Follow the source behind the result.

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

"""v479 -- SEAM.KRONHEIMER.01: the R3 "combinatorial-to-geometric bridge" of the
det K = 1 keystone (SEAM.DETK.01/v344) is an ESTABLISHED THEOREM -- Kronheimer's
ALE hyper-Kaehler quotient construction + Torelli classification (J. Differential
Geom. 29 (1989) 665-683 and 685-697) -- with ALL of its finite input data already
TFPT compiler outputs.  A citable REDUCTION in the same genre as the NPW26 / MMST /
AGT imports (v424/v336/v459); it does NOT close SEAM.EQUIV.01.

Background.  SEAM.DETK.01 (v344) ranked the hypergraph route R3 as the most
promising access to the one open keystone bit det K = 1 and named its residual:
"the combinatorial-to-geometric bridge -- the rewrite attractor (a graph) IS the
du Val singularity".  That bridge is EXACTLY Kronheimer's theorem pair:

  (K-A) CONSTRUCTION: given the McKay graph with dimension vector delta = the Kac
        marks (= irrep dimensions of Gamma), the hyper-Kaehler quotient of the
        flat double-quiver representation space by G = prod U(delta_i)/U(1) at
        moment-map level zero IS C^2/Gamma; nonzero levels give its ALE
        deformations/resolutions.
  (K-B) TORELLI/CLASSIFICATION: every ALE hyper-Kaehler 4-manifold is one of
        these quiver varieties; the minimal resolution's intersection form is
        minus the ADE Cartan matrix.

TFPT already outputs the graph and the marks: the (2,3,5) rewrite attractor is the
affine-E8 McKay graph and the Kac marks are its Perron vector (v312/v298); the
seam signature (|Z2|, N_fam, g_car) = (2,3,5) forces Gamma = 2I (v346 and the
CORR.FORCING contract).  This module machine-checks every FINITE ingredient of
Kronheimer's input data from the compiler atoms:

  [E] 1. MARKS = NULL VECTOR = PERRON VECTOR: C_aff delta = 0 and A delta =
        2 delta for the affine-E8 graph; multiset {1,2,2,3,3,4,4,5,6} = the 2I
        irrep dimensions (McKay) = the v312 attractor Perron vector.
  [E] 2. KRONHEIMER DIMENSION COUNT = 4: dim_R M - 4 dim_R G = 4(sum_edges
        d_i d_j - sum d_i^2 + 1) = 4 -- the quotient is a 4-real-dimensional
        space (the C^2/Gamma geometry); sum_edges d_i d_j = sum d_i^2 = 120 is
        FORCED by the null-vector identity.
  [E] 3. GAUGE DATUM = |2I|: sum delta_i^2 = 120 = |mu4| h(E8) = 4*30, with
        h(E8) = 30 = |Z2| N_fam g_car -- the quiver's group size is the seam's
        own binary-icosahedral order.
  [E] 4. RESOLUTION DATA: rank(E8) = 8 exceptional spheres, intersection form
        -Cartan(E8) unimodular (det = 1 -- the Poincare-sphere link, the det K=1
        face, v232/v344); hyper-Kaehler deformation space h (x) R^3, dim 3*8=24.
  [C] 5. The theorems (K-A)/(K-B) are CITED, not re-proved (same import class as
        NPW26/MMST/AGT, v424/v336/v459).
  [O] 6. The residual premise TRANSFORMS, it does not vanish: "the raw seam
        supplies ALE hyper-Kaehler data asymptotic to the (2,3,5) cone" -- the
        SAME single arrow as SEAM.EQUIV.01 (v346 L2).  NO gate closes; no marker
        moves.

NET EFFECT: R3's "combinatorial-to-geometric bridge" residual (v344/v345) is
discharged to the literature; the keystone's open content is now concentrated in
ONE premise reachable from BOTH sides (net-side: scaling limit v336/v458; geometry
side: Kronheimer datum).  Exact integer/symbolic arithmetic (sympy); the Wolfram
mirror is added to tfpt_readouts_extension.wl (counted with the next verified
engine run -- see wolfram/README.md).
"""
import sympy as sp
from sympy.matrices.normalforms import smith_normal_form

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

Z2 = 2
MU4 = 4
H_E8 = 30

# affine E8: nodes 1..8 = finite E8 (Bourbaki), node 9 = affine node on node 8.
FINITE_EDGES = [(1, 3), (3, 4), (4, 5), (5, 6), (6, 7), (7, 8), (2, 4)]
AFFINE_EDGES = FINITE_EDGES + [(8, 9)]
KAC_MARKS = sp.Matrix([2, 3, 4, 6, 5, 4, 3, 2, 1])


def adjacency(n, edges):
    A = sp.zeros(n, n)
    for a, b in edges:
        A[a - 1, b - 1] = A[b - 1, a - 1] = 1
    return A


def run():
    reset()
    print("v479  SEAM.KRONHEIMER.01: the R3 graph->geometry bridge is Kronheimer's theorem")

    # 1. marks = null vector = Perron vector
    A = adjacency(9, AFFINE_EDGES)
    C_aff = 2 * sp.eye(9) - A
    null_ok = (C_aff * KAC_MARKS) == sp.zeros(9, 1)
    perron_ok = (A * KAC_MARKS) == 2 * KAC_MARKS
    multiset_ok = sorted(KAC_MARKS) == [1, 2, 2, 3, 3, 4, 4, 5, 6]
    check("MARKS = NULL = PERRON [E]: C_aff delta = 0 and A delta = 2 delta on the "
          "affine-E8 McKay graph; multiset {1,2,2,3,3,4,4,5,6} = 2I irrep dims = the "
          "v312 rewrite-attractor Perron vector",
          null_ok and perron_ok and multiset_ok)

    # 2. Kronheimer hyper-Kaehler dimension count
    d = list(KAC_MARKS)
    e_sum = sum(d[a - 1] * d[b - 1] for a, b in AFFINE_EDGES)
    sq_sum = sum(x * x for x in d)
    dimM = 4 * e_sum                    # double quiver, quaternionic
    dimG = sq_sum - 1                   # prod U(d_i) / U(1)
    dimX = dimM - 4 * dimG
    check("KRONHEIMER DIMENSION COUNT [E]: dim_R M - 4 dim_R G = 4(sum_edges d_i d_j "
          "- sum d_i^2 + 1) = 4 -- the hyper-Kaehler quotient of the marks quiver is "
          "4-real-dimensional (the C^2/Gamma geometry); sum_edges d_i d_j = "
          "sum d_i^2 = 120 forced by the null-vector identity",
          e_sum == sq_sum == 120 and dimX == 4)

    # 3. gauge datum = |2I| = 120 = |mu4| h(E8), h = 2*3*5
    check("GAUGE DATUM = |2I| [E]: sum delta_i^2 = 120 = |mu4| h(E8) = 4*30, "
          "h(E8) = 30 = |Z2| N_fam g_car = 2*3*5 -- the quiver group size is the "
          "seam's own binary icosahedral order (regular representation)",
          sq_sum == 120 == MU4 * H_E8 and H_E8 == Z2 * N_fam * g_car)

    # 4. resolution data: -E8 intersection form, unimodular; 3*rank deformations
    Afin = adjacency(8, FINITE_EDGES)
    Cfin = 2 * sp.eye(8) - Afin
    det1 = int(Cfin.det()) == 1
    snf_id = smith_normal_form(Cfin) == sp.eye(8)
    check("RESOLUTION DATA [E]: 8 = rank(E8) exceptional spheres with intersection "
          "form -Cartan(E8), det = 1 and SNF = identity (the Poincare-sphere link = "
          "the det K = 1 face, v232/v344); deformation space h (x) R^3 has dim "
          "3*8 = 24",
          det1 and snf_id and 3 * rankE8 == 24)

    # 5./6. typing
    check("CITED THEOREMS [C]: Kronheimer (K-A) construction + (K-B) Torelli "
          "(J. Diff. Geom. 29 (1989) 665 and 685) are IMPORTED, not re-proved -- the "
          "same citation class as NPW26/MMST/AGT (v424/v336/v459)", True)
    check("RESIDUAL TRANSFORMS, NOT CLOSED [O]: the open premise becomes 'the raw "
          "seam supplies ALE hyper-Kaehler data asymptotic to the (2,3,5) cone' -- "
          "the SAME single arrow as SEAM.EQUIV.01 (v346 L2); no gate closes, no "
          "marker moves", True)

    return summary("v479 SEAM.KRONHEIMER.01: R3's combinatorial-to-geometric bridge "
                   "discharged to Kronheimer 1989 -- marks = null/Perron vector [E], "
                   "hyper-Kaehler quotient dim = 4 [E], gauge datum |2I| = 120 = 4*30 "
                   "[E], resolution = unimodular -E8 (det 1 = Poincare link = det K=1 "
                   "face) with 24 deformation parameters [E]; theorems cited [C]; the "
                   "one premise stays [O] (SEAM.EQUIV.01 unchanged)")


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

c3d627fa57516c5127778380f92d49dc87a21cb45391f5482c23995ce9ad331e

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.