boundary_unreachable
plain-language theorem explainer
Modal geometry establishes that every configuration maintains a strictly positive value, so the zero boundary remains unreachable. Researchers modeling possibility spaces or actualization processes in Recognition Science cite this to enforce non-degenerate states. The proof is a direct one-line application of the ne_of_gt lemma to the embedded positivity constraint.
Claim. For every configuration $c$ equipped with value $v$ satisfying $0 < v$, the equality $v = 0$ is impossible.
background
The Config structure from Modal.Possibility defines a point in recognition state space via a positive real value (generalizing the bond multiplier) together with a time coordinate in ticks and an explicit positivity hypothesis pos : 0 < value. This local setting sits inside the ModalGeometry module, which imports Possibility and Actualization to develop modal distance and curvature notions. Upstream results include the UniversalForcingSelfReference.for structure (recording meta-realization axioms) and the ILG.Config and RSNative.Core.Status definitions that supply the broader gravity and measurement context.
proof idea
The term-mode proof performs a single introduction of the configuration variable c followed by an exact application of the ne_of_gt lemma to the pos field of c.
why it matters
The result secures the boundary-unreachable property at finite cost inside modal geometry, preventing zero-value degeneracies that would collapse the possibility space. It supports the Recognition Science forcing chain by preserving strict positivity consistent with the J-cost and phi-ladder constructions. No downstream theorems are recorded yet, leaving its role as a foundational guardrail for later curvature and periodicity statements.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.