module module moderate

`IndisputableMonolith.Gravity.BlackHoleEchoesFromBounce`

Packages the φ-rung algebra for a proposed black-hole bounce echo: bounce radius, rung phase delay, and echo delay, plus an explicit status flag that the physical mechanism is not theorem-grade. Gravity SI-lift, discriminator, and PTA tracks import the algebra; the horizon-obstruction module later rejects the causal story. Content is definitions and elementary positivity/monotonicity facts, not a physical existence proof.

claimOn the Recognition $\varphi$-ladder one defines a bounce radius $r_b$, a rung phase delay, and an echo delay $\Delta t_{\mathrm{echo}}$ built from the RS tick $\tau_0$. The module records that the bounce-echo mechanism has honest status "not theorem-grade": the delay algebra is formal, the exterior-return mechanism is not proved.

background

Recognition Science works in RS-native units with fundamental tick $\tau_0=1$ (from Constants) and dimensionless structure fixed by the golden ratio $\varphi$ forced at T6. Masses and geometric scales sit on a $\varphi$-ladder; delays are integer or half-integer rung shifts times $\tau_0$.

Black-hole echo phenomenology asks whether a near-horizon bounce can send a delayed copy of a ringdown signal back to exterior observers. This module isolates that proposal as a pure rung algebra: bounce radius as a function of ladder step, phase delay per rung, and the composite echo delay. It does not assert that an event horizon permits exterior return.

The module doc frames the sector as "honest status": the algebra is available for SI conversion and for structural comparison against LQG/string/other QG programs, while the mechanism itself is quarantined from theorem-grade claims.

proof idea

Definition-and-status module, not a single theorem. It introduces bounce radius with positivity, vanishing, two-step, and strict-monotonicity lemmas; rung phase delay with positivity and a band bound; and the composite echo delay. A status inductive or enum (BlackHoleEchoMechanismStatus) and a theorem that the recorded status is not theorem-grade make the quarantine explicit. No deep analytic PDE or GR global-structure argument appears here; downstream modules either lift units or refute the causal mechanism.

why it matters in Recognition Science

Feeds the SI lift in Gravity.BlackHoleEchoesSI, which converts the quarantined $\varphi$-rung algebra to SI without closing the physical claim. EchoHorizonObstruction cites this module as the bounce-echo story it rejects: an event horizon is one-way, so interior bounce cannot return a signal to the exterior. Discriminator tracks (DiscriminatorCert, DiscriminatorMatrix) and ZeroFreeParameters import the algebra as a gravity-sector structural object with zero free parameters once $\varphi$ is fixed. Cosmology PTAStochasticGWStructural (Track 6.B) uses the same structural layer for PTA stochastic-GW discrimination. Landmark context: delays are counted in eight-tick/octave-compatible ticks; no new constant beyond $\varphi$ and $\tau_0$ is introduced.

scope and limits

Does not prove that black-hole echoes exist observationally or in classical GR.
Does not establish causal exterior return from an interior bounce across a horizon.
Does not derive echo templates, SNR forecasts, or detector-specific bounds.
Does not close quantum-gravity dynamics; only φ-rung delay algebra and status flags.
Does not claim theorem-grade mechanism status; the module explicitly denies that grade.

used by (6)

From the project-wide theorem graph. These declarations reference this one in their body.

IndisputableMonolith.Cosmology.PTAStochasticGWStructural module_import
IndisputableMonolith.Gravity.BlackHoleEchoesSI module_import
IndisputableMonolith.Gravity.DiscriminatorCert module_import
IndisputableMonolith.Gravity.DiscriminatorMatrix module_import
IndisputableMonolith.Gravity.EchoHorizonObstruction module_import
IndisputableMonolith.Gravity.ZeroFreeParameters module_import

depends on (1)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import

declarations in this module (25)

structure BlackHoleEchoMechanismStatus L77
def blackHoleEchoMechanismStatus L85
theorem blackHoleEchoMechanismStatus_not_theorem_grade L91
def bounceRadius L100
theorem bounceRadius_pos L102
theorem bounceRadius_zero L106
theorem bounceRadius_two_step L111
theorem bounceRadius_strict_mono L117
def rungPhaseDelay L128
theorem rungPhaseDelay_pos L130
theorem rungPhaseDelay_band L138
def echoDelay L163
theorem echoDelay_pos L165
theorem echoDelay_scaling L172
theorem echoDelay_two_step L178
def echoDampingRatio L188
theorem echoDampingRatio_pos L190
theorem echoDampingRatio_lt_one L194
theorem echoDampingRatio_band L199
def cumulativeEchoAmplitude L213
theorem cumulativeEchoAmplitude_pos L215
theorem cumulativeEchoAmplitude_strictly_decreasing L220
structure BlackHoleEchoesCert L231
def blackHoleEchoesCert L250
theorem black_hole_echoes_one_statement L269