Plain English (1): The declaration J_log_quadratic_approx states that for any real number ε satisfying |ε| < 1, the absolute difference |J_log(ε) − ε²/2| is at most |ε|⁴/20. In other words, near zero the cost function J_log behaves like a quadratic bowl with a controlled fourth-order error term.
Importance in Recognition Science (2): This bound supplies the local quadratic approximation needed to quantify how small log-space perturbations raise the recognition cost. It supports the module’s larger claim that continuous configuration spaces admit no isolated stable minima (because infinitesimal moves cost arbitrarily little), while discrete spaces can lock into finite-cost steps, thereby forcing discreteness for stable RSExists configurations.
How to read the formal statement (3): The theorem takes two arguments: a real ε and a hypothesis hε : |ε| < 1. It concludes the inequality |J_log ε − ε²/2| ≤ |ε|⁴/20. The proof first rewrites J_log ε as cosh(ε) − 1, normalizes the absolute-value expression, and invokes the supporting bound cosh_quadratic_bound.
Visible dependencies and certificates (4): The declaration lives in module IndisputableMonolith.Foundation.DiscretenessForcing. It directly depends on the definition J_log (which equals cosh(t) − 1) and on cosh_quadratic_bound proved earlier in the same file. No external axioms or sorrys appear in its proof; the module as a whole contains zero sorrys in its core forcing statements.
What this declaration does not prove (5): It does not establish the global discreteness-forcing theorems such as continuous_no_isolated_zero_defect or rs_exists_requires_discrete; those appear later in the same module and rely on additional connectedness and stability arguments. It likewise does not address the full forcing chain, Alexander duality, or constant derivations found in other supplied modules.