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A registered result in its source context

Original ledger classification: Identity/Lattice [I]/[L] (dictionary derived) - residual folded into GATE.QGEO

Recorded statement: DISCRIMINANT DICTIONARY DERIVED (P1 closed modulo GATE.QGEO): the generation space carries an INTEGER mu4 - G = T_A*Sigma acts on the cusp basis as G e1=-e1, G e3=e2, G e2=-e3 (the v69 B1+E decomposition realised integrally; G^4=I, char…

Statement, assumptions and sources

Paper context: Standard Model and flavor structure

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FLAV.SHEET.03

Original ledger classification: Identity/Lattice [I]/[L] (dictionary derived) - residual folded into GATE.QGEO

Original claim text

DISCRIMINANT DICTIONARY DERIVED (P1 closed modulo GATE.QGEO): the generation space carries an INTEGER mu4 - G = T_A*Sigma acts on the cusp basis as G e1=-e1, G e3=e2, G e2=-e3 (the v69 B1+E decomposition realised integrally; G^4=I, char (t+1)(t^2+1)); the cusp-0 eigenvector IS the unique self-conjugate Z4-class-2 line and the swapped cusp pair {1/3,2/3} IS the conjugate class pair {1,3} => the v97 dictionary 'Q_+ grading = A3 discriminant grading' is DERIVED inside the model, not assumed. T_A G T_A^-1 = G^-1 (= discriminant conjugation k->-k); q_A3(1)=q_A3(3)=3/8 (equal on the swapped pair), q_A3(2)=1/2 (fixed); the A3-side negation swaps the two Lagrangian glues (v92 re-derived). The two D4 reflection classes separate the physics: T_A e1=+e1, det=-1 (glue-SWAPPING, v92 single conjugation) vs Sigma e1=-e1, det=+1 (glue-FIXING, simultaneous); Sigma=T_A G. CONSEQUENCE: P1 carries NO separate [P] residual any more - what remains is exactly the already-open GATE.QGEO realisation step (one gate, not two); conditional on that established typing the sector->sheet selection is CLOSED (up/down = deck-odd pair, leptons on the ramification, deck involution = A3-side conjugation T_A)

Original classification detail

Identity/Lattice: P1 dissolved into the one existing Q-geometry gate

Dependencies as recorded

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

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Registered verification modules

  • v98_discriminant_dictionary.pyRegistered verification module · registry
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    Discriminant dictionary DERIVED (P1 closed modulo GATE.QGEO): the generation space carries an INTEGER μ₄ — G = T_A·Σ acts on the cusp basis as G·e₁=−e₁, G·e₃=e₂, G·e₂=−e₃ (the v69 B₁⊕E decomposition realised integrally); the cusp-0 line IS the self-conjugate Z₄-class-2 line, the swapped cusp pair IS the conjugate class pair {1,3}; T_A G T_A⁻¹ = G⁻¹ (conjugation k→−k), q(1)=q(3)=3/8 equal, q(2)=1/2 fixed; reflection classes = glue-swap (det −1) vs glue-fix (det +1) parities ⇒ P1 carries NO separate [P] — the residual folds into the one existing Q-geometry gate

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