lj_phi_connection_approx
plain-language theorem explainer
The theorem shows that the Lennard-Jones minimum-distance ratio 2^{1/6} differs from phi minus one half by less than 0.01, where phi is the golden ratio. Chemists using Recognition Science to model noble-gas van der Waals trends would cite the numerical bridge to the phi-ladder. The proof establishes separate numerical bounds on each side via square-root and sixth-power comparisons, then combines them with linear arithmetic.
Claim. $|2^{1/6} - (φ - 1/2)| < 0.01$, where $φ = (1 + √5)/2$ is the golden ratio and the left-hand side is the ratio at the Lennard-Jones potential minimum.
background
The Van der Waals module derives intermolecular forces from 8-tick ledger fluctuations that induce temporary dipoles, with interaction strength scaling as 1/r^6 and increasing with polarizability down the noble-gas group. The auxiliary definition ljRatioPhiConnection sets the Lennard-Jones minimum ratio equal to phi minus one half, where phi is the self-similar fixed point. Upstream, the definition is described as the φ-connection linking 2^{1/6} ≈ 1.122 to φ - 0.5 ≈ 1.118; related structures include nuclear densities in phi-tiers and ledger factorization.
proof idea
The tactic proof unfolds ljRatioPhiConnection to phi - 0.5, rewrites the latter as √5/2, and applies Real.sqrt_lt_sqrt together with norm_num to obtain 1.117 < phi - 0.5 < 1.119. Parallel bounds 1.122 < 2^{1/6} < 1.123 are obtained via Real.rpow_lt_rpow on sixth powers. The four inequalities are then combined by linarith to conclude the absolute difference is less than 0.01.
why it matters
The result supplies the explicit numerical link between the Lennard-Jones geometry and the phi constant arising at T6 of the forcing chain, supporting the CH-013 derivation of van der Waals forces from 8-tick ledger asymmetries and polarizability. It underpins the sibling noble-gas boiling-point monotonicity statements without invoking the Recognition Composition Law directly. No downstream theorems are recorded, leaving open whether the 0.01 tolerance can be tightened to a sharper phi-ladder identity.
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