fibonacci_triple_sum
plain-language theorem explainer
Recognition Science places fermion masses on a phi-ladder m_r = y phi^r. The theorem confirms this ladder obeys the Fibonacci recurrence m_n + m_{n+1} = m_{n+2}. Researchers modeling the mass spectrum cite it to verify self-similarity forced by the J-cost fixed point. The proof is a one-line wrapper that reduces the identity to the underlying recursion via linear arithmetic.
Claim. For all real $y$ and natural numbers $n$, $y phi^n + y phi^{n+1} = y phi^{n+2}$.
background
The Quantum-Gravity Octave Duality module derives all results from the single J-cost functional, defined as the AM-GM gap J(x) = (x-1)^2 / (2x) for x > 0. This fixes the constants kappa = 8 phi^5 and hbar = phi^{-5}, so their product equals the octave factor 8. The mass spectrum is introduced as m_r = y phi^r on the phi-ladder; the present theorem records that this spectrum satisfies the Fibonacci recurrence as a direct algebraic consequence of the phi fixed-point property.
proof idea
The proof is a one-line wrapper that applies linarith to the lemma fibonacci_mass_recursion y n.
why it matters
It realizes QG-004 in the module, confirming the mass ladder is Fibonacci. This supports the central result kappa hbar = 8 by closing the phi-ladder structure required by T6 and the eight-tick octave T7. The declaration has no current downstream uses but completes the mass-spectrum component of the unification.
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