Follow the source behind the result.

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

"""v413 -- The sheet axis is a discrete characteristic-difference calculator.
On the sheet axis M_t = C + tV (v95/v218) the three elementary-symmetric
coefficients of the characteristic polynomial encode the carrier atoms not as
VALUES but as DIFFERENCE ORDERS, and the anchor energy advances in steps of
the amplitude 11 = a^T V a (v410).  [E] algebra, [C] reading.

  [E] 1. ELEMENTARY-SYMMETRIC CALCULUS.  chi_{M_t} = lambda^3 - e_1 lambda^2 +
         e_2 lambda - e_3 with
             e_1(t) = 3t + 12,
             e_2(t) = 2t^2 + 15t + 25,
             e_3(t) = det M_t = 4t^2 + 14t + 14,
         so the carrier atoms appear as difference orders:
             Delta e_1   = 3 = N_fam        (first difference, linear trace),
             Delta^2 e_2 = 4 = |mu4|        (second difference of the
                                             surface coefficient),
             Delta^2 e_3 = 8 = rank E_8     (second difference of the volume;
                                             the curvature already in v218/v224).
         The NEW content here is the full triple (e_1, e_2, e_3) calculus:
         N_fam -> |mu4| -> rank E_8 as a discrete derivative ladder.
  [E] 2. ANCHOR-ENERGY LADDER.  The anchor energy is exactly linear with the
         amplitude 11 = a^T V a (v410):
             a^T M_t a = 52 + 11 t,
             30 --(+11)--> 41 --(+11)--> 52 --(+11)--> 63,
         the four steps reading (h(E_8), 10 b_1, dim F_4, 7 N_fam^2): the 11 is
         the STEP of the anchor energy along the sheet axis, not just the quark
         Pluecker norm.
  [E] 3. THE FOUR-ROW SHEET AXIS J,K,C,F = M(1, -2..1):
             t=-2 J: (tr,e_2,det)=(6,3,2),  a^T M a = 30
             t=-1 K: (9,12,4),              41
             t= 0 C: (12,25,14),            52
             t=+1 F: (15,42,32),            63

HONEST TYPING.  All coefficients/differences are exact [E]; Delta^2 e_3 = 8
overlaps the v218/v224 sheet-axis curvature and is cross-cited, not re-claimed.
The atom/Lie readings (10 b_1 = 41, dim F_4 = 52, 7 N_fam^2 = 63) are [C],
audit-typed.  Mirrored in wolfram/tfpt_readouts_extension.wl.
"""
import sympy as sp

from tfpt_constants import check, summary, reset, N_fam, rankE8, b1

R = sp.Matrix([[1, 3, 0], [1, 5, 2], [2, 5, 3]])
Q = sp.Matrix([[3, 1, 0], [3, 2, 0], [3, 2, 1]])
C = R + Q * sp.diag(1, 0, 0)
V = Q * sp.diag(0, 1, 1)
ONE = sp.Matrix([1, 1, 1])
A = sp.Matrix([1, 1, 2])

MU4 = 4
H_E8 = 30
DIM_F4 = 52


def run():
    reset()
    print("v413 sheet axis as a characteristic-difference calculator")

    t, lam = sp.symbols('t lambda')
    Mt = C + t * V
    chi = sp.Poly((lam * sp.eye(3) - Mt).det(), lam)
    e1 = sp.expand(-chi.nth(2))
    e2 = sp.expand(chi.nth(1))
    e3 = sp.expand(-chi.nth(0))
    d1 = lambda f: sp.expand(f.subs(t, t + 1) - f)

    check("ELEMENTARY-SYMMETRIC CALCULUS [E]: e_1(t)=3t+12, e_2(t)=2t^2+15t+25, "
          "e_3(t)=det M_t=4t^2+14t+14 => Delta e_1 = 3 = N_fam, Delta^2 e_2 = "
          "4 = |mu4|, Delta^2 e_3 = 8 = rank E_8 (the e_3 curvature cross-cited "
          "to v218/v224; the full (e_1,e_2,e_3) ladder is the new content)",
          e1 == 3 * t + 12 and e2 == 2 * t**2 + 15 * t + 25
          and e3 == 4 * t**2 + 14 * t + 14
          and d1(e1) == N_fam and d1(d1(e2)) == MU4 and d1(d1(e3)) == rankE8)

    energy = sp.expand((A.T * Mt * A)[0])
    check("ANCHOR-ENERGY LADDER [E]: a^T M_t a = 52 + 11 t (step = 11 = "
          "a^T V a, v410); 30 -> 41 -> 52 -> 63 reads (h(E_8), 10 b_1, "
          "dim F_4, 7 N_fam^2) -- the 11 is the STEP of the anchor energy "
          "along the sheet axis",
          energy == 52 + 11 * t
          and [int(energy.subs(t, tv)) for tv in (-2, -1, 0, 1)]
          == [H_E8, int(10 * b1), DIM_F4, 7 * N_fam**2]
          and int(10 * b1) == 41 and 7 * N_fam**2 == 63)

    rows = {}
    for tv, name in [(-2, 'J'), (-1, 'K'), (0, 'C'), (1, 'F')]:
        Mv = C + tv * V
        rows[name] = (Mv.trace(), int(e2.subs(t, tv)), Mv.det(),
                      int((A.T * Mv * A)[0]))
    check("FOUR-ROW SHEET AXIS J,K,C,F = M(1,-2..1): (tr,e_2,det)/a^TMa = "
          "(6,3,2)/30, (9,12,4)/41, (12,25,14)/52, (15,42,32)/63",
          rows['J'] == (6, 3, 2, 30) and rows['K'] == (9, 12, 4, 41)
          and rows['C'] == (12, 25, 14, 52) and rows['F'] == (15, 42, 32, 63))

    return summary("v413 sheet characteristic calculus (3->4->8 differences, "
                   "anchor energy 52+11t)")


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

01d451781ba19dcd0064dc85aefc2eda30fdf441d85985d2a9731902a7dc8555

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.