"""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)