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Chapter 11 — Čech Cohomology Framework
Čech cohomology as the natural language of the theory — what holds exactly in the discrete setting, what requires additional topological hypotheses, and how the cocycle condition encodes the closure of relational transit.
Chapter 12 — Three Computational Axes
The three computational axes of invariants: intrinsic cohomology, relative cohomology of the fruit–door pair, and persistence barcodes of the door set.
Ontology Neural Network and ORTSF: A Framework for Topological Reasoning and Delay-Robust Control
The advancement of autonomous robotic systems has led to significant capabilities in perception, localization, mapping, and control, yet a critical challenge re
An Axiomatic Framework for Understanding Relations via Gauge-Invariant Cohomology
What does it mean for something to exist within a web of relations? We answer this question by developing a rigorous mathematical framework in which existence e