Symmetric Signatures of Global Configurations: Topological Rigidity and the Epoch of Convergence in a Case Study of Orion–Giza Correspondence | A Unified Framework for 2D Similarity Invariants, 3D Orientation Geometry, and 4D Precessional Dynamics

We develop a geometric framework for finite triple‑point configurations embedded in heterogeneous metric environments and evolving under a time‑dependent SO (3) flow. The theory unifies similarity geometry in two dimensions, plane geometry in three dimensions, and cyclic angular structure on the circle S1 into a single invariant‑based formalism. At the planar level, any non‑collinear triple of points in the plane is uniquely determined up to similarity by a minimal invariant pair (, ), consisting of a metric ratio and an interior angle. These two quantities provide natural coordinates on the moduli space of similarity classes of ordered triples. For triples embedded in three‑dimensional space, the associated planes determine unit normal vectors. Their relative orientation is captured by the angle gamma, defined as the arccosine of the dot product between the two normals. Additional orientation invariants, denoted alpha and beta, arise from scalar products between the terrestrial plane normal and two prescribed reference directions on the celestial sphere. The ordered triple (, , ) therefore forms a coordinate‑free descriptor of the three‑dimensional embedding. Temporal evolution is introduced through a smooth one‑parameter family of rotations R (t) in SO (3) acting on the celestial configuration. The induced planar projection produces a trajectory ( (t), (t) ) in the moduli space of similarity invariants. The structural mismatch between two configurations is modeled as a time‑dependent point in the product manifold R × S1, equipped with a weighted Riemannian metric that consistently combines linear and cyclic components. Within this setting, we define a cost functional J (t) as the norm of the mismatch vector and show that critical epochs satisfy the stationarity condition dJ/dt = 0. These epochs correspond to stationary points of the induced flow on the mixed linear–cyclic configuration space. The resulting global invariant S (t) consists of the similarity invariants (t) and (t), the orientation invariants alpha, beta, and gamma, the temporal derivatives of the similarity invariants, and the combined observational uncertainty vector. This multi‑layer geometric signature is defined entirely in terms of Euclidean, spherical, and product‑manifold structures. The construction is coordinate‑free and depends only on intrinsic properties of triple‑point configurations and their evolution under SO (3) actions.