Erdos132OrderedConwayEventualNoDeepResidualPack
plain-language theorem explainer
Packages two residual hypotheses that jointly imply the ordered form of Erdős problem #132: the Conway straight-line thrackle support bound, and the eventual ban on deep-layer configurations in the low-shell regime. Anyone closing the residual path to #132 cites this pack. It is a pure Prop structure with no proof body; the work sits in the two fields and the downstream applicator.
Claim. The honest eventual residual package is the conjunction of (i) the Conway straight-line thrackle support bound: for any finite point set $A$ and edge set $E$ on $A$, if $E$ is a Conway thrackle (every pair of edges meets simply) then the unordered edge support has size at most $|A|$; and (ii) the eventual low-shell no-deep theorem: for all sufficiently large $n$, no $n$-point set whose diameter shell $\Delta$ has low-shell structure admits a deep-layer case.
background
The module physicalizes Erdős problem #132: classically a distance value is a shell in the pairwise Euclidean spectrum; here it is a two-body recognition-energy shell whose multiplicity is occupancy. Ordered pairs are used for Lean simplicity, so the classical multiplicity threshold $\le n$ becomes $\le 2n$ for positive distances.
The first field is the Conway thrackle support bound: if every pair of edges meets simply, the undirected support has at most $|A|$ edges (Lovász–Pach–Szegedy / Cairns–Nikolayevsky counting). The bound fails without the simple-meeting hypothesis (collinear overlaps).
The second field is the sharp geometric leftover after finite accounting: eventually, in the low-shell regime around a diameter shell, the residual deep-layer case cannot occur. Together these are the honest residual package for the ordered #132 statement.
proof idea
No proof body: the declaration is a Prop-valued structure that simply records two named hypotheses as fields. Downstream, the applicator theorem unpacks the pack and feeds both fields into the live combiner erdos132_from_ordered_conway_and_eventual_no_deep_layer_live, which returns the ordered Erdős #132 statement. The logical content is packaging, not derivation.
why it matters
This is the honest residual interface for the ordered #132 path in Distance Shell Multiplicity. The sole direct consumer is erdos132_from_ordered_conway_eventual_no_deep_residual_pack, whose doc-comment states that the package proves Erdős #132: it applies the live combiner to the two fields and obtains Erdos132Ordered.
In the broader RS framing, #132 is the combinatorial skeleton behind distance-shell multiplicity (two-body recognition-energy shells). Closing the residual with Conway thrackle counting plus the eventual no-deep-layer theorem is the last geometric gate before the ordered multiplicity bound. The pack keeps those two open geometric claims explicit rather than burying them inside a single opaque theorem.
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