impossible_is_non_positive
plain-language theorem explainer
Recognition Science equates a configuration x being impossible with it having non-positive ratio, hence equivalent to not being possible. Modal ontologists and philosophers of physics cite the result to ground impossibility in the recognition ledger's positivity constraint. The proof is a direct term-mode reduction that unfolds the two predicates and resolves the biconditional via linear arithmetic.
Claim. $x$ is impossible if and only if it is not possible, where impossible means $x ≤ 0$ and possible means $0 < x$, for all real $x$.
background
The ModalOntologyStructure module resolves modal metaphysics in Recognition Science via J-cost. Necessity holds solely for the unique zero-defect configuration at $x=1$. Possibility requires positive ratio with finite J-cost; impossibility captures any violation of positivity.
proof idea
The proof introduces arbitrary real $x$, simplifies the definitions of impossibility as $x ≤ 0$ and possibility as $0 < x$, then splits the biconditional. One direction assumes both and derives a contradiction by linarith. The converse negates possibility and reapplies linarith.
why it matters
This anchors the impossibility predicate in the Recognition Science modal framework and directly enables the sibling result that nothing is both possible and impossible. It realizes the module's claim (PH-013) that impossibility arises from non-positive ratios violating ledger positivity, before accessibility or actualization are addressed.
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