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A registered result in its source context

Original ledger classification: Identity/Lie [I]/[L]

Recorded statement: A cyclic element INSIDE the compiler hull (computed, not asserted): built from the E8 Cartan matrix, the Coxeter element c=s1...s8 has order EXACTLY h(E8)=30=|Z2|*N_fam*g_car=2*3*5 (verified by matrix powers, v55), and its eigenvalues are…

Statement, assumptions and sources

Paper context: Origin Theory

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SEAM.CYCLE.01

Original ledger classification: Identity/Lie [I]/[L]

Original claim text

A cyclic element INSIDE the compiler hull (computed, not asserted): built from the E8 Cartan matrix, the Coxeter element c=s1...s8 has order EXACTLY h(E8)=30=|Z2|*N_fam*g_car=2*3*5 (verified by matrix powers, v55), and its eigenvalues are primitive 30th roots exp(2pi i m/30) with m the E8 exponents {1,7,11,13,17,19,23,29} = EXACTLY the phi(30)=8 totatives of 30. Hence rank E8=8=phi(30)=#exponents=#live phases of the order-30 cycle => the 8 in c3=1/(8pi) is the live-phase count of a primitive order-30 cyclic rotation of the hull, period 30=|Z2|*N_fam*g_car. PLUS one-alpha cross-sector: S_dS*rho_Lambda=1/(128 c3^4)=32 pi^4 (de Sitter Gibbons-Hawking entropy ~ 1/Lambda), so the SAME EM fixed point alpha^-1=137.036 sets the cosmic horizon entropy and Lambda smallness ~10^-122=e^(-2 alpha^-1). FURTHER: S_dS=2^g_car*pi^|mu4|*e^(2 alpha^-1) - the entropy prefactor 32 pi^4 = (Dirac spinor dim of D5=SO(10): S+ + S- =16+16=2^g_car) x pi^|mu4|, with 128=2^7 (7=scalaron exponent g+n-1): cosmic horizon entropy = carrier Clifford dim x pi^glue x EM fixed point. [I]/[L]; cyclic-cosmology reading is [P], not included

Original classification detail

Identity/Lie [I]/[L]

Dependencies as recorded

v6; v8; v3; h(E8)=30

Open the original ledger source, row 134

Location recorded in the ledger

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Registered verification modules

  • v55_coxeter_cycle.pyRegistered verification module · horizon
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    E₈ Coxeter order 30 = 2·3·5; 8 = φ(30); S_dS·ρ_Λ = 32π⁴

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