"""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) = (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)