One boundary back-end, on the corner graph

[hdrake]

Targets topology-overhaul (the branch behind #22), not main.

Downstream of hdrake/sectionate#15, which makes corner identity come from the grid's declared metadata rather than from coordinates, and gives every grid a corner topology instead of only multi-tile ones. Needs that branch installed; the pyproject.toml floor names the release it will become.

One back-end instead of two

_single_tile_boundaries_from_mask is gone, and with it lockey — a round(xyz, 9) position on the unit sphere used as a hash bucket. Two problems with it. It meant this package and sectionate had to agree on a coincidence tolerance and did not: a 6.4 mm quantisation bin here against a 1 mm metric there. And it merged corners a grid distinguishes but happens to place at one point, which is what a tripolar cap's singular meridian does to a whole column of them.

Everything now stitches on the corner-node graph, which is uniform across a periodic wrap, a bipolar fold, rotated and reversed tile seams, and cube-vertex junctions alike.

Nothing was lost. Comparing the old coordinate-merged graph against the new declared one on ECCO LLC90: the same 105302 nodes, the same 78 corners stored on no face, zero partition mismatches in either direction, zero nodes with a different native index — and 235 nodes that gain edges.

A fix, not just a simplification

The corner frame _trace_and_drop works in used to depend on the staggering. It is now the 'outer' lattice on every grid, which is what the corner topology is indexed on.

That was a bug. On a 'right'-staggered single-tile grid, a region wrapping the periodic seam traced as two loops with a spurious seam face and a budget that did not close:

             before                              after
interior     1 loop,  8 faces, closes            1 loop,  8 faces, closes
seam-wrap    2 loops, 10 faces, DOES NOT close   1 loop,  8 faces, closes

Nothing covered that path — every single-tile fixture here is 'outer'-staggered — so there is a new 'right' fixture, and both of its tests fail against the old frame.

Consequences of the upstream API

grid_section returning the same five values on every grid removes _normalize_grid_section and the f_c is None guards. f_c is present everywhere; a single-tile grid is one face, so it is zeros rather than None. ij= correspondingly takes (i_c, j_c, f_c) — it was documented as a 2-tuple and raised IndexError on one, and now says what it wants.

curve="latitude circle" on an axis-aligned box now raises, because a meridional segment follows no parallel. That is the migration sectionate intended when it added "latitude and great circle"; the test uses it, and pins the raise.

Fixtures

fold_grid placed its seam row at s(i) = min(i, −i) in both longitude and latitude so that only genuine mirror pairs coincided. That folds the coordinates back on themselves and leaves 3 of its 24 cells inside out — sectionate now refuses that rather than signing their velocity faces backwards. None of the contrivance is needed any more, because identity comes from the declared fold, so it is a pinched cylinder whose ellipses genuinely nest. It keeps the same three mirror pairs and gains the five periodic-wrap pairs the old one was missing (it had them only at the first and last row, though a periodic grid's first and last corner columns are the same point at every row).

The ECCO loader declares LLC90's southern boundary fold, which face_connections cannot express — its schema maps a whole tile edge to a whole tile edge, and that boundary folds onto itself across four of them. Without it the southern cap's boundary runs along an unmerged lip and cannot close.

Known, documented, not fixed

_pad_center reads the grid's own metadata, so it does not see identifications declared through sectionate.topology.declare_identifications. Cells either side of such a seam look unconnected to connected_components, which would split a component spanning it. Budgets still close — the tracer annihilates the doubly-traced edges within one trace — so this is region identity, not conservation, and on LLC90 the cells concerned are Antarctic land. It becomes live for any declared seam that carries flow. The fix is to source cell adjacency from the corner topology, which has seen them; that is a change worth making on its own.

Verification

• 48 passed, 10 skipped by default; 60 passed with REGIONATE_REALDATA_TESTS=1.
• All six notebooks re-executed. No figure and no numerical output changed.
• The divergence-theorem tests are the acceptance criteria and all hold: the fold and periodic single-tile cases, the two-face and rotated-seam and cube-vertex cases, and the five real-LLC90 hard-topology cases including the southern cap.
• Two adversarial review rounds. The first refuted my diagnosis of the one failing real-data test — it was an incomplete port, not the upstream behaviour change I had blamed — and caught that a previous commit here had tracked seventeen absolute symlinks into another checkout.

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