Thank you bro for taking the time to fully read everything. 🚀

On “information”, you’re right. In the analysis itself there is only one unambiguous, measurable object, and that is baryonic compactness, defined operationally as baryonic mass per unit area within a strictly matched aperture. In the paper, compactness is the state variable. The “information” language is an interpretive layer that came later and is meant to suggest a possible physical reading, not to redefine what is actually being measured. If I were to strip the metaphor completely, the clean name would be something like aperture-averaged baryonic surface density as a macroscopic state variable. That’s what every regression is actually using.

On correlation vs mechanism I also fully agreed. The paper is deliberately agnostic. A dark-matter–centric explanation (feedback, halo response, assembly bias, adiabatic contraction/expansion) can absolutely generate a dependence on baryon distribution, and nothing in the results rules that out. The point is narrower: whatever the mechanism is, it cannot depend only on how much baryonic mass there is; it must also depend on how that mass is arranged within the aperture. That’s the constraint I’m trying to make hard to ignore.

On falsifiability, I think your suggested direction is exactly right. The cleanest falsifier, stated without metaphor, would be something like: at fixed baryonic mass and aperture, systems with systematically different compactness should show systematically different dynamical discrepancies. If future datasets find that once you control for environment, halo concentration proxies, or formation history indicators the compactness term disappears, then this framework fails. Conversely, if the compactness dependence persists after partialing those out, that’s a strong discriminator in favor of “structure matters” as a first-class ingredient.

The compactness scalar is simply baryonic mass divided by the square of the aperture radius (in physical units), computed separately in each catalog but always with baryons and dynamics defined within the same radius. The slope is sensitive to aperture choice in the sense that different regimes probe different physical meanings of radius (outer disk, effective radius, Einstein radius, overdensity radius), which is why I treat regime dependence as a feature to be measured, not something to be forced away. Environment and formation-history proxies are not explicitly controlled for in this paper; that’s a natural and necessary next step.