Foundation Pair Kernel Pair Cost Band Kernel Inverse Distance Decay

The band kernel decay

The inverse-square law of gravity and electrostatics says that the strength of a force between two objects falls off as one over the square of the distance between them. Before that continuum law can appear, a discrete version must hold on a lattice: the energy between two separated points should decay at least as fast as one over the distance. Recognition Science, a framework that derives physical structure from a ledger of recognition events, proves exactly such a discrete decay bound for a specific two-body interaction energy on a line of points.

In the framework's library, a machine-checked collection of formal theorems, the declaration band_kernel_inverse_distance_decay establishes three concrete numerical facts about the pair cost W(a,b), the minimal interaction energy of two pinned defects at positions a and b on a graph with only nearest-neighbor connections. The theorem proves that on a three-point line, the cost between adjacent points is at least 1, while the cost between the two endpoints, separated by distance 2, is at most 1/2. On a four-point line, the cost between points separated by distance 3 is at most 1/3. In each case, the energy bound is the reciprocal of the separation: 1/1, 1/2, 1/3.

This is not a numerical coincidence. The theorem is built from earlier results in the same file: the cost is always at least the direct link weight between the two points, and a specific ramp configuration of field values gives an upper bound that decreases with distance. The combination yields the general inequality Wpair (bandWeight n) 0 d ≤ 1/d for any separation d on a line of n points. The framework's library proves this bound holds for all distances, not just the three displayed cases.

What the theorem does not claim is equally important. It does not derive the continuum inverse-square law 1/(4πr²); that limit is a separate, numerically measured result, not a proved theorem here. It does not establish the full nonlinear pair kernel, which remains an open target. It does not compare the 1/r decay against a Yukawa potential or any other force law; that comparison requires a forced value of the framework's fundamental constant L0, which is still a hypothesis. The theorem works only in the quadratic, Gaussian approximation of the framework's action, and only for the specific band graph with nearest-neighbor links.

The significance is that the framework's core cost function, derived from first principles, already produces a distance-dependent interaction energy on a simple lattice. This is the first rung of a ladder: a discrete inverse-distance decay that, in principle, could grow into the familiar inverse-square laws of classical physics. The proof is axiom-clean, with zero sorry and zero new axioms, meaning the result is as solid as the framework's foundational axioms themselves.

/-- **The discrete `1/d` decay law, three points.** `W_band(0,1) ≥ 1`, `W_band(0,2) ≤ 1/2`,
    `W_band(0,3) ≤ 1/3` — the pinned-pair kernel falls off as `1/(index distance)` on the chain,
    the lattice precursor of the continuum `1/r` (L4). -/
theorem band_kernel_inverse_distance_decay :
    (1:ℝ) ≤ Wpair (bandWeight 3) (0 : Fin 3) (1 : Fin 3) ∧
    Wpair (bandWeight 3) (0 : Fin 3) (2 : Fin 3) ≤ 1 / 2 ∧
    Wpair (bandWeight 4) (0 : Fin 4) (3 : Fin 4) ≤ 1 / 3 :=
  ⟨Wpair_band_adjacent_pos, Wpair_band_far_le_half, Wpair_band_dist3_le_third⟩

/-- **The discrete `1/d` decay law, three points.** `W_band(0,1) ≥ 1`, `W_band(0,2) ≤ 1/2`,
    `W_band(0,3) ≤ 1/3` — the pinned-pair kernel falls off as `1/(index distance)` on the chain,
    the lattice precursor of the continuum `1/r` (L4). -/
theorem band_kernel_inverse_distance_decay :
    (1:ℝ) ≤ Wpair (bandWeight 3) (0 : Fin 3) (1 : Fin 3) ∧
    Wpair (bandWeight 3) (0 : Fin 3) (2 : Fin 3) ≤ 1 / 2 ∧
    Wpair (bandWeight 4) (0 : Fin 4) (3 : Fin 4) ≤ 1 / 3 :=
  ⟨Wpair_band_adjacent_pos, Wpair_band_far_le_half, Wpair_band_dist3_le_third⟩

/-- The two-body cost `Wpair` inherits the general `1/d` decay (since `Wpair = pairMin`). The kernel
    of two defects at index-distance `d` on the band chain is `≤ 1/d` for every `0 < d < n`. -/
theorem Wpair_band_le_inv_dist {n d : ℕ} (hd : 0 < d) (hdn : d < n) :
    Wpair (bandWeight n) (⟨0, by omega⟩ : Fin n) (⟨d, hdn⟩ : Fin n) ≤ 1 / (d : ℝ) := by
  rw [Wpair_eq_pairMin]; exact pairMin_band_le_inv_dist hd hdn

/-- **The discrete `1/d` decay law, three points.** `W_band(0,1) ≥ 1`, `W_band(0,2) ≤ 1/2`,
    `W_band(0,3) ≤ 1/3` — the pinned-pair kernel falls off as `1/(index distance)` on the chain,
    the lattice precursor of the continuum `1/r` (L4). -/
theorem band_kernel_inverse_distance_decay :
    (1:ℝ) ≤ Wpair (bandWeight 3) (0 : Fin 3) (1 : Fin 3) ∧
    Wpair (bandWeight 3) (0 : Fin 3) (2 : Fin 3) ≤ 1 / 2 ∧
    Wpair (bandWeight 4) (0 : Fin 4) (3 : Fin 4) ≤ 1 / 3 :=
  ⟨Wpair_band_adjacent_pos, Wpair_band_far_le_half, Wpair_band_dist3_le_third⟩

What this page does not claim

The theorem does not derive the continuum inverse-square law 1/(4πr²), which remains a separately measured numerical result. The theorem does not establish the full nonlinear pair kernel, which is an open target. The theorem does not compare the 1/r decay against a Yukawa potential or any other force law, which requires a still-hypothetical forced value of L0.

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/Foundation/PairKernelPairCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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