Explanation of J_log_quadratic_approx

(1) Plain English: The theorem asserts that for any real number ε with |ε| < 1, the absolute difference |J_log(ε) − ε²/2| is at most |ε|⁴/20. In other words, near the origin the cost function J_log is quadratically approximated by ε²/2, with a concrete fourth-order error bound.

(2) Relevance to Recognition Science: In RS the cost J(x) = (x + x⁻¹)/2 − 1 quantifies configuration defect. Its log-coordinate form J_log(t) = cosh(t) − 1 forms a convex bowl whose curvature at the minimum sets the minimal step cost. The quadratic bound shows that infinitesimal perturbations in a continuous space incur only O(ε²) cost, allowing drift and preventing stable isolated minima. This is the local analytic ingredient used to argue that stable RS configurations require discrete configuration spaces.

(3) Reading the formal statement:

theorem J_log_quadratic_approx (ε : ℝ) (hε : |ε| < 1) :
    |J_log ε - ε^2 / 2| ≤ |ε|^4 / 20

It takes a real ε together with the hypothesis |ε| < 1 and returns the stated inequality. The proof first rewrites J_log ε via J_log_eq_J_exp to Real.cosh ε − 1, normalizes the absolute-value term, then invokes cosh_quadratic_bound.

(4) Visible dependencies and certificates: The declaration lives in module IndisputableMonolith.Foundation.DiscretenessForcing. It directly depends on the definition J_log and the auxiliary bound cosh_quadratic_bound (both in the same module). It also uses J_log_eq_J_exp for the cosh rewriting. No external axioms are introduced; the proof is grounded in Mathlib’s Real analysis plus the module’s own lemmas J_log_nonneg and J_log_eq_zero_iff.

(5) What it does not prove: The theorem supplies only a local Taylor-style error bound. It does not establish the global instability result continuous_no_isolated_zero_defect, the definition of stability IsStable, or any claim that RS existence forces discreteness. It likewise does not address the broader forcing chain or constants derived in other modules.