Virtual Invasion Dynamics: From Attribute Cluster to Entity — Invasion Ontology with Modular Ground and VID 3.0 Condensation Theory

The Virtual Invasion Dynamics (VID) framework rests on four core claims: interaction precedes entities; time is not a background parameter but the accumulation of irreversible marks left by invasion; boundaries are products of condensation rather than prerequisites; and the virtual/actual distinction is observation-section dependent.

Version 2.0 sank one layer beneath interaction: the chain difference → selection-potential → selection (singularity) → interaction → entity.

Version 3.0 sinks beneath difference itself. An attribute cluster is no longer a primitive element of a poset: it is redefined as a response-equivalence class of operator-algebraic pairs (U, ω), and the entire ground layer is reconstructed on Tomita–Takesaki modular theory.

The v3.0 ground carries: an intrinsic time flow (the modular flow, no external time parameter); a boundary-as-duality theorem (the modular conjugate J exchanges a system with its commutant, realizing single-sidedness as an operator identity); a condensation criterion as a spectral bifurcation (continuous-to-discrete transition of the modular spectrum); an existence criterion for stable entities as KMS fixed points of self-generated dynamics; a two-tier conservation law (cocycle chain rule for invasion flux; cohomology class for knot charge); a spectral dimension dS read directly from relative-entropy paths without presupposing space; a dimension-locking bridge theorem (candidate) reducing the restriction of 3+1-dimensional statistics to bosons/fermions to the simple-connectedness of high-dimensional configuration spaces; a homotopy-type account of fusion coherence (candidate) under which pentagon and hexagon equations become theorems about deformation spaces rather than axioms; and a gerbe-theoretic portrait of the attribute-cluster universe.

The v3.0 ground is realized explicitly in the Bost–Connes system, where the selection problem admits a precise solution: selection as orbit under a shuttle group action, time as knot on the type III₁ substrate.

Every item is labeled proved / imported / conjecture / suspended. The v2.0 ground layer is retained as the frozen-section projection of the v3.0 modular ground; the bridge is tabulated.