erdos132_from_conway_counting_screening_residual_pack
plain-language theorem explainer
A residual package of support-level Conway thrackle counting plus pointwise deep-layer screening already forces the ordered form of Erdős problem #132: every large enough finite planar set has two distinct sparse distance shells. Discrete geometers on the thrackle route to #132, and RS workers tying shell occupancy to recognition energy, would cite it. The proof is a one-line unpack of the package into the live assembly lemma.
Claim. If a residual package supplies both a support-level Conway thrackle bound and a pointwise deep-layer screening certificate, then Erdős problem #132 holds in ordered-pair normalization: for all sufficiently large $n$, every planar $n$-point set $A$ admits distances $r \neq s$ such that both the $r$-shell and the $s$-shell of $A$ are sparse.
background
The module records the Recognition Science physicalization of Erdős problem #132. Classically a distance value is a shell in the pairwise Euclidean spectrum; physically it is a two-body recognition-energy shell, and multiplicity is shell occupancy. Ordered pairs are used for Lean simplicity: for positive distances, ordered multiplicity is twice the unordered count, so the classical threshold $\le n$ becomes $\le 2n$.
The ordered target asserts that for every sufficiently large finite planar set there exist two distinct sparse shells. After the full diameter-side Conway condition is closed, the live residual package retains exactly two inputs: a Conway thrackle support bound on the support, and a pointwise deep-layer screening certificate.
Upstream, the live assembly result states that those two residual inputs alone already imply the ordered Erdős claim once diameter-side local geometry is finished.
proof idea
One-line term-mode wrapper. Unpack the residual package into its two fields (support-level Conway thrackle bound; pointwise deep-layer screening certificate) and apply the live assembly theorem that takes exactly those two hypotheses and returns the ordered Erdős statement. No extra algebra or case splits occur at this layer.
why it matters
This is the packaging theorem for the current live residual route to Erdős #132 inside the distance-shell module: once support Conway counting and pointwise deep-layer screening are granted, the ordered two-sparse-shells claim follows. It sits after the diameter-side Conway geometry has been closed, so the residual surface is minimal and classical in form (thrackle bound plus deep screening). No downstream consumers are wired yet; the declaration is the top-level entry for this residual pack. In RS language, sparse shells are low-occupancy recognition-energy levels in the planar two-body spectrum, so the result ties a classical discrete-geometry problem to the shell-flux reading used elsewhere in the monolith. The stronger RS suggestion that the number of sparse shells diverges remains a separate target.
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