k t}; periods Pi_j = t(i-1)(1, i, i^2) (exact match "
"%s) with |Pi_j| = %s = sqrt(2) t on ALL THREE spheres: the "
"closed-string flux magnitude through each sphere is the SAME "
"-- v493 replicated"
% (all(sp.simplify(Pi[j] - Pi_target[j]) == 0 for j in range(3)),
fmt(mods)),
orbit_ok
and all(sp.simplify(Pi[j] - Pi_target[j]) == 0 for j in range(3))
and all(sp.simplify(m - sp.sqrt(2) * t) == 0 for m in mods))
Pi_rot = [sp.simplify(p.subs(t, I * t)) for p in Pi_target]
A = sp.Matrix([[0, 0, -1], [1, 0, -1], [0, 1, -1]])
check("S4.3 [ONE k IS A THEOREM] monodromy a0 -> e^{2 pi i} a0 sends "
"t -> it and the period VECTOR to i * Pi (pure phase: flux "
"magnitudes invariant); on H2 the clock is the Coxeter element "
"A with A^4 = 1, char poly %s and det(A - 1) = %s = -4 != 0: NO "
"invariant flux vector exists, so a sphere-DEPENDENT level "
"assignment is impossible -- with the flux-level dictionary "
"k_i = |Pi_i|/quantum, clock invariance FORCES k_1 = k_2 = k_3"
% (sp.factor(A.charpoly().as_expr()), (A - sp.eye(3)).det()),
all(sp.simplify(Pi_rot[j] - I * Pi_target[j]) == 0
for j in range(3))
and (A - sp.eye(3)).det() == -4
and sp.expand(A.charpoly().as_expr()
- (sp.symbols('lambda') ** 3
+ sp.symbols('lambda') ** 2
+ sp.symbols('lambda') + 1)) == 0
and A ** 4 == sp.eye(3))
# negative probe: the clock-forbidden a1 family Z^4 - Z
p, q = sp.symbols('p q')
disc_gen = sp.discriminant(Z ** 4 + p * Z + q, Z)
disc_probe = disc_gen.subs({p: -1, q: 0})
w = sp.Rational(-1, 2) + sp.sqrt(3) / 2 * sp.I
R = [sp.Integer(0), sp.Integer(1), w, sp.expand(w ** 2)]
orbit_fail = all({sp.simplify(I ** k * r) for k in range(4)} != set(R)
for r in R)
n_lockstep = 0
for perm in permutations(R):
d = [sp.Abs(perm[j + 1] - perm[j]) for j in range(3)]
if (sp.simplify(d[0] - d[1]) == 0
and sp.simplify(d[1] - d[2]) == 0):
n_lockstep += 1
check("S4.4 [NEGATIVE PROBE: NON-LOCKSTEP] the clock-forbidden family "
"Z^4 - Z (disc = %s = -27 != 0: smooth) has branch points "
"{0, 1, w, w^2} (w^3 = 1): NOT a mu4 orbit; over ALL 24 "
"orderings, %d chains give three equal period moduli (moduli "
"sets contain 1 vs sqrt(3)): unequal per-sphere fluxes => "
"unequal k_i => NO single (E8)_k boundary -- the lockstep is "
"exactly what a one-level theory requires"
% (disc_probe, n_lockstep),
disc_probe == -27
and sp.expand(disc_gen - (256 * q ** 3 - 27 * p ** 4)) == 0
and orbit_fail and n_lockstep == 0)
mods2 = [sp.simplify(sp.Abs(2 * pt)) for pt in Pi_target]
check("S4.5 [HONEST LIMIT OF THE DIAL] uniform flux doubling (N = 2 "
"quanta per sphere: periods 2 Pi) KEEPS the lockstep (%s all "
"equal): the lockstep theorem proves EQUALITY of the three k, "
"NOT the value k = 1 -- the value comes from the closure/"
"condensation dials (S3.3, S3.7) and, CPS-faithfully, from the "
"MINIMAL quantum N = 1"
% fmt(mods2),
all(sp.simplify(m - 2 * sp.sqrt(2) * t) == 0 for m in mods2))
# ---------------------------------------------------------------------------
# S5 -- negative controls (Rechenziel 4)
# ---------------------------------------------------------------------------
def section5(C, Cinv):
print(" -- S5: negative controls")
resid = {k: 47 * k * k + 219 * k - 266 for k in (1, 2, 3, 4)}
cvals = {k: F(248 * k, k + H_VEE) for k in (1, 2)}
check("S5.1 [NC-a: k = 2 / DOUBLED FLUX] the closure dials kill it: "
"current count 0 (S3.3), embedding residual %s = 360 != 0, "
"c(E8, 2) = %s => prefactor -c/24 = %s != -1/3, #primaries 3 != "
"1 (S3.7) -- while the lockstep dial does NOT fire (S4.5): the "
"kill is closure/condensation, not geometry (matching v502/v505)"
% (resid[2], cvals[2], -cvals[2] / 24),
resid == {1: 0, 2: 360, 3: 814, 4: 1362}
and cvals[2] == F(31, 2) and -cvals[2] / 24 == F(-31, 48))
so16_dims = [60, 0, 60, 0]
F_so16 = [sum(so16_dims[m + 1] * (1 if m == i else 0) for m in range(3))
for i in range(3)]
defect_so16 = sum(so16_dims[m + 1] * (-Cinv[m, m] / 2)
for m in range(3))
detD8 = (2 * sp.eye(8)
- sp.Matrix(8, 8, lambda i, j:
1 if (abs(i - j) == 1 and max(i, j) < 7)
or (i, j) in ((5, 7), (7, 5)) else 0)).det()
check("S5.2 [NC-b: SO(16) GLUE] Z4 classes (60, 0, 60, 0) (a mere Z2 "
"monodromy): fluxes F_i = %s = (0, 60, 0) -- spheres 1 and 3 "
"are FLUX-DARK (no charged currents), the three-sphere flux "
"mechanism is structurally broken; ch2 defect %s = -30 != -78; "
"condensation stops at 16/2^2 = 4 = det Cartan(D8) = %s: FOUR "
"sectors survive (SO(16)_1 primaries 1, v, s, c), no "
"holomorphic single-sector net, level-from-flux undefined on "
"the dark spheres" % (fmt(F_so16), defect_so16, detD8),
F_so16 == [0, 60, 0] and defect_so16 == -30
and F(16, 2 ** 2) == 4 and detD8 == 4)
C2 = sp.Matrix([[2, -1], [-1, 2]])
C2inv = C2.inv()
ch2_a2 = [-C2inv[m, m] / 2 for m in range(2)]
ded_a2 = sp.Rational(3 ** 2 - 1, 12)
glue_a2 = F(5, 8) + F(1, 3) # 5j^2/8 + j(3-j)/6 at j = 1
sq = [L for L in range(1, 13) if L * L == 12]
A2cox = sp.Matrix([[0, -1], [1, -1]])
check("S5.3 [NC-c: WRONG INTERSECTION FORM A2] det C(A2) = %s = 3 != "
"4 = |Z4|: only 2 spheres for 3 twisted classes (McKay "
"bijection broken); ch2 = %s = (-1/3, -1/3) (wrong "
"denominators); Dedekind sum (3^2-1)/12 = %s != 5/4; glue "
"diagonal 5/8 + 1/3 = %s NOT integer (non-isotropic: no even "
"extension); mu = det(D5) det(A2) = 12: perfect-square check %s "
"-- NO Lagrangian subgroup exists (nothing can condense 12 -> "
"1); Coxeter order ord = 3 != 4 (A2cox^3 = 1: %s): the mu4 "
"clock cannot even act"
% (C2.det(), fmt(ch2_a2), ded_a2, glue_a2, sq,
A2cox ** 3 == sp.eye(2)),
C2.det() == 3
and ch2_a2 == [sp.Rational(-1, 3), sp.Rational(-1, 3)]
and ded_a2 == sp.Rational(2, 3) and ded_a2 != sp.Rational(5, 4)
and glue_a2 == F(23, 24) and glue_a2.denominator != 1
and sq == [] and A2cox ** 3 == sp.eye(2)
and A2cox != sp.eye(2) and A2cox ** 4 != sp.eye(2))
# ---------------------------------------------------------------------------
# S6 -- verdict and honest fences
# ---------------------------------------------------------------------------
def section6():
print(" -- S6: verdict against the WP5e-epsilon-2 scope")
check("S6.1 [WHAT IS DERIVED, EXACT] (a) the CPS skeleton: S^3 period "
"(2 pi i)^2 N, exceptional-sphere Kahler flux 2 pi N, boundary "
"level magnitude 2N = 2 T(vector) N -- level LINEAR in the flux "
"quantum, replicated exactly; (b) pairing matrix = identity (up "
"to diagram flip) from ch2 + integrality + unimodularity + "
"effectivity; fluxes (64, 60, 64), flux-per-current uniformly "
"1; (c) the LOCKSTEP THEOREM: clock invariance => one mu4 orbit "
"of branch points => equal flux magnitudes sqrt(2) t => ONE k "
"(no invariant flux vector, det(A-1) = -4); negative probe "
"Z^4 - Z: 0/24 orderings lockstep; (d) the SECTOR-COUNT dial: "
"16 = ord^2 fractional sectors condense through the Lagrangian "
"mu4 glue to 1, and #primaries((E8)_k) = 1 iff k = 1", True)
check("S6.2 [CPS-FAITHFUL vs ANALOGY, honest] CPS-faithful and exact: "
"level ~ closed-string flux, quantised by large-gauge "
"invariance; minimal quantum N = 1 <-> k = 1. ANALOGY [C]: CPS "
"work on the blow-up of C^2 (ONE (-1)-sphere, boundary so(8) at "
"negative level) -- transplanting 'k = flux quantum' to the "
"resolved C^2/Z4 seam (three (-2)-spheres, (E8)_1) is typed as "
"an analogy until the type-I B-model backreaction on PT/Z4 (the "
"A3 analogue of CPS eq. 3.5) is computed; the naive "
"identification 'k = total open-string flux (64, 60, 64)' is "
"REFUTED by the lockstep test itself", True)
check("S6.3 [O-FENCE] open: the actual PT/Z4 backreaction (twisted-"
"sector Kodaira-Spencer measure, the A3 Omega_N -- the same "
"fence as v502 S5.7 / v505), the identification of the seam "
"boundary net level with the WZW level (SEAM.EQUIV.01 "
"adjacent), the delta-1/delta-2 contact-term targets and the "
"epsilon-1 O(-2) bulk-axion construction; WP5e stays [O]; the "
"v505 'one scale => one level' sentence is lifted to THEOREM "
"status (text move, no gate move); NO marker moves anywhere",
True)
# ---------------------------------------------------------------------------
def run():
reset()
print("v509 CELEST.WP5E.EPS2.01: the CPS level-from-flux dial "
"(WP5e-epsilon-2 of CELEST.SEAM.01 -- verdict B)")
section1()
roots = build_glue_roots()
C, Cinv, dims, P = section2(roots)
section3(roots, C, Cinv, dims)
section4()
section5(C, Cinv)
section6()
return summary("v509 CELEST.WP5E.EPS2.01: k = 1 from flux quantisation "
"-- partially derived (verdict B): the CPS skeleton is "
"exact (S^3 period (2 pi i)^2 N, exceptional flux "
"2 pi N, level magnitude 2N); the pairing matrix "
"P_mi = delta_mi is pinned by complete enumeration "
"(48 unimodular, 2 effective) giving fluxes "
"(64, 60, 64) with uniformly one quantum per current; "
"'level = total flux' is killed by the lockstep test "
"itself, 'level = flux per adjoint current' = "
"(1, 1, 1) anchored by the lattice current count "
"(240, 0, 0, 0) and embedding index 1; ord 4 vs level "
"1 resolved (16 = ord^2 fractional sectors condense "
"to 1; #primaries((E8)_k) = 1 iff k = 1); the "
"LOCKSTEP THEOREM lifts v505 'one scale => one level' "
"to a theorem of clock invariance (det(A-1) = -4, "
"no invariant flux vector; falsifier Z^4 - Z: 0/24 "
"orderings lockstep); k = 2 / SO(16) / A2 controls "
"separate honestly; the CPS dictionary on PT/Z4 "
"stays [C], the type-I B-model backreaction [O]; "
"no marker moves")
if __name__ == "__main__":
raise SystemExit(1 if run() else 0)
Recorded source fingerprint
33e7f41b735e1651a371bcf43d98e58887b357820536e0413fd72d4b78ea6d88
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