Foundation Recognition Signature Gauge Signature Projection Injective Of Separat

Separating recognizers

A recognition event is a discrete record of an observation, and a recognizer is a function that assigns a value to each state. In Recognition Science, the full recognition signature of a state is the tuple of all values assigned by a chosen family of recognizers. Two states are considered physically identical when they share the same full signature, meaning no recognizer in the family can tell them apart.

The declaration signature_projection_injective_of_separating establishes a precise condition for when this identification is lossless. It proves: if a family of recognizers is separating, meaning that whenever two states have the same full signature they are actually the same state, then the projection from the original state space onto the quotient of physical states is injective. In plain terms, the projection maps distinct states to distinct equivalence classes, so no information is lost in the identification. This is a THEOREM in the machine-checked library, with zero axioms beyond the standard three.

The proof relies on the definition of SameRecognitionSignature and the quotient construction. The key insight is that the separating condition is exactly the hypothesis needed to rule out collisions in the projection. The library also shows a concrete example: the family containing both coordinate recognizers on the two-bit state space Bool × Bool is separating, and its projection is injective. In contrast, the family containing only the first bit is not separating, because the states (false, false) and (false, true) share the same signature under that family.

The declaration does not claim that scalar-cost equality is a complete gauge invariant. A separate completeness hypothesis is required for that, and the library shows that a single Boolean coordinate is not complete in general. It also does not claim that any arbitrary family of recognizers is separating; the condition must be verified for each family. The theorem provides a sufficient condition for injectivity, not a characterization of all injective projections.

What this fact changes: it gives a clean criterion for when a recognition-based identification is faithful. If you can exhibit a separating family, you know the quotient projection loses nothing. This is the foundation for later claims about completeness and the boundary between one-bit and two-bit observations.

/-- If the full recognition signature separates states, the quotient projection
is injective.  This is the precise "complete recognizer family" condition. -/
theorem signature_projection_injective_of_separating (F : Set (X → C))
    (hsep : ∀ x y, SameRecognitionSignature F x y → x = y) :
    Function.Injective (proj F) :=
  proj_injective_of_separating F hsep

/-- Therefore the physical quotient by the two-coordinate family is injective:
two Boolean recognizers recover the whole two-bit toy state. -/
theorem pairBitFamily_projection_injective :
    Function.Injective (proj pairBitFamily) :=
  signature_projection_injective_of_separating pairBitFamily pairBitFamily_separating

/-- One Boolean coordinate fails to separate the two states with the same first
bit and different second bit. -/
theorem one_boolean_coordinate_not_complete :
    ∃ x y : PairBoolState,
      x ≠ y ∧ SameRecognitionSignature firstBitFamily x y := by
  refine ⟨(false, false), (false, true), ?_, ?_⟩
  · decide
  · intro f hf
    have hf' : f = firstBit := hf
    rw [hf']
    rfl

/-- Under the explicit completeness hypothesis, the scalar-cost kernel matches
the full recognition-signature equivalence.  Without this hypothesis, scalar
cost equality is only a cost observable. -/
theorem scalar_cost_kernel_eq_signature_of_complete
    (F : Set (X → C)) (cost : X → ℝ) (hcomplete : ScalarCostCompleteFor F cost)
    (x y : X) :
    cost x = cost y ↔ SameRecognitionSignature F x y :=
  hcomplete x y

What this page does not claim

The theorem does not assert that every family of recognizers is separating; it provides a sufficient condition, not a necessary one. The declaration does not establish that scalar-cost equality alone is a complete gauge invariant without the separate completeness hypothesis. It does not claim that a single Boolean coordinate is complete for any state space, only that the two-bit example shows incompleteness in general.

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

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