FCC Lattice Code & Proton-Electron Mass Ratio Analysis

From the FCC Lattice Code to the Proton-Electron

Mass Ratio

Raghu Kulkarni

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

raghu@idrive.com

Abstract

The

[[192, 130, 3]]

CSS code on the Face-Centered Cubic (FCC) lattice has weight-

3 logical operators at tetrahedral voids. We show that the QCD baryon Y-junction

three color ux tubes meeting at a pointis realized as an interstitial defect at these

voids: one quark per orthogonal sheet of the triad decomposition, junction at the

interstitial node, color singlet realized as the weight-3 logical acting on the three

bounding edges. The code's coupling matrix

B ∈ {0, 1}

36×51

has fault-tolerant

syndrome extraction cost

dim(B) = 36 × 51 = 1836

. By Landauer's principle,

specifying 1836 bits requires energy

1836 × kT ln 2

; by

E = mc

2

, this energy is

mass. In a vacuum with no independent storage mediumwhere the code state

is

the vacuumthe information content of the baryon is its mass:

m

p

/m

e

= 1836

,

matching experiment to

0.008%

.

1 Introduction

The proton-to-electron mass ratio

m

p

/m

e

= 1836.15267343(11)

[11] is computed numeri-

cally by lattice QCD [12] but has no known closed-form expression. We present a deriva-

tion chainseven steps from established QCD to

m

p

/m

e

= 1836

in which every link is

either established physics, a veried computation, or a geometric theorem.

The chain begins with the

[[192, 130, 3]]

CSS quantum error-correcting code on the FCC

lattice [2]. This code has weight-3 logical operators at tetrahedral voids. We show that the

QCD baryon Y-junction [3, 5] maps onto these voids as an interstitial defect: one quark

per orthogonal sheet, junction at the inserted node, color singlet realized as the weight-3

logical. The code's coupling matrix

B

has fault-tolerant verication cost

dim(B) = 1836

.

By Landauer's principle [8] and

E = mc

2

[10], this information content is massand the

ratio

m

p

/m

e

= 1836/1

follows, with

kT ln 2/c

2

cancelling exactly between numerator and

denominator.

The single hypothesis is that the physical vacuum is the CSS code state on the FCC

latticethe unique densest sphere packing in 3D [1]. We do not claim to supersede

lattice QCD; the two approaches are complementary.

1

2 The code

The FCC lattice is the unique densest sphere packing in 3D [1]. It supports a CSS stabilizer

code with parameters

[[3L

3

, 2L

3

+2, 3]]

and encoding rate

67.7%

, veried computationally

in [2]. At

L = 4

:

n = 192

physical qubits on edges,

k = 130

logical qubits, distance

d = 3

.

The code's parity check matrices are the vertex-edge and face-edge incidence matrices of

the FCC lattice. Its weight-3 logical operators act on three edges bounding a tetrahedral

void.

3 The coordination cluster

The 13-node coordination cluster

C

13

(origin + 12 nearest neighbors) has

f

-vector

(f

0

, f

1

, f

2

) = (13, 36, 38)

, with

f

2

= 32

triangles

+6

squares and Euler characteristic

χ = 15

[2]. An interactive 3D visualization of the cluster, its three sheets, the baryon

Y-junction, and the electron is available online.

1

Denition 1.

The coupling matrix

B ∈ {0, 1}

f

1

×(f

0

+f

2

)

= {0, 1}

36×51

has

B

ec

= 1

if edge

e

participates in constraint

c

(vertex or face stabilizer).

Proposition 2.

(i)

BB

T

= ∆

1

, the edge Hodge Laplacian. (ii)

∥B∥

2

F

= nnz(B) = 192 =

n

. (iii)

dim(B) = 36 × 51 = 1836

.

Proof.

(i)

(BB

T

)

ee

′

=

P

c

B

ec

B

e

′

c

counts shared constraints, which is the unsigned Hodge

Laplacian. (ii) Each edge participates in 2 vertex stabilizers plus incident face stabilizers;

summing gives

2 × 36 + 3 × 32 + 4 × 6 = 192

. (iii)

36 × 51 = 1836

.

4 The baryon is an interstitial defect at the tetrahedral

void

Step 1: QCD baryon = Y-junction.

In the strong-coupling expansion of lattice

QCD [3, 4], the baryon is a Y-shaped string junction: three quarks (one red, one green,

one blue) connected by three color ux tubes meeting at a junction point. Lattice mea-

surements conrm this geometry [5].

Step 2: Y-junction = interstitial defect at tetrahedral void on FCC.

The

K =

12

coordination decomposes uniquely into three orthogonal sheets of 4 neighbors each:

(±1, ±1, 0)

,

(±1, 0, ±1)

,

(0, ±1, ±1)

. Each tetrahedral void is bounded by the origin and

three shell nodes, one per sheet [2]. The baryon is an extra node inserted into this

voidan interstitial defect bonded to the four bounding atoms, disrupting the vacuum

entanglement pattern. The mapping is:

1

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

2

QCD FCC lattice

Quark of color

a ↔

Shell node on sheet

a

Color ux tube

↔

Bond from interstitial to shell node

Junction point

↔

Interstitial node at void center

Color singlet

ε

abc

↔

Weight-3 logical

Z

e

1

Z

e

2

Z

e

3

Baryon

↔

Entanglement defect at void

The weight-3 logical acts on one edge per sheet, producing zero syndrome (gauge invari-

ance = color singlet) while being topologically distinct from the vacuum.

5 Fault-tolerant verication cost

In fault-tolerant syndrome extraction [6, 7], the code state is veried by checking every

(qubit, constraint) pair. Active pairs (

B

ec

= 1

) are checked via CNOT gates. Inactive

pairs (

B

ec

= 0

) must also be veried because hook errorserrors introduced by the

extraction circuitcan corrupt qubits outside a stabilizer's support [7]. Each check costs

the same circuit depth.

The total verication cost for the baryon state (spanning all three sheets, coupling to all

f

0

+ f

2

= 51

constraints) is:

C

p

= dim(B) = f

1

× (f

0

+ f

2

) = 36 × 51 = 1836

(1)

The electron cost.

The electron is a single edge error on one sheet. On a single sheet,

F

s

= 0

(Theorem in [2]: no two shell nodes within a sheet are adjacent, so no faces ex-

ist). Without faces, there are no cross-sheet stabilizers through which hook errors could

propagate. Fault-tolerant verication of

B

ec

= 0

entries is unnecessary on a faceless

graphnon-interactions between decoupled sheets are automatic. The electron's veri-

cation cost is therefore its syndrome change alone: one edge error producing one ipped

bit.

C

e

= 1

(2)

This is not an assumptionit follows from

F

s

= 0

. On the full 3D cluster (

F = 38

,

all cross-sheet), every

B

ec

= 0

must be checked because hook errors propagate through

faces. On a single 2D sheet (

F

s

= 0

), no propagation path exists, so only active changes

contribute.

6 Information, energy, mass

Link 1: Information has energy.

Specifying

N

bits requires energy

≥ N × kT ln 2

(Landauer 1961 [8], experimentally veried [9]).

Link 2: Energy has mass.

E = mc

2

(Einstein 1905 [10]).

Link 3: The vacuum has no substrate.

For a hard drive, the Landauer energy

(

∼ 10

−25

kg/bit) is negligible next to the substrate mass (

∼ 0.1

kg). But if the vacuum

3

E

s

= 4

,

V

s

= 5

,

F

s

= 0

ELECTRON

(1 edge, 1 sheet)

xy-sheet (red)

E

s

= 4

,

V

s

= 5

,

F

s

= 0

xz-sheet (green)

E

s

= 4

,

V

s

= 5

,

F

s

= 0

yz-sheet (blue)

1.5

1.0

0.5

0.0

0.5

1.0

1.5

x

1.5

1.0

0.5

0.0

0.5

1.0

1.5

y

1.5

1.0

0.5

0.0

0.5

1.0

1.5

z

3D projection: 3 sheets coupled

38 faces emerge, all cross-sheet

0 10 20 30 40 50

51 constraints

0

5

10

15

20

25

30

35

36 edges

V=13 F_tri=32 F_sq=6

Coupling matrix B (36 x 51)

dim(B) = 1836, nnz = 192

ELECTRON (1 sheet,

F

s

= 0

):

1 edge error, 1 syndrome bit

No faces no hook errors

C

e

= 1

PROTON (3 sheets,

F

= 38

):

Interstitial at tet void

All faces cross-sheet

C

p

= dim(

B

) = 36 × 51 = 1836

MASS RATIO:

m

p

m

e

=

C

p

kT

ln 2 /

c

2

C

e

kT

ln 2 /

c

2

=

1836

1

The mass ratio

3D projection: sheets couple through 38 emergent faces

Figure 1: Top row: the three orthogonal 2D sheets of the FCC triad

τ = (4, 4, 4)

. Each is

a star graph

K

1,4

with

F

s

= 0

faces (dashed lines show non-existent shellshell edges). The

electron (orange highlight, top left) is a single edge error on one sheetno faces means no

hook error propagation, so

C

e

= 1

. Bottom left: the 3D FCC cluster formed by coupling

all three sheets; 38 cross-sheet faces emerge, creating the constraint structure that gives

dim(B) = 1836

. The baryon (gold diamonds) is an interstitial defect at the tetrahedral

void, spanning all three sheets. Bottom center: the coupling matrix

B

(colored = active,

gray = protected non-interactions). Bottom right: the mass ratio with explicit

kT ln 2/c

2

cancellation.

is

the code state, there is no separate substrate. An excitation above the vacuum is a

pattern of syndrome bits with no independent carrier. Its information content is its only

mass. The FCC lattice, being the unique densest 3D packing [1], is the maximally ecient

encodingno lighter alternative exists.

Combining Links 13: the baryon has information content

C

p

= 1836

bits; the electron

has

C

e

= 1

bit. Their masses are

m

p

= C

p

× kT ln 2/c

2

and

m

e

= C

e

× kT ln 2/c

2

. The

ratio:

m

p

m

e

=

C

p

× kT ln 2 / c

2

C

e

× kT ln 2 / c

2

=

C

p

C

e

=

dim(B)

1

=

1836

1

= 1836

(3)

The temperature

T

, Boltzmann constant

k

, and speed of light

c

cancel exactly. While

absolute masses may depend on the vacuum temperature, the

ratio

is a pure topological

invariant of the FCC coordination clusterindependent of temperature, energy scale, or

cosmological epoch.

4

Experimental:

m

p

/m

e

= 1836.153

[11]. Deviation:

0.008%

.

7 Summary

Step Statement Source Status

1

[[192, 130, 3]]

CSS code on FCC [2] Veried

2 Weight-3 logicals at tet voids [2] Computed

3 QCD baryon = Y-junction [3, 5] Established

4 Y-junction = interstitial at void Geometry + [2] Proved

5 FT verication cost

= dim(B)

[6, 7] Established

6 Information

→

energy [8, 9] Veried

7 Energy

→

mass [10] Established

The single hypothesis is that the physical vacuum is the CSS code state on the FCC

lattice. Every other step is either established physics or a computation veried in [2].

This hypothesis is motivated by the FCC lattice being the unique densest packing, and

is testable: it predicts

m

p

/m

e

= dim(B) = 1836

, which matches experiment to

0.008%

.

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