IndisputableMonolith.Gravity.EchoReflectionCoefficient
At a self-similar barrier with scale ratio equal to the golden ratio, incident energy splits into reflected share φ^{-2} and transmitted share φ^{-1}. Echo and barrier-scattering calculations in the RS gravity stack cite these fractions and the derived amplitude and damping factor. All identities are pure algebra from φ² = φ + 1.
claimLet $\varphi>1$ satisfy $\varphi^2=\varphi+1$. At a self-similar barrier of scale ratio $\varphi$, the energy partition is reflected fraction $R=\varphi^{-2}$ and transmitted fraction $T=\varphi^{-1}$, so $R+T=1$, with $0<R,T<1$. The reflection amplitude is $\sqrt{R}$, and an echo damping factor is built from these shares.
background
Recognition Science forces the golden ratio $\varphi$ as the unique self-similar fixed point (forcing step T6). Dividing the defining relation $\varphi^2=\varphi+1$ by $\varphi^2$ yields the unit partition $1=\varphi^{-1}+\varphi^{-2}$. This module sits in the Gravity domain and imports only Mathlib plus RS Constants; it treats that algebraic split as the energy budget at a $\varphi$-scaled barrier.
Objects introduced: reflected fraction $\varphi^{-2}$, transmitted fraction $\varphi^{-1}$, completeness of the partition, positivity and strict bounds below one for each share, reflection amplitude (square root of the reflected fraction), amplitude squared recovering the reflected fraction, and an echo damping factor for successive reflections.
proof idea
Mostly definitions plus short algebraic lemmas. The partition identity is the golden-ratio equation rewritten as $1=\varphi^{-1}+\varphi^{-2}$. Positivity and the bounds $R,T\in(0,1)$ follow from $\varphi>1$. Amplitude-squared is definitional recovery of the reflected fraction. No analytic estimates or dynamics; only field arithmetic on $\varphi$.
why it matters in Recognition Science
Gives the $\varphi$-native reflection and transmission coefficients used in RS gravitational echo and barrier models. The graph currently lists no downstream consumers, but the named fractions, amplitude, and echo damping factor are the natural inputs to successive-reflection templates in the Gravity stack. The construction sits on T6 ($\varphi$ forced) and the self-similarity theme of the forcing chain; it does not itself advance T7/T8 or the mass ladder.
scope and limits
- Does not derive the barrier from Einstein equations or RS field dynamics.
- Does not fix absolute cross-sections, phases, or frequencies, only energy fractions.
- Does not prove the partition is unique beyond the φ algebraic identity.
- Does not connect fractions to GW echo searches or data reduction.
- Does not model multi-barrier interference beyond a single damping factor.
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