The Freezing Point of Space:
Deriving the Spectral Index (n
s
≈ 0.96) from the
Thermodynamics of Vacuum Crystallization
Raghu Kulkarni
Independent Researcher, Calabasas, CA
February 13, 2026
Abstract
Standard Inflationary Cosmology predicts a nearly scale-invariant spectrum of primordial
fluctuations (n
s
≈ 1), but the observed “Red Tilt” (n
s
≈ 0.96) requires fine-tuning the potential
of a hypothetical Inflaton field. We propose that this tilt is a **Geometric Invariant** of
the vacuum’s phase transition. In the Selection-Stitch Model (SSM), the universe began as
a metastable “Fluid Foam” (N = 4 degrees of freedom) which relaxed into a stable “Solid
Lattice” (K = 12, Stiffness=108). We derive the Spectral Index as the ratio of the remnant
disorder (Foam Entropy) to the final lattice capacity (Crystal Stiffness). Using the integer values
established in previous SSM papers (N
foam
= 4 and E
stiff
= 108), we derive n
s
= 1 − 4/108 ≈
0.9630. This zero-parameter prediction matches the Planck 2018 observation (0.9649 ± 0.0042)
within 0.5σ. We show that this geometric damping factor (1/27) corresponds to an exact
effective e-fold count of N = 54 when compared to standard inflationary models. Finally, we
identify the **Latent Heat** of this crystallization as the physical source of Cosmic Reheating.
1 Introduction
The Cosmic Microwave Background (CMB) provides a snapshot of the universe at the moment of
recombination. A key parameter describing this epoch is the **Scalar Spectral Index (n
s
)**, which
measures how density fluctuations vary with scale (P (k) ∝ k
n
s
−1
). A perfectly scale-invariant
universe would have n
s
= 1. Observations by the Planck satellite measure n
s
= 0.9649 ± 0.0042
[1], a deviation known as the “Red Tilt.”
In the standard ΛCDM model, this tilt is explained by Slow-Roll Inflation, where the value
depends on the slope of an arbitrary scalar field potential V (ϕ). There is no fundamental reason
why n
s
should be 0.96 rather than 0.99 or 0.92; it is a fitted parameter.
We present a geometric derivation of n
s
within the **Selection-Stitch Model (SSM)** [2]. We
posit that “Inflation” was not driven by a scalar field, but by the geometric instability of a **Tetra-
hedral Foam**. The “Big Bang” was the phase transition where this foam crystallized into the
**Face-Centered Cubic (K = 12)** vacuum we inhabit today. We show that the spectral tilt arises
naturally from the entropy ratio of these two geometric phases.
2 The Geometry of the Tilt
The Spectral Index n
s
represents the “purity” of the vacuum structure. The deviation (1 − n
s
)
represents the magnitude of the primordial defects frozen into the lattice structure. We define the
Tilt as the ratio of the **Initial Disorder** to the **Final Stiffness**:
1
1 − n
s
=
Entropy of the Foam (S
foam
)
Stiffness of the Crystal (E
stiff
)
(1)
2.1 The Foam Factor (N
foam
= 4)
As established in the foundational SSM paper [2], the pre-geometric phase (Inflation) was defined
by random fluctuations lifting nodes into 3D space. The fundamental simplex of 3D space is the
**Tetrahedron**. Unlike the crystal phase, the foam phase is disordered. In the SSM, nodes are the
fundamental degrees of freedom (each node is a stitching event), so the foam’s disorder is measured
by its vertex count.
N
foam
= 4 (Nodes per Simplex) (2)
This integer represents the “noise” input—the number of ways a local geometry can fluctuate before
locking.
2.2 The Crystal Capacity (E
stiff
= 108)
The vacuum eventually froze into the densest possible packing: the Face-Centered Cubic (FCC)
lattice (K = 12). The rigidity of this lattice is defined by its tension capacity. In our derivation
of the Proton Mass [3], we established that the total binding energy of a unit cell is defined by
the locking of its chiral faces: 3 strands × 3 steps × 12 neighbors = 108. Physical Justification:
The Crystal Stiffness (E
stiff
) and the Proton Anchoring Energy (E
anchor
) are identical because
they measure the same physical quantity: the energy required to distort the **Triangular Face**
interface. The proton anchors by twisting this face; the crystal resists melting by holding this face
rigid. Thus, the denominator is uniquely determined as E
stiff
= 108.
2.3 The Damping Mechanism
Crystallization is a time-dependent process. The “freezing front” propagates through the universe.
1. **Early Times (Large Scales):** The lattice is soft. The “Disorder” term (N = 4) is significant
relative to the nascent tension. Fluctuations are preserved.
2. **Late Times (Small Scales):** The lattice stiffens as it locks into the K = 12 configuration.
The “Stiffness” term (E = 108) dominates, damping out the fluctuations.
The ratio δ = 4/108 acts as the **Damping Coefficient**. It defines how much power is lost as the
lattice refines itself from foam to crystal.
n
s
= 1 −
N
foam
E
stiff
= 1 −
4
108
(3)
Simplifying the fraction:
n
s
= 1 −
1
27
≈ 0.96296... (4)
3 Results and Comparison
We compare this zero-parameter geometric prediction with the latest cosmological data.
Implication: The “Red Tilt” is not a random number. It is the rational fraction 26/27. This
suggests that the primordial power spectrum is simply the ratio of the old geometry (Tetrahedral)
to the new geometry (Cuboctahedral Tension).
2
Source Method Value (n
s
) Uncertainty Sigma
SSM (This Work) Geometric (26/27) 0.9630 0 –
Planck 2018 [1] CMB (TT,TE,EE) 0.9649 ±0.0042 0.45σ
ACT + WMAP [4] CMB Lensing 0.9649 ±0.0044 0.43σ
Table 1: Comparison of the SSM prediction with observational constraints. The geometric value
26/27 lies squarely within the 0.5σ confidence interval of the Planck measurements.
3.1 Relation to Standard Inflation
In standard Slow-Roll Inflation models (e.g., Starobinsky or ϕ
2
), the spectral index scales as n
s
≈
1 − 2/N, where N is the number of e-folds. Identifying the SSM damping factor 1/27 with this
leading-order approximation:
2
N
=
1
27
=⇒ N = 54 (5)
This implies that the SSM crystallization process corresponds to an effective duration of **54 e-
folds**, which aligns perfectly with the physically required range (N ≈ 50−60) to solve the Horizon
Problem.
4 The Thermodynamics of Genesis
Standard cosmology requires a mechanism for **Reheating**: converting the potential energy of
Inflation into the hot particle bath of the Big Bang. In the SSM, this is a thermodynamic phase
transition.
4.1 Latent Heat of Crystallization
When water freezes into ice, it releases latent heat. When the vacuum froze from the “Gappy
Foam” (K = 4) to the “Solid Crystal” (K = 12), the system moved from a high-energy disordered
state to a low-energy ordered state. The energy density released is proportional to the change in
binding energy density:
ρ
heat
∝
E
stiff
− N
foam
L
4
p
(6)
This energy difference was released as high-frequency lattice vibrations. We identify these primor-
dial vibrations as **Photons**. The “Reheating” of the universe was the exothermic crystallization
of space itself.
5 Summary of SSM Constants
This result joins a growing set of SSM zero-parameter predictions, all derived from the same integers
(4, 12, 17, 108).
6 Falsifiability
This derivation is rigid. It depends on the integer values 4 and 108. The theory is falsified if:
3
Constant SSM Derivation Predicted Observed
Spectral Index (n
s
) 1 − 4/108 0.9630 0.9649
Inv. Fine Structure (α
−1
) 136 + 1 137 137.036
Dark Matter Ratio (Ω
R
) 5 × 1.08 5.40 5.36
Hubble Boost (H
0
) 13/12 1.083 1.086
Proton Mass Ratio (µ) 12
3
+ 108 1836 1836.15
Table 2: The Zero-Parameter Cluster. Five fundamental constants derived from the integer prop-
erties of the Cuboctahedral vacuum.
1. Future CMB experiments (e.g., CMB-S4, LiteBIRD) measure n
s
with high precision to be
significantly different from 0.9630 (e.g., 0.968 ± 0.001).
2. The unit cell of the vacuum is proven to be something other than the Cuboctahedron (e.g.,
E8 or dodecahedral).
7 Conclusion
We have derived the Cosmic Spectral Index n
s
≈ 0.96 as a geometric invariant of the vacuum’s
phase transition. By modeling the Big Bang as the freezing of a Tetrahedral Foam (N = 4) into
a Cuboctahedral Crystal (E = 108), we show that the spectral tilt corresponds to the rational
fraction 26/27. This result eliminates the need for an arbitrary Inflaton potential and unifies the
origin of the CMB with the origin of Mass and Coupling constants.
References
1. Planck Collaboration, “Planck 2018 results. VI. Cosmological parameters,” Astron. Astro-
phys. 641, A6 (2020).
2. R. Kulkarni, “The Selection-Stitch Model (SSM),” Zenodo (2026).
3. R. Kulkarni, “The Geometric Origin of Mass,” Zenodo (2026).
4. Aiola, S., et al. (Atacama Cosmology Telescope), JCAP 12, 047 (2020).
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