The Coxeter clock and its quarter action
Original ledger classification: Identity (exact)
Recorded statement: COXETER TOTATIVE CLOCK (2026-06-16): the mu4 clock is the order-4 character of the E8 Coxeter cycle. (Z/30)^x={1,7,11,13,17,19,23,29}=the E8 exponents=live Coxeter phases; phi(30)=8=rank E8, 30=|Z2|*N_fam*g_car=2*3*5. (Z/30)^x ~ Z/4 x Z/2,…
Statement, assumptions and sources
Paper context: Origin Theory
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Read the exact registered claim and its dependencies
COX.CLOCK.01
Original ledger classification: Identity (exact)
Original claim text
COXETER TOTATIVE CLOCK (2026-06-16): the mu4 clock is the order-4 character of the E8 Coxeter cycle. (Z/30)^x={1,7,11,13,17,19,23,29}=the E8 exponents=live Coxeter phases; phi(30)=8=rank E8, 30=|Z2|*N_fam*g_car=2*3*5. (Z/30)^x ~ Z/4 x Z/2, order-4 generator 7 (7^2=19,7^4=1 mod 30): a literal mu4 clock <7>={1,7,13,19} on the exponents; conjugate pairs m+(30-m)=30 give |mu4|=4 invariant 2-planes the clock permutes. Only E8 has phi(h)=rank AND h=2*3*5. Complements MCKAY.E8.01 (McKay gives (2,3,5) from the finite group, this gives mu4 from the Coxeter cycle).
Original classification detail
[E] (Z/30)^x = E8 exponents, order-4 element 7 = mu4 clock on the exponents, 4 invariant planes; seam-clock identification [C]
Dependencies as recorded
SEAM.COX.01; AX.P2.01
Open the original ledger source, row 308Location recorded in the ledger
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- Origin Theory
Sections that explicitly cite this claim
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- 2026-06-16 (the icosahedral bedrock, CM-norm duality, and a structural-finds round)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.
- 2026-06-21 (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
- 2026-06-17 (CP phases as [formula in source] powers — a reduction of red-team Target D)Research changelog; TFPT — Introduction and reading guide
- The whole integer skeleton from one pair [formula in source]input 4 origin theory
- Family [formula in source] from [formula in source] (proved, with the metric settled)Architecture, E8 and the electromagnetic fixed point
- The Sheet Diamond and the Winding Line: masses and transport from one surfacetfpt 2 standard model
- Family [formula in source] from [formula in source] (proved, with the metric settled)tfpt 1 architecture e8
- The whole integer skeleton from one pair [formula in source]input 4 origin theory
- The compiler: why [formula in source]TFPT — Introduction and reading guide
All 15 linked sections
- The whole integer skeleton from one pair [formula in source]
- Outcome
- 2026-06-22 (bandwidth search of the unmapped [formula in source] region, with a strict forced-identity discriminator: the genuine overlooked structure is COLLECTIVE ( [formula in source] , [formula in source] , exponents [formula in source] totatives of [formula in source] ); the per-degree atom matches are unforced and DECLINED; no new physical hit – reconciling v66 and v354: v355)
- Front matter / support
- Front matter / support
- Family [formula in source] from [formula in source] (proved, with the metric settled)
- The whole integer skeleton from one pair [formula in source]
Registered verification modules
- v223_coxeter_totative_clock.pyRegistered verification module · horizon
Read the description recorded in the module registry
The μ₄ clock is the order-4 character of the E₈ Coxeter cycle. (ℤ/30)ˣ={1,7,11,13,17,19,23,29}=the E₈ exponents=live Coxeter phases; φ(30)=8=rank E₈, 30=2·3·5. (ℤ/30)ˣ≅ℤ/4×ℤ/2 with order-4 generator 7 (7²=19,7⁴=1 mod 30): a literal μ₄ clock ⟨7⟩={1,7,13,19} on the exponents; conjugate pairs m+(30−m)=30 give |μ₄|=4 invariant planes the clock permutes. Only E₈ has φ(h)=rank AND h=2·3·5.
A registered module is a source reference. Its presence does not assert that the module was executed in this snapshot.