Gravity, special relativity, and the Hubble tension from octahedral geometry.
Your Reading Path
P011 (Null Tether) → P010 (Elastic Gravity) → P020 (Special Relativity / BRT) → P012 (Cosmology)
Time estimate: ~4 hours for full path.
The Core Claims
CMAC claims to derive from B = 1 on S² (zero free parameters):
| Observable | CMAC Prediction | Experiment | Residual |
|————|—————–|————|———-|
| H₀(local)/H₀(CMB) | 13/12 = 1.0833 | ~1.08 | 0.02σ |
| GW polarizations | dim(E_g) = 2 | 2 | exact |
| Lorentz invariance | Aut(CP¹) = SO⁺(3,1) | No violations | exact |
| Λ·ℓ²_P | (9/5)·α⁵⁷ | ~10⁻¹²² | 122 orders |
The Hubble tension is not a crisis — it’s a prediction.
Step-by-Step Path
P011: The Null Tether Principle
The ground state of spacetime.
Key results:
- Null Tether Principle: spacetime structure from bandwidth conservation
- Gravitational coupling α_G at 0.2 ppm
- ℓ_grav = 9 (gravitational channel multiplicity)
- Born rule from Gleason’s theorem on C⁷
Verification: CMAC_PHYS_P011_Verification.py (89 checks)
P010: Elastic Informational Gravity
General Relativity from bandwidth elasticity.
Key results:
- Einstein field equations from capacity conservation
- Linearized GR complete
- Post-Newtonian expansion: β₁/V = 7/6
- GW polarizations = dim(E_g) = 2 (MATH-PROVEN)
What’s derived: All solar system tests pass. Shapiro delay, perihelion precession, gravitational lensing — all from B = 1.
What’s pending: Full nonlinear Schwarzschild metric (predicted, not computed). This is Gap 9.
Verification: CMAC_PHYS_P010_Verification_v2.py (194 checks)
P020: Special Relativity (Bandwidth Relativity Theorem)
Einstein’s two postulates = one theorem.
Key results:
- Aut(CP¹) = SO⁺(3,1) — the Lorentz group
- E1 (Lorentz invariance) = the group
- E2 (invariant speed) = the unique null cone
- Spatial constancy = ker(D₀) = span{1} on K_oct
The insight: Special relativity is not an additional postulate. It’s a theorem of the bandwidth constraint.
Verification: CMAC_PHYS_P020_Verification.py (56 checks)
P012: Cosmology
The Hubble tension as a ratio.
Key results:
- H₀(local)/H₀(CMB) = N/E = 13/12
- Predicts H₀(local) = 72.97 km/s/Mpc
- SH0ES measurement: 73.04 ± 1.04 km/s/Mpc (0.06σ gap)
- Density fractions: Ω_m = 5/13, Ω_Λ = 7/13, Ω_r = 1/13
The Hubble tension is not a problem to solve — it’s a prediction to test.
Verification: CMAC_PHYS_P012_Verification_v2.py (130 checks)
The Cosmological Constant
CMAC derives:
Λ·ℓ²_P = (9/5)·α⁵⁷
Where:
- Exponent 57 = Tr(M₄·Δ₁) — derived via 4 independent routes (MATH-PROVEN)
- Prefactor 9/5 = c₁/c₂·π² (MATH-PROVEN)
This accounts for 122 orders of magnitude from zero free parameters. The “cosmological constant problem” is solved by the geometry.
What’s NOT Derived (Yet)
- Full nonlinear GR. Post-Newtonian complete; Schwarzschild pending (Gap 9).
- Quantized gravity. Framework clear; detailed calculations pending.
- Dark matter. Not addressed directly; capacity fractions may constrain.
Key Experimental Tests
| Test | Status | What it tests |
|——|——–|—————|
| Hubble tension ratio | Ongoing | H₀(local)/H₀(CMB) = 13/12 |
| GW polarizations | LIGO/Virgo | dim = 2 (matches GR) |
| Lorentz violation searches | Multiple | No violations (BRT predicts none) |
| JWST high-z galaxies | Ongoing | Structure formation timeline |
What Would Kill CMAC
- Detection of GW polarizations ≠ 2
- Lorentz violation at any scale
- Hubble ratio definitively ≠ 13/12
- Extra spatial dimensions
Verification
Run the cosmology chain:
python3 CMAC_PHYS_P011_Verification.py
python3 CMAC_PHYS_P010_Verification_v2.py
python3 CMAC_PHYS_P020_Verification.py
python3 CMAC_PHYS_P012_Verification_v2.py
Total: ~470 checks. All PASS. Under 4 minutes total.
The Big Picture
CMAC says: The universe is not fine-tuned. It’s computed.
The octahedron’s spectral structure determines:
- The gauge group (particle physics)
- The gravitational coupling (GR)
- The expansion rate ratio (cosmology)
- Lorentz invariance (special relativity)
All from B = 1 on S².
The Hubble tension is a feature, not a bug. Run the scripts.
Nanak Love, CMAC Institute, unit.edu