echoPhaseSeparation
plain-language theorem explainer
Total recognition phase after n rung crossings equals n times log φ. Gravity and black-hole echo modelers cite it when converting rung count into cumulative phase (hence echo delay). The body is the one-line product of the natural number with the fixed per-rung phase.
Claim. For each natural number $n$, the cumulative echo phase separation is $\Delta\Phi(n) := n\,\log\varphi$, where $\varphi$ is the golden ratio and $\log\varphi$ is the recognition phase accumulated on a single rung crossing.
background
The module treats the near-horizon recognition structure as a φ-self-similar potential barrier. Energy at each rung splits by the golden-ratio partition $1=\varphi^{-1}+\varphi^{-2}$, so the single-rung reflection coefficient is forced: $|R|^2=\varphi^{-2}$ and $|R|=\varphi^{-1}$.
Upstream, the phase per rung is defined as the logarithm of the scale ratio: crossing from scale $\ell$ to $\varphi\ell$ accumulates phase $\log(\varphi\ell/\ell)=\log\varphi$. That constant is positive because $\varphi>1$. Echo $n$ is then expected to arrive with amplitude $\varphi^{-n}$ and delay proportional to the cumulative phase.
The surrounding development links cumulative phase to the observable echo delay via $\Delta t_{\mathrm{echo}}=(\mathrm{phase\ per\ rung})/(\pi f_{\mathrm{ring}})$, with $f_{\mathrm{ring}}$ the fundamental ringdown frequency.
proof idea
Pure definitional abbreviation: the cumulative phase after $n$ crossings is written as the product of the natural number $n$ with the already-defined per-rung phase $\log\varphi$. No lemmas or tactics are required; the body is the arithmetic expression itself.
why it matters
This definition is the additive generator used by the successor identity echoPhaseSeparation_succ, which states that advancing one rung adds exactly one more $\log\varphi$. Together they turn the geometric rung ladder into a linear phase clock for successive echoes.
In the Recognition framework the construction is forced by φ-self-similarity (T6) rather than by dimensional analysis: the same $\varphi$ that fixes the energy partition also fixes the phase step. The module status is structural (zero sorry, no RS-internal axiom), so the cumulative phase sits inside the proved echo-reflection package that underwrites the QG-paper echo prediction.
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