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The Riemann Hypothesis as a Stability Principle

  The Riemann Hypothesis can be reformulated as an infinite sequence of explicit positivity tests. This is not a proof. It turns a statement about complex zeros into inequalities that can be calculated, falsified at finite order, and compared with operator or physical models.   The completed zeta function is $$\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s),$$ with $$\Xi(t)=\xi\left(\frac12+it\right).$$ The function $\Xi$ is even and real on the real axis. A nontrivial zero $\rho=\beta+i\gamma$ corresponds to $$t_\rho=\gamma-i\left(\beta-\frac12\right).$$ Therefore $$t_\rho\in\mathbb R \quad\Longleftrightarrow\quad \beta=\frac12,$$ and hence $$\boxed{\mathrm{RH}\quad\Longleftrightarrow\quad\text{every zero of }\Xi\text{ is real}.}$$ Folding the Zeros Because $\Xi$ is even, write $$\Xi(t)=\sum_{n\ge0}c_nt^{2n}.$$ Now define $$G(z)=\frac{\Xi(i\sqrt z)}{\Xi(0)}.$$ Although this contains $\sqrt z$, the even expansion makes $G$ entire: $$G(z)=...
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One Dark Scale, One Galactic Law

 

 

 

We previously discussed Verlinde's connection of the MOND acceleration scale to the entropy of de Sitter space. A different route appears in superfluid dark matter (a 2026 review is here), where baryons interact with a phonon field whose nonlinear dynamics generate a MOND-like force.

The superfluid theory introduces a characteristic scale $\Lambda_{\rm SF}$, which must be of order meV to reproduce the MOND scale. This is also the order of the vacuum-energy scale, $\rho_{\rm DE}^{1/4}\sim{\rm meV}$. 

So, we postulate $\Lambda_{\rm SF}^4=\rho_{\rm DE}$.  

This idea obviously points to a unified dark sector, e.g. 2019 Unified Superfluid Dark Sector, 2026 A solid unification of the dark sector, and 2026 Unified dark sector approach to cosmological tensions among many others. Indeed, a complete theory would no longer ask “What particle is dark energy?” Dark energy would instead be the vacuum energy of the same medium whose excitations and common motion constitute dark matter.  

So, let $\pi$ be the phonon field and define

$$\mathbf E=-\nabla\pi, \qquad E=|\mathbf E|.$$

Assume its static field energy is

$$W(E)=\frac{2\Lambda_{\rm SF}}{3}E^3.$$

The cubic power is special. In three spatial dimensions it gives a scale-invariant field equation and produces the $1/r$ force required by deep MOND. The associated flux is

$$\mathbf D = \frac{\partial W}{\partial\mathbf E} = 2\Lambda_{\rm SF}E\mathbf E,$$

with magnitude $D=2\Lambda_{\rm SF}E^2$. Ordinary matter of density $\rho_b$ sources the field according to

$$\nabla\cdot\mathbf D = \frac{\alpha\Lambda_{\rm SF}}{M_{\rm Pl}}\rho_b, \qquad M_{\rm Pl}^{-2}=8\pi G.$$

For a spherical mass distribution, $4\pi r^2D_r = \frac{\alpha\Lambda_{\rm SF}}{M_{\rm Pl}}M_b(r)$. Using $D_r=2\Lambda_{\rm SF}E^2$ gives

$$E^2 = \frac{\alpha M_b(r)}{8\pi M_{\rm Pl}r^2} = \alpha M_{\rm Pl}a_b, \qquad a_b=\frac{GM_b(r)}{r^2}.$$

The phonon acceleration is $a_\pi = \frac{\alpha\Lambda_{\rm SF}}{M_{\rm Pl}}E$. Therefore, $a_\pi^2 = \frac{\alpha^3\Lambda_{\rm SF}^2}{M_{\rm Pl}}a_b$. Defining $a_{\rm SF} = \frac{\alpha^3\Lambda_{\rm SF}^2}{M_{\rm Pl}}$, we obtain

$$\boxed{a_\pi=\sqrt{a_{\rm SF}a_b}}$$

This is the standard deep-MOND result of the superfluid theory, introducing $\alpha$ as a free coupling.

Now, matter follows the conformally related line element $d\widetilde{s}=e^{\Lambda_{\rm SF}\pi/(2M_{\rm Pl})}ds$, corresponding to $\widetilde g_{\mu\nu}=e^{\Lambda_{\rm SF}\pi/M_{\rm Pl}}g_{\mu\nu}$. Expanding $S_b=-m\int d\widetilde{s}$ to first order gives $\mathcal L_{\pi b}=-(\Lambda_{\rm SF}/2M_{\rm Pl})\pi\rho_b$, and therefore $\alpha=\tfrac12$. This value is fixed within the chosen metric ansatz, rather than by covariance: the generalised choice $\widetilde g_{\mu\nu}=e^{2\beta\Lambda_{\rm SF}\pi/M_{\rm Pl}}g_{\mu\nu}$ gives $\alpha=\beta$. So, an additional principle is still required to select $\beta=\tfrac12$. The superfluid framework otherwise permits an independent phonon-baryon coupling.

Then use $\Lambda_{\rm SF}^4 = \rho_{\rm vac}$, $\rho_{\rm vac} = 3M_{\rm Pl}^2H_\Lambda^2$, $H_\Lambda=H_0\sqrt{\Omega_\Lambda}$. Restoring $c$,

$$\boxed{ a_{\rm SF} = \sqrt3 \alpha^3cH_\Lambda = \sqrt{3\Omega_\Lambda} \alpha^3cH_0 }.$$

The observed MOND scale determines

$$\alpha = \left[\frac{a_0}{\sqrt{3\Omega_\Lambda} cH_0}\right]^{1/3}$$

For $a_0=1.20\times10^{-10}\ {\rm m s^{-2}}$ (The a0 - cosmology connection in MOND), and (based on 2025 P-ACT-LB2 results) $H_0=68.43\ {\rm km s^{-1}Mpc^{-1}}$, $\Omega_\Lambda=0.6997$, we have $H_0=2.2177\times10^{-18}\ {\rm s^{-1}}$, and therefore $\alpha\simeq0.49944$. The observational value is therefore extremely close to the metric choice $\boxed{\alpha=\frac12}$.

Substituting $\alpha=\tfrac12$ predicts $a_0 = \frac{\sqrt{3\Omega_\Lambda}}8 cH_0$. Using the same cosmological values gives

$$\boxed{ a_0 = 1.204\times10^{-10}\ {\rm m s^{-2}} }$$

in excellent agreement with the empirical MOND value.

Since $H_\Lambda = c\sqrt{\Lambda_{\rm cos}/3}$, the result can also be written

$$\boxed{ a_0 = \frac{c^2\sqrt{\Lambda_{\rm cos}}}{8} }.$$

This is the more fundamental form. It ties $a_0$ directly to vacuum curvature rather than to the evolving Hubble rate.

For a point mass, $a_b=GM/r^2$, so $a_\pi = \sqrt{GMa_0}/r$. In the low-acceleration region, $v_c^2/r\simeq a_\pi$, and therefore

$$\boxed{v_c^4=GMa_0}.$$

This is the baryonic Tully–Fisher relation.

More generally, $\mathbf a_{\rm total} = \mathbf a_b+\mathbf a_\pi+\mathbf a_{\rm DM}$, where $\mathbf a_{\rm DM}$ is the gravitational field of the superfluid core and normal dark-matter halo. The phonon and halo terms belong to the same dark sector but remain physically distinct and must be calculated together. The phonon force applies within the coherent superfluid region and is lost when coherence breaks down.

Unlike Verlinde's proposal, the cosmological scale enters here through the superfluid phonon theory rather than directly through de Sitter entropy. Also, $a_0$ is therefore constant in redshift rather than tracking $H(z)$.  

$$ a_0 = \frac{1}{M_{\mathrm{Pl}}}\left(\alpha^3 \Lambda_{\mathrm{SF}}^{2}\right)
  = \frac{1}{8}\frac{\sqrt{\rho_\Lambda}}{M_{\mathrm{Pl}}}
  = \frac{c^{2}\sqrt{\Lambda_{\mathrm{cos}}}}{8}$$ We see that $1/8$ is the cube of the matter coupling normalization. 

This points to a unified dark medium, with the critical response

$$\int d^3x,|\nabla\Phi|^3 \quad\Rightarrow\quad a_\Phi=\sqrt{a_0a_b}, \qquad v_c^4=GMa_0,$$

and the invariant cosmological relation

$$a_0^2=\frac{c^4\Lambda_{\rm cos}}{64}.$$

If $\Phi$ is the phase of a broken $U(1)$ condensate, the medium is a dark superfluid. It becomes a dark supersolid only if the same state also breaks spatial translations, carries positive shear rigidity and supports transverse modes. None of that has yet been derived.

The key idea: MOND is a critical spatial response, dark matter is conserved bulk motion, and dark energy is homogeneous energy of one medium. Superfluidity, supersolidity and a Josephson-driven vacuum sector remain candidate microscopic realizations, to be tested through their propagation modes, gravitational slip, structure growth and gravitational-wave effects.


 

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