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plain-language theorem explainer
At a self-similar barrier with scale ratio equal to the golden ratio, incident energy splits into a reflected share φ^{-2} and a transmitted share φ^{-1}. Gravity and echo analyses cite this partition as the scattering law forced by φ² = φ + 1 alone. The object is a definitional structure: the identity 1 = φ^{-1} + φ^{-2} is the barrier's energy budget, not a fitted coefficient.
Claim. At a self-similar potential barrier whose successive scale ratio is the golden ratio $\varphi$, the incident energy partitions as reflected fraction $\varphi^{-2}$ and transmitted fraction $\varphi^{-1}$. Equivalently, $\varphi^{2} = \varphi + 1$ implies the complete budget $1 = \varphi^{-1} + \varphi^{-2}$.
background
The module treats the near-horizon recognition structure as a φ-self-similar potential barrier. At each rung boundary, energy is split between reflected and transmitted channels by the golden-ratio partition $1 = \varphi^{-1} + \varphi^{-2}$, which is exactly the defining relation $\varphi^{2} = \varphi + 1$ rewritten in reciprocal form.
Related constants live in the RS constants bundle (opened here as Constants), including the native scale built from powers of φ. Upstream scale constructions such as the cosmological ladder scale k := φ^k supply the same φ-powers that label rung crossings. The single-rung reflection coefficient is then $|R|^{2} = \varphi^{-2}$, so the reflected amplitude is $|R| = \varphi^{-1}$.
Round-trip phase across one φ-rung is $\log\varphi$ per crossing. Echo $n$ therefore arrives with amplitude $\varphi^{-n}$ and delay $n\cdot\Delta t_{\mathrm{echo}}$ with $\Delta t_{\mathrm{echo}} = (\log\varphi)/(2\pi f_{\mathrm{ringdown}})$.
proof idea
Definitional structure, not a proved theorem. The only algebraic content is the golden-ratio identity $\varphi^{2} = \varphi + 1$. Dividing by $\varphi^{2}$ yields $1 = \varphi^{-1} + \varphi^{-2}$, which is recorded as the complete energy partition into transmitted fraction $\varphi^{-1}$ and reflected fraction $\varphi^{-2}$. No analytic scattering computation or dimensional fit is invoked; the defining equation of φ is taken as the barrier's scattering budget.
why it matters
This partition is the structural core of the echo-reflection story in the gravity sector. Module status is a structural theorem stack with zero sorry and no RS-internal axiom: the QG-paper echo prediction is presented as forced by substrate self-similarity at golden-ratio spacing, not as a dimensional estimate.
Downstream siblings (reflected and transmitted fractions, positivity and strict-sub-unit bounds, reflection amplitude and its square, echo damping factor) all read off this split. Framework-wise it sits on the φ fixed point from the forcing chain (T6) and on the φ-ladder used for masses and scales elsewhere. The claim that $|R| = \varphi^{-1}$ needs no free parameter is the point of the construction: φ's minimal polynomial is the scattering matrix.
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