PAPER–Σ: The Geometric Foundations of Matter and Spacetime: Rigidity, Visibility, and the Topology of Compactification

This paper addresses three foundational questions within the Fracture–Berry–Tension (FBT) framework: (i) how symplectic rigidity serves as a unifying geometric principle underlying both quantum uncertainty and gravitational regularity; (ii) how visible and dark matter sectors are to be distinguished geometrically; and (iii) how black holes should be understood in a six-dimensional symplectic ontology. 

We show that Gromov’s non–squeezing theorem provides a common geometric explanation for the existence of a minimal action quantum and for the impossibility of true geometric singularities. Building on this, we classify matter sectors in terms
of two geometric invariants of compactified surfaces

Σ2 ↩→M6 : AΣ =󰁝Σ2 Ω, cind 1 (Σ).

Visible matter requires both nonzero symplectic area and nontrivial induced gauge topology, together with admissible observable phase locking. Dark sectors arise either when the induced gauge topology is trivial or when coherent observable
readout fails despite gravitational activity. Measure arguments suggest that dark compactifications are generic, whereas visible compactifications are structurally selective.

Black holes are reinterpreted as regions where regular observable projection fails, rendering the interior observationally inaccessible while the underlying six-dimensional geometry remains regular.