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A registered result in its source context

Original ledger classification: Identity [I] (index-11 congruence + parity + reductions) + [P]/[A] (establishment of the two line pairings)

Recorded statement: FRAME INTEGRALITY (R4' narrowed: three pairings -> two): the pairing triples (m.1, m.a, m.sigma) of INTEGER covectors against the frame (1,a,sigma) form an index-11 sublattice of Z^3, cut out by the single congruence x-3y+z = 0 (mod 11) -…

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GATE.UWALL.12

Original ledger classification: Identity [I] (index-11 congruence + parity + reductions) + [P]/[A] (establishment of the two line pairings)

Original claim text

FRAME INTEGRALITY (R4' narrowed: three pairings -> two): the pairing triples (m.1, m.a, m.sigma) of INTEGER covectors against the frame (1,a,sigma) form an index-11 sublattice of Z^3, cut out by the single congruence x-3y+z = 0 (mod 11) - the index IS 11 = ||Pl(K)||_1 = det(1|a|sigma) (full 11^3 scan, exact). With the two line pairings at their established atoms (x,y)=(|Z2|,rank E8)=(2,8), integrality FORCES the sigma-pairing z = 0 (mod 11): the v139 line z(t)=11(11+t) is exactly the integer line. PRIMITIVITY forces t EVEN (odd t makes every entry of n(t) even); t=0 is the unique even t with z a perfect square for |t|<88 (next square at t=88; global pick still needs the v139 square selector - recorded honestly). CRAMER REDUCTIONS (restructuring, not establishment): n.a = det R = 8 is the Laplace expansion along R's anchor column (v121/v122); n.1 = 8(R^-1 1)_1 = 2 with R^-1 1 = (1,1,-1)/4 (v134) - both consume R. STATUS: R4' reduces from 'derive three pairings' to 'derive the two line pairings (|Z2|, rank E8)'; typing [A]/[P] unchanged, content strictly smaller

Original classification detail

Identity [I]: third pairing integrality-forced mod 11; R4' = two line pairings

Dependencies as recorded

GATE.UWALL.11; GATE.UWALL.10; FLAV.H2.05; v134; v139

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Registered verification modules

  • v142_frame_integrality.pyRegistered verification module · registry
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    frame integrality (R4′): integer covectors in the (1,a,σ) dual frame = index-11 sublattice (x−3y+z ≡ 0 mod 11); with (2,8) established the σ-pairing is forced ≡ 0 mod 11; primitivity forces even line parameter — three pairings reduce to two (since v145: the values are atom identities — residue = one lift map)

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