Cross-Ratio Invariance and the Necessity of Scale-Dependent Expansion

Title: Cross-Ratio Invariance and the Necessity of Scale-Dependent Expansion Authors: Oscar Wojciechowski-Prill Abstract: This paper investigates the fundamental role of cross-ratio invariance in projective geometry and its implications for physical distance measures. We prove that a scale-independent distance function is incompatible with cross-ratio invariance under the full projective group PGL(2,R)\mathrm{PGL}(2,\mathbb{R})PGL(2,R). By defining a projective metric of the form dproj(x,y)=Σ(λ) Δ(x,y),d_{\mathrm{proj}}(x, y) = \Sigma(\lambda)\,\Delta(x, y),dproj(x,y)=Σ(λ)Δ(x,y), where Δ(x,y)\Delta(x,y)Δ(x,y) is a projective interval and λ=ln⁡(L/L0)\lambda = \ln(L/L_0)λ=ln(L/L0) is a logarithmic scale parameter, we demonstrate that Σ\SigmaΣ must be scale-dependent for invariance to hold. This result establishes an unavoidable scale-dependence in fundamental geometric structures, independent of specific physical theories. Beyond its mathematical implications, we discuss potential applications in physics, including consequences for metric theories of gravity, quantum geometry, and cosmic expansion. A concrete experimental test is proposed via high-precision spectroscopic measurements, which could detect subtle deviations arising from this geometric constraint. Keywords: Cross-ratio invariance, projective geometry, scale-dependent metrics, projective transformations, mathematical physics, quantum geometry, cosmology. License: CC BY 4.0 DOI:10.5281/zenodo.14868549 Citation: Oscar Wojciechowski-Prill, Zenodo, 044.2025, DOI: 10.5281/zenodo.14868549