Mathematical Underpinnings of the Multi-Wavelength Structure of the TRGB
  We consider the application of the tip of the red giant branch (TRGB) in the
optical and in the near infrared for the determination of distances to nearby
galaxies. We analyze ACS VI (F555W & F814W) data and self-consistently
cross-calibrate WFC3-IR JH (F110W & F120W) data using and absolute magnitude
calibration of M_I = -4.05 mag as determined in the LMC using detached
eclipsing binary star geometric parallaxes. We demonstrate how the optical and
near-infrared calibrations of the TRGB method are mathematically
self-consistent, and illustrate the mathematical basis and relations amongst
these multi-wavelength calibrations. We go on to present a method for
determining the reddening, extinction and the true modulus to the host galaxy
using multi-wavelength data. The power of the method is that with
high-precision data, the reddening can be determined using the TRGB stars
themselves, and decreases the systematic (albeit generally small) uncertainty
in distance due to reddening for these halo stars.
