CDM3Y-IVF1 Equations of State

• CDM3Y-IVF1 equations of state are semi-microscopic models derived from the M3Y-Paris interaction, designed to simulate dense nuclear matter in neutron stars.
• They feature independent isoscalar and isovector density dependencies, calibrated to symmetric matter saturation and stiff symmetry energy at supra-saturation densities.
• The model incorporates higher-order symmetry-energy expansions and extends to cold beta-equilibrated npeμ matter to predict core-crust transitions, moment of inertia, and direct Urca thresholds.

CDM3Y-IVF1 equations of state are a family of semi-microscopic nuclear equations of state derived from the M3Y-Paris nucleon-nucleon interaction within a non-relativistic Hartree-Fock framework and extended to cold, charge-neutral, $\beta$-equilibrated $npe\mu$ matter relevant to neutron-star cores. Their defining feature is a density-dependent isoscalar sector calibrated to symmetric nuclear matter saturation and a separately tuned isovector sector that produces a stiff symmetry energy at supra-saturation density. In recent neutron-star applications, this framework has been used to span very soft to extremely stiff nuclear matter through variations of the saturation incompressibility $K_0$, while simultaneously retaining a full hierarchy of higher-order symmetry-energy coefficients. That structure makes CDM3Y-IVF1 especially suitable for systematic studies of core-crust transition properties, crustal moment of inertia, direct Urca thresholds, adiabatic indices, compactness, and tidal observables (Seif et al., 24 Jul 2025).

1. Microscopic definition and interaction structure

The CDM3Y construction starts from the semi-realistic M3Y-Paris interaction, decomposed into central isoscalar and isovector components with direct and exchange terms. Medium effects are incorporated through explicit density-dependent factors multiplying the finite-range interaction, so that the effective interaction is written in isoscalar and isovector channels as

$$v^{D,Ex}_{00(01)}(\rho,r)=F_{0(1)}(\rho)\,v^{D,Ex}_{00(01)}(r),$$

with

$$F_{0(1)}(\rho)=C_{0(1)}\left(1+\alpha_{0(1)}e^{-\beta_{0(1)}\rho}-\gamma_{0(1)}\rho\right).$$

A given set of parameters $\{C_i,\alpha_i,\beta_i,\gamma_i\}$ defines one member of the CDM3Y-$K_0$ family (Seif et al., 24 Jul 2025).

In asymmetric nuclear matter with baryon density $\rho$ and proton fraction $x_p=\rho_p/\rho$, the Hartree-Fock energy per nucleon is

$$\begin{aligned} E_A(\rho,x_p) &= \frac{3\hbar^2 k_F^2}{20m} \Big[(2-2x_p)^{5/3} + (2x_p)^{5/3}\Big] \ &\quad + \frac{\rho}{2} \Big[ F_0(\rho) J^{D}_{00} + (1-2x_p)^2 F_1(\rho) J^{D}_{01} \Big] \ &\quad + \frac{\rho}{8} \int d\vec r \Big[ F_0(\rho) v^{Ex}_{00}(r) B_0^2(r,x_p) + F_1(\rho) v^{Ex}_{01}(r) B_1^2(r,x_p) \Big], \end{aligned}$$

where $npe\mu$0, $npe\mu$1, $npe\mu$2, and

$npe\mu$3

with $npe\mu$4 (Seif et al., 24 Jul 2025).

Within the IVF1 parametrization, the isoscalar density dependence $npe\mu$5 is adjusted to reproduce symmetric nuclear matter saturation in Hartree-Fock, with $npe\mu$6, $npe\mu$7, and a prescribed incompressibility

$npe\mu$8

The isovector density dependence $npe\mu$9 is fitted independently by matching Brueckner-Hartree-Fock neutron optical potentials, which yields a stiff symmetry energy at supra-saturation density; for Paris-IVF1, the quoted saturation values are $K_0$0 and $K_0$1 (Seif et al., 24 Jul 2025).

The broader CDM3Y literature clarifies the structural meaning of this construction. The underlying finite-range M3Y radial interaction is retained, while the density dependence is applied separately in the isoscalar and isovector channels. In earlier CDM3Y-n studies, the stiff isovector sector was likewise tied to Brueckner-Hartree-Fock or JLM isovector optical-potential information, whereas soft variants were generated by taking the isovector density dependence proportional to the isoscalar one (Loan et al., 2011). This establishes CDM3Y-IVF1 as part of the stiff CDM3Y branch rather than the soft CDM3Y-ns branch.

2. Symmetry-energy expansion and higher-order coefficients

A central feature of CDM3Y-IVF1 is that it retains not only the quadratic symmetry energy but also higher even powers in the isospin asymmetry

$K_0$2

The nucleonic energy per particle is expanded as

$K_0$3

with

$K_0$4

Here $K_0$5 is the usual symmetry energy, while $K_0$6 and $K_0$7 encode higher-order isovector structure (Seif et al., 24 Jul 2025).

Around saturation density, the expansion is organized through the scaled variable

$K_0$8

The symmetric-matter sector is written as

$K_0$9

while the quadratic and quartic symmetry-energy contributions are expanded as

$$v^{D,Ex}_{00(01)}(\rho,r)=F_{0(1)}(\rho)\,v^{D,Ex}_{00(01)}(r),$$0

$$v^{D,Ex}_{00(01)}(\rho,r)=F_{0(1)}(\rho)\,v^{D,Ex}_{00(01)}(r),$$1

The coefficients $$v^{D,Ex}_{00(01)}(\rho,r)=F_{0(1)}(\rho)\,v^{D,Ex}_{00(01)}(r),$$2, $$v^{D,Ex}_{00(01)}(\rho,r)=F_{0(1)}(\rho)\,v^{D,Ex}_{00(01)}(r),$$3, $$v^{D,Ex}_{00(01)}(\rho,r)=F_{0(1)}(\rho)\,v^{D,Ex}_{00(01)}(r),$$4, $$v^{D,Ex}_{00(01)}(\rho,r)=F_{0(1)}(\rho)\,v^{D,Ex}_{00(01)}(r),$$5, $$v^{D,Ex}_{00(01)}(\rho,r)=F_{0(1)}(\rho)\,v^{D,Ex}_{00(01)}(r),$$6, and $$v^{D,Ex}_{00(01)}(\rho,r)=F_{0(1)}(\rho)\,v^{D,Ex}_{00(01)}(r),$$7 are, respectively, slope, curvature, skewness, kurtosis, fifth-order, and sixth-order density derivatives at $$v^{D,Ex}_{00(01)}(\rho,r)=F_{0(1)}(\rho)\,v^{D,Ex}_{00(01)}(r),$$8 (Seif et al., 24 Jul 2025).

The incompressibility of asymmetric matter at its own saturation density is written as

$$v^{D,Ex}_{00(01)}(\rho,r)=F_{0(1)}(\rho)\,v^{D,Ex}_{00(01)}(r),$$9

where $$F_{0(1)}(\rho)=C_{0(1)}\left(1+\alpha_{0(1)}e^{-\beta_{0(1)}\rho}-\gamma_{0(1)}\rho\right).$$0, $$F_{0(1)}(\rho)=C_{0(1)}\left(1+\alpha_{0(1)}e^{-\beta_{0(1)}\rho}-\gamma_{0(1)}\rho\right).$$1, and $$F_{0(1)}(\rho)=C_{0(1)}\left(1+\alpha_{0(1)}e^{-\beta_{0(1)}\rho}-\gamma_{0(1)}\rho\right).$$2 are isobaric incompressibility coefficients (Seif et al., 24 Jul 2025).

The recent higher-order analysis identifies two opposite classes of coefficients. EOS stiffening, in the sense of increasing high-density pressure, is correlated with increasing $$F_{0(1)}(\rho)=C_{0(1)}\left(1+\alpha_{0(1)}e^{-\beta_{0(1)}\rho}-\gamma_{0(1)}\rho\right).$$3 and $$F_{0(1)}(\rho)=C_{0(1)}\left(1+\alpha_{0(1)}e^{-\beta_{0(1)}\rho}-\gamma_{0(1)}\rho\right).$$4, and with less negative values of $$F_{0(1)}(\rho)=C_{0(1)}\left(1+\alpha_{0(1)}e^{-\beta_{0(1)}\rho}-\gamma_{0(1)}\rho\right).$$5, $$F_{0(1)}(\rho)=C_{0(1)}\left(1+\alpha_{0(1)}e^{-\beta_{0(1)}\rho}-\gamma_{0(1)}\rho\right).$$6, $$F_{0(1)}(\rho)=C_{0(1)}\left(1+\alpha_{0(1)}e^{-\beta_{0(1)}\rho}-\gamma_{0(1)}\rho\right).$$7, $$F_{0(1)}(\rho)=C_{0(1)}\left(1+\alpha_{0(1)}e^{-\beta_{0(1)}\rho}-\gamma_{0(1)}\rho\right).$$8, $$F_{0(1)}(\rho)=C_{0(1)}\left(1+\alpha_{0(1)}e^{-\beta_{0(1)}\rho}-\gamma_{0(1)}\rho\right).$$9, and $\{C_i,\alpha_i,\beta_i,\gamma_i\}$0. By contrast, increasing $\{C_i,\alpha_i,\beta_i,\gamma_i\}$1, $\{C_i,\alpha_i,\beta_i,\gamma_i\}$2, $\{C_i,\alpha_i,\beta_i,\gamma_i\}$3, $\{C_i,\alpha_i,\beta_i,\gamma_i\}$4, $\{C_i,\alpha_i,\beta_i,\gamma_i\}$5, and $\{C_i,\alpha_i,\beta_i,\gamma_i\}$6, or making $\{C_i,\alpha_i,\beta_i,\gamma_i\}$7 and $\{C_i,\alpha_i,\beta_i,\gamma_i\}$8 less negative, produces systematically opposite neutron-star trends relative to increasing $\{C_i,\alpha_i,\beta_i,\gamma_i\}$9 (Seif et al., 24 Jul 2025). This hierarchy is one of the main reasons CDM3Y-IVF1 is not reducible to a single-parameter incompressibility model.

Earlier work on CDM3Y-type neutron-star matter often relied on the parabolic approximation

$K_0$0

and treated higher-order terms as small in many applications (Loan et al., 2011). The more recent CDM3Y-IVF1 treatment shows that explicit fourth- and sixth-order sectors can systematically shift observable thresholds and crustal properties (Seif et al., 24 Jul 2025). A common misconception is therefore that only $K_0$1, $K_0$2, and $K_0$3 control neutron-star observables in this family; the recent results show that high-order isoscalar, isovector, and isobaric coefficients modify several trends in a non-negligible way.

3. Extension to cold $K_0$4-equilibrated $K_0$5 matter

For neutron-star cores, CDM3Y-IVF1 is extended to uniform cold matter composed of neutrons, protons, electrons, and muons. The total energy density is

$K_0$6

with

$K_0$7

Electrons and muons are treated as free relativistic Fermi gases with

$K_0$8

and the total pressure is

$K_0$9

The equilibrium conditions at $\rho$0 are

$\rho$1

together with charge neutrality

$\rho$2

These relations lead to an implicit equation for the proton fraction $\rho$3 (Seif et al., 24 Jul 2025).

In this framework, the isovector sector directly controls the composition of the stellar core. For CDM3Y-Paris-IVF1, the stiff symmetry energy produces higher proton content at intermediate densities than IVF0, but at very high densities the proton fraction decreases with increasing $\rho$4 (Seif et al., 24 Jul 2025). Quantitatively, for a maximum-mass star with soft EOS $\rho$5 MeV, the central proton fraction reaches $\rho$6 at $\rho$7, whereas for a very stiff EOS $\rho$8 MeV, $\rho$9 at $x_p=\rho_p/\rho$0 (Seif et al., 24 Jul 2025).

For a fixed $x_p=\rho_p/\rho$1, the central proton fraction increases with stellar mass; one quoted example rises from $x_p=\rho_p/\rho$2 at $x_p=\rho_p/\rho$3 to $x_p=\rho_p/\rho$4 at its own $x_p=\rho_p/\rho$5 (Seif et al., 24 Jul 2025). Higher-order coefficients modify these trends: more negative $x_p=\rho_p/\rho$6 and more positive $x_p=\rho_p/\rho$7 decrease $x_p=\rho_p/\rho$8, whereas larger $x_p=\rho_p/\rho$9, $$\begin{aligned} E_A(\rho,x_p) &= \frac{3\hbar^2 k_F^2}{20m} \Big[(2-2x_p)^{5/3} + (2x_p)^{5/3}\Big] \ &\quad + \frac{\rho}{2} \Big[ F_0(\rho) J^{D}_{00} + (1-2x_p)^2 F_1(\rho) J^{D}_{01} \Big] \ &\quad + \frac{\rho}{8} \int d\vec r \Big[ F_0(\rho) v^{Ex}_{00}(r) B_0^2(r,x_p) + F_1(\rho) v^{Ex}_{01}(r) B_1^2(r,x_p) \Big], \end{aligned}$$0, $$\begin{aligned} E_A(\rho,x_p) &= \frac{3\hbar^2 k_F^2}{20m} \Big[(2-2x_p)^{5/3} + (2x_p)^{5/3}\Big] \ &\quad + \frac{\rho}{2} \Big[ F_0(\rho) J^{D}_{00} + (1-2x_p)^2 F_1(\rho) J^{D}_{01} \Big] \ &\quad + \frac{\rho}{8} \int d\vec r \Big[ F_0(\rho) v^{Ex}_{00}(r) B_0^2(r,x_p) + F_1(\rho) v^{Ex}_{01}(r) B_1^2(r,x_p) \Big], \end{aligned}$$1, $$\begin{aligned} E_A(\rho,x_p) &= \frac{3\hbar^2 k_F^2}{20m} \Big[(2-2x_p)^{5/3} + (2x_p)^{5/3}\Big] \ &\quad + \frac{\rho}{2} \Big[ F_0(\rho) J^{D}_{00} + (1-2x_p)^2 F_1(\rho) J^{D}_{01} \Big] \ &\quad + \frac{\rho}{8} \int d\vec r \Big[ F_0(\rho) v^{Ex}_{00}(r) B_0^2(r,x_p) + F_1(\rho) v^{Ex}_{01}(r) B_1^2(r,x_p) \Big], \end{aligned}$$2, $$\begin{aligned} E_A(\rho,x_p) &= \frac{3\hbar^2 k_F^2}{20m} \Big[(2-2x_p)^{5/3} + (2x_p)^{5/3}\Big] \ &\quad + \frac{\rho}{2} \Big[ F_0(\rho) J^{D}_{00} + (1-2x_p)^2 F_1(\rho) J^{D}_{01} \Big] \ &\quad + \frac{\rho}{8} \int d\vec r \Big[ F_0(\rho) v^{Ex}_{00}(r) B_0^2(r,x_p) + F_1(\rho) v^{Ex}_{01}(r) B_1^2(r,x_p) \Big], \end{aligned}$$3, $$\begin{aligned} E_A(\rho,x_p) &= \frac{3\hbar^2 k_F^2}{20m} \Big[(2-2x_p)^{5/3} + (2x_p)^{5/3}\Big] \ &\quad + \frac{\rho}{2} \Big[ F_0(\rho) J^{D}_{00} + (1-2x_p)^2 F_1(\rho) J^{D}_{01} \Big] \ &\quad + \frac{\rho}{8} \int d\vec r \Big[ F_0(\rho) v^{Ex}_{00}(r) B_0^2(r,x_p) + F_1(\rho) v^{Ex}_{01}(r) B_1^2(r,x_p) \Big], \end{aligned}$$4 and less negative $$\begin{aligned} E_A(\rho,x_p) &= \frac{3\hbar^2 k_F^2}{20m} \Big[(2-2x_p)^{5/3} + (2x_p)^{5/3}\Big] \ &\quad + \frac{\rho}{2} \Big[ F_0(\rho) J^{D}_{00} + (1-2x_p)^2 F_1(\rho) J^{D}_{01} \Big] \ &\quad + \frac{\rho}{8} \int d\vec r \Big[ F_0(\rho) v^{Ex}_{00}(r) B_0^2(r,x_p) + F_1(\rho) v^{Ex}_{01}(r) B_1^2(r,x_p) \Big], \end{aligned}$$5 increase the central proton fraction, opposite to increasing $$\begin{aligned} E_A(\rho,x_p) &= \frac{3\hbar^2 k_F^2}{20m} \Big[(2-2x_p)^{5/3} + (2x_p)^{5/3}\Big] \ &\quad + \frac{\rho}{2} \Big[ F_0(\rho) J^{D}_{00} + (1-2x_p)^2 F_1(\rho) J^{D}_{01} \Big] \ &\quad + \frac{\rho}{8} \int d\vec r \Big[ F_0(\rho) v^{Ex}_{00}(r) B_0^2(r,x_p) + F_1(\rho) v^{Ex}_{01}(r) B_1^2(r,x_p) \Big], \end{aligned}$$6 (Seif et al., 24 Jul 2025).

The contrast with soft CDM3Y variants is important. In the older stiff-versus-soft classification, stiff CDM3Y-n interactions produce a symmetry energy that rises with density and correspondingly large proton fractions, while soft CDM3Y-ns variants can drive the proton fraction down to a few percent and eventually toward pure neutron matter at high density (Loan et al., 2011). CDM3Y-IVF1 belongs to the stiff class in this sense, but the higher-order analysis shows that even within the stiff class the composition is not monotonic in all stiffness indicators.

4. Core-crust transition and crustal structure

The core-crust transition in CDM3Y-IVF1 is determined by the thermodynamic instability of uniform $$\begin{aligned} E_A(\rho,x_p) &= \frac{3\hbar^2 k_F^2}{20m} \Big[(2-2x_p)^{5/3} + (2x_p)^{5/3}\Big] \ &\quad + \frac{\rho}{2} \Big[ F_0(\rho) J^{D}_{00} + (1-2x_p)^2 F_1(\rho) J^{D}_{01} \Big] \ &\quad + \frac{\rho}{8} \int d\vec r \Big[ F_0(\rho) v^{Ex}_{00}(r) B_0^2(r,x_p) + F_1(\rho) v^{Ex}_{01}(r) B_1^2(r,x_p) \Big], \end{aligned}$$7-equilibrated matter. The relevant quantity is the generalized compressibility at fixed lepton chemical potential,

$$\begin{aligned} E_A(\rho,x_p) &= \frac{3\hbar^2 k_F^2}{20m} \Big[(2-2x_p)^{5/3} + (2x_p)^{5/3}\Big] \ &\quad + \frac{\rho}{2} \Big[ F_0(\rho) J^{D}_{00} + (1-2x_p)^2 F_1(\rho) J^{D}_{01} \Big] \ &\quad + \frac{\rho}{8} \int d\vec r \Big[ F_0(\rho) v^{Ex}_{00}(r) B_0^2(r,x_p) + F_1(\rho) v^{Ex}_{01}(r) B_1^2(r,x_p) \Big], \end{aligned}$$8

which can be expressed as

$$\begin{aligned} E_A(\rho,x_p) &= \frac{3\hbar^2 k_F^2}{20m} \Big[(2-2x_p)^{5/3} + (2x_p)^{5/3}\Big] \ &\quad + \frac{\rho}{2} \Big[ F_0(\rho) J^{D}_{00} + (1-2x_p)^2 F_1(\rho) J^{D}_{01} \Big] \ &\quad + \frac{\rho}{8} \int d\vec r \Big[ F_0(\rho) v^{Ex}_{00}(r) B_0^2(r,x_p) + F_1(\rho) v^{Ex}_{01}(r) B_1^2(r,x_p) \Big], \end{aligned}$$9

Uniform matter is stable for $npe\mu$00. The transition density $npe\mu$01 and transition pressure $npe\mu$02 are defined where $npe\mu$03 first vanishes upon decreasing density (Seif et al., 24 Jul 2025).

Within the CDM3Y-Paris and CDM3Y-Reid families, $npe\mu$04 increases with increasing symmetry energy at saturation $npe\mu$05. For nearly equal $npe\mu$06, $npe\mu$07 decreases as $npe\mu$08 increases, while $npe\mu$09 and the transition proton fraction $npe\mu$10 vary only weakly with $npe\mu$11 and $npe\mu$12, at the level of $npe\mu$13 in absolute fractions (Seif et al., 24 Jul 2025). Along the CDM3Y-Paris-IVF1 sequence, increasing $npe\mu$14 raises both $npe\mu$15 and $npe\mu$16, and also increases $npe\mu$17 more mildly (Seif et al., 24 Jul 2025).

The higher-order coefficients introduce a more differentiated pattern. Increasing $npe\mu$18 or $npe\mu$19, and making $npe\mu$20, $npe\mu$21, $npe\mu$22, $npe\mu$23, $npe\mu$24, and $npe\mu$25 less negative, raises $npe\mu$26 and $npe\mu$27 and slightly increases $npe\mu$28. Conversely, increasing $npe\mu$29, $npe\mu$30, $npe\mu$31, $npe\mu$32, $npe\mu$33, $npe\mu$34, or making $npe\mu$35 and $npe\mu$36 less negative, modifies $npe\mu$37 and $npe\mu$38 in the direction opposite to the change in $npe\mu$39 (Seif et al., 24 Jul 2025). This is one of the clearest demonstrations that high-order coefficients cannot be collapsed into a single effective stiffness label.

A frequent oversimplification is that a stiffer EOS always implies a thicker crust. The CDM3Y-IVF1 results are more specific. A higher transition pressure is associated with a thicker crust and larger radius for a given mass, but a larger transition density implies that the uniform core extends farther outward, which can reduce fractional crust thickness in very massive configurations (Seif et al., 24 Jul 2025). This distinction becomes central once rotational observables are considered.

Earlier CDM3Y-based neutron-star studies combined the uniform core EOS with crust models such as the compressible liquid drop model or the Douchin-Haensel inner crust, mainly to assess global stellar properties and the impact of soft versus stiff symmetry energy [(Loan et al., 2011); (Seif et al., 15 Jul 2025)]. The recent CDM3Y-IVF1 analysis brings the transition itself into the foreground by connecting it directly to the high-order symmetry-energy hierarchy (Seif et al., 24 Jul 2025).

5. Moment of inertia, crustal fraction, and compactness

Static stellar configurations are obtained from the Tolman-Oppenheimer-Volkoff equations,

$npe\mu$40

$npe\mu$41

For slowly rotating stars, the total moment of inertia $npe\mu$42 is calculated in the Hartle-Thorne formalism through the frame-dragging function $npe\mu$43 or, equivalently, through the accumulated moment-of-inertia function $npe\mu$44 (Seif et al., 24 Jul 2025).

The crust is defined as the region between the core radius $npe\mu$45 at density $npe\mu$46 and the stellar radius $npe\mu$47. Its contribution to the moment of inertia is

$npe\mu$48

The same formalism yields the fractional crust thickness $npe\mu$49 with $npe\mu$50 (Seif et al., 24 Jul 2025).

For the CDM3Y-Paris-IVF1 family, the total moment of inertia $npe\mu$51 increases with increasing $npe\mu$52 at fixed stellar mass, and this sensitivity becomes stronger with increasing mass. The maximum mass $npe\mu$53 also increases with stiffness; soft EOS with $npe\mu$54 MeV may fail to support $npe\mu$55 (Seif et al., 24 Jul 2025). The crust thickness $npe\mu$56 and its fractional value $npe\mu$57 are strongly anti-correlated with mass, and their dependence on $npe\mu$58 at fixed mass is weak, weakening further for very stiff EOS (Seif et al., 24 Jul 2025).

The behavior of the crustal moment-of-inertia fraction is more nuanced than the behavior of the total moment of inertia. For fixed mass, increasing $npe\mu$59 generally increases $npe\mu$60 and can also increase $npe\mu$61. For the maximum-mass configuration supported by a given EOS, however, increasing $npe\mu$62 yields a larger $npe\mu$63 but a very thin crust, causing both $npe\mu$64 and $npe\mu$65 to decrease (Seif et al., 24 Jul 2025). This distinction directly addresses a common misconception: within CDM3Y-IVF1, EOS stiffening does not have a single universal effect on the crustal fraction; the sign of the trend depends on whether the comparison is made at fixed mass or along the maximum-mass sequence.

The glitch-motivated criterion

$npe\mu$66

is satisfied for neutron-star masses up to $npe\mu$67 in this family (Seif et al., 24 Jul 2025). The same work states that both total $npe\mu$68 and crustal fraction $npe\mu$69 show approximately linear decrease with compactness

$npe\mu$70

and that the glitch condition implies $npe\mu$71 and $npe\mu$72 within the CDM3Y-Paris-IVF1 family (Seif et al., 24 Jul 2025).

The higher-order coefficient dependence again separates into two groups. Increasing $npe\mu$73 or $npe\mu$74, and reducing the magnitude of negative $npe\mu$75, $npe\mu$76, $npe\mu$77, $npe\mu$78, $npe\mu$79, and $npe\mu$80, increases $npe\mu$81 and $npe\mu$82 for a given mass, but decreases $npe\mu$83 and $npe\mu$84 for the maximum-mass configuration. Increasing $npe\mu$85, $npe\mu$86, $npe\mu$87, $npe\mu$88, $npe\mu$89, $npe\mu$90, or making $npe\mu$91 and $npe\mu$92 less negative, reduces $npe\mu$93 and $npe\mu$94 for fixed mass, yet increases $npe\mu$95 and $npe\mu$96 for the maximum-mass configuration (Seif et al., 24 Jul 2025). The paper explicitly notes that these patterns are consistent with independent constraints from Skyrme-based meta-modeling and relativistic mean-field studies.

The broader semi-microscopic CDM3Y program gives compatible global scales. In an earlier stiff-versus-soft survey, the stiff CDM3Y3, CDM3Y4, and CDM3Y6 interactions yielded maximum masses of $npe\mu$97, $npe\mu$98, and $npe\mu$99, respectively, with radii near $K_0$00 km and moments of inertia increasing with stiffness, whereas the soft CDM3Y3s, CDM3Y4s, and CDM3Y6s gave smaller maximum masses and systematically lower moments of inertia (Loan et al., 2011). This older dichotomy provides the global background against which the finer IVF1 higher-order trends should be read.

6. Direct Urca thresholds, adiabatic response, and astrophysical constraints

The nucleonic direct Urca process,

$K_0$01

requires the proton fraction to exceed the threshold

$K_0$02

When muons are absent, $K_0$03; when muons are present, $K_0$04 (Seif et al., 24 Jul 2025).

For CDM3Y-Paris-IVF1 over $K_0$05 MeV, the threshold proton fraction grows weakly from $K_0$06 to $K_0$07, the threshold density increases from $K_0$08 to $K_0$09, and the threshold pressure rises strongly from $K_0$10 to $K_0$11 MeV fm$K_0$12. For still stiffer EOS with $K_0$13 MeV, the proton fraction may never reach the direct-Urca threshold before the central density is exceeded, so direct Urca does not occur (Seif et al., 24 Jul 2025). IVF1 yields lower $K_0$14, $K_0$15, and $K_0$16 than IVF0, while CDM3Y-Reid-IVF1 behaves similarly but typically with slightly higher DU thresholds than Paris-IVF1 (Seif et al., 24 Jul 2025).

The dependence on the higher-order hierarchy is parallel to the crustal analysis. Increasing $K_0$17 or $K_0$18, and making $K_0$19, $K_0$20, $K_0$21, $K_0$22, $K_0$23, and $K_0$24 less negative, raises $K_0$25, $K_0$26, and $K_0$27. Increasing $K_0$28, $K_0$29, $K_0$30, $K_0$31, $K_0$32, $K_0$33, or making $K_0$34 and $K_0$35 less negative, lowers these thresholds (Seif et al., 24 Jul 2025). A plausible implication is that modest changes in high-order symmetry coefficients can move DU cooling from the domain of only very massive stars into densities realized by lighter objects.

The local stiffness of cold matter is characterized by the adiabatic index

$K_0$36

For non-rotating stars with CDM3Y-Paris-IVF1, the central adiabatic index $K_0$37 decreases with increasing stellar mass. For $K_0$38 and $K_0$39 stars, $K_0$40 exhibits a minimum at intermediate $K_0$41 and then increases with further stiffening. Along the maximum-mass sequence, $K_0$42 decreases with $K_0$43 up to about $K_0$44 MeV and increases beyond this value (Seif et al., 24 Jul 2025). Less negative $K_0$45 increases $K_0$46, whereas less negative $K_0$47 or larger $K_0$48 decreases it, mirroring the two opposite higher-order coefficient classes identified earlier (Seif et al., 24 Jul 2025).

Recent CDM3Y studies have also connected the same semi-microscopic EOS family to tidal observables. For CDM3Y parameterizations with $K_0$49 MeV, the predicted canonical radii are $K_0$50 km for Paris and $K_0$51 km for Reid, while the corresponding maximum masses are $K_0$52 and $K_0$53 (Seif et al., 15 Jul 2025). The same work reports $K_0$54 to $K_0$55 for Paris and $K_0$56 to $K_0$57 for Reid when $K_0$58 varies from 200 to 330 MeV, and states that the $K_0$59 sets reproduce most NICER and GW-inferred mass-radius constraints (Seif et al., 15 Jul 2025). It also gives an approximately EOS-insensitive relation for $K_0$60,

$K_0$61

with coefficient of determination $K_0$62 across the CDM3Y-Paris and Reid sets with $K_0$63 MeV (Seif et al., 15 Jul 2025).

Taken together, these results place CDM3Y-IVF1-type EOSs in a constrained but nontrivial region of parameter space. They must be stiff enough to sustain heavy pulsars, yet their higher-order symmetry structure controls whether the same EOS produces a sufficiently large crustal moment-of-inertia fraction, permits or suppresses direct Urca cooling, and shifts tidal deformabilities within observationally allowed bands. The recurring theme is that the macroscopic neutron-star phenomenology of CDM3Y-IVF1 is governed not only by $K_0$64 and $K_0$65, but by the full coupled hierarchy of isoscalar, isovector, and isobaric coefficients carried by the Hartree-Fock interaction (Seif et al., 24 Jul 2025).