Information Emlfrom Recognition Reciprocal Cost Forgets Orientation

The orientation blind spot

In mathematics, a function that assigns the same value to a number and its reciprocal is said to be symmetric under reciprocation. For example, a function with this property treats 2 and 1/2 as equivalent. The Recognition Science framework's central cost function, written J(x) = (x + 1/x)/2 - 1, has exactly this symmetry. Its theorem reciprocal_cost_forgets_orientation states this formally: for any positive number u, the cost of u equals the cost of its reciprocal, J(u) = J(1/u). In log coordinates, where a number is represented by its logarithm, this means the cost cannot distinguish a positive log coordinate from its negative counterpart.

This symmetry is a direct consequence of the cost function's defining property of reciprocal symmetry, one of the five plain conditions from which J is uniquely derived. The theorem is proved in the framework's machine-checked library of formal theorems, meaning the statement is verified by a computer from the axioms. The proof is short: it applies the symmetry property of J to the positive number exp(u), then rewrites the expression using the fact that exp(-u) is the reciprocal of exp(u).

What this theorem does not claim is more important than what it does. It does not say that the reciprocal cost function alone can derive the EML gate, an operator that combines exponentiation and logarithms. That would be false, because the cost function's symmetry means it has forgotten which direction is forward. The framework's own documentation is explicit on this point: the reciprocal cost J alone does not derive the EML gate, and any claim that it does is incorrect.

In Recognition Science, the framework models an oriented recognition ledger before reciprocal symmetrization. This ledger has an additive log coordinate, and its combiner is subtraction. The maps between the ledger and positive ratios are exponentiation and logarithm. From this oriented data, the framework derives the EML gate as exp(x) - log(y). The reciprocal cost function, by contrast, is what remains after the orientation is quotiented away. The theorem reciprocal_cost_forgets_orientation marks exactly where that forgetting happens, and the framework keeps the oriented layer separate precisely because of it.

The practical consequence is a division of labor. The reciprocal cost function J is the unique cost forced by the five conditions, but it is blind to direction. The EML gate, which needs to know which way exponentiation goes, requires the oriented ledger that the cost function cannot provide. The framework's library proves that the oriented compiler gate is exactly EML, and that from EML one can recover exponentiation, logarithms, and subtraction. The reciprocal cost theorem is the boundary marker: it shows why the oriented layer must exist, and why it cannot be derived from the cost function alone.

/-- Reciprocal cost forgets the orientation of the log coordinate. -/
theorem reciprocal_cost_forgets_orientation (u : ℝ) :
    Jlog u = Jlog (-u) := by
  have h := Jcost_symm (Real.exp_pos u)
  simpa [Jlog, Real.exp_neg] using h

/-- Reciprocal cost forgets the orientation of the log coordinate. -/
theorem reciprocal_cost_forgets_orientation (u : ℝ) :
    Jlog u = Jlog (-u) := by
  have h := Jcost_symm (Real.exp_pos u)
  simpa [Jlog, Real.exp_neg] using h

/-- The oriented ledger combiner before reciprocal symmetrization. -/
def orientedSub (a b : ℝ) : ℝ :=
  a - b

/-- The induced oriented compiler gate is exactly EML. -/
theorem oriented_compiler_gate_eq_eml (x y : ℝ) :
    orientedCompilerGate x y = eml x y := by
  rfl

What this page does not claim

The reciprocal cost function J alone derives the EML gate. The theorem establishes that the oriented ledger is unique. The theorem applies to the physical recognition-to-linking bridge, which remains open.

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

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