erdos132_from_current_large_nonstar_residual
plain-language theorem explainer
Assuming the large non-star Conway thrackle support bound and the convex-layer screening bridge, every sufficiently large finite planar point set carries two distinct sparse distance shells (ordered-pair Erdős #132). Discrete geometers closing the thrackle/Conway residual of #132 would cite this reduction. The proof is a short term application that converts the large-nonstar bound and feeds both residual halves into the live Conway-plus-screening assembly.
Claim. If the large non-star Conway thrackle support bound holds and the convex-layer screening bridge holds, then for all sufficiently large $n$, every finite planar set $A$ with $|A|=n$ admits two distinct distances $r\neq s$ that are both sparse shells of $A$ (ordered-pair form of Erdős problem #132).
background
This module is the Recognition Science physicalization of Erdős problem #132. Classically a distance value is a shell in the pairwise Euclidean distance set; here it is a two-body recognition-energy shell whose multiplicity is shell occupancy. The development uses ordered pairs for Lean simplicity: for a positive distance, ordered multiplicity is exactly twice the unordered multiplicity, so the classical threshold $\le n$ becomes $\le 2n$.
The target statement Erdős #132 in ordered normalization asserts that eventually every $n$-point planar set has two distinct sparse shells. The hypothesis packages the current live residual after finite bookkeeping: all small and star Conway cases are closed, leaving only the large non-star Conway thrackle support bound on the counting side, conjoined with the convex-layer screening bridge.
A sibling lemma converts that large-nonstar bound into the ordered Conway support bound expected by the live assembly. The module also records a stronger flux-style target (sparse shells diverge), but this declaration only discharges the two-shell existence claim.
proof idea
Term-mode composition, not a tactic script. Project the residual hypothesis into its two conjuncts. On the first conjunct, apply the in-module conversion that turns the large non-star Conway thrackle support bound into the ordered Conway support bound. Pass that converted bound together with the second conjunct (the convex-layer screening bridge) to the live assembly lemma that already knows how to conclude ordered Erdős #132 from Conway support plus convex-layer screening. No further case splits or arithmetic appear at this node.
why it matters
This is a residual-closure node on the RS reading of Erdős #132: distance shells as recognition-energy shells, multiplicity as occupancy. It records that once small and star Conway cases are finished by finite bookkeeping, the single remaining counting obligation is the large non-star thrackle support bound, and that bound plus convex-layer screening already yields two distinct sparse shells for all large planar sets.
The declaration is explicitly a legacy reduced assembly kept for comparison with the pre-Conway-correction proof graph; the live final endpoint named in the module is the fuller ordered Conway–convex-layer residual pack. No downstream consumers are wired yet (used_by is empty), so its role is archival and comparative inside the Distance Shell Multiplicity development rather than a load-bearing step of a later theorem. It does not itself touch the forcing chain T0–T8, RCL, or the $\varphi$-ladder constants; those enter only insofar as the broader RS physicalization frames shells as recognition energy.
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