Gravity Recognition Ledger Boundary Cost Nonneg

The boundary cost

In the Recognition Science framework, a ledger is a discrete record of costs: it assigns a non-negative number to every pair of cells in a finite lattice. The framework proves that this number, the recognition cost of comparing two cells, is always at least zero. The declaration boundaryCost_nonneg extends that fact to a specific quantity: the boundary cost. Given any way of splitting the lattice into an inside and an outside, the boundary cost is the sum of all recognition costs between a cell in the inside and a cell in the outside. The theorem states that this sum is always non-negative, for any ledger and any split.

This is a structural result, not a numerical prediction. It does not say how large the boundary cost is, nor how it behaves under dynamics. It only says that a certain sum of non-negative terms is itself non-negative. The proof is direct: each term in the sum is non-negative by the definition of a ledger, and a sum of non-negative numbers is non-negative. The theorem is checked in the machine-checked library of formal theorems, with no axioms beyond the standard logical ones.

The boundary cost is one of the derived quantities in the framework's model of gravity. The framework models gravitational action as the continuum limit of the total ledger cost restricted to hinges, and the boundary cost is the portion of that total cost that crosses a chosen dividing surface. The non-negativity result is a consistency condition: it guarantees that the cost of comparing two regions never subtracts from the total, which is a prerequisite for interpreting the ledger as a bookkeeping of actual costs.

What the theorem does not claim is just as important. It does not claim that the boundary cost is positive, only that it is not negative. A flat ledger, where every cost is zero, has a boundary cost of zero, and the theorem allows that. It does not claim that the boundary cost is a physically meaningful observable, nor that it equals any measured quantity. It is a formal statement about a defined sum within the framework's model, and its role is to keep the model internally consistent.

The practical consequence is that any calculation using boundary costs within the framework starts from a guaranteed non-negative base. This is a small but necessary step for the framework's gravitational action program, where the total cost must be bounded below to serve as an action. The theorem is a foundation stone, not a finished tower.

/-- Boundary cost is non-negative. -/
theorem boundaryCost_nonneg {Λ : Type*} [Fintype Λ] [DecidableEq Λ]
    (L : RecognitionLedger Λ) (P : SubstrateBipartition Λ) :
    0 ≤ boundaryCost L P := by
  unfold boundaryCost
  apply Finset.sum_nonneg
  intro i _
  apply Finset.sum_nonneg
  intro j _
  exact L.nonneg i j

/-- The boundary cost of a bipartition: sum of costs between interior
and exterior cells.  This is the ledger analogue of the boundary term
in the gravitational action. -/
noncomputable def boundaryCost {Λ : Type*} [Fintype Λ] [DecidableEq Λ]
    (L : RecognitionLedger Λ) (P : SubstrateBipartition Λ) : ℝ :=
  ∑ i ∈ P.interior, ∑ j ∈ P.exterior, L.cost i j

/-- Boundary cost is non-negative. -/
theorem boundaryCost_nonneg {Λ : Type*} [Fintype Λ] [DecidableEq Λ]
    (L : RecognitionLedger Λ) (P : SubstrateBipartition Λ) :
    0 ≤ boundaryCost L P := by
  unfold boundaryCost
  apply Finset.sum_nonneg
  intro i _
  apply Finset.sum_nonneg
  intro j _
  exact L.nonneg i j

What this page does not claim

The boundary cost is positive, only non-negative. The boundary cost equals any measured physical quantity. The theorem establishes the physical recognition-to-linking bridge for gravity.

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

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