Koide's Relation as a Kinematic Effect in a D4 Model

Koide’s relation as a kinematic eﬀect

in a D

lattice model of lepton mass

Raghu Kulkarni

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

raghu@idrive.com

Abstract

Koide’s relation among the charged-lepton masses, K = (m

)/(

√

)

= 2/3, holds to one part in 10

and has no accepted derivation. We examine it in a

lattice model of particle mass [2, 1], in which a particle is a topological defect in a quantum

error-correcting vacuum and its rest mass is the cost of the syndrome measurements that

keep the defect veriﬁed.

The model satisﬁes the relation, and does so through its kinematic term rather than its

static counts. The bare lattice products give K = 0.6631; subtracting the D

trajectory

correlations required of a free worldline, a term ﬁxed elsewhere in the model, gives K =

0.6667. Treated as a continuous parameter, the subtraction that saturates the relation

exactly is 8.86038, and the model independently requires D

= 9, the nearest integer to

it. In this description Koide’s relation is not a static packing identity but a relation among

dynamically realized costs. We also show that the angle it requires cannot come from the

lattice geometry.

The result depends on the tauon footprint, where Ref. [1] records two undischarged

conventions. We show the ﬁrst is not a convention at all: the muon and tauon occupy

diﬀerent supports, and one induced-subcomplex rule applied to each gives 36 and 96 with

no choice made. What remains is whether the site the tauon’s support surrounds but does

not contain enters the footprint. We propose that it does. Its magnitude, one unit rather

than a number tied to the 24 edges involved, follows from the stabilizer algebra. Its weight

in the cost functional does not, and that is the principal gap. The prediction of Ref. [1] is

unchanged at 3447; these are candidate resolutions of an open problem, not corrections.

1 What is claimed

Two things are reported here, and they depend on each other. The ﬁrst is a counting rule for the

tauon footprint, which resolves one of two conventions Ref. [1] leaves undischarged. The second

is that, with that rule in place, Koide’s relation is satisﬁed, and satisﬁed by the kinematic term

rather than by the static lattice counts. Section 2 sets out the model from scratch, for readers

meeting it here. Sections 4 and 5 give the rules, each with its own open edges, and Section 6

gives the Koide result.

2 The model in brief

This section is self-contained. Readers who know Refs. [2, 1] can skip it.

Particles as defects. The model treats the vacuum as a quantum error-correcting code laid

on a lattice. Qubits sit on the edges, and stabilizer measurements detect errors, as in any

stabilizer code. A particle is a stable topological defect in that code: a localized conﬁguration

of edges the stabilizers cannot remove.

Mass as a veriﬁcation cost. Landauer’s principle [4] assigns an energy k

T ln 2 to each bit

of information erased at temperature T . Reference [2] takes the rest mass of a defect to be

the thermodynamic cost of the syndrome measurements needed to keep it veriﬁed against the

vacuum. That cost is a bit count,

= E

× C

, (1)

in which E

is the number of lattice edges in the defect’s footprint and C

is the number of

stabilizers able to detect an error on it. Because T and k

are common to every defect, they

cancel in any ratio, so the model predicts mass ratios and not masses. All quantities below are

ratios to the electron.

The lattice. Reference [2] works on the face-centered cubic lattice, written D

: the integer

3-vectors of even coordinate sum, the densest sphere packing in three dimensions [5], in which

each site has 12 nearest neighbors arranged as a cuboctahedron. Reference [1] lifts this to D

the integer 4-vectors of even coordinate sum, the densest packing in four dimensions, in which

each site has 24 nearest neighbors arranged as a 24-cell. One coordinate is selected as time, and

the spatial slice of D

is exactly D

, so the three-dimensional results carry over.

A worked example. The proton is a defect occupying the full centered D

coordination

cluster: one central site with its 12 neighbors. The induced subcomplex has 36 edges, so

= 36. Its active sector, the set of stabilizers that detect it in the sense of Ref. [2], comprises

the 13 vertices and the 38 faces of that cluster, so C

= 51. Equation (1) gives

= 36 × 51 = 1836, against m

= 1836.15,

a deviation of 0.008 % with nothing ﬁtted. The electron, pion, muon and neutron follow in the

same way, each within 0.12 %.

Which stabilizers count. Not every stabilizer detects every defect. The active sector, in the

terminology of Refs. [2, 1], is derived from the defect’s physical state, whether it is pinned or

free, conﬁned or deconﬁned, charged or neutral, by four rules stated in Ref. [1]. A pinned defect

activates the vertex stabilizers; a conﬁned one activates the triangular faces; a free charged

one activates the square circuits. This is what makes C

diﬀer between particles that share a

footprint: the proton and the muon both have E

= 36, but C

= 51 against C

= 6.

The axioms, and Axiom 4 in particular. Reference [2] states four axioms constraining

which lattice conﬁgurations are admissible defects: Minimum Topological Dimension, Sector

Completeness, Boundary Closure and Kinematic Shedding. Reference [1] adds a ﬁfth, Foliation

Selection, and generalizes the fourth. Only the fourth is used below.

Its content is that a defect free to move sheds the part of its syndrome that reports position

rather than structure. Reference [1] states it as follows:

Axiom 4 (Redundancy Shedding). A defect sheds the D spatial translations

for each independent vector required to ﬁx its macroscopic embedding in the spatial

slice.

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

= 9 bits. These are trajectory correlations, not structure, and are subtracted:

= E

× C

− D

(free worldlines). (2)

Two conditions restrict this. Shedding applies only to free defects, so a pinned one keeps its

vertex syndromes; and a defect can shed only what it has, which excludes the electron, whose

sector carries a single bit. Equation (2) is what separates the muon, 36 ×6 −9 = 207, from the

proton at 36 × 51, which is pinned and sheds nothing. The value D

is ﬁxed by the axiom and

is not adjusted anywhere in this note.

Five hypotheses. Reference [1] states ﬁve structural hypotheses in advance and tags every

result with those it needs. Two are relevant below. H3 posits that the 72 square circuits of the

24-cell are identiﬁed in antipodal pairs. H4 of Ref. [1] concerns how logical operators are graded

by support dimension. The status of each is recorded there; this note concerns conventions that

H3 leaves open.

3 The two conventions

Five of the six predictions listed in Section 2 follow without adjustable choices. The tauon does

not. Its cost,

= E

× F

□

− D

= 96 × 36 − 9 = 3447, (3)

rests on two conventions the paper states plainly and does not discharge. Equation (3) is the

quantity at issue throughout.

Convention A, the footprint. Reference [1] records this as a departure: the tauon uses

= 96, the 24-cell’s shell edges alone, whereas applying the Part I rule to the centered D

cluster would give the induced-subcomplex count E

= 120.

We observe that the departure is apparent rather than real, and that it dissolves once the

supports are compared. The two defects do not occupy the same object. Reference [1] identiﬁes

the depth-2 support as the D

3-sheet, which is the centered cuboctahedron, 13 sites including

the center; and the depth-3 support as the full 24-cell, which is the shell, 24 sites and no center.

Taking the induced subcomplex of each, by the same rule and with no choice made,

depth 2, centered D

cluster (13 sites) −→ E

ind

= 36,

depth 3, 24-cell shell (24 sites) −→ E

ind

= 96.

The count 120 arises only for the centered D

cluster of 25 sites, which is not the support the

ladder names. The muon’s spokes are counted because its support contains the center; the

tauon’s are not because its support does not. A defect occupies the object it occupies, and the

footprint follows.

What this leaves open is narrower, and it is the subject of Section 4. The tauon’s shell

surrounds a site it does not contain. Whether that site enters the footprint is not settled by

the induced subcomplex, which by construction sees only what lies inside the support.

Convention B, the halving. The 24-cell has 72 planar 4-circuits, identiﬁed in antipodal

pairs to give F

□

= 36. Reference [1] records the objection to this itself. The cuboctahedron

is equally centrally symmetric, and its six squares form three antipodal pairs. Halving applied

uniformly would therefore replace the muon’s F

□

= 6 by 3, destroying a prediction accurate to

0.11 %.

Section 4 proposes a rule for A and Section 5 for B. Everything reported is reproduced by

koide 17 enclosed voids.py, included with this note.

4 Convention A: enclosed sites

The rule. A lattice site is enclosed by a defect when it does not lie in the support and every

edge incident to it terminates in the support. The proposal is that the footprint counts enclosed

sites alongside induced edges:



edges of the induced subcomplex



fully enclosed sites



. (4)

Why an enclosed site would be counted. Every qubit incident to an enclosed site has

both endpoints inside the defect, so no stabilizer outside the defect acts on any of them and a

syndrome at that site cannot be read from outside. The site is causally interior: its veriﬁcation

falls to the defect.

Why the contribution is 1 and not 24. The center of the D

cluster has 24 incident edges,

so an argument resting on inaccessible incident qubits might seem to suggest a contribution of

24 rather than 1. It does not. Those 24 edges are the support of a single operator, the vertex

stabilizer at the enclosed site, and one counts the operator rather than the qubits it acts on.

Figure 1 shows the distinction between containment and enclosure.

This is veriﬁable in the code. Writing V (x) for the vertex stabilizer at site x, the rank of

the stabilizer set grows by exactly one when the enclosed site is included:

rank



V (x) : x ∈ Σ ∪ {c}



− rank



V (x) : x ∈ Σ



= 1, (5)

with Σ the 24-cell shell and c the enclosed center. So the enclosed site carries one independent

syndrome degree of freedom, not 24.

The count is per enclosed site and not per enclosed region. Constructing supports that

enclose regions of one, two and three sites, the rank rises by one, two and three respectively,

while the number of bounded components of the complement is one in all three cases. The

tauon’s enclosed region contains exactly one site, which is why N

enclosed

= 1 there. Equation (5)

and the multi-site test are both in koide 17 enclosed voids.py.

What remains unestablished. The magnitude is ﬁxed by Eq. (5), but the weighting is not.

Equation (4) adds quantities of two combinatorial types, 96 edges and one syndrome degree of

freedom, and Ref. [2] introduces E

as an edge count.

One reading under which it is dimensionally coherent is that E

counts independently veriﬁed

footprint elements rather than edges literally. Ordinary supports contribute through their edge-

qubits; a fully enclosed site contributes the single inaccessible syndrome degree of freedom it

carries. We do not establish that reading here, and it would have to be checked against Ref. [2],

where every E

is an edge count and the reinterpretation must leave all ﬁve earlier predictions

unchanged.

Two further things would put the rule on ﬁrmer ground. It has one non-trivial application

in the framework, and that instance is the quantity it was sought to explain; a second defect

class with a non-zero enclosed term, and an independently checkable prediction, would settle

whether the rule is general. And Eq. (4) is stated rather than derived: obtaining it from the

veriﬁcation postulate of Ref. [2] would remove the question of weighting altogether. The object

is not new to the framework. In Part I [2] the Z-stabilizers of the D

code sit on the octahedral

voids of the lattice, so a void is already a veriﬁcation site. Equation (4) adds only that a void

enclosed by a defect belongs to that defect’s footprint.

What it gives. Applied to every defect of the framework, each on its own coordination

cluster, the rule changes one number. The enclosed count is zero for every D

support, because

the octahedral void at distance one requires a second-shell site no ﬁrst-shell cluster contains. It

is one for the 24-cell shell, and the enclosed site is the origin.

The tauon moves from −0.87 % to +0.17 %. It ceases to be a seven-fold outlier and joins

the band occupied by the other ﬁve, with residuals mixed in sign rather than one being system-

atically low.

support includes the centre;

its spokes are internal edges

muon: centre contained

= 36 + 0 = 36

every edge at the centre ends in the support,

so no outside stabilizer reaches it

tauon: centre enclosed, not contained

= 96 + 1 = 97

Figure 1: Containment against enclosure. Left: the muon’s support is the centered D

cluster; it

contains the center, so the twelve spokes are internal edges and nothing is enclosed, E

= 36+0.

Right: the tauon’s support is the 24-cell shell; it surrounds the center without containing it,

every edge at the center terminates in the support, and the site is enclosed, E

= 96 + 1.

Table 1: The rule applied to every defect. Only the tauon row changes.

Particle Cluster E

ind

encl. E

Deviation

Electron D

, 1-edge 1 0 1 1 exact

Pion D

, 2-sheet 16 0 16 273 −0.048 %

Muon D

, 3-sheet 36 0 36 207 +0.112 %

Proton D

, 3-sheet 36 0 36 1836 −0.008 %

Neutron D

, 3-sheet 36 0 36 1839 +0.017 %

Tauon D

, 24-cell 96 1 97 3483 +0.166 %

Checks.

1. The enclosure is unique. A search over the neighborhood of the 24-cell shell ﬁnds exactly

one enclosed site, the origin.

2. No D

support encloses anything. The octahedral void at (1, 0, 0) requires all six surrounding

sites; (2, 0, 0) is second shell.

3. No double counting with the pion. The pion’s +1 in 16 ×17 + 1 is the closing string required

by Axiom 3 of Ref. [2], which enforces Gauss’s law on the lattice. There are 129 distinct

supports with E

ind

= 16, and every one encloses zero sites, so the two unit terms are diﬀerent

objects.

4. Vertex–plaquette incidence. Repeating the computation on the D

torus at L = 6 (648

sites, 7776 edge-qubits), with enclosure deﬁned purely by edges and their endpoints, gives

96 induced edges and one enclosed site for the shell, and 120 with none enclosed for the

centered cluster—the latter reproducing the ﬁgure of Ref. [1].

The sieve. Reference [1] enumerates every candidate defect conﬁguration and eliminates those

that violate an axiom, leaving the observed particles as survivors. Two rows of that sieve use

filled

6 squares, all of them faces

identifying them would quotient the complex

cuboctahedron

the square is a 2-face

not a cell

72 circuits, none a face; each spanned by 4 triangles

identifying them leaves the complex unchanged

24-cell

the square is only a circuit

Figure 2: The cuboctahedron’s six squares are 2-faces of the polytope. The 24-cell’s seventy-two

are closed paths in the 1-skeleton and are not cells; each is spanned by four triangular 2-faces.

the 24-cell footprint: the tauon row, 3447 → 3483, and the static row rejected by Axiom 4,

3459 → 3495. No rejection changes, because every rejection follows from the derived sector or

from an axiom applied to the state label, and the footprint enters neither. No cost collides with

another row: the nearest is the static row at 3495, and every other diﬀers by more than 1600.

The survivor set is unaﬀected.

5 Convention B: faces against circuits

The objection recorded in Ref. [1] treats the cuboctahedron’s squares and the 24-cell’s as the

same kind of object. They are not, as Figure 2 shows.

A supporting-hyperplane test gives the counts exactly. All six of the cuboctahedron’s 4-

circuits are 2-faces. None of the 24-cell’s seventy-two is: its only 2-faces are the 96 triangles,

and each square circuit is spanned by four of them. The antipodal orbit counts are 3 and 36

respectively.

4-circuits of which 2-faces antipodal orbits F

□

used

Cuboctahedron 6 6 3 6

24-cell 72 0 36 36

A structural observation, not yet a rule. One could count 2-faces where the squares

are faces, and antipodal orbits of circuits where they are not. This gives 6 and 36, the values

the framework uses. The asymmetry is structural rather than chosen: identifying antipodal

faces quotients the cell complex, so the cuboctahedron would cease to be the cuboctahedron;

identifying antipodal circuits leaves the complex untouched, because circuits are not cells.

The tension. A square circuit of the 24-cell and its antipode are spanned by disjoint sets

of triangles in 54 of the 72 cases, and overlapping sets in the remaining 18; in no case are the

spanning sets identical. If a circuit is detected only through the faces that span it, then antipodal

circuits are detected by disjoint cells, and identifying them needs a justiﬁcation beyond their

not being faces. We state this as the open edge of the observation rather than resolve it.

A second objection is structural. The rule as written is disjunctive, two clauses with a

condition selecting between them, and a single principle would be better. The most plausible

form of one would be framed in terms of what the code detects rather than what the polytope

contains: a derivation of 72 → 36 from operator equivalence, stabilizer redundancy or logical

homology would dispose of the convention entirely.

6 Koide’s relation

Koide’s empirical relation among the charged-lepton masses [3],

K =

+ m



√



, (6)

holds to one part in 10

and has no accepted derivation. It is equivalent to a statement about a

single angle. Writing u = (

√

) and d = (1, 1, 1), the angle between them satisﬁes

cos

θ = 1/(3K), so K = 2/3 is exactly θ = 45

◦

Applied to C

= 1 and C

= 207, Eq. (6) demands C

= 3481. The three costs give

K θ

measured leptons 0.666661 44.9997

◦

= 3447 (Ref. [1]) 0.665682 44.9576

◦

= 3483 (Section 4) 0.666724 45.0025

◦

so the rule lands within three thousandths of a degree of the required angle, from lattice counting

and without reference to Eq. (6).

The angle does not come from the lattice. It lives in the three-dimensional ﬂavour space

spanned by the charged-lepton costs, whereas the lattice’s own threefold structure is a 120

◦

rotation, and such a rotation cannot produce 45

◦

at all:

Let a 120

◦

rotation be generated by a pair of roots at 120

◦

, with Gram matrix



2 −1

−1 2



. Any other root of a simply-laced system has integer inner products with

those two, so its projection onto the rotation plane satisﬁes

cos

θ ∈

, 0

, (7)

that is 0

◦

, 54.74

◦

or 90

◦

. The value 1/2 does not occur.

We veriﬁed Eq. (7) numerically (koide 18 a2 angles.py) for A

, A

, D

, . . . , D

, E

and E

: every one returns the identical spectrum {0

◦

, 54.74

◦

, 90

◦

}, independent of dimension

or of the number of roots. Root lattices are organized around 1/

√

3, the angle of a diagonal in

three dimensions; Koide requires 1/

√

Where in the framework the agreement enters

The costs of Table 1 are not bare lattice counts. By Axiom 4 of Ref. [2], a free worldline

sheds D

= 9 spatial trajectory correlations; the electron, below the shedding threshold, sheds

nothing. It is worth separating the two contributions.

static counts E

× C

1 216 3492 0.663084

after the Axiom 4 shedding 1 207 3483 0.666724

The static counts miss Eq. (6) by −0.54 %. The shedding carries them to +0.009 %. The

agreement therefore enters through the kinematic term and not through the static geometry.

The shedding value is not free. It is ﬁxed at D

by Axiom 4 of Ref. [1], independently of

anything in this note, and no neighboring framework quantity does the same work:

shed C

deviation of K from 2/3

none 216 3492 −0.537 %

D = 3 213 3489 −0.358 %

6 210 3486 −0.176 %

= 9 207 3483 +0.009 %

12 204 3480 +0.196 %

(D+1)

= 16 200 3476 +0.450 %

= 27 189 3465 +1.177 %

Treated as a continuous parameter, the shedding value that saturates Eq. (6) exactly is s =

8.86038. The value the framework independently requires, s = D

= 9, is the nearest integer to

it, at a distance of 0.140, and yields K = 0.6667244, a relative deviation of 8.65 × 10

−5

. The

window in which K falls within 0.1 % of 2/3 runs from s = 7.240 to s = 10.468, so no other

quantity the framework makes available, D = 3 or (D+1)

= 16 or D

= 27, lies inside it.

The two rules are needed together. The shedding alone does not suﬃce. With the

published footprint E

= 96 the static counts are (1, 216, 3456) with K = 0.662044, and the

shedding carries them only to 0.665682, still −0.148 % short. With E

= 97 the same shedding

carries (1, 216, 3492) to 0.666724, which is +0.009 %. The enclosed site of Section 4 and the

kinematic subtraction of Axiom 4 are jointly necessary, and neither was obtained with reference

to the other.

The electron’s exemption is forced. Axiom 4 of Ref. [2] applies only where dim(B

sector

)

exceeds the quantity to be shed. The electron, with dim(B) = 1 < 9, cannot shed, so the

subtraction acts on two of the three leptons and not on all three. This asymmetry is what makes

the triple inequivalent. Applied uniformly it would not: three costs related by an unbroken

symmetry are equal, and equal costs give K = 1/3, the minimum of the range [1/3, 1] in which

K takes values.

What we conclude. Within the description set out here, Eq. (6) is not a static packing

identity. It is a relation among dynamically realized lepton costs. The static lattice counts do

not satisfy it, missing by −0.54 %; the subtraction of D

spatial trajectory correlations carries

them to +0.009 %; and that subtraction is ﬁxed by Axiom 4 independently of anything in this

note. Read together with the degenerate value K = 1/3 that an unbroken threefold symmetry

would give, K measures how far the dynamics displace the spectrum from degeneracy, and 2/3

sits at the midpoint of the available range.

Data availability

Every number in this note is reproduced by the scripts below, which are available at github.

com/raghu91302/ssmtheory. They require only the standard library, apart from the ﬁgure gen-

erators, which need numpy and matplotlib, and koide 18 a2 angles.py, which needs numpy.

The suite runs in about a minute.



koide 17 enclosed voids.py — the counting rule of Section 4: the table of all six de-

fects, the four checks, the stabilizer-rank result of Eq. (5) with its multi-site test, the sieve

comparison, and the static-against-shed tables of Section 6.



koide 18 a2 angles.py — Eq. (7) across A

, A

, D

, . . . , D

, E

and E



koide gen fig5 enclosed.py and koide gen fig6 faces.py — Figures 1 and 2. Both

compute their own counts and print them rather than taking them as given.



koide note check.py — an end-to-end audit of this note: that every reference resolves,

that the numbers agree throughout, and that the scripts run.

References

[1] R. Kulkarni, The Mass–Energy–Information Equivalence Extended: D

Lattice, Triality, and the

Three-Generation Structure of Matter, submitted to Physics Open (2026). doi.org/10.5281/

zenodo.20323298

[2] R. Kulkarni, The Mass–Energy–Information Equivalence: A bottom-up identiﬁcation of the parti-

cle spectrum via FCC lattice error correction, Phys. Open 27 (2026) 100414. doi.org/10.1016/j.

physo.2026.100414

[3] Y. Koide, New view of quark and lepton mass hierarchy, Phys. Rev. D 28 (1983) 252. doi.org/10.

1103/PhysRevD.28.252

[4] R. Landauer, Irreversibility and heat generation in the computing process, IBM J. Res. Dev. 5 (1961)

183. doi.org/10.1147/rd.53.0183

[5] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed., Springer (1999).

doi.org/10.1007/978-1-4757-6568-7

Koide's relation as a kinematic effect in a D4 lattice model of lepton mass