module module high

`IndisputableMonolith.RecogGeom.Connectivity`

The Connectivity module defines recognition-connected sets in Recognition Geometry as collections where every element is mutually indistinguishable under a given recognizer. Researchers building geometric models of recognition would cite these predicates when establishing local uniformity before full framework integration. The module supplies predicate definitions together with elementary lemmas on empty sets, singletons, and subsets.

claimA set $S$ is recognition-connected for recognizer $r$ when $x \sim_r y$ holds for all $x,y \in S$, where $\sim_r$ is the indistinguishability relation of the recognition quotient.

background

Recognition Geometry begins with the quotient construction $C_R = C/\sim$ in the Quotient module, which collapses configurations that cannot be told apart by the recognizer. The Connectivity module introduces the predicate IsRecognitionConnected, which encodes the strong condition that every point in $S$ sees the same event. It further defines the regularity predicates IsLocallyRegular, IsRegular, and SatisfiesRG5, along with lemmas such as locally_regular_cell_connected.

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

The module supplies the connectivity notions required by the Integration module, which assembles all components of Recognition Geometry into a complete summary. It fills the geometric layer that precedes any appeal to the forcing chain or the recognition composition law.

scope and limits

Does not construct or redefine the indistinguishability quotient.
Does not prove global topological properties of the recognition space.
Does not link connectivity to the phi-ladder or mass formulas.
Does not address the eight-tick octave or spatial dimension forcing.

used by (1)

From the project-wide theorem graph. These declarations reference this one in their body.

IndisputableMonolith.RecogGeom.Integration module_import

depends on (1)

Lean names referenced from this declaration's body.

IndisputableMonolith.RecogGeom.Quotient module_import

declarations in this module (11)

def IsRecognitionConnected L44
theorem isRecognitionConnected_empty L48
theorem isRecognitionConnected_singleton L54
theorem isRecognitionConnected_resolutionCell L62
theorem isRecognitionConnected_subset L69
def IsLocallyRegular L82
def IsRegular L86
structure SatisfiesRG5 L94
theorem locally_regular_cell_connected L101
theorem constant_recognizer_regular L115
def connectivity_status L138