PossibilityBoundary
plain-language theorem explainer
PossibilityBoundary marks the origin as the unreachable limit point of possible configurations in modal geometry. Researchers analyzing J-cost divergence in Recognition Science modal models would cite this when bounding finite-cost recognition events. The declaration is a direct set definition with no lemmas or reductions required.
Claim. The boundary of possibility is the set $P = {0} subset mathbb{R}$, the limit point approached as the positive argument tends to zero at which the J-cost diverges to infinity.
background
In the ModalGeometry module, which imports possibility and actualization structures, the J-cost is the non-negative recognition cost induced by multiplicative recognizers and observer forcing events. Upstream definitions establish cost as the derived comparator output on positive ratios and confirm non-negativity for any recognition event. The local setting treats the boundary as a limit of configurations rather than a reachable state, consistent with the module's focus on modal distances and curvature.
proof idea
The declaration is a direct definition that constructs the singleton set containing only the real number zero via set comprehension. No upstream lemmas are invoked; the body is an immediate abbreviation of the mathematical limit point described in the module documentation.
why it matters
This definition supplies the boundary marker required for modal geometry constructions such as possibility balls and curvature statements. It aligns with the Recognition Science forcing chain at the J-uniqueness step, where cost diverges at the origin, and supports the eight-tick octave periodicity. No direct downstream uses are recorded, leaving open its integration into explicit modal distance theorems.
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