A registered result in its source context
Original ledger classification: Arithmetic hull [E]/[C] (the three facet fields compose to K=Q(i,sqrt-3,sqrt5): degree 2^3=8=rank E8, Galois (Z/2)^3, 7 subfields = 4 imaginary (|mu4|) + 3 real (N_fam), ramified only at {2,3,5}; 2^omega(h)=rank is E8-special) -- a synthesis, no new number
Recorded statement: The three number-field facets COMPOSE to a degree-8 = rank(E8) abelian field (2026-06-24): K=Q(i,sqrt-3,sqrt5), the arithmetic hull dual to the E8 lattice hull, composing the v390/v394 facets that were only ever used one at a time. [E]…
Statement, assumptions and sources
Paper context: Origin Theory
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ARITH.HULL.01
Original ledger classification: Arithmetic hull [E]/[C] (the three facet fields compose to K=Q(i,sqrt-3,sqrt5): degree 2^3=8=rank E8, Galois (Z/2)^3, 7 subfields = 4 imaginary (|mu4|) + 3 real (N_fam), ramified only at {2,3,5}; 2^omega(h)=rank is E8-special) -- a synthesis, no new number
Original claim text
The three number-field facets COMPOSE to a degree-8 = rank(E8) abelian field (2026-06-24): K=Q(i,sqrt-3,sqrt5), the arithmetic hull dual to the E8 lattice hull, composing the v390/v394 facets that were only ever used one at a time. [E] INDEPENDENCE: the squarefree parts -1,-3,5 are multiplicatively independent in Q*/(Q*)^2 (no nonempty subset product is a square), so the three quadratics are linearly disjoint and [K:Q]=2^3. [E] DEGREE = rank E8: [K:Q]=2^3=8=rank(E8)=g_car+N_fam, Galois group (Z/2)^3 of order 8; the relation 2^omega(h)=rank is E8-special (the E7 control h=18, omega=2, 2^2=4!=7). [E] SEVEN SUBFIELDS: exactly 7 quadratic subfields {-1,-3,5,3,-5,-15,15}. [E] 4+3 SPLIT = |mu4|+N_fam: exactly 4 imaginary (-1,-3,-5,-15)=|mu4| (CM/dynamic) + 3 real (3,5,15)=N_fam (RM/static). [E] RAMIFIED ONLY AT THE ATOMS: every quadratic subfield ramifies only over {2,3,5}=the atoms=the primes of h(E8)=30. [E] CM/RM=DYNAMIC/STATIC: 2,3 imaginary-quadratic (disc -4,-3) carry the {2,3}-arithmetic dynamic rates (v394), 5 real-quadratic (disc +5) carries the static golden carrier (no independent prime-5 rate). [C] K is the abelian (Z/2)^3 'arithmetic hull' dual to the E8 lattice hull; the 7 quadratic subfields are a candidate for the 7 E8 audit slices.
Original classification detail
[E] the field arithmetic (independence => degree 8 = rank E8; Galois (Z/2)^3; the 7 subfields = 4 imaginary (|mu4|) + 3 real (N_fam); ramified only at {2,3,5}=the atoms; the E8-special 2^omega(h)=rank with the E7 negative control); [C] the arithmetic-hull reading and the 7-subfields<->7-slices candidate. A synthesis composing the established facet fields (v390/v394), introduces no new number; the prime set {2,3,5} forced by h(E8)=30.
Dependencies as recorded
COXETER.PRIME2.01; COXETER.ATOMS.01; v23; v390; v394; v222
Open the original ledger source, row 473Location recorded in the ledger
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- Origin Theory
Sections that explicitly cite this claim
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- 2026-06-24 (The arithmetic hull + the [formula in source] -character CM duality: v403, v404)Research changelog; TFPT — Introduction and reading guide
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
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- The whole integer skeleton from one pair [formula in source]input 4 origin theory
- The unique attractor: a gapped transport selects ONE fixed pointorigin theory
- The unique attractor: a gapped transport selects ONE fixed pointinput 4 origin theory
- The whole integer skeleton from one pair [formula in source]input 4 origin theory
- The whole integer skeleton from one pair [formula in source]Origin Theory
- The unique attractor: a gapped transport selects ONE fixed pointinput 4 origin theory
- 2026-06-25 (The Gaussian operator [formula in source] the atom trichotomy: v415, v416)Research changelog; TFPT — Introduction and reading guide
- The whole integer skeleton from one pair [formula in source]origin theory
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Registered verification modules
- v403_facet_compositum.pyRegistered verification module · frontier
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ARITH.HULL.01: the three number-field facets (v390) COMPOSE to K=Q(i,sqrt-3,sqrt5), a degree-8=rank(E8) abelian field -- the arithmetic hull dual to the E8 lattice hull. [E] -1,-3,5 independent in Q*/(Q*)^2 so [K:Q]=2^3=8=rank E8=g_car+N_fam, Galois (Z/2)^3 order 8; 2^omega(h(E8))=2^3=8=rank E8 is E8-special (E7: h=18, omega=2, 2^2=4!=7). [E] exactly 7 quadratic subfields {-1,-3,5,3,-5,-15,15} = 4 imaginary (|mu4|, CM/dynamic) + 3 real (N_fam, RM/static). [E] ramified only at {2,3,5} = the atoms = primes of h(E8)=30. [C] K is the abelian (Z/2)^3 arithmetic hull dual to E8; the 7 subfields a candidate for the 7 E8 audit slices. A synthesis composing v390/v394's facets, no new number.
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