Gravity Recognition Ledger Rcl Gate Zero Right

The zero-cost shortcut

The Recognition Science framework models comparison as a ledger, a discrete record of costs assigned to every pair of cells in a finite lattice. The cost of comparing a cell with itself is zero, and the cost of comparing two cells is the same in either direction. The framework's central question is how these pairwise costs must combine when a comparison passes through an intermediate cell. The answer is forced by a function called the recognition gate, written R(u, v) = 2uv + 2u + 2v, which bounds the cost of a direct comparison by the costs of the two steps that could replace it.

What happens when one of those steps costs nothing? The declaration rclGate_zero_right answers exactly that. It proves, as a theorem in the framework's machine-checked library of formal theorems, that R(u, 0) = 2u. If comparing cell A with cell B costs u, and comparing B with C costs zero, then the gate says the direct cost from A to C is at most 2u. The zero-cost step does not vanish; it doubles the remaining cost. The same identity holds with the zero on the left, since the gate is symmetric, and the framework proves that symmetry separately. These are not assumptions; they are consequences of the gate's defining formula, checked by the library's kernel.

The identity matters because it is the sharpest bound the framework's structure allows. The ledger's defining property is subadditivity: the direct cost from A to C never exceeds the gate applied to the two intermediate costs. When one intermediate cost is zero, the theorem pins down exactly what that bound becomes. It is a small algebraic fact, but it is the kind of fact that makes the whole ledger structure usable: without it, every application of the gate would carry an unresolved inequality.

What the theorem does not claim is just as precise. It does not say that a zero-cost step is free, or that it can be ignored. It does not say anything about what the direct cost actually is, only that it is bounded by 2u. It does not assert that such zero-cost steps exist in any particular physical situation; the theorem is conditional on the costs u and 0 being given. And it says nothing about the fine-structure constant, the Riemann Hypothesis, or any other quantity outside the ledger's own definitions. The declaration is a lemma about an algebraic function, proved from its definition, and it earns its place by being exactly that.

theorem rclGate_zero_right (u : ℝ) : rclGate u 0 = 2 * u := by
  unfold rclGate; ring

theorem rclGate_symmetric (u v : ℝ) : rclGate u v = rclGate v u := by
  unfold rclGate; ring

/-- A recognition ledger on a finite substrate lattice `Λ`. -/
structure RecognitionLedger (Λ : Type*) [Fintype Λ] [DecidableEq Λ] where
  /-- The cost function assigning recognition cost to each cell pair. -/
  cost : Λ → Λ → ℝ
  /-- Symmetry: ℒ(i,j) = ℒ(j,i). -/
  symmetric : ∀ i j, cost i j = cost j i
  /-- Diagonal zero: a cell has zero cost of comparison with itself. -/
  diagonal_zero : ∀ i, cost i i = 0
  /-- Non-negativity: all costs are non-negative. -/
  nonneg : ∀ i j, 0 ≤ cost i j
  /-- RCL subadditivity: the cost from i to k is bounded by the RCL gate
  applied to the costs from i to j and j to k, for every intermediate j. -/
  rcl_subadditive : ∀ i j k, cost i k ≤ rclGate (cost i j) (cost j k)

What this page does not claim

A zero-cost step is free or can be ignored in the ledger. Zero-cost comparisons exist in any particular physical substrate. The theorem determines the actual direct cost, which remains bounded but unspecified.

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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