  Star formation theories have struggled to reproduce binary brown dwarf
population demographics (frequency, separation, mass-ratio). Kernel-phase
interferometry is sensitive to companions at separations inaccessible to
classical imaging, enabling tests of formation at new physical scales below the
hydrogen burning limit. We analyze the detections and sensitivity limits from
our previous kernel-phase analysis of archival HST/NICMOS surveys of field
brown dwarfs. After estimating physical properties of the 105 late M to T
dwarfs using Gaia distances and evolutionary models, we use a Bayesian
framework to compare these results to a model companion population defined by
log-normal separation and power-law mass-ratio distributions. When correcting
for Malmquist bias, we find a companion fraction of $F=0.11^{+0.04}_{-0.03}$
and a separation distribution centered at $\rho=2.2^{+1.2}_{-1.0}$ au, smaller
and tighter than seen in previous studies. We also find a mass-ratio power-law
index which strongly favors equal-mass systems:
$\gamma=4.0^{+1.7}_{-1.5}-11^{+4}_{-3}$ depending on the assumed age of the
field population ($0.9-3.1$ Gyr). We attribute the change in values to our use
of kernel-phase interferometry which enables us to resolve the peak of the
semimajor axis distribution with significant sensitivity to low-mass
companions. We confirm the previously-seen trends of decreasing binary fraction
with decreasing mass and a strong preference for tight and equal-mass systems
in the field-age sub-stellar regime; only $0.9^{+1.1}_{-0.6}$ % of systems are
wider than 20 au and $<1.0^{+1.4}_{-0.6}$% of systems have a mass-ratio
$q<0.6$. We attribute this to turbulent fragmentation setting the initial
conditions followed by a brief period of dynamical evolution, removing the
widest and lowest-mass companions, before the birth cluster dissolves.
