A registered result in its source context

PG32.FLAGS.01

Original ledger classification: Exact lattice/identity-grade compression with the kill typed as a split [Identity/Lattice]: 60-sector exact via the symplectic polarity (equivariant incidence dictionary), 45-sector obstructed on the doily (typed negative); compresses v819 S4 / v821 / v822 RM surfaces into the PG(3,2) normal form; no marker move (Python-only per GATE.WOLFRAM.02)

Original claim text

PG32.FLAGS.01: the PG(3,2) flag normal form for the 105 + 35 vacuum completion, with the incidence kill test typed as a split, ONE module from one probe (26/26 checks, verdict PG32-FLAGS-EXACT + INCIDENCE-SPLIT; discovery probe pg32_flag_completion_probe.py, 2026-08-07, re-run identically at promotion, promoted VERBATIM, < 1 s). COUNTING NORMAL FORM: [4 choose 2]_2 = 35; 140 affine 2-flats of AG(4,2) = 35 through the origin + 105 not; PG(3,2) has 15 points, 35 lines, 105 = 35 x 3 point-line flags, canonically bijective to the C(15,2) = 105 point pairs via {x,y} <-> (x+y, {x,y,x+y}). RM LAYER (v819 S4 / v821 / v822 rebuilt, all set equalities): [15,10,4] weight enumerator 1 + 105z^4 + 280z^6 + 435z^8 + 168z^10 + 35z^12 EXACT; the 105 weight-4 words ARE the flats avoiding 0; the 140 weight-4 words of RM(2,4) ARE all flats; Grams 22I + 6J, 6I + J, 28I + 7J entrywise. VACUUM READING: origin planes puncture to weight 3, NONE in the shortened code, ONE unshortens dim 10 -> 11 -- the vacuum coordinate is ORIGIN RESTORATION. THE INCIDENCE SPLIT (the kill, typed): with the corpus symplectic form (v774) the 105 = 45 + 60 split is the Sp(4,2)-ORBIT decomposition on BOTH sides (|Sp(4,2)| = 720 brute force); on the 60 NON-ISOTROPIC rungs the polarity map PHI: (U,t) -> ((t+U) n U^omega, U^omega) is well-defined, bijective, Sp(4,2)-EQUIVARIANT (all 60 x 720) and INCIDENCE-PRESERVING; on the 45 DOILY rungs the obstruction is typed: U^omega = U and (t+U) n U = EMPTY for 45/45 flats, the unique abstract equivariant bijection matches flags with support n line = 0, and the flag Gram is 18I + 3J != 22I + 6J. TYPED CONCLUSION: '105 checks = 105 flags' is exact as a deck-equivariant BIJECTION, NOT an incidence isomorphism -- the obstruction sits exactly on the doily (the typed reason v822's continuum dilation died). VACUUM CARRIER CONTENT: the zero class carries NO E8 root (0/240 rebuilt, v689 [E]) -- the vacuum point repairs the label Gram but adds no carrier content; v822's frozen continuum kill cited as the companion negative. Controls fire (scrambled plane row, degenerate form, the 45-sector obstruction as its own control). Exact integer; no floats, no RNG, no fit. Python-only per GATE.WOLFRAM.02.

Original classification detail

PG32-FLAGS-EXACT + INCIDENCE-SPLIT: 105 = 45 + 60 is the Sp(4,2)-orbit decomposition on both sides; the polarity dictionary is exact, equivariant and incidence-preserving on the 60 non-isotropic rungs; the 45 doily rungs carry the typed obstruction ((t+U) n U empty 45/45, flag Gram 18I+3J != 22I+6J) -- '105 checks = 105 flags' is a deck-equivariant bijection, NOT an incidence isomorphism

Dependencies as recorded

v689; v774; v819; v821; v822