PossibilitySpace
plain-language theorem explainer
The PossibilitySpace definition constructs the transitive closure of the Possibility relation as the set of all configurations reachable from a starting configuration c via finite paths of allowed transitions. Modal logicians in the Recognition Science framework cite this when analyzing reachable states from any given point in configuration space. The definition is supplied directly by set comprehension over path length n and Fin-indexed sequences that satisfy the step condition at each stage.
Claim. Let $c$ be a configuration. The set of reachable configurations from $c$ is defined as the collection of all $y$ such that there exists $n$ in the natural numbers and a map path from Fin$(n+1)$ to configurations satisfying path at the initial index equals $c$, path at the final index equals $y$, and for each intermediate index $i$ the successor configuration lies in the possibility set of the predecessor.
background
A configuration is a structure consisting of a positive real value, a natural-number time coordinate, and a logarithmic bound ensuring the value lies between roughly $10^{-7}$ and $10^{7}$. The possibility operator maps each configuration to the set of successor configurations whose time coordinate advances by exactly eight ticks while obeying the cost inequality that combines the J function at the initial and final values with the transition cost. This construction occurs inside the modal logic module, which imports arithmetic foundations and observer forcing to support modal notions built on the recognition state space.
proof idea
The definition is supplied directly by set comprehension. It existentially quantifies over the path length $n$ and a function from Fin$(n+1)$ to Config. The initial and terminal conditions on the path are discharged using the zero_lt_succ and le_succ arithmetic lemmas to validate the Fin bounds, while the universal step condition requires each consecutive pair to satisfy the possibility predicate.
why it matters
This definition supplies the reachability relation required for modal development inside Recognition Science. It is invoked by the theorem establishing that the identity configuration belongs to every possibility space, obtained by exhibiting a one-step path to the identity at the next octave tick. In the larger framework the construction supports the possibility operator that appears in the forcing chain and in the analysis of observer events.
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