# 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.