Simultaneous identities of the golden ratio and Euler's identity, formally verified in Lean 4 over the known Mersenne prime exponents

Keywords: Golden ratio, Mersenne primes, Euler’s identity, Cyclotomic fields, Gelfond–Schneider theorem, Formal verification, Lean 4, Mathlib4, Fibonacci numbers, Modular arithmetic.

We present a sustained study of a single object followed across representations, where the content lives in the equivalences between them, not in any representation in isolation. We prove and formally verify in Lean 4 that the golden ratio $\varphi=(1+\sqrt{5})/2$ satisfies simultaneously twelve identities, inclusions, and identifications across five canonical structures: the complex rotor of Euler’s identity, the real trigonometric and real-multiplicative lines, the arithmetic of Mersenne numbers modulo 20, and the Galois group $(\mathbb{Z}/20\mathbb{Z})^\times$ of $\mathbb{Q}(\zeta_{20})/\mathbb{Q}$.

The central clause is the identity $3\varphi^{\sigma(p)}=2^p$, with $\sigma(p)=p\lambda-\log_\varphi 3$ and $\lambda=\log_\varphi 2$ transcendental by Gelfond–Schneider. Specialised to the 52 known Mersenne prime exponents (GIMPS, 1952–2024), it holds exactly for each, and every pair of Mersenne numbers, in their binary form $2^p=M_p+1$, stands in an exact golden ratio.

The factor 3 coincides with $M_2$, with the number of admissible residue classes of $M_p \bmod 20$, and with the numerator of $M_2/|(\mathbb{Z}/20\mathbb{Z})^\times|=3/8$. The verification is conducted against Mathlib4 with 0 sorry, taking as external axioms only the Gelfond–Schneider theorem and the 44 large GIMPS primalities.