There is a little bit of geometry and group theory jargon here in my description of 'c' in General Relativity, but no equations:
The plane tangent to the surface of the ball in this diagram is a tangent space.
https://upload.wikimedia.org/wikipedia/commons/thumb/6/66/Im...
A metric on a space determines lengths (both spatial lengths and time durations).
The geometry of spacetime induces the Minkowski metric on tangent spaces.
At every point in a Minkowski space lengths are invariant under the Poincaré isometry group, which means that lengths stay the same under any or all of three spatial translations along orthogonal axes, three rotations about these axes, three Lorentz boosts, and one time translation.
There is one free parameter in the Poincaré group, which is "c", which physically corresponds to a particle with zero invariant mass. Photons are expected to have zero invariant mass, and experiment sets a very very very small upper bound on their mass.
Observers in the same tangent space as a photon will measure it as massless and moving at 'c'.
Minkowski space is flat spacetime, and in flat spacetime each tangent space covers the whole spacetime; they overlap completely.
As you add curvature^2 to the geometry of spacetime, the tangent spaces shrink and are oriented to each other at different angles.
Observers in one tangent space measuring a photon in another tangent space may not agree on its speed. In fact, in General Relativity, it is mostly forbidden to talk about the relative velocities of objects in other tangent spaces -- that's a dramatic departure from Special Relativity, but then SR is the theory of Minkowski space, where all tangent spaces cover the same region of spacetime, and so they are indistinguishable.
Spacetime curvature around us is pretty gentle, so within a small laboratory test apparatus everything is in the same tangent space or so close to it that differences aren't detectable. We have excellent and readily reproduced data on the speed of light in such setups.
The Poincaré isometry group is a subgroup of the group theory of the Standard Model. All the scatterings allowed under the Standard Model are in a region of spacetime where curvature doesn't matter, and we have many exabytes of evidence supporting the Standard Model in terrestrial experiments, in experimental systems elsewhere in the solar system, and in astronomical observations. Indeed, we can use the repeatability of the Standard Model and its built-in Poincaré invariance to measure spacetime curvature, and we do so with GNSS systems like GPS.
A violation of Poincaré invariance would produce different behaviours in microscopic systems, and so far they have not been detected. The Lorentz group is a subgroup of the Poincaré group, and there have been many tests specifically probing for violations of Lorentz invariance near us and in distant parts of the observable universe, and as of today none have been found after a century of trying.
So unless there is a low energy breakdown of the Standard Model, we can be pretty confident that the photon is a massless particle and moves at 'c' for all observers close to the photon.
