Information Emlfrom Recognition Eml Recovers Exp

Exponentiation from a ledger

The exponential function is the map that turns addition into multiplication: exp(a + b) = exp(a) · exp(b). It is one of the most used functions in mathematics, appearing in compound interest, radioactive decay, and the growth of populations. The question here is whether something as abstract as a recognition ledger, a discrete record of events with a forced cost, can produce it. The answer is yes, with one condition: the ledger must remember which way it is oriented.

The key object is the EML operator, named after the physicist Andrzej Odrzywolek, which takes two real numbers x and y and returns exp(x) − log(y). This is not a random combination. If the ledger records events in an additive coordinate, then exponentiation lifts those coordinates to positive ratios, and the logarithm projects positive ratios back to ledger coordinates. The EML operator is the compiler gate induced by these two charts and subtraction. In the machine-checked library of formal theorems, this gate is defined as eml x y = exp x − log y.

The theorem eml_recovers_exp states that feeding the identity terminal, the number 1, to the logarithmic channel recovers the exponential function exactly: eml x 1 = exp x for every real x. The proof is a single simplification step, because log(1) = 0 and subtracting zero changes nothing. The same mechanism recovers the constant e by setting x = 1, and a longer cancellation loop recovers the logarithm from EML. The certificate structure EMLFromRecognitionCert packages these recoveries together with the claim that the reciprocal cost J forgets orientation, since J(x) = J(1/x).

In Recognition Science, this result is a compiler-layer bridge. It does not claim that the reciprocal cost alone derives the EML gate; that would be false, because the cost function J(x) = J(x⁻¹) forgets orientation. The oriented ledger before reciprocal quotienting carries the additive log coordinate, and that orientation is what makes exponentiation recoverable. The framework's library proves the recovery as a theorem, not as a definitional choice, and the certificate holds in the ambient type theory with no RS-specific axioms.

What this changes is the status of the exponential function inside the framework. It is not an arbitrary primitive; it is what an oriented recognition ledger produces when the logarithmic channel is fed the identity. The same construction recovers the logarithm and subtraction on positive ratios, so the whole arithmetic of positive reals sits on the oriented ledger. The reciprocal cost, by contrast, sees only the symmetric picture, which is why the orientation must be kept separate.

/-- EML recovers exponentiation by feeding the identity terminal to the
logarithmic channel. -/
theorem eml_recovers_exp (x : ℝ) :
    eml x 1 = Real.exp x := by
  simp [eml]

/-- The compiler gate induced by the oriented exp/log chart and subtraction. -/
def orientedCompilerGate (x y : ℝ) : ℝ :=
  orientedSub (orientedToRatio x) (orientedFromRatio y)

/-- 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

What this page does not claim

The reciprocal cost J alone derives the EML gate. The exponential function is defined by the ledger rather than by its usual series or differential equation. The orientation of the ledger is physically observable in any known experiment.

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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