RSImpossible
plain-language theorem explainer
RSImpossible classifies a real number x as impossible precisely when x is non-positive. Modal metaphysicians and philosophers of physics cite this to anchor impossibility in the positivity constraint of the recognition ledger. The definition directly encodes the ledger rule that negative or zero ratios are excluded from consideration.
Claim. A configuration $x$ is impossible in Recognition Science if and only if $x ≤ 0$, violating the positivity constraint of the J-cost ledger.
background
Recognition Science grounds modal notions in the J-cost functional, where defect(x) equals J(x) for positive x and J obeys the Recognition Composition Law. The module PH-013 defines necessity as the unique J-minimizer at x=1, possibility as positive ratio with finite cost, and impossibility as non-positive ratio. Upstream defect is defined as J x in LawOfExistence, supplying the zero-defect condition at unity.
proof idea
This is a direct definition that equates impossibility with the predicate x ≤ 0. It serves as the foundational encoding of ledger positivity violation, enabling the equivalence theorems that follow.
why it matters
RSImpossible supplies the base case for the modal resolution in PH-013, feeding directly into impossible_is_non_positive which establishes the equivalence with negation of possibility, and into possible_not_impossible. It completes the triad of necessity (unique minimizer), possibility (positive ratio), and impossibility (non-positive), aligning with the eight-tick octave and D=3 framework landmarks by enforcing positivity in the ledger.
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