Mathematics Bipartite Distance Spectrum Two Channel Range Experiment

A two-channel range experiment

A two-channel range experiment is a formal object that holds two finite collections of points in the plane, called sources and detectors. The structure records only those two sets, nothing else. Its purpose is to study the cross-coupling spectrum: the set of all distances between a source and a detector. The size of that set is the alphabet size, a count of how many distinct separation lengths occur. This is a classical combinatorial geometry question, and the framework's declaration merely packages it into a named structure.

The definition is deliberately minimal. A finite set of sources, a finite set of detectors, and the induced set of squared distances between them. The squared distance alphabet is defined first because squaring avoids square-root arithmetic; the unsquared alphabet carries the same asymptotic information since distances are nonnegative. A simple argument proves the alphabet can have at most one value per source-detector pair, so its size is bounded by the product of the two set sizes. That is the only result proved about this structure in the module.

The connection to Erdős problem #661 is what gives the structure its interest. The positive direction asks whether there exist two n-point planar sets whose cross-distance alphabet size is little-o of n divided by the square root of the natural logarithm of n. The framework's physical reading suggests the opposite: a rigidity target claiming that in one planar carrier, the square-grid scale n over sqrt(log n) is a lower bound for the alphabet size. This target is stated as a definition, not proved. The module also defines a four-dimensional collapse condition where all cross-distances are equal, as a contrast showing why the planar case is hard.

In Recognition Science, the experiment models a physical readout: sources and detectors coupled by Euclidean propagation delay, with the distance count as the alphabet. The framework's library records this as a machine-checked formal structure. The inverse target, that a small alphabet at the Landau-Ramanujan scale forces the points to lie approximately on a Gaussian-integer-like lattice, is explicitly left unproved. It is named as the exact classical lemma needed to turn the physical statement into a conventional proof.

What the declaration does not claim is substantial. It does not prove the rigidity target, does not prove the inverse target, and does not solve Erdős #661. It establishes only the vocabulary: a structure for two point sets, a spectrum of cross-distances, and a formal statement of the open problems. The value is in the precise framing, not in a result.

/-- A two-channel range experiment: sources and detectors in the visible plane. -/
structure TwoChannelRangeExperiment where
  sources : Finset Point2
  detectors : Finset Point2

/-- 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

/-- Positive formulation of Erdős #661: there are two `n`-point planar channels
whose cross-distance alphabet is little-o of the square-grid scale. -/
def Erdos661Positive : Prop :=
  ∃ P Q : ℕ → Finset Point2,
    (∀ᶠ n in atTop, (P n).card = n ∧ (Q n).card = n) ∧
      Tendsto
        (fun n : ℕ =>
          ((crossDistSpectrum (P n) (Q n)).card : ℝ) /
            ((n : ℝ) / Real.sqrt (Real.log n)))
        atTop
        (𝓝 0)

/-- Negative RS target: planar cross-spectrum rigidity at the grid scale. -/
def PlanarCrossSpectrumRigidity : Prop :=
  ∃ c : ℝ, 0 < c ∧
    ∀ᶠ n in atTop,
      ∀ P Q : Finset Point2,
        P.card = n →
        Q.card = n →
          c * ((n : ℝ) / Real.sqrt (Real.log n)) ≤
            ((crossDistSpectrum P Q).card : 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)

What this page does not claim

This answer does not claim that the rigidity target or the inverse target is proved. This answer does not claim that Erdős problem #661 is solved. This answer does not claim that the two-channel range experiment is a physical measurement apparatus.

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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