Appendix X｜Lag-Projection Equivalence in Galactic-Center Dynamics
Recent analyses of S-star dynamics (e.g., Crespi et al. 2026, MNRAS 546) explore the observational indistinguishability between black holes and fermionic cores.
1. Orbital Motion in Central Potentials
For a test mass orbiting a spherically symmetric source, the radial dynamics is governed by
\[\frac{1}{2}\dot r^2 + V_{\mathrm{eff}}(r) = E,\]with
\[V_{\mathrm{eff}}(r) = \Phi(r) + \frac{L^2}{2r^2} + \delta V_{\mathrm{GR}}(r).\]In the Schwarzschild weak-field limit,
\[\delta V_{\mathrm{GR}}(r) = -\frac{GM L^2}{c^2 r^3}.\]Thus the effective potential difference between two models reduces to
\[\delta V_{\mathrm{eff}}(r) = \delta\Phi(r) + \delta V_{\mathrm{GR}}(r).\]2. Competing Potentials
Black-hole model:
\[\Phi_{\mathrm{BH}}(r) = -\frac{GM}{r}.\]Fermionic dark-matter core:
\[\Phi_{\mathrm{DM}}(r) = -\frac{G M(r)}{r},\]where $M(r)$ transitions from $M(r)\propto r^3$ near the core
to $M(r)\to M$ asymptotically.
Define:
\[\delta\Phi(r) = \Phi_{\mathrm{BH}}(r) - \Phi_{\mathrm{DM}}(r).\]3. Observational Scale: S2 Constraint
The strongest current dynamical constraint is set by the S2 orbit, with pericenter distance
\[r_p \approx 120\mathrm{au} \approx 1.8\times 10^{13}\mathrm{m}.\]Therefore,
\[\sup_{r\in\text{orbit}}|\delta\Phi(r)| \approx |\delta\Phi(r_p)|.\]4. Fermionic Core Compactness
For fermionic dark-matter cores reproducing a central mass
$M \sim 4\times 10^6 M_\odot$,
particle masses typically lie in
\[mc^2 \sim 56\text{–}300\mathrm{keV}\]This implies core radii
\[r_\mathrm{core} \sim \mathcal{O}(0.1\text{–}1)r_g, \quad r_g = \frac{2GM}{c^2}.\]Since
\[r_p \gg r_\mathrm{core},\]we have
\[|\delta\Phi(r_p)| \ll \frac{GM}{r_p}.\]For reference, the gravitational radius of a $M \sim 4\times 10^6 M_\odot$ object is
\[r_g = \frac{2GM}{c^2} \approx 1.2\times10^{10}\,\mathrm{m}, \qquad \frac{r_g}{r_p} \sim 10^{-3}.\]5. Including Schwarzschild Correction
The relativistic correction term scales as
\[\delta V_{\mathrm{GR}}(r_p) \sim \frac{r_g}{r_p} \frac{L^2}{r_p^2}.\]Since
\[\frac{r_g}{r_p} \ll 1,\]both the GR correction and the fermionic-core deviation remain subleading at the S2 scale.
Thus,
\[|\delta V_{\mathrm{eff}}(r_p)| < \epsilon_{\mathrm{obs}}.\]6. Translation into Lag Formalism
Define lag variable:
\[\lambda(t) = S'(t) - O'(t).\]Observables access only projected configurations:
\[\mathcal{Z}_0(t) = \mathcal{P}(\lambda(t;\Phi)).\]Introduce closure exposure parameter:
\[\Delta Z_0 \sim \frac{\delta V_{\mathrm{eff}}(r_p)}{V_{\mathrm{eff}}(r_p)}.\]Given the hierarchy
\[r_\mathrm{core} \lesssim r_g \ll r_p,\]we obtain
\[\Delta Z_0 \ll 1.\]Therefore,
\[\mathcal{P}(\lambda_{\mathrm{BH}}) \approx \mathcal{P}(\lambda_{\mathrm{DM}}).\]The models are lag-projection equivalent within current resolution.
7. Structural Statement
The Galactic-center problem reduces to:
\[\text{Observational distinguishability} \propto \sup_{r\in\text{S2}} |\delta V_{\mathrm{eff}}(r)|.\]When this lies below observational resolution, ontological distinctions collapse into lag-equivalent projections.
The black-hole vs fermionic-core distinction can therefore be recast, within this framework, as a question of closure configuration in lag topology.
We adopt their mass range (56–300 keV) and orbital scale for the S2 star as representative reference scales.
Crespi, M. et al. 2026, MNRAS, 546, staf1854
https://doi.org/10.1093/mnras/staf1854
── Lag–Projection Reframing
“We merely reformulate the indistinguishability condition in lag-theoretic terms.”
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| Drafted Feb 11, 2026 · Web Feb 11, 2026 |