# Build handoff: T477

The filled draft is https://numberdb.org/T477, **Best known packings of equal circles in a square**. It remains unpublished, unreviewed, and not offered for review. The controller should run the separate critique and repair stages before offering it. No published table was edited, and no queue item was marked built merely because this draft was filled.

## Scope and decisions

Family #202 was read first. Live and repository skills were read; the stricter local restrictions on parameter selection were followed. Screening with `already_here`, `already_asked`, text searches, and numerical `/api/lookup` calls found related tables T394 and T449 but no duplicate square-packing table. The title claim created T477, so no title collision was bypassed.

There are 198 entries: both `separation` and `container` for every integer n from 2 through 100, ordered by n and then separation before container. The range covers the small classical configurations, all cited proven cases, and a continuous run of best known larger cases. The parameter domain remains n >= 2. The one-circle container is omitted because its pairwise separation is undefined.

The two normalisations are required by the family: d_n is minimum point separation in the unit square, and S_n = 2 + 2/d_n is the enclosing square side for circles of **radius** 1. The inverse is d_n = 2/(S_n - 2). They are one packing value in two invertible conventions. The table does not store a third radius or density column.

The source index is Packomania's 10-Sep-2026 snapshot, retrieved with coordinates on 25-Oct-2026. Formula sources are Erich Friedman's collection and David W. Cantrell's explicit shift/compression constructions. Three numerical enclosures come directly from Markót (2021), Section 7.5. The draft cites primary proof papers separately from these numerical sources.

The 25 explicit-form configurations comprise five exact square-grid cases and 20 radical cases. Their 50 rows include 10 exact integer/rational values and 40 values computed with Arb to 100 significant digits. The widest working enclosure was less than 4e-118 with 64 guard bits; all were recomputed at 160 digits. The six rows for n=31,32,33 retain 14 significant digits supported by the published certified intervals. The remaining 142 rows retain 10 significant digits. Table-wide rigour is `heuristic (agreement-checked)`, as the general entries are best known optimisation results. Numerical error is explicitly distinguished from a possible improvement to an unproved optimum.

Tags are the existing `discrete geometry` and `sequence`. The proposed `extremal configurations` tag is absent. It would fit T394, T449, T446 and this T477. Adding it only to this draft would make a singleton; this run did not edit published siblings to introduce it. Similar-table slugs were read from actual corpus search results.

## Checks and evidence

- `dry-run.log`: standard NumberDB exactness, ring-import, disguised-exact-value and prose checks pass for all 198 entries. The measured block is 30.2 KB; the longest raw ball representation is 137 characters. Final stored values are at most 102 characters.
- `local-checks.json`: recomputes minimum pairwise center distances and bounding-square spans for all 99 coordinate files, independently of the printed radius. Every retained numerical interval contains the coordinate-derived interval when each printed coordinate is allowed 1e-12 rounding uncertainty. All source radius, separation and container columns agree with those enclosures. All explicit formulas and certified bounds agree with the source and geometry.
- Precision selection was measured: 10 significant digits has zero containment failures; 11 has four (container rows n=10,19,21,22); 12 has 139. No extra digits were invented from rounded coordinates.
- Controls run first: an exact 2-by-2 grid gives d=1 and S=4; two opposite corners give sqrt(2) and 2+sqrt(2); a perturbed grid strictly worsens the value.
- Both conversion directions and monotonicity are checked over the full range.
- `stored-checks.log` and `stored-checks.json`: `verify(sample=None)` reports **198/198 matched, 0 differing, 0 missing, 0 extra**. Independent geometric, exact-form, certified-bound, conversion and monotonicity checks then pass on the values fetched from the database. The subsequent prose edit left `Numbers` identical.
- Generator, independent checks, source snapshot, and local evidence are attached to the draft. Stored verification and audit evidence are attached as well. `artifact-sha256.json` records local evidence hashes. Sage 10.9 and numberdb 0.1.10 were used through `agents/sage.sh`.

## Audit disposition

`audit.json` has `clean: false` with **one structural finding**: the symbolic `normalisation` parameter has two complete groups. This is the family-mandated, explicitly permitted invertible-normalisation exception. The definition states it, the header names d_n and S_n, and the conversion and inverse are supplied and checked on every stored pair. The audit implementation deliberately does not inspect prose to suppress this question. This finding must remain visible for the separate critique; the run does not describe the audit as clean.

The earlier audit also flagged definition length. The definition was shortened from 345 to 262 characters without changing the numbers, and that finding disappeared. `audit-before-definition-edit.json` preserves the earlier report.

## Source findings for the critic

Packomania's HTML and `distance.txt` disagree at n=2: HTML gives 1.414213562373, while the text file gives 1.414213562368. Converting the rounded radius 0.292893218813 via 2r/(1-2r) yields 1.4142135623678206..., consistent with the text file's lower precision. The exact sqrt(2), HTML, and coordinate enclosure agree. This run uses the HTML index and exact small formulas, preserving both downloaded source versions.

Packomania marks n<=30 and n=36 optimal, but its markings omit the 2021 certified global-optimum results for n=31,32,33. Those primary-paper enclosures were transcribed and checked independently against the coordinates. The comments distinguish those numerical global certificates from the exact structural proof for n=28,29,30.

This is a build report, not the separate critique. No publication, offering, human review, deployment, push, commit, skill edit or pipeline-source edit was performed.