Invariant Tensor Validation of Kalimuthu's Degenerate Spherical Triangle Theorems: Topological Phase Transitions and Dimensional Collapse Across Reparametrised Physical Manifolds

Abstract

This revised paper addresses the reviewer clarifications concerning Kalimuthu's degenerate spherical triangle theorems where the interior angle sum Σ α_i approaches the boundary values 2π (360°) and 3π (540°). We explicitly distinguish between coordinate-induced degeneration and genuine curvature singularities. Employing invariant tensor calculus, we compute metric tensor components, Christoffel symbols, and coordinate-independent curvature invariants (Ricci scalar R and Kretschmann scalar K = R {μνρσ}R^{μνρσ}) on S². We show that the underlying manifold S² maintains constant Gaussian curvature K_G = 1/R² and finite scalar invariants at all points including the equatorial limit θ = π/2. The reported vanishing of Christoffel symbols Γ^λ_{μν} → 0 at θ = π/2 is shown to be a coordinate-dependent property of the standard polar chart, not a tensorial invariant, but its geometric interpretation as local affine trivialization along the great-circle geodesic is preserved. The apparent contradiction between non-zero Girard area and dimensional collapse is resolved: the area formula Area = R²(Σα_i - π) yields πR² for Σ=2π and 2πR² for Σ=3π, which correspond to regularized limits where the interior domain tends to a hemispherical domain with its boundary compressed onto a closed geodesic. Applications to hyperbolic space, BSSN formalism, cosmic strings, loop quantum gravity, wormholes, and AdS/CFT are now explicitly classified as mathematical analogies and regularisation models with stated limitations, not direct physical proofs of new phenomena.