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v495_celestial_wp3_gs_alignment.py
verification/v495_celestial_wp3_gs_alignment.py
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"""v495 -- CELEST.WP3.01: the Green-Schwarz axion coefficient lambda_g of celestial
SDYM (Costello) vs the TFPT seam constant c3 = 1/(8 pi) -- WP3 of the research
contract CELEST.SEAM.01 (the pre-registered "alignment test only"). Exact
arithmetic (Fraction/sympy, no floats).
Question: Costello's twistorial SDYM theory is quantum-consistent exactly on the
algebras with no independent quartic Casimir (arXiv:2111.08879, eq. 1.2.2/2.4.2:
Tr_adj X^4 = lambda_g^2 (tr X^2)^2), and the forced Green-Schwarz axion coupling
is kappa = lambda_g/(8 pi sqrt 3) (his eq. 1.2.4). WP3 asks ONLY: does kappa
align arithmetically with c3 = 1/(8 pi) for e8 -- and, brutally honestly, is any
such alignment e8-SELECTIVE at all?
[E] S1. ROOT DATA + KILLING. Exact root systems for all 8 algebras on
Costello's list (sl2, sl3, g2, so8, f4, e6, e7, e8; e6/e7 carved out of
the v492 E8 root set); |roots| = dim - rank; the Killing normalisation
Sum_alpha <alpha,x>^2 = 2 h_vee <x,x> as an exact POLYNOMIAL identity
(theta^2 = 2) pins h_vee = (2,3,4,6,9,12,18,30).
[E] S2. OKUBO IDENTITY DERIVED, NOT QUOTED. Sum_alpha <alpha,x>^4 =
[5/(2(dim g + 2))] * (Sum_alpha <alpha,x>^2)^2 holds as an exact
polynomial identity for ALL 8 algebras ("no independent quartic
Casimir"); negative control: sl4 (which HAS one) fails the
proportionality (5/32 on s0^4 vs 3/32 on s0^2 s2^2) -- the list is sharp.
[E] S3. LAMBDA IN ALL CONVENTIONS. Deligne-series dimension formula
dim g = 2(5h_vee-6)(h_vee+1)/(h_vee+6) exact for all 8; unit-trace
lambda~^2 = 10 h_vee^2/(dim+2) = h_vee + 6 EXACTLY (8,9,10,12,15,18,24,
36) -- fixes Costello-Paquette eq. (1.5); fundamental Dynkin indices
i_R = (1,1,2,2,6,6,12,60) from explicit weight systems; E8: lambda~ = 6
and lambda_fund = 1/10 EXACT. Cross-checks: Costello's sl2 = 8, sl3 = 9
reproduced; his PRINTED lambda^2(so8) = 3/2 is a factor-2 slip (his own
weight content gives Sum w^4 = 192, not 96: lambda^2 = 3, Okubo-
consistent 12/4); the CP formula reading disambiguated (h_vee OUTSIDE
the square root).
[E] S4. ALIGNMENT vs c3 = 1/(8 pi). kappa = lambda_g/(8 pi sqrt 3) contains
c3 with (kappa/c3)^2 = lambda^2/3: e8 unit-trace 36/3 = 12 = |mu4| N_fam
EXACT; adjoint-trace 1/300 EXACT. kappa/c3 ITSELF is irrational
(2 sqrt 3 resp. 1/(10 sqrt 3)): the pre-registered "lambda/sqrt3
rational?" reading FAILS; only the squared alignment is exact. Anchor
decomposition: h_vee = 30 = |Z2| N_fam g_car (all three atoms once),
phi(30) = 8 = rank, dim+2 = 250 = 2 * 5^3, lambda~ = 6 = |Z2| N_fam,
lambda_fund = 1/10 = 1/(|Z2| g_car).
[E] S5. LOOK-ELSEWHERE (the HARD FENCE of this module -- alignment != selection).
All 8 candidates through the same battery:
test passes who
(kappa/c3)^2 an exact rational 8/8 everyone
h_vee {2,3,5}-smooth 8/8 everyone
lambda~ integer (h_vee+6 square) 2/8 sl3, e8
lambda_fund rational 2/8 sl3, e8
g_car = 5 | h_vee 1/8 e8 ONLY
phi(h_vee) = rank 4/8 sl2,sl3,g2,e8
phi(h) = rank 5/8 +f4
The alignment test AS SUCH has ZERO selective power (8/8); the only
e8-selective single test is "the atom 5 = g_car divides h_vee" (1/8);
the conjunction that isolates e8 (1/8) was assembled POST HOC from
correlated sub-tests -- it quantifies where e8 sits, it is NOT a
preregistered discriminator.
[E] S6. KILL-TEST K3 + VERDICT. K3 ("no exact alignment => the c3-connection
of the celestial route dies") does NOT fire: exact alignment EXISTS.
BUT by S5 the alignment format itself passes 8/8, so the c3-connection
survives ONLY as convention-level compatibility plus genuine
lambda-arithmetic -- it is NOT e8-selective evidence; K3's scope is
DEMOTED accordingly in the contract.
[C] TYPED: the 8 pi in Costello's (1.2.4) is the 2 pi-convention of the
anomaly polynomial (his A.0.10 constant descended to spacetime), NOT an
independent occurrence of the seam constant; the sqrt 3 is the
anomaly's own (sqrt 48 = 4 sqrt 3 from the twistor box diagram,
48 = 2*4!). WP3 is an alignment yes/no -- it can never derive c3 from
celestial data (contract wording, honoured here).
[O] FENCE: no marker moves anywhere; WP5 (constructive holography) and
SEAM.EQUIV.01 (the continuum (E8)_1 net on the seam) stay [O],
untouched. Verdict B: exact arithmetic alignment, honestly typed as
non-selective.
Status: [E] exact arithmetic (Fraction/sympy, no floats); [C] the convention
typing of the 8 pi and sqrt 3; [O] the celestial programme (WP5). Python;
Wolfram-mirrorable (exact), counted with the next verified engine run."""
from collections import Counter
from fractions import Fraction as F
from itertools import combinations, combinations_with_replacement, product
from math import factorial
import sympy as sp
from tfpt_constants import check, summary, reset, g_car, N_fam, rankE8
Z2 = 2 # sheet order |Z2|
MU4 = 4 # |mu4|
# ---------------------------------------------------------------------------
# exact multivariate polynomials: dict {sorted index tuple: Fraction}
# ---------------------------------------------------------------------------
def dot(u, v):
return sum(x * y for x, y in zip(u, v))
def multinom(combo, p):
c = Counter(combo)
r = factorial(p)
for v in c.values():
r //= factorial(v)
return r
def add_pow(out, a, p):
"""out += (sum_j a_j s_j)^p as an exact polynomial dict."""
idxs = [j for j in range(len(a)) if a[j] != 0]
for combo in combinations_with_replacement(idxs, p):
coef = F(multinom(combo, p))
for j in combo:
coef *= a[j]
out[combo] = out.get(combo, F(0)) + coef
def poly_square(K):
out = {}
items = list(K.items())
for i in range(len(items)):
k1, c1 = items[i]
key = tuple(sorted(k1 + k1))
out[key] = out.get(key, F(0)) + c1 * c1
for j in range(i + 1, len(items)):
k2, c2 = items[j]
key = tuple(sorted(k1 + k2))
out[key] = out.get(key, F(0)) + 2 * c1 * c2
return out
def poly_eq_scaled(P, Q, num, den):
"""den * P == num * Q exactly (both polynomial dicts)."""
keys = set(P) | set(Q)
return all(den * P.get(k, F(0)) == num * Q.get(k, F(0)) for k in keys)
def power_sums(vectors, basis):
"""(Sum <v,x>^2, Sum <v,x>^4) as polynomial dicts in x = sum s_j b_j."""
K, Q = {}, {}
for v in vectors:
a = [F(dot(v, b)) for b in basis]
add_pow(K, a, 2)
add_pow(Q, a, 4)
return K, Q
def gram_quad(basis):
"""<x,x> as a polynomial dict (same key convention as add_pow)."""
out = {}
m = len(basis)
for j in range(m):
for k in range(j, m):
g = F(dot(basis[j], basis[k]))
if g:
out[(j, k)] = out.get((j, k), F(0)) + g * (1 if j == k else 2)
return out
# ---------------------------------------------------------------------------
# root systems (theta^2 = 2 normalisation throughout)
# ---------------------------------------------------------------------------
def roots_sl2():
return [(1, -1), (-1, 1)]
def roots_sl3():
out = []
for i, j in product(range(3), repeat=2):
if i != j:
v = [0] * 3
v[i], v[j] = 1, -1
out.append(tuple(v))
return out
def roots_g2():
lng = []
for i, j in product(range(3), repeat=2):
if i != j:
v = [F(0)] * 3
v[i], v[j] = F(1), F(-1)
lng.append(tuple(v))
sht = []
for i in range(3):
v = [F(-1, 3)] * 3
v[i] = F(2, 3)
sht.append(tuple(v))
sht.append(tuple(-x for x in v))
return lng, sht
def roots_so8():
out = []
for i, j in combinations(range(4), 2):
for si in (1, -1):
for sj in (1, -1):
v = [0] * 4
v[i], v[j] = si, sj
out.append(tuple(v))
return out
def roots_f4():
lng = roots_so8()
sht = []
for i in range(4):
for s in (1, -1):
v = [F(0)] * 4
v[i] = F(s)
sht.append(tuple(v))
for signs in product((1, -1), repeat=4):
sht.append(tuple(F(s, 2) for s in signs))
return lng, sht
def roots_sl4():
out = []
for i, j in product(range(4), repeat=2):
if i != j:
v = [0] * 4
v[i], v[j] = 1, -1
out.append(tuple(v))
return out
def roots_e8():
"""v128/v492 construction in D5 (+) A3 glue coordinates, flattened to R^9
(A3 block sum-zero); all 240 roots have norm 2."""
HALF = F(1, 2)
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.append(r + z4)
for r in a3_roots:
roots.append(z5 + r)
for d in d5_s:
for w in wclass(1):
roots.append(d + w)
for d in d5_v:
for w in wclass(2):
roots.append(d + w)
for d in d5_c:
for w in wclass(3):
roots.append(d + w)
return roots
def basis_sumzero(n, ambient):
"""simple-root-style basis (1,-1,0,..) of the sum-zero subspace of R^n,
padded to the ambient dimension."""
out = []
for i in range(n - 1):
v = [F(0)] * ambient
v[i], v[i + 1] = F(1), F(-1)
out.append(tuple(v))
return out
# ---------------------------------------------------------------------------
# the algebra table (name, dim, h_vee, h, rank, roots, basis, fund weights, i_R)
# ---------------------------------------------------------------------------
def build_algebras():
E8 = roots_e8()
e8_basis = []
for i in range(5):
v = [F(0)] * 9
v[i] = F(1)
e8_basis.append(tuple(v))
for i in range(3):
v = [F(0)] * 9
v[5 + i], v[6 + i] = F(1), F(-1)
e8_basis.append(tuple(v))
delta1 = tuple([F(1), F(1)] + [F(0)] * 7) # D5 root (1,1,0,0,0)
delta2 = tuple([F(0), F(-1), F(1)] + [F(0)] * 6) # D5 root (0,-1,1,0,0)
e7_roots = [r for r in E8 if dot(r, delta1) == 0]
e7_basis = [tuple([F(1), F(-1)] + [F(0)] * 7)] + \
[e8_basis[i] for i in (2, 3, 4, 5, 6, 7)]
e7_56 = [r for r in E8 if dot(r, delta1) == 1]
e6_roots = [r for r in E8
if dot(r, delta1) == 0 and dot(r, delta2) == 0]
e6_basis = [tuple([F(1), F(-1), F(-1)] + [F(0)] * 6)] + \
[e8_basis[i] for i in (3, 4, 5, 6, 7)]
e6_27 = [r for r in E8 if dot(r, delta1) == 1 and dot(r, delta2) == 0]
g2_long, g2_short = roots_g2()
f4_long, f4_short = roots_f4()
sl2_fund = [(F(1, 2), F(-1, 2)), (F(-1, 2), F(1, 2))]
sl3_fund = [(1, 0, 0), (0, 1, 0), (0, 0, 1)]
so8_vec = []
for i in range(4):
for s in (1, -1):
v = [0] * 4
v[i] = s
so8_vec.append(tuple(v))
return [
("sl2", 3, 2, 2, 1, roots_sl2(), basis_sumzero(2, 2), sl2_fund, 1),
("sl3", 8, 3, 3, 2, roots_sl3(), basis_sumzero(3, 3), sl3_fund, 1),
("g2", 14, 4, 6, 2, g2_long + g2_short, basis_sumzero(3, 3),
g2_short, 2),
("so8", 28, 6, 6, 4, roots_so8(),
[tuple(F(1) if j == i else F(0) for j in range(4))
for i in range(4)], so8_vec, 2),
("f4", 52, 9, 12, 4, f4_long + f4_short,
[tuple(F(1) if j == i else F(0) for j in range(4))
for i in range(4)], f4_short, 6),
("e6", 78, 12, 12, 6, e6_roots, e6_basis, e6_27, 6),
("e7", 133, 18, 18, 7, e7_roots, e7_basis, e7_56, 12),
("e8", 248, 30, 30, 8, E8, e8_basis, E8, 60),
]
def section1(algs):
print(" -- S1: exact root systems + Killing normalisation (theta^2 = 2)")
counts = {name: len(roots)
for (name, _, _, _, _, roots, _, _, _) in algs}
ok_counts = all(len(roots) == dim - rank
for (_, dim, _, _, rank, roots, _, _, _) in algs)
check("S1 [E]: root counts = dim - rank for all 8 algebras on Costello's "
"list: %s" % counts, ok_counts)
ok_e67 = (counts["e6"] == 72 and counts["e7"] == 126
and len(algs[5][7]) == 27 and len(algs[6][7]) == 56)
check("S1 [E]: e6/e7 carved out of the v492 E8 root set: 72 = |e6 roots|, "
"126 = |e7 roots|, branching slices |27| = 27 (e6), |56| = 56 (e7) "
"-- adjoint e8 = (78,1)+(1,8)+(27,3)+(27b,3b) resp. "
"(133,1)+(1,3)+(56,2) at the root level", ok_e67)
ok_norm = True
for (name, dim, hv, _, _, roots, basis, _, _) in algs:
K, _ = power_sums(roots, basis)
Gq = gram_quad(basis)
ok_norm &= poly_eq_scaled(K, Gq, 2 * hv, 1)
check("S1 [E]: KILLING = 2 h_vee <.,.> as an exact POLYNOMIAL identity in "
"x for all 8 (h_vee = 2,3,4,6,9,12,18,30) -- the theta^2 = 2 "
"normalisation and the dual Coxeter numbers are pinned", ok_norm)
def section2(algs):
print(" -- S2: the quartic (Okubo) identity, derived exactly")
ok_okubo = True
for (name, dim, hv, _, _, roots, basis, _, _) in algs:
K, Q = power_sums(roots, basis)
K2 = poly_square(K)
ok_okubo &= poly_eq_scaled(Q, K2, 5, 2 * (dim + 2))
check("S2 [E]: OKUBO IDENTITY DERIVED -- Sum_alpha <alpha,x>^4 = "
"[5/(2(dim g+2))] * (Sum_alpha <alpha,x>^2)^2 holds as an exact "
"polynomial identity for ALL 8 algebras (= no independent quartic "
"Casimir; the anomaly is a perfect square)", ok_okubo)
sl4_basis = basis_sumzero(4, 4)
K4, Q4 = power_sums(roots_sl4(), sl4_basis)
K42 = poly_square(K4)
k1 = (0, 0, 0, 0) # monomial s0^4
k2 = (0, 0, 2, 2) # monomial s0^2 s2^2
r1 = F(Q4[k1]) / F(K42[k1])
r2 = F(Q4[k2]) / F(K42[k2])
check("S2 [E]: NEGATIVE CONTROL -- sl4 (HAS an independent quartic "
"Casimir) fails the proportionality: Q/K^2 = %s on s0^4 but %s on "
"s0^2 s2^2: Costello's list is sharp, the identity is not a generic "
"accident" % (r1, r2), r1 == F(5, 32) and r2 == F(3, 32))
def section3(algs):
print(" -- S3: lambda_g in all conventions (exact)")
rows = {}
ok_dl = ok_lam = ok_idx = True
for (name, dim, hv, h, rank, roots, basis, fund, i_exp) in algs:
ok_dl &= (dim * (hv + 6) == 2 * (5 * hv - 6) * (hv + 1))
lam2_unit = F(10 * hv * hv, dim + 2)
ok_lam &= (lam2_unit == hv + 6)
Kf, _ = power_sums(fund, basis)
Gq = gram_quad(basis)
ok_idx &= poly_eq_scaled(Kf, Gq, i_exp, 1)
rows[name] = (hv, h, rank, lam2_unit, i_exp, F(hv + 6, i_exp * i_exp))
check("S3 [E]: DELIGNE-SERIES DIMENSION FORMULA -- dim g = 2(5h_vee-6)"
"(h_vee+1)/(h_vee+6) exactly for all 8: Costello's list IS the "
"Deligne exceptional series", ok_dl)
check("S3 [E]: UNIT-TRACE LAMBDA -- lambda~^2 = 10 h_vee^2/(dim g+2) = "
"h_vee + 6 EXACTLY for all 8: (8,9,10,12,15,18,24,36); fixes "
"Costello-Paquette eq. (1.5) [lambda = sqrt(10) h_vee/sqrt(dim+2)] "
"and gives Tr_adj X^4 = [(h_vee+6)/(4 h_vee^2)] (Tr_adj X^2)^2",
ok_lam)
check("S3 [E]: FUNDAMENTAL DYNKIN INDICES from explicit weight systems -- "
"tr_R X^2 = i_R <X,X> with i_R = (1,1,2,2,6,6,12,60) for "
"(2, 3, 7, 8v, 26, 27, 56, 248): exact polynomial identities (for "
"e8 the smallest faithful rep IS the adjoint, i = 2 h_vee = 60)",
ok_idx)
check("S3 [E]: LAMBDA_E8 EXACT -- fundamental(=adjoint)-trace convention: "
"lambda_e8^2 = 36/3600 = 1/100, lambda_e8 = 1/10; unit-trace "
"convention: lambda~_e8^2 = 36, lambda~_e8 = 6. Both perfect "
"squares of rationals",
rows["e8"][5] == F(1, 100) and rows["e8"][3] == 36)
check("S3 [E]: COSTELLO CROSS-CHECK -- his appendix-A samples "
"lambda^2(sl2) = 32/4 = 8 and lambda^2(sl3) = 36/4 = 9 reproduced "
"exactly in his own (fundamental-trace) convention",
rows["sl2"][5] == 8 and rows["sl3"][5] == 9)
w4_adj_so8 = 12 * 2 ** 4 # six weights +2, six weights -2
w2_fund_so8 = 8 # four weights each of +-1
lam2_so8 = F(w4_adj_so8, w2_fund_so8 ** 2)
check("S3 [E]: COSTELLO SO8 SLIP FLAGGED -- with HIS stated weight "
"content (fundamental: four each of +-1; adjoint: six each of +-2) "
"the sums are Sum w^4 = 192, (Sum w^2)^2 = 64: lambda^2(so8) = 3 "
"exactly (= (h_vee+6)/i^2 = 12/4, Okubo-consistent); the PRINTED "
"96/64 = 3/2 undercounts Lambda^2 C^4_+ (+) Lambda^2 C^4_- by a "
"factor 2",
lam2_so8 == 3 and rows["so8"][5] == 3 and lam2_so8 != F(3, 2))
misread = F(10 * 2, 3 + 2) # the "sqrt(10 h_vee)" reading for sl2
check("S3 [E]: CP FORMULA DISAMBIGUATED -- the reading lambda^2 = "
"10 h_vee/(dim+2) gives %s for sl2, contradicting the exact 8; the "
"reading lambda = sqrt(10) h_vee/sqrt(dim+2) gives h_vee+6 = 8 -- "
"the latter is the correct normalisation (h_vee outside the root)"
% misread, misread == 4 and misread != 8)
return rows
# ---------------------------------------------------------------------------
# S4/S5 -- alignment + look-elsewhere
# ---------------------------------------------------------------------------
def phi(n):
return sum(1 for k in range(1, n + 1) if sp.gcd(k, n) == 1)
def smooth235(n):
for p in (2, 3, 5):
while n % p == 0:
n //= p
return n == 1
def is_square(q):
q = F(q)
return sp.sqrt(sp.Rational(q.numerator, q.denominator)).is_rational
def section4(rows):
print(" -- S4: alignment against c3 = 1/(8 pi) (typed brutally honestly)")
hv, _, _, lam2_unit, _, lam2_fund = rows["e8"]
ratio2_unit = F(lam2_unit, 3) # (kappa/c3)^2, unit-trace CS
ratio2_fund = F(lam2_fund, 3) # (kappa^/c3)^2, adjoint-trace CS
check("S4 [E]: the GS coupling kappa = lambda_g/(8 pi sqrt 3) contains "
"c3 = 1/(8 pi) with (kappa/c3)^2 = lambda^2/3: e8 unit-trace = "
"36/3 = 12 EXACT; adjoint-trace = (1/100)/3 = 1/300 EXACT -- both "
"anchor rationals (12 = |mu4| N_fam, 300 = 2^2 3 5^2)",
ratio2_unit == 12 and ratio2_fund == F(1, 300)
and 12 == MU4 * N_fam and 300 == 2 ** 2 * 3 * 5 ** 2)
check("S4 [E]: kappa/c3 ITSELF IS IRRATIONAL -- 12 and 1/300 are not "
"squares of rationals (kappa/c3 = 2 sqrt3 resp. 1/(10 sqrt3)): the "
"pre-registered 'lambda_e8/sqrt(3) rational?' reading FAILS; only "
"the SQUARED alignment is exact. The sqrt(3) is the anomaly's own "
"(sqrt 48 = 4 sqrt 3 from the twistor box diagram, 48 = 2*4!)",
(not is_square(12)) and (not is_square(F(1, 300)))
and 48 == 2 * factorial(4))
check("S4 [E]: ANCHOR DECOMPOSITION OF THE E8 DATA -- h_vee = 30 = "
"|Z2|*N_fam*g_car = 2*3*5 (squarefree, all three TFPT atoms once); "
"phi(30) = 8 = rank(E8); dim+2 = 250 = 2*5^3; lambda~ = 6 = "
"|Z2|*N_fam; lambda_fund = 1/10 = 1/(|Z2|*g_car); lambda~^2 = 36 = "
"h_vee + 6",
hv == Z2 * N_fam * g_car and phi(30) == rankE8
and 250 == 2 * 5 ** 3 and lam2_unit == (Z2 * N_fam) ** 2
and lam2_fund == F(1, (Z2 * g_car) ** 2) and 36 == hv + 6)
check("S4 [C, TYPED]: the 8 pi in Costello's (1.2.4) is the 2 pi-"
"convention of the anomaly polynomial (his A.0.10 constant "
"C_YM = 1/((2 pi i)^{3/2} sqrt 48) descended to spacetime), NOT an "
"independent occurrence of the seam constant: the WP3 test is "
"'exact arithmetic alignment yes/no' ONLY -- it can never derive "
"c3 from celestial data (contract wording, honoured here)", True)
def section5(rows):
print(" -- S5: look-elsewhere -- ALL 8 candidates through the same "
"battery (the hard fence: alignment != selection)")
names = ["sl2", "sl3", "g2", "so8", "f4", "e6", "e7", "e8"]
tab = {}
for n in names:
hv, h, rank, lam2u, i_r, lam2f = rows[n]
tab[n] = (smooth235(hv), is_square(lam2u), is_square(lam2f),
hv % g_car == 0, phi(hv) == rank, phi(h) == rank)
n_smooth = sum(1 for n in names if tab[n][0])
check("S5 [E]: LOOK-ELSEWHERE (the brutal one) -- '(kappa/c3)^2 = "
"(h_vee+6)/3 is an exact rational' passes 8/8 BY CONSTRUCTION and "
"'h_vee is {2,3,5}-smooth' passes %d/8: the alignment test AS SUCH "
"has ZERO selective power for e8 -- every algebra on Costello's "
"list aligns this way" % n_smooth, n_smooth == 8)
ints = [n for n in names if tab[n][1]]
fundrats = [n for n in names if tab[n][2]]
check("S5 [E]: 'lambda~ integer' (h_vee+6 a perfect square) passes 2/8: "
"%s; 'lambda_fund rational' passes the SAME 2/8: %s -- e8 shares "
"this distinction with sl3 (9 = 3^2, 36 = 6^2 are the only squares "
"in (8,9,10,12,15,18,24,36))" % (ints, fundrats),
ints == ["sl3", "e8"] and fundrats == ["sl3", "e8"])
five = [n for n in names if tab[n][3]]
check("S5 [E]: 'g_car = 5 divides h_vee' passes 1/8: %s -- the ONLY test "
"in the battery that singles out e8 alone (30 is the unique h_vee "
"on the list containing the atom 5)" % five, five == ["e8"])
phivs = [n for n in names if tab[n][4]]
phihs = [n for n in names if tab[n][5]]
check("S5 [E]: 'phi(h_vee) = rank' passes 4/8 (%s); 'phi(h) = rank' "
"passes 5/8 (%s): the totient test is real but NOT e8-specific"
% (phivs, phihs),
phivs == ["sl2", "sl3", "g2", "e8"]
and phihs == ["sl2", "sl3", "g2", "f4", "e8"])
units = sorted(k for k in range(1, 30) if sp.gcd(k, 30) == 1)
exps = [1, 7, 11, 13, 17, 19, 23, 29]
prodd = 1
for e in exps:
prodd *= e + 1
check("S5 [E]: E8 EXTRAS (exact arithmetic; exponent list [C] standard "
"data) -- exponents of e8 = %s = the units mod 30 (the phi-set of "
"h_vee), sum = 120 = #positive roots; degrees product = %d = "
"|W(E8)|" % (exps, prodd),
units == exps and sum(exps) == 120 and prodd == 696729600)
joint = [n for n in names if tab[n][1] and tab[n][3] and tab[n][4]]
check("S5 [E]: JOINT TEST TYPED HONESTLY -- the conjunction (lambda~ "
"integer AND g_car | h_vee AND phi(h_vee) = rank) passes 1/8 (%s), "
"but the conjunction was assembled POST HOC from correlated "
"sub-tests: it quantifies where e8 sits, it is NOT a preregistered "
"discriminator. Honest summary: alignment alone 8/8, sharpest "
"single natural test 1/8 (atom 5 = g_car in h_vee)" % joint,
joint == ["e8"])
def section6():
print(" -- S6: kill-test K3 evaluation + verdict")
check("S6 [E] KILL-TEST K3 (contract: 'no exact alignment => the c3-"
"connection of the celestial route dies') -- K3 does NOT fire: "
"exact arithmetic alignment EXISTS ((kappa/c3)^2 = 12 resp. 1/300, "
"lambda_e8 = 6 resp. 1/10, all exact anchor rationals). BUT the "
"look-elsewhere battery (S5) shows the alignment format itself "
"passes 8/8: the c3-connection SURVIVES only as convention-level "
"compatibility plus genuine lambda-arithmetic -- it is NOT "
"e8-selective evidence; K3's scope is demoted accordingly", True)
check("S6 [C]: WHAT IS NEW AND EXACT HERE -- (i) the Okubo coefficient "
"5/(2(dim+2)) DERIVED as a polynomial identity for all 8 algebras "
"with a sharp sl4 negative control; (ii) the closed form lambda~^2 "
"= h_vee + 6 across the Deligne series; (iii) lambda_e8 = 1/10 "
"(adjoint units) resp. 6 (unit-trace) EXACT; (iv) the so8 factor-2 "
"slip in the printed source localised; (v) the honest pass-count "
"table (S5)", True)
check("S6 [O] FENCE -- WP3 verdict B: exact arithmetic alignment, "
"honestly typed as NON-SELECTIVE (the e8-specific content is "
"h_vee = 2*3*5 with phi = 8 = rank and lambda-integrality shared "
"only with sl3); no marker moves anywhere; WP5 (constructive "
"holography) and SEAM.EQUIV.01 stay [O], untouched", True)
def run():
reset()
print("v495 CELEST.WP3.01: Green-Schwarz axion coefficient lambda_g "
"(Costello) vs the TFPT seam constant c3 = 1/(8 pi) (WP3 of "
"CELEST.SEAM.01 -- alignment test only)")
algs = build_algebras()
section1(algs)
section2(algs)
rows = section3(algs)
section4(rows)
section5(rows)
section6()
return summary("v495 CELEST.WP3.01: Okubo coefficient 5/(2(dim+2)) derived for all 8 "
"algebras (sl4 negative control); lambda~^2 = h_vee + 6 across the Deligne "
"series; lambda_e8 = 6 (unit-trace) resp. 1/10 (adjoint-trace) exact; "
"(kappa/c3)^2 = 12 = |mu4| N_fam resp. 1/300 exact, kappa/c3 irrational; "
"so8 factor-2 slip flagged; look-elsewhere: alignment 8/8 (zero selectivity), "
"g_car | h_vee 1/8 (e8 only), lambda-integrality 2/8 (with sl3), joint test "
"post hoc. K3 does not fire, scope demoted: compatibility, not e8-selective "
"evidence. Verdict B; SEAM.EQUIV.01 untouched")
if __name__ == "__main__":
raise SystemExit(1 if run() else 0)
Recorded source fingerprint
5c99ef09765e691924fd15ec669526137500ebddc1a9fa88464cab64329d9685
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