Topological Void Analysis: A Mathematical Framework for Systematic Technical Innovation Discovery in Knowledge Spaces

Identifying where to innovate in a dense technical domain such as operating systems or hardware/software co-design-is fundamentally a search problem in a high-dimensional knowledge space. Existing approaches rely on keyword search, citation proximity, or human intuition, none of which formalise the notion of an unexplored region that is simultaneously relevant to a target goal and absent from prior art. We present Topological Void Analysis (TVA), a mathematical framework that defines topological voids as triads (A, B, C) in a dense-sparse hybrid embedding space. A void requires three conditions: (i) both concepts A and B are semantically cohesive with domain anchor C; (ii) their pairwise similarity falls within a calibrated marginality band-avoiding both obvious combinations and unrelated noise; and (iii) they share a sparse lexical bridge while the geodesic midpoint on the embedding hypersphere is unoccupied. Applied to ~140k indexed documents, TVA generates 2,128 invention candidates across 96 targets; 90% survive automated quality filtering, yielding 191 REVISE and 1 APPROVE verdict from four-specialist adversarial review (0.05% end-to-end). Two case studies demonstrate the framework surfaces non-obvious connective tissue rather than merely obvious related pairs.