The Mass–Energy–Information Equivalence Extended:
D4 Lattice, Triality, and the
Three-Generation Structure of Matter
Raghu Kulkarni
*
1
1
SSMTheory Group, IDrive Inc, Calabasas, CA 91302, USA
Abstract
Part I derived the rest-mass spectrum of the five lightest non-strange particles—electron, muon, pion,
proton, neutron—from a substrate-free CSS stabilizer code on the FCC lattice, with predicted mass ratios
matching experiment to within 0.12 % and no fitted parameters. The framework left several phenomena
outside its scope: the tauon, all three neutrinos, the strange and charm sectors, the electroweak gauge
bosons, and the Higgs. We address each of these by lifting the construction to four dimensions on the D4
lattice—the densest 4D lattice packing—with one direction selected as time. The spatial slice of D4 is
exactly FCC, so the five Part I predictions are preserved unchanged. The derivation is conditional: we
state five structural hypotheses (H1–H5) up front and derive the first-shell spectrum from the D4 code
structure given them, with each result’s dependence recorded; the five Part I predictions depend only
on H1 (foliation selection). The hypotheses are then confronted computationally. The sector counts of
Parts I–II are shown to be exact stabilizer counts of a vertex–plaquette code on the same lattice, and the
construction extends to four chain-complex codes on D4 whose logical spaces realize the Betti numbers
of the 4-torus
(4,6,4,1)
: one time class and three spatial winding classes—with every particle worldline
shown to inhabit the time class, so that generation structure is finer than homology and H2 remains
a hypothesis—six worldsheets, four membranes, and a unique extensive condensate class, settling the
dimensional grading (H4a) as cellular homology, with the physical identification of the graded sectors
(H4b) remaining a hypothesis. The Higgs cost is treated as a vacuum subtraction: extensivity motivates
it, a factor-seven obstruction excludes uniform-count products, and an exhaustive scan of a stated rule
grammar contains no value near the target; under the proposed kissing-resolution measurement model
the ansatz
C
H
= K
2
3
(K
D
3
−D
D
) = 244,944
matches to 0.07 %, with
K
3
= 12
the kissing number of the
spatial slice fixed by the packing axiom. Using the shell-edge convention
E
s
= 96
and conditional on H3
(antipodal pairing), the tauon cost is
C
τ
= 96×36 −9 = 3447
, agreeing with experiment to 0.87 %. Three
neutrino flavors are proposed as time-only worldline defects below the spatial topological floor, and the
gauge bosons are given a worldsheet correspondence; both remain qualitative. A fifth axiom—Foliation
Selection (hypothesis H1)—spontaneously selects a preferred time direction, reducing the emergent
rotational invariance from SO(4) to SO(3), with translations along the selected coordinate remaining
available; the emergence of Lorentzian dynamics on the selected foliation is stated as the open component
of H1. The lift also determines what the 3D framework could not: the stabilizer sector assignments follow
from worldline kinematics (pinned worldlines count vertex syndromes, free worldlines shed them as
trajectory correlates), reproducing all seven assignments while being provably underivable from spatial
footprints alone, and the resulting derived-sector sieve yields two new negative predictions—no neutral
lepton exists at the muon or tauon mass scale—consistent with observation.
1. Introduction
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 [
1
]. The construction sat on the FCC lattice—the densest
*
raghu@idrive.com
1
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. 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.
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 FCC construction to the D4 lattice, provably the densest lattice packing in four
dimensions and the densest sphere packing known there. We show in Section 3 that the same CSS-code
structure carries over, with weight-24 stabilizers and an asymptotic encoding rate of
5/6
. 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), reducing the lattice’s emergent rotational invariance from SO(4)
to SO(3), with translations along the selected coordinate remaining available. The spatial slice of D4 is then
exactly FCC, and the five Part I predictions survive the lift unchanged (Section 7). A complementary account
of the same FCC lattice, addressing the ontology of a trapped-defect picture rather than verification cost,
appears in a separate publication [
3
]; the two descriptions contact at the proton-to-electron ratio, as discussed
there, and the present paper is logically independent of that work.
The new content is structural. The local 24-cell of nearest neighbors 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—and 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 D4 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 D4 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 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
2
than diffused through the framework.
H1 (Foliation).
A single direction of the D4 lattice is selected as time; the three orthogonal directions
constitute space, and the spatial sub-lattice is exactly FCC (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: the triality worldlines are homologous representatives within 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
e
,
µ
,
τ
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
F
□
= 36
for the 24-cell electromagnetic sector. The quotient applies to the 24-cell and nowhere else in
the first shell: the 2-sheet and 3-sheet face counts of Parts I–II, whose faces are not centrosymmetric
substructure circuits, are unquotiented.
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
H
1
, gauge worldsheets to
H
2
, and the Higgs condensate to
H
4
.
Section 4.2 settles H4a in two parts: the grading is refuted for the void 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
K
3
= 12
distinguishable levels in the syndrome-extraction sense, so that its
joint configuration count is
K
2
3
. Section 12.1 grounds the resolution count in the packing axiom through
the kissing resolution principle—
K
3
= 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. The table states the hypotheses as posed; their final statuses, established in the body of the paper, are
uneven by design: H4 is split by the ladder—its grading component H4a is discharged as a theorem on the
mass-bearing structure, while the physical identification H4b remains a hypothesis—and H2 is sharpened
and resolved 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), and 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 FCC spatial slice; the new results of this paper draw on
H2–H5 as indicated.
3
Table 1: Dependency and falsifiability of the structural hypotheses.
Hypothesis Results conditional on it Falsified by
H1 all (foliation; FCC slice) failure of emergent Lorentz invariance
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
τ
= 3447 an axiom-derived face count = 36
H4a dimensional grading of logical classes — (established; Section 4.2)
H4b worldline/worldsheet/condensate a graded sector whose physical
classification; Higgs support identification fails
H5 Higgs factor K
2
3
a derived condensate resolution = K
3
3. The D4 Lattice and Its CSS Code
3.1. Geometry
Consider the set of integer 4-vectors whose coordinates sum to an even number:
D4 = {x ∈Z
4
: x
1
+ x
2
+ x
3
+ x
4
≡ 0 (mod 2)}.
Each lattice point has
K
4
= 24
nearest neighbors at distance
√
2
, obtained by taking any two of the four
coordinates to be
±1
with the other two zero. There are
4
2
×4 = 24
such sign and permutation combinations.
Throughout this paper
K
d
denotes the kissing number of
d
-dimensional space:
K
4
= 24
for the D4 bulk and
K
3
= 12 for the spatial FCC slice. The mass formulas of Part I and of this paper consume K
3
.
Three structural facts will matter throughout. First, D4 is the densest lattice packing in four dimensions [
4
],
a result analogous to Hales’s Kepler-conjecture proof for FCC in 3D [
5
]. The Part I stability argument carries
over without modification: maximum packing density implies maximum stability, and any departure from
D4 coordination is a topological mismatch whose cost is the defect’s mass. Second, D4 is self-dual: the
dual lattice
D4
∗
equals D4 up to scaling. This means the X- and Z-stabilizers of the CSS code can sit on
geometrically equivalent objects, unlike in FCC where they sit on vertices and octahedral voids respectively.
Third, the Voronoi cell of D4—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. Lattice isotropy and emergent SO(4)
The Part I derivation of emergent Lorentz invariance from
S
µν
= 4δ
µν
required a continuum limit. For D4,
the same calculation gives a cleaner result. The structure tensor over the 24 nearest-neighbor bonds is
S
µν
=
24
∑
j=1
n
µ
j
n
ν
j
.
For diagonal components
S
µµ
, the three coordinate pairs
(µ,∗)
involving index
µ
each contribute 4 bonds
with
(n
µ
)
2
= 1
, giving
S
µµ
= 12
exactly. For off-diagonal
S
µν
with
µ = ν
, only one coordinate pair
(µ,ν)
contributes, and the four sign combinations sum to zero. Putting these together,
S
µν
= 12 δ
µν
[exact, by enumeration]. (1)
4
Centrosymmetry of the bond set forces
T
µνλ
= 0
exactly. The scalar-field dispersion relation is then isotropic
in all four directions at the lattice level:
ω(k)
2
= c
2
lat
|k|
2
+ O(k
4
a
4
), c
lat
= a
√
6κ.
The lattice symmetry itself is the discrete automorphism group of D4; what the identity
S
µν
= 12δ
µν
supplies
exactly is an isotropic second moment of the bond set, and with it an SO(4)-invariant leading-order continuum
dispersion—the same emergent status as the FCC result of Part II, with anisotropic corrections entering only
at
O(k
4
a
4
)
. Selecting a time direction then defines a preferred foliation with spatial rotation symmetry SO(3)
at the emergent level (Section 5); the emergence of Lorentzian dynamics on that foliation is an additional
step, discussed there.
3.3. The CSS code
Place one physical qubit on each edge of the D4 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
∑
x
i
≡ 1
(mod 2)
—and act on the 24 edges among the 8 D4 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 FCC’s octahedral
void, which had 6 surrounding vertices and 12 edges among them.
Both stabilizer types have uniform weight 24. The CSS condition
H
X
H
T
Z
= 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 an L
4
torus with L even, the parameters are
n = 6L
4
qubits, n
Z
= L
4
/2 Z-stabilizers, n
X
= L
4
/2 X-stabilizers,
and accounting for one global product relation per stabilizer type,
k ≥6L
4
−2(L
4
/2 −1) = 5L
4
+ 2.
The asymptotic encoding rate is therefore
k/n →5/6 ≈83.3%
, somewhat higher than FCC’s
2/3
. Minimum
distance is at least 3 by the same exhaustive weight-
≤ 2
argument that worked for FCC [
2
]; whether the D4
geometry supports distance 4 or higher is open.
3.4. The FCC sub-lattice and the role of time
Pick one coordinate—call it
x
0
, with the others spatial
(x
1
,x
2
,x
3
)
. The 24 nearest-neighbor bonds split cleanly
into two sets. The 12 bonds with
n
x
0
j
= 0
involve only spatial coordinates and are exactly the permutations of
(±1,±1,0)
in
(x
1
,x
2
,x
3
)
. These are the FCC nearest-neighbor bonds of Part I. The remaining 12 bonds have
one component in x
0
and one in a spatial direction, connecting adjacent time-slices.
This is the structural fact that makes the lift possible. Figure 1(a) shows the projection: the 12 spatial NN
form a cuboctahedron in 3-space (the FCC coordination shell), while the 12 time-mixed NN project pairwise
onto the six
±
axis points. A particle “at rest” is a worldline aligned along
x
0
, and its spatial cross-section
at any instant is an FCC defect of Part I. The five known masses survive the lift trivially; we verify this in
Section 7.
5
−1
0
1
x
−1
0
1
y
−1
0
1
z
(a) D4 nearest neighbors projected to 3D:
spatial (FCC) vs. time-mixed
Spatial NN (FCC, 12)
Time-mixed NN (12, projected to 6)
−1
0
1
x
−1
0
1
y
−1
0
1
z
(b) Triality decomposition:
three inscribed 16-cells
16-cell A
16-cell B
16-cell C
Figure 1: The 24 nearest neighbors of a D4 vertex, projected to 3D by dropping the time coordinate. (a) Spatial
bonds (blue circles, 12 vertices) form the FCC cuboctahedron of Part I. Time-mixed bonds (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.
3.5. Logical operators in a 4D CSS code
The CSS code of Section 3 lives in four lattice dimensions, and its logical-operator structure is qualitatively
richer than the 2D case that 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
2
), closed 3-volumes (
∼ L
3
), or the full spacetime extent (
∼ L
4
). 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 D4 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.
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
d
st
= 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
6
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 void 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 of Parts I–II consume f-vector data of the coordination cluster: vertex counts and face
counts. The CSS code of Section 3, like its FCC predecessor, carries vertex stabilizers and void stabilizers—
no face stabilizers. This raises a question the earlier papers did not pose: 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 code
Place 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. 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 FCC at
L = 6
this yields a
[[648,3]]
code whose three logical qubits are the torus cycles (verified:
rank(H
Z
) = 107
,
rank(H
plaq
) = 538 = 541 −3; script
09_code_functional_validation.py
).
On this code, define the detecting-stabilizer count of a defect with edge support
S
as the number of
stabilizers associated with the cells of the induced subcomplex of
S
: vertex stabilizers incident to
S
, and
plaquette stabilizers whose support is contained in
S
, 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
6
(six contained squares), the proton’s
13 + 38 = 51
(thirteen vertices, thirty-two triangles, six squares), with the electron’s
C
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.
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 void 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 void 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 D4
The plaquette construction lifts to D4 and extends to a family. The Delaunay decomposition of D4 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
△
,F
□
,C
3
,C
4
) =
L
4
2
, 6L
4
, 16L
4
, 9L
4
, 12L
4
,
3L
4
2
,
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
7
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
: 1 time + 3 spatial winding classes (wt. 4)
faces [[6400,6]] 6 H
2
: 3 + 3 worldsheets under the foliation (wt. 8)
tetrahedra [[3072,4]] 4 H
3
: membranes (wt. 192 representative)
16-cells [[384,1]] 1 H
4
: 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
H
1
(T
4
) = Z
4
2
: one time class and three spatial
winding classes, computed and exhausted. Two equivalence relations must be kept distinct here: inequivalence
under the void 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. (2) separate them. Their homology vectors, computed from their step displacements, are:
operator (w
t
,w
x
,w
y
,w
z
)
wl
A
(1,0,0,0)
wl
B
(1,0,0,0)
wl
C
(1,0,0,0)
All three wrap the time direction with vanishing net spatial winding: they are inequivalent under the void
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 void 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 void code of
Section 3, the dimensional grading is refuted: a complete census of its
1282
logical classes (script
08
) 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 void 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.
8
5. Foliation Selection: The Fifth Axiom
The four axioms of Part I—Minimum Topological Dimension, Sector Completeness, Boundary Closure,
Kinematic Shedding—carry through, with one generalization. Axiom 1 was originally a spatial statement
(
d
spatial
top
≥ 1
), since the FCC framework knew no time dimension. With time now a fourth lattice direction,
Axiom 1 reads as a spacetime statement:
d
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 full enumeration of which
candidate defects survive all five axioms and which fail—the D4 analog of Part I’s 25-candidate sieve—is
collected in Section 13.
The remaining three axioms apply to spatial defects exactly as in Part I, 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 D4 lattice as the time axis, defining a preferred foliation. The emergent rotational invariance
of Section 2.2 reduces from SO(4) to SO(3); translations along the selected coordinate remain available.
The 12 nearest-neighbor bonds of the resulting spatial sub-lattice are the FCC bonds of Part I, while the 12
time-mixed bonds carry energy and momentum between time-slices and do not contribute to spatial sector
counts.
The D4 lattice Hamiltonian carries the discrete D4 automorphism symmetry, with SO(4) invariance emerg-
ing at leading order in the continuum limit (Section 2.2). 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. The order parameter is a condensate of link variables along a single
direction:
⟨U
t
⟩ = 0, ⟨U
x
i
⟩ = 0 (i = 1, 2,3),
where
U
µ
is the lattice link variable in direction
µ
. 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.
Below the condensation scale, the four directions split as
12+12
(spatial
+
time-mixed), and the dispersion
relation reads
ω
2
= m
2
+ c
2
s
|k
s
|
2
with k
s
the spatial momentum and
m
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(∂
t
φ)
2
−B|∇φ|
2
with A,B > 0, a step not derived here and listed among the open problems.
Two points deserve emphasis. First, the axiom does not pick out 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
D
2
= 9
rather than rising to
16
, because the redundancy being shed is
9
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 D4 is the correct lift.
6. The 24-Cell, Triality, and the Three-Generation Structure
The 24 nearest neighbors of a D4 vertex form the vertex set of a 24-cell centered on that vertex. Its f-vector is
( f
0
, f
1
, f
2
, f
3
) = (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 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
A
form a 16-cell; the same for
B
and
C
. Triality is
the order-3 symmetry that cyclically permutes these three 16-cells. As a Lie-algebra fact, it is the outer
automorphism of Spin(8) [
6
], and it has no analog in any other dimension. The geometric realization in the
24-cell of D4 is its only finite-dimensional manifestation.
Figure 1(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
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. We also note that the automorphism
realized computationally is a lattice coordinate 3-cycle; its identification with the outer-automorphism triality
of Spin(8), in the representation-theoretic sense, is a further open step.
A 16-cell has 8 vertices, 24 edges, 32 triangular faces, and 16 tetrahedral cells. The verification cost at
depth
d
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.
10
LEP’s precision measurement of the invisible Z width gives
N
ν
= 2.996 ±0.007
[
7
], 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.
We verify this computationally. Let
π : (x
0
,x
1
,x
2
,x
3
) → (x
0
,x
2
,x
3
,x
1
)
be 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 D4 CSS code: the lattice permutations on D4 vertices (vert_perm) and
X-stabilizer sites (xsite_perm) and the induced permutation on edges (edge_perm) satisfy
H
Z
[vert_perm][:
,edge_perm] = H
Z
and
H
X
[xsite_perm][:,edge_perm] = H
X
. That is, applying triality permutes the Z-
and X-stabilizer groups among themselves rather than mapping any stabilizer outside the group.
3.
The orbit structure of
π
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
A
,wl
B
,wl
C
, each anchored in one of the three triality sets and wrapping the time
direction on the L = 4 torus via four nearest-neighbor steps:
wl
A
: (1,1,0,0) →(1,−1,0, 0) →(1,1, 0,0) →(1,−1,0,0); wl
B
: (pair (0,2) steps); wl
C
: (pair (0,3) steps).
(2)
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 (
H
X
·wl
i
= 0 (mod 2)
), so they are valid logical
Z candidates.
• None lies in the row-span of H
Z
, so all three are genuine logical operators rather than stabilizers.
• The triality permutation cycles them: π(wl
A
) = wl
C
, π
2
(wl
A
) = wl
B
, π
3
(wl
A
) = wl
A
.
•
The pairwise differences
wl
A
+ wl
B
,
wl
A
+ wl
C
,
wl
B
+ wl
C
are not in the row-span of
H
Z
. 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
11
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 distinct labels are in play in the generation sector, and the mapping among them must be kept
explicit: 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 7–8
consume, and the ladder terminates at depth 3 because full support is maximal: there is nothing further to
engage. 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.
(2)
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 FCC 3-sheet 207
tauon all three, spatial and time-mixed (symmetric) 3 24-cell tube 3447
The table makes the open structure visible, and hypothesis H2 resolves into the three components stated
in Section 2: whether this three-level ladder exhausts the admissible structures (H2a); why flavor tracks
depth—why symmetry-equivalent transverse labels acquire unequal engagement levels at all (H2b); and the
status of the depth-1 triple (H2c): the code contains three inequivalent depth-1 logical operators, and H2c
postulates that they do not represent three distinct physical electron species—no identifying gauge redundancy
has been constructed, so the postulated physical equivalence is a quotient coarser than the void-code logical
one. All three are discussed further in Sections 4.2 and 15.
7. Worldlines and the Part I Predictions
In Part I a particle at time
t
was a defect geometry in 3-space, and its rest mass was the verification cost of
that geometry. In D4 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.
The verification cost of a worldline is defined over a tubular spacetime neighborhood of one proper-
time step: the edges entering
E
s
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
12
spatial, the tube count reduces to the fixed-time spatial cross-section—exactly an FCC defect—and the cost
reduces to the Part I formula
C
worldline at rest
x
= C
FCC spatial cross-section
x
.
The Landauer–Einstein relation [
11
] and the cancellation of
kT ln 2
in mass ratios both carry through
unchanged. So all five Part I predictions are preserved:
C
e
= 1, C
µ
= 207, C
π
±
= 273, C
p
= 1836, C
n
= 1839,
matching the experimental ratios m
x
/m
e
to within 0.008–0.12 % as before.
The critical consistency check concerns Axiom 4. The muon’s formula reads
C
µ
= E
s
×C
s
−D
2
=
36 ×6 −9 = 207
. If
D
2
had to be re-interpreted as
4
2
= 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
D
2
= 9
is therefore strong internal evidence that the FCC sub-lattice is
the correct “spatial slice” of D4.
7.1. The sector map from worldline kinematics
Parts I–II assign each particle an active stabilizer sector—which of
V
,
F
△
,
F
□
enter
C
s
—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.
The D4 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. V 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
D
2
= 9
dimension check
above, this constitutes strong internal evidence that D4 is the correct lift: the sector map is underivable in the
3D framework and determined in the 4D one.
Two qualifications. 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
C
s
= 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.
13
8. The Tauon and the Three Charged Leptons
8.1. Electron: depth 1
The electron is a worldline whose spatial cross-section is a single FCC edge—the minimum 1-sheet defect of
Part I. By triality, the cross-section is localized within a single 16-cell sub-structure of the local 24-cell. With
E
s
= 1 and trivial sector C
s
= 1,
C
e
= 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 FCC 3-sheet (the 13-node cluster), with only the EM sector active and the state deconfined so
Axiom 4 applies:
C
µ
= E
s
×C
s
−D
2
= 36 ×6 −9 = 207.
This is identical to the Part I derivation; the new interpretation is that the FCC 3-sheet defect engages the
spatial halves of all three triality sub-structures simultaneously. The 12 FCC 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 D4—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 24-cell has 96 triangular 2-faces but no square 2-faces. The FCC
F
□
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
(in FCC this happens to coincide with the square 2-faces of the cuboctahedron, which is why both counts
gave 6 in Part I). 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 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
F
(24-cell)
□
=
72
2
= 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 the tauon prediction below is conditional
14
on it. H3 also fixes where the quotient applies: only to the 24-cell, whose squares are centrosymmetric
substructure circuits; the 2-sheet and 3-sheet face counts of Parts I–II are unquotiented.
With edges E
s
= 96 and the same Axiom 4 subtraction:
C
τ
= 96 ×36 −9 = 3447. (3)
The experimental ratio is m
τ
/m
e
= 3477.23 [8], giving a deviation of 30.23/3477.23 = 0.87%.
Two observations on this result. 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 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 FCC 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
Part I had no place for neutrinos. The electron was the minimum stable defect at
C
e
= 1
, and any neutrino
with
m
ν
/m
e
< 2 ×10
−6
would require
C
ν
< 10
−6
, far below the topological floor of one bit. The Part I defect
classification contained no configuration in this regime.
D4 with a selected time axis offers a new class of defect: a worldline whose spatial cross-section is empty.
Such a defect has no FCC 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
C
ν
(τ) = α
t
τ,
with
α
t
a rate set by the time-direction detection cost. Because the time direction is condensed (Axiom 5),
α
t
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
α
t
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:
m
ν
m
e
∼ α
t
≪ 1.
15
The specific value of
α
t
depends on the condensate scale—the Higgs 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:
ν
e
↔ 16-cell A, ν
µ
↔ 16-cell B, ν
τ
↔ 16-cell C.
The 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
N
ν
= 2.996 ±0.007
. (ii) All three are far lighter than
the electron,
m
ν
≪ m
e
, consistent with current bounds
m
ν
< 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 SM gauge boson count structurally: the 12 FCC nearest-neighbor bonds partition as
K
3
= 8 + 4
, matching 8 gluons plus the four electroweak bosons. The masses themselves were excluded
as “involving the Higgs mechanism.” The D4 construction permits treating the gauge bosons as genuine
topological defects.
A gauge field
A
µ
is a 1-form. Its field strength
F
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. D4 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
γ
= 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,
C
γ
= 0,
predicting
m
γ
= 0
exactly. This matches the experimental bound
m
γ
< 10
−18
eV
and is a genuine structural
consequence of trivial-bundle topology, not a fitted result.
16
10.2. Gluons: confined SU(3) octet
The eight gluon worldsheets occupy the triangular-plaquette bonds (
S
TOR
= 8
in the FCC sub-lattice), 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
S
TR
in the FCC
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
m
W
/m
e
≈ 1.575 ×10
5
and
m
Z
/m
e
≈ 1.785 ×10
5
, with the ratio
m
W
/m
Z
≈
0.882 = cos θ
W
identifying the Weinberg angle. A topological calculation predicting these masses to percent-
level requires extending the f-vector formalism to 2-cells in the D4 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 D4 framework reproduces this
count via the same
K
3
= 12 = 8 + 4
partition of the FCC sub-lattice; the time-mixed bonds contribute to
Wilson-line phases [
10
] 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 FCC—12 nearest neighbors at distance
√
2
—and
accounted for the five lightest non-strange particles. The heavier hadrons require defects anchored beyond
the first shell.
The second coordination shell of D4 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.
The natural identification is that strangeness, charm, bottomness, and topness are radial shell indices:
Shell Distance Flavor index
1
√
2 u, d (non-strange)
2 2 s (strange)
3
√
6 c (charm)
4
√
8 b (bottom)
5
√
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
17
states (
K
±
,
K
0
,
¯
K
0
). The
Λ
baryon is the
uds
configuration with one second-shell quark, and the
Σ
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 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
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 SO(4) lattice
symmetry breaks to the observed Lorentz structure. The Higgs boson itself is a localized fluctuation of
Φ
: 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
m
H
/m
e
≈ 2.45 ×10
5
, in the same
order of magnitude as
m
W,Z
/m
e
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.
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
Φ
. 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
3
2
L
4
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
L
4
: 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.
C
H
= 244,944 = 2
4
·3
7
·7
, while every cell invariant of the honeycomb
is
{2,3}
-smooth (
1
2
,6,16, 9,12,
3
2
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:
K
3
3
−D
3
= (K
3
−D)(K
2
3
+ K
3
D + D
2
) = 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.
18
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
14
)—contains nothing within
5%
of
C
H
,
and its maximum value is
167,281
. The binary complex cannot reach the Higgs scale by any rule in its own
grammar.
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
a
K
3
-ary (qudit) link model makes the cost evaluable, and the evaluation is fixed by two principles already
validated elsewhere in the framework:
Condensate shedding rule. A localized fluctuation of the condensate must verify its link
resolution—
K
3
levels per real degree of freedom, jointly
K
2
3
for the complex link variable—over
its spatial coordination volume
K
D
3
; from this, the spatial translation redundancy
D
D
is shed at
the same resolution:
C
H
= K
2
3
(K
D
3
−D
D
) = 144 ×1701 = 244,944. (4)
Each symbol is anchored.
D = 3
is the foliation. The spatial-only character of the shedding—
D
D
, not
(D + 1)
D+1
—is the same principle that fixes the muon at
D
2
= 9
rather than
4
2
= 16
, the framework’s
strongest internal check (Section 7); the checked alternatives fail (
D
4
gives
237,168
; resolution
K
1
3
gives
20,412
; coordination the full coordination
K
4
= 24
gives
1.99 ×10
6
). And
K
3
= 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 FCC 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
3
= 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 245,113; the deviation is 0.07%.
One step in this chain is a model, not a theorem, and we state it plainly. “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
2
12
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
6
2
(6
2
−2
2
) = 1152 electron masses.
Status of the derivation. The natural alternative identification—the Higgs as an intrinsic extended logical-
operator class of the void code—is refuted by direct computation: the complete census of that code’s logical
space (script
08
, 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
19
the muon precedent;
K
3
and
D
are fixed by the packing axiom and the foliation. Under the proposed kissing-
resolution measurement model these yield the ansatz
C
H
= K
2
3
(K
D
3
−D
D
)
; the numerical match is reported in
Table 3.
13. The Extended Defect Sieve
Part I’s central structural argument enumerated twenty-five candidate spatial defects on the FCC 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 D4
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 Part I plus
Foliation Selection) apply uniformly, with results conditional on the hypotheses of Section 2 as marked. We
organize the candidates by spacetime dimension d
st
of the defect’s support.
Two structural changes distinguish the D4 sieve from Part I’s. First, Part I distinguished static from moving
defects as separate candidate rows; in D4, 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 via four rules: R1,
V ∈ σ
iff the worldline is pinned
(Section 7.1); R2,
F
△
∈ σ
iff the state is confined (color flux present); R3, for free worldlines,
F
□
∈ σ
iff
the state is electrically charged; R4, confined 3-sheet states activate the full
V + F
sector by Axiom 2, since
the square faces detect hook errors from the color-flux tubes (Part II, Axiom 2), while confined 2-sheet
states leave
F
□
decoupled—an asymmetry inherited from Parts I–II whose connecting derivation 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 2 collects the complete accounting.
d
st
= 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 Part I, with the dimension count now over four directions rather than three. One
candidate, no survivor.
d
st
= 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.
20
Empty spatial cross-section (time-only). A worldline with no FCC 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 2:
ν
e
,
ν
µ
,
ν
τ
. 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 FCC 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—
F
□
would be active by R3, but a single
edge bounds no faces (
F
□
= 0
), leaving only the existence bit
C
s
= 1
: the electron at
C
e
= 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 Part I). 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 = FCC 2-sheet and 3-sheet. The 12 FCC 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 Part I (open color-flux
boundary; C
x
= 272).
•
2-sheet confined, neutral: the physical counterpart is the
π
0
, 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 Part I).
•
3-sheet free, neutral: derived sector empty (
V
shed by R1,
F
△
absent by R2,
F
□
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
C
x
= 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.
21
Cross-section = full 24-cell. The defect engages all 24 D4 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
C
x
= 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.
d
st
= 2: worldsheets (gauge bosons)
A 2D defect in 4D spacetime represents a gauge field strength
F
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
γ
= 0, m
γ
= 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
g
= 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
+
,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.
22
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 EW), matching the Standard Model
gauge boson count.
d
st
= 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.
d
st
= 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 D4 as time; the Higgs boson is a localized excitation of
Φ
around its vacuum value, with
quantitative verification cost
C
H
= K
2
3
(K
3
3
−D
3
) = 244,944
matching experiment to
0.07%
(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).
23
Tally and rejection summary
The exhaustive enumeration produces the following surviving first-shell spectrum:
Class Count Identification
Time-only worldlines 3 ν
e
,ν
µ
,ν
τ
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
+
,W
−
,Z
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
u
and
d
quarks; the quark sector itself appears via second-shell extensions to
s,c,b,t
.
Table 2 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, but the candidate
space 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 (
π
0
), 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
C
x
= 219
(as in Part I) 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 (
π
0
) 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
p
,
n
, and
π
entering as effective composites
24
Table 2: 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.
d
st
Candidate (cross-section, dynamics, charge) σ (derived) C
x
Verdict
0 spacetime point — — × Ax. 1
1 time-only, depth 1/2/3 ∅ (no spatial support) ≪ 1 ✓
†
ν
e
,ν
µ
,ν
τ
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
0
)
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.) ✓
‡
W
±
,Z
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
25
e
μ
π
±
τ
p
n
H
10
0
10
1
10
2
10
3
10
4
10
5
10
6
Mass ratio m
x
/m
e
(log scale)
new
new
(a) Predicted vs experimental mass ratios
Predicted C
x
Experimental m
x
/m
e
10
−3
10
−2
10
−1
10
0
|deviation| (%)
e
μ
π
±
τ
p
n
H
exact
0.112%
0.048%
0.869%
0.0083%
0.017%
0.069%
(b) Deviation from experiment
Figure 2: (a) Predicted topological costs
C
x
versus experimental mass ratios
m
x
/m
e
for the seven mass-bearing
particles on a log scale. The tauon at
C
τ
= 3447
and the Higgs at
C
H
= K
2
3
(K
3
3
−D
3
) = 244,944
are the two new
predictions of this work; the other five are inherited from Part I [
1
]. (b) Absolute deviation
|m
exp
−C
x
|/m
exp
in
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 sits at 0.07%, comparable to the muon.
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.
14. Comparison with Experiment
Table 3 and Figure 2 summarize the quantitative predictions. The five Part I particles are inherited; the tauon
and the Higgs are the new entries.
Table 3: Predicted topological costs versus experimental mass ratios. The electron is exact by definition; the
muon, pion, proton, and neutron values are from Part I [
1
]; the tauon prediction is from Equation 3; the Higgs
prediction is from Equation 4. Experimental values from CODATA-22 [8] and the Particle Data Group [9].
Particle Formula Predicted C
x
Experimental m
x
/m
e
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
2
3
(K
3
3
−D
3
) 244,944 245,113 0.07 %
14.1. Structural predictions
The framework also makes a set of qualitative structural predictions whose status is summarized below.
26
Prediction Experimental status
Exactly 3 charged lepton generations ✓ (e, µ, τ observed)
Exactly 3 light neutrino species ✓ (N
ν
= 2.996 ±0.007)
3 neutrino flavors, all m
ν
≪ m
e
✓ (m
ν
< 1 eV)
3 quark generations from second-shell triality ✓ (CKM is 3 ×3)
Photon associated with untwisted U(1) sector (masslessness pro-
posed)
✓ (m
γ
< 10
−18
eV)
Gluons confined, bare m
g
= 0 ✓ (QCD confinement)
8 gluons + 4 electroweak bosons ✓ (12 gauge bosons in SM)
Single scalar Higgs, mass C
H
= K
2
3
(K
3
3
−D
3
) = 244,944 ✓ (m
H
/m
e
= 245,113, 0.07 %)
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, the W and Z masses to percent precision, or any cosmological parameters. 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
E
s
×C
s
+ κ
, with
E
s
drawn from the edge counts of the framework’s sub-structures
{1,4,16, 36,96,K
2
3
}
,
C
s
from the stabilizer-count sums appearing in the f-vector arithmetic, and
κ
from
the correction terms the axioms generate {0,±1,±D, ±D
2
,±D
3
,±K
2
3
D
3
}. 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,
k
the number of achievable integers matching at least as well,
ρ
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 4: 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.
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 169.0 1 0.0002 0.076 0.073
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
27
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, under an independence approximation the joint probability
of all six matches arising by chance is
8 ×10
−7
; enlarging the formula space to an adversarial one—any
product of two structural integers appearing anywhere in Parts I–II, 1048 achievable values—weakens this
only to 5 ×10
−6
.
Three caveats bound what the joint figure means. The proton and neutron predictions are not independent
trials (
C
n
= C
p
+ D
); treating them as one trial raises the joint probability by roughly a factor of
20
. 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
2
K
3
. The subtraction structure, the shedding grammar, and the values of
K
3
and
D
are independently anchored; the measurement-model lemma is not, and the formula is conditional on
it.
The tauon edge-count convention. The tauon formula uses
E
s
= 96
, the 24-cell’s shell edges alone, whereas
the Part I convention (applied to the muon and proton, whose
E
s
= 36
includes the 12 spokes) would give the
induced-subcomplex count
E
s
= 120
. The departure is not derived; under the uniform convention the depth-3
lepton formula would not reproduce the tauon mass, so the published value depends on this choice together
with H3.
The antipodal identification. The tauon formula depends on halving the 72 planar squares of the 24-cell to
F
□
= 36
via antipodal pairing, stated as hypothesis H3. The chirality reading offered in Section 8 is physically
plausible but is not derived from the axioms; an alternative identification of the EM sector on the 24-cell
would change the prediction, so the tauon mass is a conditional prediction, not a direct one.
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(∂
t
φ)
2
−B|∇φ|
2
—is not established by selecting a preferred axis.
Verification scale. All D4 computations are at
L = 4
(the FCC plaquette-code validation is at
L = 6
). The
code parameters
k = 5L
4
+ 2
and the distance bound
d ≥ 3
are confirmed there; the asymptotic claims (rate
→ 5/6, distance scaling) are extrapolations pending larger-L verification (Section 15).
Sieve completeness. The derived-sector architecture (Section 7.1) removes the largest source of enumer-
28
ation freedom—the sector is computed, not assumed—and Table 2 accounts for every candidate state so
generated. Residual caveats remain: the cross-section axis rests on the classification by support dimension
d
st
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
π
0
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
s
= 1, await first-principles calculations (Section 15).
Interpretive distance. Finally, the framework’s central identification—rest mass as fault-tolerant verifica-
tion 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
x
< m
e
—would also falsify the model, since the electron is by construction the
minimum spatial defect. No such particle has been observed at the LHC or anywhere else [9].
14.6. Computational verification
Every numerical claim in this paper is reproduced by a verification suite of fourteen scripts, executed by the
master script
d4_run_all.py
in under three minutes on a standard laptop. The scripts and the claims they
verify are:
•
01_structure_tensor.py
(Section 3):
S
µν
= 12δ
µν
exactly;
T
µνλ
= 0
by centrosymmetry; emer-
gence of the FCC 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 FCC 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,
H
X
H
T
Z
= 0 (mod 2)
,
k = 5L
4
+ 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
C
H
= K
2
3
(K
3
3
−D
3
) = 244,944, and their deviations from experiment.
•
05_triality_code_automorphism.py
(Sections 6.3 and 12.1): the coordinate 3-cycle
π
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 4, and the joint chance probabilities.
29
•
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 void code’s 1282 logical
classes; every class reduces to weight ≤ 9.
•
09_code_functional_validation.py
(Section 4.1): the FCC 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 D4 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
2
(T
4
).
•
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. 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: showing that the mutual information
between successive-round vertex syndromes and the worldline tilt saturates the syndrome entropy for a
minimal moving defect (so that vertex bits of free worldlines are trajectory correlates and nothing more), and
deriving the electron’s residual
C
s
= 1
as an existence bit rather than interpreting 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 derived-sector sieve exposes the neutral confined 2-sheet as a candidate row,
whose physical counterpart, the
π
0
, is a flavor superposition of quark–antiquark states rather than a single
defect. A defect-level treatment of superposed flavor states—and with it a prediction for the
π
0
/π
±
mass
splitting—lies outside the present single-defect classification and 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
m
W
/m
Z
should reproduce cosθ
W
at percent level.
30
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
C
H
= K
2
3
(K
3
3
−D
3
)
formula derivation. A larger L would verify the asymptotic operator-weight scalings.
Second-shell hadrons. Mass predictions for
K
,
Λ
,
Σ
,
D
, and
B
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 D4 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(∂
t
φ)
2
−B|∇φ|
2
, 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
K
3
-outcome measurement against the kissing configuration—rather than by
K
3
binary
interrogations, which would resolve
2
K
3
levels. An argument selecting this model from the verification axioms
(or an information-theoretic optimality principle) would complete the derivation; its failure would leave
C
H
unexplained. The rule’s dimension-portability prediction (
6
2
(6
2
−2
2
) = 1152
for a
(2+1)
-dimensional
vacuum) provides an internal consistency target for any such argument.
The generation sector. The three triality worldlines are homologous representatives within the time class
of the mass-bearing complex (Section 4.2), so generation structure is finer than homology, carried by the
transverse triality label. Three questions are open. First, exhaustion: whether the three-level engagement
ladder exhausts the admissible transverse worldline structures within the time class (H2a). Second, symmetry
breaking: the depth-1 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 transverse structure with engagement depth (H2b). Third, the H2c quotient: constructing the
physical equivalence—coarser than the void-code logical quotient—under which the three inequivalent
depth-1 operators represent one electron rather than three degenerate species; the Standard Model’s own
unexplained Yukawa hierarchy is the second of these problems in a different vocabulary, here relocated onto
a concrete combinatorial structure. The physical identification of the three spatial winding classes of
H
1
(T
4
)
is likewise 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.
31
Neutrino mass scale. The time-direction detection rate
α
t
should be computable from the Higgs VEV and
the D4 lattice spacing.
Numerical verification at larger
L
. The CSS-code script
03_d4_css_code.py
already confirms
k =
5L
4
+ 2
and
d ≥ 3
at
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 and whether the distance grows with
L
in a manner
consistent with topological scaling. 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 FCC to D4—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,
reducing D4’s emergent rotational invariance from SO(4) to SO(3) while retaining translations along the
selected coordinate; the emergence of Lorentzian dynamics on that foliation remains open. Beneath the
selected foliation the spatial sub-lattice is exactly FCC, 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. Rigorous and computationally verified: the iden-
tification of the mass-bearing vertex–plaquette code, on which the sector counts of Parts I–II are exact
stabilizer counts; the homological ladder of four chain-complex codes on D4, whose logical spaces realize the
Betti numbers of the 4-torus (4,6,4, 1) with every class explicitly constructed; the theorem-grade statement
of the dimensional grading H4a (refuted for the void code, exact as cellular homology on the ladder; its
physical identification H4b remains a hypothesis), together with the homological placement of the particle
worldlines (all within the single time class, so that generation structure is finer than homology and H2 remains
a hypothesis); the sector-map result (sectors provably underivable from spatial footprints, determined by
worldline kinematics); and the three scoped obstructions motivating the Higgs vacuum subtraction. Proposed
as an ansatz under the kissing-resolution measurement model: the Higgs cost
C
H
= K
2
3
(K
D
3
−D
D
) = 244,944
,
matching experiment to 0.07 %, with
K
3
= 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
C
τ
= 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.
What the framework now reproduces, at the topological-structural level, is the full Standard Model field
content: 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
32
exact), a single scalar Higgs (mass matching to
0.07%
), 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 D4 lattice topology.
The deepest claim is the structural one. If Axiom 5 can be derived from a code-state condensate rather than
taken as input, the framework would explain why spacetime has three spatial dimensions and one temporal
dimension rather than four spatial. That would lift the signature of spacetime from a brute fact to a topological
consequence, which is a much stronger result than mass ratios. The path to that derivation, through the
link-variable condensation mechanism sketched in Section 5, is concrete; it remains to be executed.
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
(fourteen scripts, mas-
ter runner, figure generators, and a README mapping each script to the claims it verifies). An interactive
3D visualization of the D4 lattice and its 24-cell is hosted at
raghu91302.github.io/ssmtheory/d4_
interactive.html
.
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33
