Possible
The definition introduces the modal possibility operator in Recognition Science. A property p holds possibly at configuration c precisely when some reachable future y satisfies p. Cosmologists cite it when arguing that observed states like flatness or specific spectra are reachable under cost minimization. The definition is a direct existential quantification over the Possibility set.
claimLet $p$ be a predicate on configurations and $c$ a configuration. Then $p$ is possible at $c$ if and only if there exists a configuration $y$ reachable from $c$ such that $p(y)$ holds.
background
In the Modal.Possibility module, a configuration is a point in recognition state space carrying a positive real value, a time coordinate in ticks, and a logarithmic bound ensuring physical scale. A ConfigProp is any predicate on such configurations. The upstream Possibility operator builds the set of futures reachable from $c$ within one octave step while keeping total J-cost finite and non-increasing.
proof idea
One-line definition that applies existential quantification directly to the Possibility set constructed in the same module.
why it matters in Recognition Science
This definition supplies the possibility operator used by downstream cosmology results including flat_minimizes_cost and cosmological_constant_problem. It operationalizes the Recognition Science claim that physical outcomes are those reachable at finite cost, linking to the J-cost framework and the eight-tick octave. The definition closes the modal pair with Necessary and feeds directly into applications that treat observed cosmology as cost-minimizing reachable states.
scope and limits
- Does not compute explicit reachability for concrete numerical values.
- Does not incorporate full LedgerState details from the complete theory.
- Does not address infinite-cost transitions.
- Does not derive specific physical predictions on its own.
formal statement (Lean)
323def Possible (p : ConfigProp) (c : Config) : Prop :=
proof body
Definition body.
324 ∃ y ∈ Possibility c, p y
325
326/-- Modal notation -/
327notation:50 "□" p => Necessary p
328notation:50 "◇" p => Possible p
329
330/-- Duality: □p ⟺ ¬◇¬p -/