The Mass–Energy–Information Equivalence Extended: D4 Lattice, Triality, and the Three-Generation Structure of Matter

The Mass–Energy–Information Equivalence Extended:

Lattice, Triality, and the

Three-Generation Structure of Matter

Raghu Kulkarni

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

Abstract

Part I derived the rest-mass ratios of the electron, muon, pion, proton and neutron from a Calderbank–

Shor–Steane stabilizer code on the face-centered cubic lattice, the root lattice

, matching experiment to

within 0.12 % with no ﬁtted parameters. It could not reach the tauon, the neutrinos, the gauge bosons

or the Higgs. We lift the construction to the

lattice, the densest four-dimensional packing, with one

coordinate selected as time; the spatial slice is exactly

, so the earlier predictions are unchanged. Five

structural hypotheses are stated in advance, and every result is tagged with those it requires.

Three results are computational. The sector counts used by the mass formulas are exact stabilizer

counts of a vertex–plaquette code on the same lattice, which extends to four chain-complex codes

whose logical spaces realize the Betti numbers of the 4-torus,

(4,6,4,1)

. The

code has parameters

[[1536,1282,3]]

with uniform weight-24 stabilizers. The 24 nearest-neighbor bonds form a spherical

5-design, isotropic through fourth order, where the

shell is isotropic through second order only.

Tree-level direction-dependence of the lattice dispersion is therefore suppressed by

(ka)

rather than

(ka)

. Emergent Lorentz invariance is left open, alongside the connection to the Standard Model gauge

group.

Two mass predictions are not parameter-free. The tauon cost is

3447

against a measured

3477.23

given two stated counting conventions, and the Higgs cost of

244,944

follows from the same shedding

axiom, within experimental uncertainty. Neutrinos and gauge bosons are treated qualitatively.

1. Introduction

1.1. The framework in brief

This subsection sets out the results of Refs. [1, 2] that the present paper uses.

The starting idea. Landauer’s principle says that erasing or irreversibly generating one classical bit costs

at least

kT ln 2

of energy [

]. Einstein’s relation says energy has mass. Put together, one bit carries a mass

kT ln 2/c

. In any laboratory object this mass is hopeless to see, because the hardware holding the bit weighs

times more. The framework asks what happens if there is no hardware: if the vacuum itself is a quantum

error-correcting code, a localized excitation of it has no substrate, and its information content is the only

thing it can weigh.

raghu@idrive.com

Particles as defects. The vacuum is modeled as a Calderbank–Shor–Steane (CSS) stabilizer code on a

lattice [

]; the construction is set out in Section 3.3. Physical qubits sit on the edges. Stabilizer measurements

check the state and return a syndrome. A particle is a localized defect: a pattern in the lattice that the

stabilizers detect. Detecting it costs bits, and by the paragraph above those bits cost mass.

The cost formula. The central quantity is the fault-tolerant veriﬁcation cost

of a defect

. It is the size

of the coupling sub-matrix that the syndrome-extraction circuit must handle:

= E

×C

where

is the number of edges in the defect’s geometric footprint and

is the number of stabilizers capable

of detecting it. Both are counted from the lattice, not ﬁtted. The mass follows as m

= C

kT ln 2/c

Why only ratios are predicted. The temperature

is unknown and the framework does not ﬁx it. But

every particle is an excitation of the same vacuum at the same

, so in a ratio of two masses the factor

kT ln 2/c

cancels exactly:

= C

. All predictions in this line of work are therefore dimensionless

integer ratios, taken relative to the electron (C

= 1).

What has been derived so far. Part I [

] placed the code on the densest three-dimensional lattice—the face-

centered cubic lattice, written

throughout this paper for the reason given in Section 1.2—and counted ﬁve

defect geometries that survive four topological axioms. Their costs are

207

273

1836

and

1839

, matching

the measured electron, muon, pion, proton and neutron ratios to better than

0.12%

, with no ﬁtted parameters.

The proton is the clearest case: its footprint has

= 36

edges,

= 13 + 38 = 51

detecting stabilizers, and

36 ×51 = 1836

against a measured

1836.15

. The underlying code,

[[192,130,3]]

with uniform weight-12

stabilizers, is constructed and veriﬁed separately [2].

What was left out, and why this paper exists. The three-dimensional construction had no room for the

tauon, the neutrinos, the gauge bosons, the heavier hadrons, or the Higgs. Section 1.2 explains why each

omission traces to the same limitation, and the rest of the paper adds one lattice dimension to address them.

1.2. Scope of the present paper

Part I argued that a particle’s rest mass is, at root, the thermodynamic cost of detecting the topological defect

it represents in a quantum error-correcting vacuum [

]. The construction sat on the

lattice (the densest

packing in three dimensions) and reproduced the masses of the electron, muon, pion, proton, and neutron as

small integers derived from f-vector enumeration. The accuracy was high (sub-0.12 % across all ﬁve), but

the scope was narrow. The tauon did not appear: the muon and the electron are distinguished in Ref. [

] by

the size of their spatial footprint, which the

coordination shell supplies in two sizes and no more, so a

third charged lepton had nowhere to sit (the point becomes hypothesis H2b, stated in Section 2 and used in

Section 8). Neutrinos sat below the topological threshold of one bit and could not be reached. Strangeness,

charm, bottom, and top were absent, as were the W and Z masses. The Higgs was set aside as a condensate

mode outside the defect classiﬁcation.

What the omissions have in common. These omissions all share a feature: each involves either an extra

generation, a sub-threshold defect, an extended (2D) object, or a vacuum condensate. None of these is

well-described by static 3D geometry alone. Adding a single dimension (time, treated lattice-theoretically

rather than as a continuum parameter) enlarges the space of geometrically available defect classes. Defects can

now be worldlines, worldsheets, or condensates, and triality, a symmetry that exists only in four dimensions,

provides the framework with a candidate threefold generation label.

This paper lifts the construction to the

lattice, provably the densest lattice packing in four dimensions

and the densest sphere packing known there.

A note on notation. Throughout this paper the face-centered cubic lattice is written

. The two names

describe one object from the crystallographic and the lattice-theoretic side. Crystallographically, take the

conventional cubic cell of side 2 and place points at its eight corners and six face centers: every such point has

even coordinate sum, the nearest-neighbor distance is

√

, and each point has 12 nearest neighbors arranged

as a cuboctahedron. Lattice-theoretically,

x ∈ Z

∑

even

, (1)

and

is exactly that set of corner-and-face-center points [

, §6.3]. (In three dimensions

∼

, so the

same lattice appears in the literature under the latter name as well.) Writing

lets the lift of this paper read

→ D

(one family, one dimension apart, Eq.

(1)

n = 3

and

n = 4

) rather than as a change of setting.

We show in Section 3 that the same CSS-code structure carries over, with weight-24 stabilizers, an asymptotic

encoding rate of

5/6

, and minimum distance exactly 3. The derivation is organized as a conditional one:

Section 2 states ﬁve structural hypotheses, and every result is derived from the code given them, with each

result’s dependence recorded. We add one axiom in Section 5: spontaneous selection of a time direction

(hypothesis H1), breaking the lattice’s coordinate-permutation symmetry

, with translations along

the selected coordinate remaining available. Section 3.2 states carefully what the lattice does and does not

supply on the question of continuum symmetry. The spatial slice of

is then exactly

, and the ﬁve Part I

predictions survive the lift unchanged (Section 7).

What is new here. The new content is structural. The local 24-cell of nearest neighbors (the polytope

whose geometry is set out in Section 3, with its incidence structure collected in Figure 1) decomposes

into three interpenetrating 16-cells under a symmetry called triality, which permutes the three components

cyclically (Section 6). The engagement depth of a lepton worldline is its level of participation in the triality

decomposition: localized within one component, the spatial halves of all three, or full spacetime support.

It takes three values; under hypothesis H2, that these levels exhaust the admissible lepton structures and

correlate with ﬂavor, the lepton sector admits exactly three generations, and a fourth-generation observation

would falsify H2. Section 4 establishes the structural foundation for all of this: the sector counts of Parts I–

II are exact stabilizer counts of a vertex–plaquette code on the lattice, and the construction extends to a

family of chain-complex codes on

whose logical spaces realize the cellular homology of the 4-torus.

From this structure we extract a quantitative prediction for the tauon (Section 8), the three-ﬂavor neutrino

sector (Section 9), the gauge bosons as 2D worldsheets (Section 10), the heavier hadron families through

second-shell defects (Section 11), and the Higgs cost as a vacuum subtraction grounded in the packing axiom

(Section 12). Section 14 reports the comparison with experiment; Section 15 lists the open problems with

concrete resolution paths; Section 16 concludes.

2. The Conditional Derivation: Five Structural Hypotheses

This paper derives the ﬁrst-shell particle spectrum from the

CSS code together with ﬁve explicitly stated

structural hypotheses. We state them here, before any construction, so that the logical status of every result is

ﬁxed in advance: everything that follows is derived unconditionally from the code given H1–H5, and nothing

in the derivation is adjusted to the experimental targets. This is the standard architecture of ﬁrst-principles

mass computations—lattice quantum chromodynamics (QCD) derives the hadron spectrum conditional on

the QCD action and its discretization; grand uniﬁed models derive fermion mass relations conditional on

assumed representation content—and it has a speciﬁc virtue: if a prediction fails, the failure is localized to a

named hypothesis rather than diffused through the framework.

H1 (Foliation).

One of the four coordinate directions of the

lattice is selected as time (the four are

equivalent under the lattice’s coordinate-permutation symmetry; see Section 3.4 for the resulting split

of the 24 bonds); the three orthogonal directions constitute space, and the spatial sub-lattice is exactly

(Section 3). The selection mechanism is modeled as a condensate and discussed in Section 5; the

emergence of Lorentzian dynamics on the selected foliation is not derived in this paper and is an open

problem of the framework.

H2 (Triality exhaustion).

The three worldline logical classes constructed in Section 6.3, which form a

single orbit under the order-three code automorphism, exhaust the admissible charged-lepton sectors

of the code: no logical class outside this orbit satisﬁes Axioms 1–5 as a charged-lepton worldline.

The constructed orbit is veriﬁed computationally; the exhaustion claim is the hypothesis. Section 4.2

sharpens its content. That section builds what we call the ladder: a sequence of four chain-complex

codes on the same lattice, each placing qubits on cells of one higher dimension than the last, so that their

logical spaces reproduce the homology of the 4-torus rank by rank. Within it, the triality worldlines are

homologous representatives inside the single time class of the mass-bearing complex, so generation

structure is ﬁner than homology. H2 resolves into three components, each independently falsiﬁable:

H2a (exhaustion)—the admissible charged-lepton structures terminate at the three listed engagement

depths; H2b (ﬂavor–depth identiﬁcation) (the physical

correspond to depths 1, 2, 3; H2c (orbit

selection)) the code contains three inequivalent depth-1 logical operators, and H2c postulates that they

do not represent three distinct physical electron species; no gauge redundancy identifying them has

been constructed, so the postulated physical equivalence, if it holds, is a quotient coarser than the

void-code logical quotient. All three remain hypotheses.

H3 (Antipodal pairing).

The 72 square circuits of the 24-cell are identiﬁed in antipodal pairs, giving

□

= 36

, where

□

counts the planar 4-circuits of the polytope, for the 24-cell electromagnetic (EM)

sector. The halving is applied to the 24-cell and to nothing else in the ﬁrst shell. No principle we

can state ﬁxes that scope. The cuboctahedron is equally centrally symmetric, and its six square faces

form three antipodal pairs, ﬁxed-point-free under v

7→ −

v, exactly as the 24-cell’s 72 squares form

36 pairs (veriﬁed in

02_24cell_triality.py

). Applied uniformly, the halving would replace the

muon’s

□

= 6

by 3. H3 is therefore a bare hypothesis, and the tauon prediction inherits that status

(Section 14.4).

H4 (Dimensional grading).

Two claims, stated separately. H4a (grading)—the logical operators of the D4

code admit representatives graded by support dimension: line-like, surface-like, volume-like, and

bulk. H4b (physical identiﬁcation)—the physical defect classes of Sections 7–12 correspond to these

graded representatives: worldlines to

, gauge worldsheets to

, and the Higgs condensate to

Section 4.2 settles H4a in two parts: the grading is refuted for the D4 code of Section 3 (a complete

census of its logical space ﬁnds no extended class) and is exactly realized as cellular homology on the

chain-complex codes, where all four classes are explicitly constructed. H4b remains a hypothesis: the

chain complex supplies the graded classes, not their particle identiﬁcation.

H5 (Condensate resolution).

The time-axis link condensate carries two real degrees of freedom, modulus

and phase, each resolving at

= 12

distinguishable levels in the syndrome-extraction sense, so that

its joint conﬁguration count is

. Section 12.1 grounds the resolution count in the packing axiom

through the kissing resolution principle (

= 12

is the kissing number of the spatial slice) conditional

on one stated measurement-model lemma.

Table 1 records which results are conditional on which hypotheses and what observation would falsify

each. One entry needs a word of explanation. H1 has no independent empirical test, because the framework

supplies no derived continuum dynamics connecting the assumed foliation to preferred-frame observables.

Failure of a prediction conditional on H1 would reject the conjunction of H1 with the relevant auxiliary

assumptions; it would not identify H1 as the cause. Such a test would need the continuum-symmetry question

settled (Section 3.2): a selected time axis is a preferred frame, and preferred-frame bounds are where it would

live. The independent falsiﬁers of the framework are those listed for H2a, H2b and H3, together with the

negative predictions of Section 14.5. The table states the hypotheses as posed. Their ﬁnal statuses, established

in the body of the paper, are uneven by design. The ladder splits H4: its grading component H4a is discharged

as a theorem on the mass-bearing structure, while the physical identiﬁcation H4b remains a hypothesis. H2

is sharpened into three components (H2a–H2c: exhaustion, ﬂavor–depth identiﬁcation, orbit selection), all

of which remain hypotheses (Section 4.2). H5 is grounded in the packing axiom up to one stated lemma

(Section 12.1). H1 and H3 remain hypotheses. The ﬁve Part I mass predictions (

e, µ,π, p, n

) depend only on

H1, since their derivations live entirely in the

spatial slice; the new results of this paper draw on H2–H5

as indicated.

Table 1: Dependency and falsiﬁability of the structural hypotheses.

Hypothesis Results conditional on it Falsiﬁed by

H1 all (foliation; D

slice) no independent test within the present framework

H2a exactly three generations a fourth admissible transverse worldline structure

H2b ﬂavor–depth identiﬁcation; tauon at depth 3 a charged lepton whose mass matches no depth cost

H2c one physical state per depth degenerate within-depth partner states

fourth-generation exclusion satisfying Axioms 1–5

H3 tauon mass C

= 3447 an axiom-derived face count = 36

H4a dimensional grading of logical classes — (refuted for the D4 code, exact on the ladder; Section 4.2)

H4b worldline/worldsheet/condensate a graded sector whose physical

classiﬁcation; Higgs support identiﬁcation fails

H5 Higgs factor K

a derived condensate resolution = K

3. The D

Code

3.1. Geometry

as a point set. Consider the set of integer 4-vectors whose coordinates sum to an even number:

D4 = {x ∈Z

: x

+ x

≡ 0 (mod 2)}.

Each lattice point has

= 24

nearest neighbors at distance

√

, obtained by taking any two of the four

coordinates to be

±1

with the other two zero. There are





×4 = 24

such sign and permutation combinations.

Throughout this paper

denotes the kissing number of

-dimensional space:

= 24

for the

bulk and

= 12 for the spatial D

slice. The mass formulas of Ref. [1] and of this paper consume K

Figure 1 collects the incidence structure of the 24-cell once, so that the counts used later in the paper can

be read off rather than reconstructed. It also makes explicit a distinction that is easy to lose: the 96 triangles

n = 24

m = 1

n = 8

m = 2

n = 3

m = 3

n = 2

m = 8

n = 1

m = 1

Face lattice of the 24-cell

n: how many of the lower rank each upper element contains.

m: how many of the upper rank each lower element lies in.

The 72 squares are not faces

2-faces of the polytope: 96, all triangles

planar 4-circuits of the edge graph: 72

A square circuit is a closed 4-loop of

nearest-neighbor bonds. It is not a face:

the 24-cell has triangular 2-faces only.

The squares therefore do not appear at

any rank of the lattice at left.



1 the 24-cell

octahedral cells (f

)

triangular 2-faces (f

)

edges (f

)

vertices (f

)



0 ∅

Figure 1: Hasse diagram of the face lattice of the 24-cell, with the element count of each rank at the right of

each box. On every covering relation

a < b

the labels read:

, how many elements of the lower rank each upper

element contains;

, how many elements of the upper rank each lower element lies in. All incidence numbers are

computed from the polytope by

gen_fig1_hasse.py

, not asserted. The box at right records why the 72 square

circuits appear at no rank of the lattice.

are 2-faces of the polytope and appear at rank 2 of the face lattice, whereas the 72 “squares” are closed

4-circuits of the edge graph and are not faces at all. The 24-cell has no square 2-faces.

Density, self-duality, and the Voronoi cell. Three structural facts will matter throughout. First,

is the

densest lattice packing in four dimensions [

], a result analogous to Hales’s Kepler-conjecture proof for

in 3D [

]. The Part I stability argument carries over without modiﬁcation: maximum packing density implies

maximum stability, and any departure from

coordination is a topological mismatch whose cost is the

defect’s mass. Second,

is self-dual: the dual lattice

∗

equals

up to scaling. This means the X- and

Z-stabilizers of the CSS code can sit on geometrically equivalent objects, unlike in

where they sit on

vertices and octahedral voids respectively. Third, the Voronoi cell of

(the region of 4-space closer to a

given lattice point than to any other) is the regular 24-cell. The 24-cell has no analog in any other dimension;

this fact is exploited in Section 6.

3.2. How isotropic is the lattice?

A lattice picks out directions. Before anything else is claimed, it is worth asking exactly how much direction-

dependence the D

bond set carries, and stating the answer in a form that can be checked.

The question, posed as a statement about moments. Let n

,.. .,

be the unit vectors along the 24

nearest-neighbor bonds. For each even integer p deﬁne the p-th moment of the bond set,

(k) =

∑



k ·n



. (2)

depends on k only through

, the bond set has no preferred direction at order

. A set for which

this holds for every

p ≤ t

is called a spherical

-design, a standard notion from the geometry of point

conﬁgurations [

]. Odd moments vanish identically here because the bond set is centrally symmetric: every

has a partner −n

We stress what this is and is not.

is a polynomial in k built from a ﬁxed ﬁnite set of lattice vectors. It is

invariant under the point group of the lattice and nothing larger. In particular the second moment, written as

an array

µν

∑

, is not a tensor with respect to SO(4) or the Lorentz group; calling it one, as Part I did,

invites a stronger reading than the object supports. The statements below are statements about the moments

of a ﬁnite vector set, and are exact in that sense only.

is a 3-design;

is a 5-design. Direct computation of Eq.

(2)

gives, for the 12

bonds of the spatial

shell,

= 4 |k|

, M

= 3



|k|



−

∑

, (3)

and, for the 24 bonds of D

= 6 |k|

, M

= 3



|k|



exactly. (4)

Both identities are veriﬁed symbolically in

15_spherical_design.py

. The

shell is a spherical 3-design

and no more: its fourth moment carries the anisotropic term

−

∑

, which is largest along a coordinate axis

and smallest along a body diagonal. The

shell is a spherical 5-design: its second and fourth moments are

direction-independent, and the ﬁrst anisotropy appears only at sixth order. Figure 2(a) shows the two fourth

moments over directions.

The

shell is therefore isotropic at second order only, and its fractional direction-dependence enters

(ka)

. This supersedes the stronger reading of

µν

∝ δ

µν

offered in Part I [

], where isotropy was stated

at all orders in

; a second moment does not constrain a fourth. The compatibility conclusion drawn there

is unaffected. At optical energies, with

the Planck mass,

(E/M

)

≈ 3 ×10

−56

, far below the

−20

−40

constraints of Refs. [5, 6].

What this buys for the dispersion relation. For a scalar ﬁeld with nearest-neighbor couplings the exact

lattice dispersion is

ω(

)

= κ

∑

[1 −cos(

a)]

. Expanding the cosine expresses

as a series in

the moments

,.. .

with coefﬁcients

−a

/24

/720

. The leading term is isotropic on

both lattices, giving

ω = c

lat

with

lat

= a

κ M

/2|k|

. The ﬁrst direction-dependent term is the ﬁrst

anisotropic moment, so the fractional spread of the propagation speed over directions scales as

∆(ω/|k|)

ω/|k|

∼

(

(ka)

on D

(3-design),

(ka)

on D

(5-design).

(5)

Figure 2(b) conﬁrms both exponents numerically from the exact dispersion, over more than a decade in

; the ﬁtted exponents are

2.00

and

4.00

. At

ka = 0.1

the

spread is

7 ×10

−5

and the

spread is

2 ×10

−8

. With

a ∼ ℓ

so that

ka ∼E/M

, tree-level direction-dependence is suppressed by

(E/M

)

in the

three-dimensional construction and by (E/M

)

in the four-dimensional one.

What this does not establish. Equation

(5)

is a tree-level, kinematic statement about one scalar ﬁeld on a

ﬁxed lattice. Three gaps separate it from emergent Lorentz invariance.

No renormalization procedure is supplied. Recovering a continuum symmetry from a lattice theory

requires a deﬁned renormalization-group ﬂow toward a ﬁxed point at which the symmetry is restored.

Nothing of the sort is constructed here.

Loop corrections are not suppressed by the design property. Collins, Pérez, Sudarsky, Urrutia and

Vucetich [

] showed that in an interacting theory with a Planck-scale preferred frame, radiative

corrections lift Planck-suppressed Lorentz violation to the percent level unless bare parameters are

tuned to roughly one part in

. A spherical design constrains the tree-level kinematics of the bond

set; it says nothing about loop integrals cut off at

1/a

. The ﬁne-tuning problem identiﬁed in [

]

therefore applies to this construction as it stands, and is not addressed by Eqs. (4)–(5).

Isotropy is not boost invariance. Even an exactly isotropic dispersion is a statement about a single

frame. Lorentzian signature and boost invariance require, in addition, that the effective quadratic action

develop the sign structure

A(∂

φ)

−B|∇φ|

with

A,B > 0

; selecting a preferred axis (Section 5) does

not accomplish this.

How the shell compares with the lattice where emergence is established. One comparison is worth

making, because it bounds the obstruction from the other side. Lattice quantum chromodynamics is formulated

on the hypercubic lattice

, whose 8 nearest-neighbor bonds

(±1,0,0, 0)

and permutations form a spherical

3-design with

)

(k) = 2

∑

, (6)

which is as anisotropic as a fourth moment can be: it is extremal along the coordinate axes and takes its

minimum along the body diagonal, with no isotropic part to cancel the variation. Full Lorentz invariance is

nonetheless recovered in the continuum limit of that theory. The

shell, by Eq.

(4)

, has no fourth-order

anisotropy at all. Both statements are veriﬁed in 15_spherical_design.py.

So the tree-level obstruction is strictly weaker here than in a setting where the recovery is a matter of record.

We are careful about how much this is worth. The recovery in lattice QCD rests on renormalization-group

ﬂow to a critical point, at which the symmetry-breaking operators are irrelevant, and not on the design

strength of the bond set; a better shell does not supply that ﬂow. It also applies to modes with

ka ≪1

, and

this is the point at which the comparison stops being reassuring for the present framework. The defects whose

costs are computed in Sections 7–12.1 are lattice-scale objects, with footprints of between 1 and 96 edges and

correlation lengths of order the lattice spacing. They live at

ka ∼1

, not

ka ≪1

. Even a complete emergence

argument for the long-wavelength sector would not cover them, and we record that as a distinct limitation in

Section 14.4.

The connection to continuum spacetime symmetry is therefore left as an open problem (Section 15), on

the same footing as the connection to the gauge group SU(3)

SU(2)

U(1) in Section 10. What the design

property does provide is a concrete internal reason for working in four lattice dimensions rather than three:

the lift from the cuboctahedron to the 24-cell raises the design strength from 3 to 5, and with it the order at

which direction-dependence can ﬁrst appear.

0.9

1.0

1.1

varies with

direction

exactly

constant

(a) Fourth moment ∑

(k ⋅ n

)

, normalized

shell (12 bonds)

shell (24 bonds)

−1

dimensionless momentum ka

−10

−9

−8

−7

−6

−5

−4

−3

spread of ω/|k| over directions

∝ (ka)

(b) Direction-dependence of the propagation speed

shell

Figure 2: Direction-dependence of the two bond sets. (a) The fourth moment

(

)

over directions in a coordinate

plane, each curve normalized by its own mean. The

shell (12 bonds) varies with direction; the

shell (24

bonds) is exactly constant, because the 24-cell is a spherical 5-design and the cuboctahedron only a 3-design.

(b) Fractional spread of the phase speed

ω/|

over directions, from the exact lattice dispersion, against the

dimensionless momentum

. The measured exponents are

2.00

(

) and

4.00

(

), matching Eq.

(5)

. Both

panels are computed by 15_spherical_design.py and gen_fig2_isotropy.py; neither is drawn by hand.

3.3. The D4 code as a CSS code

Where the qubits and stabilizers sit. Place one physical qubit on each edge of the

lattice. Z-stabilizers

sit at the lattice vertices, each one acting on the 24 incident edges. X-stabilizers sit at the odd-sum integer

points (those satisfying

∑

≡ 1 (mod 2)

) and act on the 24 edges among the 8

vertices that surround

each such point through axis-aligned displacements

(±1,0,0, 0)

and permutations. The 8 surrounding

vertices form the edge skeleton of a 16-cell (the 4D cross-polytope), with 24 edges among them. This is

the four-dimensional analog of

’s octahedral void, which had 6 surrounding vertices and 12 edges among

them.

Why the stabilizers commute. Both stabilizer types have uniform weight 24. The CSS condition

= 0

follows from the fact that any X-site and Z-vertex share either 0 or 6 edges—a Z-vertex is either one of the 8

vertices surrounding an X-site (sharing 6 edges to the other 7 vertices in that 16-cell) or none of them. The

count is even in both cases, so the supports commute modulo 2.

On the D

lattice reduced modulo an even positive number L (that is, on an L

torus) the parameters are

n = 6L

qubits, n

= L

/2 Z-stabilizers, n

= L

/2 X-stabilizers,

and accounting for one global product relation per stabilizer type,

k ≥6L

−2 (L

/2 −1) = 5L

+ 2.

The asymptotic encoding rate is therefore k/n →5/6 ≈83.3%, somewhat higher than D

’s 2/3.

The distance is exactly 3. Both halves of the argument are short, so we give both. No weight-1 or weight-2

vector lies in either kernel, checked exhaustively on both sides (script

), so

d ≥3

. For the upper bound,

every triangle of nearest-neighbor bonds is a closed loop and therefore meets each vertex star in an even

number of edges, so it lies in

ker(H

)

; and no triangle lies in the row space of

. At

L = 4

all

4096

triangles

of the D

lattice are weight-3 logical operators (script 16_distance_exact.py), giving d ≤ 3. Hence

D4 code at L = 4 : [[1536, 1282, 3]]. (7)

The same argument applies to the

code of Refs. [

], where the 256 triangles play the identical role: that

code is

[[192,130,3]]

with

d = 3

exact, which upgrades the lower bound reported there. The distance is the

length of the shortest closed loop of nearest-neighbor bonds and does not grow with

. Raising it requires

adding face stabilizers (a different code on the same lattice, of the kind constructed in Section 4.2) not a

larger lattice.

3.4. The D

sub-lattice and the role of time

Splitting the 24 bonds. Pick one coordinate—call it

, with the others spatial

)

. The 24 nearest-

neighbor bonds split cleanly into two sets. The 12 bonds with

= 0

involve only spatial coordinates

and are exactly the permutations of

(±1,±1,0)

)

. These are the

nearest-neighbor bonds

of Ref. [

]. The remaining 12 bonds have one component in

and one in a spatial direction, connecting

adjacent time-slices.

Why this matters. This is the structural fact that makes the lift possible. Figure 3(a) shows the projection:

the 12 spatial nearest neighbors (NN) form a cuboctahedron in 3-space (the

coordination shell), while

the 12 time-mixed NN project pairwise onto the six

axis points. A particle “at rest” is a worldline aligned

along

, and its spatial cross-section at any instant is an

defect of Ref. [

]. The ﬁve known masses survive

the lift trivially; we verify this in Section 7.

3.5. Logical operators in the D4 code

Classes by dimension. The CSS code of Section 3 lives in four lattice dimensions, and its logical-operator

structure is qualitatively richer than the 2D case that quantum error correction (QEC) vocabulary usually

defaults to. In 2D toric codes, logical operators are strings on a 1-cycle of the torus; logical X and logical Z

are dual 1-cycles, and the only operator classes are strings and their products. In 4D codes the analogous

duality lifts to a richer cellular structure. Logical operators admit representatives supported on closed paths

(line-like, mass

∼ L

), closed surfaces (surface-like,

∼ L

), closed 3-volumes (

∼ L

), or the full spacetime

extent (

∼ L

). That this grading by representative support corresponds to intrinsic logical classes (rather than

merely to chosen representatives) is hypothesis H4, and it is settled in Section 4.2: refuted for the present

code, and realized exactly as cellular homology on the chain-complex codes of the mass-bearing structure.

The

CSS code at

L = 4

has 1282 logical qubits, large compared with the four nontrivial 1-cycle classes of

a bare 4-torus—the bulk of the logical space arises from the cellular structure of the lattice itself, including

non-cycle stabilizer dependencies that have no 2D analog. The volume-like (3D) classes do not exist on the

D4 code. A complete census of all 1282 logical classes (Section 4.2, script

) ﬁnds every one reducible

to weight at most 9, well below the

= 64

scale a volume class would require. The 3D classes appear

−1

(a) D

nearest neighbors projected to 3D:

spatial (D

) vs. time-mixed

Spatial NN (D

, 12)

Time-mixed NN (12, projected to 6)

−1

(b) Triality decomposition:

three inscribed 16-cells

16-cell A

16-cell B

16-cell C

Figure 3: The 24 nearest neighbors of a

vertex, projected to 3D by dropping the time coordinate. (a) Spatial

bonds (blue circles, 12 vertices) form the

cuboctahedron of Ref. [

]. Time-mixed nearest neighbors (red

squares, 12 vertices projecting onto 6 axis points) connect adjacent time-slices. (b) The same 24 vertices recolored

by triality. The three inscribed 16-cells (green, orange, purple) interpenetrate at the origin and are permuted

cyclically by triality. The order-three symmetry supplies a distinguished threefold decomposition, proposed as

the transverse generation structure (hypothesis H2, Section 4.2); the mass hierarchy does not follow from this

geometry alone.

only on the tetrahedron-qubit code of the ladder, where they realize

) = Z

and their minimum-weight

representative has weight 192.

What counts as a defect. This matters for what counts as a “defect” inside the framework. In a 2D reading,

a defect is necessarily a string-supported logical operator with point endpoints carrying syndrome charge.

In the 4D reading, under the dimensional grading (H4a, settled in Section 4.2) together with its proposed

physical identiﬁcation (H4b), defects of dimension

= 1,2, 3,4

correspond to logical-operator classes

whose veriﬁcation costs scale differently with the lattice parameters. The Part I rest-mass particles (leptons,

hadrons) are worldline-class (1D) operators. The gauge bosons of Section 10 are worldsheet-class (2D). The

Higgs of Section 12 sits in the 4D class (the unique extensive top class of the ladder) and its cost is treated in

Section 12.1 as a vacuum subtraction rather than a support volume. The veriﬁcation scripts construct explicit

logical operators of dimension 1 (the three triality worldlines of Section 6.3) and verify they are inequivalent

under the stabilizer group. The full classiﬁcation across all 1282 logical qubits is carried out in Section 4.2,

with a result that reshapes this picture: on the D4 code every logical class reduces to a local representative,

and the dimensional grading is instead realized exactly on the chain-complex codes of the mass-bearing

structure.

4. The Mass-Bearing Code and the Homological Ladder

The veriﬁcation costs consume f-vector data of the coordination cluster: the sub-complex formed by one

lattice site together with its nearest neighbors and all the cells they span—13 nodes on

, 25 on

. What

the cost formulas read off that cluster is its vertex count and its face count. The D4 code of Section 3.3, like its

predecessor, carries vertex stabilizers and void stabilizers—no face stabilizers. This raises a question not

raised in the earlier papers: are the sector counts functionals of any code at all, or are they combinatorial data

external to the error-correcting structure? The answer, established computationally in this section, is that they

are exact stabilizer counts of a second code on the same lattice, and that the two codes play complementary

roles.

4.1. The vertex–plaquette (VP) code

On the vertex–plaquette (VP) code, deﬁne qubits on the lattice edges as before, Z-stabilizers on the vertex

stars, and X-stabilizers on all plaquettes—the triangular and square circuits of nearest-neighbor bonds. We

abbreviate it the VP code throughout, to keep it distinct from the D4 code of Section 3.3, which carries

no face stabilizers at all. Because every plaquette is a closed loop, it shares an even number of edges with

every vertex star, so the construction is CSS-valid automatically. On

L = 6

this yields a

[[648,3]]

code

whose three logical qubits are the torus cycles (veriﬁed:

rank(H

) = 107

rank(H

plaq

) = 538 = 541 −3

; script

09_code_functional_validation.py).

The detecting-stabilizer count. On this code, deﬁne the detecting-stabilizer count of a defect with edge

support

as the number of stabilizers associated with the cells of the induced subcomplex of

: vertex

stabilizers incident to

, and plaquette stabilizers whose support is contained in

, ﬁltered by the active sector

of Section 7.1. Evaluated on the Part I defect geometries, this functional reproduces the sector counts exactly:

the pion’s

9 + 8 = 17

(nine incident vertices, eight contained triangles), the muon’s

(six contained squares),

the proton’s

13 + 38 = 51

(thirteen vertices, thirty-two triangles, six squares), with the electron’s

= 1

as the zero-contained-cells ﬂoor. No parameter mediates the comparison. The sector counts of Parts I–II,

introduced there as combinatorial data of the coordination cluster, are thereby shown to be code-native: they

are the literal stabilizer content of the vertex–plaquette code.

How the two codes relate. The two codes on the lattice are directly related. A void stabilizer’s support is an

even subgraph (every vertex of the octahedral frame has even degree), hence a sum of plaquette boundaries:

the D4 code’s X-group is a subgroup of the plaquette code’s. The high-rate code of Section 3 is the plaquette

code with most face stabilizers omitted, and its logical count measures the omission. The division of labor is

then clean, and we adopt it for the remainder of the paper: the D4 code is the topological substrate, carrying

the encoding-rate and distance structure; the plaquette code is the metric structure, on which veriﬁcation

costs (masses) are literal stabilizer counts.

4.2. The homological ladder on D

The ladder. The plaquette construction lifts to

and extends to a family. The Delaunay decomposition of

is the 16-cell honeycomb: at

L = 4

, exactly

384

cross-polytope 4-cells (

128

centered at odd integer points,

256 at half-odd points), with cell counts

(V,E, F

△

□

) =



, 6L

, 16L

, 9L

, 12L



every tetrahedron shared by exactly two 4-cells and every triangle by exactly three (veriﬁed; scripts 10–13).

Placing qubits on the cells of each dimension in turn, with stabilizers on the adjacent dimensions, yields four

CSS codes whose logical spaces are the cellular homology of the 4-torus:

Qubits on Code k Logical content (= Betti number)

edges [[1536,4]] 4 H

: 1 time + 3 spatial winding classes (wt. 4)

faces [[6400,6]] 6 H

: 3 + 3 worldsheets under the foliation (wt. 8)

tetrahedra [[3072,4]] 4 H

: membranes (wt. 192 representative)

16-cells [[384,1]] 1 H

: the fundamental class (wt. 384, extensive)

Every parameter was predicted from the honeycomb structure before computation and conﬁrmed by it,

including the diagnosis and elimination of a

2304

-qubit excess traceable to the square-faced 3-cells (script

12). Three consequences restructure the hypotheses of Section 2.

First, the homological placement of the particle worldlines is settled, and it sharpens H2 rather than

discharging it. The logical space of the edge code is exactly

) = Z

: one time class and three spatial

winding classes, computed and exhausted. Two equivalence relations must be kept distinct here: inequivalence

under the D4 code’s stabilizer group (the relation veriﬁed in Section 6.3) and homology in the mass-bearing

chain complex (equivalence modulo plaquette boundaries). These are different quotients, and the three triality

worldlines of Eq. (8) separate them. Their homology vectors, computed from their step displacements, are:

operator (w

)

(1,0,0,0)

All three wrap the time direction with vanishing net spatial winding: they are inequivalent under the D4 code

yet homologous in the chain complex, all representing the single time class. This is the physically correct

placement (a massive particle at rest persists in time, so its worldline wraps the time circle with zero net

spatial displacement) and it holds for every particle worldline in the taxonomy. The neutrino of Section 9

is the bare time wrap; the charged leptons are time wraps carrying transverse structure, distinguished by

which spatial axis the worldline oscillates along (the triality label). Generation structure is therefore ﬁner

than homology: it lives within the time class, at the level the D4 code resolves and the chain complex does

not. The three spatial winding classes, by contrast, are spacelike torus cycles, not particle worldlines; their

physical identiﬁcation is open. Two consequences follow for H2. Its exhaustion component (H2a) (that the

triality orbit exhausts the admissible transverse worldline structures within the time class) is not settled by

the homological computation and remains a hypothesis. And its ﬂavor component (H2b, together with the

within-depth redundancy H2c) faces the obstruction of symmetry: the three transverse structures form a

single orbit of the coordinate 3-cycle and carry identical veriﬁcation cost, so the observed hierarchy requires

a mechanism that breaks the axis-permutation symmetry and correlates each structure with the engagement

depth that the mass formulas of Sections 7–8 consume. Both questions are stated in Section 15.

Second, the grading component of H4 (H4a) is corrected and then upgraded. For the D4 code of Section 3,

the dimensional grading is refuted: a complete census of its

1282

logical classes (script

) shows every class

reduces to weight

≤ 9

—no volume-like or bulk-like class exists there. On the ladder, the grading is exact

cellular homology: line, surface, volume, and bulk classes all exist, explicitly constructed. The grading H4a

postulated is real; it lives on the chain complex, not on the D4 code. The identiﬁcation of the graded classes

with the physical sectors (H4b) is not settled by this computation and remains a hypothesis.

Third, the vertex ﬁgure of the honeycomb, computed from the complex, is

(24,96,96, 24)

—the f-vector

of the 24-cell. The polytope on which Sections 6–8 are built is not an auxiliary construction; it is the local

structure of the mass-bearing complex at every vertex.

5. Foliation Selection: The Fifth Axiom

The four earlier axioms still hold. The four axioms of Ref. [

] (Minimum Topological Dimension, Sector

Completeness, Boundary Closure, Kinematic Shedding) carry through, with two generalizations. Axiom 1

was originally a spatial statement (

spatial

top

≥ 1

), since the

framework knew no time dimension. With time

now a fourth lattice direction, Axiom 1 reads as a spacetime statement:

spacetime

top

≥ 1

. A defect must extend

in at least one direction, but it can extend purely in time. This generalization is the one new ingredient that

admits time-only worldlines (the neutrino class of Section 9) which would have been rejected by the strict

spatial form. Spatial defects that satisﬁed the original axiom satisfy the generalized version automatically.

The second generalization concerns Axiom 4. In Ref. [

] it was stated for moving point defects, which

shed

trajectory correlations. The underlying quantity is not speciﬁc to trajectories, and we state it in the

form the D

setting requires:

Axiom 4 (Redundancy Shedding). A defect sheds the

spatial translations for each indepen-

dent vector required to ﬁx its macroscopic embedding in the spatial slice.

Two cases arise in this paper, and the vector count is ﬁxed by the defect’s geometry rather than chosen. A

worldline is ﬁxed by two vectors, a spatial position and a kinematic tangent, and sheds

. A condensate

spans the whole spatial slice: translating it is the identity, so no position vector enters, and its embedding

is ﬁxed by the

orthogonal basis vectors that span its extent, shedding

. The count is a property of the

embedding, not of the particle.

Two conditions carry over unchanged from Ref. [

]. Shedding applies only to free defects: a pinned

worldline’s vertex syndromes report internal structure, not position, and are counted. And a defect can shed

only what it has, so the shedding applies only when

dim(B

sector

)

exceeds the quantity to be shed; the electron,

with

dim(B) = 1

, sheds nothing and keeps

= 1

. At

D = 3

the worldline case gives

= 9

, reproducing the

muon and tauon costs of Section 8 unchanged, and the condensate case gives

= 27

, used in Section 12.1.

The full enumeration of which candidate defects survive all ﬁve axioms and which fail (the

analog of

Ref. [1]’s 25-candidate sieve) is collected in Section 13.

The remaining three axioms apply to spatial defects exactly as in Ref. [

], with “spatial” now meaning the

three directions orthogonal to the selected time axis. To make that selection well-deﬁned, we add one new

axiom.

Axiom 5 (Foliation Selection; hypothesis H1). The vacuum code state spontaneously selects a single

direction in the

lattice as the time axis, deﬁning a preferred foliation. The symmetry that breaks is

combinatorial: the symmetric group

permuting the four lattice coordinates is reduced to the subgroup

permuting the three coordinates orthogonal to the selected one. Translations along the selected coordinate

remain available. The 12 nearest-neighbor bonds of the resulting spatial sub-lattice are the

bonds

of Ref. [

], while the 12 time-mixed bonds carry energy and momentum between time-slices and do not

contribute to spatial sector counts.

The

lattice Hamiltonian carries the discrete automorphism symmetry of the lattice, which contains the

coordinate permutations

. A vacuum state need not share its Hamiltonian’s symmetry—this is standard

spontaneous breaking, here of the discrete direction-permutation symmetry among the four equivalent lattice

axes,

→ S

. We describe the breaking in these combinatorial terms throughout; the relation between the

lattice and any continuum symmetry is left open (Section 3.2). The order parameter is a condensate of link

variables along a single direction:

⟨U

⟩ = 0, ⟨U

⟩ = 0 (i = 1,2,3),

where

is the lattice link variable in direction

. For gauge link variables the local expectation value is

gauge-dependent and vanishes without gauge ﬁxing, so this condensate must ultimately be characterized

gauge-invariantly (for instance, through direction-dependence of gauge-invariant plaquette averages); that

characterization is left open, and the condensate here is a model of the selection, not a derivation of it. We

identify its localized excitation with the Higgs ﬁeld; the discussion is deferred to Section 12.

What the split gives. Below the condensation scale, the four directions split as

12 + 12

(spatial

time-

mixed), and the dispersion relation reads

= m

+ c

with k

the spatial momentum and

set by the condensate strength. A Wick rotation connects this to the

Euclidean dispersion as an analytic continuation. Selecting the time axis deﬁnes the preferred foliation;

obtaining Lorentzian dynamics on it requires, in addition, that the effective quadratic action develop the sign

structure A(∂

φ)

−B|∇φ|

with A,B > 0, a step listed among the open problems.

Orientation is arbitrary; the shedding count is not. First, the axiom leaves open which direction is time;

before symmetry breaking, the four candidates are equivalent. The axiom asserts only that some direction is

selected, which is the generic outcome of spontaneous breaking in a system with discrete rotational symmetry.

The orientation relative to any external frame is arbitrary, but the existence of a time axis is forced. Second, the

kinematic shedding count of Axiom 4 stays at

= 9

rather than rising to

, because the redundancy being

shed is purely spatial trajectory correlations. Time translation generates energy, not a kinematic redundancy.

This is checked directly in Section 7; it constitutes the strongest piece of internal evidence that

is the

correct lift.

6. The 24-Cell, Triality, and the Three-Generation Structure

The local polytope. The 24 nearest neighbors of a

vertex form the vertex set of a 24-cell centered on

that vertex. Its f-vector is

( f

, f

) = (24,96, 96,24),

with Euler characteristic

24 −96 +96 −24 = 0

as required for a closed 4-polytope. Each of the 24 octahedral

3-cells has 6 vertices, 12 edges, and 8 triangular faces. Two octahedral cells meet at each triangular face,

and the 24-cell is the unique regular 4-polytope that is both self-dual and admits the additional symmetry

described below.

6.1. Triality

The threefold split. The 24 vertices partition uniquely into three subsets of 8, each subset forming the

vertex set of an inscribed 16-cell (the regular 4D cross-polytope). Concretely, the six coordinate pairs {i, j}

with i < j split into three pairs of complementary pairs:

A =



{1,2},{3, 4}



, B =



{1,3},{2, 4}



, C =



{1,4},{2, 3}



The 8 vertices with nonzero coordinates in pair set

form a 16-cell; the same for

and

. Triality is the

order-3 symmetry that cyclically permutes these three 16-cells. The corresponding statement in Lie theory is

this:

names the rank-4 simple Lie algebra

so(8)

, whose simply connected group is

Spin(8)

, and triality

is the order-three outer automorphism of that group [

]. It has no analog in any other dimension. We use

“

” for the lattice throughout and name the algebra and group explicitly whenever they are meant, since

the same symbol serves both. The geometric realization in the 24-cell of

is its only ﬁnite-dimensional

manifestation.

Figure 3(b) shows the three 16-cells in the 3D projection, with their edges drawn in different colors. Each

16-cell is at distance 0 from the origin (they share the common center) and is mapped to the next by an outer

triality rotation.

6.2. Why exactly three generations

Three levels, three generations. A worldline defect can participate in the triality decomposition at three

levels: localized within a single 16-cell component, engaging the spatial halves of all three components,

or engaging all three in both their spatial and time-mixed halves. These are the three engagement depths

realized by the constructed classes; the third level engages the full 24-cell, so no further level exists within

this ladder. Conditional on hypothesis H2a (that this ladder exhausts the admissible lepton structures) the

framework predicts exactly three generations of leptons (and, by analogous reasoning at the hadronic level,

three generations of quarks). The orbit itself is veriﬁed computationally (Section 6.3); H2 is what converts

the existence of a threefold orbit into an exhaustive generation count; Section 4.2 shows the orbit lives within

the single time class of the mass-bearing complex, so the exhaustion question is ﬁner than homology and

remains open. The automorphism realized computationally is a lattice coordinate 3-cycle; whether it is the

Spin(8) triality in the representation-theoretic sense is open (Section 15).

A 16-cell has 8 vertices, 24 edges, 32 triangular faces, and 16 tetrahedral cells. The veriﬁcation cost at

depth

scales roughly with the f-vector size of the engaged sub-structure: a single 16-cell at depth 1, the

muon’s 3-sheet at depth 2, and the full 24-cell at depth 3. We work out the quantitative numbers in Section 8.

The Large Electron–Positron collider (LEP) precision measurement of the invisible Z width gives

2.996 ±0.007

[

], which is incompatible with two or four generations and consistent with three to within

experimental precision. So the framework’s structural prediction is empirically tight: any future evidence of a

fourth light neutrino would falsify it.

6.3. Triality as a code automorphism

The three-generation prediction of Section 6.2 is structurally clean but invites a sharper question. The 24-cell

has an order-3 symmetry that permutes three inscribed 16-cells; that is a fact about the polytope. But the

framework’s physical content sits in the CSS code of Section 3, not in the bare polytope. To upgrade “triality

has order three” to “the code has three generations” we need triality to act as a symmetry of the CSS code,

not just of its stabilizer support geometry.

Verifying the automorphism. We verify this computationally. Let

π : (x

) → (x

)

the coordinate 3-cycle ﬁxing the time axis. The veriﬁcation script

05_triality_code_automorphism.py

conﬁrms the following at L = 4:

1. π

has order three, ﬁxes the time coordinate selected by Axiom 5, and cycles the three 16-cell triality

sets as

A → C → B → A

with exactly 8 nearest-neighbor edges in each transition (matching the

8 + 8 + 8 = 24 decomposition).

2. π

is a code automorphism of the

CSS code: the lattice permutations on

vertices (vert_perm) and

X-stabilizer sites (xsite_perm) and the induced permutation on edges (edge_perm) satisfy

[vert_perm][:

,edge_perm] = H

and

[xsite_perm][:,edge_perm] = H

. That is, applying triality permutes the Z-

and X-stabilizer groups among themselves rather than mapping any stabilizer outside the group.

The orbit structure of

on the qubit set has no ﬁxed points: every one of the 1536 qubits sits in a size-3

orbit, for a total of 512 orbits. Triality is non-trivial on every physical qubit.

The three-generation claim then becomes a concrete prediction about logical operators. We construct three

explicit closed worldlines

,wl

, each anchored in one of the three triality sets and wrapping the time

direction on the L = 4 torus via four nearest-neighbor steps:

: (1,1,0, 0) →(1,−1, 0,0) →(1,1, 0,0) →(1, −1,0,0); wl

: (pair (0,2) steps); wl

: (pair (0,3) steps).

(8)

Each is a Z-string of weight 4, supported on edges of a single triality set. The veriﬁcation script conﬁrms:

•

All three worldlines commute with every X-stabilizer (

·wl

= 0 (mod 2)

), so they are valid logical

Z candidates.

• None lies in the row-span of H

, so all three are genuine logical operators rather than stabilizers.

• The triality permutation cycles them: π(wl

) = wl

, π

(wl

) = wl

, π

(wl

) = wl

•

The pairwise differences

+ wl

are not in the row-span of

. The three

worldlines therefore represent three inequivalent logical operator classes of the CSS code, related by a

code automorphism.

The computation establishes three inequivalent void-code logical operators forming a single orbit under the

order-three automorphism: this is the code-level structure the framework associates with the three lepton

generations. Two cautions bound the claim. The order of the automorphism ﬁxes the size of this orbit, not the

number of admissible worldline structures, so exhaustion is not established by the construction. And in the

mass-bearing chain complex of Section 4.2, all three operators are homologous representatives of the single

time class (their homology vectors are computed there), so the structure that distinguishes them is ﬁner than

homology. Their interpretation as the three generations, and the claim that no other admissible transverse

worldline structure exists, constitute hypotheses H2b and H2a respectively (Section 2).

The construction above is at

L = 4

with one speciﬁc choice of worldline anchor; the same operators arise

for any anchor in the same triality orbit by lattice translation invariance. A complete classiﬁcation of all

logical operators by their triality orbit and cellular-dimension class remains an open problem (Section 15).

Three labels, and how they map. Three distinct labels are in play in the generation sector, and the

mapping among them must be kept explicit. They are: the triality label indexes which 16-cell component(s)

a worldline’s support selects; the engagement depth is its level of participation in the triality decomposition—

localized support within one component (depth 1), spatial-half support across all three (depth 2), or full

spatial-plus-time-mixed support across all three (depth 3); the ﬂavor names the physical particle. Depth is

what the mass formulas of Sections 7–12.1 consume; Table 2 gives the three values and why the ladder stops

there. Two threefold structures are in play here and are related but not identical: the triality decomposition

supplies three components, while the support hierarchy supplies three engagement levels. The coincidence

of the two counts is a property of the constructed ladder, not a consequence of the automorphism’s order;

why the decomposition admits precisely this nested support ladder, rather than other engagement patterns, is

part of the exhaustion question H2a. The triality label acts differently at each level: the depth-1 localization

selects one component (three symmetry-equivalent choices, permuted by the automorphism—the operators

of Eq.

(8)

are exactly such a degenerate triple), while the depth-2 and depth-3 cross-sections engage all three

components symmetrically and are triality-invariant. The assignments:

object triality label depth spatial support cost

electron one component (3 degenerate choices, unassigned) 1 single edge 1

muon spatial halves of all three (symmetric) 2 D

3-sheet 207

tauon all three, spatial and time-mixed (symmetric) 3 24-cell tube 3447

The table makes the open structure visible: each of the three components of H2 (Section 2) corresponds to a

column that the construction does not ﬁll. They are taken up again in Sections 4.2 and 15.

7. Worldlines and the Spatial Predictions

From static defects to worldlines. In Part I a particle at time

was a defect geometry in 3-space, and its

rest mass was the veriﬁcation cost of that geometry. In

a particle is a worldline through 4-space. A particle

at rest corresponds to a worldline aligned with the time axis; a particle with 3-velocity v corresponds to a

worldline tilted by tan

−1

(|v|/c) relative to that axis.

Cost of a worldline. The veriﬁcation cost of a worldline is deﬁned over a tubular spacetime neighborhood

of one proper-time step: the edges entering

are those of the defect’s coordination structure within the tube.

The tube formulation is needed for uniformity, because depth-3 defects engage time-mixed bonds that lie in

no ﬁxed-time slice (Section 8). For a worldline aligned with the time axis whose coordination structure is

purely spatial, the tube count reduces to the ﬁxed-time spatial cross-section (exactly an

defect) and the

cost reduces to the Part I formula

worldline at rest

= C

spatial cross-section

The Landauer–Einstein relation [

] and the cancellation of

kT ln 2

in mass ratios both carry through

unchanged. So all ﬁve Part I predictions are preserved:

= 1, C

= 207, C

= 273, C

= 1836, C

= 1839,

matching the experimental ratios m

to within 0.008–0.12 % as before.

The consistency check. It concerns Axiom 4. The muon’s formula reads

= E

×C

−D

= 36×6 −9 =

207

. If

had to be re-interpreted as

= 16

in the 4D theory, the muon would come out at

216 −16 = 200

missing the empirical value of 206.77 by

3.3%

—far outside the framework’s accuracy. The foliation-selection

axiom is what keeps

D = 3

: kinematic shedding subtracts spatial trajectory correlations, not time translations.

The match at

= 9

is therefore strong internal evidence that the

sub-lattice is the correct “spatial slice”

of D

7.1. The sector map from worldline kinematics

Parts I–II assign each particle an active stabilizer sector (which of

△

□

enter

) but do not derive the

assignment. In three dimensions no derivation is possible: the muon and the proton occupy the identical

36-edge coordination footprint, with identical boundary structure (12 odd-incidence vertices; veriﬁed by

07_sector_map.py

), yet carry different sectors. Any rule that reads the sector off the spatial footprint must

assign them the same one.

Pinned and free worldlines. The

lift supplies the missing information. A defect’s worldline is either

pinned to the time axis (conﬁned states, whose color-ﬂux structure anchors them) or free (deconﬁned states,

whose worldline tilt encodes motion, as above). For a free worldline, the vertex syndromes recorded at

successive extraction rounds report only the defect’s changing position—they are trajectory correlates in

precisely the sense of Axiom 4, which mandates that such syndromes be shed from the veriﬁcation cost. For

a pinned worldline they report internal color structure and must be counted. This yields the

Worldline sector rule. The vertex sector

belongs to the active sector

if and only if the

worldline is pinned; for free worldlines the vertex syndromes are kinematic and are shed.

The rule reproduces all seven sector assignments of Parts I and III—electron, muon, tauon, pion, proton,

neutron, neutrinos (veriﬁed by

07_sector_map.py

). The diagnostic pattern is sharper than mere agreement:

the two particles that every footprint-based rule misclassiﬁes, the electron and the muon, are exactly the

free-worldline states—the cases in which the trajectory-shedding mechanism operates. The 3D projection

fails precisely where the 4D kinematics carries the information. Together with the

= 9

dimension check

above, this constitutes strong internal evidence that

is the correct lift: the sector map is underivable in the

3D framework and determined in the 4D one.

What the sector rule does not yet prove. The rule is established here as a consistency result across the

seven assignments; a ﬁrst-principles calculation showing that the vertex bits of a tilted worldline carry position

information only (for instance, that the mutual information between successive-round vertex syndromes and

the worldline tilt saturates the syndrome entropy for a minimal moving defect) remains open and is listed in

Section 15. And the residual

= 1

of the electron, the single bit that survives shedding, is interpreted as the

defect’s existence bit; the rule motivates this interpretation but does not force it.

8. The Tauon and the Three Charged Leptons

Before the three derivations, Table 2 ﬁxes what the three depths mean, since the word “depth” is doing real

work and its content is otherwise spread across three subsections.

The ladder terminates at depth 3 because full support is maximal: there is nothing further to engage. That

termination is a property of the constructed ladder; whether the ladder exhausts the admissible structures is

hypothesis H2a, and the identiﬁcation of depths with

is H2b. Both remain open (Section 15). The

depth-3 row also carries the two undischarged conventions discussed in Section 14.4: the edge count

= 96

and the antipodal halving of the squares.

Table 2: The three engagement depths. “Cross-section” is the slice through the worldline at an instant of proper

time; “16-cells engaged” counts how many of the three triality components the tube section touches, and whether

it touches their spatial halves only or both halves. The cost column applies the worldline sector rule of Section 7.1

together with Axiom 4. That rule ﬁxes the vertex sector; the full set of four sector rules, which the sieve needs, is

stated in Section 13.

Depth Tube section 16-cells engaged Active sector C

1 single D

edge one, fractionally trivial (C

= 1) 1

2 D

3-sheet all three, spatial halves F

□

= 6 36 ×6 −9 = 207

3 full 24-cell all three, both halves F

□

= 36 (H3) 96 ×36 −9 = 3447

8.1. Electron: depth 1

The electron is a worldline whose spatial cross-section is a single

edge—the minimum 1-sheet defect of

Ref. [

]. By triality, the cross-section is localized within a single 16-cell sub-structure of the local 24-cell.

With E

= 1 and trivial sector C

= 1,

= 1 ×1 = 1.

8.2. Muon: depth 2

The muon worldline engages the spatial halves of all three 16-cell sub-structures. Its spatial cross-section is

the full

3-sheet (the 13-node cluster), with only the EM sector active and the state deconﬁned so Axiom 4

applies:

= E

×C

−D

= 36 ×6 −9 = 207.

This is identical to the Part I derivation; the new interpretation is that the

3-sheet defect engages the

spatial halves of all three triality sub-structures simultaneously. The 12

nearest-neighbor bonds split as 4

bonds in each of the three triality sets—the spatial pair from each pair-of-pairs

{(0,1),(2, 3)}

{(0,2),(1, 3)}

{(0,3),(1, 2)}

contributes its purely-spatial coordinate pair

(2,3)

(1,3)

(1,2)

respectively. The 3-sheet

therefore touches all three 16-cells but only on their spatial halves, leaving the time-mixed halves uninvolved.

We call this conﬁguration “depth 2” to mark its intermediate position between depth 1 (a single edge touching

one 16-cell fractionally) and depth 3 (the full 24-cell touching every 16-cell in both halves).

8.3. Tauon: depth 3

The tauon worldline engages all three 16-cell sub-structures simultaneously. Its proper-time tube section

is the full 24-cell of

—all 24 nearest neighbors of the central node, including the 12 time-mixed bonds,

which lie in the spacetime tube though in no ﬁxed-time slice (Section 7). This is why the tube formulation of

the worldline cost is required: the electron and muon tube sections happen to be purely spatial; the tauon’s is

not. We need to identify the EM sector in the 24-cell.

The square count does not lift as a face count. The 24-cell has 96 triangular 2-faces, but no square

2-faces. The cuboctahedron’s

□

count of 6, which gave the muon’s factor of 6, does not lift as a face

count—one needs a different identiﬁcation of the EM sector. The natural generalization is the count of planar

4-vertex conﬁgurations whose four sides are polytope edges (on the cuboctahedron they coincide with its

square 2-faces, which is why both counts gave 6 in Ref. [

]). Direct enumeration on the 24-cell yields 72

such squares: each pair of vertices at distance 2 has exactly four common nearest neighbors among the

24-cell skeleton, two of which themselves form a distance-2 pair, giving 2 squares per diagonal pair and

72 ×2/2 = 72 distinct squares in total. The script 02_24cell_triality.py enumerates these explicitly.

The antipodal halving. The 24-cell is centrally symmetric, and the 72 squares partition into 36 antipodal

pairs: each square has an antipodal partner obtained by negating all four vertex coordinates. The independent

contribution to the EM sector is therefore

(24-cell)

□

= 36,

giving the prediction below. The antipodal identiﬁcation is hypothesis H3 (Section 2). It has a natural physical

reading (in the unbroken-parity phase the two members of each pair carry opposite chirality and contribute

as one effective square) but it is not derived from the axioms, and its scope is restricted to the 24-cell for

no reason we can state (Section 2). The tauon prediction below is conditional on it. Figure 1 sets out the

incidence structure the counts are drawn from, including the fact that the 72 squares are 4-circuits of the edge

graph rather than 2-faces of the polytope.

With edges E

= 96 and the same Axiom 4 subtraction:

= 96 ×36 −9 = 3447. (9)

The experimental ratio is m

= 3477.23 [10], giving a deviation of 30.23/3477.23 = 0.87%.

Size of the tauon deviation. First, the deviation is roughly eight times larger than the largest Part I

deviation (0.11 % for the muon). That is still under one percent, comparable in spirit to the kind of sub-integer

corrections that Part I attributed to quantum ﬁeld theory (QFT) effects outside the topological framework (the

proton-neutron splitting was 18.5 % off in absolute terms). Second, the formula structure (96 edges times 36

squares minus 9 kinematic checks) is identical to the muon’s, with the

sub-structure replaced by the full

24-cell. No new parameters are introduced.

8.4. Summary

The three charged-lepton generations correspond to triality depths 1, 2, and 3 with veriﬁcation costs 1, 207,

and 3447. Within this classiﬁcation the depth variable terminates at 3—the third level engages the full 24-cell,

leaving nothing further to engage; but exhaustion of the admissible transverse worldline structures (that no

state exists outside this classiﬁcation) is precisely hypothesis H2a (Section 4.2). The prediction is therefore

conditional and stated as such: under H2a, the framework admits exactly three charged-lepton generations,

and the discovery of a fourth would falsify H2a, consistent with the falsiﬁability accounting of Section 14.5.

9. Neutrinos as Time-Only Defects

Why neutrinos were missing. Part I had no place for neutrinos. The electron was the minimum stable

defect at

= 1

, and any neutrino with

< 2 ×10

−6

would require

< 10

−6

, far below the topological

ﬂoor of one bit. The Part I defect classiﬁcation contained no conﬁguration in this regime.

Time-only worldlines.

with a selected time axis offers a new class of defect: a worldline whose spatial

cross-section is empty. Such a defect has no

presence at any instant, but it represents a real topological

obstruction in the time direction. It engages only the 12 time-mixed bonds, not the 12 spatial ones. The

instantaneous spatial veriﬁcation cost is zero. The accumulated cost over a worldline segment of proper time

τ scales as

(τ) = α

τ,

with

a rate set by the time-direction detection cost. Because the time direction is condensed (Axiom 5),

is suppressed relative to the unbroken-phase rate by the condensate density. The status of this construction

should be stated precisely. The time-only worldline operator itself is explicitly constructed: the time-axis

cycle of the mass-bearing edge code (Section 4.2) is a weight-4, spatially localized, time-wrapping logical

operator. Because it is logical, it commutes with every stabilizer and ﬁres no local syndrome—the same

structural fact that sets its spatial veriﬁcation cost to zero also renders it nearly invisible to the code, a

suggestive match to neutrino phenomenology. But it also means that detectability rests entirely on the

time-direction channel, and

is not computed. The neutrino sector is therefore a qualitative proposal within

the framework—its three-ﬂavor count follows from the triality anchoring (conditional on H2), but its mass

mechanism is a program, not a prediction. The neutrino mass is parametrically small:

∼ α

≪ 1.

The speciﬁc value of

depends on the condensate scale (the Higgs vacuum expectation value (VEV)) and is

not determined by topology alone. So the framework structurally predicts that neutrinos are much lighter

than the electron without ﬁxing the absolute scale.

By triality, time-only defects come in three ﬂavors, one per 16-cell:

↔ 16-cell A, ν

↔ 16-cell B, ν

↔ 16-cell C.

The Pontecorvo–Maki–Nakagawa–Sakata (PMNS) mixing matrix is proposed to arise from triality rotations

between these three 16-cells; a derivation of the mixing angles requires ﬁxing the gauge in which the rotation

operators act, and we defer it.

Three qualitative expectations follow under H2 and the hypothesized time-direction channel. (i) There are

exactly three light neutrino species—matching LEP’s

= 2.996 ±0.007

. (ii) All three are far lighter than

the electron,

≪ m

, consistent with current bounds

< 1 eV

. (iii) Generation mixing is nontrivial, with

mixing angles related to triality rotations. The framework cannot yet predict the absolute mass scale, but the

qualitative picture is ﬁxed.

10. Gauge Bosons as 2D Worldsheets

Part I recovered the Standard Model (SM) gauge boson count structurally: the 12

nearest-neighbor bonds

partition as

= 8 + 4

, matching 8 gluons plus the four electroweak bosons. The masses themselves were

excluded as “involving the Higgs mechanism.” The

construction permits treating the gauge bosons as

genuine topological defects.

Why gauge bosons are 2D objects. A gauge ﬁeld

is a 1-form. Its ﬁeld strength

µν

is a 2-form. The

natural lattice realization of a 2-form is a two-dimensional defect—a worldsheet, not a worldline. Worldsheets

exist as proper extended objects only in spacetime dimension

≥ 4

: in 3D a 2D defect has codimension 1 and

partitions space into halves, which is too restrictive.

is the minimum-dimensional setting in which gauge

bosons can be realized as topological defects in the same sense that fermions are realized as worldlines.

The scope of this section should be ﬁxed before the constructions: what follows is a proposed geometric

correspondence, conditional on the sector identiﬁcation H4b, not a derivation of the gauge sector. Representa-

tions, structure constants, interactions, helicities, and gauge redundancies are not obtained here; the

8 + 4

bond partition matches the gauge boson count structurally but does not derive the group SU(3)

SU(2)

U(1);

and the photon’s masslessness is argued from bundle triviality at the level of this correspondence, not proven

from the code.

10.1. Photon: C

= 0

The photon worldsheet sits in the electromagnetic sector with trivial bundle topology—the U(1) gauge ﬁeld

is the trivial circle bundle over the lattice, with no monodromy around any cycle. An untwisted worldsheet

has no topological boundary requiring veriﬁcation. Its cost is exactly zero,

= 0,

predicting

= 0

exactly. This matches the experimental bound

< 10

−18

and is a genuine structural

consequence of trivial-bundle topology, not a ﬁtted result.

10.2. Gluons: conﬁned SU(3) octet

The eight gluon worldsheets occupy the triangular-plaquette bonds (

TOR

= 8

in the

sub-lattice, the

notation of Ref. [

], where

TOR

and

count the bonds carried by triangular and square plaquettes

respectively and

TOR

+ S

= K

= 12

), lifted to 2D worldsheets in the 4D lattice. By Axiom 3 (Boundary

Closure), they are conﬁned: an isolated gluon worldsheet has an open color ﬂux boundary that cannot close

at ﬁnite cost. The bare gluon mass is zero, but free gluons are not asymptotic states. This is the topological

version of QCD conﬁnement.

10.3. W and Z: twisted worldsheets

The W

and Z worldsheets occupy the electroweak sector (the four square-plaquette bonds

in the

sub-lattice) with non-trivial bundle topology. The SU(2)

U(1) gauge group introduces a twist that creates a

topological obstruction. The veriﬁcation cost of detecting this twist is the boson mass.

The experimental ratios are

≈ 1.575 ×10

and

≈ 1.785 ×10

, with the ratio

≈

0.882 = cos θ

identifying the Weinberg angle. A topological calculation predicting these masses to percent-

level requires extending the f-vector formalism to 2-cells in the

chain complex, with kinematic shedding

adapted to 2D extended states. We mark this as a concrete open problem solvable within the framework—a

ﬁnite enumeration, not a search for new physics.

10.4. Count

The SM has 12 gauge bosons: 8 gluons, W

, W

−

, Z, and the photon. The

framework reproduces this

count via the same

= 12 = 8 + 4

partition of the

sub-lattice; the time-mixed bonds contribute to

Wilson-line phases [

] rather than new gauge bosons. The Higgs is treated separately as a scalar order

parameter (Section 12).

11. Heavier Hadrons via Second-Shell Defects

Part I’s enumeration covered the ﬁrst coordination shell of

(12 nearest neighbors at distance

√

) and

accounted for the ﬁve lightest non-strange particles. The heavier hadrons require defects anchored beyond

the ﬁrst shell.

The second shell. The second coordination shell of

contains 24 next-nearest neighbors at distance 2,

consisting of 8 axis-aligned sites (

±2

in one coordinate) and 16 fully-diagonal sites (

±1

in all four coordinates

with even sum). These 24 second-shell sites form their own 24-vertex sub-structure with rich topological

content. A quark defect anchored to a second-shell site instead of a ﬁrst-shell site engages this structure and

has a larger veriﬁcation cost.

What the shell index means. The natural identiﬁcation is that strangeness, charm, bottomness, and topness

are radial shell indices:

Shell Distance Flavor index

√

2 u, d (non-strange)

2 2 s (strange)

√

6 c (charm)

√

8 b (bottom)

√

10 t (top)

A kaon is a quark-antiquark pair with one quark from shell 1 and one from shell 2, giving four strange meson

states (

). The

baryon is the

uds

conﬁguration with one second-shell quark, and the

triplet is

similar.

Triality at the second shell again supplies the threefold label proposed for the three quark generations

(u,d)

(c,s)

(t,b)

, under the same H2-type hypothesis as the lepton sector. The Cabibbo–Kobayashi–Maskawa

(CKM) matrix corresponds to triality rotations between these three generations, analogous to PMNS for

neutrinos. Quantitative predictions for the strange and charm hadron masses require an enumeration analogous

to Part I’s Section 6 applied at the second shell—a substantial combinatorial task that we defer to a follow-up

paper dedicated to the heavier hadron spectrum.

12. The Higgs as Time-Axis Condensate

The condensate. Axiom 5 models the selection of the time axis through a condensate

⟨Φ⟩ = 0

along

that axis, subject to the gauge-invariance caveat of Section 5. We identify the localized excitation of this

condensate with the Higgs.

The Higgs VEV

v ≈ 246

GeV sets the condensation scale—the energy below which the coordinate-

permutation symmetry

of the lattice breaks to

and a time axis is singled out. The Higgs boson itself is a

localized ﬂuctuation of

: a region of spacetime where the time-axis direction is locally perturbed. Detecting

such a ﬂuctuation requires the full 4D coordination cluster around the perturbation site, giving the Higgs a

veriﬁcation cost comparable to but distinct from the W and Z. The experimental ratio is

≈2.45 ×10

in the same order of magnitude as

W,Z

but without the twisted-bundle topology of the gauge bosons.

We model the order parameter as a single real scalar, matching the one physical Higgs that survives in the

Standard Model after symmetry breaking; recovering the full SU(2)-doublet structure with its Goldstone

modes from the lattice construction is open. A speciﬁc veriﬁcation-cost formula follows from the CSS-code

structure of Section 3, derived next.

Fermion masses. Mass generation for fermions follows the standard SM picture, now with a topological

interpretation. A fermion worldline aligned with the time axis acquires veriﬁcation cost per unit length

proportional to its Yukawa coupling to

. When

⟨Φ⟩= 0

in the symmetric phase, all worldlines are equivalent

and all fermions are massless. The framework therefore recovers spontaneous symmetry breaking as the

mechanism of mass generation, while interpreting “Yukawa coupling” as the strength of a defect’s engagement

with the time-axis condensate.

12.1. The Higgs cost as a vacuum subtraction

The condensate’s home in the ladder is unambiguous: the unique logical class of the top code

[[384,1]]

, the

fundamental class—all

sixteen-cells, closed, extensive, ﬁlling spacetime. This uniqueness matches the

observed Higgs sector (one scalar), but it immediately implies that the Higgs cost cannot be a count. Three

independent results make this precise.

Extensivity. The validated cell-count functional, applied to the fundamental class, scales as

: a con-

densate’s raw veriﬁcation cost is proportional to the volume of spacetime, which is physically correct for a

condensate and useless as a mass. The Higgs boson is a localized ﬂuctuation of the condensate, so its cost, if

ﬁnite, is a difference: perturbed vacuum minus unperturbed vacuum, with the extensive parts canceling.

The factor-seven obstruction.

= 244,944 = 2

·3

·7

, while every cell invariant of the honeycomb

{2,3}

-smooth (

,6,16, 9,12,

per site). No product of uniform cell counts of this complex, at any

lattice size, can carry the factor of seven. In the formula itself the seven arises only through the difference:

−D

= (K

−D)(K

+ K

D + D

) = 9 ·189 = 9 ·27 ·7

. Within the space of uniform-count products no

route to the number exists; a difference structure is the remaining avenue.

The scale obstruction. An exhaustive scan of the subtraction rules constructible from the binary complex—

every product of induced cell counts and their partial sums over the natural perturbation supports, minus

every Axiom-4-type shedding term;

37,334

generated values (script

)—contains nothing within

and its maximum value is

167,281

. The binary complex cannot reach the Higgs scale by any rule in its own

grammar.

The resolution. The resolution is already written in the condensate’s deﬁnition. A defect is binary (present

or absent) which is why the binary codes of the ladder exhaust the particle taxonomy. The condensate variable

U = |U|e

iθ

is not binary: it is a link degree of freedom with a resolution. Lifting the condensate sector to

-ary (qudit) link model makes the cost evaluable, and the evaluation is ﬁxed by two principles already

validated elsewhere in the framework:

Condensate cost. A localized ﬂuctuation of the condensate must verify its link resolution (

levels per real degree of freedom, jointly

for the complex link variable) over its spatial

coordination volume

. The cost is a difference, perturbed condensate minus unperturbed,

so what is veriﬁed is the condensate’s embedding; by Axiom 4 that embedding is ﬁxed by

orthogonal basis vectors, and the corresponding

spatial translations are shed at the same

resolution:

= K

−D

) = 144 ×1701 = 244,944. (10)

Each symbol is anchored.

D = 3

is the foliation. The exponent is not a free choice: Axiom 4 assigns two

vectors to a worldline and

to a volume, so the muon sheds

and the condensate

by one rule. The

spatial-only character of the shedding (

, not

(D + 1)

D+1

) is the same principle that ﬁxes the muon at

= 9

rather than

= 16

, the framework’s strongest internal check (Section 7); the checked alternatives fail

(

gives

237,168

; resolution

gives

20,412

; coordination the full coordination

= 24

gives

1.99 ×10

And

= 12

is not an assumption imported from outside: it is the kissing number of three-dimensional

space—the maximum number of unit spheres that can touch one, proven optimal and realized by the

slice.

The founding axiom of the framework, densest packing, ﬁxes the number of local references available for

comparison at exactly twelve; a locally veriﬁed degree of freedom can be resolved no more ﬁnely than the

references it is compared against. We name this the kissing resolution principle: a locally veriﬁed degree of

freedom is kissing-resolved, distinguishable at exactly as many levels as the kissing number of the spatial slice.

The condensate link variable is a kissing-resolved qudit,

d = K

= 12

; defects are binary. The dichotomy is

categorical, not incidental. Under this principle the entire formula descends from the packing axiom and the

foliation: the Higgs-to-electron mass ratio is a function of the kissing number and the dimension of space.

The observed ratio is

245,010 ±215

(Particle Data Group, PDG, 2024;

= 125.20 ±0.11

GeV [

]); the

prediction sits inside that interval, so the agreement is at the level the measurement can resolve and no tighter

claim should be made.

The step that is a model. “Resolution equals reference count” presumes that each extraction round

performs a nearest-reference classiﬁcation—a single twelve-outcome measurement of which neighbor the

link best aligns with—rather than twelve binary interrogations, which would resolve

levels. Nearest-

reference classiﬁcation is the natural measurement of a directional order parameter, but the selection of

this measurement model over alternatives is the open lemma on which the derivation rests, and we list it in

Section 15. The rule is falsiﬁable beyond its target: it is dimension-portable, assigning a

(2+1)

-dimensional

vacuum (triangular lattice, kissing number 6) a condensate excitation at

−2

) = 1152

electron masses.

Status of the derivation. The natural alternative identiﬁcation—the Higgs as an intrinsic extended logical-

operator class of the D4 code—is refuted by direct computation: the complete census of that code’s logical

space (script

, Section 4.2) contains no extended class, as every one of its 1282 logical classes reduces to a

local representative. The construction above is therefore the route that remains open within the examined

grammar. Its epistemic ledger: extensivity motivates a vacuum subtraction; two scoped impossibility results

(uniform-count products, and the stated rule grammar) constrain the alternatives; the shedding pattern follows

the muon precedent;

and

are ﬁxed by the packing axiom and the foliation. Under the proposed

kissing-resolution measurement model these yield

= K

−D

)

; the numerical match is reported in

Table 4.

13. The Extended Defect Sieve

Part I’s central structural argument enumerated twenty-ﬁve candidate spatial defects on the

lattice, applied

the four axioms to each, and rejected twenty. The ﬁve survivors matched the electron, muon, pion, proton, and

neutron to within 0.12 %. The rejections are at least as informative as the survivors: each rejected candidate

is a particle the framework forbids, and the empirical absence of those particles constitutes the negative

prediction expected of a falsiﬁable theory. This section carries out the same exercise for the D

framework.

The candidate space is larger here, on three counts. Time as a fourth lattice direction enlarges every

spatial defect class by a triality-depth dimension. Extended objects (2D worldsheets) are now classiﬁable

as gauge-ﬁeld defects rather than excluded as “non-particle” conﬁgurations. And vacuum condensates (4D

defects in the strict sense) enter the classiﬁcation as the Higgs sector. The ﬁve axioms (the four of Ref. [

]

plus Foliation Selection) apply uniformly, with results conditional on the hypotheses of Section 2 as marked.

We organize the candidates by spacetime dimension d

of the defect’s support.

Two structural changes distinguish the

sieve from Part I’s. First, Part I distinguished static from moving

defects as separate candidate rows; in

, motion is a worldline tilt (Section 7) (a continuous parameter, not a

separate conﬁguration) so the static/moving axis disappears. Second, and more consequentially, the sector is

no longer enumerated as a free axis. By the worldline sector rule of Section 7.1, the active sector is derived

from a candidate’s physical state label. Four rules do this work:

• R1: V ∈ σ if and only if the worldline is pinned.

• R2: F

△

∈ σ if and only if the state is conﬁned, that is, carries color ﬂux.

• R3: for free worldlines, F

□

∈ σ if and only if the state is electrically charged.

•

R4: conﬁned 3-sheet states activate the full

V + F

sector by Axiom 2, because the square faces detect

hook errors from the color-ﬂux tubes. Conﬁned 2-sheet states leave

□

decoupled. This asymmetry is

inherited from Parts I–II; a derivation connecting the two cases is open (Section 15).

A worldline candidate is therefore a physical state label (cross-section, conﬁned or deconﬁned, charged or

neutral, triality depth). The Axiom 2 rejection is scoped by its own logic: a state with spatial engagement

whose derived sector is empty carries veriﬁcation cost with no detecting stabilizer (cost without channel) and

is rejected as vacuum-indistinguishable. A time-only worldline, having no spatial engagement to detect, falls

outside the scope of this rule; its survival is conditional on the time-direction channel of Section 9 and is

marked accordingly (†) in the table. Table 3 collects the complete accounting.

= 0: spacetime points

A defect localized to a single spacetime event has no topological extent in any direction. Axiom 1 rejects it

on the same grounds as in Ref. [

], with the dimension count now over four directions rather than three. One

candidate, no survivor.

= 1: worldlines

A worldline carries the rest-mass particles. We classify by spatial cross-section (the slice through the worldline

at any instant of proper time) and by triality structure where it bears on the veriﬁcation cost. The ﬁve Part I

cross-sections (point, 1-edge, 2-sheet, 3-sheet, larger sheets) are joined by one new case, empty cross-section.

Empty spatial cross-section (time-only). A worldline with no

presence at any instant. The generalized

Axiom 1 is satisﬁed through temporal extent. The defect is distinguished only by which of the three 16-cell

time-halves the worldline is anchored to. Three survivors, marked conditional in Table 3:

. The mass

scale is set by the time-axis condensate strength rather than by topology (see Section 9), so the framework

predicts three light neutrino ﬂavors without ﬁxing the absolute scale.

Cross-section = single

edge (1-edge). The defect occupies one bond and lies in one triality set’s spatial

half. The charged deconﬁned state has derived sector trivial—

□

would be active by R3, but a single edge

bounds no faces (

□

= 0

), leaving only the existence bit

= 1

: the electron at

= 1

. The neutral deconﬁned

variant has empty derived sector and is vacuum-indistinguishable (Axiom 2). The conﬁned variant fails

Axiom 1 (conﬁnement requires at least two sheets, as in Ref. [

]). The variants at depth 2 or depth 3 fail

Axiom 2—the cross-section is too small to engage more than one 16-cell. One survivor (electron) out of

ﬁve 1-edge candidates.

Cross-section =

2-sheet and 3-sheet. The 12

nearest-neighbor bonds split as 4–4–4 across the three

triality sets (veriﬁed by

02_24cell_triality.py

), so a 3-sheet defect inherently engages the spatial halves

of all three 16-cells; the pion’s 2-sheet cross-section is grouped here as its conﬁned companion. The depth

label is 2 by our convention. Four physical states survive, with sectors derived by R1–R4, matching the four

remaining Part I particles:

State (dynamics, charge) Derived sector Formula Particle

3-sheet free, charged F

□

= 6 (R1, R3) 36 ×6 −9 = 207 µ

2-sheet conﬁned, charged, string V +F

△

= 17 (R1, R2, R4) 16 ×17 + 1 = 273 π

3-sheet conﬁned, charged full V +F = 51 (R1, R2, R4) 36 ×51 = 1836 p

3-sheet conﬁned, neutral full + probe 36 ×51 + 3 = 1839 n

The rejected states:

•

2-sheet conﬁned, charged, no closing string: rejected by Axiom 3, as in Ref. [

] (open color-ﬂux

boundary; C

= 272).

•

2-sheet conﬁned, neutral: the physical counterpart is the

, a ﬂavor superposition of quark–antiquark

states rather than a single defect, and lies outside the single-defect classiﬁcation; a defect-level treatment

of ﬂavor superpositions is an open problem (Section 15).

•

2-sheet free: rejected by Axiom 1 (deconﬁned states spread beyond a 2-sheet footprint, as in Ref. [

]).

•

3-sheet free, neutral: derived sector empty (

shed by R1,

△

absent by R2,

□

inactive by R3)—

vacuum-indistinguishable under Axiom 2. This is a negative prediction the sector-enumerated sieve

could not express: no neutral lepton exists at the muon mass scale, consistent with observation.

•

3-sheet free, charged, static: Axiom 4 requires colorless deconﬁned states to move. The static value

= 216 + 3 = 219 matches no observed particle. (Part I row 18.)

•

Depth 1 or depth 3 (2 rows each for the 2-sheet and 3-sheet): rejected by Axiom 2. The sheet structure

ﬁxes the depth at 2; depth 1 would require dropping bonds, depth 3 adding the 12 time-mixed bonds.

•

Heavier-quark composites: rejected because second-shell quarks are required, and second-shell defects

belong to a different shell index (Section 11).

Four survivors out of fourteen 2-/3-sheet candidates.

Cross-section = full 24-cell. The defect engages all 24

nearest-neighbor bonds (the 12 spatial bonds of

the 3-sheet plus the 12 time-mixed bonds) and so engages all three 16-cells in both halves. Depth 3. One

physical state survives:

State (dynamics, charge) Derived sector Formula Particle

24-cell free, charged F

□

= 36 (R1, R3) 96 ×36 −9 = 3447 τ

Rejected 24-cell states:

•

24-cell free, neutral: derived sector empty—vacuum-indistinguishable under Axiom 2. As with the

3-sheet analog, this is a negative prediction: no neutral lepton exists at the tauon mass scale, consistent

with observation.

•

24-cell conﬁned: Axiom 3 rejects. The 24-cell already engages the full 4D coordination shell;

conﬁnement creates a color boundary that cannot close at ﬁnite cost. The heavy proton analog is

likewise excluded: its constituent quarks would need to be second-shell, but ﬁrst-shell quarks are used

by the 3-sheet baryons.

•

24-cell free, charged, static: Axiom 4 requires colorless deconﬁned states to move; the static value

= 96 ×36 + 3 = 3459 matches no observed particle.

•

24-cell at depth 1 or 2: geometrically impossible—the 24-cell’s bond count ﬁxes the depth at 3—and

therefore not counted as candidates.

One survivor (tauon) out of four 24-cell candidates.

Larger spatial sheets (5-sheet, 7-sheet, ...). Inherited rejections from Part I. The boundaries of larger

sheets cannot close at ﬁnite cost (Axiom 3). No survivors at the ﬁrst coordination shell. Second-shell defects,

treated in Section 11, are a separate branch of the classiﬁcation.

= 2: worldsheets (gauge bosons)

A 2D defect in 4D spacetime represents a gauge ﬁeld strength

µν

. Worldsheets exist as proper topological

objects only in dimension

≥ 4

, so this class is genuinely new compared to Part I. We classify by sector and

by bundle topology.

Photon: trivial U(1) bundle, EM sector. No monodromy around any cycle. The Berry phase around every

closed loop is the identity; the defect has zero veriﬁcation cost. Survivor: C

= 0, m

= 0 exactly.

Gluons: SU(3) bundles on triangular plaquettes. The 8 triangular-plaquette bonds carry color-sector

bundles. Boundary closure (Axiom 3) is satisﬁed only inside color-singlet combinations; isolated gluons

have open color ﬂux. Eight survivors (the SU(3) gluon octet) all conﬁned, all with bare m

= 0.

W, Z: SU(2)

U(1) twisted bundles on square plaquettes. The 4 square-plaquette bonds carry electroweak

bundles. The non-trivial twists give three survivors (W

−

,Z); the trivial twist combination is the photon

(above). The W/Z masses come from the twist contribution to veriﬁcation cost, ﬁnite but not enumerated

quantitatively here.

Rejected worldsheet variants.

•

Trivial U(1) bundle on color sector: rejected by Axiom 2; SU(3) is non-abelian, so a trivial bundle

carries no ﬁeld strength and is not a gauge boson.

•

Non-trivial U(1) bundle on EM (would be a heavy photon): excluded at the level of the proposed

correspondence, which assigns the EM channel a single untwisted sector; the construction generates no

second species. This is a correspondence-level exclusion (Section 10 scope), not a derived one.

•

Worldsheets on time-mixed plaquettes: these contribute to Wilson-line phases (which set the gauge-

coupling running) but do not represent independent gauge bosons. Rejected by Axiom 5—foliation

selection breaks the time direction’s gauge structure into condensate modes.

Twelve proposed correspondences (1 photon + 8 gluons + 3 electroweak, EW), matching the Standard

Model gauge boson count.

= 3: worldvolumes

A 3D defect in 4D spacetime has codimension one: it partitions spacetime into two half-spaces. Axiom 3

(Boundary Closure) cannot be satisﬁed at ﬁnite cost without ﬁlling one half-space, in which case the defect is

no longer a defect but a phase boundary. No survivors at the ﬁrst shell.

A subtlety: some authors treat domain walls as legitimate topological objects, but in our framework

they violate the ﬁnite-cost requirement for a single localized particle, and so are excluded from the particle

classiﬁcation. They could in principle appear as cosmological objects, but that is outside the scope here.

= 4: vacuum condensates

A 4D defect ﬁlls spacetime—an order parameter with non-zero expectation value across the entire lattice. By

gauge choice, any such condensate reduces to a real scalar ﬁeld once gauge phases are absorbed.

Higgs: time-axis condensate. The unique survivor. The order parameter

⟨Φ⟩

of Axiom 5 picks out

one direction in

as time; the Higgs boson is a localized excitation of

around its vacuum value, with

quantitative veriﬁcation cost

= K

−D

) = 244,944

, inside the experimental uncertainty on

(Section 12.1). One survivor.

Rejected condensate variants.

•

Multi-axis condensates: rejected by Axiom 5. Foliation selection picks exactly one direction; conden-

sates along two or more axes over-specify the breaking and would leave residual SO(2) symmetry not

observed in nature.

•

Higher-rank tensor condensates: rejected by Axiom 1 generalized—tensor condensates break spatial

SO(3), contradicting the observed isotropy of space at the laboratory scale.

•

Spatial-only condensate (no time component): rejected by Axiom 5; without a time-axis selection, the

framework has no mechanism for fermion mass generation, and the observed mass spectrum requires

one.

One survivor (the Higgs scalar).

Tally and rejection summary

The exhaustive enumeration produces the following surviving ﬁrst-shell spectrum:

Class Count Identiﬁcation

Time-only worldlines 3 ν

,ν

1-edge worldlines 1 electron

3-sheet worldlines 4 µ,π

, p,n

24-cell worldlines 1 τ

Trivial U(1) worldsheet 1 photon

Color worldsheets 8 8 gluons (conﬁned)

Twisted EW worldsheets 3 W

−

4D vacuum condensate 1 Higgs scalar

Total survivors 22 —

This is the full Standard Model ﬁeld content modulo the heavy quark families, which sit at second-shell

defect levels and reach the same triality-driven generation count (Section 11). The

π, p,n

entries are ﬁrst-shell

effective composites of

and

quarks; the quark sector itself appears via second-shell extensions to

s,c,b,t

Table 3 summarizes the result. With sectors derived rather than enumerated, the sieve comprises 48

distinct physical-state candidates, of which 22 survive and 26 are rejected. Of the 22 non-rejected rows,

seven are mass-bearing survivors, three are conditional time-only states (

†

), and twelve are proposed gauge

correspondences (

‡

). The survivor set is identical to that of a sector-enumerated sieve. The candidate space,

however, differs in kind: sector-variant rows no longer exist (a sector is not something a candidate can “have

wrongly”), and the derived architecture exposes physical rows a sector enumeration cannot see—the neutral

conﬁned 2-sheet (

), the neutral deconﬁned 3-sheet and 24-cell states, and the static 24-cell. The two starred

rows are negative predictions: the derived sector of a neutral deconﬁned heavy lepton is empty, so no such

particle exists at the muon or tauon mass scale, consistent with observation.

The rejections group by axiom as follows:

• Axiom 1 rejected the spacetime point, the conﬁned 1-edge, the free 2-sheet, and tensor condensates.

•

Axiom 2 rejected every state whose derived sector is empty (the neutral 1-edge and the neutral

deconﬁned 3-sheet and 24-cell states—the latter two as starred negative predictions), all off-depth

variants, second-shell composites, and trivial color bundles.

•

Axiom 3 rejected the unconﬁned-gluon limit (gluons survive only conﬁned), the string-free conﬁned

2-sheet, the conﬁned 24-cell, larger sheets, and worldvolumes whose boundaries cannot close.

•

Axiom 4 rejected the static charged movers—the 3-sheet at

= 219

(as in Ref. [

]) and its 24-cell

analog at 3459.

•

Axiom 5 rejected multi-axis and spatial-only condensates and time-mixed gauge bosons (which become

Wilson-line phases rather than independent particles).

•

Outside the axioms, the neutral conﬁned 2-sheet (

) is excluded as a ﬂavor superposition rather than a

single defect (Section 15).

The structural claim, stated with its scope: within the enumeration grammar of this section, the surviving

spectrum accounts for the observed ﬁrst-shell particle content, with

, and

entering as effective composites

rather than fundamental ﬁelds, and with antiparticles, spin, and chirality not enumerated as separate axes.

Within that grammar there are no orphan survivors predicting unobserved particles, and no observed ﬁrst-shell

particle lacks a survivor. The correspondence can be falsiﬁed by any future discovery that violates either side

of this accounting.

Table 3: The derived-sector sieve. Sectors are computed from rules R1–R4, not assumed; a spatially engaged

state with empty derived sector (

σ = ∅

) carries cost without channel and is vacuum-indistinguishable (Axiom 2).

Starred rejections are negative predictions of the derived-sector architecture.

†

Conditional survivors: time-only

worldlines have no spatial engagement and fall outside the Axiom 2 rule; their survival rests on the hypothesized

time-direction channel (Section 9).

‡

Proposed gauge correspondences (Section 10): correspondence-level entries,

not derived survivors.

Candidate (cross-section, dynamics, charge) σ (derived) C

Verdict

0 spacetime point — — × Ax. 1

1 time-only, depth 1/2/3 ∅ (no spatial support) ≪ 1 ✓

†

,ν

1 1-edge, free, charged trivial (F

□

absent) 1 ✓ electron

1 1-edge, free, neutral ∅ — × Ax. 2 (vacuum-indist.)

1 1-edge, conﬁned — — × Ax. 1

1 1-edge, depth 2 or 3 — — × Ax. 2 (2 rows)

1 2-sheet, conﬁned, charged, string V +F

△

= 17 16×17+1 = 273 ✓ pion π

1 2-sheet, conﬁned, charged, no string V +F

△

272 × Ax. 3

1 2-sheet, conﬁned, neutral V +F

△

— × ﬂavor superposition (π

)

1 2-sheet, free — — × Ax. 1

1 2-sheet, depth 1 or 3 — — × Ax. 2 (2 rows)

1 3-sheet, conﬁned, charged full V +F = 51 36×51 = 1836 ✓ proton

1 3-sheet, conﬁned, neutral full + probe 1836+3 = 1839 ✓ neutron

1 3-sheet, free, charged F

□

= 6 216−9 = 207 ✓ muon

1 3-sheet, free, neutral ∅ — ×

∗

Ax. 2 (no neutral µ-analog)

1 3-sheet, free, charged, static F

□

+ probe 219 × Ax. 4

1 3-sheet, depth 1 or 3 — — × Ax. 2 (2 rows)

1 3-sheet, second-shell quarks — — × shell index (Sec. 11)

1 24-cell, free, charged, depth 3 F

□

= 36 96×36−9 = 3447 ✓ tauon

1 24-cell, free, neutral ∅ — ×

∗

Ax. 2 (no neutral τ-analog)

1 24-cell, conﬁned V +F

△

— × Ax. 3

1 24-cell, free, charged, static F

□

+ probe 3459 × Ax. 4

1 sheets ≥ 5 — — × Ax. 3 (class)

2 U(1) trivial bundle, spatial — 0 ✓

‡

photon

2 SU(3) octet, conﬁned — 0 ✓

‡

8 gluons

2 EW twisted bundle — (Higgs mech.) ✓

‡

2 SU(3) trivial bundle — — × Ax. 2

2 U(1) nontrivial bundle — — × single sector in proposed corr.

2 time-mixed plaquettes — — × Ax. 5

3 worldvolumes — — × Ax. 3 (class)

4 single-axis condensate — 244,944 ✓ Higgs

4 multi-axis condensate — — × Ax. 5

4 tensor condensate — — × Ax. 1

4 spatial-only condensate — — × Ax. 5

Totals: 48 candidates 22 26 rejected

Mass ratio m

(log scale)

new

(a) Predicted vs experimental mass ratios

Predicted C

Experimental m

−3

−2

−1

|deviation| (%)

exact

0.112%

0.048%

0.869%

0.0083%

0.017%

0.027%

(b) Deviation from experiment

Figure 4: (a) Predicted topological costs

versus experimental mass ratios

for the seven mass-bearing

particles on a log scale. The tauon at

= 3447

and the Higgs at

= K

−D

) = 244,944

are the two new

predictions of this work; the other ﬁve are inherited from Part I [

]. (b) Absolute deviation

exp

−C

|/m

exp

percent on a log scale. The electron is exact by deﬁnition. Five of the seven particles agree to better than

0.12%

;

the tauon is the largest deviation at 0.87%; the Higgs agrees within the PDG uncertainty on m

14. Comparison with Experiment

Table 4 and Figure 4 summarize the quantitative predictions. The ﬁve Part I particles are inherited; the tauon

and the Higgs are the new entries.

Table 4: Predicted topological costs versus experimental mass ratios. The electron is exact by deﬁnition; the

muon, pion, proton, and neutron values are from Part I [

]; the tauon prediction is from Equation 9; the Higgs

prediction is from Equation 10. Experimental values from CODATA-22 [10] and the Particle Data Group [11].

Particle Formula Predicted C

Experimental m

Deviation

Electron 1 ×1 1 1.000 exact

Muon µ 36 ×6 −9 207 206.768 0.11 %

Pion π

16 ×17 + 1 273 273.132 0.05 %

Tauon τ 96 ×36 −9 3447 3477.23 0.87 %

Proton p 36 ×51 1836 1836.153 0.008 %

Neutron n 36 ×51 + 3 1839 1838.684 0.017 %

Higgs H K

−D

) 244,944 245,010 0.03 %

14.1. Structural predictions

The framework also makes a set of qualitative structural predictions whose status is summarized below.

Prediction Experimental status

Exactly 3 charged lepton generations ✓ (e, µ, τ observed)

Exactly 3 light neutrino species ✓ (N

= 2.996 ±0.007)

3 neutrino ﬂavors, all m

≪ m

✓ (m

< 1 eV)

3 quark generations from second-shell triality ✓ (CKM is 3 ×3)

Photon associated with untwisted U(1) sector (masslessness pro-

posed)

✓ (m

< 10

−18

eV)

Gluons conﬁned, bare m

= 0 ✓ (QCD conﬁnement)

8 gluons + 4 electroweak bosons ✓ (12 gauge bosons in SM)

Single scalar Higgs, mass C

= K

−D

) = 244,944 ✓ (within the PDG uncertainty)

PMNS, CKM from triality rotations Qualitative match

14.2. Quantities not predicted

The framework does not yet ﬁx the absolute neutrino mass scale, the individual quark masses, the numerical

CKM and PMNS entries, or the W and Z masses to percent precision. These are open enumeration problems

within the framework rather than failures of principle: each requires extending the f-vector formalism to a

speciﬁc sub-structure (second-shell hadrons, twisted worldsheets for the electroweak gauge bosons), and we

list them as concrete problems in Section 15.

14.3. Statistical signiﬁcance of the mass matches

Integer mass formulas invite the objection that with enough structural constants available, some product of

them will land near any target. We quantify this look-elsewhere effect directly. Deﬁne the achievable set as

every integer of the form

×C

+ κ

, with

drawn from the edge counts of the framework’s sub-structures

{1,4,16, 36,96, K

}

from the stabilizer-count sums appearing in the f-vector arithmetic, and

from

the correction terms the axioms generate {0,±1,±D, ±D

,±D

,±K

}. This yields 523 distinct integers

(script

06_statistics.py

). For each particle, let

∆

be the achieved absolute deviation of the paper’s

prediction from the experimental ratio,

the number of achievable integers matching at least as well,

the

local density of the achievable set within

±10%

of the target, and

λ = 2ρ∆

the expected number of chance

matches of the achieved quality under a uniform-density null. The chance probability of at least one match

this good is p = 1 −e

−λ

Table 5: Look-elsewhere analysis of the six predicted mass ratios (the electron is the normalization and is

excluded).

k = 1

means the paper’s prediction is the unique achievable integer matching that well.

∗

The Higgs row

is reported but excluded from the joint ﬁgure. Its subtraction grammar is ﬁxed by Axiom 4, but the

multiplier

rests on H5, which was introduced for this condensate; the row is excluded conservatively (see the caveats below).

Particle ∆ k ρ (per unit) λ p

Muon 0.23 1 0.484 0.224 0.201

Pion 0.13 1 0.293 0.077 0.074

Tauon 30.23 8 0.026 1.565 0.791

Proton 0.15 1 0.071 0.022 0.021

Neutron 0.32 1 0.071 0.045 0.044

Higgs

∗

66.0 1 0.0002 0.030 0.029

Three conclusions follow. First, the tauon match carries little statistical weight on its own: its

0.87%

window admits eight achievable integers, with

λ = 1.6

expected by chance, so agreement at this level

is unremarkable. The tauon’s evidential contribution rests on the triality derivation of its existence and

generation index, not on the precision of its mass. Second, the muon, pion, proton, neutron, and Higgs

predictions are each the unique achievable integer matching as well as observed (

k = 1

), with individual

chance probabilities between

0.02

and

0.20

. Third, an independence approximation over all six rows would

give a joint probability of

8 ×10

−7

, and

5 ×10

−6

over an adversarial space of 1048 achievable values. We

report these numbers for completeness but do not rely on them: the caveats below remove two of the six rows

and raise the ﬁgure by three orders of magnitude.

Three caveats bound what the joint ﬁgure means, and the ﬁrst two are severe enough that we no longer

quote the joint number in the abstract or the conclusions. The proton and neutron predictions are not

independent trials (

= C

+ D

); treating them as one trial raises the joint probability by roughly a factor of

. The Higgs entry needs a different qualiﬁcation. Its subtraction grammar is not free: Axiom 4 (Section 5)

mandates

for a volume defect, exactly as it mandates

for the muon, and imposing that constraint

together with the

resolution of H5 leaves

= K

−D

)

as the only expression in the framework’s

inventory within

3σ

of the measured ratio. What is not pre-speciﬁed is H5 itself, which was introduced

to evaluate this condensate. We therefore exclude the Higgs row from the joint ﬁgure, conservatively, to

avoid look-elsewhere bias on the multiplier. Removing the Higgs and collapsing the proton and neutron

into one trial raises the narrow-space joint probability from

8 ×10

−7

to roughly

3 ×10

−4

, with the tauon

(

p = 0.79

) contributing nothing. More fundamentally, the null model conditions on the framework’s formula

grammar as given: it does not charge for the freedom exercised in constructing that grammar—the choice

of lattice, polytopes, arithmetic operations, correction terms, the allocation of formulas to particles, and the

development history of the tauon and Higgs expressions. The true look-elsewhere space includes model

construction itself and is not enumerable, so the joint ﬁgure is not interpretable as a conventional statistical

signiﬁcance, and we do not present it as one. What the table does establish, as a descriptive robustness check,

is narrower and solid: within the framework’s own formula grammar, ﬁve of the six predictions are the unique

achievable integers matching as well as observed, and the tauon match is individually unremarkable and is

identiﬁed as such rather than counted as evidence.

14.4. Limitations

The framework carries several methodological limitations beyond the unpredicted quantities above.

Derivational status of the Higgs formula. The derivation of Section 12.1 rests on one stated assumption:

the nearest-reference measurement model, under which a link degree of freedom resolves at the kissing

number of levels rather than at

. The subtraction structure, the shedding grammar, and the values of

and

are independently anchored; the measurement-model lemma is not, and the formula is conditional on

it.

The tauon edge-count convention, and its interaction with H3. The tauon formula uses

= 96

, the

24-cell’s shell edges alone, whereas the Part I convention (applied to the muon and proton, whose

= 36

counts the cuboctahedron’s 24 shell edges together with the 12 spokes to the center) would give the induced-

subcomplex count

= 120

(conﬁrmed by

10_d4_plaquette_code.py

). Neither this departure nor the

scope of H3 is derived. The two choices are not independent, and the cleanest way to see the exposure is to

tabulate all four combinations against the two leptons that use them:

squares unhalved squares halved (H3)

shell edges only (24 / 96) 135 / 6903 63 / 3447

shell + spokes (36 / 120) 207 / 8631 99 / 4311

Each cell gives (muon

tauon) under

×F

□

−9

. The muon requires the lower-left cell; the tauon requires

the upper-right. They sit on opposite corners: the tauon needs both conventions ﬂipped relative to the muon,

in opposite directions, and no other cell is near either target. The published

= 3447

is therefore not a

parameter-free consequence of the same formula applied to a larger polytope: it rests on two undischarged

choices. A single principle that ﬁxes

and the square count consistently across both polytopes would settle

this; the search for one is recorded as an open problem (Section 15).

The tauon residual. The prediction falls 0.87 % below experiment, roughly eight times the largest Part I

deviation. Whether this gap is a QFT-level correction (as conjectured) or a defect of the topological count is

not established.

Status of Foliation Selection. Axiom 5 (hypothesis H1) is assumed, not derived, and carries two open

components: the condensate picture requires a gauge-invariant characterization (Elitzur’s theorem forbids a

nonvanishing gauge-noninvariant local order parameter), and the emergence of Lorentzian signature on the

selected foliation (the sign structure A(∂

φ)

−B|∇φ|

) is not established by selecting a preferred axis.

Scale separation. The emergent-symmetry arguments available for lattice theories apply to long-wavelength

modes,

ka ≪1

. The defects of this paper are lattice-scale,

ka ∼1

. Whatever is eventually established about

the continuum limit of the

lattice (Section 3.2) will therefore not automatically transfer to the objects

whose veriﬁcation costs give the mass spectrum. Closing this gap requires a treatment of Planck-scale defects

that does not rely on a long-wavelength expansion, which we do not have.

Veriﬁcation scale. All

computations are at

L = 4

(the

plaquette-code validation is at

L = 6

). The

parameters

k = 5L

+ 2

and

d = 3

are conﬁrmed there; the asymptotic rate claim (

→ 5/6

) is an extrapolation

pending larger-

veriﬁcation (Section 15). The distance is not an extrapolation: it is ﬁxed at 3 by the triangles

for every L.

Sieve completeness. The derived-sector architecture (Section 7.1) removes the largest source of enu-

meration freedom (the sector is computed, not assumed) and Table 3 accounts for every candidate state so

generated. Residual caveats remain: the cross-section axis rests on the classiﬁcation by support dimension

and sheet structure, rule R4’s asymmetry between the conﬁned 2-sheet and 3-sheet is inherited from Parts I–II

rather than derived, and the

is excluded by a scope argument (ﬂavor superposition) rather than by an

axiom.

Status of the worldline sector rule. The rule of Section 7.1 is a consistency result across seven assignments,

not yet a derivation; the trajectory-correlate status of vertex syndromes for tilted worldlines, and the existence-

bit interpretation of the electron’s residual C

= 1, await ﬁrst-principles calculations (Section 15).

Interpretive distance. Finally, the framework’s central identiﬁcation (rest mass as fault-tolerant veriﬁcation

cost) remains an interpretive postulate inherited from Part I. The present paper extends its reach and internal

consistency but does not supply an independent derivation of the Landauer–Einstein mass formula itself.

14.5. Falsiﬁability

The framework forecloses two kinds of discoveries explicitly, the ﬁrst conditional on H2a. A fourth-generation

charged lepton or quark would require a fourth admissible transverse worldline structure, and the constructed

three-level engagement ladder has no fourth rung; the exclusion is conditional because exhaustion at that level

is precisely hypothesis H2a (Section 4.2). A stable particle below the topological ﬂoor in the spatial sector

(i.e., a non-neutrino with

0 < m

< m

) would also falsify the model, since the electron is by construction

the minimum spatial defect. No such particle has been observed at the Large Hadron Collider (LHC) or

anywhere else [11].

14.6. Computational veriﬁcation

Every numerical claim in this paper is reproduced by a veriﬁcation suite of sixteen scripts, executed by the

master script

d4_run_all.py

in under four minutes on a standard laptop. The scripts and the claims they

verify are:

• 01_structure_tensor.py

(Section 3):

µν

= 12δ

µν

exactly; the third moment

µνλ

∑

vanishes by centrosymmetry; emergence of the D

sub-lattice after foliation selection.

• 02_24cell_triality.py

(Sections 6 and 8): the 24-cell f-vector

(24,96,96, 24)

; the

8+8 +8

triality

split into three inscribed 16-cells; the 4–4–4 split of the 12

spatial bonds across triality sets; the 72

raw squares pairing antipodally into F

□

= 36.

• 03_d4_css_code.py

(Section 3): full construction of the CSS code at

L = 4

, verifying

n = 1536

uniform stabilizer weight 24,

= 0 (mod 2)

k = 5L

+ 2 = 1282

, and

d ≥ 3

by exhaustive

weight-≤ 2 elimination on both sides.

• 04_mass_spectrum.py

(this section): all seven rest-mass and Higgs formulas, the structural identity

= K

−D

) = 244,944, and their deviations from experiment.

• 05_triality_code_automorphism.py

(Sections 6.3 and 12.1): the coordinate 3-cycle

as a code

automorphism; the orbit structure on the 1536 qubits (512 orbits of size 3, none ﬁxed); explicit

construction of three triality-inequivalent worldline logical Z operators.

• 06_statistics.py

(Section 14.3): the achievable-integer spaces (523 narrow, 1048 broad), the

per-particle window densities and λ values of Table 5, and the joint chance probabilities.

• 07_sector_map.py

(Section 7.1): the muon–proton footprint degeneracy (identical 36-edge, 12-odd-

vertex boundary structure) and the reproduction of all seven sector assignments by the worldline sector

rule.

• 08_logical_classification.py

(Section 4.2): the complete census of the D4 code’s 1282 logical

classes; every class reduces to weight ≤ 9.

• 09_code_functional_validation.py

(Section 4.1): the

vertex–plaquette code

[[648,3]]

and

the exact reproduction of the sector counts 17, 6, and 51 as cell-count functionals.

• 10_d4_plaquette_code.py

(Section 4.2): the

edge code

[[1536,4]]

, the four weight-4 worldline

logicals, and the volume-scale membrane class.

• 11_face_qubit_code.py

(Section 4.2): the face-qubit code and the six coordinate worldsheet logi-

cals.

• 12_complete_cells.py

(Section 4.2): the completed 3-cell inventory and the collapse to

k = 6 =

dimH

• 13_top_rungs.py

(Section 4.2): the codes

[[3072,4]]

and

[[384,1]]

, the fundamental class, the honey-

comb incidence structure, and the factor-seven analysis.

• 14_rule_space_scan.py

(Section 12.1): the exhaustive scan of binary subtraction rules (37,334

values, maximum 167,281) and the vertex-ﬁgure veriﬁcation (24,96, 96,24).

• 15_spherical_design.py

(Section 3.2): symbolic computation of the bond-set moments; the

shell is a spherical 3-design with

= 3(k

)

−

∑

, the

shell a spherical 5-design with

3(k

)

exactly; and the measured anisotropy exponents

2.00

and

4.00

from the exact lattice dispersion.

• 16_distance_exact.py

(Section 3.3): all 4096 triangles of the

lattice at

L = 4

(and all 256 of

) are weight-3 logical operators, closing the distance from above and giving

d = 3

exactly for both

codes.

15. Open Problems

The framework leaves a list of ﬁnite, well-posed problems whose resolution would tighten it substantially.

We list them in rough order of importance.

Formalizing the worldline sector rule. Section 7.1 establishes the rule as a consistency result across

seven assignments. Two calculations would promote it to a derivation. The ﬁrst is the mutual-information

saturation described in Section 7.1; the second is a derivation of the electron’s residual

= 1

as an existence

bit rather than an interpretation of it as one. A connecting argument for rule R4’s asymmetry (why hook-error

completeness engages the square faces only when all three sheets are spanned) belongs to the same program.

Flavor superpositions. The neutral conﬁned 2-sheet row of Section 7.1 has no single-defect counterpart. A

defect-level treatment of superposed ﬂavor states (and with it a prediction for the

/π

mass splitting) is an

open extension.

W and Z masses. Enumerate the 2-cell stabilizer overlaps for SU(2)

U(1) twisted worldsheets on the

24-cell skeleton, with appropriate kinematic shedding for 2D extended states. The predicted ratio

should reproduce cosθ

at percent level.

Complete logical-operator classiﬁcation. The veriﬁcation script

05_triality_code_automorphism.py

constructs three explicit worldline (1D) logical Z operators related by triality and veriﬁes they are inequiva-

lent. The next step is an exhaustive partition of all 1282 logical qubits at

L = 4

by their minimum-weight

representative’s cellular dimension class (1D worldline, 2D worldsheet, 3D volume, 4D bulk) and triality

orbit. This would explicitly identify the Higgs’s logical-operator class and conﬁrm the

= K

−D

)

formula derivation. A larger L would verify the asymptotic operator-weight scalings.

Second-shell hadrons. Mass predictions for

, and

via second-shell defect enumeration,

analogous to Part I’s Section 6.

PMNS and CKM angles. Identifying the gauge-ﬁxing choice that translates triality rotation parameters

into observable mixing angles.

Derivation of Axiom 5 (H1) and Lorentzian emergence. Showing explicitly that the

Hamiltonian has

ground states selecting a time direction would upgrade Foliation Selection from hypothesis to theorem; a

complete derivation must also supply a gauge-invariant characterization of the condensate and a mechanism

for the Lorentzian sign structure

A(∂

φ)

−B|∇φ|

, since selecting a preferred axis accomplishes neither by

itself.

The nearest-reference lemma. The Higgs derivation of Section 12.1 is conditional on a single measurement-

model assumption: that each extraction round resolves a link degree of freedom by nearest-reference

classiﬁcation (one

-outcome measurement against the kissing conﬁguration) rather than by

binary

interrogations, which would resolve

levels. An argument selecting this model from the veriﬁcation axioms

(or an information-theoretic optimality principle) would complete the derivation; its failure would leave

unexplained. The rule’s dimension-portability prediction (

−2

) = 1152

for a

(2+1)

-dimensional

vacuum) provides an internal consistency target for any such argument.

The generation sector. H2a, H2b and H2c (Section 2) are all open, and each needs different work: an

exhaustion argument for the engagement ladder, a mechanism that breaks the axis-permutation symmetry and

correlates transverse structure with depth, and an explicit construction of the quotient identifying the three

depth-1 operators. The second of these is the Standard Model’s unexplained Yukawa hierarchy in a different

vocabulary, relocated onto a concrete combinatorial structure. The physical identiﬁcation of the three spatial

winding classes of H

) is likewise open.

A uniform convention for shell counts. The muon and the tauon are computed with different edge

conventions and different treatments of the square circuits (Section 14.4). A single rule that ﬁxes

and

the square count consistently on both the cuboctahedron and the 24-cell would either conﬁrm

= 3447

replace it. This is a ﬁnite combinatorial search and we regard it as the most pressing open item in the paper.

Continuum symmetry. Section 3.2 establishes the spherical-design property of the bond set and stops

there. Three further steps would be needed to connect the lattice to continuum spacetime symmetry: a deﬁned

renormalization-group procedure; an assessment of the radiative ﬁne-tuning problem of [

] in this speciﬁc

construction; and a mechanism for Lorentzian signature. We do not claim any of the three.

Root lattices beyond

. The reviewer’s question of whether a useful CSS code exists on the

root

lattice is well posed and open. More generally, the design strength of the shell (3 for

, 5 for

) is a lattice

invariant that could be used as a selection criterion among candidate substrates, and the

shell (a spherical

7-design) is the obvious next case.

Fiber polytopes for the gauge sector. The worldsheet correspondence of Section 10 places a continuous

SU(2)

U(1) bundle over a discrete polytope, which sits awkwardly inside an otherwise combinatorial

framework. Replacing the bundle by an appropriate ﬁber polytope would keep the construction discrete

throughout; whether the gauge-boson count survives that replacement is open.

Spin(8) triality. The order-three code automorphism is a lattice coordinate 3-cycle; establishing (or

refuting) its identiﬁcation with the outer-automorphism triality of Spin(8) in the representation-theoretic

sense (exhibiting the three corresponding representation sectors in the code) remains open.

Neutrino mass scale. The time-direction detection rate

should be computable from the Higgs VEV and

the D

lattice spacing.

Numerical veriﬁcation at larger

. Scripts

and

conﬁrm

k = 5L

+ 2

and

d = 3

L = 4

in under a

second; pushing the veriﬁcation to

L = 6,8

would test whether the asymptotic rate

k/n →5/6

is approached

uniformly. The distance needs no such test (it is pinned at 3 by the triangles at every

) but a companion

question is open and more interesting: which self-orthogonal subsets of the triangular plaquettes can be

added as stabilizers to raise the distance while retaining a high rate. Adding all of them collapses

to the

Betti numbers (Section 4.2), so the useful constructions lie strictly between the two extremes. At

L = 2

the

lattice degenerates (

±1 ≡ 1 (mod 2)

, so distinct NN displacements collapse onto coincident sites) and the

construction does not apply.

Each item is a ﬁnite calculation within the existing framework. None requires new physics.

16. Conclusions

Lifting the Part I framework from

(with one direction selected as time) gains structural access to

the phenomena that were excluded from the original scope. The derivation is conditional by construction:

ﬁve structural hypotheses are stated in Section 2, and every result carries its dependence explicitly. The

lift costs one new axiom (Foliation Selection, hypothesis H1), which spontaneously selects a preferred

time direction, breaking the lattice’s coordinate-permutation symmetry

while retaining translations

along the selected coordinate; the connection to continuum spacetime symmetry remains open (Section 3.2).

Beneath the selected foliation the spatial sub-lattice is exactly

, and the ﬁve Part I predictions for the

electron, muon, pion, proton, and neutron survive the lift unchanged, dependent on H1 alone. The lift also

resolves a question the 3D framework could not pose, let alone answer. the sector assignments, provably

underivable from spatial footprints (the muon and proton share an identical footprint), are determined by

worldline kinematics, and the derived-sector sieve built on that rule adds two negative predictions (no neutral

lepton at the muon or tauon mass scale) both consistent with observation.

The new content divides cleanly by epistemic status. Five results are rigorous and computationally veriﬁed.

The vertex–plaquette code carries the sector counts as literal stabilizer content. The homological ladder of four

chain-complex codes on

realizes the Betti numbers of the 4-torus,

(4,6,4, 1)

, with every class explicitly

constructed. The dimensional grading H4a is settled: refuted for the D4 code, exact as cellular homology

on the ladder. The particle worldlines all sit within the single time class, so generation structure is ﬁner

than homology. And the sector map is determined by worldline kinematics, having been shown underivable

from spatial footprints. Three scoped obstructions motivate the Higgs vacuum subtraction. Conditional on

the kissing-resolution measurement model of H5: the Higgs cost

= K

−D

) = 244,944

, with the

subtraction mandated by Axiom 4, agreeing with experiment within the measurement uncertainty, with

= 12

the kissing number of the spatial slice ﬁxed by the packing axiom through the kissing resolution

principle. Under the shell-edge convention and conditional on H3: the tauon cost

= 3447

, matching to

0.87 %. Proposed and qualitative: the neutrino sector as time-only worldlines, and the gauge bosons as a

worldsheet correspondence. The paper’s unconditional core is the mass-bearing structure itself—the chain

complex of the 16-cell honeycomb, whose vertex ﬁgure is the 24-cell and whose homology supplies the

dimensional grading on which the proposed particle taxonomy is built; its conditional claim is that this

structure, under the remaining hypotheses and one lemma, yields the ﬁrst-shell particle spectrum.

At the topological-structural level the framework now reproduces the full Standard Model ﬁeld content.

The items are: three charged leptons (matching to

≤ 0.87 %

); three neutrinos (count conditional on H2,

scale parametric); six quarks across three generations (structural); twelve gauge bosons partitioned

8 + 3 + 1

(count exact); a single scalar Higgs (mass matching within the experimental uncertainty); and the ﬁve lightest

non-strange hadrons (matching to

≤ 0.12 %

from Part I). The mass spectrum of the lightest charged particles

and the Higgs is predicted with no ﬁtted parameters; the discrete counts emerge as consequences of the

lattice topology.

CRediT authorship contribution statement. Raghu Kulkarni: Writing – review & editing, Writing –

original draft, Visualization, Validation, Methodology, Conceptualization.

Declaration of competing interest. The author declares no known competing ﬁnancial interests or personal

relationships that could have inﬂuenced the work reported in this paper.

Data availability. The complete veriﬁcation suite described in Section 14.6 is available as a single archive at

github.com/raghu91302/ssmtheory/blob/main/d4_extended_scripts.zip

(sixteen scripts, master

runner, ﬁgure generators, and a README mapping each script to the claims it veriﬁes). An interactive

3D visualization of the

lattice and its 24-cell is hosted at

raghu91302.github.io/ssmtheory/d4_

interactive.html.

References

[1]

R. Kulkarni, The Mass–Energy–Information Equivalence: A bottom-up identiﬁcation of the particle spectrum via

FCC lattice error correction, Phys. Open 27 (2026) 100414. doi.org/10.1016/j.physo.2026.100414

[2]

R. Kulkarni, A 67%-rate CSS code on the FCC lattice:

[[192,130,3]]

from weight-12 stabilizers, arXiv:2603.20294

(2026). arxiv.org/abs/2603.20294

[3]

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer, 3rd ed., 1999. Chapters 4 (the

family) and 7 (the 24-cell). doi.org/10.1007/978-1-4757-6568-7

[4]

P. Delsarte, J. M. Goethals and J. J. Seidel, Spherical codes and designs, Geom. Dedicata 6 (1977) 363.

doi.

org/10.1007/BF03187604

[5]

J. Collins, A. Pérez, D. Sudarsky, L. Urrutia and H. Vucetich, Lorentz invariance and quantum gravity: an

additional ﬁne-tuning problem?, Phys. Rev. Lett. 93 (2004) 191301.

doi.org/10.1103/PhysRevLett.93.

191301

[6]

S. Liberati and L. Maccione, Lorentz violation: motivation and new constraints, Ann. Rev. Nucl. Part. Sci. 59

(2009) 245. doi.org/10.1146/annurev.nucl.010909.083640

[7]

T. C. Hales, A proof of the Kepler conjecture, Ann. Math. 162 (2005) 1065.

doi.org/10.4007/annals.2005.

162.1065

[8]

J. F. Adams, Lectures on Exceptional Lie Groups, Chicago Lectures in Mathematics, University of Chicago Press,

1996. ISBN 978-0-226-00527-0. search.worldcat.org/isbn/9780226005270

[9]

ALEPH, DELPHI, L3, OPAL, SLD Collaborations, Precision electroweak measurements on the Z resonance,

Phys. Rep. 427 (2006) 257. doi.org/10.1016/j.physrep.2005.12.006

[10]

E. Tiesinga et al., CODATA recommended values of the fundamental physical constants: 2022, Rev. Mod. Phys.

97 (2024) 025002. doi.org/10.1103/RevModPhys.97.025002

[11]

R. L. Workman et al. (Particle Data Group), Review of Particle Physics, Prog. Theor. Exp. Phys. (2022) 083C01.

doi.org/10.1093/ptep/ptac097

[12] K. G. Wilson, Conﬁnement of quarks, Phys. Rev. D 10 (1974) 2445. doi.org/10.1103/PhysRevD.10.2445

[13]

R. Landauer, Irreversibility and heat generation in the computing process, IBM J. Res. Dev. 5 (1961) 183.

doi.org/10.1147/rd.53.0183