echoDelay_SI_two_step
plain-language theorem explainer
The SI echo delay doubles its rung index by multiplying by φ²: Δt(N+2) = Δt(N)·φ². Anyone citing the SI lift of the quarantined black-hole echo rung algebra needs this two-step scaling. The proof rewrites both sides into the compact Planck-time form, applies pow_add, and finishes by ring.
Claim. For every natural number $N$, the SI echo delay satisfies $\Delta t_{\mathrm{SI}}(N+2) = \Delta t_{\mathrm{SI}}(N)\,\varphi^{2}$, where $\Delta t_{\mathrm{SI}}(N) = (2\,r_{\min}(N)/c)\,\log\varphi$ is the formal delay at rung gap $N$.
background
Gravity Track 3.D lifts the φ-rung algebra of BlackHoleEchoesFromBounce into SI units. It is a structural conversion only: the native module already marks the old event-horizon escape story as rejected and leaves the horizon-consistent exterior mechanism open. No observable merger-echo claim is made here.
The SI delay is defined by $\Delta t_{\mathrm{SI}}(N) = (2,r_{\min}^{\mathrm{SI}}(N)/c_{\mathrm{SI}}),\log\varphi$, with bounce radius $r_{\min}^{\mathrm{SI}}(N) = \ell_{\mathrm{P}},\varphi^{N}$. An upstream identity rewrites this as the compact form $\Delta t_{\mathrm{SI}}(N) = 2,t_{\mathrm{P}},\varphi^{N},\log\varphi$, using $\ell_{\mathrm{P}} = t_{\mathrm{P}},c$. The two-step law is the delay analogue of the same φ² scaling already proved for the bounce radius.
proof idea
Term-mode proof. Rewrite both sides with the compact identity echoDelay_SI_eq_planckTime_form, so the claim becomes $2,t_{\mathrm{P}},\varphi^{N+2},\log\varphi = (2,t_{\mathrm{P}},\varphi^{N},\log\varphi),\varphi^{2}$. Apply pow_add to expand $\varphi^{N+2} = \varphi^{N},\varphi^{2}$, then close by ring.
why it matters
Bundled into the master certificate BlackHoleEchoesSICert and the one-statement theorem black_hole_echoes_SI_one_statement, which packages positivity of radius and delay, the two-step φ² laws, the squared Planck form, and the dimensionless damping ratio $1/\varphi\in(0.617,0.622)$. The two-step identity is the SI image of the native rung algebra driven by the golden ratio φ forced at T6; it is the delay counterpart of the radius scaling $r_{\min}(N+2)=r_{\min}(N),\varphi^{2}$. It does not close any physical echo mechanism or assert a detectable post-merger signal.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.