Foundation Pair Kernel Response Ancestry S21 Response Coordinate Quotient Projec

The quotient map

A quotient is a way of declaring some things equivalent. In mathematics, when you have a set and a family of tests, you can form a quotient by saying two elements are the same if every test gives the same result on both. The declaration quotient here is built from a family of coordinate probes: each probe is a function that reads a response and returns a real number, and two responses are equivalent exactly when all probes return the same values. The theorem injective says that this equivalence is trivial: if two responses are equivalent under all probes, then they are actually the same response. In other words, the probes separate the responses completely; no two distinct responses are indistinguishable by the full probe family.

The proof is short and rests on a stronger fact. The family of coordinate probes is separating, meaning that for any two distinct responses there is at least one probe that gives different values on them. From that, a general lemma about quotients yields that the projection map from responses to equivalence classes is injective. The declaration is a theorem in the machine-checked library of formal theorems, proved without any axiom beyond the standard logical ones. It is one small piece of a larger structure that traces physical responses back to committed recognition acts, but the theorem itself is purely about the probe family and the quotient it defines.

What the theorem does not claim is important. It does not say that the probes are the only way to distinguish responses, nor that the probes are physically realizable in every system. It does not say that any particular physical channel reads the probes correctly. Those are separate statements in the same module, and some of them are open or depend on additional assumptions. The injectivity result is a statement about the abstract probe family, not about any concrete physical implementation. It also does not claim that the quotient map is surjective, nor that the equivalence classes have any particular structure beyond being in bijection with the original responses.

/-- The observational quotient of the Recognition response basis has no
nontrivial identifications. -/
theorem responseCoordinateQuotient_projection_injective :
    Function.Injective
      (proj responseCoordinateProbeFamily3) :=
  proj_injective_of_separating
    responseCoordinateProbeFamily3
    responseCoordinateProbes_separate

/-- The committed response probes separate all five source responses. -/
theorem responseCoordinateProbes_separate
    (left right : RecognitionParentResponse3)
    (hindistinguishable :
      ObsEquiv responseCoordinateProbeFamily3
        left right) :
    left = right := by
  by_contra hne
  have hright : right ≠ left :=
    Ne.symm hne
  have hprobe :=
    hindistinguishable
      (responseCoordinateObservation3 left)
      ⟨left, rfl⟩
  simp [responseCoordinateObservation3,
    hright] at hprobe

What this page does not claim

The theorem does not claim that the coordinate probes are physically realizable in every system. It does not claim that any particular physical channel reads the probes correctly. It does not claim that the quotient map is surjective or that the equivalence classes have additional structure.

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/PairKernelResponseAncestryS21.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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: