gmor_reasonable
plain-language theorem explainer
The GMOR prediction for the pion mass lies between 100 and 100000 in the units used. Researchers modeling chiral symmetry breaking in Recognition Science would cite this bound to confirm order-of-magnitude consistency with the phi-ladder mass formula before experimental comparison. The argument reduces the claim to direct numerical simplification of the explicit expression involving light quark mass, condensate, and decay constant.
Claim. Let $m_π^{GMOR}$ denote the pion mass from the Gell-Mann–Oakes–Renner relation using Recognition Science values for the light quark mass, quark condensate, and pion decay constant. Then $100 < m_π^{GMOR} < 100000$ (in MeV).
background
The Pion Masses module derives the lightest meson masses from Recognition Science by modeling pions as quark-antiquark bound states with binding energies set by phi-ladder rungs. Chiral symmetry breaking yields Goldstone bosons whose small masses arise from explicit breaking by the light quark masses, with the Gell-Mann–Oakes–Renner relation supplying the link $m_π^2 ∝ (m_u + m_d) ⟨q̄q⟩. The module also fixes the pion at a definite rung on the phi-ladder, producing concrete predictions such as a charged pion mass near 139.6 MeV and a ratio to the electron mass near 273, comparable to φ^12/2.
proof idea
The proof is a one-line wrapper that applies simplification to unfold the GMOR prediction definition together with the numerical constants for light quark mass, condensate, and decay constant, then normalizes the resulting arithmetic inequality.
why it matters
This result verifies that the GMOR prediction sits inside a physically plausible interval, thereby anchoring the Recognition Science mass formula for the pion on the phi-ladder. It supports the module's broader predictions, including the electron mass ratio approximating φ^12/2, and connects to the forcing chain landmarks of the eight-tick octave and three spatial dimensions. The theorem closes a numerical consistency check within the pion mass derivation.
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