phi_lt_exp_0483
plain-language theorem explainer
The lemma establishes that a rational upper bound on the golden ratio plus the order-10 Taylor remainder for exp at 0.483 is strictly less than the corresponding partial sum. Numerics researchers establishing certified bounds on log of algebraic constants cite it to justify the comparison step. The proof is a single native_decide call that evaluates the rational inequality inside the kernel.
Claim. Let $x = 0.483$. Let $S_{10}(x)$ be the partial sum $1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! + x^6/6! + x^7/7! + x^8/8! + x^9/9!$ and let $E_{10}(x)$ be the remainder bound $x^{10}·11/(10!·10)$. Then $1.6180340 + E_{10}(x) < S_{10}(x)$.
background
The Numerics.Interval.Log module supplies rigorous interval bounds for the natural logarithm by combining monotonicity with Taylor series error estimates. For log(φ) it uses the substitution x = φ − 1 ≈ 0.618 < 1 and controls the remainder after n terms by |x|^{n+1}/((n+1)(1 − |x|)). The two upstream definitions exp_taylor_10_at_0483 and exp_error_10_at_0483 supply the concrete rational partial sum and remainder for exp(0.483) at order 10.
proof idea
The proof is a one-line term-mode wrapper that applies native_decide. The tactic evaluates the concrete rational comparison 16180340/10000000 + exp_error_10_at_0483 < exp_taylor_10_at_0483 directly in the kernel and returns the certified inequality.
why it matters
The lemma supplies the numerical witness required by the downstream theorem log_phi_lt_0483 that certifies log(φ) < 0.483. In the Recognition Science setting it anchors the placement of log(φ) on the phi-ladder, supporting the eight-tick octave structure and the derivation of mass and coupling scales from the self-similar fixed point φ. It closes the computational step that feeds higher claims about the fine-structure constant interval.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.