Subtractive Gap Detection in Geometric Knowledge Manifolds: Reverse Decomposition via Constraint Topology Conservation

We present a method for subtractive gap detection in geometric knowledge manifolds. Given a target concept and an existing manifold of synthesized knowledge, the method encodes the target into a high-dimensional phasor lattice, computes its geometric neighborhood, and identifies which structural prerequisites are supported by the existing manifold and which are absent. The absent regions constitute a precise geometric profile of missing knowledge — not a list of labels, but a measurable shape in the lattice whose topology specifies what must exist for the target to be achievable.

The method is subtractive: it works by eliminating geometrically invalid connections through quality gates rather than generating prerequisites additively. We report results from two proof-of-concept runs on manifolds of 3,745 and 527 nodes, achieving consistent coverage ratios (67%) and void axis counts (8) across fundamentally different domains (pharmaceutical research and philosophical inquiry).

Theoretical motivation is derived from a 748-node synthesis run in which the engine itself derived the architectural principles of reverse decomposition, finding that forward synthesis and backward decomposition are symmetric operations of constraint satisfaction on the same geometric substrate. The method requires no new mathematical machinery beyond the existing forward synthesis engine.

This is the fifth paper in the Omuo geometric knowledge synthesis series. Prior work established the synthesis engine, the 168/72 lattice invariant, recursive self-deepening, and cross-domain convergence on anomaly cancellation. This paper extends the framework from discovery (what connects known concepts?) to prescription (what is missing for a target to be achievable?).