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A registered result in its source context

Original ledger classification: Identity/Lattice [E] (mu3+Galois orbit) + [C] (verdict)

Recorded statement: ARE THE 3 GENERATIONS A mu3/GALOIS ORBIT? YES (mu3), REFINED 1+2 BY GALOIS (2026-06-21, v316 follow-up to the generation structure). [E] GENERATIONS = CUBE ROOTS: the 3 cusp weights {0,1/3,2/3} (Spec(Q+)=3a+1={1,2,3}=A3 exponents) map…

Statement, assumptions and sources

Paper context: Origin Theory

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GALOIS.FAMILY.01

Original ledger classification: Identity/Lattice [E] (mu3+Galois orbit) + [C] (verdict)

Original claim text

ARE THE 3 GENERATIONS A mu3/GALOIS ORBIT? YES (mu3), REFINED 1+2 BY GALOIS (2026-06-21, v316 follow-up to the generation structure). [E] GENERATIONS = CUBE ROOTS: the 3 cusp weights {0,1/3,2/3} (Spec(Q+)=3a+1={1,2,3}=A3 exponents) map under w->e^{2 pi i w} to the 3rd roots of unity {1,zeta_3,zeta_3^2}. [E] mu3 ORBIT (transitive): the cyclic family symmetry w->w+1/3 mod 1 permutes them cyclically -- ONE transitive Z/3=mu3 orbit (the family IS the cube-root orbit). [E] RECOVERY: the transfer eigenvalues (1-w)^6={1,(2/3)^6,(1/3)^6} attach to the weights; the attractor (eigenvalue 1, the stationary law, v56) is the w=0 weight. [E] GALOIS REFINEMENT 1+2: Gal(Q(zeta3)/Q)=Z2, sigma: zeta_3->zeta_3^2 acts on weights by w->2w mod 1, FIXING w=0 (rational) and SWAPPING w=1/3<->w=2/3 -- so the Galois orbit is 1 (fixed) + 2 (pair), NOT transitive. [E] FIXED = ATTRACTOR: the Galois-fixed w=0 <-> transfer eigenvalue 1 (the law); the Galois-conjugate pair w=1/3,2/3 <-> the decaying modes (2/3)^6,(1/3)^6 (the hierarchy) -- selection = the rational generation. [E] SAME Z2 AS CP CONJUGATION (v316): sigma: zeta_3->zeta_3^2=conj(zeta_3). [E] COUNTS: N_fam=3=|mu3|=#weights=|Spec(Q+)| (A3 exponents {1,2,3}). [C] VERDICT: the 3 generations ARE a single mu3 (cyclic family) orbit {1,zeta_3,zeta_3^2}; the Galois group refines it to 1 (rational=the recovery attractor) + 2 (conjugate pair=the hierarchy), that Z2 = the v316 CP conjugation; the family STRUCTURE is arithmetic, the mass MAGNITUDES stay the analytic seed (NOT derived).

Original classification detail

[E] the cusp-weights = cube-roots identification + the transitive mu3 orbit + the recovery-eigenvalue attachment + the Galois 1+2 refinement (fixed=attractor, pair=hierarchy) + the same-Z2-as-CP-conjugation + the counts; [C] the verdict. Shows the family STRUCTURE is the same arithmetic (mu3 orbit, Galois-refined) as the CP phase; does NOT derive the mass magnitudes (analytic seed).

Dependencies as recorded

GALOIS.READOUT.01; COX.COUPLE.01; v316; v315; v56

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Registered verification modules

  • v317_galois_family.pyRegistered verification module · frontier
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    Are the 3 generations a mu3/Galois orbit? Yes (mu3), refined 1+2 by Galois (v316 follow-up). The 3 cusp weights {0,1/3,2/3} (Spec(Q+)={1,2,3}=A3 exponents) map under w->e^{2pi i w} to the 3rd roots of unity {1,zeta_3,zeta_3^2}; the cyclic family symmetry w->w+1/3 is a transitive mu3 orbit (the family IS the cube-root orbit). The transfer eigenvalues (1-w)^6={1,(2/3)^6,(1/3)^6} attach to the weights (attractor=eigenvalue 1 at w=0). The Galois Z2 (zeta_3->zeta_3^2, = CP conjugation v316) fixes w=0 (rational) and swaps w=1/3<->2/3 -> orbit 1+2; so the Galois-FIXED generation is the recovery ATTRACTOR (the law), the Galois PAIR is the decaying hierarchy. The family structure is arithmetic; the mass magnitudes stay the analytic seed

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