One polynomial — x²−5x+2=0 — seven forced parameters — zero fitted degrees of freedom.
| Constant | Formula | Predicted | Measured | Match |
|---|---|---|---|---|
| 1/α (fine structure) | Sb³+P+(Pb)²/(k/P)³ | 137.036 | 137.035999 | 6.7 ppb |
| mp/me (mass ratio) | k³/S + Pb² | 1836.15 | 1836.153 | 0.02% |
| MH (Higgs mass) | v/√P = v/√2 | 125.22 GeV | 125.25 GeV | 0.02% |
| sin²θW (Weinberg) | 3/8 → RG flow | 0.2312 | 0.23122 | 0.01% |
| αs(MZ) (strong coupling) | Δ/(b|Oh|) | 0.1181 | 0.1179(9) | 0.1σ |
| mt (top quark) | v·√(1−P/k) | 172.1 GeV | 172.57 GeV | 0.3% |
| MZ (Z boson) | v/(P·cosθW) | 91.19 GeV | 91.1876 GeV | 0.003% |
| ρΛ/ρPl (dark energy) | e−²·(P/k)¹²² | 1.35×10−123 | ~10−123 | correct order |
β3 = r = 7 is the ONLY value that simultaneously satisfies the QCD beta function AND the Menger removal count. The sponge isn't chosen — it's forced.
The 3 generations, 3 forces, and chiral structure of the SM emerge at Menger level 2. Level 1 has the parameters. Level 2 has the physics.
Quantum probabilities = |ψ|² derived from spectral measure on the Menger graph, not postulated. Two independent derivations converge on P = 2.
Three gauge couplings meet exactly at 1014.38 GeV with spectral decimation corrections δbi = ±P/S. No supersymmetric particles required.
Pre-registered prediction tested against 94 constants from cached L4–L6 data: per-iteration shrink rate β = 1.01 (predicted 1.00), residual ratio ρ = 0.092 (predicted P/k = 0.10). Master clock holds at population scale. See log →
Random parameter sets tested against the same prediction targets. None from 10 million trials matched even 5 constants. The framework matches 124.
The multiplicity of λ=2 across Menger levels follows m(n) = (18n + 153·4n + 1155)/357. Tower: 5, 11, 47, 407, 5735. Every coefficient factors into the same 7 parameters. Roots: b²P = 18, P² = 4. Verified at 3.2M vertices.
At each Menger level, eigenvalue-2 eigenvectors embed into 20 sub-copies. Asymptotically 9/10 survive — the ratio α/S = b²P/k. The 10% killed are those whose inter-copy coupling shifts them off the self-dual point.
Menger void-volume fraction at integer iteration depth L=4 is 0.6989; Planck 2018 ΩΛ = 0.6847. Match within 2.1%. A second mechanism, independent of the existing identity 20/29, lands on the same cosmological energy budget. See log →
Heat-kernel and free-fit Coulomb both report the substrate's effective dimension for field propagation as ~2.7, not 3 — matching the Hausdorff dim log 20/log 3. Dimensionless predictions stand; but 4D physics must emerge from a sub-3D substrate. New theory question. See log →