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Original ledger classification: Dual keystone at tau=omega [E]/[C]/[O] (the family/flavor sector as the order-3 Eisenstein/A2 face: chi_E8(omega)=0, A2 Cartan det 3=N_fam reads R in E6xA2 with det R=8=dim A2, delta=the tau=omega CM phase, cusp weights=the order-3 deck; dual to the seam=(E8)_1 at tau=i). A named contract, NOT a closure

Recorded statement: The dual keystone at tau=omega (2026-06-24): the family/flavor sector as the order-3 Eisenstein/A2 face of E8, dual to the seam=(E8)_1 at tau=i (SEAM.EQUIV.01/v404). A NAMED CONTRACT (genre of v286/v398), NOT a closure. [E]…

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Paper context: Origin Theory

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SEAM.EQUIV.02

Original ledger classification: Dual keystone at tau=omega [E]/[C]/[O] (the family/flavor sector as the order-3 Eisenstein/A2 face: chi_E8(omega)=0, A2 Cartan det 3=N_fam reads R in E6xA2 with det R=8=dim A2, delta=the tau=omega CM phase, cusp weights=the order-3 deck; dual to the seam=(E8)_1 at tau=i). A named contract, NOT a closure

Original claim text

The dual keystone at tau=omega (2026-06-24): the family/flavor sector as the order-3 Eisenstein/A2 face of E8, dual to the seam=(E8)_1 at tau=i (SEAM.EQUIV.01/v404). A NAMED CONTRACT (genre of v286/v398), NOT a closure. [E] chi_E8(omega)=j(omega)^{1/3}=0 (E4 vanishes at tau=omega, v404) vs chi_E8(i)=12 -- the family sits where the holomorphic (E8)_1 structure degenerates. [E] A2 is the tau=omega Eisenstein/hexagonal lattice: Cartan det=3=N_fam, h(A2)=3=N_fam, automorphism mu6=mu3(triality) x mu2(sheet, v233). [E] the flavor matrix R lives in the E6 x A2 slice: 248=||R||_F^2+det R+2(1^T R a)N_fam=78+8+2*27*3, ||R||_F^2=78=dim E6, det R=8=dim A2=rank E8. [E] the CP phase delta_PMNS=4pi/3 (delta_CKM,lead=pi/3=arg zeta6) is the hexagonal Eisenstein CM phase at tau=omega (v220/v231/v233). [E] the cusp weights {0,1/3,2/3} are the order-3 deck = N_fam = det Q = cusp-class order (v72), fixing Spec(Q+)=(1,2,3)=A3 exponents (v69/v137). [C] the dual keystone: the raw cusp/family transport IS the order-3 Eisenstein/A2 deck at tau=omega; the two keystones sit at the only two elliptic points of PSL(2,Z). [O] residual (parallel to SEAM.EQUIV.01's continuum existence): the canonical Z3 deck identification (v140) + the absolute scale = one v_geo-class UV input (No-Unit, v153).

Original classification detail

[E] the arithmetic (chi_E8(omega)=0; A2 Cartan det 3=N_fam, h=3; the E6xA2 slice with det R=8=dim A2; delta=the tau=omega CM phase; cusp weights=the order-3 deck); [C] the dual-keystone reading; [O] the canonical-map Z3 choice (v140) + the v_geo-class scale. A named contract (genre of v286/v398), reuses established facts; NOT a closure; no new number.

Dependencies as recorded

E8.CMDUAL.01; SEAM.EQUIV.01; v404; v220; v233; v72; v69; v137; v140

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Registered verification modules

  • v405_seam_equiv_omega.pyRegistered verification module · frontier
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    SEAM.EQUIV.02: the dual keystone at tau=omega -- the family/flavor sector as the order-3 Eisenstein/A2 face of E8, dual to the seam = (E8)_1 at tau=i (v404). A named contract, NOT a closure. [E] chi_E8(omega)=0 (E8 degenerates at the family CM point) vs chi_E8(i)=12. [E] A2 = the tau=omega Eisenstein lattice (Cartan det 3=N_fam, h=3=N_fam, mu6=mu3 x mu2). [E] R lives in E6 x A2: 248=78+8+2*27*3, ||R||_F^2=78=dim E6, det R=8=dim A2=rank E8. [E] delta_PMNS=4pi/3 = the tau=omega CM phase (arg zeta6=pi/3). [E] cusp weights {0,1/3,2/3}=order-3 deck=N_fam=det Q, Spec(Q+)=(1,2,3). [C] the family sector is the order-3 Eisenstein/A2 face; the two keystones sit at the only two elliptic points of PSL(2,Z). [O] residual = the canonical Z3 map (v140) + the v_geo-class scale (No-Unit v153).

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