Ralf Meelker · 2026

Precausal Substrate Theory (PST)

The Higgs self-coupling from the substrate.

The running SM quartic λ_SM(µ) descends toward the modal scale M★, where the substrate predicts e⁻¹/4 exactly

substrate prediction e⁻¹/4 = 0.09197 (parameter-free)M★ = 4π mₕ √(2/3) ≈ 1.285 TeVλ_SM(µ), two-loop running200300500800M★2000renormalisation scale µ (GeV)0.090.100.110.120.13Higgs quartic λ_SM

The Higgs quartic λSM is one of the Standard Model’s free inputs: nothing in that theory says what it should be, so it is measured and fed in by hand. But in quantum field theory a coupling is not a fixed number, it runs: its measured strength drifts with the energy at which you probe it. The orange curve is that run, from the electroweak scale upward. PST claims to know where it is going.

  • The blue line is not fitted. The substrate’s partition function delivers a field-normalisation factor of exactly e⁻¹, and with the Landau-Ginzburg quartic b = 1/4 that predicts λSM = e⁻¹/4 = 0.09197 at the modal scale M★. The number comes out of the three postulates. Nothing was tuned to make it land anywhere.
  • M★ is also predicted. The scale where the two lines are supposed to meet is not chosen either: M★ = 4π mh √(2/3) ≈ 1.285 TeV follows from a parameter-free one-loop relation. So the theory names both the meeting place and the value to be met there.
  • And they meet. Running the measured coupling up to M★ gives 0.0927 ± 0.0007, a 0.8% central deviation from the prediction: agreement within one sigma, meaning the gap is smaller than the uncertainty in the measurement itself. The inset shows why that is not a fluke of a crude calculation, one-loop misses by 5–7%, two-loop closes to 0.8%, with the steps shrinking. The residual uncertainty is dominated by the measured top and Higgs masses, not by the theory. Tighten those measurements and the comparison becomes sharper still: a precision test of a number the theory fixed in advance.

Loop-order convergence of λ_SM(M★) — Computation 123

e⁻¹/40.091450.092340.09270123loop order → (steps shrinking)0.0910.0920.0930.094
Scale µ300 GeV
λ_SM(µ)0.1169
Prediction at M★0.09197
1.0σ match — 0.8% central deviation at two loops
Z² = λ_SM(M★)/b = e⁻¹, b = 1/4
predicted 0.09197 · run 0.0927 ± 0.0007
The substrate side of Z² = e⁻¹ is derived from P1–P3; its identification with the Standard-Model ratio rests on one premise that experiment can test, the Standard Model as PST’s effective theory below M★. Curve shape is a schematic interpolation between computed endpoints; endpoints, uncertainty budget, and loop sequence per Computation 123.
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