massRatio_21_canonical
plain-language theorem explainer
The theorem establishes that the mass ratio between second and first generation fermions equals φ to the 11th power for any rich ledger whose torsion map matches the canonical generation torsion. Researchers deriving fermion mass hierarchies from the Recognition Science φ-ladder would cite this result when converting rung differences into explicit ratios. The proof is a short term-mode reduction that unfolds the mass ratio definition, applies the rung difference lemma under the torsion hypothesis, and simplifies the offset term.
Claim. For any rich ledger $L$ whose torsion function equals the canonical generation torsion, and for any fermion sector, the mass ratio between the second and first generations computed from their rung difference equals $φ^{11}$.
background
In Recognition Science, fermion masses sit on a φ-ladder whose rung for a given sector and generation is the sector base rung plus a generation torsion offset. The RSLedger structure augments a basic ledger with a torsion map from generations to integers together with base rungs per sector (leptons at 2, up and down quarks at 4). Canonical torsion values are 0 for generation 1, 11 for generation 2, and 17 for generation 3; these arise from D=3 cube combinatorics (edge-dressed for gen 2, face-plus-edge for gen 3) as stated in the module documentation.
proof idea
The proof is a term-mode reduction. It first simplifies the massRatioFromRungs definition, rewrites the rung difference via the lemma rungDiff_canonical applied to the supplied sector and the hypothesis that torsion equals generationTorsion, then simplifies the torsionDiff term.
why it matters
This supplies one leg of the triple of ratios asserted by the downstream theorem massRatiosFromTiers_canonical, which states that the full set of inter-generation mass ratios matches the φ-powers 11, 17 and 6. It realizes the geometric torsion derivation from cube combinatorics described in the module documentation and feeds the canonical specialization canonical_massRatio_21. The result therefore closes the bridge from D=3 geometry to the φ-ladder mass hierarchy.
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