IndisputableMonolith.Gravity.EchoReflectionCoefficient
Golden-ratio energy partition at a self-similar barrier: reflected fraction φ^{-2} and transmitted fraction φ^{-1}. Gravity and black-hole echo modelers cite it for RS-native reflection coefficients and damping. All identities reduce to the single algebraic relation φ² = φ + 1.
claimAt a self-similar barrier with scale ratio $\varphi$, incident energy splits as reflected fraction $\varphi^{-2}$ and transmitted fraction $\varphi^{-1}$. These satisfy $\varphi^{-1}+\varphi^{-2}=1$, both lie in $(0,1)$, the reflection amplitude is $\varphi^{-1}$, and an echo damping factor is built from the reflected piece.
background
Recognition Science forces the golden ratio $\varphi$ as the unique self-similar fixed point (forcing chain T6). The module works in that native scale: a barrier whose successive length (or impedance) ratios equal $\varphi$ partitions energy without free parameters.
The only algebraic input is the defining equation $\varphi^2=\varphi+1$. Dividing by $\varphi^2$ yields the partition identity $1=\varphi^{-1}+\varphi^{-2}$. Constants supplies the RS time quantum $\tau_0=1$ tick, so the same $\varphi$-ladder that sets masses and $\hbar$ also sets the echo coefficients.
Named objects introduced here are the reflected and transmitted fractions, their positivity and strict sub-unity bounds, the reflection amplitude (square root of the reflected fraction), and an echo damping factor assembled from that amplitude.
proof idea
The module is a short definition-plus-lemma cluster, not a deep derivation. Fractions are defined as $\varphi^{-2}$ and $\varphi^{-1}$. Completeness of the partition is the one-line rewrite of $\varphi^2=\varphi+1$. Positivity and upper bounds follow from $\varphi>1$. The reflection amplitude is the positive square root of the reflected fraction; its square recovers the fraction by construction. The damping factor is a thin wrapper around that amplitude for multi-bounce echo trains.
why it matters in Recognition Science
Black-hole and horizon-echo phenomenology in RS needs a parameter-free reflection coefficient fixed by the same $\varphi$ that appears in the mass ladder and the eight-tick octave. This module supplies that coefficient and the associated damping, so downstream gravity developments can quote a single algebraic source rather than fitting an ad-hoc reflectivity.
No parent theorems are recorded in the dependency graph yet (used_by is empty); the natural consumers are echo-strain templates and any theorem that converts a barrier crossing into an observable ringdown modification. The construction sits downstream of T6 ($\varphi$ uniqueness) and is independent of the spatial-dimension forcing T8.
scope and limits
- Does not derive the barrier from Einstein equations or quantum gravity dynamics.
- Does not compute waveforms, time delays, or detector SNR.
- Does not treat non-self-similar or multi-scale barriers.
- Does not claim observational detection of echoes.
- Does not fix absolute normalization beyond the $\varphi$-partition.
depends on (1)
declarations in this module (29)
-
structure
at -
theorem
phi_energy_partition -
def
reflectedFraction -
def
transmittedFraction -
theorem
partition_complete -
theorem
reflectedFraction_pos -
theorem
transmittedFraction_pos -
theorem
reflectedFraction_lt_one -
theorem
transmittedFraction_lt_one -
def
reflectionAmplitude -
theorem
reflectionAmplitude_sq -
def
echoDampingFactor -
theorem
echoDampingFactor_eq_reflectionAmplitude -
def
echoAmplitude -
theorem
echoAmplitude_zero -
theorem
echoAmplitude_succ -
theorem
echo_ratio_constant -
theorem
echo_geometric -
def
phasePerRung -
theorem
phasePerRung_pos -
def
echoPhaseSeparation -
theorem
echoPhaseSeparation_succ -
structure
PhiSelfSimilarBarrier -
def
singleRungBarrier -
theorem
barrier_total_reflection -
theorem
echo_reflection_coefficient_forced -
structure
EchoReflectionCoefficientCert -
def
echoReflectionCoefficientCert -
theorem
echoReflectionCoefficientCert_inhabited