The Bernstein constant: ten rigorously certified digits

A rigorous, machine-checkable enclosure of the Bernstein constant beta, proving ten correctly-rounded decimal places: beta = 0.2801694990. The 1985 Varga-Carpenter rigorous enclosure, used here as the comparison baseline, determines five. Every bound is proved in interval arithmetic; no step relies on a numerical estimate.

BEGIN CERTIFIED ENCLOSURE
lower_endpoint: 0.280169499016595460711186
upper_endpoint: 0.28016949904799999998
rounded_value: 0.2801694990
correctly_rounded_places: 10
END CERTIFIED ENCLOSURE

A machine-checkable certificate for the Bernstein constant beta = lim 2n E_2n(|x|; [-1,1]):

0.280169499016595460711186 <= beta <= 0.28016949904799999998

Both endpoints are proved, and the printed decimals are rounded outward so that the printed endpoints are themselves valid bounds. The enclosure has width 3.140454e-11 and determines ten correctly-rounded decimal places, beta = 0.2801694990.

The upper endpoint is certified at m = 64000 via the classical inequality beta <= 2 mu_m (Varga and Carpenter, Constr. Approx. 1 (1985) 333-348, eqs. 3.5-3.10, attributing the limiting relation to Bernstein). Because mu_m is an infimum, no optimality proof is needed: a near-optimal coefficient witness is found numerically and then a single rigorous sup-norm bound is certified. The bound is established by adaptive branch-and-bound in Arb ball arithmetic at 160 bits over a partition of [0, 64000] into the 64001 pole intervals, plus a monotonicity lemma for the tail, across 1069 shards and 436,201,931 cells, with every emitted per-interval bound an exact rational.

Proved digits, not estimated ones. The rigorous enclosure of Varga and Carpenter (eq. 1.16) determines five correctly-rounded decimal places, beta = 0.28017. The same paper's widely-quoted ~50-digit value (eq. 1.18) is a Richardson extrapolation, described there as "probably accurate to 50 decimal places" -- an estimate, not a proved bound, and nothing in this release rests on it. The ten correctly-rounded places certified here are proved: every emitted per-interval bound is an exact rational, and the enclosure is narrower than an outward rendering of the published rigorous pair by a factor of 153901. No priority claim is made or implied -- see the comparison and its limits in LITERATURE.md.

This is a computational contribution. The underlying inequality is classical; this release contributes a large-scale rigorous evaluation of it and a certificate whose aggregation and digit claim can be re-checked from the raw data. It is not a new theorem in approximation theory and is not presented as one.

Verification is layered. Level 1 needs only the Python standard library: it validates every shard against a strict schema including array cardinality, re-derives the partition of all 64001 pole intervals, and recomputes the exact rational maximum from the raw shard files. A separate stdlib-only check verifies the published enclosure and the ten-place digit claim end to end. Level 2 adds mpmath for an independent evaluation of the residual that shares no code with the certifying kernel. Level 3 adds python-flint for the full strict recombination including the tail bound. An included negative-test suite demonstrates that 36 distinct mutation classes are each rejected by the level responsible for them (37 expected outcomes: one mutation is required to be accepted by aggregation and rejected by independent evaluation), so that a passing check is meaningful. The suite also documents the limits of each level: aggregation cannot detect an understated bound, because it never evaluates the residual.

The monotonicity lemmas are proved, not assumed. The branch-and-bound relies on the strict monotonicity of the residual between poles, a tail lemma, and a pole-adjacent correction. Earlier release candidates listed these as trusted-unproved inputs, which made the enclosure conditional; PROOFS.md now proves them from an integral representation of the digamma difference, and make lemmas corroborates each one numerically. The claim is therefore unconditional given the trusted inputs below, and the release records that as a checked field rather than as prose.

What is trusted, and what is unavailable — stated here and in Section 5 of the included note, not only inside the archive:

• The classical Varga-Carpenter results are used as published and are not reproved here: the upper reduction beta <= 2 mu_m and, separately, the lower bound of eq. (4.6) with its admissibility conditions. Each carries its own transcription risk, and that transcription is a common-mode input 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. That is parallelism, not independent evidence, and no independent rigorous upper-bound implementation exists.
• The production FLINT/Arb C library version was not recorded and cannot be recovered; the exact production environment is not reproducible. The versions used for the release verification are recorded in the transcript.
• No producer chain of custody is released. The 1069 manifests record hashes and the shipped files match them, but that does not by itself prove those files generated the recorded outputs. The producer git history is not included.
• There is no continuous integration. The included cleanroom transcript is one manually initiated final run, not continuous verification.

The lower endpoint, by contrast, was cross-checked by two independent backends agreeing to 19 decimals; Arb supplies the rigour and mpmath only a numerical cross-check.

No priority claim is made, and none is implied. The included LITERATURE.md records the Varga-Carpenter enclosure used as the comparison baseline and nothing else. This release makes no statement about whether later rigorous enclosures exist, about novelty, or about being best known. The author's admission rule requires a subject-matter expert to be contacted before any such statement, and that step is open.

Licensing: data and text are CC BY 4.0 (the record license shown here); code files are MIT. See LICENSES.md in the archive for the path-by-path mapping.

Documentation page: https://severinvisionary.github.io/bernstein-constant-certificate/ — the abstract, the method, the comparison against the 1985 baseline, and the trust boundary, on one page.