diff_magnitude_dist
plain-language theorem explainer
The theorem establishes that integers 2 and 3 produce distinct events under the magnitude recognizer, which maps each integer to its absolute value. Recognition geometers cite it when confirming that the magnitude map separates positive integers of unequal size. The proof reduces the negated equality to an immediate numerical falsehood by unfolding the two definitions.
Claim. Let $R:ℤ→ℕ$ be the magnitude recognizer $R(n)=|n|$. Then $R(2)≠R(3)$.
background
The module supplies concrete recognition geometries on the integers. The magnitude recognizer is the map $n↦|n|$, which partitions ℤ into infinitely many classes indexed by the nonnegative integers. Two configurations are indistinguishable under a recognizer precisely when they yield the same recognition event, i.e., when the recognizer function returns equal values. The upstream Composition axiom supplies the multiplicative consistency condition that later forces the functional form of recognition cost, but is not invoked in this elementary example.
proof idea
The proof is a one-line wrapper that applies simp to the definitions of Indistinguishable and magnitudeRecognizer, reducing the claim to the false statement 2=3.
why it matters
The result illustrates that magnitude alone distinguishes 3 from 5 while sign alone distinguishes 3 from -3, thereby motivating the composition construction that refines both. It occupies an example slot in the Recognition Geometry module and aligns with the module's stated key insight that composition of recognizers increases distinguishing power. No open question is touched.
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