echo_reflection_coefficient_forced
plain-language theorem explainer
Successive black-hole echo amplitudes are forced into the constant ratio φ^{-1}, with the same golden-ratio identity fixing the energy partition, the reflection amplitude in (0,1), and the equality of its square to the reflected fraction. No free parameter enters. Ringdown-echo analysts cite this as the structural origin of geometric decay. The proof is a five-conjunct package of four prior identities plus elementary positivity and bound facts on φ.
Claim. For every natural number $n$, the ratio of successive echo amplitudes equals $\varphi^{-1}$. In addition $\varphi^{-1}+\varphi^{-2}=1$, the square of the reflection amplitude equals the reflected energy fraction, and the reflection amplitude lies strictly between $0$ and $1$.
background
Recognition Science models the near-horizon region as a φ-self-similar potential barrier. At each rung boundary, energy splits between reflected and transmitted channels by the golden-ratio partition $1=\varphi^{-1}+\varphi^{-2}$, equivalent to the defining equation $\varphi^2=\varphi+1$. Consequently the single-rung reflected fraction is $\varphi^{-2}$ and the reflection amplitude is $\varphi^{-1}$.
Echo amplitude at reflection number $n$ is taken proportional to $\varphi^{-n}$ (equivalently $(\varphi^{-1})^n$). Successive echoes therefore form a pure geometric sequence. Upstream, the energy-partition theorem proves $1=\varphi^{-1}+\varphi^{-2}$ from $\varphi^2=\varphi+1$ alone; the constant-ratio theorem shows the amplitude ratio equals $\varphi^{-1}$ for every $n$. The module status is structural: zero sorry, no RS-internal axiom.
proof idea
Term-mode refine of a five-fold conjunction. The first three conjuncts are discharged by naming the already-proved lemmas on the constant echo ratio, the golden-ratio energy partition, and the identity that squared reflection amplitude equals reflected fraction. Positivity of the reflection amplitude is inv_pos applied to positivity of φ. The strict upper bound less than one unfolds the reflection amplitude to $\varphi^{-1}$ and applies the standard comparison inv_lt_one_of_one_lt₀ to the fact $1<\varphi$.
why it matters
Master certificate for the Gravity-domain echo-reflection story. The module states that the QG-paper echo prediction is not a dimensional-analysis estimate but a forced consequence of substrate self-similarity at golden-ratio spacing: "the golden ratio's defining equation IS the barrier's scattering matrix." It bundles constant ratio, energy partition, amplitude-fraction equality, and the physical bounds $0<|R|<1$ into one theorem. The partition is the local face of φ as the self-similar fixed point (forcing-chain T6). No downstream edges are wired yet; this remains the natural citation point for any claim that echo trains decay geometrically with ratio $\varphi^{-1}$.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.