coronal_adjacent_ratio
plain-language theorem explainer
The theorem establishes that consecutive coronal timescales on the phi-ladder stand in the exact ratio phi. Researchers modeling solar coronal reconnection would cite this result to ground Lyapunov times in the Recognition Science structure. The proof is a direct rewrite via the successor ratio lemma followed by field simplification using positivity.
Claim. For each natural number $k$, the ratio of the coronal timescale at rung $k+1$ to the coronal timescale at rung $k$ equals the golden ratio $phi$.
background
The module places solar coronal Lyapunov times on the phi-ladder of characteristic timescales. The definition coronalTime sets the timescale at rung k to referenceTime times phi to the power k. The upstream positivity result shows that this timescale is strictly positive, while the successor ratio result states that advancing one rung multiplies the timescale by phi exactly. This setup encodes the chaotic evolution of coronal magnetic fields before reconnection, with the ladder starting from the Alfvén crossing time as rung zero.
proof idea
The proof invokes the successor ratio theorem to substitute the ratio directly with phi. It then applies field_simp, using the fact that the denominator is nonzero from the positivity theorem, to conclude the equality.
why it matters
This theorem provides the adjacent ratio equality used to construct the coronal Lyapunov certificate. The certificate aggregates positivity, the successor ratio, strict increase, and this adjacent ratio to certify the full phi-ladder for coronal timescales. It realizes the module's claim that adjacent coronal timescales differ by phi, aligning with the self-similar fixed point phi in the Recognition framework.
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