fibonacci_mass_recursion
plain-language theorem explainer
Fermion masses on the phi-ladder satisfy the Fibonacci recurrence m_{r+2} = m_{r+1} + m_r for any yardstick y and rung n. Unification theorists modeling discrete spectra would cite this as the formal statement of QG-004. The proof is a one-line wrapper that invokes the pure phi Fibonacci identity and closes the equality by ring.
Claim. For any real $y$ and natural number $n$, $y phi^{n+2} = y phi^{n+1} + y phi^n$, which states that the mass ladder $m_r = y phi^r$ obeys the Fibonacci relation $m_{r+2} = m_{r+1} + m_r$.
background
The Quantum-Gravity Octave Duality module derives kappa hbar = 8 from J-cost as AM-GM gap and places the mass spectrum on the phi-ladder. Masses take the form yardstick times phi to the rung power, with phi the self-similar fixed point forced at T6. The upstream phi_fibonacci_recursion establishes the unscaled case phi^{n+2} = phi^{n+1} + phi^n via phi^2 = phi + 1 and ring.
proof idea
The term proof rewrites the scaled target by the phi_fibonacci_recursion lemma, which supplies the base Fibonacci identity for phi-powers, then applies ring to equate both sides.
why it matters
It realizes QG-004 from the module documentation, supplying the Fibonacci mass ladder to fibonacci_triple_sum (inverse form) and qg_octave_cert (full certificate). This embeds the discrete spectrum inside the Recognition phi-ladder and eight-tick octave structure.
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