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v136_dual_normal_selector.py
verification/v136_dual_normal_selector.py
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"""v136 -- The dual-normal selector: (d, n) pins R COLUMNWISE (each
column the unique lattice point in the address box), the A0* zero mode
carries the exact spectral normal (2,-9,5) with two new atoms, and the
W(A3) orbit control shows the lift sigma -> n is a genuine mechanism,
not a group element. [I] exact + brute-force census + [P] recorded
programme (external review 2026-06-12, validated and sharpened).
[I] 1. COLUMN NORMAL FORM. For the columns c_j of R:
d . c_j = a_j and n . c_j = 8 delta_{1j},
and these two covectors FORCE each column individually: the
common kernel of (d, n) is the primitive vector (6, 8, 7)
(audit: p_2, rank E8, scalaron-7), so within the address box
{0..5}^3 each line v + t(6,8,7) contains AT MOST one lattice
point. Brute-force census (216 per column): c_1 = (1,1,2)
= a, c_2 = (3,5,5), c_3 = (0,2,3) are each the UNIQUE
solution. R4' shrinks from a 6^9 matrix census (v121) to
three 1-of-216 column problems: (d, n) => R.
[I] 2. WHAT (d, n) DOES NOT DO (honesty). K is NOT pinned directly:
d^T K = (1, 1/2, 1), n^T K = (14, 1, -6) -- no clean form.
det K = 4 follows from R via the diamond (K = M(1,-1) on the
mu_4 wall, v135), not from the dual pair.
[I] 3. THE SPECTRAL NORMAL OF A0*. The zero mode of the exact
anchor residue A0* (v115) is v = (-sqrt2, 3, -sqrt5)/3; its
signed squared components x 9 give EXACTLY
sigma = (2, -9, 5) = (|Z2|, -N_fam^2, g_car).
Two new atoms: ||v||^2 = 16/9 (numerator 16 = dim S+), and
adj(A0*) = (2/9) P_v with 2/9 = |Z2|/N_fam^2
-- the rank-1 adjugate carries the SdS entropy curvature
(v101/v106) as its scale.
[I] 4. SIGMA RELATIONS. sigma.1 = -2 = -|Z2|, sigma.a = 3 = N_fam,
and n . sigma = 121 = ||Pl(K)||_1^2 (the 121 audit constant,
v119) -- the torsion normal pairs with the spectral normal
to the squared quark constant.
[I] 5. ORBIT CONTROL (the honest lift status). Exact enumeration
of W(A3) = <U, M0> (24 elements, v117): the only REAL images
of sigma are its signed copies {(2,-9,5), (2,9,-5),
(-2,5,-9), (-2,-5,9)}; NO image is proportional to
n = (5,-9,6). The naive cofactor/orbit lift FAILS, exactly
as the review predicted: the step sigma -> n (the
|Z2|-to-|Z2|N_fam = 6 torsion lift + the carrier endpoint 5)
is a genuine mechanism beyond the monodromy group.
[P] 6. R4' RESTATED (recorded): "derive the pair (d, n); then R is
forced columnwise." d is already structural (a^T R^{-1},
Sherman-Morrison invariant, v134); the residue is the
derivation of n -- now sharply bounded by sigma.1/sigma.a,
the 121 pairing, and the negative orbit control.
"""
import itertools
import sympy as sp
from tfpt_constants import check, summary, reset
II = sp.I
R = sp.Matrix([[1, 3, 0], [1, 5, 2], [2, 5, 3]])
Q = sp.Matrix([[3, 1, 0], [3, 2, 0], [3, 2, 1]])
K = sp.Matrix([[4, 2, 0], [4, 3, 2], [5, 3, 2]])
A = sp.Matrix([1, 1, 2])
ONE = sp.ones(3, 1)
D = sp.Matrix([sp.Rational(-1, 2), sp.Rational(-1, 2), 1])
N = sp.Matrix([5, -9, 6])
SIGMA = sp.Matrix([2, -9, 5])
A0 = sp.Matrix([[sp.Rational(1, 2), sp.sqrt(2) / 6, 0],
[sp.sqrt(2) / 6, sp.Rational(1, 4), sp.sqrt(5) / 12],
[0, sp.sqrt(5) / 12, sp.Rational(1, 4)]])
U_MON = sp.diag(1, II, -II)
M0 = sp.Matrix([[0, -(1 + II) / 2, (1 - II) / 2],
[-(1 + II) / 2, -II / 2, sp.Rational(-1, 2)],
[(1 - II) / 2, sp.Rational(-1, 2), II / 2]])
def _key(m):
# entries live in (1/2)Z[i]; canonicalise as exact (re, im) pairs
return tuple((sp.nsimplify(sp.re(x)), sp.nsimplify(sp.im(x)))
for x in sp.expand(m))
def weyl_a3():
elems = [sp.eye(3)]
seen = {_key(sp.eye(3))}
frontier = [sp.eye(3)]
while frontier:
new = []
for g in frontier:
for h in (U_MON, M0):
p = sp.expand(g * h)
k = _key(p)
if k not in seen:
seen.add(k)
elems.append(p)
new.append(p)
frontier = new
return elems
def run():
reset()
print("v136 dual-normal selector ((d,n) => R columnwise)")
# 1. column normal form
cols = [R[:, j] for j in range(3)]
ker = sp.Matrix([[*D.T], [*N.T]]).nullspace()[0]
ker = ker * sp.lcm([x.q for x in ker])
census = []
for aj, nj in [(1, 8), (1, 0), (2, 0)]:
sols = [v for v in itertools.product(range(6), repeat=3)
if (D.T * sp.Matrix(v))[0] == aj
and (N.T * sp.Matrix(v))[0] == nj]
census.append(sols)
check("COLUMN NORMAL FORM: d.c_j = a_j and n.c_j = 8 delta_1j for "
"the columns of R; common kernel of (d,n) = primitive "
"(6,8,7) (audit: p_2, rank E8, scalaron-7); brute-force "
"census over the address box {0..5}^3: each column is the "
"UNIQUE solution (1 of 216) -- (d,n) => R, and R4' shrinks "
"from a 6^9 census to three 1-of-216 column problems",
[(D.T * c)[0] for c in cols] == [1, 1, 2]
and [(N.T * c)[0] for c in cols] == [8, 0, 0]
and (ker == sp.Matrix([6, 8, 7])
or ker == -sp.Matrix([6, 8, 7]))
and [len(s) for s in census] == [1, 1, 1]
and [sp.Matrix(s[0]) for s in census] == cols)
# 2. honesty: K is not pinned directly
check("WHAT (d,n) DOES NOT DO (honesty): d^T K = (1, 1/2, 1) and "
"n^T K = (14, 1, -6) -- K is NOT pinned by the dual pair; "
"det K = 4 follows from R via the diamond (K = M(1,-1) on "
"the mu_4 wall, v135)",
list(D.T * K) == [1, sp.Rational(1, 2), 1]
and list(N.T * K) == [14, 1, -6]
and K == R + Q * sp.diag(1, -1, -1) and K.det() == 4)
# 3. spectral normal of A0*
v0 = A0.nullspace()[0]
v0 = sp.simplify(v0 / v0[1] * 3) # -> (-sqrt2, 3, -sqrt5)
signed_sq = sp.Matrix([sp.sign(x) * sp.simplify(x ** 2) for x in v0])
nrm = sp.simplify((v0.T * v0)[0] / 9) # back to v = v0/3
proj = sp.simplify((v0 * v0.T) / (v0.T * v0)[0])
check("SPECTRAL NORMAL OF A0*: zero mode v = (-sqrt2, 3, -sqrt5)/3; "
"signed squares = (-2, 9, -5) = -sigma with sigma = (2,-9,5) "
"= (|Z2|, -N_fam^2, g_car); NEW ATOMS: ||v||^2 = 16/9 "
"(numerator 16 = dim S+) and adj(A0*) = (2/9) P_v with "
"2/9 = |Z2|/N_fam^2 = the SdS entropy curvature (v101/v106)",
signed_sq == -SIGMA
and nrm == sp.Rational(16, 9)
and sp.simplify(A0.adjugate() - sp.Rational(2, 9) * proj)
== sp.zeros(3))
# 4. sigma relations
check("SIGMA RELATIONS: sigma.1 = -2 = -|Z2|, sigma.a = 3 = N_fam, "
"n.sigma = 121 = ||Pl(K)||_1^2 = 11^2 (the v119 audit "
"constant) -- the torsion normal pairs with the spectral "
"normal to the squared quark constant",
(SIGMA.T * ONE)[0] == -2 and (SIGMA.T * A)[0] == 3
and (N.T * SIGMA)[0] == 121 and 121 == 11 ** 2)
# 5. orbit control
grp = weyl_a3()
reals = set()
prop_to_n = False
for g in grp:
img = sp.simplify(g * SIGMA)
if all(sp.im(x) == 0 for x in img):
reals.add(tuple(img))
if img.cross(N) == sp.zeros(3, 1):
prop_to_n = True
check("ORBIT CONTROL (honest): |W(A3)| = 24 exactly; the only "
"REAL orbit images of sigma are its signed copies "
"{(2,-9,5), (2,9,-5), (-2,5,-9), (-2,-5,9)}; NO image is "
"proportional to n = (5,-9,6) -- the naive cofactor/orbit "
"lift FAILS (as predicted); the step sigma -> n is a "
"genuine mechanism beyond the monodromy group",
len(grp) == 24 and not prop_to_n
and reals == {(2, -9, 5), (2, 9, -5), (-2, 5, -9),
(-2, -5, 9)})
# 6. restated programme
check("R4' RESTATED [P] (recorded): derive the PAIR (d, n); then "
"R is forced columnwise. d is already structural (v134); "
"the residue is the derivation of n -- bounded by "
"sigma.1 = -|Z2|, sigma.a = N_fam, the 121 pairing, and "
"the negative orbit control", True)
return summary("v136 dual-normal selector")
if __name__ == "__main__":
raise SystemExit(1 if run() else 0)
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