Foundation Pair Kernel Pair Cost

The pair cost

In Recognition Science, a ledger is a discrete record of events on a graph, and a cost is a number assigned to a field configuration on that graph. The pair cost, written W(a,b), measures the interaction energy between two pinned defects: one site a is held at value 1, another site b at value 0, and the cost is the minimum possible energy of the field subject to those pins. The construction subtracts the energy of each isolated pin, but a theorem shows that subtraction is a bookkeeping step that vanishes, so W(a,b) equals the pinned minimum directly.

The main result is that this pair cost is a genuine, informative instrument. It is always nonnegative, and it is bounded below by the direct graph weight between the two sites, so a direct link always contributes. On a small three-site graph with only nearest-neighbor couplings, the cost for adjacent sites is at least 1, while the cost for the far pair is at most 1/2, proving the cost is distance-dependent and not flat. On the same three sites but with all-to-all couplings, the far pair's cost is at least 1, strictly larger than the local case. Thus the pair cost discriminates between a local, finite-range coupling and a non-local, mean-field one.

The cost also decays with distance on a band graph: for sites separated by distance d, the cost is at most 1/d, matching the inverse-distance behavior familiar from Coulomb's law. This is a leading-order result, not the full story: the cost is computed from a quadratic approximation to the exact action, and the full nonlinear pair kernel remains open.

In Recognition Science, this pair cost is a step toward deriving physics from the ledger. The framework proves the instrument is non-vacuous and sees locality, which is the groundwork for later stages that aim to derive the Green's function and interaction laws. The framework's library is a machine-checked collection of formal theorems, and the module contains zero unproved assumptions.

/-- `Wpair = pairMin`, because both isolated self-energies vanish. The subtraction is real (the
    self-energies are genuine infima), it just evaluates to zero. -/
theorem Wpair_eq_pairMin {n : ℕ} (G : WeightedLedgerGraph n) (a b : Fin n) :
    Wpair G a b = pairMin G a b := by
  unfold Wpair
  rw [isolated_eq_zero, isolated_eq_zero]; ring

theorem Wpair_nonneg {n : ℕ} (G : WeightedLedgerGraph n) {a b : Fin n} (hab : a ≠ b) :
    0 ≤ Wpair G a b := by
  rw [Wpair_eq_pairMin]; exact pairMin_nonneg G hab

/-- Band is distance-dependent: the adjacent pair strictly outcosts the far pair. -/
theorem band_distance_dependent :
    Wpair (bandWeight 3) (0 : Fin 3) (2 : Fin 3) < Wpair (bandWeight 3) (0 : Fin 3) (1 : Fin 3) := by
  have h1 := Wpair_band_far_le_half
  have h2 := Wpair_band_adjacent_pos
  linarith

/-- Locality separation: the same far pair is strictly cheaper on the local (band) graph than on
    the non-local (mean-field) graph. -/
theorem local_below_nonlocal :
    Wpair (bandWeight 3) (0 : Fin 3) (2 : Fin 3) < Wpair (meanFieldWeight 3) (0 : Fin 3) (2 : Fin 3) := by
  have h1 := Wpair_band_far_le_half
  have h2 := Wpair_meanField_far_ge_one
  linarith

/-- 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

What this page does not claim

The full nonlinear pair kernel is derived; it remains open. The continuum 1/(4πr) form is derived here; it is only measured numerically in later stages. The pair cost is a fundamental physical constant or a derivation of any specific force law.

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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