Dark Energy & Lightest Neutrino Mass: ρ

Dark energy and the lightest neutrino:

𝜌

= 64 𝑚

and a lattice reading of its factor

Raghu Kulkarni

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

raghu@idrive.com

Abstract

Among particle-physics scales, only the neutrino masses lie near the dark-energy scale of

2.24 meV. If the neutrino masses satisfy the Koide-type condition 𝐾

𝜈

= 1/2, the measured splittings

ﬁx the lightest mass at 0.794 ±0.013 meV in the normal ordering. The fourth root of the dark-energy

density is then 2

√

2 times this mass, to within 0.2𝜎. So the density is 64 times the fourth power of

the lightest mass. We ask whether a vacuum lattice can supply this factor. In the four-dimensional

lattice of the Selection–Stitch Model, a neutrino is a time-only worldline. It steps between the cells

of an orthogonal bond frame. We take the sixteen bonds inside each cell to carry the vacuum.

Their isotropy factor is 4. If the vacuum is impedance-matched to the neutrino over one cell, with a

Euclidean interval per step, 64 follows as the cube of 4. The matching is a postulate. The premises

exclude the other bond and clock choices. Given the Koide-type condition, the relation predicts the

solar splitting from the atmospheric splitting and the dark-energy density. It agrees with current

data, including the latest JUNO splittings. The test will tighten as JUNO and Hubble-constant data

improve. It requires the normal ordering, which JUNO now favors, and constant dark energy.

Keywords: dark energy; cosmological constant; neutrino masses; Koide relation; FCC lattice; 𝐷

lattice;

JUNO; Hubble tension

1 Introduction

Two energy scales of particle physics and cosmology lie near one milli-electronvolt. The ﬁrst is the

dark-energy density, 𝜌

1/4

≈ 2.24 meV in the standard cosmological model [1]. The second is the

neutrino mass scale. The solar and atmospheric splittings correspond to



Δ𝑚

≈ 8.7 meV and



Δ𝑚

≈ 50 meV [2, 4]. No other particle mass is nearby. This coincidence has led to models that tie

the two together. The best known are mass-varying neutrino models, in which a light scalar coupled to

neutrinos supplies the dark energy [8]. The simplest such models are unstable to neutrino clumping [9],

and none has observational support. The cosmological data also link the two questions. The DESI bound

on the neutrino mass sum depends on the dark-energy model. It is

𝑚

𝜈

< 64.2 meV for a cosmological

constant and < 163 meV for evolving dark energy [10].

This paper reports a numerical relation between the two scales. It asks how much of the relation

a speciﬁc vacuum model can explain. It also states how to test the relation. The relation needs the

individual neutrino masses, which oscillations do not give. We take them from the conjecture 𝐾

𝜈

= 1/2

for the neutrino analog of Koide’s charged-lepton quantity [12]. Section 2 gives the masses. Section 3

states the relation. Section 4 gives a reading of its factor in the close-packed vacuum lattice of the

Selection–Stitch Model (SSM) [16, 17]. It summarizes the model, lists the premises, and shows that the

premises select one count among the alternatives. Section 5 gives the consequences for measurements.

The factor 4 in 64 is a postulate of the lattice model. We apply it at a scale the model does not

otherwise have. For the count per four-cell, we adopt an action prescription for a time-only worldline. It

assigns an action per unit four-volume, in Euclidean form, instead of an energy per unit spatial volume.

That the vacuum’s action equals the neutrino’s action in each cell is a matching postulate. Two further

choices are stated and not derived. The vacuum’s channel lies on the corner bonds of the neutrino’s cell.

The action interval is Euclidean (Section 4.6). Among dimensions with structured spatial defects, the

factor 4 is the same for the vacuum lattice, for spatial defects and for the time-only worldline only in

three space dimensions. The same matching places the proton’s cell at the strong-interaction scale if its

four channels are saturated (Section 4.8). The premises select the count 64 from every alternative bond,

cell and clock choice considered (Section 4.6). For today’s data, the implied solar splitting agrees with

its measured value. This agreement restates the relation. The relation will become a decisive test as the

inputs improve.

2 Neutrino masses for 𝐾

𝜈

= 1/2

Koide’s quantity for three masses is

𝐾 =

𝑚

+ 𝑚



√

𝑚

√

𝑚

√

𝑚



. (1)

For the charged leptons it equals 2/3 to about 10

−5

[12, 13]. Every triple can be written as [14, 15]

√

𝑚 = 𝑎

1 + 𝑏 cos



𝛿 +

2𝜋𝑘

i

, 𝑘 = 0, 1, 2, (2)

with 𝐾 = (1 + 𝑏

/2)/3. The charged leptons have 𝑏 =

√

2. We take 𝐾

𝜈

= 1/2, that is 𝑏 = 1, as a

conjecture (premise C of Section 4.2). It lies inside the range that Li and Ma estimated for neutrinos,

0.50 < 𝑘

𝜈

< 0.85 in their normalization 𝑘 = 3𝐾/2 [11]. It is also singled out by geometry. For 𝑏 ≥ 1

the amplitude circle reaches zero. Only for 𝑏 = 1 is zero its minimum. The circle then touches zero

without crossing it, and every amplitude is non-negative for every phase. Label the masses in increasing

order. The lightest state then lies a phase distance 𝜀 = 𝜋/3 − 𝛿

𝜈

from the zero, and

𝑚

= 𝑎

(1 − cos 𝜀)

. (3)

The two oscillation splittings ﬁx 𝑎 and 𝛿

𝜈

. We use Δ𝑚

= (7.530 ± 0.097) × 10

−5

from the

global ﬁt that includes the ﬁrst JUNO data [5, 4]. For Δ𝑚

we use 2.521 ×10

−3

, the NuFIT 6.1 value

without the Super-Kamiokande atmospheric data quoted there [5, 3]. Its uncertainty of 0.020 ×10

−3

is our estimate. The normal ordering gives 𝛿

𝜈

= 0.3327, 𝑎

= 13.27 meV, 𝜀 = 0.714, and

𝑚

= 0.794 ± 0.013, 𝑚

= 8.714, 𝑚

= 50.22 meV,



𝑚

𝜈

= 59.7 meV. (4)

The uncertainty on 𝑚

includes both splittings (script nde_numbers.py). In the inverted ordering, the

same condition forces the lightest mass to 𝑚

≈ 10

−8

meV. This matters in Section 5. The sum lies below

the DESI bound for a cosmological constant, 64.2 meV [10].

3 The relation

The dark-energy density follows from the Planck 2018 parameters of the ﬂat ΛCDM model, ℎ =

0.6736 ±0.0054 and Ω

𝑚

ℎ

= 0.1430 ±0.0011 [1]. We use 𝜌

= 3(ℎ

−Ω

𝑚

ℎ

)(100 km s

−1

Mpc

−1

)

𝑀

with the reduced Planck mass. This gives 𝜌

1/4

= 2.239 ± 0.013 meV. Here we treat the two parameters

as independent. Their correlation in the Planck posterior would change this uncertainty somewhat. It

does not aﬀect what follows. The ratio to the lightest neutrino mass is

𝜌

1/4

𝑚

= 2.821 ± 0.048, (5)

within 0.2𝜎 of 2

√

2 = 2 .828. The relation we examine is

𝜌

= 64

(𝑚

𝑐

)

(ℏ𝑐)

, equivalently 𝜌

1/4

= 2

√

2 𝑎

(1 − cos 𝜀)

. (6)

In the other direction, the oscillation data and 𝐾

𝜈

= 1/2 give 2

√

2 𝑚

= 2.245 ± 0.036 meV. The Planck

value is 2.239 ± 0.013 meV.

We use 𝑚

rather than another combination of the masses because of the zero of the 𝑏 = 1 circle.

The lightest state is the one closest to zero mass. This choice is not forced. Section 6 returns to it.

4 A lattice reading of the factor 64

4.1 The Selection–Stitch Model in brief

The Selection–Stitch Model treats the vacuum as a quantum error-correcting code on a lattice. Space is

a face-centered-cubic (FCC) lattice, the densest packing of spheres in three dimensions. Each site has

twelve nearest neighbors (Fig. 1a). Qubits sit on the bonds, and stabilizer checks on the sites and on the

voids between them detect errors. A particle is a topological defect of this code. It is a pattern that the

checks detect and that the code cannot remove. Its rest mass is the cost of verifying the defect, counted

in bits. This is the mass–energy–information equivalence [18]. Counted this way, the mass ratios of

the electron, muon, pion, proton and neutron come out as small integers, and they match experiment to

within 0.12% [18]. The lattice spacing is near the Planck length. The model’s black-hole sector ﬁxes

it at 1.843 ℓ

𝑃

[19]. A kinematic postulate ﬁxes how fast the lattice transmits quanta [16]. Section 4.3

states it.

Spacetime is the four-dimensional lattice 𝐷

, the densest lattice packing in four dimensions [17].

The FCC lattice is its slice at ﬁxed time. Each 𝐷

site has 24 nearest neighbors. Twelve are spatial. The

other twelve are time-mixed. Each of these is displaced by 𝐿/

√

2 in time and by 𝐿/

√

2 in space, so in a

Lorentzian reading the time-mixed bonds are null (Fig. 1c). The 24 neighbors split into three groups of

eight. Each group is the vertex set of a 16-cell and forms a frame of four mutually perpendicular bonds

(Fig. 1b). An order-three symmetry, triality, permutes the three groups. Ref. [17] uses the three groups

to label the three particle generations. This is its generation hypothesis. Particles are worldlines that

wrap the time direction. A charged lepton’s worldline also has spatial structure. A neutrino’s worldline

has none. It is a closed string of time-mixed bonds of one frame (Fig. 1c). It is a logical operator of the

code. It ﬁres no check, so the code barely sees it. Ref. [17] leaves the neutrino mass scale open.

4.2 Premises

The reading rests on the premises below. We collect them here. Two come from the model. One is the

action prescription we adopt for the model. One is the conjecture of Section 2. Four are hypotheses of

this paper.

I1 (postulate of the model). Each bond of the vacuum lattice transmits at most one quantum ℏ𝑐/(4𝐿)

per tick, with 4 = 𝐾/𝑑 [16] (Section 4.3).

(a) FCC slice: 12 bonds

(b)

: 24 bonds, three frames

cell

(c)

(

)

section: null bonds

neutrino string (frame A)

Figure 1: The lattices of the Selection–Stitch Model. (a) The twelve bonds of an FCC site, which form a

cuboctahedron. (b) The 24 bonds of a 𝐷

site, projected to three dimensions by dropping time. Circles

with solid lines are spatial bonds. Squares with dashed lines are time-mixed bonds. Pairs of them project

onto the same point. Colors mark the three orthogonal frames (16-cells). (c) A section of 𝐷

in the

(𝑡, 𝑥) plane. Gray lines are time-mixed bonds, which are null in a Lorentzian reading. The red path is

the frame-A neutrino string of Ref. [17]. Dotted squares are the frame-A cells centered on the sites of

the string, and one is shaded. The neutrino takes one step per cell (Section 4.6).

I2 (construction of Ref. [17]). The neutrino of ﬂavor 𝑘 is the logical string that wraps the time direction

in steps along the bonds of orthogonal frame 𝑘. It has no equal-time state (Sections 4.5 and 4.6).

A (action prescription). A time-only worldline is assigned an action per unit four-volume, in Euclidean

form, not an energy per unit spatial volume (Section 4.5).

C (conjecture). 𝐾

𝜈

= 1/2 in the normal ordering (Section 2).

𝜈

1. The bond rule I1 applies to an eﬀective cell of size 𝐿

𝜈

, in which one quantum equals 𝑚

𝑐

(Section 4.4).

𝜈

2. Inside the neutrino’s four-cell, the vacuum transmits along bonds that stay inside the cell and carry

the factor 4 of I1. These bonds set the bond stiﬀness (Section 4.6).

𝜈

3. The neutrino’s quantum is held for one step per four-cell, counted once per crossing and measured

by the step’s Euclidean length (Section 4.6).

𝜈

4. The vacuum is impedance-matched to the neutrino over the four-cell (Section 4.7).

The count per four-cell follows from I2 and A. Each failed alternative in Table 1 departs from A or from

one of H

𝜈

1–H

𝜈

3 (Section 4.6). So does each count with more than one quantum in Section 4.7.

4.3 The bond rule

In the Selection–Stitch Model, the vacuum is a face-centered-cubic lattice of bonds of length 𝐿. It is the

time slice of the 𝐷

lattice [17]. Its twelve bond directions ˆ𝑛 have the second moment

ˆ𝑛 ˆ𝑛

⊤

= (𝐾/𝑑) ⊮

with 𝐾/𝑑 = 12/3 = 4. Here 𝐾 is the coordination number, not Koide’s quantity of Section 2. This

is isotropy at second order only [17]. The kinematic postulate of the model [16] has two parts. Light

crosses 𝐾/𝑑 bond lengths per lattice tick, so 𝑐 = 4𝐿/𝜏. Each bond transmits at most one quantum per

tick:

𝐸

bond

ℏ

𝜏

ℏ𝑐

4𝐿

. (7)

4.4 An eﬀective cell at the neutrino scale

The model has one lattice, and its spacing is Planckian (Section 4.1). Our reading applies Eq. (7) to an

eﬀective cell set by the neutrino’s mass. This cell is the scale on which a particle’s quanta are counted.

It is a scale in the way a Compton wavelength is a scale. It is not an object or a second lattice. In this

cell the bond quantum equals the rest energy of the lightest neutrino:

𝐿

𝜈

ℏ𝑐

4 𝑚

𝑐

= 62.1 𝜇m. (8)

Nothing in the published model supplies this scale. It is hypothesis H

𝜈

1. In this cell one light-tick,

𝑐𝜏 = 4𝐿

𝜈

, equals the reduced Compton wavelength of the lightest neutrino.

The same construction has a counterpart at the nuclear scale. There the eﬀective cell is a measured

size. With sixteen quanta, 𝑚

𝑝

𝑐

= 16 ℏ𝑐/(4𝐿), the rule gives 𝐿 = 4ℏ/(𝑚

𝑝

𝑐) = 0.8412 fm. This is within

0.1% of the measured proton charge radius, 0.84075(64) fm [24]. Take the proton as a node trapped in a

tetrahedral void and bonded to the void’s four corners. Its two-step paths number 4(3 + 1) = 16. Twelve

run from a corner along an edge of the tetrahedron, and four return to the trapped node. In units of the

electron’s cost 𝜖

𝑏

= 𝑚

𝑒

𝑐

the rule reads 1836 𝜖

𝑏

= 16𝐸

bond

. A single bond, the graph of one step along

a worldline, gives one quantum. Section 4.8 tests this case. One quantum at 𝐿

𝜈

plays the role for the

neutrino that sixteen quanta at 0.84 fm would play for the proton. We test the choice of one quantum

below.

4.5 A worldline defect, and the one cube that needs time

In 𝐷

a neutrino is a time-only worldline. It is a logical operator that wraps the time direction and ﬁres

no syndrome (Section 4.1). It is not a defect of the spatial lattice. The absence of a syndrome does not by

itself forbid an energy per unit of spatial volume. A logical operator can still act on the energy eigenstates

of a Hamiltonian. A Euclidean formulation also serves ordinary propagating ﬁelds. So premise A is a

prescription that we adopt for the model. It does not follow from the missing syndrome.

The geometry of 𝐷

ﬁxes the four-cell. No three of the twelve spatial bond vectors are mutually

perpendicular. So no cube of space has bond-length edges. The 24 bond vectors of 𝐷

fall into the three

frames of Section 4.1. Each frame has four mutually perpendicular vectors. Each frame pairs the time

axis with one spatial axis. One example is

(1, 1, 0, 0), (1, −1, 0, 0), (0, 0, 1, 1), (0, 0, 1, −1) (9)

in coordinates (𝑡, 𝑥, 𝑦, 𝑧). The other two frames pair 𝑡 with 𝑦 and with 𝑧. So every cube of edge 𝐿 in 𝐷

has an edge that leans into time (scripts frame_cube.py, lorentz_cell.py).

Each frame spans a four-cell of edge 𝐿 and four-volume 𝐿

. The cell contains two 𝐷

sites. Its two

time-bearing edges are null in Lorentzian signature. Build a three-dimensional cube on one of them. Its

three-volume is 𝐿

in Euclidean signature, 𝐿

√

2 when projected onto an equal-time slice, and zero in

Lorentzian signature. The three-dimensional count is therefore ambiguous. The four-volume of the cell

is 𝐿

in either signature. The vacuum side has the same property. An energy density with 𝑤 = −1 has a

stress tensor proportional to the metric, so it is the same for every observer. An action per four-volume

needs no choice of time slicing. So we count both sides of the relation per orthogonal-frame four-cell.

This is consistent with premise A. The duration of the neutrino’s quantum within the cell is not ﬁxed.

The model’s two clocks disagree on this duration (Section 4.6).

4.6 The neutrino’s hypercube, and the clock

Two quantities enter the count. The ﬁrst is the isotropy factor 𝛼 of the bonds that set the tick 𝛼𝐿/𝑐

through the bond rule. The second is the interval 𝛽𝐿/𝑐 over which the neutrino’s quantum is held in one

four-cell. Then 𝑚

𝑐

= ℏ𝑐/(𝛼𝐿). The neutrino’s action in the cell is 𝑚

𝑐

𝛽𝐿/𝑐. The vacuum’s action

is 𝜌

𝐿

/𝑐. If the two actions are equal, as the matching condition H

𝜈

4 requires (Section 4.7), then

𝜌

= 𝛽 𝛼

(𝑚

𝑐

)

(ℏ𝑐)

. (10)

The neutrino’s path. Ref. [17] constructs the three neutrino operators explicitly. The ﬂavor tied to

the 16-cell of frame 𝑘 is a closed string that wraps the time direction. It steps along frame 𝑘’s own

time-bearing bonds. For frame (9) these are (1, 1, 0, 0) and (1, −1, 0, 0) in alternation. Tile the lattice

with frame-𝑘 tesseracts of edge 𝐿 centered on the lattice sites. A step along a frame bond, ±𝑓

𝑖

, runs

from one tesseract’s center to the center of the next. The neutrino therefore takes one step of length 𝐿

per four-cell.

The vacuum channel inside the cell. The site at a tesseract’s center has 24 bonds. Eight are the frame

bonds ±𝑓

𝑖

that the neutrino’s path uses. They leave the cell through its faces. The other sixteen end on

the tesseract’s corners, (±𝑓

± 𝑓

)/2. All sixteen have length 𝐿. They stay inside the cell. They

are the bonds of the other two frames. We take them as the channel through which the vacuum inside the

neutrino’s cell transmits its quanta. They set the bond stiﬀness 𝐽 used in Section 4.7. Their directions

have an isotropic second moment,



corners

ˆ𝑛 ˆ𝑛

⊤

⊮ = 4 ⊮. (11)

So the bond rule inside the cell has 𝛼 = 𝐾/𝑑 = 16/4 = 4. In 𝑛 dimensions the corner directions of a

hypercube give 𝐾/𝑑 = 2

𝑛

/𝑛. This equals 4 only for 𝑛 = 4. The value matches that of the FCC slice,

12/3 (Section 4.8). So the bond quantum ℏ𝑐/(4𝐿) of Eq. (7) is the same in both descriptions, and 𝐿

𝜈

in Eq. (8) does not change. In this reading the neutrino shares no bond with the vacuum channel it is

matched to. It moves along its own frame. The vacuum of its cell transmits along the other two frames.

Hypothesis H

𝜈

2 asks for a channel that stays inside the cell and carries the factor 4. The corner bonds

are the only such set (see the selection below).

The interval. Under premise A, the neutrino’s quantum is assigned a Euclidean action instead of an

instantaneous energy. We take the Euclidean length of a step as the interval, so 𝛽 = 1. This is hypothesis

𝜈

3. The vacuum side is Euclidean in the same sense. The vacuum energy of a quantum ﬁeld is its

Euclidean eﬀective action per unit four-volume, 𝜌

vac

= −lim

𝑉

→∞

(ℏ𝑐/𝑉

) ln 𝑍

𝐸

[26]. The Euclidean

interval diﬀers from the Lorentzian time over which a neutrino is seen to propagate. The neutrino’s

Lorentzian dispersion still governs that time. With 𝛼 = 4 and 𝛽 = 1, Eq. (10) gives 64, the coeﬃcient of

Eq. (6).

Operator check. We used the torus of side 4 of Ref. [17], with 1536 edge qubits. The frame-𝑘

string commutes with every check. It is a logical operator. It is inequivalent to the strings of the other

two frames, as Ref. [17] reports. The label 𝑘 is preserved because the string uses only frame-𝑘 bonds.

The same check shows why the corner bonds cannot be the neutrino’s path. A closed string of corner

steps from one other frame is that frame’s neutrino, which is a diﬀerent ﬂavor. A string that mixes the

two other frames ﬁres four or eight checks and is not a logical operator (script nu_operator.py). A

stabilizer calculation cannot address two further properties of the neutrino operator. One is the amplitude

of a single step, which needs a Hamiltonian for the time channel. The other is the Lorentzian dispersion

𝐸

= 𝑝

𝑐

+ 𝑚

𝑐

, which must be recovered after continuation from the Euclidean description. Both

remain open.

Selection of the count. Three rules, each a premise stated in Section 4.2, select one count from the

alternatives. The rules make no use of the data.

(i) Premise A assigns the time-only worldline a Euclidean action per unit four-volume. This excludes

six choices that assign the neutrino an energy per spatial cell: the octahedron, the tetrahedron

or a single bond, the conventional FCC cube, the ⟨100⟩ cubic cell, the ⟨100⟩ network with its

own tick, and the 𝐷

null spacing with its own tick (scripts derive64.py, cubes.py). It also

excludes every Lorentzian or equal-time interval: the cell’s time extent (

√

2 𝐿/𝑐), its mean extent

(𝐿/(

√

2𝑐)), and the null-step tick (2

√

2 𝐿/𝑐).

(ii) Hypothesis H

𝜈

2 requires a vacuum channel that stays inside the cell and carries the factor 4 of

I1. This excludes the frame’s own eight bonds, which leave the cell and give 𝐾/𝑑 = 2, and all 24

bonds, which include those eight and give 𝐾/𝑑 = 6. Only the sixteen corner bonds remain.

(iii) Hypothesis H

𝜈

3 counts the quantum once per crossing. The time-axis worldline crosses one cell

per step, so charging a whole tick to each cell counts each crossing more than once. This excludes

the FCC tick 4𝐿/𝑐 and a whole null-step tick per cell.

What remains is the corner channel, 𝛼 = 4, with one Euclidean step, 𝛽 = 1. Eq. (10) then gives 64.

Table 1 lists the bond and clock choices, the rule that excludes each, and its pull against the measured

coeﬃcient (scripts hypercube.py, clocks.py). Every excluded choice but one also disagrees with the

data, by 4𝜎 or more. The exception is the whole null-step tick, which rule (iii) excludes. Fig. 3b shows

all ﬁfteen choices.

Table 1: Bond classes, ticks and intervals in Eq. (10), the rule of Section 4.6 that excludes each, and the

pull against the measured coeﬃcient 63.3 ± 4.3.

Bonds or steps setting 𝛼 Interval in one four-cell Excluded by Coeﬃcient Pull

hypercube corners, 16/4 one Euclidean step, 𝐿/𝑐 — 64 +0.2𝜎

frame bonds, 8/4 one Euclidean step (ii) 8 −12.8𝜎

all 𝐷

bonds, 24/4 one Euclidean step (ii) 216 +35𝜎

FCC slice, 12/3 cell time extent,

√

2𝐿/𝑐 (i) 90.5 +6.3𝜎

FCC slice, 12/3 cell mean extent, 𝐿/(

√

2𝑐) (i) 45.3 −4.2𝜎

FCC slice, 12/3 one tick, 4𝐿/𝑐 (iii) 256 +45𝜎

null steps, 2

√

2 cell time extent (i) 32 −7.3𝜎

null steps, 2

√

2 cell mean extent (i) 16 −11.0𝜎

null steps, 2

√

2 one tick, per cell (i), (iii) 64 +0.2𝜎

4.7 The count as an impedance-matching condition

On a lattice, a compact 𝑈(1) ﬁeld has a charging energy 𝑈 and a bond stiﬀness 𝐽. In a common

normalization its coupling is



𝑈/𝐽/4𝜋. Its wave impedance is proportional to



𝑈/𝐽 [25]. Two regions

of lattice with diﬀerent spacings join without reﬂection when their ratios 𝑈/𝐽 agree. A mismatched

boundary reﬂects part of what crosses it and stores it. We apply the same condition to the channel that a

time-only worldline uses. This channel does not propagate in space, so the speed of its waves does not

ﬁx its impedance. We compare the two stiﬀnesses as actions over the same four-cell.

Neutrino side. The lightest charged defect sets the charging energy of the photon channel. In the

same way, the lightest time-only worldline sets the charging energy of the time channel, 𝑈 = 𝑚

𝑐

. The

bond stiﬀness is the bond quantum, 𝐽 = 𝐸

bond

. Write 𝑚

𝑐

= 𝑁𝐸

bond

. With one quantum per step of

the worldline, 𝑁 = 1. The neutrino’s impedance is then 𝑍

𝜈

= 𝑈/𝐽 = 1. Its action per four-cell is one

quantum per bond-length edge, 𝐸

bond

𝐿/𝑐 = ℏ/(𝐾/𝑑) = ℏ/4.

Vacuum side. The vacuum’s only energy in the cell is 𝜌

𝐿

. Its action there is 𝜌

𝐿

/𝑐. With the

same bond stiﬀness, 𝑍

vac

= (𝜌

𝐿

/𝑐)/(ℏ/4).

Matching. Setting 𝑍

𝜈

= 𝑍

vac

gives

𝜌

𝑐

𝐿

𝜈

ℏ

𝑚

𝑐

𝐿

𝜈



𝐾

𝑑



(𝑚

𝑐

)

(ℏ𝑐)

= 64

(𝑚

𝑐

)

(ℏ𝑐)

. (12)

This is Eq. (6), and Eq. (10) with 𝛼 = 4 and 𝛽 = 1. The factor 4 is the lattice’s light-speed factor. The

exponent 3 comes from the four-cell’s three remaining edges. The cell size is thus determined twice. The

neutrino mass gives 62.1 𝜇m through Eq. (8). The dark-energy density gives 𝐿 = (ℏ𝑐/4𝜌

)

1/4

= 62.3 𝜇m

through the matching alone. The relation is the condition that the two agree. This restates Eq. (5). It is

not new evidence. For 𝑁 quanta the matching gives 𝜌

= 64 𝑚

/𝑁

. A closed cycle (𝑁 = 4) gives 1,

and the proton’s sixteen give 1/64. Both are excluded (script nu_impedance.py).

Status of the matching condition. This subsection has three limits. First, the impedance of Ref. [25]

belongs to a 𝑈(1) gauge ﬁeld whose waves propagate in space. Nobody has shown that the channel of a

time-only worldline is a ﬁeld of that kind. Here its impedance is deﬁned by analogy, as a charging energy

over a bond energy in the same four-cell. Second, the choice of one quantum (H

𝜈

1, 𝑁 = 1) ﬁxes 𝑍

𝜈

= 1.

So the matching condition 𝑍

vac

= 𝑍

𝜈

is algebraically identical to Eq. (12). It gives the count a physical

name and a reason to prefer one quantum over four or sixteen. It adds no constraint beyond the relation.

Third, the condition is postulated, not derived. It is hypothesis H

𝜈

4. A derivation would need three

steps. It would construct the operator of the time channel in the code. It would compute that channel’s

charging and bond stiﬀnesses in the vacuum. It would show that a mismatched boundary stores quanta

that the code cannot export, so that a mismatch costs rest energy. This paper does none of these.

4.8 A consistency rule across three geometries

An eﬀective cell must be a cell of the spacetime lattice itself. For a spatial defect it is a cell of the

FCC slice. For a time-only worldline it is a cell of 𝐷

that needs time. The bond quantum of Eq. (7)

then enters through three geometries. The vacuum lattice sets it. The twelve bonds of an FCC site give

𝐾/𝑑 = 12/3 (premise I1). The two eﬀective cells must reproduce it, each from inside a single cell. The

time-only worldline uses the corner channel of its hypercube, with 2

𝑛

/𝑛 = 16/4 (Eq. (11)). Those bonds

stay inside the cell. A spatial defect uses the edges of its own simplex. The proton’s cell is a tetrahedral

void (Section 4.4). Its twelve internal edge paths are the void’s six edges, each taken in both directions.

They point along the six bond directions of the FCC slice. Their second moment is



edges

ˆ𝑛 ˆ𝑛

⊤

⊮ = 4 ⊮, (13)

the value of the FCC slice. This equality is an identity. The twelve edge vectors of the tetrahedron are the

twelve bond vectors of an FCC site. The FCC lattice is the root lattice 𝐴

, whose shortest vectors are the

edges of a tetrahedron. No neighboring cell is needed. Two counts must be kept apart here. All sixteen

paths carry a quantum, so all sixteen enter the mass, 16𝐸

bond

= 𝑚

𝑝

𝑐

. Only the twelve edge paths enter

the stiﬀness. The reason is that only their second steps run along lattice bonds, the tetrahedron’s edges.

Every path starts on a spoke, the defect’s own bond from the trapped node to a corner. The four return

paths run out along a spoke and back. We take the spokes to obey the capacity of the bond rule I1, one

quantum per tick. They are not bonds of the FCC lattice, so they do not enter its isotropy factor. A spoke



3/8 = 0.61 of a bond length. The spokes alone would give 4/3. Edges and spokes together would

give 16/3. So a free proton and a single neutrino cell each carry the factor 4 internally (Fig. 2).

Both cells couple to the vacuum. If both are impedance-matched to it (H

𝜈

4, extended to spatial

defects), all three geometries must carry the same bond quantum ℏ𝑐/(4𝐿). Otherwise a boundary

between the vacuum, a spatial channel and the time channel would reﬂect. We state this as a consistency

rule. In 𝐷 space dimensions the densest lattice gives 𝐾

𝐷

/𝐷, with 𝐾

𝐷

its kissing number. The simplex

has (𝐷 +1)𝐷 directed edges, which give 𝐷 +1. The hypercube of 𝐷 +1 dimensions gives 2

𝐷+1

/(𝐷 +1).

The rule requires

𝐾

𝐷

= 𝐷 + 1 =

𝐷+1

𝐷 + 1

. (14)

The ﬁrst equality holds for 𝐷 = 1, 2 and 3, where the densest lattice is the root lattice 𝐴

𝐷

with

𝐾

𝐷

= 𝐷 (𝐷 + 1). It fails at 𝐷 = 4, where the densest lattice is 𝐷

with 24 > 20. It also fails in every

trapped

node

(a) proton: tetrahedral void

12 edge paths (bonds), 12/3 = 4; 4 spokes (not lattice bonds)

(b) neutrino: frame tesseract (projected)

16 corner bonds inside the cell, 16/4 = 4

Figure 2: The two eﬀective cells. (a) The proton’s cell, a tetrahedral void of the FCC slice with its

trapped node. The six edges are lattice bonds. Taken in both directions they give the twelve edge paths

of Eq. (13), with second moment (12/3) ⊮. The four spokes (dashed) are the defect’s own bonds, not

lattice bonds. They carry quanta for the mass but do not enter the stiﬀness. (b) The neutrino’s cell, the

frame tesseract centered on a 𝐷

site, in a two-dimensional projection. The sixteen corner bonds (dark)

stay inside the cell, with second moment (16/4) ⊮ (Eq. (11)); the projection keeps all sixteen distinct.

Thin lines are the tesseract’s 32 edges. The eight frame bonds, which leave through its faces, are not

drawn. Both panels are conceptual projections, not bond diagrams to scale.

higher dimension where the densest lattice is known. The second equality is (𝐷 +1)

= 2

𝐷+1

. On its own

it restricts agreement to 𝐷 = 1 and 3. The equation 𝑛

= 2

𝑛

has exactly two positive integer solutions,

𝑛 = 2 and 𝑛 = 4, since 2

𝑛

> 𝑛

for every 𝑛 ≥ 5. So all three geometries agree for 𝐷 = 1 and 𝐷 = 3 only

(Table 2; script nu_consistency.py). A line has no room for a spatial defect with structure. So among

the dimensions that have such defects, only three space dimensions let the vacuum, a spatial defect and

a time-only worldline share one bond quantum.

Table 2: The three sides of Eq. (14): the vacuum lattice, the simplex of a spatial defect, and the hypercube

of a time-only cell. The kissing numbers are exact for the dimensions shown.

𝐷 vacuum, 𝐾

𝐷

/𝐷 simplex, 𝐷 + 1 hypercube, 2

𝐷+1

/(𝐷 + 1) all equal

1 2 2 2 yes

2 3 3 2.67 no

3 4 4 4 yes

4 6 5 6.4 no

8 30 9 56.9 no

This rule is a consistency condition. It is not a derivation. It rests on two geometric premises, the

simplex for a spatial defect and the hypercube for the time-only cell. A cube is not an alternative for

a spatial defect. The FCC slice has no three mutually perpendicular bonds, so no cube of bonds exists

in it. A cube also has no count that gives 4. Counted by edge direction it gives 𝐾/𝑑 = 2, and counted

by directed edges, as the simplex is, it gives 8. These place sixteen quanta at 1.68 fm or 0.42 fm. The

measured value is 0.84 fm. Under either count a cube satisﬁes Eq. (14) only for 𝐷 = 1. The rule does

not show that matching is enforced.

The proton’s cell and the strong-interaction vacuum. The matching of Section 4.7 can also be applied

to the proton’s cell. The question is how many of the proton’s sixteen quanta the cell holds at one instant.

For the neutrino 𝑁 = 1, and the question does not arise. If all sixteen are present at once, 𝜌

QCD

𝐿

𝑐

= 𝑚

𝑝

𝑐

and 𝜌

1/4

QCD

= 𝑚

𝑝

𝑐

/(2

√

2) = 332 MeV. If only one is present, the result is 166 MeV. The proton model

and the bond rule decide between these. The sixteen paths all begin on the trapped node’s four spokes,

four paths on each. Each spoke carries at most one quantum per tick. The capacity follows from the

bond’s phase current. For a compact 𝑈 (1) link of stiﬀness 𝐽, the current is 𝐼

𝑏

= −(𝐽/ℏ) sin 𝐵

𝑏

[25], so

the transfer in one tick is at most 𝐽𝜏/ℏ. With 𝐽 = 𝐸

bond

= ℏ/𝜏, this is one quantum. So at most four

quanta are in transit at any instant, and the sixteen take at least four ticks. We further assume that the four

channels are saturated. Each carries one quantum in each of four consecutive ticks. The path count and

the capacity come from the proton model. Saturation does not. It is an assumed dynamical condition.

This ordering is a statement about Lorentzian time. The quanta of a spatial defect follow one another

causally. The neutrino’s single quantum, held for one Euclidean step, has no such ordering. With the

channels saturated, four quanta are in the cell, 𝜌

QCD

𝐿

𝑐

= 𝑚

𝑝

𝑐

/4, and

𝜌

1/4

QCD

𝑚

𝑝

𝑐

= 235 MeV. (15)

The energy of the strong-interaction vacuum is model dependent. The MIT bag constant gives 𝐵

1/4

145–235 MeV [27]. The gluon condensate, ⟨(𝛼

𝑠

/𝜋)𝐺

⟩ = 0.012–0.02 GeV

[28], gives |𝜀

vac

1/4

= 241–

274 MeV for three ﬂavors. Eq. (15) lies at the top of the bag range. It lies just below the condensate

range, within 3% of the standard condensate value. Of the other two readings, 332 MeV lies above the

span and 166 MeV inside the bag range. The estimates themselves spread by nearly a factor of two, so

this check is weak. Still, apply the bond rule to the proton’s own cell, with its quanta ordered in time and

its four channels saturated. The matching scale then falls at the scale of the strong interaction.

Matching and pressure balance. The bag model ﬁxes a hadron’s size by pressure balance [27].

Minimizing 𝑎ℏ𝑐/𝐿 +𝜌𝐿

over 𝐿 leaves a vacuum energy in the cell equal to one third of the conﬁnement

energy. For the neutrino this gives the coeﬃcient 64/3 and 𝜌

1/4

= 1.71 meV, a pull of −9.7𝜎. Within this

reading the data select equal energies in the cell. That is the matching condition rather than mechanical

equilibrium.

4.9 What remains open

Several points remain open. The postulates do not ﬁx the vacuum’s bond stiﬀness in the time channel,

or its assignment to the corner bonds of the neutrino’s cell (H

𝜈

2). The Euclidean interval of one step per

four-cell is assumed (H

𝜈

3). The matching condition is postulated, not derived (H

𝜈

4). The step amplitude

and the Lorentzian dispersion of the neutrino operator are not computed (Section 4.6). The proton check

of Section 4.8 adds three assumptions of its own: an eﬀective cell is a cell of the lattice, the matching

extends to spatial defects, and the four channels of the proton’s cell are saturated. The scale itself (H

𝜈

and the action prescription A are assumed throughout.

A possible origin of the matching condition is entropic. Suppose each bond between a particle’s cell

and the vacuum is in its most probable state. Its energy is then shared equally between its two ends. This

gives equal energies on the two sides of the cell boundary, which is H

𝜈

4. Minimizing the energy instead

gives the pressure balance excluded above. For the proton’s cell the bonds to the vacuum are the four

spokes. The same rule would then tie the number of quanta in ﬂight to these four bonds. We have not

derived this from lattice dynamics.

Premise Content Status

I1 𝐾/𝑑 = 4, 𝐸

bond

= ℏ𝑐/(4𝐿) postulate of the model [16]

I2 neutrino as a frame-𝑘 time string constructed in Ref. [17]; veriﬁed here

A Euclidean action per four-volume adopted prescription; not implied by the syndrome

C 𝐾

𝜈

= 1/2, hence 𝑚

= 0.794 meV conjecture; inside the range of Ref. [11]

𝜈

1 bond rule in an eﬀective cell, 𝐿

𝜈

= 62 𝜇m hypothesis; the lattice itself is Planckian

𝜈

2 internal channel with the factor 4 of I1 hypothesis; met only by the corner bonds

𝜈

3 one Euclidean step per crossing hypothesis; with A, selects 𝛽 = 1

𝜈

4 impedance matching hypothesis; equivalent to the relation

Rule Eq. (14): vacuum, space and time agree consistency condition; holds for 𝐷 = 1, 3 only

5 What the relation implies

Given 𝐾

𝜈

= 1/2, any two of Δ𝑚

, Δ𝑚

and 𝜌

ﬁx the two neutrino parameters 𝑎 and 𝛿

𝜈

of Eq. (2).

Eq. (6) then links the three. We solve it for the solar splitting, using Δ𝑚

and the Planck 𝜌

Δ𝑚

= (7.55 ± 0.11) × 10

−5

. (16)

The measured value is (7.530 ± 0.097) × 10

−5

[5, 4]. The diﬀerence is +0.2 𝜎 (script

nde_numbers.py). The masses become 0.792, 8.726 and 50.22 meV. This agreement is alge-

braically the same statement as the ratio in Eq. (5). For today’s data it is not independent evidence. The

coeﬃcient is ﬁxed at 64 and has no free parameter. So future data test it without adjustment. A ﬁxed

coeﬃcient that agreed only by accident would fall within 1𝜎 with probability 1.5% at present precision.

With JUNO’s six-year data and the Planck Hubble constant, the ratio’s uncertainty falls to about 0.7%.

That probability then falls to 0.6% (script lookelsewhere_model.py). Most of the uncertainty in

Eq. (16) comes from Δ𝑚

(0.093) and from ℎ (0.049). The prediction moves by 0.047 × 10

−5

for each 0.01 × 10

−3

change in Δ𝑚

. The NuFIT 6.1 value that includes the atmospheric data,

2.511 × 10

−3

[5], gives 7.51.

The JUNO data from 207 days give Δ𝑚

= (7.388 ± 0.078) × 10

−5

. For the normal ordering

they give Δ𝑚

= (2.509

+0.027

−0.025

) × 10

−3

[6]. With this Δ𝑚

the relation implies Δ𝑚

= (7.50 ±

0.13) × 10

−5

. That is +0.7𝜎 from the JUNO value. With our input Δ𝑚

the diﬀerence is +1.2𝜎.

Taken alone, the JUNO splittings give 𝑚

= 0.802 ± 0.014 meV and 𝜌

1/4

/𝑚

= 2.794 ± 0.051, which is

−0.7𝜎 from 2

√

2. The same data favor the normal ordering at 2.1𝜎 or more [6].

The constraint becomes a test as its inputs improve. It has four consequences.

It depends on the Hubble constant. At ﬁxed Ω

𝑚

ℎ

, 𝜌

grows with ℎ, and the implied splitting

falls (Fig. 3a). It is 7.55 for the Planck ℎ and 7.32 for ℎ = 0.70. For the distance-ladder value

ℎ = 0.730 ±0.010 [20] it is 7.08±0.12, which is 3.0𝜎 below the measurement. In the other direction, the

value 𝜌

1/4

= 2.245±0.036 meV of Section 3, with the Planck Ω

𝑚

ℎ

, gives 𝐻

= 67.6±1.5 km s

−1

Mpc

−1

This agrees with the Planck value and lies 3.0𝜎 below the distance-ladder value. With the JUNO

splittings, the relation reproduces the JUNO Δ𝑚

at ℎ = 0.686. This result depends on the relation. It

is not independent evidence. If the relation holds, a high 𝐻

is disfavored. The distance-ladder case also

pairs ℎ with the Planck Ω

𝑚

ℎ

, which is not a self-consistent cosmology.

It needs better inputs. After six years, JUNO projects 0.3% on Δ𝑚

and 0.2% on Δ𝑚

[7]. The

prediction’s present 1.4% uncertainty would then be limited by the Planck ℎ, about 0.7%, and by the

dark-energy model. A sub-percent test needs both JUNO’s Δ𝑚

and a resolved Hubble constant.

It excludes the inverted ordering. With 𝐾

𝜈

= 1/2, the inverted ordering has a lightest mass near

−8

meV (Section 2). Eq. (6) would then put 𝜌

1/4

about eight orders of magnitude below its value. A

conﬁrmed inverted ordering rules the relation out.

65 70 75

(km s

−

Mpc

−

)

7.0

7.2

7.4

7.6

7.8

(10

−

)

Planck SH0ES

(a) the solar splitting predicted from

predicted,

= 64

measured (global

ﬁ

t with JUNO)

JUNO 6-year precision (illustrative)

0.8 1 1.5 2 3 4

1/4

implied (meV)

corner bonds, Euclidean step (Eq. 12)

frame bonds, Euclidean step [ii]

all 24 D4 bonds, Euclidean step [ii]

FCC tick, cell time extent [i]

FCC tick, cell mean extent [i]

FCC tick, one tick [iii]

null steps, cell time extent [i]

null steps, cell mean extent [i]

null steps, one tick per cell [i, iii]

octahedron volume [i]

tetrahedron volume / bond [i]

conventional FCC cube [i]

⟨

100

⟩

cell, FCC tick [i]

⟨

100

⟩

network, own tick [i]

D4 null spacing, own tick [i]

(b) each choice, with its excluding rule

Figure 3: (a) The solar splitting implied by Eq. (6) with 𝐾

𝜈

= 1/2 and the measured Δ𝑚

, as a function

of the Hubble constant that sets 𝜌

at ﬁxed Ω

𝑚

ℎ

. The blue band is the 1𝜎 range from Δ𝑚

; the gray

bands are the global-ﬁt measurement and, for illustration, the 0.3% precision JUNO projects for six

years [7]. (b) The value of 𝜌

1/4

implied by each bond, cell and clock choice of Section 4.6, with the

Planck value (gold band). The upper nine are the choices of Table 1; the selected count is in red. The

lower six, in gray, assign an energy per spatial cell and are excluded by rule (i).

It needs a constant dark energy. The DESI data mildly prefer evolving dark energy [10]. If dark

energy evolves, a ﬁxed relation to 𝑚

would require the neutrino mass to evolve with it, as in mass-varying

models [8]. A conﬁrmed evolution of the dark-energy density with a constant neutrino mass rules the

relation out.

Some quantities follow from 𝐾

𝜈

= 1/2 alone and do not change. They are

𝑚

𝜈

= 59.7 meV and the

eﬀective mass for beta decay, 𝑚

𝛽

= 8.9 meV. For Majorana neutrinos, 𝑚

𝛽𝛽

lies between 1.0 and 4 .3 meV

over all Majorana phases. These values use sin

𝜃

= 0.3096 and sin

𝜃

= 0.0220 [5, 3]. They lie far

below the present limits [21, 22]. The length 𝐿

𝜈

= 62 𝜇m of Eq. (8) is close to the separations at which

the inverse-square law has been tested, down to 52 𝜇m [23]. The relation predicts no deviation there.

Table 3 collects these consequences. Its ﬁrst four rows state one constraint in four ways. The rows

below the rule follow from 𝐾

𝜈

= 1/2 alone. They do not test the relation.

Three results would rule the relation out: a conﬁrmed inverted ordering, a conﬁrmed evolution of the

dark-energy density with a constant neutrino mass, or a resolved Hubble constant near the distance-ladder

value with the CMB matter density unchanged. Two other results would rule out 𝐾

𝜈

= 1/2, and with

it the masses the relation uses: a cosmological

𝑚

𝜈

well above 60 meV, or a double-beta signal above

about 5 meV.

6 Discussion

What would make it a derivation. Each open point of Section 4.9 would have to follow from the code’s

dynamics. If all of them follow from the model, every factor of 𝜌

= (𝐾/𝑑)

𝑚

comes from the lattice.

The relation then becomes a falsiﬁable Koide-type law. Ref. [17] proposes that the time-axis condensate

Table 3: What the relation and 𝐾

𝜈

= 1/2 imply, and what is measured. The ﬁrst four rows are equivalent;

the last three follow from 𝐾

𝜈

= 1/2 alone.

Quantity Implied Measured or limit Status

𝜌

1/4

from neutrino data 2.245 ±0.036 meV 2.239 ± 0.013 meV [1] +0.2𝜎

Δ𝑚

(10

−5

) 7.55 ±0.11 7.530 ±0.097 [5] +0.2𝜎

Δ𝑚

, JUNO Δ𝑚

7.50 ±0.13 7.388 ±0.078 [6] +0.7𝜎

𝐻

(km s

−1

Mpc

−1

) 67.6 ±1.5 67.4 ±0.5 [1]; 73.0 ± 1.0 [20] agrees; −3.0𝜎

𝑚

from 𝜌

alone 0.792 ±0.005 meV not measurable at present untested

mass ordering normal normal favored, ≥ 2.1𝜎 [6] consistent

dark energy constant, 𝑤 = −1 evolution mildly preferred [10] open

inverse-square law at 62 𝜇m no deviation none down to 52 𝜇m [23] consistent

𝑚

𝜈

59.7 meV < 64.2 meV [10] consistent

𝑚

𝛽

8.9 meV < 0.45 eV [21] below reach

𝑚

𝛽𝛽

(Majorana) 1.0–4.3 meV < 28–122 meV [22] below reach

sets the neutrino mass scale. It presents this as a program, not a result. A mechanism there that set both

𝑚

and a residual vacuum energy would be a natural place to look.

Why the lightest neutrino. The 𝑏 = 1 family is the only Koide family whose amplitude circle has

zero as its minimum. Its member closest to that zero is 𝑚

. In the matching reading of Section 4.7, the

relation matches the vacuum to its lowest excitation. This is an interpretation, not a derivation.

Relation to the cosmological constant problem. The relation does not explain why the vacuum

energy is not Planckian. It relates the observed value to a particle mass. In the mass–energy–information

equivalence, a particle’s mass is the veriﬁcation cost of a defect, and the defect-free ground state has

none [18]. A residual energy tied to the lowest excitation ﬁts this picture. Holographic arguments give the

same scale from 𝑀

𝐻

[29]. The relation here ties it to 𝑚

instead. With general relativity it then ﬁxes

the asymptotic de Sitter horizon. Its expansion rate is 𝐻



512 𝜋/3 𝑚

/𝑀

, about 56 km s

−1

Mpc

−1

Its entropy is 𝑆 = (3/512)(𝑀

/𝑚

)

≈ 3 ×10

122

in units of 𝑘

𝐵

. For this reason it constrains the neutrino

sector. The relation concerns the vacuum’s time channel, with one quantum per cell. It does not concern

a population of neutrinos. It holds in empty space, and it does not dilute as the universe expands. In

this it diﬀers from mass-varying neutrino models, in which the dark energy is carried by a ﬁeld coupled

to relic neutrinos [8]. The relation explains why the dark-energy scale lies near the neutrino masses. It

does not explain why the dark-energy density is comparable to the matter density today, which depends

on the matter content of the universe.

Limitations. The relation rests on the conjecture 𝐾

𝜈

= 1/2, on the normal ordering, and on a constant

dark energy. It also rests on the action prescription A and the hypotheses H

𝜈

1–H

𝜈

4 of Section 4. The

present agreement of Eq. (16) is not independent of it.

7 Conclusion

For the 𝐾

𝜈

= 1/2 neutrino spectrum, the dark-energy density and the lightest neutrino mass satisfy

𝜌

= 64 𝑚

to 0.2 𝜎. In the close-packed lattice of the Selection–Stitch Model, the factor can be read

as (𝐾/𝑑)

. This reading has three parts. The bond rule applies in an eﬀective cell at the neutrino scale.

This cell is not a second lattice. The neutrino is a time-only worldline in an orthogonal-frame four-cell

of 𝐷

. The vacuum is impedance-matched to it over that cell. The neutrino steps along its own frame’s

bonds, and this path is a veriﬁed logical operator of the model’s code. The factor 4 is the isotropy factor

of the sixteen corner bonds of its cell. These bonds are taken as the vacuum’s channel, and their factor

equals that of the FCC slice. That equality is a consistency rule across the vacuum, a spatial defect and

a time-only worldline. Among dimensions with structured spatial defects, it holds only in three space

dimensions. The proton uses the same quantum in its spatial form. The quantum is held for one Euclidean

step per cell. The scale is an assumption. The matching is a postulate equivalent to the relation. The

channel and the interval are stated and not derived. Lorentzian intervals give 16 to 256. A whole null tick

charged to each cell gives 64, but it counts each crossing twice. The cell’s other bonds give 8 or 216. The

premises select this count from every alternative considered. With 𝐾

𝜈

= 1/2 the relation ﬁxes the solar

splitting from the atmospheric splitting and the dark-energy density, Δ𝑚

= (7.55 ± 0.11) × 10

−5

It agrees with the 207-day JUNO data at 0.7𝜎. It also requires the normal ordering and a constant dark

energy. It becomes a precise test once JUNO’s six-year data and a resolved Hubble constant bring the

inputs below one percent.

Data availability

The scripts that reproduce every number are archived at https://github.com/raghu91302/ss

mth eory/bl ob/main/neutrino_ dark_en ergy_scripts_v2.zip. The script nde_numbers.py

covers Sections 2–5, including the Hubble dependence, the error budget, the sensitivity to Δ𝑚

and

the comparison with the 207-day JUNO data. It also gives the de Sitter rate and entropy of Section 6.

The script koide_zero.py checks Eq. (3). Section 4 uses lorentz_cell.py, frame_cube.py,

nu_impedance.py, clocks.py, hypercube.py, derive64.py, cubes.py, nu_operator.py and

nu_consistency.py. Section 5 uses lookelsewhere_model.py, and the script predict_dm21.py

is an independent check of Eq. (16). The script make_figs.py draws Fig. 3, make_fig_ssm.py draws

Fig. 1, and make_fig_cells.py draws Fig. 2. All scripts use numpy. Four also use scipy, and the three

ﬁgure scripts use matplotlib.

CRediT authorship contribution statement

Raghu Kulkarni: Conceptualization, Methodology, Writing – review & editing, Validation.

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.

Funding

This research did not receive any speciﬁc grant from funding agencies in the public, commercial, or

not-for-proﬁt sectors.

References

[1] Planck Collaboration, N. Aghanim et al., Planck 2018 results. VI. Cosmological parameters, Astron. Astro-

phys. 641 (2020) A6, doi:10.1051/0004-6361/201833910.

[2] I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, I. Martinez-Soler, J. P. Pinheiro and T. Schwetz,

NuFit-6.0: updated global analysis of three-ﬂavor neutrino oscillations, JHEP 12 (2024) 216,

doi:10.1007/JHEP12(2024)216.

[3] NuFIT 6.1 (2025), www.nu-ﬁt.org.

[4] A. Abusleme et al. (JUNO Collaboration), Measurement of reactor neutrino oscillation with the ﬁrst JUNO

data, Nature 654 (2026) 343, doi:10.1038/s41586-026-10538-z; arXiv:2511.14593.

[5] I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, I. Martinez-Soler, J. P. Pinheiro and T. Schwetz, Lessons

from the ﬁrst JUNO results, arXiv:2601.09791 (2026).

[6] JUNO Collaboration, Measurement of neutrino oscillation parameters at JUNO and indication of the neutrino

mass ordering, arXiv:2609.37895 (2026).

[7] JUNO Collaboration, Sub-percent precision measurement of neutrino oscillation parameters with JUNO,

Chin. Phys. C 46 (2022) 123001, doi:10.1088/1674-1137/ac8bc9.

[8] R. Fardon, A. E. Nelson and N. Weiner, Dark energy from mass varying neutrinos, JCAP 10 (2004) 005,

doi:10.1088/1475-7516/2004/10/005.

[9] N. Afshordi, M. Zaldarriaga and K. Kohri, On the stability of dark energy with mass-varying neutrinos, Phys.

Rev. D 72 (2005) 065024, doi:10.1103/PhysRevD.72.065024.

[10] DESI Collaboration, Constraints on neutrino physics from DESI DR2 BAO and DR1 full shape, Phys. Rev.

D 112 (2025) 083513, doi:10.1103/w9pk-xsk7; arXiv:2503.14744.

[11] N. Li and B.-Q. Ma, Estimate of neutrino masses from Koide’s relation, Phys. Lett. B 609 (2005) 309,

doi:10.1016/j.physletb.2005.01.066; arXiv:hep-ph/0505028.

[12] Y. Koide, New view of quark and lepton mass hierarchy, Phys. Rev. D 28 (1983) 252,

doi:10.1103/PhysRevD.28.252.

[13] S. Navas et al. (Particle Data Group), Review of Particle Physics, Phys. Rev. D 110 (2024) 030001,

doi:10.1103/PhysRevD.110.030001.

[14] C. A. Brannen, Koide mass formula for neutrinos (2006), https://brannenworks.com/MASSES.pdf.

[15] A. Rivero and A. Gsponer, The strange formula of Dr. Koide, arXiv:hep-ph/0505220 (2005).

[16] R. Kulkarni, A two-step quantum–classical threshold at 13.1–22.6 𝜇g with exact ratio

√

3 from a close-packed

vacuum lattice, Quantum Rep. 8 (2026) 78, doi:10.3390/quantum8030078.

[17] R. Kulkarni, The mass–energy–information equivalence extended: 𝐷

lattice, triality, and the three-

generation structure of matter, Phys. Open (2026) 100499, doi:10.1016/j.physo.2026.100499.

[18] R. Kulkarni, The mass–energy–information equivalence: a bottom-up identiﬁcation of the particle spectrum

via FCC lattice error correction, Phys. Open 27 (2026) 100414, doi:10.1016/j.physo.2026.100414.

[19] R. Kulkarni, Black holes in the FCC selection–stitch model: Bekenstein–Hawking entropy, geometric evap-

oration, and primordial-black-hole signatures, Eur. Phys. J. Plus 141 (2026) 916, doi:10.1140/epjp/s13360-

026-08148-9.

[20] A. G. Riess et al., A comprehensive measurement of the local value of the Hubble constant with 1 km s

−1

Mpc

−1

uncertainty from the Hubble Space Telescope and the SH0ES team, Astrophys. J. Lett. 934 (2022) L7,

doi:10.3847/2041-8213/ac5c5b.

[21] M. Aker et al. (KATRIN Collaboration), Direct neutrino-mass measurement based on 259 days of KATRIN

data, Science 388 (2025) 180, doi:10.1126/science.adq9592.

[22] S. Abe et al. (KamLAND-Zen Collaboration), Search for Majorana neutrinos with the complete KamLAND-

Zen dataset, Phys. Rev. Lett. 135 (2025) 262501, doi:10.1103/jkf6-48j8; arXiv:2406.11438.

[23] J. G. Lee, E. G. Adelberger, T. S. Cook, S. M. Fleischer and B. R. Heckel, New test of the gravitational 1/𝑟

law

at separations down to 52 𝜇m, Phys. Rev. Lett. 124 (2020) 101101, doi:10.1103/PhysRevLett.124.101101;

arXiv:2002.11761.

[24] P. J. Mohr, D. B. Newell, B. N. Taylor and E. Tiesinga, CODATA recommended values of the fundamental

physical constants: 2022, Rev. Mod. Phys. 97 (2025) 025002, doi:10.1103/RevModPhys.97.025002.

[25] J. Kogut and L. Susskind, Hamiltonian formulation of Wilson’s lattice gauge theories, Phys. Rev. D 11 (1975)

395, doi:10.1103/PhysRevD.11.395.

[26] S. Coleman, Aspects of Symmetry (Cambridge University Press, Cambridge, 1985), chapter 7, The uses of

instantons, doi:10.1017/CBO9780511565045.

[27] A. Chodos, R. L. Jaﬀe, K. Johnson, C. B. Thorn and V. F. Weisskopf, New extended model of hadrons, Phys.

Rev. D 9 (1974) 3471, doi:10.1103/PhysRevD.9.3471.

[28] M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, QCD and resonance physics. Theoretical foundations,

Nucl. Phys. B 147 (1979) 385, doi:10.1016/0550-3213(79)90022-1.

[29] A. G. Cohen, D. B. Kaplan and A. E. Nelson, Eﬀective ﬁeld theory, black holes, and the cosmological

constant, Phys. Rev. Lett. 82 (1999) 4971, doi:10.1103/PhysRevLett.82.4971.

Dark Energy & Lightest Neutrino Mass: ρ_Λ = 64 m₁⁴ Factor