ProjectionMultiplicityCandidate
plain-language theorem explainer
Four-field diagnostic record marking when a classical extremal problem is ripe for a projection-multiplicity attack: hidden carrier, projected visible relation, growing fibers, and controlled carrier geometry. Extremal combinatorialists checking whether visible-dimensional counting can be beaten by fiber multiplicity would cite it. Pure structure definition packing four Props; no proof content.
Claim. A projection-multiplicity candidate is a record of four propositions: (i) a hidden carrier exists; (ii) the visible relation is the projection of a carrier relation; (iii) fibers can grow; (iv) carrier geometry stays controlled. The candidate is ready when all four hold simultaneously.
background
The module abstracts the method exposed by the Erdős unit-distance miss: lift a classical extremal problem to a richer carrier, produce many hidden carrier events, then project those events back to the visible classical surface. Visible-dimensional counting can fail when a high-rank carrier has many distinct events that share one low-dimensional invariant.
The four fields name the preconditions of that attack. A hidden carrier is the richer ambient relation; the visible relation must be its projection; fiber growth is what multiplies events beyond the visible dimension; controlled carrier geometry keeps the lift from becoming an unconstrained higher-dimensional problem.
The companion readiness predicate is simply the conjunction of the four fields. The point is diagnostic naming, not a formalization of any particular published proof.
proof idea
Definitional only: a structure with four Prop fields and a one-line readiness predicate equal to their conjunction. No lemmas are applied and no tactics run.
why it matters
Records the abstract failure mode missed by classical visible-dimensional counting, so future extremal problems can be screened before a full lift-and-project attack. Sits beside sibling scaffolding (ClassicalExtremalProblem, LiftData, FiniteWindow, BeatsLinearBy, ProjectionMultiplicityCertificate) that will turn the diagnostic into certificates of polynomial gain. No downstream theorems yet consume it; it is the named entry gate for the method. Not tied to a T0–T8 forcing step; it is methodological infrastructure for classical extremal arguments inside the monolith.
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