Recognition Geometry

Recognition Geometry is a new geometric framework that inverts the traditional relationship between space and measurement. In classical geometry, space is primitive and measurements are operations performed on a pre-existing spatial substrate. Recognition Geometry reverses this: recognition maps are primitive, and space emerges as a quotient structure.

This paper presents the complete axiomatic foundations (RG0–RG7), proves the fundamental theorems including the Universal Property of the Recognition Quotient, develops the theory of recognition charts and dimension, and establishes the bridge to Recognition Science physics.

We formally define a Configuration Space (RG0) and a Locality Structure (RG1), upon which Recognizers (RG2) operate to produce observable Events. We define the fundamental Indistinguishability Relation (RG3) and construct the Recognition Quotient, proving that it captures exactly the observable structure of reality. We further develop the theory of Finite Resolution (RG4), Local Regularity (RG5), Composite Recognizers (RG6), and Comparative Recognizers (RG7), demonstrating how continuous geometric structure emerges from discrete recognition processes.

Finally, we show that this framework provides a rigorous mathematical foundation for Recognition Science, explaining the dimensionality of spacetime, the nature of gauge symmetries, and the origin of physical metrics. All definitions and 50+ theorems have been formalized and verified in the Lean 4 interactive theorem prover (approximately 3,100 lines of code across 16 modules).