Theory Comparison Matrix
A systematic comparison of theoretical frameworks in fundamental physics
| Aspect | TCR | Standard Model + GR | String Theory | Loop Quantum Gravity | Causal Set Theory |
|---|---|---|---|---|---|
| Foundational Structure | |||||
| Core Principle | Single variational principle: coherence cost minimization | Separate postulates for QM, GR, and gauge theory | Fundamental extended objects (strings) in target spacetime | Background-independent quantization of GR | Discrete causal structure as fundamental |
| Free Parameters | 1 (Hubble constant H0) | ~25 (19 SM + 6 cosmological) | ~10^500 vacua (landscape problem) | 1 (Barbero-Immirzi parameter, fitted) | Few (discreteness scale) |
| Mathematical Foundation | Spectral geometry, octonions, information theory | Differential geometry, Lie groups, QFT | Conformal field theory, algebraic geometry | Spin networks, category theory | Order theory, measure theory |
| Spacetime Nature | Emergent from coherence constraints | Fundamental continuum (assumed) | Fundamental continuum with extra dimensions | Emergent from spin foam dynamics | Fundamentally discrete (causal sets) |
| Spacetime and Gravity | |||||
| Spacetime Dimension | Derived: d=4 uniquely stable | Assumed (empirical input) | 10 or 11 required; 4 via compactification | Assumed to be 4 | 4 emerges statistically |
| Gravity Mechanism | Entropic force from KL divergence minimization | Fundamental field (metric tensor) | Closed string excitation (graviton) | Quantum geometry dynamics | Emergent from causal structure |
| Einstein Equations | Derived from maximum entropy | Postulated (action principle) | Low-energy effective limit | Semiclassical limit (in progress) | Emergent in continuum limit |
| Quantum Gravity | Unified: both emerge from same principle | Incompatible (non-renormalizable) | UV-finite perturbative gravity | Non-perturbative quantization | Discrete structure regularizes |
| Area Quantization | Derived: A_min = (pi-3)hG, gamma derived | None (continuous) | None in standard formulation | Yes, but gamma fitted empirically | Yes (fundamental discreteness) |
| Singularity Resolution | Bandwidth saturation prevents infinities | Singularities unavoidable (Penrose-Hawking) | String scale provides cutoff | Bounce replaces singularity | Discrete structure prevents |
| Quantum Mechanics | |||||
| Hilbert Space | Derived from coherence kernel (RKHS) | Postulated | Postulated | Postulated | Postulated |
| Born Rule | Derived via Gleason's theorem | Postulated | Postulated | Postulated | Postulated |
| Uncertainty Principle | Derived from kernel structure | Postulated (canonical commutation) | Postulated | Modified at Planck scale | Modified at discreteness scale |
| Measurement Problem | Decoherence from bandwidth constraints | Unresolved (interpretation-dependent) | Unresolved | Unresolved | Unresolved |
| Wave Function | Epistemic (coherence state) | Interpretation-dependent | Ontic (string state) | Spin network state | Quantum measure |
| Particle Physics | |||||
| Gauge Group Origin | Derived from octonion automorphisms | Empirical input | From compactification geometry | Not addressed | Not addressed |
| U(1) x SU(2) x SU(3) | Uniquely derived from G2 structure | Assumed | One of many possibilities | External input required | External input required |
| Number of Generations | Derived: 3 from octonion triality | Empirical input (unexplained) | Depends on compactification | Not addressed | Not addressed |
| Fermion Masses | Hierarchical structure from Jordan algebra | Yukawa couplings (free parameters) | From moduli (landscape) | Not addressed | Not addressed |
| Cabibbo Angle | Derived: theta = pi/14 (1.3% accuracy) | Free parameter | Moduli-dependent | Not addressed | Not addressed |
| Top Quark Mass | Derived: m = v/sqrt(2) (0.6% accuracy) | Free parameter | Moduli-dependent | Not addressed | Not addressed |
| Proton/Electron Mass Ratio | Derived: ~e^7.5 (1.5% accuracy) | Unexplained | Anthropic selection | Not addressed | Not addressed |
| Anomaly Cancellation | Derived: kappa1/kappa2 = 5/3 | Required for consistency (verified) | Automatic in some constructions | Not applicable | Not applicable |
| Cosmology | |||||
| Dark Energy | Derived: Omega_Lambda = 0.690 (exact match) | Cosmological constant (10^120 fine-tuning) | Landscape selection (anthropic) | Not directly addressed | Emergent from discreteness |
| Dark Matter | Resolution field saturation (no new particles) | Unknown particle (WIMP, axion, etc.) | Moduli, gravitino, or other BSM | Not addressed | Not addressed |
| MOND Phenomenology | Emergent in low-acceleration regime | Not explained | Not explained | Not addressed | Not addressed |
| Cosmological Constant Problem | Resolved: Landauer erasure cost | Unresolved (worst prediction in physics) | Landscape/anthropic (controversial) | Partially addressed | Addressed via discreteness |
| Inflation | Not required (coherence initialization) | Required (separate mechanism) | Various mechanisms possible | Bounce cosmology alternative | Not required |
| Arrow of Time | Derived from Landauer erasure | Assumed (past hypothesis) | Not addressed | Not addressed | Built into causal structure |
| Unification | |||||
| Gravity + QM | Unified at foundational level | Incompatible | Unified (perturbatively) | Gravity quantized, QM separate | Gravity from causality, QM separate |
| Gravity + Gauge Forces | Same geometric origin (spectral) | Separate theories | Unified in higher dimensions | Gauge theory added externally | Gauge theory added externally |
| Matter + Geometry | Both from Dirac operator fluctuations | Matter on fixed geometry | Strings in target space | Matter coupled to spin networks | Matter on causal sets |
| GUT Embedding | Natural: G2 contains SU(5), SO(10) | Possible but not required | Various GUT groups possible | Not addressed | Not addressed |
| Experimental Status | |||||
| Confirmed Predictions | Cabibbo angle, top mass, dark energy density, d=4 | Extensive (Higgs, W/Z, etc.) | None unique to strings | None yet | None yet |
| Testable Predictions | Sub-linear decoherence, quantum processor anomalies | Continued precision tests | Requires Planck-scale energies | Planck-scale discreteness effects | Lorentz violation at high energy |
| Near-Term Tests | Quantum computing platforms (IBM, Google) | LHC, precision measurements | None feasible | Gamma-ray dispersion (marginal) | Gamma-ray dispersion (marginal) |
| Falsifiability | Multiple quantitative predictions | Incremental tests | Difficult (landscape) | Limited | Limited |
| Theoretical Status | |||||
| Mathematical Rigor | Lean 4 formalization (15,917 lines) | Well-established (QFT caveats) | Partial (no complete formulation) | Rigorous in kinematic sector | Mathematically well-defined |
| Conceptual Economy | Single principle derives all structure | Multiple independent postulates | Single object, complex dynamics | Focused on gravity only | Minimal assumptions |
| Open Problems | Full CKM matrix, neutrino sector | Hierarchy, strong CP, unification | Landscape, moduli stabilization | Semiclassical limit, matter coupling | Dynamics, continuum limit |
| Development Stage | Emerging framework with formal proofs | Mature, precision-tested | 40+ years, no unique predictions | 30+ years, limited phenomenology | 30+ years, foundational stage |