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# Build handoff: T468
Draft: https://numberdb.org/T468
Title: Probability that $n$ uniform random points of a convex body are in convex position.
Family: numberdb-data issue #200.
Run: 20261024T211700Z-build.
## Result and scope
The draft holds 18 entries: every n from 3 through 8 for triangles,
parallelograms and ellipses. All regions use independent uniform area sampling
in the interior. Each region is an affine class, with the family's canonical
keys `equilateral-triangle`, `unit-square`, `unit-disk`. The ordered parameters
are `(region, n)`, and the value column is headed `p_K(n)`.
Thirteen entries are exact rationals, including three exact values of 1.
Five disk/ellipse probabilities are certified 100-significant-digit decimal
enclosures. The largest serialized entry is 104 characters; the annotated
entries block measured approximately 2.38 KiB. The generator, reusable check
helper, independent checks, and numerical evidence are attached to the draft.
No publication, review, or offer-for-review action was taken. The controller
must run the separate critique and repair stages before offering the draft.
The controller also records the successful new build in the proposal queue.
## Decisions that the proposal left to this run
- The scope is planar convex bodies with nonempty interior. The literature
and the family's three canonical affine examples agree on this convention.
- The stored range is the common rectangle n=3,...,8, covering all five
nontrivial disk expressions explicitly displayed by Marckert. Larger
rational-only tails were not added. The parameter domain remains n>=3.
- Exact rational values are stored exactly, superseding the older family's
suggestion to turn them into 100-digit decimals. Both current skills
explicitly prefer exact rational storage; T467 uses the same mixed approach.
- The disk source displays 1-p, so the generator takes its complement.
- The body argument denotes what the probability is of; it does not select
different quantities. Thus all three regions belong in one table.
## Checks actually run
1. `already_here` and `already_asked` from the supplied screen helper;
supplementary searches for convex position, Sylvester and Valtr; direct
API lookups of 25/36, 1/2 and a disk-probability prefix. Existing hits were
unrelated families. The title-unique draft claim succeeded as T468.
2. The supplied `dry_run.py`, `exactness`, `measure`, `agrees_with`,
`names_its_rings`, and prose checks. All final checks passed.
3. Exact factorial expressions independently agree with all 12 rational
polygon entries generated by recurrences.
4. Direct Dyck-path enumeration verifies the parallelogram probabilities
through n=8. T264's actual stored q-Catalan polynomials, evaluated at q=1,
independently give the same counts for n=4,...,8. Its definition provides
C_2(q)=q+1 for n=3. Its actual slug was read from the API search result.
5. Arb evaluation uses 413 working bits, including an 80-bit guard. Maximum
radius is 1.9469747e-123; minimum relative accuracy is 120.6665 decimal
digits before rounding. An independent mpmath evaluation of the displayed
complement formulas at 160 digits lies inside every computational ball.
6. Marckert's segment convolution integral, using L0,...,L5, was independently
integrated at 130 and 170 decimal digits for n=3,...,7. The normalization
control n=3 gives 1. All results agree with the closed forms to at least
120 and 160 decimal places respectively.
7. A separate exact line-coordinate calculation of mean random-triangle area
gives 35/(48*pi^2) after normalizing by disk area, hence the disk n=4
probability via p(4)=1-4*mean-area-ratio. The line-coordinate Jacobian's
normalization was checked against the product of two disk areas first.
8. Fixed-seed uniform sampling used 500000 samples per region and tested all
six sample sizes. Every probability estimate was within 1.976 standard
errors of its formula. Tests included known convex/nonconvex configurations,
sampler coordinate moments, normalized triangle areas and affine images of
1000 samples per region and sample size.
9. `verify(sample=None)` reported 18/18 matched, zero differing, missing or
extra. Readback checks test exact rationals, polygon recurrences, every
stored disk interval against the independent formula evaluation, and the
disk integral through n=7 against those actual stored intervals.
10. The final prose-only edit shortened the definition. Its Numbers block is
asserted identical to the verified snapshot; no numerical recomputation
or resubmission was needed for that edit.
## Limits of the independent checks
The disk n=8 formula has an independent geometric simulation check, but the
second high-precision evaluation rearranges the same published formula.
It is not an independent derivation at 100 digits. The draft says this
explicitly. The certified precision comes from ball arithmetic applied to the
published exact formula; neither the simulations nor quadratures certify it.
The separate critique should inspect this transcription, especially the
complement and signs in the n=8 numerator.
## Audit disposition
The final authenticated site audit has one finding and reports `clean: false`.
It flags the three symbolic region labels as possible multiple quantities.
This is an acknowledged structural warning: region is the body argument of
one probability p_K(n), explicitly permitted by the skill and required by
family #200. Both definition and comments state the quantity and argument.
The implementation flags every 2-to-4-valued nonnumeric parameter with multiple
rows, irrespective of its mathematical role. No schema workaround was used.
The initial long-definition warning was fixed; there are no remaining
reference, tag, heading, snippet, or internal-link findings. The API audit does
not check outward links. The arXiv preprint and DIMACS abstract were accessed;
the author PDF and publisher issue PDF were unavailable from the build tools.
## Tags
The draft uses the existing `probability theory` and `elementary geometry`
tags. `geometric probability` does not exist. The family proposes that tag for
T468, T467 (Mean distance between two uniform random points of a region),
T19 (Polya's random walk constants) and T283 (Mahler measures associated with
uniform random walks in the plane), as well as its remaining proposals.
No published table or tag was edited.
## Reproduction and artifacts
`generate.py` reproduces the values; `checks.py` runs the independent checks;
`check.py` is an unmodified copy of the supplied helper. Run these with Sage
or the campaign's `agents/sage.sh`. For the draft, authenticated verification
needs the authorized key. No credential is in any artifact.
`numerical-values.json` is the client-serialized input. `evidence-before-fill.json`
and `evidence-after-fill.json` hold numerical results and readback evidence.
`audit.json` is the final site response. `family-200.md`, `screening.json`,
`neighbors-lookups.json`, the source PDF, logs, document drafts and lessons
remain in the run's artifact directory. Website copies of the generator and
checks are authoritative together with the table itself.
The initial entries-only submission was atomically refused because its CITE
references were not in the title-first claim. The completed prose and checked
records were then written together once. This is recorded in `lessons.md` and
`fill-failed.log`; no incorrect entries were stored.