A registered result in its source context
Original ledger classification: Eisenstein/CP operator [E]/[C]: the family rotation P realises the hex norm 7=N_omega(3+2omega) via (3I+2P)(3I+2P^2)={7,7,25} (dual of v415's {13,13,9}); the CP clock W=-P^2 (W^2=P, W^3=-I, Spec {-1,zeta6,zeta6^-1}) gives delta_CKM=pi/3, delta_PMNS=4pi/3; all flavor clocks are zeta30-powers, the seam mu4 the Galois side phi(30)=8=rank E8; the carrier mu5 has no 3x3 operator [C]
Recorded statement: The Eisenstein/CP operator -- the hexagonal dual of the Gaussian seam operator v415 (2026-06-25): where v415 put the SQUARE seam (tau=i, mu4) on operators via J=[U,V]/3, this puts the HEX/flavor side (tau=omega, mu3/mu6) on operators and…
Statement, assumptions and sources
Paper context: Standard Model and flavor structure
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DIAMOND.EISEN.01
Original ledger classification: Eisenstein/CP operator [E]/[C]: the family rotation P realises the hex norm 7=N_omega(3+2omega) via (3I+2P)(3I+2P^2)={7,7,25} (dual of v415's {13,13,9}); the CP clock W=-P^2 (W^2=P, W^3=-I, Spec {-1,zeta6,zeta6^-1}) gives delta_CKM=pi/3, delta_PMNS=4pi/3; all flavor clocks are zeta30-powers, the seam mu4 the Galois side phi(30)=8=rank E8; the carrier mu5 has no 3x3 operator [C]
Original claim text
The Eisenstein/CP operator -- the hexagonal dual of the Gaussian seam operator v415 (2026-06-25): where v415 put the SQUARE seam (tau=i, mu4) on operators via J=[U,V]/3, this puts the HEX/flavor side (tau=omega, mu3/mu6) on operators and unifies both with the seam into ONE order-30 Coxeter clock. [E] EISENSTEIN DECK: the family rotation P=(1 2 3) has P^3=I, chi_P=x^3-1, P^2+P+I=ONES, so on the plane perp to (1,1,1) it satisfies omega^2+omega+1=0 -- the order-3 dual of J^2=-I. [E] HEX NORM 7 AS OPERATOR: (3I+2P)(3I+2P^2)=7I+6 ONES, Spec {7,7,25} -- 7=N_Z[omega](3+2omega)=scalaron on the omega-plane, 25=g_car^2 on (1,1,1); the dual of v415's (3I+2J)(3I-2J)->{13,13,9}; NEG control (5I+4P)(5I+4P^2)->{21,21,81}, N_omega(5,4)=21!=41 (v222 rigidity). [E] CP CLOCK W=-P^2 (=omega^2*(-1)=rho v233), order 6: W^3=-I, W^2=P, chi_W=x^3+1, Spec {-1,zeta6,zeta6^-1}; delta_CKM,lead=arg(zeta6)=pi/3 (eigenvalue of W), delta_PMNS=arg(zeta6^4)=4pi/3 (eigenvalue of W^4=P^2), W^3=-I the Z2 sheet flip rho^3=-1 (v231/v233/v316/v320); mu6=mu3 x mu2 in one operator. [E] ONE ORDER-30 CLOCK: all flavor-side clocks are powers of zeta30 (h(E8)=30=2*3*5=|Z2|*N_fam*g_car): zeta30^15=-1 (mu2), zeta30^10=omega (mu3), zeta30^6=zeta5 (mu5), zeta30^5=zeta6 (mu6); the SEAM mu4 (i) is NOT a power (4 nmid 30) -- the Galois side (Z/30)^x of order phi(30)=8=rank E8=mu4 x Z2 (totatives {7,13,17,23} order 4); Q(zeta30) carries BOTH E8 invariants (cyclic h=30 flavor hands + Galois rank=8 seam). [C] REALISATIONS + HONEST GAP: J(mu4)/P(mu3)/W=-P^2(mu6)/sigma(mu2) are 3x3 operators, but the CARRIER mu5 (zeta5) has none (needs dim>=4; the golden phi=2cos(pi/5) shows up only as an output-side trace, v313/v349); honest negatives: no U,V-diamond operator carries x^2-x+1 (only -P does), the seam and family clocks do NOT combine ([D,P]!=0, no mu12), and N(3+2 zeta6)=19!=7 (the clean 7 is the omega-norm). An operator lift + synthesis tying v222/v230/v233/v315/v316/v404/v415; no new number.
Original classification detail
Identity
Dependencies as recorded
DIAMOND.GAUSS.01; CMNORM.DUAL.01; CENTER.NORM.01; FLAV.CP.01; v415; v222; v230; v233; v316; v320; v404; v95
Open the original ledger source, row 485Location recorded in the ledger
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- Standard Model and flavor structure
Sections that explicitly cite this claim
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- 2026-06-25 (The Eisenstein/CP operator [formula in source] the order-30 clock: v417)Research changelog; TFPT — Introduction and reading guide
Navigation through a shared verification script
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- The Sheet Diamond and the Winding Line: masses and transport from one surfacetfpt 2 standard model
- 2026-06-25 (The cyclotomic norm triple [formula in source] the carrier-5 clock: v418)Research changelog; TFPT — Introduction and reading guide
- Front matter / supportStandard Model and flavor structure
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
Registered verification modules
- v417_eisenstein_cp_operator.pyRegistered verification module · flavor
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The Eisenstein/CP operator (hex dual of v415): the family rotation P=(1 2 3) and the CP clock W=-P^2 put the tau=omega flavor side on operators, unified with the seam into ONE order-30 clock. P^3=I, P^2+P+I=ONES (Eisenstein omega^2+omega+1=0 on the plane perp to (1,1,1)); (3I+2P)(3I+2P^2)=7I+6 ONES with Spec {7,7,25} -- 7=N_omega(3+2omega)=scalaron on the omega-plane, 25=g_car^2 on (1,1,1) -- the dual of v415's {13,13,9} (NEG: (5I+4P)(5I+4P^2)->{21,21,81}, N_omega(5,4)=21!=41); the CP clock W=-P^2 (order 6, =rho v233) has W^2=P, W^3=-I, Spec {-1,zeta6,zeta6^-1}, so delta_CKM=arg(zeta6)=pi/3 and delta_PMNS=arg(zeta6^4)=4pi/3 (W^4=P^2); all flavor clocks are powers of zeta30 (zeta30^15=-1, ^10=omega, ^6=zeta5, ^5=zeta6 = orders 2,3,5,6) while the seam mu4=i is the Galois side (Z/30)^x of order phi(30)=8=rank E8; the carrier mu5 has no clean 3x3 operator (golden phi an output-side trace, v313/v349).
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