Information Emlfrom Recognition Eml Keeps Oriented Channels

The oriented channel

The declaration eml_keeps_oriented_channels is a theorem in the machine-checked library of formal theorems. It states that for any real number x, the operation eml x 1 equals the exponential of x, and the logarithm of x equals the logarithm of x. This is a precise way of saying that the operation keeps the exponential and logarithmic channels separate and addressable.

The operation eml is defined as eml x y = exp x - log y. It is read as a compiler gate: a two-input, one-output device that combines an exponential channel and a logarithmic channel. The theorem shows that by feeding the identity terminal, the number 1, into the logarithmic channel, the gate recovers the exponential function. The logarithmic channel remains independently accessible, so the gate does not collapse the two directions into one.

This property matters because the framework's central cost function J(x) = (x + 1/x)/2 - 1 forgets orientation. The theorem reciprocal_cost_forgets_orientation proves that Jlog u = Jlog (-u), meaning the cost function cannot distinguish a positive log coordinate from its negative. The eml operation, by contrast, preserves that distinction. The theorem is a compiler-layer result: it shows that before reciprocal symmetrization, an oriented recognition ledger has an additive log coordinate, and the induced two-input gate is exactly eml.

The theorem does not claim that the cost function J alone derives the eml gate. That would be false, since J forgets orientation. The theorem is narrower: it shows that the exp/log data, before symmetrization, induces the eml gate. The same exp/log data, after reciprocal symmetrization, gives J(exp u) = cosh u - 1, a different object.

/-- EML keeps oriented channel data: the same terminal can be used to expose
the forward exp channel, while the log channel remains separately addressable. -/
theorem eml_keeps_oriented_channels (x : ℝ) :
    eml x 1 = orientedToRatio x ∧
    orientedFromRatio x = Real.log x := by
  constructor
  · simp [eml, orientedToRatio]
  · rfl

/-- 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

/-- EML keeps oriented channel data: the same terminal can be used to expose
the forward exp channel, while the log channel remains separately addressable. -/
theorem eml_keeps_oriented_channels (x : ℝ) :
    eml x 1 = orientedToRatio x ∧
    orientedFromRatio x = Real.log x := by
  constructor
  · simp [eml, orientedToRatio]
  · rfl

What this page does not claim

The theorem does not claim that the cost function J alone derives the eml gate; that would be false since J forgets orientation. The theorem does not claim that eml is the unique operation preserving oriented channels. The theorem does not claim any physical interpretation for the oriented channels beyond the formal compiler-layer statement.

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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