flameSpeed_adjacent_ratio
plain-language theorem explainer
The ratio of laminar flame speeds at consecutive rungs of the phi-ladder equals the golden ratio phi exactly. Combustion theorists would cite this when deriving parameter-free scaling laws for premixed flames across fuels. The proof is a one-line wrapper that rewrites via the successor relation and cancels the positive denominator by field simplification.
Claim. For every natural number $k$, the ratio of flame speed at rung $k+1$ to flame speed at rung $k$ equals $phi$, where flame speed at rung $k$ is defined as reference speed times $phi^k$.
background
Flame speed at rung $k$ is defined as referenceSpeed multiplied by $phi$ raised to the power $k$. The positivity theorem establishes that this quantity is strictly positive for every natural number $k$, using the positivity of $phi$ and the reference speed. The successor-ratio theorem states that flame speed at $k+1$ equals flame speed at $k$ multiplied by $phi$, obtained by unfolding the definition and applying the power-succ rule followed by ring simplification.
proof idea
The proof rewrites the target ratio by the successor-ratio lemma, introduces the positivity fact for the denominator, and applies field_simp to cancel the common positive factor.
why it matters
This supplies the adjacent-ratio component required by the flameSpeedCert structure, which bundles positivity, successor scaling, strict increase, and the exact phi ratio. It realizes the module-level claim that adjacent-rung flame speeds stand in the ratio phi with no fitted parameters. In the broader Recognition Science setting the phi-ladder originates from the self-similar fixed point T6 and the eight-tick octave T7, here specialized to combustion.
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