possible_not_impossible
plain-language theorem explainer
Recognition Science shows that no real number satisfies both the positive-ratio predicate for possibility and the non-positive predicate for impossibility. Modal metaphysicians or RS model builders would cite this to confirm internal consistency of the J-cost ontology. The proof is a direct term-mode contradiction that invokes the characterization impossible_is_non_positive to derive absurdity from the two assumptions.
Claim. For every real number $x$, if $0 < x$ then it is not the case that $x ≤ 0$.
background
The module PH-013 resolves modal metaphysics inside Recognition Science by grounding necessity, possibility, and impossibility in J-cost. RSPossible x holds precisely when the ratio is positive, i.e., $0 < x$, which also guarantees finite J-cost. RSImpossible x holds when the ratio is non-positive, i.e., $x ≤ 0$, violating ledger positivity. The local setting treats the identity configuration $x = 1$ as the unique J-minimizer and defines accessibility via J-cost decreasing paths.
proof idea
The term proof assumes RSPossible x and RSImpossible x, then applies the lemma impossible_is_non_positive (which states that impossibility implies non-positive ratio) to obtain a contradiction via absurd.
why it matters
The result supplies the mutual-exclusivity axiom required for the S5 modal fragment in PH-013. It supports the downstream claims necessity_is_unique_minimizer and actuality_is_j_minimum by keeping the three predicates disjoint. The theorem closes one consistency check in the RS modal resolution without touching the forcing chain T0-T8 or the Recognition Composition Law.
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