Foundation Primitive Recognition Calculus Valid Comparison

Valid comparison

A ledger, a discrete record of events, is only as good as its comparisons. In Recognition Science, the question is when a comparison made through a display, an intermediate representation, is valid. The framework's recognition calculus, its formal system for tracking these events, answers this with a structure called a bridge. A bridge connects a native object, the thing itself, to a display object, its representation, and an observable object, the protocol by which both are measured. The bridge's commuting law states that observing the display gives the same result as observing the native object directly. This single law is the foundation for what follows.

The core theorem, validComparison_iff_native, proves that a comparison of two displayed values is valid exactly when the corresponding native observations are equal. In plain language, if two representations look the same, they are truly the same only when the underlying realities also match. This is not a definitional choice but a proved equivalence in the machine-checked library of formal theorems. The result is stable under composition: if a native object maps to a display, and that display maps to another display, the composite bridge remains valid, and the comparison still descends to native equality.

In Recognition Science, this establishes the valid comparison doctrine: a comparison in a display carrier is legitimate precisely when it reduces to equality of the native observable protocol, and this legitimacy survives the chaining of display bridges. The practical consequence is that any layer of representation, any intermediate format, can be added or removed without corrupting the truth of a comparison, as long as each layer obeys the commuting law. This is what makes the ledger trustworthy across transformations.

theorem validComparison_iff_native {N D O : Type*} (B : Bridge N D O) (x y : N) :
    IsValidComparison B x y ↔ B.observeNative x = B.observeNative y := by
  unfold IsValidComparison
  rw [B.commutes x, B.commutes y]

theorem validComparison_compose {N D E O : Type*}
    (B₁ : Bridge N D O) (B₂ : Bridge D E O) (x y : N) :
    IsValidComparison (compose B₁ B₂) x y ↔ B₂.observeNative (B₁.display x) = B₂.observeNative (B₁.display y) :=
  validComparison_iff_native (compose B₁ B₂) x y

What this page does not claim

This module does not define what a native object or an observable protocol is in any particular physical theory. The valid comparison theorem does not by itself force the specific cost function J(x) or any derived constant.

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

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