Mathematics Projection Multiplicity Method Projection Multiplicity Certificate

The certificate shape

The projection multiplicity method is a named proof strategy for classical extremal problems, problems that ask how large a set can be when its elements are forbidden from forming too many of some relation. The method's origin is a famous miss: the Erdős unit-distance problem, which asks how many pairs of points at distance exactly one can appear among n points in the plane. The classical counting approach fails there, and the reason it fails is instructive. A richer carrier space can hold many distinct events that all project down to the same visible low-dimensional invariant, so counting by visible dimension alone undercounts the true structure.

The framework's machine-checked library records this failure mode as an abstract method rather than as a proof of the Erdős result. The central object is the certificate, a formal template for a lift-return proof. A certificate names four ingredients: a richer carrier with a projection map back to the visible problem, a relation on that carrier, a way to pick finite windows in the carrier, and a guarantee that carrier events project to valid visible events. The certificate's defining theorem, called certificate_gives_polynomial_gain, states that any certificate of this shape yields a fixed polynomial gain: the visible event count grows at least like the visible size raised to a power strictly greater than one. That is the formal output, a proof that linear growth is beaten by a definite exponent.

The certificate does not itself prove the Erdős unit-distance bound. It names the method so future problems can be checked for the same failure mode. The library also defines a diagnostic predicate, the projection multiplicity candidate, with four tests: a hidden carrier exists, the visible relation is a projection, the fibers can grow, and the carrier geometry stays controlled. A problem passing all four is ready for a full attack, but readiness is not a result. The certificate is a shape, not a theorem about any particular problem.

What the declaration does not claim is as important as what it establishes. It does not formalize the OpenAI/Sawin proof of the unit-distance problem; the docstring says so explicitly. It does not assert that any concrete problem satisfies the certificate. It does not claim the method always works, only that when a certificate exists, the polynomial gain follows. The value is diagnostic: a checklist for spotting when visible-dimensional counting can be beaten.

In plain terms, the certificate is a reusable skeleton for a family of proofs. It tells a future mathematician what to build: a carrier, a projection, a window, a gain. It does not tell them the carrier will exist. The framework's contribution is to make the method's shape precise enough to be checked by machine, so that a classical problem can be tested against the same failure mode that broke the unit-distance count.

/-- The formal output of the method: the lifted carrier gives a visible
polynomial gain. -/
theorem certificate_gives_polynomial_gain
    (C : ProjectionMultiplicityCertificate.{u, v}) :
    ∃ δ : ℝ, 0 < δ ∧ BeatsLinearBy C.visibleSize C.eventCount δ :=
  ⟨C.delta, C.delta_pos, C.polynomial_gain⟩

/-- A classical extremal problem consists of visible objects, a relation whose
instances are counted, and two numerical readouts: visible size and event count. -/
structure ClassicalExtremalProblem where
  Visible : Type u
  relation : Visible → Visible → Prop
  size : Finset Visible → ℝ
  eventCount : Finset (Visible × Visible) → ℝ

/-- Diagnostic predicate for the rule we missed.  A problem should be checked
for projection multiplicity when its visible relation can be expressed as the
projection of a carrier relation and the event count is controlled by fiber
multiplicity rather than by visible dimension alone. -/
structure ProjectionMultiplicityCandidate where
  has_hidden_carrier : Prop
  visible_relation_is_projected : Prop
  fibers_can_grow : Prop
  carrier_geometry_stays_controlled : Prop

What this page does not claim

The certificate does not formalize the OpenAI/Sawin proof of the Erdős unit-distance problem. The certificate does not assert that any concrete problem satisfies its hypotheses. The certificate does not claim the method always succeeds, only that a certificate yields a polynomial gain.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Mathematics/ProjectionMultiplicityMethod.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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