Mass-Energy-Information Equivalence: Nuclear Binding as FCC Max-Cut Deduplication

MassEnergyInformation Equivalence II:

Nuclear Binding as Max-Cut Deduplication

on the FCC Lattice Code

Raghu Kulkarni

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

raghu@idrive.com

Abstract

Part I of this framework [1] established the MassEnergyInformation (M/E/I)

equivalence: if the physical vacuum is a

[[192, 130, 3]]

CSS code on the FCC lat-

tice [2], particle mass equals fault-tolerant verication cost [3], and ve particle

mass ratios follow from the lattice geometry. This paper extends the framework

to multi-nucleon systems. When nucleons occupy adjacent FCC voids, their shared

stabilizer constraints are deduplicated by Landauer's principle [3]the verication

cost of the bound state is less than the sum of the parts. Maximizing the number

of protonneutron bonds is the Max-Cut problem [5] on the FCC cluster graph.

Running this optimization for 21 nuclei from

H to

238

U, we nd: (i) the eective

deduplication energy is

α ≈ 4.5

MeV/bond, approximately constant for all

A > 6



an emergent value, not tted; (ii) the volume and surface terms of the semi-empirical

mass formula [6] arise from the FCC cluster geometry (interior

K=12

vs boundary

K<12

); (iii) the asymmetry term is recovered within the QEC framework via the

Shannon congurational entropy of the CSS stabilizer partition: expanding





around

Z = A/2

yields a penalty

∼ (A − 2Z)

, whose coecient is the stan-

dard Weizsäcker value

= 23.2

MeVthe same status as the Coulomb coecient

0.72 MeV [7], both applied from nuclear physics. With all four terms, the model

predicts

exactly for all light elements (

A ≤ 16

) and Fe-56, within

±1

for 71%

(15/21) of nuclei, and within

±3

for 81% (17/21).

Keywords:

massenergyinformation equivalence; nuclear binding energy; Max-Cut;

FCC lattice; quantum error correction

1 Introduction

Part I of this framework [1] established the MassEnergyInformation (M/E/I) equiva-

lence on the FCC lattice: if the physical vacuum is a

[[192, 130, 3]]

CSS quantum error-

correcting code [2], then a particle's mass is its fault-tolerant verication cost in units of

Landauer's bit-erasure energy

kT ln 2

[3], converted to mass via

E = mc

[4]. Any lo-

calized excitation

in the topological vacuum requires continuous verication to prevent

code collapse. The fault-tolerant verication cost

is the total number of classical bits

generated by checking the quantum constraints associated with the defect, giving mass:

= C

kT ln 2

(1)

When computing ratios, the environmental variables (

) cancel exactly:

. Mass ratios are pure topological invariants of the vacuum network.

Part I applied four thermodynamic axioms to lter 15 candidate defect types on the FCC

coordination clusterthe 13-node cuboctahedron with

-vector

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

where

F = 32

triangles

squaresdown to exactly 5 physically stable particles. Their

verication costs

C = 1, 207, 273, 1836, 1839

match the electron, muon, pion, proton, and

neutron to within 0.12%.

For reference, the four axioms from Part I are stated in full below, as they govern the

nuclear extension developed here.

The Four Axioms of Topological Mass

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 arbitrarily trig-

ger 3D cross-sheet stabilizers, and a 3D baryon cannot be veried by a 2D sub-matrix.

The FCC coordination cluster decomposes uniquely into three orthogonal 2D sheets of 4

neighbors each; the number of sheets disrupted by the defect sets the circuit dimension.

Axiom 2

(Sector Completeness).

The active sub-matrix must fully cover all stabilizers

capable of detecting the defect.

The circuit cannot selectively ignore constraints that

intersect the defect's spatial footprint. A baryon spanning the full 3D volume must

trigger all

V + F = 51

constraints; it cannot ignore the square faces to save energy.

Axiom 3

(Boundary Closure).

The sub-matrix must form a closed gauge boundary.

Checking vertices but ignoring faces within an active sector allows topological ux to

leak, immediately deleting the defect from the code space. For a conned color pair on

a 2D sheet, the boundary remains open until closed by a 1D string to prevent monopole

leakage (

Axiom 4

(Kinematic Shedding for Rest Mass).

Mass represents verication cost at

rest.

If a closed defect is deconned (spatially extended but not pinned by color), the

lattice's

translational checks measure kinematic momentum across the

d = 3

extraction

rounds; these

= 9

dynamic trajectory checks must be subtracted to isolate rest mass

(

−d

). Kinematic shedding applies only when

dim(B

sector

) > d

. Conversely, if a defect

is perfectly neutral and topologically closed, its boundary mimics the vacuum; the circuit

must probe its internal

d = 3

geometry to conrm the defect exists (

From single particles to nuclei

The unied mass formula from Part I is

C = dim(B

sector

)

, where

sector

∈ {0, 1}

×C

is the

coupling sub-matrix of the defect. The proton mass emerges as

dim(B

full

) = E×(V +F ) =

36 × 51 = 1836 m

This paper asks: what happens when two or more defects occupy adjacent voids? If

mass is verication cost, then binding energy is verication

savings

the reduction in

total cost when two particles share stabilizer constraints that need only be checked once.

By Landauer's principle, fewer bits veried means less energy, hence less mass. Binding

energy is information deduplication.

The natural question is: which arrangement of

protons and

A − Z

neutrons in a

compact FCC cluster minimizes the total verication cost? This is equivalent to max-

imizing the number of protonneutron (



) bondsthe Max-Cut problem [5] on the

FCC subgraphsubject to classical Coulomb repulsion [7] and a congurational entropy

penalty.

The BetheWeizsäcker semi-empirical mass formula [6] describes nuclear binding with

ve tted parameters. We show that all four major terms (volume, surface, Coulomb,

asymmetry) emerge from the FCC information geometry. We report results for 21 nuclei

from

H to

238

U, including all failures. Reproducible code is in the appendix.

2 From Verication Cost to Binding Energy

2.1 The deduplication principle

Consider two nucleons at adjacent FCC void sites. Each nucleon's verication requires

checking its own coupling sub-matrix

. When their coordination shells overlap, some

stabilizer constraints appear in both matrices. By Axiom 1 (minimum thermodynamic

action), the vacuum circuit checks each shared constraint once, not twice. The shared

constraints are

deduplicated

For a protonneutron (



) pair, the shared boundary has non-trivial holonomy: the

proton carries parity bit

(non-trivial,

= −1

) and the neutron carries parity bit

(trivial,

= +1

). The combined parity is

1 ⊕ 0 = 1

(non-trivial), so Axiom 3

(Boundary Closure) is satisedthe gauge boundary survives. For a



pair,

the combined parity is

1 ⊕ 1 = 0

0 ⊕ 0 = 0

(trivial): the shared boundary registers as

vacuum and the deduplication vanishes. Only



bonds contribute binding.

Each



bond deduplicates a set of shared stabilizer constraints. By Landauer's principle

(Eq. 1), each deduplicated bit reduces the system's mass. The binding energy from

bonds is:

topo

= α · N

(2)

where

is the energy per deduplicated bond. The value of

is not ttedit is determined

self-consistently (Section 2.6).

2.2 The Max-Cut structure

Maximizing

for a given

and

on the FCC cluster graph is the Max-Cut graph

partitioning problem [5]: assign each of

nodes to one of two classes (proton or neutron),

with

nodes in one class, to maximize the number of edges crossing the partition. The

FCC lattice's

K = 12

coordination makes this a dense Max-Cut instance. We solve it via

multi-restart greedy optimization (Appendix A).

2.3 The three geometric terms

The Max-Cut on an FCC cluster naturally generates three terms of the semi-empirical

mass formula:

Volume.

Interior nodes have the full

K = 12

neighbors. A perfect alternating



coloring gives each interior node 6 opposite-parity bonds. At

α = 4.5

MeV/bond, this

yields

∼ 27

MeV per interior nucleon, close to twice the empirical volume coecient

(

2 × 15.8 = 31.6

MeV, since each bond is shared by two nodes).

Surface.

Boundary nodes have

K < 12

. The edges-per-nucleon ratio grows from 0.5

(

A = 2

) to 4.7 (

A = 238

) as the surface-to-volume ratio shrinks. This is the surface

correction, arising from cluster geometry without any tted

Coulomb.

Applied directly as

0.72 · Z(Z−1)/A

1/3

MeV [7]. This is one of two physics

inputs from outside the lattice.

2.4 The asymmetry term from CSS congurational entropy

Without an asymmetry term, the model systematically underpredicts

for

A > 20

. The

classical Max-Cut penalizes unequal partitions too weakly: a coloring with

Z = 0.35A

still achieves

∼ 90%

of the Max-Cut value at

Z = A/2

. Coulomb wins too easily, pushing

down.

We recover the asymmetry term within the QEC framework. In a CSS code,

-type and

-type errors are decoded independently by separate parity-check matrices

and

When the vacuum veries a nucleus of

defects, it measures a mixed conguration of

protons and

N = A − Z

neutrons. The number of distinct arrangements is

Ω =





. The

congurational entropy is

mix

= k





Expanding via Stirling's approximation around

Z = A/2





≈ A ln 2 −

(A − 2Z)

(3)

By Landauer's principle, any reduction in the maximum congurational entropy of the

code requires an equivalent expenditure of verication energy. The vacuum imposes an

energetic penalty for deviating from

Z = N

asym

= γ ·

(A − 2Z)

(4)

This has the exact functional form of the Weizsäcker asymmetry term. The coecient

has the same status as the Coulomb coecient: it is a standard nuclear physics input.

We adopt the Weizsäcker value directly:

γ = a

= 23.2

MeV (5)

The QEC framework derives the

functional form

(A − 2Z)

from the CSS entropy

structure; the coecient is applied from experiment, just as the Coulomb coecient

0.72 MeV is applied from electrostatics [7].

2.5 The complete score function

Combining all terms, the ground-state proton number

for a nucleus of mass

maxi-

mizes:

S = α · N

− 23.2 ·

(A − 2Z)

− 0.72 ·

Z(Z−1)

1/3

(6)

The model has one emergent constant (

, determined self-consistently) and two applied

physics inputs (

= 0.72

MeV,

= 23.2

MeV).

2.6 Self-consistent determination of

The parameter

is not tted to nuclear data. For each nucleus, we compute:

eﬀ

exp

+ E

Coulomb

+ E

asym

(7)

where

exp

is the experimental binding energy [8] and

is the computed Max-Cut at

the experimental

. If

eﬀ

is approximately constant across all

, the model is internally

consistent.

3 Results

3.1 The eective deduplication energy is constant

Figure 1(c) shows

eﬀ

for all 21 nuclei. Excluding the deuteron (

A = 2

, surface-dominated,

α = 2.2

MeV) and

He (

A = 4

, magic number,

α = 7.3

MeV), the eective energy per

bond is:

eﬀ

= 4.5 ± 0.3

MeV/bond for all

6 ≤ A ≤ 238

(8)

This constancy is not imposedit emerges from the ratio of experimental binding energies

to the computed Max-Cut bond counts. It conrms that the FCC geometry correctly

captures the volume-to-surface transition: interior bonds deduplicate more constraints

than surface bonds, and the cluster geometry accounts for this automatically through the

coordination number of each node.

3.2 Full survey: 21 nuclei

Table 1 and Figure 1 show results with

α = 4.5

MeV/bond and

γ = 23.2

MeV.

Nuclide

A Z

pred

exp

∆Z BE/A

pred

BE/A

exp

H 2 1 1 0 2.25 1.11

He 4 2 2 0 4.27 7.07

Li 6 3 3 0 5.60 5.33

Li 7 3 3 0 5.63 5.61

Be 9 4 4 0 5.75 6.46

C 12 6 6 0 6.71 7.68

N 14 7 7 0 7.78 7.48

O 16 8 8 0 8.00 7.98

Ne 20 9 10

−1

8.26 8.03

Mg 24 12 12 0 8.19 8.26

Si 28 13 14

−1

8.52 8.45

S 32 15 16

−1

8.69 8.49

Ca 40 18 20

−2

8.73 8.55

Fe 56 26 26 0 9.27 8.79

Ni 58 27 28

−1

9.12 8.73

Se 80 38 34

9.72 8.71

Zr 90 41 40

9.09 8.71

120

Sn 120 54 50

8.96 8.51

150

Sm 150 65 62

8.67 8.28

208

Pb 208 90 82

8.18 7.87

238

U 238 103 92

+11

7.76 7.57

Table 1: Full 21-nucleus survey with

α = 4.5

MeV/bond and

γ = 23.2

MeV. Exact

: 10/21 (48%), including all

A ≤ 16

, Mg-24, and Fe-56. Within

±1

: 15/21 (71%).

Within

±3

: 17/21 (81%). Deviations for

A > 100

reect the greedy Max-Cut heuristic's

convergence limitations. Results are reproducible with the appendix code (

seed=42

4 Discussion

The M/E/I chain at nuclear scale.

Part I established: mass = verication cost =

information content

× kT ln 2/c

. This paper extends the chain: binding energy = veri-

cation

savings

= deduplicated information

× kT ln 2/c

. The temperature and constants

cancel in ratios, leaving a pure graph-theoretic quantity: the Max-Cut of the FCC cluster.

All four Weizsäcker terms.

The model accounts for each term:

Term Weizsäcker FCC/QEC origin Status

Volume

Interior

K=12 × α

Emerges

Surface

−a

2/3

Boundary

K<12

Emerges

Coulomb

−a

Z(Z−1)/A

1/3

Electrostatic Applied

Asymmetry

−a

(A−2Z)

CSS entropy, Eq. (4) Form derived, coe. applied

What is derived vs applied.

The functional form

(A − 2Z)

is derived from the

CSS congurational entropy (Eq. 3). The coecient

γ = a

= 23.2

MeV is applied from

nuclear physics, with the same status as the Coulomb coecient 0.72 MeV. The model has

one emergent constant (

α ≈ 4.5

MeV/bond) and two applied physics inputs (

= 0.72

= 23.2

0 50 100 150 200

Mass number A

0.25

0.30

0.35

0.40

0.45

0.50

0.55

Z/A

(a) Proton fraction: model vs experiment

Experimental

FCC Max-Cut ( =2.0)

0 50 100 150 200

Mass number A

Bonds per nucleon

(b) Bond density: surface interior

Total edges/A

p-n bonds/A (Max-Cut)

0 50 100 150 200

Mass number A

eff

(MeV/bond)

Deuteron

(surface)

He-4

(magic)

bulk

= 4.5 MeV/bond

0 50 100 150 200 250

Mass number A

12.5

10.0

7.5

5.0

2.5

0.0

2.5

pred

exp

(d) Z prediction error (green: ±1, orange: ±3, red: >3)

Figure 1: Full survey results. (a) Proton fraction

Z/A

: model (red) vs experiment (black).

(b) Bond density: edges per nucleon and



bonds per nucleon grow as

increases

(surface-to-volume ratio shrinks). (c) Eective deduplication energy: nearly constant at

4.5

MeV/bond for

A > 6

. Deuteron (2.2) and He-4 (7.3) are outliers. (d)

prediction

error: green

|∆Z| ≤ 1

, orange

≤ 3

, red

> 3

The precision hierarchy.

The M/E/I framework shows a gradient of accuracy across

scales: isolated particles (

= 1836

, 0.008%)

nucleon splitting (

− m

= 3 m

18.5%)

bulk deduplication (

α ≈ 4.5 ± 0.3

MeV, 7%)

proton number (

exact for

10/21 including Fe-56; within

±1

for 71%).

The deuteron outlier.

A = 2

eﬀ

= 2.2

MeVroughly half the bulk value. Part I

derived

4 m

= 2.04

MeV binding for the deuteron from 4 shared coordination shell

nodes. The factor-of-two between the deuteron

and the bulk

reects the transition

from surface-dominated to bulk-dominated deduplication.

Open questions.

The pairing term (5th Weizsäcker term) is not modeled. Whether

can be derived from the code parameters (

n = 192

k = 130

d = 3

) rather than applied

from experiment is an open problem.

Declarations

Conict of interest.

None. No external funding.

Data availability.

All code is in the appendix and at

https://github.com/

raghu91302/ssmtheory/blob/main/mei2_nuclear_simulation.py

References

[1] R. Kulkarni, The MassEnergyInformation Equivalence: A Bottom-Up Identica-

tion of the Particle Spectrum via FCC Lattice Error Correction, Zenodo (2026).

doi:10.5281/zenodo.19248556

[2] R. Kulkarni, arXiv:2603.20294 (2026). arXiv:2603.20294

[3] R. Landauer, IBM J. Res. Dev.

, 183 (1961). doi:10.1147/rd.53.0183

[4] A. Einstein, Ann. Phys.

323

, 639 (1905). doi:10.1002/andp.19053231314

[5] R. M. Karp, in

Complexity of Computer Computations

, Springer (1972).

doi:10.1007/978-1-4684-2001-2_9

[6] C. F. v. Weizsäcker, Z. Phys.

, 431 (1935). doi:10.1007/BF01337700

[7] K. S. Krane,

Introductory Nuclear Physics

, Wiley (1988).

[8] E. Tiesinga

et al.

, Rev. Mod. Phys.

, 025002 (2024).

doi:10.1103/RevModPhys.97.025002

A FCC Max-Cut Nuclear Simulation

The following self-contained Python script reproduces all results in this paper. It builds

FCC clusters, optimizes the Max-Cut for each

, applies the Coulomb and Shannon asym-

metry penalties, and reports the optimal

. Requires only

numpy

and

scipy

. Full version

with documentation:

https://github.com/raghu91302/ssmtheory/blob/main/mei2_

nuclear_simulation.py

#!/usr/bin/env python3

"""

MEI Paper II: Nuclear Binding from FCC Max-Cut

Reproduces all results: alpha, gamma scan, Z predictions.

"""

import numpy as np

from scipy.spatial.distance import pdist

def fcc_cluster(A):

"""Build spherical FCC cluster of A nodes."""

pts = []

r = 1

while len(pts) < A:

for x in range(-r, r+1):

for y in range(-r, r+1):

for z in range(-r, r+1):

if (x+y+z)%2==0 and (x,y,z) not in pts:

pts.append((x,y,z))

r += 1

pts.sort(key=lambda p: p[0]**2+p[1]**2+p[2]**2)

return pts[:A]

def adj_list(coords):

"""FCC nearest-neighbor adjacency."""

cs = {c:i for i,c in enumerate(coords)}

NN = [(1,1,0),(1,-1,0),(-1,1,0),(-1,-1,0),

(1,0,1),(1,0,-1),(-1,0,1),(-1,0,-1),

(0,1,1),(0,1,-1),(0,-1,1),(0,-1,-1)]

adj = [[] for _ in range(len(coords))]

for i,(x,y,z) in enumerate(coords):

for dx,dy,dz in NN:

nb = (x+dx,y+dy,z+dz)

if nb in cs: adj[i].append(cs[nb])

return adj

def maxcut(adj, A, Z, restarts=4, iters=2500):

"""Multi-restart greedy Max-Cut."""

best = 0

for _ in range(restarts):

s = np.zeros(A, dtype=np.int8)

s[np.random.choice(A, Z, replace=False)] = 1

cut = sum(1 for i in range(A)

for j in adj[i] if j>i and s[i]!=s[j])

for _ in range(iters):

p = np.where(s==1)[0]

n = np.where(s==0)[0]

if len(p)==0 or len(n)==0: break

pi = p[np.random.randint(len(p))]

ni = n[np.random.randint(len(n))]

d = 0

for j in adj[pi]:

if j!=ni: d += 1 if s[j]==1 else -1

for j in adj[ni]:

if j!=pi: d += 1 if s[j]==0 else -1

if d > 0:

s[pi], s[ni] = 0, 1; cut += d

elif d==0 and np.random.random()<0.2:

s[pi], s[ni] = 0, 1

if cut > best: best = cut

return best

# Experimental data: A -> (Z, BE/A MeV, name)

exp = {

2:(1,1.112,'2H'), 4:(2,7.074,'4He'),

6:(3,5.333,'6Li'), 7:(3,5.606,'7Li'),

9:(4,6.463,'9Be'), 12:(6,7.680,'12C'),

14:(7,7.476,'14N'), 16:(8,7.976,'16O'),

20:(10,8.032,'20Ne'), 24:(12,8.261,'24Mg'),

28:(14,8.448,'28Si'), 32:(16,8.493,'32S'),

40:(20,8.551,'40Ca'), 56:(26,8.790,'56Fe'),

58:(28,8.732,'58Ni'), 80:(34,8.711,'80Se'),

90:(40,8.710,'90Zr'), 120:(50,8.505,'120Sn'),

150:(62,8.278,'150Sm'),208:(82,7.868,'208Pb'),

238:(92,7.570,'238U'),

}

ALPHA = 4.5; GAMMA = 23.2

np.random.seed(42)

# Pre-compute Max-Cut for all (A, Z)

mc = {}

for A in sorted(exp.keys()):

coords = fcc_cluster(A)

adj = adj_list(coords)

zmin = max(1, int(A*0.20))

zmax = min(A-1, int(A*0.55))

if A < 10: zmin, zmax = 1, A-1

for Z in range(zmin, zmax+1):

mc[(A,Z)] = maxcut(adj, A, Z, 3,

min(A*12, 3500))

# Predict Z for each nucleus

hdr = f"{'A':>4} {'name':<7} {'Zp':>3} {'Ze':>3}"

hdr += f" {'dZ':>4} {'pn':>5} {'BE/A':>6} {'exp':>6}"

print(hdr); print("-"*45)

for A in sorted(exp.keys()):

Ze, BE, nm = exp[A]

zmin = max(1, int(A*0.20))

zmax = min(A-1, int(A*0.55))

if A < 10: zmin, zmax = 1, A-1

bZ=zmin; bS=-1e9; bPN=0

for Z in range(zmin, zmax+1):

if (A,Z) not in mc: continue

pn = mc[(A,Z)]

asym = GAMMA * (A-2*Z)**2 / A

coul = 0.72*Z*(Z-1)/(A**(1/3)) if A>1 else 0

sc = ALPHA*pn - asym - coul

if sc > bS: bS=sc; bZ=Z; bPN=pn

dz = bZ - Ze

m = "ok" if dz==0 else f"{dz:+d}"

print(f"{A:>4} {nm:<7} {bZ:>3} {Ze:>3}"

f" {m:>4} {bPN:>5} {bS/A:>6.2f} {BE:>6.3f}")