plus_minus_indist
plain-language theorem explainer
The theorem establishes that 3 and -3 are indistinguishable under the magnitude recognizer on integers, since both map to the same absolute value. Researchers modeling equivalence classes in recognition geometries would cite this as a basic case where sign information is discarded. The proof is a one-line wrapper that applies simp to unfold the definitions of Indistinguishable and magnitudeRecognizer.
Claim. Let $R:ℤ→ℕ$ be the magnitude recognizer defined by $R(n)=|n|$. Then $R(3)=R(-3)$, so 3 and -3 are indistinguishable under $R$.
background
The magnitude recognizer is the function sending each integer n to its absolute value |n|. Indistinguishable configurations are those that produce identical events under a recognizer, forming the fundamental equivalence relation of recognition geometry. This example sits inside the module that develops concrete cases on ℤ, including the discrete recognizer (where the quotient recovers the original space) and the sign recognizer (with exactly three classes).
proof idea
The proof is a one-line wrapper that applies the simp tactic to the definitions of Indistinguishable and magnitudeRecognizer, reducing the claim directly to the equality of natural-number absolutes.
why it matters
This result is invoked by the downstream examples_status definition to illustrate the composition principle: sign and magnitude recognizers together distinguish more pairs than either alone. It supplies a concrete instance of the indistinguishability relation within the Recognition Geometry examples module, showing how recognizers induce equivalence classes without reference to the forcing chain or phi-ladder.
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