How the Coxeter sectors couple
Original ledger classification: Identity/Lattice [E] (cyclotomic Galois) + [C] (verdict)
Recorded statement: COUPLE OR FACTORIZE? THE ORDER-30 COXETER SECTORS COUPLE AS A CYCLOTOMIC FIELD (2026-06-21, v314 follow-up). The v314 question -- does 30=g_car*(2 N_fam)=5*6 couple the carrier (5-fold, golden, Q(sqrt5)) and family (6=2*3, rational)…
Statement, assumptions and sources
Paper context: Origin Theory
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COX.COUPLE.01
Original ledger classification: Identity/Lattice [E] (cyclotomic Galois) + [C] (verdict)
Original claim text
COUPLE OR FACTORIZE? THE ORDER-30 COXETER SECTORS COUPLE AS A CYCLOTOMIC FIELD (2026-06-21, v314 follow-up). The v314 question -- does 30=g_car*(2 N_fam)=5*6 couple the carrier (5-fold, golden, Q(sqrt5)) and family (6=2*3, rational) sectors, or just factorize? [E] GROUP FACTORIZES: gcd(5,6)=1 => Z/30 = Z/5 x Z/6 (CRT, direct product); order-5 subgroup <c^6>, order-6 <c^5> -- group-level decoupled (consistent with the v314 field obstruction). [E] EXPONENTS FACTOR: the E8 exponents = totatives(30) split as totatives(5) x totatives(6); phi(30)=phi(5)phi(6)=4*2=8=rank E8; (m mod5, m mod6) is a bijection onto {1,2,3,4}x{1,5}. [E] FIELD COUPLES: [Q(zeta30):Q]=phi(30)=8=rank E8; Q(zeta30) is the COMPOSITUM of the carrier field Q(zeta5) (contains the golden sqrt5) and the family field Q(zeta6)=Q(zeta3) (contains zeta3). [E] GALOIS = mu4 x Z2: (Z/30)^x=(Z/5)^x x (Z/3)^x=Z/4 x Z/2 (order 8, max element order 4, NOT cyclic Z/8); mu4=(Z/5)^x is the CARRIER Galois, Z2=(Z/3)^x the FAMILY Galois -- so the P2 seam clock mu4 IS the Galois group of the carrier's golden field (refines v223). [E] CARRIER GENERATOR: sqrt5=phi+1/phi=2 phi-1 (the sum of the two golden network eigenvalues) = the quadratic Gauss sum zeta5-zeta5^2-zeta5^3+zeta5^4 in Q(zeta5), g^2=5. [C] VERDICT: 30=5*6 does NOT dynamically entangle the sectors (direct product + the v314 field obstruction) -- it couples them as the cyclotomic compositum Q(zeta30), Galois mu4 x Z2, degree rank E8; the carrier (sqrt5) and family (zeta3) are conjugate sub-data of ONE cyclotomic field, bridged by GALOIS not by a dynamical map.
Original classification detail
[E] the CRT direct-product factorisation + the exponent/totative factorisation + the Q(zeta30) compositum + the Galois=mu4 x Z2 identification (seam clock = carrier-field Galois) + the Gauss-sum carrier generator; [C] the couple-vs-factorize verdict. Answers the v314 coupling question exactly (Galois compositum, not a dynamical map); does NOT add a physical readout.
Dependencies as recorded
RATE.TRANSLATE.01; COX.TOTATIVE.01; v314; v223; v55
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- Origin Theory
Sections that explicitly cite this claim
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- 2026-06-21 (couple or factorize? the order-30 Coxeter element couples the carrier and family sectors as a cyclotomic compositum, Galois [formula in source] : v315)Research changelog; TFPT — Introduction and reading guide
Navigation through a shared verification script
These sections cite a script associated with the claim. This is a source navigation join, rather than an explicit citation of the claim or a theorem equivalence.
- 2026-06-22 (Wolfram mirror of the arithmetic arc [formula in source] three sharpening steps: the CP-lock made quantitative, the Bisognano–Wichmann keystone, the minimal hypergraph fibre: v322, v323, v324)Research changelog; TFPT — Introduction and reading guide
- The whole integer skeleton from one pair [formula in source]input 4 origin theory
- The whole integer skeleton from one pair [formula in source]input 4 origin theory
- 2026-06-21 (does the Galois [formula in source] organize the readouts? Yes — the CP phases are the family-factor cyclotomic data, Galois [formula in source] CP conjugation: v316)Research changelog; TFPT — Introduction and reading guide
- The whole integer skeleton from one pair [formula in source]Origin Theory
- OutcomeRed-team audit
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
- 2026-06-22 (the translation clock: the discrete [formula in source] dynamic bridge is the order- [formula in source] Coxeter element, a clock with two coprime hands [formula in source] , read law-inclusive ( [formula in source] ) or live-only ( [formula in source] ): v319)Research changelog; TFPT — Introduction and reading guide
Registered verification modules
- v315_coxeter_coupling.pyRegistered verification module · frontier
Read the description recorded in the module registry
Couple or factorize? The order-30 Coxeter sectors couple as a cyclotomic field (v314 follow-up). GROUP FACTORIZES: gcd(5,6)=1 => Z/30 = Z/5 x Z/6 (direct product, group-decoupled). EXPONENTS FACTOR: E8 exponents=totatives(30)=totatives(5)x totatives(6), phi(30)=4*2=8=rank E8. FIELD COUPLES: [Q(zeta30):Q]=8=rank E8, Q(zeta30) is the compositum of the carrier Q(zeta5) (contains sqrt5) and the family Q(zeta3). GALOIS = mu4 x Z2: (Z/30)^x=(Z/5)^x x (Z/3)^x=Z/4 x Z/2 (order 8, max order 4); mu4=(Z/5)^x is the carrier Galois (the seam clock = the carrier golden-field Galois), Z2=(Z/3)^x the family Galois. sqrt5=phi+1/phi=Gauss sum in Q(zeta5). VERDICT: 30=5*6 does not dynamically entangle -- it couples as the cyclotomic compositum, bridge is Galois not dynamical
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