Gravity Recognition Ledger Total Cost Eq Zero Iff Flat

The flat ledger theorem

A recognition ledger is a discrete record of comparison costs between cells in a finite lattice. Each pair of cells (i, j) carries a non-negative cost J(x_ij), the price of recognizing one cell in terms of the other. The ledger is flat when every one of those costs is zero, meaning no cell demands any recognition effort from any other. The framework's machine-checked library of formal theorems proves a clean equivalence: the total cost, summed over all cell pairs, equals zero if and only if the ledger is flat. In symbols, totalCost L = 0 ↔ isFlat L.

The theorem is structural, not numerical. It does not say that flat ledgers are the only ones with low cost, nor that zero total cost is the only interesting state. It states a precise boundary: because every cost is non-negative, the sum can only vanish when each individual term vanishes. The proof uses that non-negativity and the definition of flatness directly; it requires no assumption about the size of the lattice, only that it is finite. The same library also shows the total cost is the sum of per-cell deficits, and that the flat ledger, where all costs are set to zero, indeed has total cost zero.

What the theorem does not claim is broader. It does not assert that flatness is the only way to achieve zero total cost in some other structure, nor that a flat ledger is physically preferred. It does not claim that the ledger's cost function J is the unique one satisfying the framework's axioms; that uniqueness is a separate theorem about J itself, not about this equivalence. The theorem is a formal statement about the ledger structure as defined, not a physical law about spacetime.

In the framework's account, this equivalence is one rung in a chain that connects recognition cost to geometry, but the theorem itself is a local algebraic fact. It gives a precise handle on what flatness means in ledger terms: no recognition cost anywhere. That is a useful anchor for later steps that interpret ledger cost as gravitational action, but the anchor does not by itself carry those later interpretations.

/-- The total cost vanishes iff the ledger is flat. -/
theorem totalCost_eq_zero_iff_flat {Λ : Type*} [Fintype Λ] [DecidableEq Λ]
    (L : RecognitionLedger Λ) : totalCost L = 0 ↔ isFlat L := by
  constructor
  · intro h0
    unfold isFlat
    intro i j
    unfold totalCost at h0
    have hsums : ∀ x ∈ Finset.univ, ∑ y : Λ, L.cost x y = 0 := by
      rwa [Finset.sum_eq_zero_iff_of_nonneg
        (fun x _ => Finset.sum_nonneg (fun y _ => L.nonneg x y))] at h0
    have h0i := hsums i (Finset.mem_univ i)
    have hsumj : ∀ y ∈ Finset.univ, L.cost i y = 0 := by
      rwa [Finset.sum_eq_zero_iff_of_nonneg (fun y _ => L.nonneg i y)] at h0i
    exact hsumj j (Finset.mem_univ j)
  · intro hf
    unfold totalCost
    apply Finset.sum_eq_zero
    intro i _
    apply Finset.sum_eq_zero
    intro j _
    exact hf i j

/-- Total cost is the sum of deficits. -/
theorem totalCost_eq_sum_deficits {Λ : Type*} [Fintype Λ] [DecidableEq Λ]
    (L : RecognitionLedger Λ) : totalCost L = ∑ i, deficit L i :=
  rfl

/-- The flat ledger has zero total cost. -/
theorem flatLedger_totalCost_zero (Λ : Type*) [Fintype Λ] [DecidableEq Λ] :
    totalCost (flatLedger Λ) = 0 :=
  (totalCost_eq_zero_iff_flat _).mpr (flatLedger_isFlat Λ)

What this page does not claim

The theorem does not assert that flatness is the only way to achieve zero total cost in any other structure. The theorem does not claim that a flat ledger is physically preferred or dynamically selected. The theorem does not prove uniqueness of the cost function J; that is a separate result.

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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