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5411 bytes, as of the version from 2026-10-31 21:13 (current). Recorded here, not run.
# T641 build handoff
Built **Squared Nyman–Beurling approximation distances**,
https://numberdb.org/T641, with exactly 30 entries, N=1,...,30, at 100
significant digits and rigour `proven`. The draft remains unpublished and
not offered for review. Only the controller may offer it after separate
critique and repair. No review action, publication, Git action or source edit
was performed. All retained working artifacts are in this run's build-work
directory.
## Scope and choices
Family issue #281 was read before screening. Live T639 and T640 were read,
as were T16 and T31. The unique-title claim created T641. The cheap corpus
checks (`already_here`, `already_asked`, text search, tags and `api/lookup`
at N=1 and N=2) found no existing distance table. See screening.json and
lookups.json. The lookup values were computed from explicit one- and
two-dimensional formulas. The root requests #24, #225 and #273 have broader
scope; none was closed by this build.
The family convention is retained exactly: ordinary fractional parts,
unanchored rho_n(x)={1/(n x)} in L^2((0,infinity),dx), target 1_(0,1),
unrestricted real coefficients and squared distance. The Dirichlet-polynomial
coefficients have the opposite sign, as explained in the table and derivation.
The sole parameter N is a discrete truncation index. A full prefix through 30
is a practical, noncanonical bound for dense certification and inspectable
supporting vectors. No continuous grid, reference sample, or quantile input
was introduced. No convention was left unresolved. The separate
argument-selection.md covers every free parameter and the breadth review.
Links identify the autocorrelation Gram kernel (T640), the cotangent-sum
arithmetic (T639), the Euler constant entry (T16#0), and the alternative Li
criterion (T31). T640 is linked by its verified stable T-number, which the
site's HREF resolver accepts. No companion slug or unbuilt identifier was
invented. No extra table was built.
The existing tags `number theory` and `functional analysis` are used. The
proposed tag **Nyman-Beurling criterion** is still absent. It would connect
T639 (Bettin–Conrey cotangent sums), T640 (multiplicative fractional-part
autocorrelation), and T641 (squared distances), with the separately proposed
Selberg-error table as a future fourth member. This build did not create it.
## Numerical evidence
- All 30 minima are directly enclosed by 589-bit Arb LDL^T arithmetic, with
positive pivots. The minimum reported accuracy is 537 bits and the largest
radius is less than 1.60e-164.
- An independent 720-bit integral calculation checks all 465 symmetric Gram
entries, using 278 distinct rational-dilation integrals. Piecewise quadratic
integrals, exact rational moments, Hurwitz-zeta tail coefficients and an
explicit alternating-series bound replace cotangent sums entirely.
- Independent Gaussian elimination agrees on every prefix minimum and all
465 optimal coefficients. The largest independent distance radius is less
than 3.67e-136.
- Nonzero known-answer integral and linear-solver controls pass; a deliberately
perturbed kernel is rejected. The one- and two-dimensional closed forms,
30 quadratic norms, 29 Schur decreases, normal-equation residuals, positivity,
and strict monotonicity pass.
- `dry_run.py` passes: 30 entries, longest working enclosure 194 characters,
6.7 KB working block. Stored entries have 100 significant digits.
- The Programs example agrees for all N=1,...,30.
- `verify(sample=None)` returns 30/30 matched, zero differing/missing/extra.
- Both computed enclosures lie strictly within every decimal interval read
back from the database. All 29 stored interval decreases are certified.
- The site's audit returns `{"findings": [], "clean": true}`.
The attached files generate.py, checks.py, argument-selection.md,
derivation.md, numerical-evidence.json and skill-evidence.json reproduce the
computation and explain it. stored-verification.json records the independent
checks on the actual stored document, its SHA-256, the generator/check hashes,
versions, verification report and audit. Source downloads and their hashes
are retained locally in source-archives.json. Public sources were read in this
run; no other run's transcript was used.
## Skill alignment and next stage
Canonical policy manifest:
/home/ubuntu/numberdb-campaign-20261026-hundred/repo-tables-wanted-label/agents/runs/20261031T205515Z-build-skills/manifest.json
Its SHA-256 is
2ed18a5ba549629b2149341ece7834c3c10445d5345622bccc42c66178a837ba.
Both archived versions, skill-local.md and skill-live.md, and the canonical
source snapshot agree at SHA-256
a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278.
The manifest is cited inside numerical-evidence.json through skill-evidence.json.
This is build evidence, not independent critique acceptance. The next critique
must independently assess the argument selection and numerical claims, run the
campaign fingerprint command against the current live draft and all relevant
source attachments, and create its own required arguments sidecar. There is
no build-authored acceptance sidecar. generate.py has no local source imports
other than installed numberdb and Sage; checks.py imports generate.py. Include
argument-selection.md and derivation.md when fingerprinting the supporting
selection definitions. No work or computation remains running.