Foundation Pair Kernel Lattice3 Flow Energy Eq Half Inner F

The energy identity

In the Recognition Science framework, a ledger (a discrete record of events on a lattice of sites) carries flows: numbers assigned to each ordered pair of sites, antisymmetric so that flow from i to j is the negative of flow from j to i. The energy of such a flow is defined as half the sum over all ordered pairs of the square of the flow value. The declaration flowEnergy_eq_half_innerF proves that this energy equals exactly one half of the inner product of the flow with itself, where the inner product is the sum over all ordered pairs of the product of the two flow values. This is a theorem in the machine-checked library of formal theorems, axiom-clean, and it holds for any finite lattice size n and any antisymmetric flow.

The identity is a bookkeeping convenience, not a physical law. It rewrites the energy in terms of an inner product, which makes later arguments about flows and their interactions cleaner. The proof is a direct computation: unfolding the definitions and applying the distributive law to the sums reduces both sides to the same expression. There is no hidden assumption about the lattice geometry, no appeal to three dimensions, and no physical content beyond the algebraic definitions. The theorem is general: it applies to any finite set of sites, whether the lattice is one-dimensional, three-dimensional, or a completely unstructured graph.

The declaration appears in a file that builds a three-dimensional lattice box, but the identity itself does not depend on that construction. The three-dimensional carrier, with its taxicab metric and six-neighbor coupling, is built separately. The identity is used there as a tool: when the framework computes the energy of a flow along a path in the box, this identity lets it express that energy in terms of the inner product, which then connects to the pinned-pair cost. The theorem is a stepping stone, not the destination.

What the declaration does not claim is equally clear. It does not say that any particular flow is physical, that the energy has a minimum, or that the three-dimensional lattice is the only possible carrier. It does not derive the Coulomb potential or any decay law. It only establishes an equality between two algebraic expressions, valid for all finite lattices and all antisymmetric flows. The larger claims about dimension and Green's functions in the same file are separate theorems, each with its own proof and its own conditions.

theorem flowEnergy_eq_half_innerF {n : ℕ} (θ : Fin n → Fin n → ℝ) :
    flowEnergy θ = (1 / 2) * innerF θ θ := by
  unfold flowEnergy innerF
  congr 1
  refine Finset.sum_congr rfl (fun i _ => Finset.sum_congr rfl (fun j _ => ?_))
  ring

theorem flowEnergy_eq_half_innerF {n : ℕ} (θ : Fin n → Fin n → ℝ) :
    flowEnergy θ = (1 / 2) * innerF θ θ := by
  unfold flowEnergy innerF
  congr 1
  refine Finset.sum_congr rfl (fun i _ => Finset.sum_congr rfl (fun j _ => ?_))
  ring

theorem flowEnergy_eq_half_innerF {n : ℕ} (θ : Fin n → Fin n → ℝ) :
    flowEnergy θ = (1 / 2) * innerF θ θ := by
  unfold flowEnergy innerF
  congr 1
  refine Finset.sum_congr rfl (fun i _ => Finset.sum_congr rfl (fun j _ => ?_))
  ring

What this page does not claim

The declaration does not derive the Coulomb potential or any decay law for the three-dimensional lattice. The declaration does not establish that the three-dimensional lattice is the only possible carrier for flows. The declaration does not assert that any particular flow is physical or energy-minimizing.

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/PairKernelLattice3.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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