Follow the source behind the result.
Search the paper sections, open their original context and follow the checks they actually cite. The knowledge atlas explains the result; this library locates its source.
v414_center_resolvent_portal.py
verification/v414_center_resolvent_portal.py
This is a registered verification module. Read its original code and the paper sections that cite it, then follow the linked claim to its present classification.
Read the publication code on this page
The registered Python source is shown as text. Displaying it does not execute the code.
"""v414 -- The center C is a resolvent portal G_2 -> F_4 -> E_8. The center of
the Sheet Diamond (C = M(1,0), v95) is the t = 0 evaluation of the sheet
generator V; its shifted determinants climb the exceptional resolvent ladder
14 -> 52 -> 120 = (dim G_2, dim F_4, |R^+(E_8)|). [E] algebra, [C] reading.
[E] 1. CENTER INVARIANTS (anchored to v95):
tr C = 12 = dim g_SM = chi_{E_8}(i),
det C = 14 = dim G_2,
sum_{ij} C_{ij} = 31 = 2^{g_car} - 1 = 248/8.
[E] 2. RESOLVENT LADDER. The shifted determinants give
det(C) = 14 = dim G_2,
det(I + C) = 52 = dim F_4,
det(2I + C) = 120 = |R^+(E_8)|,
i.e. the exceptional shell ladder G_2 -> F_4 -> E_8^+ as the
resolvent of one operator C at shifts 0, 1, 2.
HONEST TYPING. tr C, det C and sum C are the v95 center invariants
(cross-cited, not re-claimed); the NEW content is the resolvent ladder
det(C + kI) for k = 0,1,2. All values are exact [E]; the
"G_2 -> F_4 -> E_8 portal" reading is [C], audit-typed exactly like the G_2
center / F_4 shadow of v95/v218. Mirrored in wolfram/tfpt_readouts_extension.wl.
"""
import sympy as sp
from tfpt_constants import check, summary, reset, g_car
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)
ONE = sp.Matrix([1, 1, 1])
I3 = sp.eye(3)
DIM_GSM = 12
DIM_G2 = 14
DIM_F4 = 52
RP_E8 = 120
def run():
reset()
print("v414 center resolvent portal G_2 -> F_4 -> E_8")
check("CENTER INVARIANTS [E] (v95): tr C = 12 = dim g_SM = chi_{E_8}(i), "
"det C = 14 = dim G_2, sum C = 31 = 2^g_car - 1 = 248/8",
C.trace() == DIM_GSM and C.det() == DIM_G2
and (ONE.T * C * ONE)[0] == 31 == 2**g_car - 1 == 248 // 8)
check("RESOLVENT LADDER [E]: det(C) = 14 = dim G_2, det(I+C) = 52 = "
"dim F_4, det(2I+C) = 120 = |R^+(E_8)| -- the exceptional shell "
"ladder G_2 -> F_4 -> E_8^+ as det(C + kI), k = 0,1,2 (reading [C], "
"audit-typed)",
C.det() == DIM_G2 and (I3 + C).det() == DIM_F4
and (2 * I3 + C).det() == RP_E8)
return summary("v414 center resolvent portal (14 -> 52 -> 120)")
if __name__ == "__main__":
raise SystemExit(1 if run() else 0)
Recorded source fingerprint
5b2779034bd729184a2e40ab5304c1c473dd9c429300d2a36f584753453a6c70
Paper sections that cite this module
What this source index covers
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.