Sagittarius A* as a Homeostatic Engine? A variance-based test using simulations and surveys

Supermassive black holes (SMBHs) are often treated as self-regulating enginesthat couple accretion inflow to feedback outflow and help stabilise their hostgalaxies. In the Milky Way, Sgr~A* is currently underluminous compared toclassical active nuclei, raising the question of whether it is a pathologicaloutlier or a typical, slowly flickering regulator. This work tests this ideain a falsification-first way, using Milky Way analogues and Sgr~A*--mass blackholes in the TNG300-1 cosmological simulation together with Milky Way--massgalaxies from SDSS, GAMA and MaNGA. At z 0 a dimensionless static homeostasis proxy is defined asΛ₀ ≡ gBH/g_, the ratio of black-hole specific growthrate to stellar specific growth rate. A homeostatic band-5 ≤ λ₀ ≤ 3 in λ₀=log₁₀Λ₀ is identified forMilky Way--mass hosts in TNG300, containing $4960$ Milky Way analogues and$115$ Sgr~A* analogues. Within this band the medians areλ̃_0, MW=-1.28 andλ̃_0, SAG=-0.96, a factor 2 difference inΛ₀. Kolmogorov--Smirnov and permutation tests reject the nullhypothesis that the two subsets are drawn from the same λ₀distribution. The offset is larger and more significant in the star-formingsubsample and does not vanish when hosts are split by specific star-formationrate. Time-domain homeostasis is probed using black-hole growth tracksΛBH(t)=ṀBH(t)/MBH(t) from the TNGblack-hole details catalogue. Sgr~A*--mass and slightly more massive controlblack holes follow a common evolutionary sequence that drifts from burstyearly growth towards more negative, regulated values at late times, withoverlapping percentile bands. Typical time-domain homeostasis does not dependstrongly on the presence or absence of major mergers within currentstatistics. The central result concerns the variance of λ₀ as a functionof black-hole mass. For TNG Milky Way analogues, the variance ofλ₀ over the mass interval5 ≤ log₁₀(MBH/M₎ ≤ 9 followsVar(λ₀) a + b log₁₀(MBH/M₎ withb -0.27. More massive black holes therefore sit in a regime ofsmaller homeostatic variance: the central engine acts as a stifferthermostat. High-fidelity observational Milky Way--mass samples from GAMA andMaNGA show variance slopes that are negative but shallow, consistent in signwith the TNG prediction within uncertainties, while SDSS DR8 shows a weakpositive slope +0.10 that is plausibly dominated by proxy noise inblack-hole mass and accretion at the high-mass end. A simple summary tableshows that TNG and GAMA agree that variance declines with MBH, SDSSdoes not, and MaNGA is consistent but limited by small N. In this picture Sgr~A*, with MBH 4×10⁶\,M_,occupies the high-variance, ``loose thermostat'' regime: the Milky Way isexpected to wander further from its median homeostatic state than galaxieswith 10⁸\,M_ black holes. This reconciles its current quiescence withevidence for recent energetic activity without treating Sgr~A* as broken. Atthe same time, the absolute normalisation of Λ₀ reveals a tension:Milky Way analogues in TNG sit at much lower Λ₀ than those in SDSSand GAMA, indicating that the simulation underpredicts specific black-holegrowth at fixed stellar growth. The main conclusion is that the trendof decreasing variance with MBH is likely real and is now detected,but the absolute level of accretion in Milky Way--mass galaxies remains a keydifference between current simulations and data.