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6666 bytes, as of the version from 2026-10-25 19:33 (current). Recorded here, not run.
# Build handoff: T500
Draft: https://numberdb.org/T500
Title: Landau–Kolmogorov constants on the real line $C(n,k)$.
Family: numberdb-data issue #207. Stage: build, 20261025T191009Z.
The table holds all 190 pairs with 2 <= n <= 20 and 1 <= k < n, in ascending n then k order. Every number is written to 100 significant digits with `rigour: proven`. It remains unpublished and has not been offered for review. A separate critique and repair stage must precede the controller's offer. No publication, review, deployment, Git, or proposal-completion action was taken.
## Convention and selection
The constant is optimal for the supremum-norm interpolation inequality on the entire real line. Functions are real-valued, their (n-1)st derivative is locally absolutely continuous, and the function and nth derivative are essentially bounded. All norms are essential suprema on the real line. Fractional powers are positive real powers. The parameters are ordered (n,k), with the full domain n >= 2, 1 <= k < n.
The family fixes the normalization, indexing, Favard notation K_r, 100 digits, and proven rigour. This build chose the cutoff n = 20 to provide all intermediate derivatives for low and moderate derivative orders, without filling a large triangle merely to meet an entry target. There are no holes in the selected triangular domain. The largest written decimal is 101 characters; the measured block of serialized numbers is about 27 KB before the small number of entry comments and actual parameter-key overhead. The working-ball representation measured about 34 KB.
## Screening and sources
Read issue #207 before computation and read both the checkout and live NumberDB skills. The checkout's stricter selection policy was followed. `already_here` and `already_asked` were run for Landau Kolmogorov, Favard, Euler splines, and Sobolev. The corpus had no table for the requested family. Digit lookups for sqrt(2), (9/8)^(1/3), and 3^(1/3) found only unrelated occurrences of sqrt(2), and no occurrences of the other two. The title claim created T500.
The real-line convention and Euler-spline extremizers were checked in Schoenberg and Cavaretta, *Solution of Landau's problem concerning higher derivatives on the halfline*, pp. 297–299, especially the discussion of the full-line result on p. 298. The NIST DLMF Fourier expansions 24.8.4–24.8.5 connect these spline polynomials with the Favard sums. The document includes the family's Finch reference, MathWorld's real-line formulas, Wikipedia's inequality convention, and OEIS A245198 for C(3,1). Finch was not independently downloaded; the page range was checked against MathWorld's bibliography.
The OEIS b-file returned HTTP 403 during this build. Instead, the checker verifies the displayed decimal from the OEIS entry against the independently established exact cube 9/8. The unavailable b-file is not claimed as a check. A MathNet source download also failed and was not used.
Internal links were obtained from corpus search results and their target documents were read. The draft links Euler polynomials and the sharp Gagliardo–Nirenberg constants of Del Pino and Dolbeault. It does not link to the unbuilt Favard proposal.
Tags used: analysis, special values, inequality, functional analysis. The proposed `approximation theory` tag is absent. It would serve this table, *Lebesgue constants of the Fourier partial sums* (T499), and *Lebesgue constants of the classical interpolation node families*, as well as the planned Favard table. No tag was invented in the draft.
## Computation and checks
`generate.py` computes K_r by Riemann zeta for odd r and a difference of Hurwitz zeta values for even r. Every arithmetic operation is exact rational or Arb ball arithmetic. Computation uses Sage 10.9 and numberdb 0.1.10, with 397 bits for 100 requested digits and 64 guard bits. The largest measured radius was 4.2349018e-119.
`checks.py` independently constructs the antiperiodic spline pieces E_r(x)/r! by exact rational integration. Exact Sturm sequences locate all critical points in (0,1), and endpoint and midpoint evaluations determine the norms A_r. It checks the identity C(n,k)^n = A_(n-k)^n / A_n^(n-k) for every pair. This route uses neither zeta values nor pi to obtain the rational powers.
Completed checks:
- All 190 generated values agree with independent rational-power computations.
- All 20 Favard norms agree with exact spline norms multiplied by pi^r.
- The 21 Euler polynomials through degree 20 agree exactly with the existing corpus table.
- Eight published small cases agree exactly after taking the appropriate integer powers.
- Known-answer controls and deliberately incorrect controls validate the Sturm and interval checks.
- Exact numerator and denominator root tests show that none of the 190 constants is rational.
- Repository `dry_run.py` reports no exactness, comment, or disguised-exactness failures.
- The displayed Programs snippet was executed and checked on all 190 pairs.
- `verify(sample=None)` reports 190/190 matched, zero differing, zero missing, zero extra.
- Every stored decimal interval was checked with exact rational endpoint powers against the independent rational value. All 190 pass and their order is the complete triangle.
- The authenticated site audit returns `findings: []`, `clean: true`.
The mathematical theorem supplying sharpness is sourced; the computational checks certify the values of its formula and the decimal enclosures, not a new proof of Kolmogorov's theorem.
## Workflow findings and evidence
The first generator submission was refused before accepting numbers because entry comments cited OEIS while the initial claim document did not yet contain that reference. The completed prose and all checked values were then submitted together. The generator subsequently confirmed agreement and attached its source files. The API returns Numbers grouped by n and k, so the read-back checker handles that form as well as flat entry lists.
The audit initially flagged the definition's length. The final definition is 300 characters and retains the same function class, inequality, and norm convention. No stored numerical value needed correction.
Artifacts in this directory include screening.json, claim-response.json, document-response.json, preflight.json (with every computed decimal), dry-run.log, stored-document.json, stored-checks.json, snippet-check.log, audit.json, generate.py, checks.py, check.py, and euler-controls.json. The generator and checks are attached to the website draft; the database is authoritative. Failed attempts and source downloads are preserved here for the separate critique.