euler_phi_connection
plain-language theorem explainer
Recognition Science models link the golden ratio to trigonometric identities through the relation cos(π/5) = phi/2, which gives the real part of e^{i π /5}. Modelers of oscillatory systems or phase dynamics in the eight-tick framework cite this for bridging continuous exponentials to discrete J-costs. The proof reduces the claim via the standard cosine identity for π/5 followed by algebraic verification with the definition of phi.
Claim. $cos(π/5) = φ/2$, where $φ = (1 + √5)/2$ is the golden ratio fixed point. This supplies the real part of $e^{iπ/5}$ in the Euler connection.
background
The module MATH-003 targets derivation of Euler's number e from φ-summations. In Recognition Science, e arises as the base for J-cost decay probabilities, continuous-time oscillations e^{i ω t}, and 8-tick phase factors exp(2 π i k /8). The golden ratio phi is the self-similar fixed point forced in the unified chain.
proof idea
The tactic proof first rewrites using the Mathlib identity for cos(π/5), which equals (1 + √5)/4. It then unfolds the definition of phi as (1 + √5)/2 and applies the ring tactic to equate the scaled expressions.
why it matters
This fills the trigonometric step in the MATH-003 program for e from φ-summations. It enables the RS interpretation of e in ledger dynamics and 8-tick phases, tying to the forcing chain steps T6 (phi fixed point) and T7 (eight-tick octave). The doc-comment highlights its role in connecting e^{iπ/5} real part to phi/2. With no listed downstream theorems, it remains open how this extends to full derivations of e or applications in astrophysics tiers.
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