Galaxies

We examine the family-separated seesaw (FSS) ansatz proposed in arXiv:2605.27049v2. Combining its paired cancellations with exact partial unitarity fixes the light and heavy mixing blocks in terms of one unitary matrix and the paired mass ratios. We classify their unitary completions and reconstruct the Dirac and singlet Majorana mass matrices, distinguishing the physical heavy masses from the singlet mass parameters. This identifies FSS as an exact realization of form dominance whose conventional one-loop leptogenesis cancellation is already established. The particular sum of lepton-number-violating invariants entering the proposed decay asymmetry therefore vanishes; individual CP invariants need not vanish. We also correct the heavy-mass reconstruction formulas and quantify their errors for both light-mass orderings. These formulas are conditional algebraic relations, not independent predictions of the heavy spectrum. Numerical mass–mixing illustrations, explicit chiral projections for double-beta decay, and a leading-logarithmic estimate of radiative misalignment clarify the physical reach and limitations. The contribution is an exact consistency analysis and correction of specific proposed consequences rather than a new mass-generation mechanism or a general prohibition of leptogenesis.

For the past fifty years, the physical foundations of black hole (BH) thermodynamics have remained widely contested because their original derivations rest at the theoretical intersection of general relativity and quantum field theory (QFT). This work provides an alternative macroscopic formulation by treating the physical vacuum in and around a BH as an elastic continuum strained by the presence of mass. Utilizing Hooke’s law of elasticity, we find that the radial strain at the event horizon of every Schwarzschild BH saturates at the same constant value. Under this principle of universal maximum strain, the laws of BH thermodynamics can be formulated within macroscopic continuum mechanics. This dispenses with the conventional QFT derivation in curved spacetime, paralleling the recent macroscopic descriptions of the Casimir effect. Our calculations are carried out using the Reformulated Planck System (RPS), which relies exclusively on Planck’s constant h. This h-native approach prevents the dimensional contamination of 3D continuum mechanics caused by the 2D geometric factor inherently embedded in Dirac’s reduced constant ℏ, a pivotal misstep first recognized in the iconic definitions of the fine-structure constant $\mathrm{\alpha }$ and the gravitational coupling constant ${\mathrm{\alpha }}_{\mathrm{g}}$, which long precluded the identification of the $\mathrm{\alpha }$-dependence of the weak coupling constant ${\mathrm{\alpha }}_{\mathrm{w}}=\sqrt{\mathrm{\alpha }}$. In BH thermodynamics, this RPS-based approach yields higher Hawking temperatures and lower Bekenstein entropies, both by a factor of $2\pi$.

We study the Kretschmann invariant K of the post-Newtonian spacetime of a slowly rotating, axially symmetric clump of collisionless matter with a prescribed Miyamoto–Nagai potential, whose metric and matter content to second order in the compactness $\beta$ were constructed in a companion paper. At $O\left({\beta }^2\right)$, K is a quadratic form in the Hessian of the potential, with closed-form central and asymptotic values. At $O\left({\beta }^3\right)$ we show that the static part of K is a closed-form local function of the potential and of the matter fields at every point: the second-order metric functions, which solve a Poisson-type problem and encode the global mass distribution, enter only through the Ricci tensor, that is, only through the local matter. This follows from an exact expression of K for static spacetimes in terms of the tidal tensor of static observers and the local stresses, and holds for any regular prescribed potential. At a centre with isotropic stresses it reduces to , also for composite bulge–disc models. The only non-local contribution at this order is gravitomagnetic; it vanishes at the centre and, for an isotropic rotator, reaches within six scale radii about 10% (round clump) to 40% (thin disc) of the static third-order term. For galaxy-scale compactness, $\beta \sim {10}^{-6}$, all relativistic corrections are of relative order ${10}^{-7}$–${10}^{-5}$; the third-order term exceeds 10% of K only for compactness $\gtrsim 0.04$, the regime of relativistic star clusters. A central black hole dominates K within about 25 pc of the centre of a Milky-Way-like bulge–disc model.