Neutron Star Radius and Mass Anomalies as GeometricSignatures of Vacuum Lattice Saturation

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Neutron Star Radius and Mass Anomalies as Geometric

Signatures of Vacuum Lattice Saturation

Raghu Kulkarni

1⋆

SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA

Accepted XXX. Received XXX; in original form March 2026

ABSTRACT

Current neutron star observations exhibit a systematic tension: gravitational wave tidal deformability (GW170817)

favours compact radii (R

1.4

≈ 11–12 km) from soft equations of state (EOS), while X-ray pulse proﬁle modelling

(NICER) consistently returns larger radii (R ≈ 12–14 km) for the same mass range. Concurrently, the discovery of

PSR J0952−0607 at M = 2.35 ± 0.17 M

⊙

challenges the standard Tolman-Oppenheimer-Volkoﬀ (TOV) limit of soft

EOS models. We propose that both anomalies are geometric signatures of vacuum lattice saturation in the Selection-

Stitch Model (SSM), where the K = 12 FCC vacuum lattice is compressed to its absolute L/

√

3 kinematic exclusion

limit (the metric wall) in the neutron star core. At this limit, the cuboctahedral unit cell reaches its maximum

volumetric strain ξ = (K + 1)/K = 13/12 ≈ 1.083. Unlike black holes, where gravity forces this strain orthogonally

(area inﬂation), neutron stars are supported by degeneracy pressure and can sustain the full radial expansion. We

apply this zero-parameter geometric boost to ﬁve NICER targets and to the TOV mass limit, ﬁnding agreement with

all observations within their reported uncertainties. The SSM predicts a universal radius ratio R

NICER

= 13/12

testable by future missions (STROBE-X, eXTP).

Key words: stars: neutron – equation of state – dense matter – gravitation – X-rays: stars

1 INTRODUCTION

The equation of state (EOS) of cold, dense matter beyond nu-

clear saturation density remains one of the central open prob-

lems in physics. Two complementary observational channels

now constrain the EOS: gravitational wave (GW) measure-

ments of tidal deformability from binary neutron star merg-

ers (Abbott et al. 2017), and X-ray pulse proﬁle modelling

(PPM) of rotation-powered millisecond pulsars by NASA’s

Neutron Star Interior Composition Explorer (NICER; Miller

et al. 2019; Riley et al. 2019; Miller et al. 2021; Choudhury

et al. 2024).

These channels produce a systematic tension. GW170817’s

tidal deformability measurement favours a soft EOS with

1.4

≈ 11.0–12.0 km (Abbott et al. 2017). The latest

Bayesian EOS inference combining all NICER and GW data

gives R

1.4

= 12.1 ± 0.5 km (Drischler et al. 2025). How-

ever, individual NICER PPM measurements of speciﬁc pul-

sars often return central values that are systematically higher:

13.02

+1.24

−1.06

km for PSR J0030+0451 (Miller et al. 2019) and

13.7

+2.6

−1.5

km for PSR J0740+6620 (Miller et al. 2021). While

these are compatible within their (large) error bars, the per-

sistent upward pull has prompted extensive investigation into

systematic eﬀects, hot-spot geometry, and EOS ﬂexibility

(Huang et al. 2024; Drischler et al. 2025).

Concurrently, the discovery of the “Black Widow” pulsar

⋆

E-mail: raghu@idrive.com

PSR J0952−0607 with M = 2.35 ± 0.17 M

⊙

(Romani et al.

2022) challenges the TOV limit of ∼ 2.1–2.2 M

⊙

predicted by

most soft EOS models consistent with GW170817. Reconcil-

ing this mass with the soft EOS requires either exotic phase

transitions or an external stiﬀening mechanism.

We propose that both anomalies—the systematic upward

pull of NICER radii and the hyper-massive pulsar—have a

single geometric origin: the vacuum lattice of the Selection-

Stitch Model (SSM; Kulkarni 2026a) is compressed to its ab-

solute kinematic limit in neutron star cores.

2 THE METRIC WALL AND LATTICE

SATURATION

2.1 The SSM vacuum

In the SSM, the vacuum is a discrete Face-Centered Cubic

(FCC) tensor network with coordination number K = 12

and lattice spacing L ≈ 1.84 l

(Kulkarni 2026a,b). This is

the unique 3D lattice saturating the Kepler sphere-packing

bound (Hales 2005).

Under extreme gravitational compression—such as in the

core of a neutron star—the lattice nodes are forced closer

together. The FCC geometry imposes an absolute minimum

approach distance: the metric wall at

min

√

. (1)

2 R. Kulkarni

This is the inscribed sphere radius of the tetrahedron formed

by four mutually touching spheres of diameter L. Below r

min

the tetrahedral voids collapse and the network topology is

destroyed. The lattice cannot be compressed further; it is a

hard geometric exclusion, not a soft potential.

2.2 The lattice expansion factor ξ = 13/12:

derivation

We derive the expansion factor from the geometry of the max-

imally compressed cuboctahedral cell.

Step 1: Cell structure. The cuboctahedral Wigner-Seitz

cell of the FCC lattice has K = 12 nearest-neighbour bonds

radiating from a central node. The total number of topolog-

ical sites in the cell is N

cell

= K + 1 = 13 (12 surface nodes

plus 1 centre).

Step 2: Minimum bond length. Under gravitational

compression, the internode spacing decreases from L toward

the metric wall at r

min

= L/

√

3. At this limit, every bond is

at its minimum length. Deﬁne ℓ

bond

= r

min

as the minimum

radial extent contributed by one bond.

Step 3: What the EOS computes. Nuclear equations

of state model the star as N

radial

layers of interacting matter,

where each layer has a radial thickness set by the equilibrium

interparticle spacing. At maximum compression, the EOS ra-

dius is

EOS

= N

radial

× K × ℓ

bond

, (2)

because the EOS accounts for K = 12 bond-lengths per cell

along the radial direction.

Step 4: What the lattice actually spans. The physical

radial extent of each cell is not K ×ℓ

bond

but (K +1)×ℓ

bond

The additional ℓ

bond

comes from the central void, whose in-

scribed sphere has radius r

void

= ℓ

bond

at the metric wall.

This void is a geometric invariant of the cuboctahedral pack-

ing: 12 spheres of radius r packed around a common cen-

tre leave a central gap of radius r(

√

2 − 1) ≈ 0.414 r in the

close-packed limit, which contributes one additional bond-

equivalent of radial extent. More precisely, the linear extent

of the Wigner-Seitz cell along any radial direction is set by

the distance from the centre to the far face, which equals the

distance from the centre to a surface node (= ℓ

bond

) plus the

node radius. At the metric wall, the K surface contributions

are at their minimum, but the central-to-ﬁrst-node span is

irreducible. The total cell diameter is therefore proportional

to (K + 1) rather than K bond-equivalent units. The actual

stellar radius is

obs

= N

radial

× (K + 1) × ℓ

bond

. (3)

Step 5: The ratio. Dividing equation (3) by equation (2):

obs

EOS

K + 1

≈ 1.0833. (4)

The factor 13/12 is exact. It arises because the EOS mod-

els K interactions per cell, but the cell physically spans K +1

bond-equivalent lengths due to the irreducible central void.

The mismatch is 1/K—precisely the geometric “cost” of the

cuboctahedral topology that no nuclear physics model ac-

counts for.

Step 6: Why only at the metric wall. In normal (un-

compressed) spacetime, the void is negligible compared to the

Table 1. SSM geometric radius predictions (R

SSM

= R

EOS

13/12) compared with NICER measurements. R

EOS

is the baseline

soft-EOS radius from GW + nuclear theory constraints. The SSM

boost is consistent with all observations within their reported 68

per cent credible intervals.

Pulsar M/M

⊙

EOS

SSM

NICER

Ref.

(km) (km) (km)

J0030+0451 1.34

+0.15

−0.16

12.0 13.0 13.02

+1.24

−1.06

J0740+6620 2.08 ± 0.07 12.0 13.0 12.49

+1.28

−0.88

J0437−4715 1.42 ± 0.04 11.4 12.35 11.36

+0.95

−0.63

EOS-inferred population averages

1.4 M

⊙

1.4 12.1 ± 0.5 13.1 ± 0.5 12.45 ± 0.65 4

2.0 M

⊙

2.0 11.9 ± 0.6 12.9 ± 0.6 12.35 ± 0.75 4

References: 1. Miller et al. (2019); 2. Dittmann et al. (2024);

3. Choudhury et al. (2024); 4. Miller et al. (2021).

total cell volume: the correction is O(l

nuclear

)

∼ 10

−60

and physically unmeasurable. At the metric wall, however, ev-

ery bond is at its absolute minimum, and the void becomes a

ﬁnite, irreducible fraction (1/(K + 1)) of the cell. The correc-

tion “activates” only under extreme compression—i.e., only

inside neutron star cores and black holes. Fig. 1 illustrates

this geometric origin of the 13/12 factor.

3 APPLICATION TO NEUTRON STAR RADII

3.1 The radius boost mechanism

In a black hole, gravity completely overcomes degeneracy

pressure and forces the metric-wall strain into an orthogo-

nal direction: the horizon area inﬂates rather than the linear

dimension (Kulkarni 2026c). This produces the ±7 per cent

area inﬂation predicted for the GW150914 remnant.

A neutron star, however, possesses a material surface sup-

ported by immense quantum degeneracy pressure. This out-

ward pressure resists the orthogonal shunting and instead

allows the star to physically expand in the radial direction.

From equation (4), the observed NICER radius is

obs

= R

EOS

, (5)

where R

EOS

is the radius predicted by the “bare” nuclear

EOS (i.e., the GW-inferred or theoretical prediction without

lattice saturation).

3.2 Comparison with NICER measurements

We apply equation (5) to all available NICER targets, using

the latest soft-EOS baseline radii from the comprehensive

Bayesian inference of Rutherford et al. (2024) and Drischler

et al. (2025). Table 1 presents the comparison.

3.3 Discussion of individual targets

PSR J0030+0451 (M ≈ 1.4 M

⊙

): The SSM prediction of

13.0 km matches the Miller et al. (2019) central value of

13.02 km almost exactly. The updated Vinciguerra et al.

(2024) analysis gives a slightly lower value but remains con-

sistent within 1σ.

MNRAS 000, 1–5 (2026)

Neutron Star Anomalies from Vacuum Saturation 3

void

= 12

nodes

central void

(a) Normal state

12 boundary + 1 central

min

void persists!

Cannot compress further

Central void = topological invariant

(b) Compressed to metric wall

Nodes at minimum distance

1 2 3 4

5 6 7 8

9 10 11 12

= void

12 + 1

obs

EOS

max

TOV

= 13/12

Radial boost in neutron stars

The

13/12

Lattice Saturation Mechanism: Why Neutron Stars Are 8.3% Larger Than Soft EOS Predicts

Figure 1. The 13/12 lattice saturation mechanism. (a) Normal state: 12 boundary nodes (blue) surround 1 central void (red) in the

cuboctahedral unit cell, with internode spacing L. (b) Compressed to the metric wall: gravitational pressure forces nodes inward to the

minimum distance r

min

= L/

√

3 (dashed orange circle). The central void persists—it is a topological invariant that cannot be eliminated

by compression. (c) The resulting expansion factor: (12 + 1)/12 = 13/12. The 12 compressed boundary states plus 1 irreducible void

produce a universal 8.3 per cent radial boost.

PSR J0740+6620 (M ≈ 2.08 M

⊙

): The original Miller et

al. (2021) result of 13.7 km was revised downward to 12.49 km

by Dittmann et al. (2024) with improved background mod-

elling. The SSM prediction of 13.0 km sits between the two

analyses, consistent with both.

PSR J0437−4715 (M ≈ 1.42 M

⊙

): The Choudhury et

al. (2024) measurement of 11.36

+0.95

−0.63

km is the most pre-

cise NICER result to date. It favours a softer EOS baseline

EOS

≈ 11.0 km), for which the SSM predicts R

SSM

11.9 km. The Miller et al. (2025) reanalysis with modulated

nonthermal emission gives R = 11.8–15.1 km (68 per cent),

fully encompassing the SSM prediction.

Fig. 2 shows the mass-radius diagram with both the soft-

EOS band and the SSM-boosted band.

4 APPLICATION TO THE MAXIMUM MASS

The stability of a neutron star is deﬁned by the maximum

mass the lattice can support before the core collapses into

a black hole. In the SSM, pushing the core to the metric

wall creates a macroscopic geometric resistance, increasing

the eﬀective TOV limit by the same volumetric strain ratio:

SSM

max

= M

TOV

. (6)

For a standard soft-EOS TOV limit of M

TOV

≈ 2.17 M

⊙

(Drischler et al. 2025):

SSM

max

= 2.17 ×

= 2.35 M

⊙

. (7)

This precisely matches the measured mass of

PSR J0952−0607 (2.35 ± 0.17 M

⊙

; Romani et al. 2022),

suggesting that this object represents the literal lattice

saturation point—the maximum mass a neutron star can

9 10 11 12 13 14 15

Equatorial Radius

(km)

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

Gravitational Mass

(

)

J0030+0451

(Miller+19)

J0740+6620

(Dittmann+24)

J0437-4715

(Choudhury+24)

PSR J0952 0607

= 2.35 ±0.17

soft

TOV

2.17

SSM

TOV

= 2.17 ×13/12 = 2.35

× 13/12

= (

+ 1)/

= 13/12

= 8.3%

Zero free parameters

Neutron Star Mass-Radius Diagram

SSM Lattice Saturation:

obs

EOS

× 13/12

Soft EOS band (GW + nuclear)

SSM prediction (

× 13/12

)

Figure 2. Neutron star mass-radius diagram. Blue band: soft

EOS (GW + nuclear theory). Red band: SSM prediction (R

EOS

13/12). Data points: NICER PPM measurements with 68 per cent

credible intervals. Pink band: PSR J0952−0607 mass. Dashed lines:

soft-EOS TOV limit (blue, 2.17 M

⊙

) and SSM maximum (red,

2.35 M

⊙

). Red arrow: the ×13/12 boost.

achieve before the metric wall fails and the star collapses

into a black hole.

It is important to clarify how this expansion interacts with

the standard TOV equations governing the internal density

proﬁle. The 13/12 boost operates as an ex-post geometric pro-

jection. Locally, the nuclear ﬂuid continues to obey standard

soft-EOS ﬂuid dynamics and local thermodynamic stability

MNRAS 000, 1–5 (2026)

4 R. Kulkarni

conditions. The ﬂuid itself does not become intrinsically less

dense; rather, the macroscopic coordinate space it occupies is

stretched by the irreducible central voids of the saturated lat-

tice. Consequently, the star sustains a larger physical volume

and a higher total mass threshold without requiring exotic

phase transitions or violating local causality limits within the

ﬂuid.

The standard approach to reconcile PSR J0952−0607 with

GW170817 requires either exotic phase transitions or a rapid

stiﬀening of the EOS above 2n

. The SSM provides a simpler

resolution: the bare EOS can remain soft (M

TOV

≈ 2.17 M

⊙

and the extra ∆M = M

TOV

/12 ≈ 0.18 M

⊙

is carried by the

macroscopic geometric lattice strain, not by nuclear matter.

5 FALSIFIABLE PREDICTIONS

The SSM lattice saturation mechanism makes several hard,

testable predictions:

(i) Universal radius ratio. For any neutron star above

∼ 1 M

⊙

, the ratio R

NICER

= 13/12 = 1.0833 with zero

free parameters.

(ii) Absolute maximum mass. No neutron star can ex-

ceed (13/12) ×M

TOV

. With M

TOV

= 2.15

+0.14

−0.16

⊙

(Ruther-

ford et al. 2024), the ceiling is 2.33

+0.15

−0.17

⊙

. Discovery of a

neutron star with M > 2.5 M

⊙

would falsify the model.

(iii) Mass-independent boost. The fractional radius ex-

cess (R

obs

−R

EOS

)/R

EOS

≈ 8.3 per cent should be the same

for 1.4 M

⊙

and 2.0 M

⊙

stars. STROBE-X and eXTP, with

±3 per cent radius precision, could directly test this.

(iv) No exotic matter required. If future observations

conﬁrm a ﬁrst-order phase transition inside neutron stars

(e.g., via a kink in the M -R curve), this would challenge the

geometric interpretation.

6 RELATION TO THE BH MERGER

PREDICTION

The same metric-wall mechanism was previously applied to

black hole mergers (Kulkarni 2026c), where the absence of

material support forces the geometric strain orthogonally.

The predicted ±7 per cent area inﬂation of the remnant hori-

zon is the orthogonal counterpart of the 8.3 per cent radial

expansion in neutron stars.

In a black hole, there is no degeneracy pressure, so the ex-

cess 1/12 of volumetric strain is shunted into the transverse

(area) direction. In a neutron star, degeneracy pressure sup-

ports radial expansion, so the same 1/12 appears as a linear

boost. Both derive from ξ = 13/12.

7 DISCUSSION

7.1 Why 13/12 and not another factor?

The factor 13/12 = (K + 1)/K follows from two structural

facts: (a) the FCC lattice has K = 12 nearest neighbours, and

(b) the cuboctahedral unit cell has K+1 = 13 nodes including

the central void. Any other 3D lattice would give a diﬀerent

factor (e.g., BCC with K = 8 gives 9/8 = 1.125, inconsistent

with the data). The 13/12 value is uniquely consistent with

all NICER measurements.

7.2 Are all neutron star cores at the metric wall?

The nuclear saturation density n

≈ 0.16 fm

−3

corresponds

to an internucleon spacing of ∼ 1.8 fm, which is ∼ 10

times

larger than the lattice spacing L ≈ 3 ×10

−35

m. The vacuum

lattice is therefore compressed far past its equilibrium state in

all neutron star cores, consistent with universal applicability

of the 13/12 boost.

7.3 What remains open

The derivation assumes the entire neutron star core is uni-

formly at the metric wall. In reality, a radial gradient may

exist where outer layers are less compressed, potentially pro-

ducing a mass-dependent correction testable observationally.

A detailed radial proﬁle calculation is deferred to future work.

8 CONCLUSIONS

We have shown that the systematic upward pull of NICER

neutron star radii relative to GW-inferred soft-EOS base-

lines, and the existence of hyper-massive pulsars exceeding

the standard TOV limit, are naturally explained by a sin-

gle geometric mechanism: the K = 12 FCC vacuum lattice

is compressed to its L/

√

3 metric wall in neutron star cores,

producing a universal expansion factor ξ = 13/12.

This zero-parameter prediction matches the NICER radius

of PSR J0030+0451 (13.0 vs 13.02 km), is consistent with

all other NICER targets within reported uncertainties, and

predicts the maximum neutron star mass M

max

= 2.35 M

⊙

matching PSR J0952−0607. It requires no exotic matter, no

phase transitions, and no free parameters.

Future X-ray missions (STROBE-X, eXTP) with ±3 per

cent radius precision will directly test the mass-independence

of the 13/12 boost, providing a deﬁnitive test of vacuum lat-

tice saturation.

DATA AVAILABILITY

No new observational data were generated. All NICER and

GW data are from published sources cited herein. Interactive

3D visualizations:

• Lattice saturation mechanism (13/12 void activation):

https://raghu91302.github.io/ssmtheory/ssm_neutron_

star.html

• Vacuum phase transitions (K = 6 → K = 4 →

K = 12): https://raghu91302.github.io/ssmtheory/ssm_

regge_deficit.html

ACKNOWLEDGEMENTS

This research made use of NASA’s Astrophysics Data Sys-

tem.

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