  Clear windows onto emergent hadron mass (EHM) and modulations thereof by
Higgs boson interactions are provided by observable measures of pion and kaon
structure, many of which are accessible via generalised parton distributions
(GPDs). Beginning with algebraic GPD Ansaetze, constrained entirely by
hadron-scale $\pi$ and $K$ valence-parton distribution functions (DFs), in
whose forms both EHM and Higgs boson influences are manifest, numerous
illustrations are provided. They include the properties of electromagnetic form
factors, impact parameter space GPDs, gravitational form factors and associated
pressure profiles, and the character and consequences of all-orders evolution.
The analyses predict that mass-squared gravitational form factors are stiffer
than electromagnetic form factors; reveal that $K$ pressure profiles are
tighter than $\pi$ profiles, with both mesons sustaining near-core pressures at
magnitudes similar to that expected at the core of neutron stars; deliver
parameter-free predictions for $\pi$ and $K$ valence, glue, and sea GPDs at the
resolving scale $\zeta=2$GeV; and predict that at this scale the fraction of
meson mass-squared carried by glue and sea combined matches that lodged with
the valence degrees-of-freedom, with a similar statement holding for
mass-squared radii.
