Bernstein's constant: β = 0.2801694990, ten rigorously certified digits

A machine-checkable enclosure of the constant of best uniform polynomial approximation to |x|, proved in interval arithmetic. Ten correctly-rounded decimal places, where the 1985 Varga–Carpenter rigorous enclosure determines five.

Abstract

Bernstein (1913) proved that the limit

β = lim_{n → ∞}  2n · E_{2n}( |x| ; [−1, 1] )

exists, where E_k(f; K) is the error of best uniform polynomial approximation of degree at most k. This release is a machine-checkable certificate for

0.280169499016595460711186  ≤  β  ≤  0.28016949904799999998

Both endpoints are proved, and the printed decimals are rounded outward, so the printed endpoints are themselves valid bounds. The enclosure has width 3.140454e−11 and determines ten correctly-rounded decimal places.

Against the published baseline

Proved digits, not estimated ones. The same 1985 paper gives a widely-quoted ~50-digit value (eq. 1.18) obtained by Richardson extrapolation and described there as “probably accurate to 50 decimal places” — an estimate, not a proved bound, and nothing here rests on it. The ten places certified here are proved: every emitted per-interval bound is an exact rational, and the enclosure is 153901× narrower than an outward rendering of the published rigorous pair. That is a comparison against one baseline, not a claim of novelty.

How it is certified

• Upper endpoint. The classical reduction β ≤ 2μ_m (Varga–Carpenter eqs. 3.5–3.10) at m = 64000. Because μ_m is an infimum, no optimality proof is needed: a near-optimal coefficient witness is found numerically, then a single rigorous sup-norm bound is certified by adaptive branch-and-bound in Arb ball arithmetic at 160 bits over the 64001 pole intervals of [0, 64000], plus a proved monotonicity lemma for the tail — 1069 shards, 436,201,931 cells, every emitted bound an exact rational.
• Lower endpoint. Varga–Carpenter eq. (4.6) at m = 1000, decimal floored, cross-checked by two independent backends (Arb and mpmath.iv) agreeing to 19 decimals.
• The monotonicity lemmas are proved, not assumed, from an integral representation of the digamma difference; they were listed as trusted inputs in earlier release candidates, which made the enclosure conditional.
• Layered verification. Level 1 needs only the Python standard library and re-derives the partition and the exact rational maximum from the raw shards; Level 2 adds an independent mpmath evaluation sharing no code with the certifying kernel; Level 3 adds python-flint for the strict recombination and the tail bound. A negative suite of 36 mutation classes, 37 expected outcomes shows each level can fail.

What is trusted, and what is not established

• The classical Varga–Carpenter results are used as published and not reproved. That transcription is common-mode to every backend, so agreement between backends cannot detect an error in it.
• The upper endpoint rests on a single interval-kernel implementation, executed in parallel by eleven workers. That is parallelism, not independent evidence; no independent rigorous upper-bound implementation exists.
• The production FLINT/Arb C library version was not recorded and cannot be recovered, so the exact production environment is not reproducible.
• No producer chain of custody is released, and there is no continuous integration — the shipped transcript is one manually initiated clean-room run.
• No priority claim is made, and none is implied. The comparison baseline is the Varga–Carpenter enclosure and nothing else; this release makes no statement about whether later rigorous enclosures exist. Readers aware of intervening work are asked to open an issue.
• An eleventh place is out of reach with this witness at m = 64000 for a provable reason rather than a budgetary one, and that statement is deliberately witness-specific.

Verify it yourself

git clone https://github.com/SeverinVisionary/bernstein-constant-certificate
cd bernstein-constant-certificate
make stdlib-only      # headline claim, Python standard library only, ~1 min

The headline number is re-derivable with no third-party packages and no network. make verify adds the independent evaluator and the negative suite; make release-gate runs everything and writes one machine-readable summary. What each level does and does not establish is tabulated in VERIFY.md.

Read further

• The computational note — method, the digit criterion, what is proved and what is trusted
• The computational note as a PDF — a typeset rendering of that same NOTE.md, from main at commit b14c5e8, with a title block added and the text unchanged; not a separate document
• VERIFY.md — what each verification level establishes, and what it does not
• PROOFS.md — proofs of the monotonicity, tail and pole-correction lemmas
• TRUSTED_INPUTS.md — everything relied on but not established here
• LITERATURE.md — the comparison baseline, and what is deliberately not claimed
• Zenodo deposit — the archive, its checksum, the dependency wheels and the clean-room transcript

Cite

Yang, Hanyu (2026). The Bernstein constant: ten rigorously certified digits
(Version 1.0.0) [Data set]. Zenodo. https://doi.org/10.5281/zenodo.22106774