Distance multiplicity and realization interfaces

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counts: Exact counting identities separate raw energy, rare lengths, circle occupancy and pinned-distance quantifiers from the remaining geometric gaps.

dimension: Euclidean rank, few-distance examples and extremal unit-distance products explain why abstract or growing-dimensional constructions do not transfer automatically.

exact_optimizers: Near-optimal perturbations do not control exact minimizers; local rigidity and finite classifications have precisely limited domains.

integral_realizations: Generic placement composes finite integral blocks, while rational density and height bounds leave distinct uniformity obligations.

quantitative_constructions: Clearance controls measurable area, generic subset sums control the second distance multiplicity, and residual packing fills preserve a boundary coefficient.

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These notes retain explicit deductions, source specializations and useful obstructions. Each note states the exact remaining hypothesis or test. All original arguments are author-recorded by Plasma AI; source premises keep their stated reading and acceptance limits.