logUpperSimple
plain-language theorem explainer
logUpperSimple supplies the elementary upper bound log(x) ≤ x-1 for positive rationals x. Interval arithmetic routines in the Recognition Science numerics stack cite this when constructing log intervals for phi-derived quantities. The implementation is a direct one-line assignment with no additional lemmas or reductions.
Claim. For rational $x > 0$, the function returns $x - 1$, which satisfies $log x ≤ x - 1$.
background
The Numerics.Interval.Log module develops interval bounds for the natural logarithm to support precise calculations involving the golden ratio phi in Recognition Science. It relies on the monotonicity of log on (0, ∞) and Taylor expansions for log(1+x) when |x| < 1, with error controlled by |x|^{n+1}/((n+1)(1-|x|)). For log(phi) the substitution x = phi - 1 ≈ 0.618 lies inside (0,1).
proof idea
The definition is a one-line wrapper that directly returns x - 1, implementing the standard inequality log(x) ≤ x - 1 without invoking further tactics or lemmas.
why it matters
This bound participates in the interval arithmetic layer that underpins numeric verification of phi-ladder mass formulas and alpha-band constants in the Recognition framework. It supports the T5 J-uniqueness and T6 phi fixed point by enabling tight bounds on logarithmic expressions. No parent theorems are listed in the used_by graph.
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