Explain the Lean theorem `log_phi_gt_0481` in module `IndisputableMonolith.Numerics.Interval.Log`. Write for an educated reader who knows science and programming but may not know Lean. Cover: (1) what the declaration says in plain English, (2) why it matters in Recognition Science, (3) how to read the formal statement, (4) visible dependencies or certificates in the supplied source, and (5) what this declaration does not prove. Cite only declarations present in the supplied Recognition source.

1. Plain English Statement

The theorem log_phi_gt_0481 formally establishes that the natural logarithm of the golden ratio, $\ln(\phi)$, is strictly greater than the real number $0.481$.

2. Importance in Recognition Science

In the RS framework, $\ln(\phi)$ is not an arbitrary mathematical value; it defines the fundamental bit cost of the universe, established as MODEL J_bit. It represents the minimum non-trivial energy/cost increment in the discrete interaction ledger. Bounding this constant rigorously is essential because RS derives downstream physics parameters (such as the fine structure constant $\alpha$ and physical masses) mathematically. To make verified empirical predictions, the formal system needs certified rational bounds (via interval arithmetic) rather than relying on unverified floating-point approximations.

3. Reading the Formal Statement

theorem log_phi_gt_0481 : (0.481 : ℝ) < log Real.goldenRatio

• (0.481 : ℝ): The literal numeric value 0.481 explicitly cast to the type of real numbers (ℝ).
• <: The strict "less than" relation.
• log: Mathlib's natural logarithm function.
• Real.goldenRatio: Mathlib's standard definition of $\phi = \frac{1+\sqrt{5}}{2}$.

4. Visible Dependencies and Certificates

The proof relies on exact rational arithmetic and calculus bounds to avoid floating-point errors:

• Monotonicity: It first transforms the goal to $\exp(0.481) < \phi$ using Mathlib's Real.lt_log_iff_exp_lt.
• Taylor Series: It computes a 10-term Taylor series expansion of $\exp(0.481)$ using exact rational fractions (encoded as the private definition exp_taylor_10_at_0481).
• Error Bounding: It calculates the maximum possible truncation error for those 10 terms (exp_error_10_at_0481) via Real.exp_bound'.
• Rational Comparison: It adds the Taylor sum and the error bound, then verifies via Lean's native_decide that the total is strictly less than $1.61803395$ (the private lemma exp_0481_lt_phi). Since $\phi > 1.61803395$, transitivity proves the theorem.

5. What this Declaration Does Not Prove

• Upper Bound: It only provides a mathematical floor. The ceiling is handled by parallel theorems, such as log_phi_lt_0483.
• Physical Origin: This is purely a numerical utility theorem. It does not prove why $\phi$ governs physics. The structural derivation of $\phi$ from discrete ledger self-similarity is proved separately by THEOREM phi_forced and THEOREM phi_forcing_complete.