Foundation Boolean Projection From Mark Bool Projection Not Canonical Without Ma

The need for a mark

A Boolean value is the simplest possible piece of information: a two-way choice, often written as true or false. The Recognition Science framework studies how such minimal choices emerge from larger structures, and one of its machine-checked theorems concerns exactly when that emergence is unambiguous. The theorem bool_projection_not_canonical_without_mark establishes that a carrier with more than one element does not, by itself, determine a unique Boolean projection, a way of mapping every element of the carrier to either true or false. The proof constructs two different marked pairs on the same three-point carrier that yield different projections, showing that non-singletonness alone leaves the choice underdetermined.

The resolution is that a canonical projection exists only after a distinguishing mark has been chosen. A marked pair, a structure consisting of a base point and an alternative point together with a proof that they are distinct, determines a projection uniquely: the base maps to false, and every other element maps to true. The framework's library proves this as boolProjection_canonical_given_mark, which states that for any marked pair, the base point is sent to false and the alternative point to true. This is the sense in which a mark is not optional decoration but a necessary ingredient for a well-defined two-valued shadow of a larger structure.

The theorem does not claim that Boolean projection is impossible without a mark, only that it is not canonical. A larger carrier always supplies at least one two-point shadow, but it does not choose which one. The theorem also does not claim that the mark itself is derived from anything more basic; it is simply assumed as part of the structure. Within the framework, this result supports the broader principle that recognition, the act of distinguishing one thing from another, requires a prior commitment to what counts as distinct.

/-- Without a mark, a three-point carrier has multiple inequivalent Boolean
shadows. This witnesses that non-singletonness alone does not canonically
select a Boolean floor projection. -/
theorem bool_projection_not_canonical_without_mark :
    ∃ (K : Type) (m1 m2 : MarkedPair K),
      boolProjection m1 ≠ boolProjection m2 := by
  classical
  let m1 : MarkedPair (Fin 3) :=
    { base := 0
      alt := 1
      distinct := by decide }
  let m2 : MarkedPair (Fin 3) :=
    { base := 1
      alt := 0
      distinct := by decide }
  refine ⟨Fin 3, m1, m2, ?_⟩
  intro h
  have h0 := congrArg (fun f : Fin 3 → Bool => f 0) h
  simp [boolProjection, m1, m2] at h0

/-- Given a marked pair, the induced Boolean projection sends the marked base
to `false` and the marked alternative to `true`. -/
theorem boolProjection_canonical_given_mark {K : Type*} (m : MarkedPair K) :
    boolProjection m m.base = false ∧ boolProjection m m.alt = true := by
  classical
  constructor
  · simp [boolProjection]
  · have halt_ne_base : m.alt ≠ m.base := fun h => m.distinct h.symm
    simp [boolProjection, halt_ne_base]

What this page does not claim

Boolean projection is impossible without a mark. The mark itself is derived from more basic principles. The theorem applies to carriers with exactly two elements in a way that differs from larger carriers.

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

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