A registered result in its source context

WOIT.GAMMA.TOY.01

Original ledger classification: Float-track certificates (tolerances declared) + exact combinatorics + [C] toy fence (numpy/sympy)

Original claim text

The interacting gamma slice at toy level (one module, 11 checks, ~2 s; discovery probe gamma_toy_interacting_probe.py 11/11, verdict GAMMA-TOY-LANDED). The gamma battery executed ON the RP-surviving interaction class (the alignment bit: delta = pi/2, positive coupling, v534) of the 16-Majorana FK seam toy (v529). TWO PARENTS [float]: chiral (2-point I + iC, the v519 kernel) and MIRROR (I - iC), both gap 1. SURVIVOR ANCHOR [float, v534 regression]: chiral interacting Grams PD (37,0,0) at the forced twist eta = +i on all four admissible cuts for g in {1/32, 1/4, 1, 8}. THE INTERACTING MIRROR FLIP [float, central]: at the SAME eta the mirror state splits sector-exactly on EVERY cut and EVERY g -- even (29,0,0) PD, odd (0,8,0) STRICTLY ND with max odd eigenvalue <= -1.5e-3 (bounded away from zero over the whole grid): every mirror fermion vector has strictly negative OS norm AT THE INTERACTING LEVEL -- in the doubled vector-like system any OS-positive subspace meets the mirror odd sector in {0}: KILL TEST (3) CANNOT FIRE on the surviving interaction class (upgrade of v608 G3 from g = 0 to the full coupling grid). CONVENTION CONTROL [float]: eta = -i swaps the roles exactly -- the datum is the RELATIVE chirality. MARK TRANSPORT INTERACTING [exact + float]: alpha_4 maps the mark-A quartet algebra exactly into the mark-B sites (16/16), and both the mark Gram G_A and the transport form T_4 are Hermitian PD (16,0,0) at every g -- KILL TEST (7) DOES NOT FIRE at the interacting toy level. MULTIPLICITY-FREE [float]: the generalized odd-block transfer spectrum has 8 SIMPLE positive eigenvalues at every g (min pairwise gap 2.3e-5 > 1e-5; honest finding: strong-coupling near-doubling at g = 8 without closing); g = 0 baseline locked (top 1.641438 -- numerically the v524 compressed-clock top: the quarter-turn transfer datum recurs across N). DEAD-MEMBER CONTROL [float]: (m = 2, s = +, g = 1/2) is indefinite (21,16,0) on its straddled cut while the avoiding cut stays PD -- off the survivor the gamma battery has no RP home: the dynamical bit selection (v534) and the chirality data CO-LOCATE on the same member. FIREWALL [C]: one toy, one interaction class (FK quartets, flat-band parent), deg <= 2 cut bases + the full 16-dim mark algebra, finite grid -- NO continuum claim, NO statement about A_hol beyond this model class; kill tests (3)/(6)/(7) stay formally live on A_hol; kill test (6) untouched; WOIT.OS.TWISTOR.01 does not move. [2026-08-01 XXXV: the last untouched kill test gets its first slice -- v616: the internal SU(2) separates from the Euclidean rotation group at the kinematic twistor level (rho_tw = right-j, clock = left-u o right-u, deck = left-i, internal SU(2) broken to U(1) by the complex choice with the mark torsor as its mu4); kill test (6)'s kinematic branch does not fire, the dynamical half stays live on A_hol; see WOIT.SU2.KIN.01.] [2026-08-01 XXXVI: the net-dynamics slice lands -- v621: the Klein-Landau semigroup contracts at g > 0 to the mu4-commensurate steps {4, 6, 7} (the survivor interaction is alpha_4-invariant exactly, breaking translation invariance to the quartet stabilizer), and the clock step k = 4 stays a positive transfer ((11,0,0)) over the whole grid: the reconstructed rotation group of the interacting toy is the clock tower; see WOIT.GAMMA.SEMI.01.]

Original classification detail

GAMMA-TOY-LANDED: interacting mirror flip sector-exact on every cut/coupling (odd strictly ND, bounded away from zero -- kill test (3) cannot fire on the survivor); mark transport Hermitian PD at every g (kill test (7) does not fire); odd transfer spectrum multiplicity-free; dead member has no RP home (bit selection and chirality co-locate); 11/11 checks

Dependencies as recorded

WOIT.GAMMA.FREE.01; SEAM.STRADDLE.CONE.01; SEAM.INT.FKTOY.01; WOIT.THETA.FREE.01; WOIT.BETA2.OS.01; v519; v524; v529; v534; v608