The Spectral Origin of the Cosmic Energy Budget: Dark Energy from the Cluster Laplacian of the FCC Tensor Network

The Spectral Origin of the Cosmic Energy Budget:

Dark Energy from the Cluster Laplacian

of the FCC Tensor Network

Raghu Kulkarni

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

raghu@idrive.com

Abstract

We derive the cosmological dark energy fraction from the Laplacian spectrum of the

13-node structural cluster in the FCC tensor network. The cluster Laplacian has ex-

act eigenvalues

{0, 3

(×3)

, 5

(×3)

, 7

(×5)

, 13

(×1)

}

, which decompose under the octahedral

group

into the translational triplet

(

λ = 3

), the torsional triplet

(

λ = 5

and the Cosserat-inert modes

+ T

+ A

(

λ = 7, 13

). The vacuum is a sea of

entanglement: every lattice cell carries 72 units of spectral weight. Of these, only

the 24 carried by Cosserat-active modes (

+ T

) can be redirected into matter

by a topological defect; the remaining 48 are structurally locked as vacuum entan-

glement energydark energy. This gives

Ω

≥ 2/3

as a spectral ceiling. Geometric

frustration from the Regge decit further reduces the matter coupling eciency,

returning frustrated entanglement to the vacuum budget. The corrected predic-

tion is

Ω

≈ 0.687

(0.3% from Planck 2018). Combined with

Ω

/Ω

= 5.3595

from the torsional entanglement cascade, the full cosmic pie follows:

Ω

≈ 0.049

Ω

≈ 0.264

Ω

≈ 0.687

, summing to unity. Dark energy is not a tted parameter

but the spectral weight of the entanglement that matter can never convert.

Keywords:

dark energy, vacuum entanglement, Laplacian spectrum, FCC lattice,

Triadic Orthogonal Calculus, Cosserat elasticity

1 Introduction

The cosmological energy budget

Ω

≈ 0.049

Ω

≈ 0.265

Ω

≈ 0.685

is measured

to percent-level precision [1] but has no theoretical derivation within

CDM. All three

fractions are treated as independent parameters tted to CMB data.

We show that the dark energy fraction emerges from the

spectrum

of a specic graph

Laplacian: the 13-node structural cluster of the FCC tensor network. The derivation

proceeds in three steps: (i) compute the exact eigenvalues, (ii) classify the eigenmodes

by their coupling character under Cosserat elasticity, (iii) identify the fraction of the

vacuum's entanglement budget that is structurally inaccessible to matter. The result is a

Hamiltonian-based spectral ceiling, not a counting argument.

Interactive 3D visualizations:



Entanglement Defect

 the 13-node structural cluster and its torsional

disruption halo:

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

html



Lattice Structure

 the

K = 12

cuboctahedral shell and three orthogonal

triad sheets:

https://raghu91302.github.io/ssmtheory/ssm_regge_deficit.html

2 Triadic Orthogonal Calculus

The FCC nearest-neighbor vectors decompose uniquely into three orthogonal 4-bond

sheets [2]:

XY:

(±1, ±1, 0) (τ

= 4)

XZ:

(±1, 0, ±1) (τ

= 4)

YZ:

(0, ±1, ±1) (τ

= 4)

(1)

Denition 1

(Triad)

The entanglement state of an FCC node is the triad

τ = (τ

, τ

) ∈

, with norm

|τ | = τ

+ τ

and dimension

dim(τ ) = 3

. For the vacuum,

vac

= (4, 4, 4)

The derived quantities relevant to the spectral derivation:

|τ | = 12

(coordination number)

dim(τ ) = 3

(orthogonal sheets)

= 4

(bonds per sheet)

|τ |+ 1 = 13

(structural cluster size)

dim(τ ) × |τ | = 36

(internal bonds of the cluster) (2)

An interstitial defect at a tetrahedral void creates a structural cluster of

|τ |+ 1 = 13

nodes (one interstitial plus 12 shell nodes). Each shell node has degree 5 within the cluster

(one bond to the origin plus four cuboctahedral edges), giving

(12+12×5) = 36

internal

bondspartitioned

3 × 12

by sheet [2].

The Regge decit

δ = 2π−5 arccos(τ

/|τ |) = 2π−5 arccos(1/3) ≈ 0.128

rad arises from

the impossibility of tiling

with regular tetrahedra whose dihedral angle is

arccos(τ

/|τ |)

3 The Cluster Laplacian Spectrum

Theorem 1

(Cluster Laplacian spectrum)

The graph Laplacian of the 13-node structural

cluster (origin + cuboctahedral shell) has eigenvalues:

σ(L) = {0

(×1)

, 3

(×3)

, 5

(×3)

, 7

(×5)

, 13

(×1)

}

(3)

Proof.

The cluster graph has one origin node of degree 12 and twelve shell nodes of degree

5 (one bond to origin

four cuboctahedral edges). The Laplacian

L = D − A

, where

D =

diag

(12, 5, . . . , 5)

and

is the adjacency matrix.

1.5

1.0

0.5

0.0

0.5

1.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

(a) = (4, 4, 4): Three sheets

XY sheet (

= 0): 4 bonds

XZ sheet (

= 0): 4 bonds

YZ sheet (

= 0): 4 bonds

1.5

1.0

0.5

0.0

0.5

1.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

XY: 12 bonds

XZ: 12 bonds

YZ: 12 bonds

(b) 36 bonds = 3 × 12

= (4, 4, 4)

| | = 12

dim = 3

= 4

| |

= 8

Rule 1:

= | |

= 144

Rule 2:

= 36

Rule 3:

= | |+1 = 13

= 1836

= 5.36

1836.15 (expt)

0.008% match

5.364 (Planck)

0.09% match

Zero free parameters

All from

= (4, 4, 4)

Figure 1: Triadic Orthogonal Calculus. (a) The triad

τ = (4, 4, 4)

: three orthogonal

4-bond sheets. (b) The 36 bonds of the 13-node cluster, partitioned

3 × 12

by sheet.

The cuboctahedron is 4-regular with adjacency eigenvalues

(×1)

, 2

(×3)

, 0

(×3)

, −2

(×5)

}

Eigenvectors of the cuboctahedral adjacency matrix orthogonal to the all-ones vector

remain eigenvectors of the cluster Laplacian with eigenvalue

5 − µ

µ = 2 =⇒ λ = 3 (×3)

µ = 0 =⇒ λ = 5 (×3)

µ = −2 =⇒ λ = 7 (×5)

(4)

The two-dimensional subspace spanned by

= (1, 0, . . . , 0)

and

u = (0, 1, . . . , 1)/

√

is invariant under

. In this basis:



,u}



12 −

√

−

√

12 1



(5)

with eigenvalues

λ(λ − 13) = 0

, giving

λ = 0

(constant mode) and

λ = 13

(breathing

mode, eigenvector

(−12, 1, . . . , 1)

Trace check:

0 + 3 × 3 + 3 × 5 + 5 × 7 + 13 = 72 = 12 + 12 × 5 =

(D)

4 Octahedral Decomposition and Cosserat Classica-

tion

The 13-node cluster has

symmetry (the octahedral group of the cuboctahedron, with

the origin at the center of symmetry). The 12-dimensional representation on the shell

vertices decomposes as:

= A

⊕ E

⊕ T

(6)

Adding the origin (

), the full 13-dimensional representation is

⊕E

⊕T

The Laplacian eigenvalues map onto these irreps:

Mult.

irrep Physical character

Constant (trivial)

Translations (

x, y, z

)

Torsions (

xy, xz, yz

)

⊕ T

Quadrupolar + odd torsion

Breathing (origin vs. shell)

In Cosserat micropolar elasticity [3, 4], the fundamental degrees of freedom are transla-

tional displacements

and microrotations

. The representation-theoretic identication

is:

(

λ = 3

×3

): These modes transform as the vector representation (

x, y, z

). They

describe rigid translations of the clusterthe modes that couple to external translational

forces. In the Cosserat framework, translational modes carry the elastic energy of charged

fermions (baryonic matter).

(

λ = 5

×3

): These modes transform as the symmetric traceless tensor (

xy, xz, yz

)

the three triad sheet indices. They describe pure torsional deformations of the cluster.

In the Cosserat framework, torsional modes carry the elastic energy of microrotational

defects (dark matter) [2].

The remaining modes (

+ T

λ = 7

and

λ = 13

) do not couple to

either translational or torsional Cosserat elds. They are

inert

: they describe internal

quadrupolar deformations and the isotropic breathing of the cluster. These modes carry

the vacuum's zero-point entanglement energythe dark energy.

5 The Spectral TOC Theorem

Every eigenvalue of the cluster Laplacian is a TOC expression:

TOC expression Mult. TOC multiplicity

trivial 1 

3 dim(τ )

dim(τ )

5 τ

+ 1

dim(τ )

7 2τ

− 1

dim(τ ) + 2

13 |τ |+ 1

1 

This is not a relabelling. The spectral weights collapse to closed-form TOC identities:

Theorem 2

(Spectral partition identity)

For the 13-node cluster Laplacian with Cosserat

classication

(

matter

= T

+ T

, vacuum

remainder

)

matter

= dim(τ ) × (|τ | − τ

) = dim(τ ) × S

tors

(7)

vacuum

= τ

× |τ |

(8)

total

= 2 × dim(τ ) × |τ |

(9)

Proof.

From the eigenvaluemultiplicity pairs:

matter

= dim(τ )

+ (τ

+ 1) dim(τ ) = dim(τ )



dim(τ ) + τ

+ 1



(10)

The key identity for the vacuum triad

τ = (4, 4, 4)

dim(τ ) + τ

+ 1 = 3 + 4 + 1 = 8 = |τ | − τ

= S

tors

(11)

Therefore

matter

= dim(τ ) × S

tors

= 3 × 8 = 24

For the vacuum:

vacuum

= (2τ

− 1)(dim(τ ) + 2) + (|τ | + 1) = 7 × 5 + 13 = 48 = τ

× |τ |

(12)

Verication:

total

= 24 + 48 = 72 = 2 × 36 = 2 dim(τ )|τ |

The dark energy fraction follows immediately:

Ω

(0)

|τ |

2 dim(τ ) |τ |

2 dim(τ )

(13)

and the matter fraction:

Ω

(0)

|τ | − τ

2 |τ |

tors

2 |τ |

(14)

This spectral weight characterizes the entanglement structure of

every

vacuum cell

not just cells containing defects. The FCC vacuum is a sea of entanglement, and

total

2 dim(τ )|τ | = 72

is the entanglement budget per cell. Of this,

|τ | = 48

is carried by

Cosserat-inert modes that no translational or torsional defect can access. Dark energy is

the entanglement that matter can never convert.

6 Regge Decit Correction

The spectral ceiling

Ω

≥ τ

/(2 dim(τ )) = 2/3

assumes all Cosserat-active spectral weight

is fully redirected into matter. In the real lattice, geometric frustration prevents this.

The Regge decit

δ = 2π−(τ

+1) arccos(τ

/|τ |) ≈ 0.128

rad exists at every tetrahedral

edge. Each Cosserat mode propagates through all

dim(τ ) = 3

sheets, encountering the

decit at each boundary. The frustration fraction per mode:

f = dim(τ ) ×

2π

= 3 × 0.02043 = 0.06130

(15)

Frustrated entanglement is not destroyedit remains as vacuum entanglement. The

corrected spectral weights:

eﬀ

matter

= dim(τ ) · S

tors

· (1 − f) = 24 × 0.939 = 22.53

(16)

eﬀ

vacuum

= τ

|τ | + dim(τ ) · S

tors

· f = 48 + 1.47 = 49.47

(17)

The corrected dark energy fraction:

Ω

|τ | + dim(τ ) S

tors

2 dim(τ ) |τ |

≈ 0.687

(18)

Planck 2018:

Ω

= 0.6847 ± 0.0073

. The spectral prediction matches to 0.3%, within

the measurement uncertainty.

matter

= 24

= dim( ) ×

tors

= 3 × 8

vacuum

= 48

× | |

= 4 × 12

total

= 72

= 2dim( )| |

: = 3, ×3

: = 5, ×3

: = 7, ×5

: = 13, ×1

(0)

2 dim( )

spectral

gap

(a) Spectral weight partition

Cosserat-active (matter)

Cosserat-inert (dark energy)

68.7%

26.4%

4.9%

Planck 2018: = 0.685,

= 0.265,

= 0.049

All from

= (4, 4, 4)

zero free parameters

(b) Cosmic energy budget ( 0.687)

Figure 2: The cosmic energy budget from the Spectral TOC Theorem. (a) Spectral weight

partition: the Cosserat-inert modes (

, dark blue) carry

vacuum

= τ

|τ | = 48

while the Cosserat-active modes (

+ T

, red) carry

matter

= dim(τ ) × S

tors

= 24

yielding

Ω

(0)

= 2/3

. The spectral gap at

λ = 6

separates the two sectors. (b) Regge-

corrected cosmic pie:

Ω

≈ 0.687

Ω

≈ 0.264

Ω

≈ 0.049

7 The Complete Cosmic Pie

The dark-matter-to-baryon ratio

Ω

/Ω

= 5.3595

is independently derived from the

torsional entanglement cascade [2]. Combining:

Ω

1 + 5.3595

0.313

6.3595

≈ 0.0492

(19)

Ω

= 0.0492 × 5.3595 ≈ 0.2637

(20)

Ω

≈ 0.687

(21)

Component Spectral prediction Planck 2018 Match

Ω

0.049 0.0493 ± 0.0006

0.2%

Ω

0.264 0.265 ± 0.007

0.4%

Ω

0.687 0.6847 ± 0.0073

0.3%

Ω

total

1.000 1.000

exact

Table 1: The cosmic energy budget from the cluster Laplacian spectrum. The dark energy

fraction derives from the Cosserat partition of the spectral weight (Section 5) with Regge

correction (Section 6). The matter split uses

Ω

/Ω

= 5.3595

[2]. Zero free parameters.

8 Dark Energy as Irreducible Vacuum Entanglement

The identication of the

+ T

+ A

modes with dark energy rests on three physical

facts.

Dark energy exists everywhere.

The Laplacian spectrum

{0, 3

, 5

, 7

, 13}

is a property

of every vacuum cell, not just cells containing defects. In an empty cell, all 72 units of

spectral weight constitute vacuum entanglement energy. When a defect forms, some

of this entanglement is redirected into matter. But the 48 units carried by Cosserat-

inert modes cannot be redirected by any translational or torsional defectthey remain

as vacuum entanglement in every cell of the universe, occupied or empty. This is the

physical origin of the cosmological constant: it is the entanglement energy that matter

structurally cannot access.

The inert modes are high-frequency.

Their eigenvalues (

λ = 7

and

λ = 13

) are the

highest in the spectrum, contributing the most zero-point energy per mode. The spectral

gap between the matter sector (

λ ≤ 5

) and the inert sector (

λ ≥ 7

) ensures that these

modes are not thermally accessible at cosmological temperaturesthey constitute the

irreducible ground-state entanglement of the lattice.

The inert modes are invisible to Cosserat probes.

They do not transform as translations

(

) or microrotations (

) under

. Any physical detector built from Cosserat elds

which includes all known matter and radiationis structurally unable to couple to these

modes. The inert entanglement gravitates (it contributes to

) but does not interact with

matter or photons. This is precisely the phenomenology of dark energy.

9 Relation to the Friedmann Equations

In standard cosmology, matter density dilutes as

−3

while

remains constant, so

Ω

/Ω

changes with redshift. This is not in conict with the spectral partition.

The spectral result

Ω

(0)

= 2/3

is a

ceiling

, not a xed ratio. It states that two-thirds

of the vacuum's entanglement budget per cell is carried by modes that cannot couple to

Cosserat defects. The vacuum entanglement energy density

is constant (it is a ground-

state property of every cell), while matter energy density

dilutes as

−3

. At early

times,

≫ ρ

and

Ω

≈ 0

not because the inert modes are absent, but because the

matter-active modes are enormously excited above their ground state. As the universe

expands and cools,

falls, and

Ω

rises. The spectral ceiling

2/3

is the asymptotic value

/ρ

→ 0

The present epoch (

Ω

≈ 0.685

) is close to but slightly above the uncorrected ceiling

2/3 = 0.667

. The Regge correction resolves this: frustrated entanglement shifts the

eective ceiling to

∼ 0.687

, and the observed value sits just below this corrected ceiling

consistent with matter having nearly but not fully diluted.

Caveat.

A full derivation of the dynamical approach to the spectral ceiling requires

coupling the Laplacian spectrum to the Friedmann equations, which is beyond the scope

of this paper. We derive the ceiling, not the trajectory.

10 Falsiable Predictions

(a)

Ω

≈ 0.687

The spectral prediction is 0.3% above Planck's central value. If

improved CMB or BAO measurements narrow the uncertainty and converge below

0.675, the spectral partition is falsied.

(b)

w = −1

(true cosmological constant).

The inert modes carry zero-point energy

that does not dilute. If dark energy is dynamical (

w = −1

), the identication with

vacuum modes is wrong.

(c)

Ω

/Ω

= 5.3595

This is independently derived and independently testable [2].

(d)

The spectral gap at

λ = 6

The Laplacian has no eigenvalue between 5 and

7. This gap separates matter from dark energy. If future lattice simulations or

analytic calculations nd additional modes in this gap (e.g., from perturbations to

the cluster), the clean partition is disrupted.

11 Limitations

The Cosserat classication of Laplacian eigenmodes (

+ T

matter-coupling, re-

mainder

inert) is motivated by representation theory and the Cosserat Lagrangian, but

it is an

assignment

, not derived from rst principles of quantum gravity. The eigenvalue-

weighting (

W ∝

× m

) assumes that each mode's contribution to the entanglement

budget scales with its eigenvalue; other weightings (e.g.,

√

, thermal at specic

) give

dierent ratios (the zero-point weighting

∝

√

gives

Ω

≈ 0.59

; the equipartition limit

gives

Ω

≈ 0.50

; the eigenvalue weighting is the natural choice for a graph Laplacian

where

is the vibrational frequency squared). The Regge frustration correction assumes

independent frustration at each sheet boundary; correlated eects could modify the fac-

tor of

dim(τ ) = 3

. The accounting that frustrated entanglement returns to the vacuum

budget is a consequence of energy conservation (entanglement not redirected into mat-

ter remains as vacuum entanglement) but the precise mechanism is not derived from a

dynamical equation.

12 Conclusion

The dark energy fraction emerges from the Spectral TOC Theorem (Theorem 2): the

Cosserat-inert spectral weight of the 13-node cluster Laplacian is

vacuum

= τ

|τ | =

, and the total budget is

total

= 2 dim(τ )|τ | = 72

, giving a spectral ceiling

Ω

≥

/(2 dim(τ )) = 2/3

. The Regge decit frustration returns uncoupled entanglement to

the vacuum budget, yielding

Ω

≈ 0.687

(0.3% from Planck).

The physical picture is that the FCC vacuum is a sea of entanglement. Of the 72 units

of spectral weight per cell,

|τ | = 48

are carried by modes that no Cosserat defect can

couple to. Dark energy is not a tted parameter but the entanglement that matter can

never convert.

References

[1] Aghanim N. et al. (Planck Collaboration),

Astron. Astrophys.

641

, A6 (2020).

[2] Kulkarni R., Matter as an Entanglement Defect,

https://doi.org/10.5281/

zenodo.18933667

(2026).

[3] Cosserat E., Cosserat F.,

Théorie des corps déformables

, Hermann (1909).

[4] Eringen A. C.,

Microcontinuum Field Theories I

, Springer (1999).