Appendix C: Ringdown Signal Modeling

This appendix describes the modeling of gravitational-wave ringdown signals used to test the informational framework against black-hole merger observations. The goal is to provide a transparent and reproducible prescription for computing expected deviations from general relativity.

C.1 Standard Ringdown in General Relativity

In general relativity, the late-time gravitational-wave signal following a black-hole merger is described as a superposition of quasinormal modes:

where:

• $\omega_n^{\rm GR}$ are the mode frequencies,
• $\tau_n^{\rm GR}$ are the damping times,
• $A_n$ and $\phi_n$ depend on merger dynamics.

The frequencies and damping times depend only on the final black-hole mass and spin.

C.2 Informational Suppression Near the Horizon

Within the informational framework, the formation of a black hole corresponds to a regime in which the effective diffusion coefficient vanishes:

This suppression alters the boundary conditions governing perturbations near the horizon. Instead of a perfectly absorbing horizon, the system exhibits a gradual reduction in information propagation.

This effect is modeled as a smooth modification of the effective potential felt by perturbations, without introducing reflective surfaces or additional degrees of freedom.

C.3 Modified Quasinormal Modes

To leading order, horizon-scale suppression produces a small shift in the quasinormal mode spectrum. We parameterize this as:

(ratio inverted relative to an earlier version, so that the suppression term vanishes for $a_{\rm hor}\gg a_*$ – ordinary stellar/intermediate-mass black holes – and grows as $a_{\rm hor}\to a_*$, matching the claim that deviations increase with mass; see the companion patch to File 22, Section 23.1) where:

• $a_{\rm hor}$ is the characteristic acceleration at the horizon,
• $a_*$ is the universal acceleration scale extracted from galactic dynamics,
• $\epsilon,\epsilon'$ are order-unity coefficients,
• $p$ controls the sharpness of the suppression.

The exponential form ensures that deviations are negligible in weak-field regimes and become relevant only near the horizon.

C.4 Synthetic Signal Generation

Synthetic ringdown signals are generated by substituting the modified frequencies and damping times into the waveform model:

function generate_ringdown(params):
    for mode in modes:
        omega = omega_GR * suppression_factor
        tau = tau_GR * damping_factor
        h += A * exp(-t/tau) * cos(omega*t + phi)
    return h

Noise consistent with detector sensitivity curves is added to assess detectability.

C.5 Parameter Estimation and Model Comparison

Bayesian parameter estimation is performed by comparing synthetic signals to observed data. The likelihood is evaluated under both general relativity and the informational model.

logL = -0.5 * sum((data - model)**2 / noise_variance)

Evidence ratios are used to assess whether the data favor horizon-scale suppression.

C.6 Detectability Thresholds

Simulations indicate that deviations at the level:

are marginally accessible to current detectors and well within reach of next-generation observatories.

The presence or absence of such deviations provides a clean and decisive test of the framework.

Note: All modeling choices are minimal and conservative. No additional degrees of freedom or ad hoc boundary conditions are introduced.