A registered result in its source context

ALPHA.QUILLEN.DETLINE.01

Original ledger classification: Numerical [N] (finite Quillen shadow; gate unchanged)

Original claim text

THE OCCUPIED BERRY LINE OVER THE U(1)-TWIST MODULI HAS CHERN NUMBER = THE BLOCH RESPONSE LEVEL (2026-07-03) -- the finite Berry-Chern response, COMPUTED at the collar model; the Quillen/Dai-Freed comparison remains a separate target; does NOT close ALPHA.QUILLEN.EXACT.01. [E] BLOCH LEVEL RE-VERIFIED: BZ Chern C(M=1) = 1, C(M=3) = 0, C(M=-1) = -1 (the v470/v367 integers). [E] THE OCCUPIED BERRY LINE COMPUTED: the same collar h(k) = sin kx SX + sin ky SY + (M - cos kx - cos ky) SZ in real space on an L x L torus with twisted boundary conditions -- the twist torus (theta_x, theta_y) IS the moduli space of flat U(1) connections -- and the Fukui-Hatsugai-Suzuki curvature of the DETERMINANT LINE of the occupied one-particle frame (the free-fermion many-body ground-state LINE; its normalized vectors are chosen locally) over that torus integrates to 2 pi x 1 exactly (integer to 1e-9) at M = 1, for L = 4 AND L = 6 -- size-independent, as an integer must be. [E] CONTROLS: trivial collar M = 3 gives 0; orientation flip M = -1 gives -1 -- the curvature tracks the phase, not the mesh. [E] TWO TORI, ONE LEVEL: the twist-moduli integer equals the Bloch-BZ integer for all three M -- the Niu-Thouless-Wu identification (PRB 31 (1985)), exhibited on the collar model that realises S3 (v460/v461). [E] GAP: the sampled Fermi gap is open at M=1; analytically |d(k)|^2 = 1 + 2(1-cos kx)(1-cos ky) >= 1 gives the global gap >= 2. This defines a smooth occupied projector, not a global nonvanishing section of its topologically nontrivial line. No Dai-Freed filling or global-section test is performed. [C] INTEGRATED RESPONSE: k0 = |C_FHS| = 1 in the collar model. The FHS continuum convention C_Berry=integral F_occ/(2pi i), with nabla=d+A_occ and c1=iF_occ/(2pi), gives c1[B]=-C_Berry; finite-mesh identification and theta-line sign/duality must be checked separately. This integrated integer does not identify individual holonomies or the exact zeta-determinant variation. The open comparison requires a separately defined chiral Fredholm/Quillen line with its zero locus, a common gauge base, relative connection-level comparison and uniform ultraviolet controls; a Dai-Freed vector requires filling data and takes values in the pulled-back inverse boundary line (Quillen 1985; Bismut-Freed CMP 106 (1986); Dai-Freed JMP 35 (1994); FHS JPSJ 74 (2005)). [O] NOT CLOSED: the abstract-seam zeta-determinant identification (the continuum leg, = the SEAM.EQUIV.01 face); ALPHA.QUILLEN.EXACT.01 stays [O]; alpha^-1 = 137.0359992168 stays [E] regardless. Structurally the same manoeuvre as v471 for SEAM.THEOREM.01's replica side: the named open lemma exercised on a real finite operator.

Original classification detail

Det-line/moduli shadow [E]/[C]/[O]: the Quillen-shaped object of ALPHA.QUILLEN.EXACT.01 (det line over the U(1) moduli) computed on the S3 collar -- FHS curvature of the occupied-frame det line over the twist torus = 1 = the inflow level, size-independent, controls 0/-1, twist==BZ integer (Niu-Thouless-Wu); the finite integrated response is exhibited; no global nonvanishing section, individual-holonomy identity or continuum Quillen comparison is established; ALPHA.QUILLEN.EXACT.01 stays [O]

Dependencies as recorded

ALPHA.QUILLEN.EXACT.01; ALPHA.QUILLEN.INFLOW.01; EM.WARD.01; v367; v460; v461; v470; v471