Parameter Free Unification of Quantum Mechanics and General Relativity - ħ, G, c, α

Update: Revised English version. Revised German version Appendix SimulationsThis work presents a parameter-free unification of quantum mechanics and general relativity based on the Planck two–measureσₚ = ℓₚ·tₚ = ħ·G / c⁴. The framework defines spacetime itself as a quantized field, where each Planck cell (σₚ) acts as an elementary unit of geometric action.By expressing the Einstein field equations through a Planck–covariant source averaging, ultraviolet divergences and classical singularities are removed without introducing any new fields or parameters. The effective cosmological constant arises naturally asΛ_eff = 3 / (c·R·t),linking the microscopic and macroscopic domains through the dimensionless invariantα_σ = σₚ / (R·t).This relation implies Λ_QFT / Λ_obs = 1, showing that the cosmological constant is not a fine-tuned parameter but the macroscopic expression of quantum gravity itself. The theory connects the constants {ħ, G, c, α} across scales via the logarithmic invariantδ = log₁₀(λ̄_p² / [α·c·R·t]),which is independent of G and stable within ±0.3 dex over admissible cosmic windows.It provides a falsifiable numerical bridge between quantum physics and cosmology. Benchmark simulations (SN Ia, BAO, CMB angle, fσ₈, weak lensing) confirm that the σₚ framework reproduces ΛCDM observables within 2% accuracy — without free parameters.Linear weak-lensing power spectra show a natural 10–20% suppression, consistent with current low-S₈ tensions. In this model, gravity is not a separate force but the emergent curvature of a σₚ–quantized spacetime.Quantum mechanics and general relativity appear as two limits of the same geometric principle:ħ quantizes action, G quantizes curvature, and their product ħ·G = c⁴·σₚ defines the invariant measure of quantum gravity. The work completes the triad of Minkowski, Einstein, and Planck–Heisenberg — uniting geometry, curvature, and action in a single, self-consistent spacetime structure. Related Works: Zander, A. (2025). Parameter Free Unification of Quantum Mechanics and General Relativity - ħ, G, c, α. Zenodo. https://doi.org/10.5281/zenodo.17533233 Zander, A. (2025). The Memory of Spacetime - Black Holes and the End of Infinity - Solving Hawkings Info-Paradox. Zenodo. https://doi.org/10.5281/zenodo.17548544 Zander, A. (2025). Hierarchical Black Hole Mergers. Zenodo. https://doi.org/10.5281/zenodo.17495572 Zander, A. (2025). The Natural Structure of Spacetime: From Quantum Mechanics, Relativity, and Gravitation. Zenodo. https://doi.org/10.5281/zenodo.17357165 Zander, A. (2025). From Quantum to Cosmos: About Space and Time, Mass and Inertia. Zenodo. https://doi.org/10.5281/zenodo.16942321