Follow the source behind the result.

The registered Python source is shown as text. Displaying it does not execute the code.

"""v149 -- BOTH dual normals are anchor data: in the cusp frame the
torsion normal pairs to (p_2, p_0, e_2)(a) = (6, 3, 5) -- pure anchor
invariants, one per Q_+ line -- just as d = (3/2)a - 2*1 is anchor data
in the generation frame.  R4' collapses to ONE discrete assignment
(which anchor invariant sits on which cusp line); no continuous or
integer freedom remains.  [I] exact; the assignment derivation is the
residue [P].

v139 derived d as pure anchor data and left n as 'frame data plus
three atoms'; v142/v145 reduced the values to atom identities given a
lift map.  Reading n in the CANONICAL frame -- the cusp eigenbasis
(e_1, e_2, e_3) of the established integer geometry (v98) -- removes
the remaining opacity:

  [I] 1. THE CUSP WEIGHTS OF n ARE ANCHOR INVARIANTS.  The pairings of
         n = (5,-9,6) with the cusp eigenbasis are EXACTLY
             n.e_1 = 6 = p_2(a),   n.e_2 = 3 = p_0(a),
             n.e_3 = 5 = e_2(a)
         -- the anchor's second power sum, zeroth power sum and second
         elementary symmetric function, one per Q_+ degree; and a
         covector is uniquely determined by its three cusp pairings
         (the cusp matrix is invertible over Z).
  [I] 2. BOTH NORMALS ARE ANCHOR DATA.  d = (3/2) a - 2*1 in the
         generation frame (v139); n = (p_2, p_0, e_2)(a) in the cusp
         frame: the dual normal PAIR of the flavor boundary (v134) is
         anchor data through and through -- one normal per frame.
  [I] 3. EQUIVALENCE.  The cusp data (6,3,5) reproduce the
         (1, a, sigma)-frame pairings (2, 8, 121) (v139/v142) and the
         w_0-lift normal form (v145) exactly -- three presentations of
         one covector, all now atomic.
  [I] 4. AUDIT observations (recorded, not promoted): the spectral
         normal has cusp pairings sigma -> (5, 1, 2); the alternate
         atomic reading (6,3,5) = (|R+(A3)|, N_fam, g_car) (hexagon /
         family / carrier); the cusp-weight sum 6+3+5 = 14 = dim G_2
         (the v80 motif).
  [P] 5. RESIDUE (recorded): R4' = derive the ASSIGNMENT -- why the
         Q_+ degrees (1, 2, 3) carry (p_2, p_0, e_2) in that order.
         One discrete assignment; zero continuous freedom; all values
         are anchor invariants.
"""
import sympy as sp

from tfpt_constants import check, summary, reset, N_fam, g_car

N = sp.Matrix([5, -9, 6])
D = sp.Matrix([sp.Rational(-1, 2), sp.Rational(-1, 2), 1])
SIGMA = sp.Matrix([2, -9, 5])
A = sp.Matrix([1, 1, 2])
ONE = sp.Matrix([1, 1, 1])
E1, E2, E3 = sp.Matrix([0, 0, 1]), sp.Matrix([0, 1, 2]), sp.Matrix([1, 0, 0])
P = sp.Matrix.hstack(E1, E2, E3)


def run():
    reset()
    print("v149 cusp normal (both dual normals are anchor data)")

    p2 = sum(x ** 2 for x in A)
    p0 = sp.Integer(3)
    e2 = A[0] * A[1] + A[0] * A[2] + A[1] * A[2]
    cusp = (P.T * N)
    check("THE CUSP WEIGHTS OF n ARE ANCHOR INVARIANTS: n.e_1 = 6 = "
          "p_2(a), n.e_2 = 3 = p_0(a), n.e_3 = 5 = e_2(a) -- one "
          "anchor invariant per Q_+ degree; the cusp matrix is "
          "unimodular-invertible, so the three pairings determine n "
          "uniquely",
          cusp == sp.Matrix([6, 3, 5])
          and (p2, p0, e2) == (6, 3, 5)
          and (P.T).solve(sp.Matrix([6, 3, 5])) == N
          and abs(P.det()) == 1)

    check("BOTH NORMALS ARE ANCHOR DATA: d = (3/2)a - 2*1 in the "
          "generation frame (v139); n = (p_2, p_0, e_2)(a) in the "
          "cusp frame -- the dual normal pair (v134) is anchor data "
          "through and through",
          D == sp.Rational(3, 2) * A - 2 * ONE)

    frame_pairings = (N.dot(ONE), N.dot(A), N.dot(SIGMA))
    w0 = sp.Matrix([SIGMA[2], SIGMA[1], SIGMA[0]])
    check("EQUIVALENCE: the cusp data reproduce the (1,a,sigma) "
          "pairings (2,8,121) (v139/v142) and the w_0-lift "
          "n = w_0(sigma) + |mu_4| e_3 (v145) exactly -- three "
          "presentations of one covector, all atomic",
          frame_pairings == (2, 8, 121)
          and w0 + 4 * sp.Matrix([0, 0, 1]) == N)

    check("AUDIT (recorded, not promoted): sigma -> cusp pairings "
          "(5, 1, 2); alternate reading (6,3,5) = (|R+(A3)|, N_fam, "
          "g_car); cusp-weight sum 6+3+5 = 14 = dim G_2 (v80 motif)",
          (P.T * SIGMA) == sp.Matrix([5, 1, 2])
          and (6, 3, 5) == (6, N_fam, g_car)
          and 6 + 3 + 5 == 14)

    check("RESIDUE [P] (recorded): R4' = derive the ASSIGNMENT (why "
          "degrees (1,2,3) carry (p_2, p_0, e_2) in that order) -- "
          "one discrete assignment, zero continuous freedom, all "
          "values anchor invariants", True)

    return summary("v149 cusp normal")


if __name__ == "__main__":
    raise SystemExit(1 if run() else 0)

8d9b60844576b7477e44ec4e87639c4bfc45d387551f186d2e63e0fedbd8b7ab

The public registry includes 116 document views, 6,693 section records and 985 physical LaTeX sources. Active publication documents, research material and archives retain their recorded roles. This is source navigation, not a new semantic review or proof run.