A Zero-Parameter Derivation of the Scalar
Spectral Index (n
s
)
from the Regge Deficit Angle of a
Tetrahedral Vacuum Phase Transition
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
raghu@idrive.com
March 2026
Abstract
Standard inflationary cosmology [1, 2] requires a scalar inflaton field with a
tuned potential to reproduce the observed scalar spectral index n
s
≈ 0.965. We
derive n
s
from pure geometry without free parameters. In the Selection-Stitch
Model (SSM) [4], the early universe undergoes a thermodynamic phase transition
from an amorphous K = 4 tetrahedral network to a crystalline K = 12 Face-
Centered Cubic (FCC) lattice. The pre-transition phase is a 3D simplicial complex
composed of irregular tetrahedra. At the crystallization front, the local geometry
regularizes: five regular tetrahedra pack around each shared edge, the maximum
permitted by the dihedral angle constraint. This packing leaves a Regge deficit
angle δ = 2π −5 arccos(1/3) ≈ 0.1284 rad per hinge, generating a constant positive
curvature that drives De Sitter expansion. We compute the primordial curvature
perturbation by projecting this 3D deficit onto the 2D holographic boundary via
the Regge action. The projection introduces a
√
3 Jacobian from the equilateral tri-
angular face geometry. The resulting spectral index is n
s
= 1 −
√
3(δ/2π) ≈ 0.9646,
matching the Planck 2018 measurement (n
s
= 0.9649 ± 0.0042) [3] with zero ad-
justable parameters.
1 Introduction
The inflationary paradigm [1,2] resolves the horizon, flatness, and monopole problems by
positing a brief epoch of exponential expansion. Quantum fluctuations during this epoch
seed the primordial density perturbations observed in the CMB. The Planck satellite
measures the scalar spectral index as n
s
= 0.9649 ±0.0042 [3], confirming a slight red tilt
of the primordial power spectrum.
To reproduce this specific value, standard cosmology requires a scalar field (the in-
flaton) with an engineered potential V (ϕ) that rolls at precisely the right speed. This
construction introduces free parameters (the potential shape and initial conditions) that
1
are tuned to match n
s
. The value 0.965 is an output of the tuning, not a prediction of
the framework.
In this paper, we derive n
s
from the geometry of a discrete vacuum undergoing a
structural phase transition. The derivation uses three ingredients, each independently
established: (i) the dihedral angle of a regular tetrahedron (Section 2.1), (ii) the Regge
deficit angle from optimal packing around a shared edge (Section 2.2–2.3), and (iii) the
holographic projection Jacobian from a triangular boundary face (Section 3). No free
parameters enter at any stage.
2 The Phase Transition and the Regge Deficit Angle
2.1 The K = 4 → K = 12 Phase Transition
In the SSM [4, 5], the pre-Big-Bang vacuum is a 3D amorphous simplicial complex with
mean coordination K = 4. Each node bonds to approximately 4 nearest neighbors, form-
ing an irregular tetrahedral network—topologically equivalent to an amorphous solid or
a random Delaunay tessellation. This is the pre-existing 3D medium. It is not embedded
in a higher-dimensional space; it is the space.
The phase transition converts this disordered K = 4 network into an ordered K = 12
FCC lattice [4]. This is a bulk 3D crystallization, analogous to the freezing of a supercooled
liquid into a crystal. The crystallization front propagates through the pre-existing 3D
medium, converting amorphous regions into crystalline regions. No additional embedding
dimension is required—the front is a 2D surface moving through 3D space, exactly as a
solidification front moves through a liquid.
The question addressed in this paper is: what is the geometry of the crystallization
front itself, and what curvature does it generate?
2.2 The Dihedral Angle of the Regular Tetrahedron
At the crystallization front, the local geometry transitions from irregular to regular tetra-
hedra. The dihedral angle of a regular tetrahedron (the angle between two faces sharing
an edge) is:
θ = arccos
1
3
≈ 70.528
◦
≈ 1.2310 rad (1)
This is a fixed number determined by the Euclidean geometry of the regular tetrahedron.
It requires no model assumptions.
2.3 Why Five Tetrahedra: The Optimal Packing Constraint
A central question is: why exactly 5 tetrahedra? We derive this from the dihedral angle
constraint. When regular tetrahedra pack around a common edge in 3D, the total angle
consumed is N ×θ, where N is the number of tetrahedra and θ is the dihedral angle (Eq.
1). The packing must satisfy N ×θ ≤ 2π (the total angle around the edge cannot exceed
360
◦
).
The maximum integer N satisfying this is:
N
max
=
2π
θ
=
2π
arccos(1/3)
= ⌊5.1043⌋ = 5 (2)
2
Five regular tetrahedra fit around a shared edge; six do not. The number 5 is not a
model parameter—it is the floor function of 2π divided by the dihedral angle of a regular
tetrahedron. This is the unique maximal packing in 3D.
Physically, this constraint operates at the crystallization front: as the K = 4 amor-
phous phase regularizes into well-formed tetrahedra, the densest local configuration around
each hinge (shared edge) accommodates exactly 5 simplices. This is analogous to icosa-
hedral short-range order in supercooled metallic liquids, where 5-fold local symmetry
dominates the solidification front [8].
2.4 The Regge Deficit Angle
The total angle consumed by 5 tetrahedra around a shared edge is:
5θ = 5 × arccos
1
3
≈ 5.1551 rad = 352.64
◦
(3)
The deficit angle is the gap between this and the full 2π:
δ = 2π − 5θ = 2π − 5 arccos
1
3
≈ 0.12838 rad ≈ 7.356
◦
(4)
In Regge calculus [7], a positive deficit angle at a hinge corresponds to positive scalar
curvature localized at that hinge. The Regge action for a simplicial manifold is [7]:
S
Regge
=
1
8πG
X
hinges
A
h
δ
h
(5)
where A
h
is the area of the triangular face dual to hinge h and δ
h
is the deficit angle at
that hinge. For a homogeneous tetrahedral foam where every hinge has the same deficit
angle δ, this reduces to:
S
Regge
=
1
8πG
δ
X
A
h
=
δ
8πG
A
total
(6)
Comparing with the Einstein-Hilbert action S = (1/16πG)
R
R
√
−g d
4
x, the effective
scalar curvature of the tetrahedral foam is:
R
eff
=
2δ
A
cell
(7)
where A
cell
is the area per hinge. This is a constant, positive curvature that is independent
of the cell size—it depends only on the deficit angle, which is a topological invariant of
the regular tetrahedron. A constant positive scalar curvature is the defining property
of De Sitter spacetime. The tetrahedral foam therefore generates De Sitter expansion
geometrically, without requiring a scalar inflaton field.
3 Derivation of the Spectral Index
3.1 The Curvature Perturbation per Hinge
In the continuum limit, the primordial curvature perturbation ζ measures the fractional
deviation of the local scalar curvature from the background. In the Regge discretization,
3
the curvature is concentrated at the hinges. The fractional curvature perturbation per
hinge is:
ζ
hinge
=
δR
R
=
δ
2π
(8)
The denominator 2π represents the full rotational angle of flat space (zero deficit). This
ratio is the natural dimensionless measure of the curvature deviation from flatness at each
hinge. It is a pure number determined by the geometry:
ζ
hinge
=
2π − 5 arccos(1/3)
2π
≈ 0.02043 (9)
3.2 The Holographic Projection: 3D Hinge to 2D Boundary
The curvature perturbation ζ
hinge
is a 3D bulk quantity—it describes the angular deficit
around an edge in the simplicial complex. However, the primordial power spectrum P
R
(k)
is defined on the 2D boundary (the last scattering surface). To obtain the spectral tilt,
we must project the bulk perturbation onto the boundary.
Each hinge (shared edge) in the tetrahedral foam is dual to a triangular face. The
holographic projection [6,11] maps the 1D hinge (an edge of length L) onto the 2D trian-
gular face (area
√
3L
2
/4). The Jacobian of this projection is the ratio of the transverse
extent to the longitudinal extent of the face.
For an equilateral triangle with edge L: the base (longitudinal, along the hinge) has
half-length L/2. The altitude (transverse, perpendicular to the hinge) is h = L
√
3/2. The
Jacobian of the projection from the 1D hinge to the 2D face is:
J =
h
L/2
=
L
√
3/2
L/2
=
√
3 (10)
This factor is the aspect ratio of the equilateral triangle—the ratio of the perpendicular
extent to the parallel extent relative to the hinge. It is a fixed geometric property of the
equilateral triangle and contains no free parameters.
Equivalently, this Jacobian appears in the Regge action (Eq. 5). The area element A
h
dual to each hinge scales as
√
3L
2
/4, while the hinge itself scales as L. The ratio of the
2D fluctuation amplitude to the 1D hinge amplitude is
√
3, independent of L.
3.3 The Spectral Index
The spectral index n
s
measures the scale dependence of the primordial power spectrum:
P
R
(k) ∝ k
n
s
−1
. A perfectly scale-invariant spectrum has n
s
= 1 (Harrison-Zel’dovich).
Any departure from n
s
= 1 requires a physical mechanism that breaks scale invariance.
In the SSM, the pre-transition K = 4 phase is an amorphous network with no char-
acteristic scale—its correlation function is scale-invariant. The departure from scale in-
variance is introduced by the crystallization front, which imprints a specific curvature
perturbation at each hinge. The amplitude of this perturbation on the 2D boundary is:
ζ
boundary
= J × ζ
hinge
=
√
3 ×
δ
2π
(11)
The spectral tilt is the fractional departure from scale invariance caused by this curvature
imprint:
n
s
− 1 = −
√
3
δ
2π
(12)
4
The negative sign indicates a red tilt: the crystallization front introduces positive cur-
vature (deficit angle > 0), which reduces power at small scales relative to large scales.
This is because larger wavelength modes span more hinges and accumulate more curva-
ture perturbation coherently, while shorter wavelength modes sample individual hinges
incoherently.
Therefore:
n
s
= 1 −
√
3
δ
2π
= 1 −
√
3
(2π − 5 arccos(1/3))
2π
(13)
Evaluating numerically:
δ = 2π − 5 arccos
1
3
= 0.12838 rad
√
3 ×
δ
2π
= 1.7321 × 0.02043 = 0.03539
n
s
= 1 − 0.03539 = 0.9646 (14)
The Planck 2018 measurement is n
s
= 0.9649 ± 0.0042 [3]. The prediction n
s
= 0.9646
lies 0.07σ from the central value.
Every number entering this calculation is a geometric constant: arccos(1/3) is the
dihedral angle of a regular tetrahedron, 5 is the maximum integer packing around a
shared edge, 2π is the angle of flat space, and
√
3 is the aspect ratio of the equilateral
triangle. No parameter is adjusted.
4 Physical Mechanism
4.1 De Sitter Expansion from the Deficit Angle
In standard inflation, exponential expansion is driven by the potential energy of a scalar
field. In the SSM, the same exponential expansion is driven by the constant positive
curvature of the tetrahedral foam (Eq. 7). The deficit angle δ > 0 at every hinge
produces a constant R
eff
> 0 throughout the crystallizing region. By the Friedmann
equation, a constant positive curvature implies exponential scale factor growth a(t) ∝ e
Ht
with H
2
∝ R
eff
. Inflation ends when the crystallization completes and the lattice relaxes
to K = 12 FCC (which has zero deficit angle in the continuum limit).
4.2 Why 3D and Not 4D
A natural question is why the crystallization front is 2D rather than 3D (which would
require 4D space). The answer is that the phase transition operates within a pre-existing
3D amorphous network. The crystallization front is the 2D boundary between the disor-
dered (K = 4) and ordered (K = 12) phases of this 3D medium. No higher-dimensional
embedding space is needed, for the same reason that the solidification front in a freezing
liquid does not require a fourth spatial dimension. The 3D medium already exists; the
front propagates through it.
The 4D Lorentzian structure (3+1 spacetime) emerges after crystallization. In the
SSM, the Cosserat Lagrangian of the K = 12 lattice generates the Schr¨odinger equation
via ψ = u + iθ [5], from which time evolution emerges as the phase rotation of the
coupled translational-rotational degrees of freedom. The crystallization itself is a 3D
5
spatial process; the time direction emerges from the lattice dynamics after the transition
completes.
4.3 End of Inflation and Reheating
Inflation ends when the K = 4 → K = 12 transition completes. The final FCC lattice
has K = 12 nearest neighbors per node, which tile 3D space without deficit angles (the
cuboctahedral coordination shell closes perfectly). The effective curvature drops to zero,
ending exponential expansion. The latent heat of the phase transition—the energy dif-
ference between the K = 4 and K = 12 configurations—is released as lattice vibrations
(the translational u-field), constituting reheating.
Incomplete crystallization leaves frozen K = 4 tetrahedral defects, which constitute
baryonic matter [9].
5 Predictions and Falsifiability
The derivation yields three testable consequences:
(a) n
s
= 0.9646 exactly. The prediction has zero error bars. If future CMB measure-
ments (CMB-S4, LiteBIRD) constrain n
s
to a value outside 0.9646±0.001, this derivation
is falsified.
(b) Tensor-to-scalar ratio r. The Regge deficit angle generates scalar perturbations
but does not produce tensor modes (gravitational waves) at leading order, because the
deficit angle is a scalar quantity. The prediction is r ≪ 0.01, consistent with current
bounds (r < 0.036 [3]) but distinguishable from large-field inflation models that predict
r ∼ 0.01 − 0.1.
(c) Running of the spectral index. Because n
s
is a geometric constant (not a
function of the number of e-folds), the running is exactly dn
s
/d ln k = 0 to leading order.
Planck measures dn
s
/d ln k = −0.0045 ± 0.0067 [3], consistent with zero.
6 Conclusion
We derived the scalar spectral index from three geometric inputs: the dihedral angle
of the regular tetrahedron (θ = arccos(1/3)), the maximum integer packing around a
shared edge (N = 5), and the aspect ratio of the equilateral triangle (
√
3). The resulting
Regge deficit angle δ = 2π − 5θ, projected onto the holographic boundary with Jacobian
√
3, yields n
s
= 1 −
√
3(δ/2π) = 0.9646. This matches the Planck 2018 central value
(0.9649 ± 0.0042) with zero free parameters.
The physical mechanism is the K = 4 → K = 12 vacuum crystallization, in which
the positive Regge curvature of the tetrahedral foam drives De Sitter expansion and the
deficit angle imprints a specific red tilt on the primordial power spectrum.
A SSM Conceptual Summary
For completeness, we outline the specific SSM mechanical constraints driving this deriva-
tion.
A.1. The K = 12 FCC Lattice. The SSM posits the vacuum as a discrete, saturated
tensor network, not a smooth continuum. Its stable ground state is the Face-Centered
6
Cubic (FCC) lattice, which physically maps the mathematically unique densest packing
of uniform spheres in 3D (the Kepler conjecture [4, 10]). Here, every node commands
exactly K = 12 nearest-neighbor bonds. The structure tensors partition strictly into a
translational sector (S
trans
= 4) and a torsional sector (S
tors
= 8), explicitly locking the
vacuum’s energy partition to a 1:2 geometric ratio.
A.2. The Cosserat Lagrangian. This discrete geometry follows the Chiral Cosserat
Lagrangian. Unlike classical elasticity, a Cosserat continuum equips every node with both
a translational vector (u) and an independent microrotation (θ). The Lagrangian features
a mechanical chiral cross-coupling, Ω(u
˙
θ − θ ˙u). This continuous cross-coupling functions
directly as the complex phase in quantum mechanics, yielding the emergent Schr¨odinger
equation when defining the wave function as ψ = u + iθ [5].
A.3. Matter as Frozen Topological Defects. The pre-Big-Bang state models an
amorphous simplicial complex with mean coordination K = 4 (a tetrahedral foam). The
Big Bang represents a bulk thermodynamic crystallization where this disordered foam
regularizes into the ordered K = 12 FCC ground state. Topological frustration impedes
total conversion; localized regions failing to crystallize remain permanently trapped as
K = 4 tetrahedral defects. These unyielding stress-centers within the FCC grid manifest
physically as baryonic matter (quarks, leptons) [9].
A.4. Torsional Dark Matter. A trapped K = 4 defect only connects to 4 of
the 12 available surrounding FCC bonds, leaving 8 structural connections permanently
orphaned. This topological mismatch generates a long-range torsional strain field that
twists the surrounding Cosserat continuum. The resulting macroscopic elastic strain forces
a ρ ∝ 1/r
2
effective density profile, perfectly matching flat galactic rotation curves without
demanding exotic dark matter particles [12].
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