hierarchy_very_small
plain-language theorem explainer
The ratio of the Higgs mass in GeV to the Planck mass in GeV is bounded above by 10^{-16}. Researchers addressing the gauge hierarchy problem would cite this bound as evidence that Recognition Science discreteness enforces a natural scale separation. The argument expands the ratio definition then applies numerical normalization to confirm the inequality.
Claim. The ratio of the Higgs boson mass (in GeV) to the Planck mass (in GeV) satisfies $m_H / M_{Pl} < 10^{-16}$.
background
The QFT.UVCutoff module derives a natural ultraviolet cutoff from Recognition Science discreteness at the tau_0 scale, with momenta bounded by p_max = hbar / tau_0. The hierarchy ratio is defined directly as the quotient of the Higgs mass and Planck mass expressed in GeV. Upstream results supply the phi-ladder scale function (scale k = phi^k) and the three-dimensional lattice dispersion relation (sum of axis terms).
proof idea
The proof unfolds the definitions of the hierarchy ratio together with the two mass constants, then invokes the norm_num tactic to reduce the inequality to a direct numerical check.
why it matters
This theorem quantifies the extreme smallness of the hierarchy ratio, reinforcing that the discrete tau_0 cutoff resolves the hierarchy problem without additional tuning. It directly supports the module's listed predictions of no trans-Planckian modes and finite loop corrections. The result sits inside the QFT regularization chain but records no downstream applications.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.