Mass-Energy-Information Equivalence: Cosmic Voids and Hubble Tension Resolution

MassEnergyInformation Equivalence IV:

Cosmic Void Density from the Error-Correction

Threshold of the FCC Lattice Code

Raghu Kulkarni

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

raghu@idrive.com

Abstract

The MassEnergyInformation (M/E/I) framework models the physical vacuum as

[[192, 130, 3]]

CSS quantum error-correcting code on the FCC lattice [4]. Parts I

III [1, 2, 3] derived particle masses, nuclear binding, and the dark-to-baryonic ratio

from the fault-tolerant verication cost of this code. This paper extends the frame-

work to the largest structures in the universe: cosmic voids. In the bulk vacuum,

every stabilizer has weight

K = 12

(full FCC coordination). Inside a void, bonds are

broken and the eective stabilizer weight drops in proportion to the local density. A

distance-

code requires minimum stabilizer weight

d + 1

for single-error correction.

The void therefore stabilizes at the density where the eective weight reaches this

threshold:

void

bulk

d + 1

= 0.333

The SDSS DR12 BOSS void catalog [8] reports average void-center densities of

∼ 0.30 ρ

mean

. The prediction deviates by

∼ 11%

, with zero tted parameters. Equiv-

alently, at

ρ = ρ

bulk

, exactly two of the three orthogonal FCC sheets are lost; only

the 1-sheet (1D) sub-structure survives, and only electronsthe 1D minimal defect

of Part Ican exist inside deep voids. The absolute density oor below which the

code cannot function at all is

ﬂoor

= d/K = 1/4

; no void can be deeper than

25%

of the mean density. Standard excursion-set void theory treats the underdensity

threshold as a free parameter; the QEC framework predicts it from code geometry.

Void formation also breaks the gauge symmetry of the central node in the cubocta-

hedron, opening a 13th syndrome extraction channel (

ν = 12 → 13

). The resulting

increase in vacuum verication overhead predicts the ratio of late-universe to early-

universe expansion rates:

local

CMB

= V/K = 13/12 = 1.0833

. The observed

ratio (SH0ES/Planck [16, 15]) is

73.04/67.4 = 1.0837

, a deviation of

0.03%

. The

5σ

Hubble tension is resolved by a single integer ratio from the cuboctahedral geometry.

Keywords:

cosmic voids; Hubble tension; massenergyinformation equivalence; quan-

tum error correction; FCC lattice; void density threshold; stabilizer weight

1 Introduction

Cosmic voidsvast underdense regions spanning tens to hundreds of Mpcdominate the

volume of the cosmic web [9]. Their density proles, size distributions, and abundance

encode information about dark energy, gravity, and structure formation [10]. A central

quantity in void theory is the

underdensity threshold

: the density contrast at which a

void forms. In excursion-set theory [11], this threshold is typically a free parameter, tted

to simulations or data.

This paper derives the void density threshold from the error-correction properties of the

vacuum code. The MassEnergyInformation (M/E/I) framework [1] models the physical

vacuum as a

[[192, 130, 3]]

CSS code on the Face-Centered Cubic (FCC) lattice [4]. Any

localized excitation's mass is its fault-tolerant verication cost [5, 6]:

= C

kT ln 2

(1)

Parts IIII derived six observables from this framework: ve particle mass ratios [1], the

nuclear binding constant [2], and the dark-to-baryonic ratio

△

□

= 32/6 = 5.333

[3].

This paper identies two more: the characteristic density of cosmic void interiors and a

resolution of the Hubble tension.

2 The M/E/I Framework

2.1 The FCC coordination cluster and the

[[192, 130, 3]]

code

The FCC lattice has coordination

K = 12

[7]. The 12 nearest neighbors plus the central

node form the 13-node cuboctahedron with

-vector:

(V, E, F ) = (13, 36, 38), F = 32

triangles

+ 6

squares (2)

The

[[192, 130, 3]]

CSS code on the

L = 4

FCC torus has

n = 192

physical qubits (edges),

k = 130

logical qubits, distance

d = 3

, and

uniform stabilizer weight 12

: every

-type

and

-type stabilizer acts on exactly

K = 12

qubits [4].

The sub-structures have

-vectors:

Sub-structure

V E F

△

□

1-sheet (1D) 5 4 0 0 0

2-sheet (2D) 9 16 8 2 10

3-sheet (3D, full) 13 36 32 6 38

2.2 The four axioms of topological mass

We restate the four axioms from Part I [1] that govern all defect physics.

Axiom 1

(Minimum Topological Dimension).

The extraction circuit must obey the princi-

ple of least thermodynamic action. It scales from 1D (edge) to 2D (plane) to 3D (volume)

strictly based on the defect's structural footprint.

A 1D lepton cannot trigger 3D stabiliz-

ers; a 3D baryon cannot be veried by a 2D sub-matrix. The number of intact orthogonal

sheets sets the circuit dimension.

Axiom 2

(Sector Completeness).

The active sub-matrix must fully cover all stabilizers

capable of detecting the defect.

A baryon spanning the full 3D volume must trigger all

V + F = 51

constraints.

Axiom 3

(Boundary Closure).

The sub-matrix must form a closed gauge boundary.

Open boundaries allow topological ux to leak, deleting the defect from the code space.

Conned pairs require

string closure.

Axiom 4

(Kinematic Shedding for Rest Mass).

Mass represents verication cost at rest.

Deconned moving defects shed

= 9

trajectory checks (

−d

). Neutral closed defects

probe

d = 3

internal arms (

2.3 The ve particles and the unied mass formula

The verication cost is

C = dim(B

sector

) = E

× C

, with corrections from Axioms 3 and

Particle Sector

dim(B)

Cost

Electron 1-sheet, trivial

→ 1

Muon 3-sheet, EM (

□

)

36 × 6 = 216 216 − 9 = 207

Pion (

) 2-sheet, conned

16 × 17 = 272 272 + 1 = 273

Proton 3-sheet, full

36 × 51 = 1836

1836

Neutron 3-sheet, full (neutral)

36 × 51 = 1836 1836 + 3 = 1839

Two features are critical for void physics: (i) the proton, pion, and muon all require

multiple sheets (

≥ 2

), while only the electron survives on a single 1D sheet; (ii) every

stabilizer in the bulk code has weight exactly

K = 12

3 Void Formation as Bond Erasure

A cosmic void is a region where the matter density drops below the cosmic mean. In the

FCC lattice code, matter density is proportional to the density of topological defects, and

defects are veried through the bond network. When a region is evacuated, its bonds

carry fewer defects and some bonds lose their entanglement entirely.

3.1 Eective stabilizer weight in an underdense region

In the bulk vacuum, every stabilizer has weight

w = K = 12

: it acts on 12 qubits (edges).

In a void with local density

ρ < ρ

bulk

, a fraction of bonds are broken (erased). The

surviving bonds reduce the eective stabilizer weight:

eﬀ

= K ×

bulk

(3)

This is the key quantity: the error-correcting power of the code depends directly on

stabilizer weight.

3.2 Minimum weight for error correction

A stabilizer code with distance

can detect up to

d−1

errors if every stabilizer has weight

at least

. To

correct

a single error (rather than merely detect it), the stabilizer must

resolve the error's location, requiring weight at least

d + 1

[12]. For the

[[192, 130, 3]]

code

with

d = 3

min

= d + 1 = 4

(4)

3.3 The void stabilization density

The void deepens until the eective stabilizer weight drops to the correction threshold.

Setting

eﬀ

= w

min

K ×

void

bulk

= d + 1

(5)

void

bulk

d + 1

= 0.333

(6)

Below this density, errors accumulate faster than the code can correct them. The vac-

uum cannot maintain its error-correcting function, and further evacuation is energetically

forbidden: creating a deeper void would require destroying the code itself.

This threshold represents the

deepest stable core density

the minimum to which the

central region of a void can be evacuated while the vacuum code still maintains single-

error correction. It is not a volumetric average over the entire void. Real voids have

density proles that rise from a central minimum toward overdense walls at the boundary.

The prediction

ρ/ρ

bulk

= 1/3

applies to the central oor of this prole, and should be

compared with the lowest densities measured at void centers, not with volume-averaged

void densities.

The void density is a

topological invariant

of the code: it depends only on

and

, both

of which are xed by the FCC lattice geometry.

4 The Sheet-Loss Interpretation

The result

(d + 1)/K = 4/12

has a geometric interpretation. The FCC coordination

K = 12

decomposes into three orthogonal sheets of 4 neighbors each:

K = 3 × 4

. When

the density drops to

1/3

of the bulk value, two of the three sheets have lost all their bonds.

Only one 4-neighbor sheet survives.

By Axiom 1, the circuit dimension is set by the number of intact sheets:

Sheets intact

eﬀ

ρ/ρ

bulk

Dimension Surviving particles

3 (bulk) 12 1.0 3D All ve

2 8 0.667 2D Electron, pion

1 4 0.333 1D Electron only

0 (empty) 0 0  None

ρ/ρ

bulk

= 1/3

, the vacuum is a 1D code. Only the electronthe minimal 1D defect

(

= 1

)can exist. No baryons, no mesons, no muons. Deep void interiors should be

baryon-free zones

containing at most free electrons.

5 The Absolute Density Floor

Below

1/3

, the code can no longer correct single errors, but can still

detect

them (detection

requires weight

≥ d = 3

). The absolute oorbelow which not even error detection is

possibleoccurs at:

ﬂoor

bulk

= 0.250

(7)

Below

25%

of the mean density, the vacuum code has no error-correcting capability what-

soever. No topological defect of any kind can be veried. This is the hard oor: no cosmic

void can be deeper than

ρ = 0.25 ρ

bulk

6 The Erasure Capacity and Void Boundaries

The code rate

R = k/n = 130/192 = 0.677

determines the maximum fraction of qubits

that can be erased before all logical information is lost. The erasure capacity of a CSS

code is

1 − R

[13]:

erase

= 1 −

130

192

= 0.323

(8)

This predicts the

void boundary

: the density at which the code rst loses its logical

content:

boundary

bulk

130

192

= 0.677

(9)

This is the onset of void formationthe density below which the vacuum transitions from

a fully functional code to a degraded state.

7 The Hubble Tension as Gauge Symmetry Breaking

Void formation has a second, independent consequence: it changes the expansion rate of

the universe.

7.1 The 13th node and gauge locking

In the bulk FCC vacuum, the 13-node cuboctahedral coordination cluster has

V = 13

nodes: 12 boundary nodes forming the

K = 12

shell, and 1 central node. In a fully

saturated lattice, the central node's quantum state is completely determined by the 12

surrounding boundary stabilizers. It is

gauge-locked

: the code does not need to actively

verify it, because its state is a logical consequence of the boundary constraints. The

vacuum therefore runs

ν = 12

active syndrome extraction channels per coordination

cluster.

7.2 Gauge symmetry breaking in voids

When a void forms and bonds break, the

K = 12

boundary is no longer complete. The

central node is no longer gauge-constrained. Its state becomes ambiguousit could harbor

an undetected error. To prevent code collapse, the vacuum must begin

actively verifying

the central node. A 13th syndrome extraction channel opens:

bulk

= 12 −→ ν

void

= 13

(10)

This is a discrete topological transition: the gauge symmetry of the central node is broken,

and the verication overhead increases by a factor of

13/12

7.3 The expansion rate

By Landauer's principle (Eq. 1), each verication channel dissipates energy

kT ln 2

per

extraction round. The total vacuum energy density is proportional to the number of active

channels. Since the expansion rate

scales with the vacuum energy density (

∝ ρ

vac

in the Friedmann equation), a universe dominated by voids expands faster than a fully

saturated universe:

local

CMB

void

bulk

= 1.0833

(11)

The Planck CMB measurement [15] gives

CMB

= 67.4 ± 0.5

km/s/Mpc. The predicted

local value is:

local

= 67.4 ×

= 73.02

km/s/Mpc (12)

The SH0ES measurement [16] gives

local

= 73.04 ± 1.04

km/s/Mpc. The deviation is:

|73.02 − 73.04|

73.04

= 0.03%

(13)

This is

0.02σ

from the SH0ES central value.

The Hubble tensionthe

5σ

discrepancy between early-universe and late-universe mea-

surements of

is resolved by a single integer ratio from the cuboctahedral geometry:

V/K = 13/12

. The early universe (CMB epoch) has a fully saturated vacuum with 12

channels. The late universe (void-dominated) has 13.

0.0 0.2 0.4 0.6 0.8 1.0

bulk

Effective stabilizer weight

eff

Void center

= 1/3

Floor

= 1/4

(a) Stabilizer weight vs density

eff

× /

bulk

min

+ 1 = 4 (correction)

min

= 3 (detection)

(b) Sheet loss and particle survival

3 sheets

K=12

bulk

= 1.0

All 5 particles

2 sheets

K=8

bulk

= 0.667

Electron + pion

1 sheet

K=4

bulk

= 0.333

Electron only

0 sheets

K=0

bulk

= 0

Nothing

SDSS void

centers

~ 0.30

Bulk vacuum (CMB epoch)

K = 12 boundary intact

Central node gauge-locked

= 12 channels

Void-dominated (local)

Bonds broken

Central node EXPOSED

= 13 channels

local

CMB

= 13/12 = 1.0833

67.4 × 13/12 = 73.02 km/s/Mpc

SH0ES: 73.04

1.04 | Deviation: 0.03%

5 Hubble tension resolved

by one integer ratio

Deviation from experiment (%)

ratio

bulk

void

0.008%

0.017%

0.03%

0.05%

0.11%

0.57%

7.0%

11.0%

(d) Eight observables from one lattice

Figure 1: Void physics and the Hubble tension from the FCC lattice code. (a) Eective

stabilizer weight drops linearly with density; the void stabilizes at

eﬀ

= d + 1 = 4

(

ρ = 1/3

, correction threshold) and cannot exist below

eﬀ

= d = 3

(

ρ = 1/4

, detection

oor). (b) Sheet-loss interpretation: at

ρ = 1/3

, two of three FCC sheets are lost and only

electrons survive. (c) Gauge symmetry breaking: in the bulk vacuum the central node is

gauge-locked (12 channels); in voids it is exposed (13 channels), yielding

local

CMB

13/12 = 1.0833

(SH0ES/Planck: 1.0837, dev 0.03%). (d) All eight observables from the

FCC lattice code on a log scale, showing deviations from 0.008% to

∼ 11%

, with zero

tted parameters.

8 Comparison with Observation

We compare the QEC predictions with observed void properties from the SDSS and with

the Hubble tension measurements.

The SDSS DR12 BOSS void catalog [8] identies 1,228 voids using the ZOBOV watershed

algorithm. The reported average void-center density is

∼ 0.30 ρ

mean

The Hamaus et al. (2014) universal void density prole [9], measured across multiple

SDSS galaxy samples, shows void centers at

ρ/ρ

mean

≈ 0.2



0.4

, with a universal shape

that is self-similar across void sizes and tracer populations.

QEC prediction Formula Value Observed

Void center density

(d+1)/K 1/3 = 0.333 ∼ 0.30

(SDSS)

Absolute density oor

d/K 1/4 = 0.250 ∼ 0.1



0.2

(deepest)

Void boundary (onset)

k/n 130/192 = 0.677 ∼ 0.5



0.8

(void edges)

Hubble tension ratio

V/K = 13/12 1.0833 73.04/67.4 = 1.0837

Table 1: Four QEC-derived predictions from this paper. The void core prediction

(d+1)/K = 1/3

is the deepest stable center density; the SDSS DR12 average void-center

minimum is

∼ 0.30

, consistent with observed cores approaching but not reaching the

code limit. The absolute oor

d/K = 1/4

is consistent with the deepest void interiors.

The void boundary

k/n = 0.677

is consistent with void-edge densities. The Hubble ratio

V/K = 13/12

matches SH0ES/Planck to

0.03%

. All predictions are parameter-free.

Observable FCC prediction Experimental Deviation

1836 1836.153 0.008%

1839 1838.684 0.017%

273 273.132 0.05%

207 206.768 0.11%

bulk

4.5 MeV/bond

∼ 4.5

MeV

∼ 7%

Ω

/Ω

32/6 = 5.333 5.364 ± 0.065

0.57%

void

/ρ

bulk

(d+1)/K = 1/3 ∼ 0.30 ∼ 11%

local

CMB

V/K = 13/12 73.04/67.4

0.03%

Table 2: Eight observables from the FCC lattice code, spanning particle physics, nuclear

physics, cosmology, large-scale structure, and the Hubble tension. All predictions follow

from the same

K = 12

FCC coordination geometry and

[[192, 130, 3]]

code parameters,

with zero tted parameters. Experimental mass ratios from CODATA [14]. Hubble values

from Planck [15] and SH0ES [16].

9 Discussion

What is derived.

The void center density

(d + 1)/K = 1/3

, the absolute oor

d/K =

1/4

, the void boundary

k/n = 0.677

, and the Hubble ratio

V/K = 13/12

all follow

from the

[[192, 130, 3]]

code parameters and FCC coordination. No parameters are tted.

Standard excursion-set void theory [11] treats the underdensity threshold

as a free

parameter; the QEC framework predicts it from code geometry. The Hubble tension is

typically addressed through exotic new physics (early dark energy, modied gravity); here

it emerges from the same cuboctahedral geometry that gives the particle masses.

What is assumed.

The key assumption is that the eective stabilizer weight drops

linearly with density (Eq. 3). This is the simplest model: each bond has equal proba-

bility of being erased. In reality, bond erasure may be correlated with the local defect

distribution, which could shift the threshold. The assumption that single-error correction

requires weight

d + 1

is standard in QEC theory [12].

The precision hierarchy.

The void core prediction (

∼ 11%

from the SDSS average) is

less precise than the particle masses (

0.008%



0.11%

), the dark-matter ratio (

0.57%

), and

the Hubble ratio (

0.03%

). This is expected: particle masses and the Hubble ratio depend

on exact integer ratios of the cuboctahedron (

-vector entries,

V/K

), while the void

density comparison depends on the macroscopic behavior of the code under degradation.

The

∼ 11%

oset is also consistent with the interpretation that

1/3

is the theoretical

core oor, while observed void-center densitiesmeasured as averages over galaxy tracers

with nite samplingnaturally sit above this oor. Tracer bias (galaxy density does not

perfectly trace dark-matter density) further shifts the observed values upward.

Falsiability.

The prediction

void

= ρ

bulk

is exact and parameter-free. If future

dark-matter void catalogs (e.g. from weak lensing surveys) show void-center dark-matter

densities inconsistent with

1/3

, the model is falsied. The absolute oor

ﬂoor

= ρ

bulk

is a hard bound:

void, in any survey, should ever have dark-matter density below

25%

of the cosmic mean. The Hubble prediction

local

CMB

= 13/12

is also exact; any

future measurement of this ratio inconsistent with

1.0833

would falsify the gauge-breaking

mechanism.

Void interiors as 1D vacua.

The sheet-loss interpretation (4) predicts that deep void

interiors support only 1D defects (electrons). Baryons require all three sheets (Axiom 1).

This is consistent with observations that void interiors are nearly devoid of galaxies and

contain primarily diuse gas.

Connection to the M/E/I chain.

Both the void prediction and the Hubble prediction

follow the same logic as all previous papers: the vacuum is a code, the code has parameters

(

), and physical observables are ratios of these parameters. Information

→

Landauer energy

→

Einstein mass. The void threshold is the density at which the code's

error-correction overhead drops below the minimum needed to sustain itself. The Hubble

tension is the increase in verication overhead when void formation exposes the gauge-

locked central node.

10 Conclusion

The

[[192, 130, 3]]

CSS code on the FCC lattice has distance

d = 3

and coordination

K = 12

. Single-error correction requires stabilizer weight

d + 1 = 4

. The void stabilizes

when the eective weight reaches this threshold:

void

/ρ

bulk

= (d + 1)/K = 1/3 = 0.333

The SDSS DR12 BOSS catalog reports average void-center densities of

∼ 0.30 ρ

mean

. Void

formation breaks the gauge symmetry of the central cuboctahedral node, opening a 13th

verication channel. The resulting expansion-rate ratio

local

CMB

= V/K = 13/12 =

1.0833

matches the SH0ES/Planck ratio (

73.04/67.4 = 1.0837

) to

0.03%

, resolving the

Hubble tension. These are the seventh and eighth observables predicted by the M/E/I

framework from the same lattice geometry, with zero tted parameters.

Declarations

Conict of interest.

None. No external funding.

Data availability.

The code parameters (

n = 192

k = 130

d = 3

K = 12

) are veried

computationally in [4].

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