TO: Wendy L. Freedman, Kavli Institute for Cosmological Physics
Astronomy & Astrophysics University of Chicago, Chicago, Illinois
FM: Bruce E. Camber
RE: Articles about your work like Cosmology at a crossroads (Nature, May 2017 PDF) and your articles within arXiv (152) such as The Hubble Constant with Barry F. Madore, August 2010 and Mathematical Underpinnings of the Multi-Wavelength Structure of TRGB, August 2020; plus your books like Measuring and Modeling the Universe: V2, Carnegie Observatories (2004); and your homepage(s) including Gruber, KICP, Hubblesite, inSPIREHEP, NASA Hubble (July 16, 2019), even Wikipedia and YouTube: Misel Family Lecture Series (2017)
This page: https://81018.com/freedman/
Email: 12 July 2026
RE: The Hubble tension as a ΔN discrepancy — a geometric perspective
Dear Professor Dr. Wendy Freedman,
I have followed your work on the Hubble constant for many years and wrote to you once before, some years ago, to thank you for your work. I am writing again because something specific has emerged from our geometric framework that connects directly to the tension your measurements have helped define.
A base-2 doubling framework anchored at the Planck scale produces the following exact identity:
ΔN = log₂[ I₁(ΩΛ) / I₂(ΩΛ) ]
where ΔN is the discrepancy between length and time doubling counts from the Planck scale to the present universe, and I₁, I₂ are the standard ΛCDM integrals for the comoving horizon and the age of the universe. The identity is exact — a consequence of ℓ_P = c · t_P — and was confirmed as mathematically sound by Alain Blanchard (Toulouse) in private correspondence this June.
The connection to your work is this: ΔN is sensitive to H₀. Different values of H₀ produce different values of ΔN, all recovering ΩΛ ≈ 0.685 through the same identity. Specifically:
Using Planck 2018 parameters (H₀ = 67.4): ΔN ≈ 1.754
Using the Wright calculator with standard parameters (closer to your TRGB value): ΔN ≈ 1.729
The difference of 0.025 doublings between these two values is the geometric signature of the Hubble tension in the base-2 framework. The tension that appears in your measurements as a disagreement about km/s/Mpc appears in our framework as a disagreement about how many doublings separate the Planck scale from the present universe.
This reframing does not resolve the tension. But it may offer a new observable: if ΔN can be measured independently — from quantum gravity signatures at the Planck scale rather than from the distance ladder — it would provide a geometric check on H₀ that bypasses the systematic uncertainties in both the CMB and distance ladder approaches.
Whether that independent measurement is achievable is a question I cannot answer. But you are better placed than almost anyone to assess whether it would be meaningful if it were.
The full paper: https://81018.com/dark-energy-from-scale-invariance/
I would be grateful for your reaction — even a sentence telling me whether the geometric reframing of the tension seems physically meaningful, or where you see an immediate problem with the argument.
With respect for your work,
Bruce
Bruce E. Camber
Independent Researcher, 81018 Science Education Project
camber@81018.com | 81018.com | 214-801-8521