Mathematics Bipartite Distance Spectrum Gaussian Like Lattice

A lattice for distance spectra

A Gaussian-like lattice, as defined in the Recognition Science framework's machine-checked library of formal theorems, is a structure that records a way to place integer-coordinate points in the plane. It consists of a map from pairs of integers to points, a positive scaling factor, and an integer-valued quadratic form that is nonnegative. The defining property is that the squared distance of any lattice point from the origin equals the scaling factor times the value of that quadratic form. This is a formal way of saying the lattice behaves like the usual grid of Gaussian integers, where squared distances are sums of two squares, but with a possible scaling and a more general quadratic shape.

The definition exists to state a precise inverse theorem target related to Erdős problem #661. The classical problem asks whether two sets of n points in the plane can have a cross-distance alphabet, the set of all distances between a point of one set and a point of the other, whose size is much smaller than n divided by the square root of the logarithm of n. The framework's physical reading suggests the opposite: that this square-grid scale is rigid, and that any pair of sets achieving a small alphabet must be approximately carried by such a Gaussian-like lattice. The formal statement, PlanarNormFiberRigidityTarget, says that if the squared cross-norm alphabet is at most the grid scale, then all but a small fraction of the points of both sets must lie in the image of a single Gaussian-like lattice.

The library explicitly does not prove this target. The docstring for the target states plainly that it is not asserted as proved; it is the exact classical lemma needed to turn the physical statement into a conventional proof. What the library does prove is a trivial upper bound: the size of the squared cross-distance alphabet is at most the number of source-detector pairs, which is the product of the two set sizes. It also defines a comparison structure in four-dimensional space where all cross-distances can be fixed to a single value, a mechanism that is unavailable in the plane.

In Recognition Science, this definition acts as a bridge. It takes a physical intuition about two-channel range experiments and expresses it as a concrete mathematical object that a conventional proof could use. The framework models the problem as two finite planar channels coupled by Euclidean propagation delay, and the Gaussian-like lattice is the proposed carrier for the inverse theorem. The definition is deliberately weak, allowing future work to replace the approximation notion with stronger variants without changing the physical statement.

/-- A rank-two lattice-like carrier in the visible plane, recorded only at the
level needed for the inverse theorem target.  The `basis` map is the proposed
Gaussian-integer coordinate chart; `integral_norm` says visible squared norm is
read from an integral positive binary quadratic form after scaling. -/
structure GaussianLikeLattice where
  basis : (Fin 2 → ℤ) → Point2
  scale : ℝ
  scale_pos : 0 < scale
  qform : (Fin 2 → ℤ) → ℤ
  qform_nonneg : ∀ z, 0 ≤ qform z
  integral_norm :
    ∀ z, normSq (basis z) = scale * (qform z : ℝ)

/-- A rank-two lattice-like carrier in the visible plane, recorded only at the
level needed for the inverse theorem target.  The `basis` map is the proposed
Gaussian-integer coordinate chart; `integral_norm` says visible squared norm is
read from an integral positive binary quadratic form after scaling. -/
structure GaussianLikeLattice where
  basis : (Fin 2 → ℤ) → Point2
  scale : ℝ
  scale_pos : 0 < scale
  qform : (Fin 2 → ℤ) → ℤ
  qform_nonneg : ∀ z, 0 ≤ qform z
  integral_norm :
    ∀ z, normSq (basis z) = scale * (qform z : ℝ)

/-- The inverse theorem target left by the RS physicalization:
small cross-norm alphabet at the Landau-Ramanujan scale forces the visible
points to be approximately carried by a Gaussian-integer-like lattice.

This is not asserted as proved here; it is the exact classical lemma needed to
turn the RS physical statement into a conventional proof. -/
def PlanarNormFiberRigidityTarget : Prop :=
  ∀ᶠ n in atTop,
    ∀ P Q : Finset Point2,
      P.card = n →
      Q.card = n →
      ((crossNormSqAlphabet P Q).card : ℝ) ≤
          (n : ℝ) / Real.sqrt (Real.log n) →
        ∃ Λ : GaussianLikeLattice,
          ApproxContainedInGaussianLikeLattice P Λ (n / 100) ∧
          ApproxContainedInGaussianLikeLattice Q Λ (n / 100)

/-- Trivial upper bound: the cross spectrum has at most one value per
source-detector pair. -/
theorem crossDistSqSpectrum_card_le_pairs (P Q : Finset Point2) :
    (crossDistSqSpectrum P Q).card ≤ P.card * Q.card := by
  classical
  unfold crossDistSqSpectrum
  calc
    ((P.product Q).image (fun pq => dist pq.1 pq.2 ^ 2)).card ≤
        (P.product Q).card := Finset.card_image_le
    _ = P.card * Q.card := by simp

What this page does not claim

The inverse theorem target is not proved in the library. The definition does not assert that every point of the sets lies in the lattice; it allows for an error term. The framework does not claim that the physical reading of Erdős #661 is a theorem; it is a target.

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/BipartiteDistanceSpectrum.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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