Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

=== Assistant: Nice — I ran a quick, focused numerical survey to check whether a simple QAT-style model energy density built from many… ===
Nice — I ran a quick, focused numerical survey to check whether a simple QAT-style model (energy density built from many photon–electron events) could plausibly reproduce a cosmological reference density. I used a compact, testable formula:

u = n · E₀ · (ν · τ)   (J / m³)
ρ = u / c²

so the required product (ν·τ) to reach a reference mass density ρ_ref (I used a typical critical density ≈ 9×10⁻²⁷ kg/m³) is

ν·τ = (ρ_ref · c²) / (n · E₀).

I evaluated that threshold for four characteristic length scales (that set a characteristic energy E₀ ≈ ħc / scale): Bohr radius, electron Compton wavelength, proton radius, and the Planck length — and for four number-density regimes (cosmic baryon, ISM, warm/dense nebula, dense plasma).

What I produced for you
* An interactive table showing the required ν·τ thresholds for each (scale, density) pair (you can edit/copy it).
* Four plots (one per length scale) showing log10(required ν·τ) across the four density regimes (so you can see how thresholds fall as density rises).
* A CSV saved to /mnt/data/QAT_thresholds_required_nu_tau.csv so you can download and re-use the numbers.

Key takeaways (high-level)
* The required product ν·τ is very sensitive to the characteristic energy scale E₀ (i.e., the length scale you assign). Using atomic scales (Bohr, electron Compton) gives far smaller thresholds than ultra-tiny Planck length scales (where energy per event would be enormous, so required ν·τ is tiny).
* For plausible atomic E₀ (Bohr radius or Compton), reaching cosmological mass densities at very low particle densities (like cosmic baryons ~0.25 m⁻³) requires very large ν·τ — e.g., log10(ν·τ) positive (meaning ν·τ ≫ 1). That says either events must be extremely frequent (very large ν) or very long-lived (large τ) per particle to matter cosmologically.
* At higher physical number densities (e.g., 10¹²–10¹⁸ m⁻³) the threshold falls dramatically — so in plasma or stellar regions the same per-event energy could more plausibly contribute noticeably to local mass/energy density.

Important caveats / interpretation guidance
* This is not a proof of QAT, just a rapid quantitative sanity-check of the idea "many small photon–electron events could sum to a cosmological-scale energy density." The result depends strongly on three choices: 1. the characteristic energy per event E₀ (I used ħc/scale), 2. the event rate per particle ν, and 3. the effective event duration τ (or more generally, how much of the event's energy counts toward the instantaneous energy density).
* The model algebraically reduces to a required ν·τ (not two separate unknowns). That makes it easy to explore parameter space, but still requires physical grounding for ν and τ. Reasonable guesses (Planck time, atomic timescales, plasma collision/radiation times) change conclusions a lot.
* I used an approximate rho_crit; you may want to replace with a different target density (e.g., baryonic density, local ISM density, dark-matter inferred density) depending on which question you're testing.

If you want to continue (two clear next steps I recommend)
# Pick some realistic physical guesses for ν and τ across regimes: - e.g. atomic photon emission/absorption rates (10⁶–10¹⁵ s⁻¹?), plasmas collisional/radiative times (10⁻⁹–10⁻³ s?), or Planck time (~5.39×10⁻⁴⁴ s) if you’re exploring extreme scales. We can then compute resulting ρ and compare to observations directly.
# Turn the ν·τ survey into contour plots across (ν, τ) for one or two representative density/scale combinations so you can see full (ν, τ) regions that meet, exceed, or fall short of a target ρ. I can make those plots and annotate physically plausible regions.

Would you like me to:
* (A) Produce contour plots of ρ/ρ_ref over a grid of ν and τ for one or two chosen (scale, density) pairs, or
* (B) Use some physically motivated example values for ν and τ (you or I pick) and compute actual ρ numbers and compare them to e.g. baryonic density, dark-matter-inferred density, or local ISM density?

Tell me which scales/densities or which values of ν and τ you’d like explored and I’ll run the follow-up visuals/calculations and explain what they imply for QAT.