projectSet
plain-language theorem explainer
projectSet defines the image of any subset U of configurations under the canonical projection to the recognition quotient. Researchers constructing induced structures on collapsed configuration spaces cite it when transferring sets across indistinguishability classes. The definition is realized directly as a set comprehension that collects every quotient element possessing at least one representative inside U.
Claim. Let $C_R = C/!sim$ be the recognition quotient obtained by collapsing configurations that the recognizer maps to the same event. For any subset $U subseteq C$, the projected set is defined by $pi(U) := { pi(c) mid c in U }$, where $pi$ denotes the canonical projection sending each configuration to its equivalence class.
background
The Recognition Geometry module builds the quotient space $C_R$ by identifying configurations $c_1 sim c_2$ precisely when the recognizer assigns them identical events. This yields the collapsed space $C_R = C/!sim$ on which further geometric operations are performed. Upstream, the neighborhood definition supplies finite-resolution collections of cells sharing a common observed label, while RS-native units fix the gauge with unit tick and unit voxel.
proof idea
The definition is a direct set comprehension that collects every quotient element q for which there exists at least one c in U whose image under the canonical projection equals q. It applies the projection map to test membership in the image without additional lemmas or reductions.
why it matters
projectSet supplies the image operation required by quotientNeighborhoods, which constructs the induced neighborhood structure on the quotient. It completes the geometric transfer step inside the Recognition Geometry module, allowing local configuration neighborhoods to descend to the collapsed space. The construction supports the Recognition Science derivation of spatial structure from recognition relations and aligns with the forcing chain that produces three-dimensional geometry.
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