fermionMassAt_adjacent_ratio
plain-language theorem explainer
The ratio of fermion masses at consecutive rungs on the phi-ladder equals the golden ratio phi. Model builders deriving SM masses from the J-cost recognition structure would cite this to confirm the geometric progression of the fermion spectrum. The proof is a one-line wrapper that rewrites via the successor ratio lemma and cancels the common factor using positivity.
Claim. Let $m_0 > 0$ be real and $k$ a natural number. Then the ratio of the fermion mass at rung $k+1$ to the mass at rung $k$ equals phi, where the mass at rung $k$ is defined as $m_0 phi^k$.
background
The fermion mass at rung k unfolds directly to the product of the base mass m0 and phi raised to k. Positivity of this quantity follows from the base mass being positive and the positivity of phi. The successor ratio states that the mass at rung k+1 equals the mass at rung k multiplied by phi. In the fermion kinetic sector the mass term maps to J-cost on the recognition ratio r, while the structural prediction requires masses to form a geometric sequence with common ratio phi on the ladder.
proof idea
The proof applies the successor ratio theorem to replace the numerator by the denominator times phi. Field simplification then cancels the common positive factor, using the non-zero denominator supplied by the positivity theorem, to reach phi.
why it matters
This supplies the adjacent mass ratio required by the fermion kinetic certificate, which packages the per-generation count, positivity, and ratio into the full kinetic sector certificate. It closes the mass-ratio step for the SM fermion spectrum on the phi-ladder, consistent with the self-similar fixed point forced at T6.
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