To Count a Stone with Six Birds: A Mathematics is A Theory

Many “higher” mathematical objects are introduced as definitions (limits, completions, analytic continuations), but it is often unclear _when_ a large discrete protocol admits a stable continuous closure and _how_ to test that stability without overclaiming. This question matters because premature or untested closure claims can propagate errors through dependent results. We instantiate the Six-Birds Theory (SBT) framework of [1]—which provides a general audit methodology for validating compressions of discrete data into continuous or idealized objects—as a closure-and-audit method for mathematics: a discrete substrate is staged by refinement, compressions are assessed by an explicit defect ledger, and a packaged object is accepted only when defects shrink or stabilize under refinement. Operationally, we (i) formalize two anchor statements in Lean/mathlib [2], [3]—the exact finite-difference Leibniz identity with its explicit remainder term and an algebraic uniqueness anchor for derivations on \(R{\lbrack X\rbrack}\)—and (ii) implement falsification-first Python diagnostics that compare competing closure routes under matched controls and an artifact contract (tables and macros are generated from snapshot-visible pointer JSONs). Across four exhibits we observe controlled, checkable separations. First, stability-only stencil filtering selects order-\(0\) closures, while adding a Leibniz-defect gate selects derivative-like order-\(1\) closures. Second, route mismatch (RM, measuring disagreement between closure paths) under a small coordinate change decays with refinement (power-law fit exponent \(p \approx 1.46\)). Third, for prime-based closures in a convergence control regime \({\Re{(s)}} > 1\), route mismatch \({RM}_{2}\) decreases with sample size \(N\) (e.g. \({{RM}_{2}{(800)}} \approx {5 \times 10^{- 2}}\)), whereas the same diagnostic in the critical strip exhibits mismatch growth by many orders of magnitude under naive staging (e.g. \({{RM}_{2}{(800)}} \approx {5 \times 10^{12}}\)). Fourth, in a self-dual toy family, tightening a positivity constraint sharply confines zeros to the symmetry locus (mean radial deviation drops from \(\approx 0.87\) to \(\approx {3 \times 10^{- 4}}\)). These results provide a reproducible methodology to separate feasible closures from protocol artifacts and to compare closure proposals across regimes. We emphasize that this paper does not establish new theorems about \(\zeta\) or zero distributions; we report audit-style diagnostics under explicit controls and document the upgrade points required for stronger claims.