A Forced Geometric Invariant for Discrete–Continuum Compatibility (COSMIC-273: Paper 1)

This work introduces a geometric solution to the incompatibility between discrete lattice substrates and continuous rotational field theories. A discrete lattice possesses only finite rotational symmetry, while physical fields transform under the continuous group SO(2). Reconciling these two regimes requires a geometric correction that allows a discrete boundary to support continuous curvature without distortion. By analysing the minimal region in which discrete and continuous descriptions must coincide, the paper derives a unique dimensionless stiffness invariant, Λ₀ = 4/π ≈ 1.2732. This value is not assumed or fitted; it is forced by geometric necessity. Λ₀ quantifies the ratio between the minimal orthogonal boundary capable of enclosing a unit interaction and the maximal rotationally invariant bulk region at the same scale. The result establishes Λ₀ as the fundamental modulus governing discrete–continuum compatibility. Once Λ₀ is fixed, it propagates through the geometry of the lattice and generates a natural hierarchy of higher‑order corrections. As an example, the paper applies Λ₀ to the torsion action of a minimal non‑contractible vortex. The resulting stiffness polynomial, S = 100Λ₀ + 6Λ₀², evaluates to S ≈ 137.05, matching the inverse fine‑structure constant α⁻¹ with high precision. This suggests that electromagnetic coupling strengths may emerge from geometric constraints rather than being arbitrary parameters. The work forms the first part of the COSMIC‑273 programme, which develops a geometric field theory built on discrete substrates. Future papers extend the framework to operator structures, spectral hierarchies, and long‑range geometric residues. The central claim is that Λ₀ is not merely a vacuum parameter but the generator of a broader geometric architecture with physical consequences. This paper presents a geometric resolution to the incompatibility between discrete lattice substrates and continuous rotational gauge fields. By analysing the boundary conditions required for a rotationally invariant field to propagate coherently on a hexagonal lattice, we derive a dimensionless stiffness invariant, Λ₀ = 4/π ≈ 1.2732. This invariant is shown to be a forced geometric necessity rather than an adjustable parameter: it quantifies the minimal correction required for discrete–continuum compatibility. Applying Λ₀ to a topological vortex configuration yields a stiffness polynomial whose numerical value reproduces the inverse fine‑structure constant α⁻¹ ≈ 137 with high precision. The result demonstrates that Λ₀ imposes a natural spectral hierarchy on vacuum torsion and suggests that fundamental coupling constants may emerge from geometric constraints of the discrete vacuum.