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A registered result in its source context

Original ledger classification: Identity/Lattice [I]/[L] (uniqueness, D4, index 2) + [P] (the one lattice dictionary)

Recorded statement: SHEET-CONJUGATION BRIDGE (P1 -> ONE [P] residual): the family-side conjugation (cusp negation 0 fixed, 1/3<->2/3 on the Q_+ eigenspaces e1=(0,0,1), e2=(0,1,2), e3=(1,0,0)) has a UNIQUE anchor-compatible generation-space realisation - T.1=1…

Statement, assumptions and sources

Paper context: Standard Model and flavor structure

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FLAV.SHEET.02

Original ledger classification: Identity/Lattice [I]/[L] (uniqueness, D4, index 2) + [P] (the one lattice dictionary)

Original claim text

SHEET-CONJUGATION BRIDGE (P1 -> ONE [P] residual): the family-side conjugation (cusp negation 0 fixed, 1/3<->2/3 on the Q_+ eigenspaces e1=(0,0,1), e2=(0,1,2), e3=(1,0,0)) has a UNIQUE anchor-compatible generation-space realisation - T.1=1 and T.a=a BOTH force scale 1 => integer involution T_A=[[0,1,0],[1,0,0],[2,-2,1]], det=-1; the two alternative pairings are REJECTED (T.a=a unsolvable). ANCHOR = CONJUGATION-SYMMETRIC VECTOR [I]: a=e2+e3, 1=a-e1, odd line e2-e3=(-1,1,2). ONE DECK ACTION [I]: T_A=-1 on R^3/span{1,a} exactly like sigma_12; the v96 collapse direction (-1,1,0) is odd for both - the cover cannot distinguish them. PARITY [I]: det T_A=-1 = single spinor conjugation (= the v92 glue swap), det Sigma=+1 = simultaneous (glue-fixing). D4 CLOSURE [L]: <T_A,Sigma> order 8, G=T_A*Sigma order 4, char (t+1)(t^2+1) = the v69 D4 and the v70 mu4 char poly. SHEET-INDEX LEMMA [I]: G and the v70 homology mu4 are rationally but NOT GL(3,Z)-conjugate - det S = 2 s8 (s6^2+(s7+2 s8)^2) always EVEN, minimal |det|=2=|Z2| (one sheet doubling; Q det=N_fam does not intertwine - independent of v70). REMAINING [P]: the single lattice dictionary 'Q_+ eigenspace grading of the generations = A3 discriminant grading' (then T_A IS the v92 A3-side conjugation and P1 closes)

Original classification detail

Identity/Lattice: one deck involution; one [P] dictionary left

Dependencies as recorded

FLAV.SHEET.01; FLAV.QGEO.01; GATE.QGEOM.02 (v70); GATE.METRIC.07 (v92 glues)

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Registered verification modules

  • v97_sheet_conjugation_bridge.pyRegistered verification module · registry
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    Sheet-conjugation bridge (P1 → one [P]): the cusp negation (0 fixed, 1/3↔2/3 on the Q₊ eigenspaces) has a UNIQUE anchor-compatible integer realisation T_A=[[0,1,0],[1,0,0],[2,−2,1]] (det −1; the two alternative pairings rejected); a = e₂+e₃ — the anchor IS the conjugation-symmetric vector; T_A = −1 on R³/span{1,a} exactly like σ₁₂ (one deck action); ⟨T_A,Σ⟩ = the v69 D₄; sheet-index lemma: generation-μ₄ vs homology-μ₄ intertwiner dets always EVEN, min |det| = 2 = |Z₂|; remaining [P]: Q₊ grading = A₃ discriminant grading

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