IndisputableMonolith.Masses.MassRatiosProved
This module calculates bounds on φ^6 to constrain mass ratios in the Recognition Science framework. From the identity φ^3 = 2φ + 1 it expands to φ^6 = (2φ + 1)^2 and inserts the interval 1.5 < φ < 1.62 to obtain 17 < φ^6 < 18. Mass spectroscopists and RS modelers cite these bounds when placing particles on the phi-ladder. The derivation is a short algebraic calculation with interval evaluation.
claim$17 < φ^6 < 18$, where φ satisfies φ^3 = 2φ + 1, supplying the numerical window for mass-ratio theorems on the phi-ladder.
background
The module sits in the Masses domain and imports the RS time quantum τ₀ = 1 tick from Constants together with the proof that φ is forced by self-similarity in a discrete ledger with J-cost from PhiForcing. J-cost is the functional J(x) = (x + x^{-1})/2 - 1 that encodes the recognition cost in the ledger. The phi-ladder assigns masses via yardstick × φ^(rung - 8 + gap(Z)), so bounds on φ^6 translate directly into ratio limits between consecutive rungs.
proof idea
The argument begins with the cubic identity φ^3 = 2φ + 1 that holds for the golden ratio. Squaring both sides yields φ^6 = (2φ + 1)^2. Substituting the interval bounds 1.5 < φ < 1.62 and evaluating the resulting quadratic produces the strict inequality 17 < φ^6 < 18. No external lemmas beyond the algebraic identity are required.
why it matters in Recognition Science
The bounds close the numerical gap between the abstract phi-forcing chain (T5–T6) and concrete mass predictions. They are used by sibling declarations such as mass_ratio_from_rung_difference and mass_ordering_from_rungs to certify ordering and ratios on the ladder. The module therefore supplies the quantitative bridge from the self-similar ledger to observable mass hierarchies.
scope and limits
- Does not establish the value of φ from the ledger axioms.
- Does not derive specific particle masses such as the electron or proton.
- Does not extend the bounds to include quantum or relativistic effects.
- Does not connect to the fine-structure constant alpha band.