6.1 KiB
Hawking Analog Phonon Loss Refinement
Purpose
This note refines the finite, lossy throat / phonon-only transport model using Hawking-style horizon equations and analogue-gravity results.
The intent is not to claim a physical astrophysical black hole or a literal engineered wormhole. The equations are used as an analogue-horizon accounting layer for phonon emission, entropy flow, greybody filtering, and finite information recovery inside a lossy metamaterial throat.
Core Statement
forming throat / active material horizon
-> surface-gravity-like gradient
-> effective Hawking temperature
-> phonon emission spectrum
-> greybody / bandgap filtering
-> entropy and information budget
-> finite recoverable active events
This strengthens the selector by giving the phonon-only throat a thermodynamic emission model instead of only a threshold model.
Hawking Baseline Equations
For a gravitational black hole, Hawking temperature is:
T_H = hbar * kappa / (2*pi*k_B*c)
where kappa is surface gravity.
For analogue acoustic horizons, the equivalent form replaces light-speed geometry with an effective flow/sound-speed gradient. In simplified form:
T_H,acoustic ~ hbar / (2*pi*k_B) * |d(v_flow - c_s)/dx|_horizon
In the project model, this becomes a material-throat emission temperature:
T_eff(i,t) = hbar/(2*pi*k_B) * kappa_eff(i,t)
where:
kappa_eff(i,t) = |partial_x(v_shock(i,t) - v_mode(i,t))| at the active horizon/gate
Phonon Emission Spectrum
The emitted phonon occupation can be modeled by a thermal-like factor:
n_omega(i,t) = Gamma_i(omega,t) / (exp[hbar*omega/(k_B*T_eff(i,t))] - 1)
where:
Gamma_i(omega,t) = greybody / bandgap / transmission coefficient
This coefficient is the bridge to the existing bandgap phonon-dump model:
Gamma_i small -> propagation blocked / localized / dissipated
Gamma_i large -> mode transmits through the gate
Entropy and Finite Information Budget
Use a finite entropy budget for each timelike emission window:
Delta S_emit(N) = integral_{T_emit(N)} integral d_omega s[n_omega(i,t)] dt
where a bosonic mode entropy proxy is:
s(n) = (1+n)log(1+n) - n log n
Recoverable information is bounded by the channel capacity after loss:
I_recoverable <= C_channel(T_emit, Gamma, T_eff, noise)
This prevents treating the throat as a lossless or infinite data source.
Page-Curve / Recovery Analogy
The Page-curve lesson is not imported as a theorem about the material. It is used as an accounting discipline:
early radiation alone may be insufficient
late radiation may be correlated with earlier modes
recovery requires collecting enough finite emitted modes
For the phonon throat:
recoverable mode set = finite subset of emitted phonons whose correlations survive torsion and greybody loss
Updated Selector Gate
The active-cell gate becomes:
chi_N(i,t) = 1 iff
t in T_emit(N)
and |u_x(i,t;nu_N)| >= theta_A(N)
and T_eff(i,t) >= T_min(N)
and omega_shock(i,t) in BandGap_i(theta_i,N)
and n_omega(i,t) >= n_min(N)
and mode_overlap(phi_in,phi_out) >= eta_min(N)
and Survivability_i(t) >= I_min(N)
The projected active set remains:
I_active(N) = { i <= N : exists t in T_emit(N), chi_N(i,t)=1 }
Density target:
|I_active(N)| / sqrt(N) -> 1 in limsup
Loss Functional Update
The throat loss functional now includes Hawking/analogue-horizon emission terms:
L_total = L_throat + L_greybody + L_entropy + L_mode_mixing
with:
L_greybody = integral -log Gamma_i(omega,t) d_omega dt
L_entropy = lambda_S * Delta S_emit
L_mode_mixing = lambda_M * (1 - mode_overlap)
and prior throat loss:
L_throat = integral_gamma [
lambda_T ||T||^2
+ lambda_kappa |kappa|
+ lambda_chi chi_mismatch
+ lambda_mu memory_strain
+ lambda_beta boundary_stress
] dp dt
Recoverability:
I_out = I_in * exp(-L_total) * R_repair
Admissibility:
I_out/I_in >= I_min(N)
T_emit(N) finite
Delta S_emit(N) finite
RepairRate_N > DegradationRate_N
Interpretation for Virtual Sidon Forcing
The Hawking analogue layer does not create Sidon pair-sum injectivity. It refines the active event accounting:
Hawking analogue equations -> finite phonon emission budget
phonon/bandgap/greybody loss -> recoverable active cells
Burgers shock -> alignment clock
nonseparable algebraic encoding -> Sidon lock
The virtual Sidon pair state remains:
S_N(a,b) = Phi_N(a) + Phi_N(b) + Lambda_N(a,b)
where Lambda_N can include a nonseparable, finite-window, emission-correlation term derived from surviving phonon correlations.
Evidence Anchors
Analogue black hole and acoustic horizon work supports using Hawking-style equations as a phonon emission model. Acoustic holes can spontaneously emit phonons at a Hawking temperature, and analogue systems have observed thermal Hawking-like spectra with temperature tied to surface gravity.
These are analogues, not evidence for literal spacetime throat transport.
Audit Classification
Receipt: HawkingAnalogPhononLossRefinement
Status: FORMAL_ANALOGY_DRAFT
Gate: U_scope
Reason: Hawking/analogue-horizon equations provide a strong phonon-emission and entropy-accounting model, but project-specific surface-gravity analogue, greybody/bandgap factor, finite emission budget, and active-cell counting proof remain required.
Required Receipts
AnalogHorizonReceipt
EffectiveSurfaceGravityReceipt
HawkingTemperatureReceipt
GreybodyBandgapReceipt
FiniteEntropyBudgetReceipt
ModeCorrelationReceipt
RecoverableEmissionReceipt
FiniteWindowActiveCountingReceipt
NonseparableEncodingReceipt
CompactDensityReceipt
Boundary
This note does not claim:
literal black holes
literal spacetime wormhole engineering
lossless information transfer
DNA or molecular transport through a throat
It claims only:
Hawking-style analogue-horizon equations are useful for modeling finite phonon emission, entropy flow, greybody filtering, and recoverable active events in a lossy metamaterial throat.