stateQuality
stateQuality composes a display channel's observation map with its quality map to produce a direct real-valued quality function on states. Researchers working on RRF channel optimization cite it when establishing orderings or optimality criteria across different projections. The definition is realized by direct composition of the two component functions supplied by the DisplayChannel structure.
claimFor a display channel $C$ with observation map $observe : State → Obs$ and quality map $quality : Obs → ℝ$, the function $stateQuality(C)$ sends each state $x$ to $quality(observe(x))$.
background
DisplayChannel is a structure that pairs an observation function from states to an observation space with a quality function from observations to reals. The module presents display channels as projections that turn state spaces into quality measures for comparison in the RRF setting. Upstream components supply the concrete State type as the discrete 2D Galerkin state from CPM2D and auxiliary notions such as active edge count A from IntegrationGap and Anchor.
proof idea
This definition is a one-line wrapper realizing the composition of the quality map with the observation map.
why it matters in Recognition Science
This definition supplies the state-to-quality map used by isOptimal to characterize minimal-quality states and by QualityEquiv to compare channels. It underpins the channel_quality_transfer theorem in OctaveTransfer and the channel_mono_transfer theorem in OrderPreservation, which preserve quality orderings under morphisms and monotone transformations. In the Recognition Science framework it supports analysis of quality orderings within the eight-tick octave structure.
scope and limits
- Does not specify the concrete form of the quality function on observations.
- Does not address how channels are constructed or chosen.
- Does not prove any optimality properties.
- Does not connect to specific physical constants or the phi-ladder.
formal statement (Lean)
23def stateQuality (C : DisplayChannel State Obs) : State → ℝ :=
proof body
Definition body.
24 C.quality ∘ C.observe
25
26/-- A state is optimal in channel C if it minimizes quality. -/