A Unified Framework for Fundamental Constants: Proposal for a Research Program

Three of the most important numbers in physics—the cosmological constant \Lambda, the fine-structure constant \alpha, and the duration of cosmic inflation as measured by the number of e-folds N—have resisted explanation from first principles. The cosmological constant is 10^{120} times smaller than quantum field theory predicts. The fine-structure constant \alpha \approx 1/137 has been called "a magic number" with no accepted explanation. The number of e-folds N \sim 55 requires an inflaton field with a carefully tuned potential. This paper presents a unified framework that yields all three numbers from a common set of principles. It is a proposal for a research program—not a claim of completed derivations. It demonstrates that the framework predicts specific numerical values that agree with observation, while honestly identifying which steps are derived, which are motivated by O(1) Planck-scale inputs, and which remain open problems. What this paper provides: · A complete axiomatic framework built on eight primitives (Order, Amplitude, Acceleration, Polarity, Dimension, Angle, Chirality, Charge) with negative counterparts under a symmetry operator \mathcal{S} (\mathcal{S}^2 = I), governed by four equations: the Unified Wave Equation (generating temporal dynamics), the Threshold Condition (selecting stable structures via local potential well formation), the Eigenvalue Equation (organizing spectra), and the Feed mechanism (driving systems toward \mathcal{S}-invariant equilibrium in meta-time).· A derivation of the cosmological constant: Baseline subtraction eliminates the 10^{120} vacuum energy discrepancy. The residual asymmetry from finite information capacity yields \Lambda = 3/(\pi R_H^2) \approx 5.6 \times 10^{-53} m^{-2}, matching observation within a factor of approximately 2. The scaling \Lambda \propto 1/R_H^2 is the robust prediction. The dark energy equation of state is predicted to be w_0 \approx -0.93, evolving to w \approx -0.73 at z \sim 2—testable with DESI, Euclid, and Roman.· A derivation of the fine-structure constant: The fundamental coupling unit is \alpha_0 = 1/\ln(R_H/\ell_P) \approx 1/140. The gauge coupling ratios are 3:2:1 from the subspace geometry of the internal 3D space. Threshold corrections and Standard Model infrared running yield \alpha^{-1} \approx 137, matching observation to 0.03\%. The framework predicts time variation \dot{\alpha}/\alpha \sim 10^{-17} yr^{-1}—testable with atomic clocks.· A derivation of the number of e-folds of inflation: Before any spacetime voxels exist, the natural state is exponential expansion at the Planck rate. Inflation ends when the first voxel nucleates. The nucleation probability follows a Gaussian product distribution, giving N = \exp(T_{ST}/\sigma^2). With T_{ST}/\sigma^2 \sim 4, N \sim 55, consistent with cosmological requirements. The spectral index n_s \approx 0.964 follows as a consistency check. The tensor-to-scalar ratio is predicted to be r \ll 0.01—testable with B-mode experiments.· A comprehensive Status of Results table as the centerpiece of the paper, explicitly distinguishing derived results from motivated inputs, effective parameterizations, Standard Model inputs, and open problems. This honest accounting maps what has been achieved and what remains to be done.· Three falsifiable predictions: (1) dark energy equation of state w_0 \approx -0.93; (2) time variation of the fine-structure constant \dot{\alpha}/\alpha \sim 10^{-17} yr^{-1}; (3) tensor-to-scalar ratio r \ll 0.01. These will be tested by DESI, Euclid, atomic clocks, and B-mode experiments within the coming decade.· The integers \{1,2,3,4,6\}—the dimensions of the gauge subspaces of a 3D internal space—appear throughout the framework's predictions. If space had D dimensions, the integers would be different. The observed pattern is a fingerprint of D = 3. Why this matters: The framework does not require supersymmetry, extra dimensions, an inflaton field, or anthropic selection. It derives specific numerical values for fundamental constants from a small set of primitives and O(1) Planck-scale inputs. The same fundamental coupling \alpha_0 = 1/\ln(R_H/\ell_P) that determines the fine-structure constant also sets the SU(3) threshold and the Majorana mass scale. The integers \{1,2,3,4,6\} are not inputs—they are the dimensions of subspaces of a 3D space. This is a research program proposal. The Status of Results table maps what has been achieved and invites the community to address the remaining open problems: a rigorous 1/I_{\text{max}} proof, a first-principles calculation of T_{ST}/\sigma^2, a complete canvas-specific RG analysis, a 1+1D toy model demonstrating closed wave formation, and a proper canvas-specific derivation of the spectral index. Keywords: cosmological constant, fine-structure constant, inflation, unified framework, eight primitives, \mathcal{S}-invariance, Feed mechanism, threshold condition, information bound, research program, falsifiable predictions, dark energy equation of state, time-varying alpha, tensor-to-scalar ratio