ClimatePhasePoint
ClimatePhasePoint supplies the type of an N-component real vector as the state space for climate phase dynamics. Recognition Science climate analyses cite it when constructing the J-cost functional and proving the vacuum achieves the global minimum. The declaration is a direct type abbreviation with no proof obligations or additional structure.
claimFor $N$ a natural number, a climate phase point is a function $s :$ Fin $N$ $→$ ℝ.
background
The Climate.AttractorStructure module formalizes the Recognition Science claim that the climate manifold possesses an attractor on which the long-term trajectory spends most time and whose J-cost is the global minimum of the energy/entropy field. ClimatePhasePoint defines the underlying state space as an idealized real vector with N components. Upstream imports supply the J-cost from Cost.Jcost (derived from the multiplicative recognizer comparator) together with non-negativity results from ObserverForcing and PhiForcingDerived.
proof idea
One-line type abbreviation with no computational content or proof obligations.
why it matters in Recognition Science
ClimatePhasePoint is the domain type for the master certificate AttractorStructureCert and the supporting theorems climateJCost, climateJCost_nonneg and vacuum_is_global_minimum. It supplies the concrete state space needed to state that the zero vector realizes the global J-cost minimum, consistent with the Recognition Science prediction that attractors occupy the lowest-cost configuration on the climate field.
scope and limits
- Does not encode climate dynamics, differential equations or time evolution.
- Does not fix the value of N or assign physical meaning to vector components.
- Does not reference the phi-ladder, forcing chain or specific constants.
- Does not prove existence or convergence of an attractor.
formal statement (Lean)
30abbrev ClimatePhasePoint (N : ℕ) := Fin N → ℝ
proof body
Definition body.
31
32/-- The J-cost of a climate state (sum of per-component costs). -/