Information Emlfrom Recognition Eml Recovers Sub

The recovery of subtraction

Subtraction is the operation that answers "how much more is this than that?" Given two positive quantities, x and y, the difference x - y tells you the gap between them. The theorem eml_recovers_sub establishes that this familiar operation can be rebuilt from a single expression, eml (a recognition-based operation defined as exp(x) - log(y)), by feeding it the right inputs: eml (log x) (exp y) = x - y, provided x is positive. The expression's name, EML, comes from the physicist Andrzej Odrzywolek, who introduced it as a kind of exponential-logarithmic gate.

The theorem is a formal statement in the framework's machine-checked library of formal theorems, meaning its proof has been verified step by step by a computer. The proof itself is short: it unfolds the definition of eml, uses the fact that exp(log x) = x for positive x, and the fact that log(exp y) = y. The condition that x must be positive is not a technicality; it is essential, because the logarithm is only defined for positive inputs in the real numbers. Without that condition, the statement would not even make sense.

The deeper point of the theorem is about what the framework calls orientation: the distinction between a forward direction and a backward direction. The framework's central cost function, J(x) = (x + 1/x)/2 - 1, has a symmetry: J(x) = J(1/x). This means the cost of a recognition event does not care which way the recognition goes. The theorem eml_recovers_sub depends on breaking that symmetry. It uses the exponential and logarithm as separate, oriented channels, so that going forward and going backward are different operations. The framework's library proves this explicitly: the reciprocal cost forgets orientation, while eml keeps the oriented channel data.

In Recognition Science, the theorem is one piece of a larger claim: that the structure of recognition events, when kept oriented, can generate the basic operations of arithmetic. The framework models a ledger, a discrete record of recognition events, and shows that before the reciprocal symmetrization, the ledger has an additive log coordinate. The eml operation is the compiler gate that this oriented structure induces. The theorem eml_recovers_sub is the final step in a small chain: once eml is available, it can recover exponentiation, then logarithm, and then subtraction.

What the theorem does not claim is just as important as what it proves. It does not claim that the reciprocal cost function J alone derives subtraction. That would be false, because J(x) = J(1/x) forgets orientation. It does not claim that subtraction is physically fundamental in the framework; it claims only that subtraction can be expressed in terms of the oriented exp-log structure. It also does not claim that the framework has derived subtraction from nothing; the theorem assumes the real numbers, the exponential function, and the logarithm as given mathematical objects.

/-- Once `exp` and `log` have been recovered, EML recovers subtraction on
positive ratios. -/
theorem eml_recovers_sub (x y : ℝ) (hx : 0 < x) :
    eml (Real.log x) (Real.exp y) = x - y := by
  unfold eml
  rw [Real.exp_log hx, Real.log_exp]

/-- 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

/-- Oriented additive ledger coordinates lift to positive ratios by exponentiation. -/
def orientedToRatio (u : ℝ) : ℝ :=
  Real.exp u

What this page does not claim

This answer does not claim that the reciprocal cost function J alone derives subtraction, since J(x) = J(1/x) forgets orientation. This answer does not claim that eml_recovers_sub holds without the positivity condition on x. This answer does not claim that the framework derives the real numbers, the exponential function, or the logarithm from scratch.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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