Observational Implications

The preceding sections established that the regular interior geometry introduces corrections to strong-field observables that are parametrically suppressed by positive powers of $g/GM$, where $g$ is the core regularization scale. This section makes that suppression quantitative by confronting each observable channel with the precision of current data, in order to place an explicit, if approximate, upper bound on $g$ and to identify which observational program is most likely to detect or exclude a non-zero core scale first.

Weak-Field and Solar-System Constraints

Because $f(r) \to 1 - 2M/r$ at large $r$ with corrections of order $\mathcal{O}(g^3/r^3)$, solar-system tests probe the theory only through the extremely suppressed tail of the core correction at $r \sim 1\,\mathrm{AU}$. The tightest such test, the Cassini bound on the parametrized post-Newtonian parameter $\gamma$ from Shapiro time delay, constrains deviations from general relativity at the level of $|\gamma - 1| \lesssim 2\times10^{-5}$. Since the core correction to $f(r)$ scales as $(g/r)^3$ at these radii, this bound translates into

where $r_\odot$ is a characteristic solar-system length scale. Given that $GM_\odot \sim 1.5\,\mathrm{km}$ while $r_\odot \sim 1\,\mathrm{AU} \sim 1.5\times10^{8}\,\mathrm{km}$, this bound is enormously weaker, in units of $GM$, than the bounds obtainable from compact-object observations discussed below. Solar-system tests therefore constrain the model only in the trivial sense already guaranteed by construction: they confirm that the exterior metric is observationally indistinguishable from Schwarzschild, without meaningfully constraining $g$ itself.

Black Hole Shadow Imaging

The Event Horizon Telescope (EHT) images of M87* and Sagittarium A* measure angular shadow diameters consistent with the Kerr/Schwarzschild prediction to within approximately $10$–$17\%$, depending on target and calibration assumptions. Since the shadow radius correction in this framework enters at

(Section 6), a fractional shadow-size bound $\epsilon_{\mathrm{EHT}} \sim 0.1$–$0.17$ implies

This is the strongest direct imaging bound available at present, and it already excludes core scales comparable to the horizon itself, while leaving substantial room for cores an order of magnitude or more below $GM$. Because the correction enters at cubic order in $g/GM$, an order-of-magnitude improvement in shadow-diameter precision (expected from next-generation very-long-baseline arrays and space-VLBI proposals) improves the bound on $g$ by only a factor of order $2$, a direct consequence of the same structural suppression that guarantees consistency with existing images.

Accretion-Disk and ISCO-Based Constraints

Independent of direct imaging, the innermost stable circular orbit (ISCO) is probed observationally through relativistic disk-reflection spectroscopy (profile of the broadened iron K$\alpha$ line) and through the continuum-fitting method applied to thermal disk spectra. Both techniques currently constrain the ISCO radius of accreting stellar-mass and supermassive black holes to within roughly $10$–$20\%$ of the Schwarzschild or Kerr value, depending on spin and inclination degeneracies. Using the ISCO shift derived in Section 5,

a $10$–$20\%$ bound on the ISCO location similarly yields $g \lesssim (0.1\text{--}0.2)^{1/3}\,GM \sim 0.5$–$0.6\,GM$, consistent with, and of the same order as, the shadow-imaging bound above. The rough agreement between two independent observational channels – photon-sphere imaging and accretion-disk spectroscopy – despite probing physically distinct aspects of the strong-field metric, is a direct consequence of both corrections being controlled by the same underlying scale $g$ in this framework, and constitutes a consistency check rather than two independent constraints.

Binary Pulsars and Strong-Field Timing

Relativistic binary pulsars, most notably the double pulsar PSR J0737$-$3039, provide tests of strong-field gravity through periastron advance, Shapiro delay, and orbital decay measured to high timing precision. These systems, however, probe the metric at orbital separations $r \sim 10^{6}$–$10^{9}\,GM$ for the component masses involved, placing them firmly in the regime where the core correction $\mathcal{O}(g^3/r^3)$ is utterly negligible, far more so than the solar-system case above. Binary pulsar timing therefore offers essentially no leverage on $g$ in the present static model; its relevance would grow only if the regularization scale were instead tied to a compact companion's own near-horizon structure, which is not assumed here.

Prospects: Ringdown and Rotating Extensions

Gravitational-wave ringdown – the relaxation of a perturbed compact object through a discrete spectrum of quasinormal modes – is, in principle, one of the most sensitive probes of near-horizon structure, since quasinormal frequencies are set by the photon-sphere region whose properties are directly modified by $g$ (Section 6). Extracting a quantitative bound from ringdown data requires the quasinormal-mode spectrum of the regular, rotating generalization of this metric, which has not yet been constructed: the present work is restricted to static, spherically symmetric spacetimes (Section 10.3). We therefore do not report a ringdown-based bound on $g$ here. We note, for completeness, that if the fractional shift in the fundamental quasinormal frequency scales with the same cubic suppression found for the shadow and ISCO, current ringdown measurements of stellar-mass merger remnants (consistent with the Kerr spectrum at the $\sim 10\%$ level) would be expected to yield a bound of comparable order to Sections 9.2–9.3, but this expectation should be treated as a target for future work rather than a result established here.

Summary of Observational Status

No existing observation excludes a structurally stabilized, singularity-free core at scales an order of magnitude or more below the gravitational radius. The theory is therefore currently viable across every tested channel, and its cubic suppression of exterior corrections means that ruling out, or detecting, a core in the range $g \sim 0.1$–$0.5\,GM$ is squarely a problem for next-generation shadow imaging and disk spectroscopy rather than for solar-system or binary-pulsar precision timing. This sets a concrete empirical target for the observational programs discussed in the concluding section.