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Original ledger classification: Prime-2 facet of the Coxeter clock [E]/[C] (h(E8)=30=2*3*5; one quadratic facet field per prime, ramified at it; the prime-2/Gaussian/Z4-seam facet completes v383's golden and family facets)

Recorded statement: The prime-2 facet of the order-30 Coxeter clock, completing v383 (2026-06-24). The clock 30=2*3*5=h(E8) had two named number-field facets (the golden phi in Q(sqrt5), prime-5/carrier; (2/3)^6 in Q, prime-3/family); this names the prime-2…

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Paper context: Origin Theory

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COXETER.PRIME2.01

Original ledger classification: Prime-2 facet of the Coxeter clock [E]/[C] (h(E8)=30=2*3*5; one quadratic facet field per prime, ramified at it; the prime-2/Gaussian/Z4-seam facet completes v383's golden and family facets)

Original claim text

The prime-2 facet of the order-30 Coxeter clock, completing v383 (2026-06-24). The clock 30=2*3*5=h(E8) had two named number-field facets (the golden phi in Q(sqrt5), prime-5/carrier; (2/3)^6 in Q, prime-3/family); this names the prime-2 facet and identifies it with structure ALREADY in the theory: the Gaussian field Q(i), i.e. the CM point tau=i and the order-4 mu4=Z/4 square seam deck (v282/v288/v333). [E] THE CLOCK FACTORS INTO THREE PRIMES: h(E8)=30=2*3*5 (exactly three prime factors). [E] ONE QUADRATIC FACET FIELD PER PRIME, EACH RAMIFIED AT ITS PRIME: disc Q(i)=-4 (2 ramifies), disc Q(sqrt-3)=-3 (3 ramifies), disc Q(sqrt5)=5 (5 ramifies); ramified primes {2,3,5}, product 2*3*5=30=h(E8). [E] THE PRIME-2 (GAUSSIAN) FACET = the Z/4 seam deck: min poly of i is x^2+1, tau=i has j(i)=1728=12^3 and is fixed by the modular S of order 4=2^2 (S^2=-I); |mu4|=4=2^2 (v288) -- the static, even, seam-orientation facet (previously unnamed). [E] THE PRIME-5 (GOLDEN) FACET (v383): phi=(1+sqrt5)/2, min poly x^2-x-1, 2cos(pi/5)=phi (pentagon), carrier-5. [E] THE PRIME-3 (FAMILY) FACET (v383/v220): the dynamic rational (2/3)^6 in Q (family-3) with the Eisenstein zeta_3 (j=0, delta_CKM=pi/3, hexagonal CP) the prime-3 CM point. [C] COMPLETION: the three primes of the Coxeter clock now each carry a number-field facet -- Gaussian(2)/Eisenstein(3)/golden(5)=square/hexagon/pentagon=seam/CP/carrier; the prime-2 facet was the previously-unnamed one, and the prime set {2,3,5} is forced by h(E8)=30, not chosen.

Original classification detail

[E] the exact arithmetic (h(E8)=30=2*3*5; discriminants -4,-3,5; ramified-prime set {2,3,5} with product 30; j(i)=1728=12^3; 2cos(pi/5)=phi); [C] the completion/identification reading. A synthesis (like v383); reuses the three smallest CM/real-quadratic fields, tau=i (v282) and the Z/4 deck (v288), introduces no new number.

Dependencies as recorded

v220; v282; v288; v314; v333; v383; v387

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  • v390_prime2_facet.pyRegistered verification module · frontier
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    COXETER.PRIME2.01: the prime-2 facet of the order-30 Coxeter clock, completing v383. The clock 30=2*3*5=h(E8) had two named number-field facets (golden phi in Q(sqrt5), prime-5/carrier; (2/3)^6 in Q, prime-3/family); this names the prime-2 facet, identified with the Gaussian field Q(i) -- tau=i and the order-4 mu4=Z/4 square seam deck (v282/v288/v333), structure already in the theory. [E] h(E8)=30 factors into exactly three primes. [E] one quadratic facet field per prime, each ramified at its prime: disc Q(i)=-4, disc Q(sqrt-3)=-3, disc Q(sqrt5)=5; ramified primes {2,3,5}, product=30. [E] prime-2 Gaussian facet: min poly x^2+1, tau=i, j(i)=1728=12^3, |mu4|=4=2^2 (the static seam-orientation facet). [E] prime-5 golden (v383): 2cos(pi/5)=phi (pentagon). [E] prime-3 family (v383/v220): (2/3)^6 + Eisenstein zeta_3 (hexagonal CP pi/3). [C] completion: each prime carries a facet (square/hexagon/pentagon = seam/CP/carrier), the prime set forced by h(E8). A synthesis, no new number.

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