mH_rs_level3_pos
plain-language theorem explainer
The theorem establishes that the Recognition Science level-three Higgs mass prediction is strictly positive. Researchers completing the phi-ladder mass table for the Standard Model would cite this to anchor the Higgs assignment before interval verification. The proof is a term reduction that unfolds the mass expression and applies positivity of multiplication to a numerical factor and the square root of the Q3 correction term.
Claim. $0 < m_{H,RS,level 3}$, where $m_{H,RS,level 3}$ is the Higgs mass obtained from the phi-ladder at rung three after the Q3 geometry correction to the vacuum expectation value.
background
The module derives the Higgs mass from the phi-ladder using Q3 geometry to complete the RS particle mass table. The core hypothesis states that the Higgs mass lies in (120,130) GeV, obtained from $m_H^2 = 2 lambda_RS v^2$ with $lambda_RS$ incorporating the Weinberg angle $(3-phi)/6$ and a one-loop correction factor. The physical Higgs is the remaining scalar after three Goldstone modes are absorbed into the W and Z bosons. The J-cost potential curvature forces the tree-level lambda to 1/2, but the observed mass discrepancy is resolved by the Q3 correction of order 1/16, yielding the level-three prediction as the vacuum expectation value scaled by the square root of that factor.
proof idea
The proof is a term-mode reduction. It unfolds the definitions of the level-three Higgs mass and the vacuum expectation value, then applies the lemma that the product of two positive reals is positive. The first factor is verified by numerical normalization and the second follows from the positivity of the square root applied to the positive Q3 correction term.
why it matters
This result supports the parent theorem that places the Higgs rung between 21 and 22 on the phi-ladder. It contributes to closing the Higgs mass question in the Recognition Science program by confirming positivity ahead of the (120,130) GeV interval check. The derivation rests on the phi-ladder mass formula and the J-cost potential, addressing the open requirement for the full one-loop electroweak correction before an exact value can be claimed.
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