Certification Complexity: A Computable Precision Lower Bound for Computer-Assisted Proofs

Computer-assisted proofs establish upper bounds: a certificate at arithmetic precision p shows that a solution exists. We make rigorous the dual — a computable precision below which no radii-polynomial certificate can close — and show that the resulting certification cost χ is unbounded within a fixed finite-dimensional problem class.

Four ingredients. (i) A three-line impossibility theorem: if the roundoff-inflated interval Jacobian contains a singular matrix then Z₀ ≥ 1 for every approximate inverse, so no certificate of that shape closes — with a trigger decided exactly by interval evaluation. (ii) A two-sided sandwich in arbitrary-precision interval arithmetic: for a near-singular family the exhibited certificate (upper) and the impossibility theorem on the p-bit enclosure (lower) bracket χ to within one bit at p ≈ log₂κ — for instance χ ∈ (82, 83], i.e. χ = 83 exactly, at κ ≈ 2⁸² — rising without bound as κ → ∞. (iii) A controlled nonlinear instance, a cubic limit-cycle family whose collocation conditioning diverges as the cycle approaches non-hyperbolicity, where the standard certifier's 53-bit reach ends at κ ≈ 1/u, and whose non-hyperbolic endpoint has χ = ∞ outright. (iv) An intrinsic/artifact dichotomy separating walls that no certifier in a precisely delimited query model escapes from walls that are properties of a representation — with a sharp invariant, the integral cancellation condition number κ_Σ, deciding which.

The dichotomy is the paper's main corrective contribution, and it exists because an earlier version of this work claimed the opposite and was refuted by adversarial review. We drive the machinery into Shi Songling's four-limit-cycle quadratic configuration (1980), where κ_Σ → ∞ forces every rounding-incurring certifier to wall, and separate two family-indexed upper bounds — ≈419 bits for the divergence family, ≈665 for the multiplier family — neither of which is a class minimum. Along the way we certify calibration ground: Kuramoto–Sivashinsky equilibria within 3.6–7.4× of published enclosures, the shortest Lorenz orbit to period width ≤ 3.4×10⁻¹¹, and van der Pol cycles.

Reproducibility. Every number in the paper regenerates from the artifact repository arperon-labs/certification-precision-lower-bounds at the commit pinned in §8 (snapshot attached to this record as a zip; code license Apache-2.0): cargo test --release reproduces every quoted figure and python verify_claims.py re-derives the headline numbers from that run.

Methodology disclosure. The results were produced using an agentic AI system directed and adjudicated by the author, under pre-registered escalation budgets and per-phase adversarial review; Section 7 documents the methodology, including the reviews that refuted the author's own earlier claims.