Dark energy and the lightest neutrino:
𝜌
Λ
= 64 𝑚
4
1
and a lattice reading of its factor
Raghu Kulkarni
SSMTheory Group, IDrive Inc, Calabasas, CA 91302, USA
raghu@idrive.com
Abstract
Of all particle-physics scales, only the neutrino masses lie near the dark-energy scale, 𝜌
1/4
Λ
=
2.24 meV. Suppose the neutrino masses satisfy the Koide-type condition 𝐾
𝜈
= 1/2. Then the
measured splittings fix the lightest mass at 𝑚
1
= 0.794 ±0.013 meV in the normal ordering. We find
𝜌
1/4
Λ
/𝑚
1
= 2.821 ± 0.048, within 0.2𝜎 of 2
√
2. In natural units this is 𝜌
Λ
= 64 𝑚
4
1
. We ask how
much of the factor a close-packed vacuum lattice can supply. In the face-centered-cubic lattice of
the Selection–Stitch Model, each bond carries one quantum ℏ𝑐/(4𝐿) per tick, with 4 = 𝐾/𝑑 = 12/3.
We apply this rule at the spacing where the quantum equals 𝑚
1
𝑐
2
. The model has no such scale of its
own. In the model’s 𝐷
4
extension, a neutrino is a time-only worldline with no equal-time presence.
It is therefore counted per four-cell. The only bond-length cubes of 𝐷
4
are the four-cells of its three
orthogonal bond frames, and each one needs the time direction. The neutrino steps from cell to cell
along its own frame’s bonds. The sixteen bonds inside the cell that its path does not use are taken
as the vacuum’s channel. Their isotropy factor is 16/4 = 4. Each step is held for its Euclidean
length. We then require the vacuum to be impedance-matched to the neutrino over the cell. This
gives 𝜌
Λ
= (𝐾/𝑑)
3
𝑚
4
1
= 64 𝑚
4
1
. The matching is a postulate. It is defined by analogy, and it is
algebraically equivalent to the relation. Other channels give 8 or 216. A Lorentzian count with null
steps gives 16 to 90. Of fifteen bond, cell and clock choices, two give 64. A chance match within
0.3% has a probability near 0.6. Given 𝐾
𝜈
= 1/2, the relation links Δ𝑚
2
21
, Δ𝑚
2
31
and 𝜌
Λ
. It gives
Δ𝑚
2
21
= (7.56 ± 0.11) × 10
−5
eV
2
, against 7.53 ± 0.10 measured. Today this agreement restates the
coincidence. It becomes a test as JUNO data and the Hubble constant improve. The relation requires
the normal ordering and a constant dark energy.
Keywords: dark energy; cosmological constant; neutrino masses; Koide relation; FCC lattice; 𝐷
4
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
Λ
≈ 2.24 meV in the standard cosmological model [1]. The second is the
neutrino mass scale. The solar and atmospheric splittings correspond to
Δ𝑚
2
21
≈ 8.7 meV and
Δ𝑚
2
31
≈ 50 meV [2, 3]. 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 [6]. The simplest such models are unstable to neutrino clumping [7],
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
Í
𝑚
𝜈
< 64 meV for a cosmological
constant and < 163 meV for evolving dark energy [8].
This paper reports a numerical relation between the two scales. It asks how much of the relation a
specific vacuum model can explain, and it states how the relation can be tested. The relation needs the
1
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 [10]. 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) [14, 15]. It summarizes the model and then lists the premises. Section 5
gives the consequences for measurements. Section 6 lists every alternative we tried and estimates the
chance of an accidental match.
We state the status of the result at the outset. The factor 4 in 64 is a postulate of the lattice model.
We apply it at a scale the model does not otherwise have. The count per four-cell follows from the
model’s description of the neutrino as a time-only worldline. The equality of the vacuum’s action and
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, and the action interval is Euclidean
(Section 4.7). By look-elsewhere standards the closeness of the match is unremarkable (Section 6). The
agreement of the implied solar splitting with its measured value is the same fact for today’s data. The
paper offers a definite relation, a candidate origin for its factor, and a test. The test becomes decisive as
the inputs improve.
2 Neutrino masses for 𝐾
𝜈
= 1/2
Koide’s quantity for three masses is
𝐾 =
𝑚
1
+ 𝑚
2
+ 𝑚
3
√
𝑚
1
+
√
𝑚
2
+
√
𝑚
3
2
. (1)
For the charged leptons it equals 2/3 to about 10
−5
[10, 11]. Every triple can be written as [12, 13]
√
𝑚 = 𝑎
h
1 + 𝑏 cos
𝛿 +
2𝜋 𝑘
3
i
, 𝑘 = 0, 1, 2, (2)
with 𝐾 = (1 + 𝑏
2
/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). It lies inside the range that Li and Ma estimated for neutrinos,
0.50 < 𝑘
𝜈
< 0.85 in their normalization 𝑘 = 3𝐾/2 [9]. 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
1
= 𝑎
2
(1 − cos 𝜀)
2
. (3)
The two oscillation splittings fix 𝑎 and 𝛿
𝜈
. We use Δ𝑚
2
21
= (7.530 ± 0.097) × 10
−5
eV
2
from the
global fit that includes the first JUNO data [4, 3]. For Δ𝑚
2
31
we use 2.521 ×10
−3
eV
2
, the NuFIT 6.1 value
without the Super-Kamiokande atmospheric data quoted there [4, 2]. Its uncertainty of 0.020 ×10
−3
eV
2
is our estimate. The normal ordering gives 𝛿
𝜈
= 0.3327, 𝑎
2
= 13.27 meV, 𝜀 = 0.715, and
𝑚
1
= 0.794 ± 0.013, 𝑚
2
= 8.714, 𝑚
3
= 50.22 meV,
𝑚
𝜈
= 59.7 meV. (4)
The uncertainty on 𝑚
1
includes both splittings (script nde_numbers.py). In the inverted ordering, the
same condition forces the lightest mass to 𝑚
3
≈ 10
−8
meV. This matters in Section 5. The sum lies below
the DESI bound for a cosmological constant, 64.2 meV [8].
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
2
= 0.1430 ±0.0011 [1]. We use 𝜌
Λ
= 3(ℎ
2
−Ω
𝑚
ℎ
2
)(100 km s
−1
Mpc
−1
)
2
𝑀
2
Pl
,
2
with the reduced Planck mass. This gives 𝜌
1/4
Λ
= 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
Λ
𝑚
1
= 2.821 ± 0.048, (5)
within 0.2𝜎 of 2
√
2 = 2 .828. The relation we examine is therefore
𝜌
Λ
= 64
(𝑚
1
𝑐
2
)
4
(ℏ𝑐)
3
, equivalently 𝜌
1/4
Λ
= 2
√
2 𝑎
2
(1 − cos 𝜀)
2
. (6)
In the other direction, the oscillation data and 𝐾
𝜈
= 1/2 give 2
√
2 𝑚
1
= 2.245 ± 0.036 meV. The Planck
value is 2.239 ± 0.013 meV.
We use 𝑚
1
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 reports how other
combinations fare.
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 [16]. 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% [16]. The lattice spacing is near the Planck length. The model’s black-hole sector fixes
it at 1.843 ℓ
𝑃
[17]. A kinematic postulate fixes how fast the lattice transmits quanta [14]. Section 4.3
states it.
Spacetime is the four-dimensional lattice 𝐷
4
, the densest lattice packing in four dimensions [15].
The FCC lattice is its slice at fixed time. Each 𝐷
4
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. [15] uses the three groups
to label the three particle generations, as its hypothesis H2. 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), and it is a logical operator of the code. It
fires no check, which is why the code barely sees it. Ref. [15] leaves the neutrino mass scale open.
4.2 Premises
The reading rests on the premises below. We state them before we use them. Two come from 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 [14] (Section 4.3).
3
(a) FCC slice: 12 bonds
(b)
D
4
: 24 bonds, three frames
cell
t
x
(c)
(
t
,
x
)
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
4
site, projected to three dimensions by dropping time. Circles
with solid lines are spatial bonds. Squares with dashed lines are time-mixed bonds, and pairs of them
project onto the same point. Colors mark the three orthogonal frames (16-cells). (c) A section of 𝐷
4
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. [15]. 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 4.7).
I2 (construction of Ref. [15]). 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 4.7).
C (conjecture). 𝐾
𝜈
= 1/2 in the normal ordering (Section 2).
H
𝜈
1. The bond rule I1 applies at the spacing 𝐿
𝜈
at which one quantum equals 𝑚
1
𝑐
2
(Section 4.4).
H
𝜈
2. Inside the neutrino’s four-cell, the vacuum transmits along the sixteen bonds from the cell’s center
to its corners. These bonds set the bond stiffness (Section 4.7).
H
𝜈
3. The neutrino’s quantum is held for the Euclidean length of one step per four-cell (Section 4.7).
H
𝜈
4. The vacuum is impedance-matched to the neutrino over the four-cell (Section 4.6).
The count per four-cell is not a separate premise. It follows from I2. Each failed alternative in Tables 1
and 3 departs from I2 or from one of H
𝜈
1–H
𝜈
3. So does each count with more than one quantum in
Section 4.6.
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
4
lattice [15]. Its twelve bond directions ˆ𝑛 give an isotropic structure tensor,
Í
ˆ𝑛 ˆ𝑛
⊤
= (𝐾/𝑑) ⊮ with 𝐾/𝑑 = 12/3 = 4. The kinematic postulate of the model [14] 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 A neutrino-scale spacing
The model’s vacuum spacing is Planckian. Its black-hole sector fixes it at 1.843 ℓ
𝑃
[17]. Our reading
applies Eq. (7) at a second, much larger spacing. At this spacing the bond quantum equals the rest energy
4
of the lightest neutrino:
𝐿
𝜈
=
ℏ𝑐
4 𝑚
1
𝑐
2
= 62.1 𝜇m. (8)
Nothing in the published model supplies this scale. It is hypothesis H
𝜈
1. At this spacing one light-tick,
𝑐𝜏 = 4𝐿
𝜈
, equals the reduced Compton wavelength of the lightest neutrino.
A structure that carries the lattice at its own spacing, set by its rest energy in bond quanta, has
a counterpart at the nuclear scale. With sixteen quanta, 𝑚
𝑝
𝑐
2
= 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 [22].
Sixteen is the number of two-step walks from one node of the complete graph on five nodes. These five
nodes are the four vertices of a tetrahedral void and the node it traps. A single bond, the graph of one
step along a worldline, gives one. We use this comparison only as motivation. 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 the extension of the model to 𝐷
4
[15], a neutrino is not a defect of the spatial lattice. It is a time-only
worldline. It is an operator that wraps the time direction. It has no presence in any equal-time slice. It
commutes with every stabilizer, so it fires no local syndrome. It is labeled by one of the three 16-cells of
the 𝐷
4
shell. Its energy per unit of spatial volume is therefore not defined at any instant. Its action per
unit of four-volume is defined.
The geometry of 𝐷
4
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
4
do fall into
exactly three frames of 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
4
has an edge that leans into time. The eight vectors ±𝑣 of one frame are the vertices of one of the
three inscribed 16-cells of the 𝐷
4
shell. Ref. [15] assigns the three flavors to these 16-cells through its
hypothesis H2 (scripts frame_cube.py, lorentz_cell.py).
Each frame spans a four-cell of edge 𝐿 and four-volume 𝐿
4
. The cell contains two 𝐷
4
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
3
in Euclidean signature, 𝐿
3
/
√
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
4
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. We therefore count both sides of the relation per orthogonal-frame
four-cell. The four-volume of the cell is fixed. The duration of the neutrino’s quantum within it is not
fixed. This duration is where the model’s two clocks disagree (Section 4.7).
4.6 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 normal-
ization its coupling is 𝛼 =
𝑈/𝐽/4𝜋. Its wave impedance is proportional to
𝑈/𝐽 [23]. 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.
5
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
1
𝑐
2
. The
bond stiffness is the bond quantum, 𝐽 = 𝐸
bond
. Write 𝑚
1
𝑐
2
= 𝑁𝐸
bond
. With one quantum per step of
the worldline, 𝑁 = 1. The neutrino’s impedance is then 𝑍
2
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
Λ
𝐿
3
. Its action there is 𝜌
Λ
𝐿
4
/𝑐. With the
same bond stiffness, 𝑍
2
vac
= (𝜌
Λ
𝐿
4
/𝑐)/(ℏ/4).
Matching. Setting 𝑍
𝜈
= 𝑍
vac
gives
𝜌
Λ
=
𝑐
𝐿
4
𝜈
·
ℏ
4
=
𝑚
1
𝑐
2
𝐿
3
𝜈
=
𝐾
𝑑
3
(𝑚
1
𝑐
2
)
4
(ℏ𝑐)
3
= 64
(𝑚
1
𝑐
2
)
4
(ℏ𝑐)
3
. (10)
This is Eq. (6). The factor 4 is the lattice’s light-speed factor. The exponent 3 comes from the four-cell’s
three remaining edges. The spacing is thus determined twice. The neutrino mass gives 62 .1 𝜇m through
Eq. (8). The dark-energy density gives 𝐿 = (ℏ𝑐/4𝜌
Λ
)
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
Λ
= 64 𝑚
4
1
/𝑁
3
. 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. [23]
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 window fixes 𝑍
2
𝜈
= 1. So the matching condition
𝑍
vac
= 𝑍
𝜈
is algebraically identical to Eq. (10). 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.
4.7 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
1
𝑐
2
= ℏ𝑐/(𝛼𝐿). The neutrino’s action in the cell is 𝑚
1
𝑐
2
𝛽𝐿/𝑐. The vacuum’s action
is 𝜌
Λ
𝐿
4
/𝑐. Equating them gives
𝜌
Λ
= 𝛽 𝛼
3
(𝑚
1
𝑐
2
)
4
(ℏ𝑐)
3
. (11)
The neutrino’s path. Ref. [15] 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
1
± 𝑓
2
± 𝑓
3
± 𝑓
4
)/2. All sixteen have length exactly 𝐿, and 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 of Section 4.6. Their
6
directions give an isotropic structure tensor,
corners
ˆ𝑛 ˆ𝑛
⊤
=
16
4
⊮ = 4 ⊮. (12)
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. 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. Assigning the
corner bonds to the vacuum side is hypothesis H
𝜈
2.
The interval. The neutrino has no equal-time state. Its quantum is therefore assigned a microscopic
action, not an instantaneous energy. The natural setting for such an action is Euclidean. We take the
Euclidean length of a step as the interval, so 𝛽 = 1. This is hypothesis H
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
4
→∞
(ℏ𝑐/𝑉
4
) ln 𝑍
𝐸
[24]. The Euclidean interval is not the Lorentzian
time over which a neutrino is seen to propagate. That time is still governed by the neutrino’s Lorentzian
dispersion. With 𝛼 = 4 and 𝛽 = 1, Eq. (11) gives 64. This is Eq. (10).
Operator check. We used the 𝐿 = 4 torus of Ref. [15], 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. [15] 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
2
= 𝑝
2
𝑐
2
+ 𝑚
2
1
𝑐
4
, which must be recovered after continuation from the Euclidean description. Both
remain open.
Other bonds and other clocks. Table 1 lists the alternatives (scripts hypercube.py, clocks.py).
If the vacuum channel were the frame’s own eight bonds, then 𝐾/𝑑 = 2. If it were all 24 bonds, then
𝐾/𝑑 = 6. With one step per cell, these give 8 and 216. Both are excluded. In a Lorentzian count the
time-bearing bonds are null, and the quantum is held for a time interval of the cell. This count fails with
either clock. Take the bond rule’s tick, 4𝐿/𝑐. The cell’s time extent (
√
2 𝐿/𝑐, two slices) gives 90.5, and
its mean extent (𝐿/(
√
2𝑐), one slice) gives 45.3. Take instead a tick of four null steps, 2
√
2 𝐿/𝑐. The
same two intervals give 32 and 16. Holding the quantum for a whole null-step tick gives 64. But this
charges a full tick to each cell, while the time-axis worldline crosses two cells per tick. Counted per
crossing, it gives 32.
Table 1: Bond classes, ticks and intervals in Eq. (11), against the measured coefficient 63.3 ± 4.4.
Bonds setting 𝛼 = 𝐾/𝑑 Interval in one four-cell Coefficient Pull
hypercube corners, 16/4 one Euclidean step, 𝐿/𝑐 (Eq. 10) 64 +0.2𝜎
frame bonds, 8/4 one Euclidean step 8 −12.6𝜎
all 𝐷
4
bonds, 24/4 one Euclidean step 216 +35𝜎
FCC slice, 12/3 cell time extent,
√
2𝐿/𝑐 90.5 +6.2𝜎
FCC slice, 12/3 cell mean extent, 𝐿/(
√
2𝑐) 45.3 −4.1𝜎
FCC slice, 12/3 one tick, 4𝐿/𝑐 256 +44𝜎
null steps, 2
√
2 cell time extent 32 −7.1𝜎
null steps, 2
√
2 cell mean extent 16 −10.7𝜎
null steps, 2
√
2 one tick, per cell 64 +0.2𝜎
7
4.8 What remains open
The equal-time alternatives count energy per section of the four-cell. They are the FCC-slice rows of
Table 1. The mid-time section
√
2 𝐿
3
gives 45.3. The section 𝐿
3
/
√
2, averaged over the cell’s duration,
gives 90.5. So 𝜌
1/4
Λ
comes out 8.0% low or 9.3% high. The FCC volume per site, 𝐿
3
/
√
2, also gives
90.5 (Section 6). Using the 𝐷
4
null spacing as both cell and tick gives a coefficient of 1, which fails by
65% (Table 3).
Four 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 matching condition is postulated,
not derived (H
𝜈
4). The Euclidean interval of one step per four-cell is assumed (H
𝜈
3). The step amplitude
and the Lorentzian dispersion of the neutrino operator are not computed (Section 4.7).
Premise Content Status
I1 𝐾/𝑑 = 4, 𝐸
bond
= ℏ𝑐/(4𝐿) postulate of the model [14]
I2 neutrino as a frame-𝑘 time string constructed in Ref. [15]; re-verified here
C 𝐾
𝜈
= 1/2, hence 𝑚
1
= 0.794 meV conjecture; inside the range of Ref. [9]
H
𝜈
1 bond rule at 𝐿
𝜈
= 62 𝜇m, one quantum hypothesis; the vacuum spacing is Planckian
H
𝜈
2 vacuum channel on the cell’s corner bonds hypothesis; other bonds give 8 or 216
H
𝜈
3 one Euclidean step per four-cell hypothesis; Lorentzian intervals give 16–90
H
𝜈
4 impedance matching hypothesis; equivalent to the relation
5 What the relation implies
Given 𝐾
𝜈
= 1/2, any two of Δ𝑚
2
31
, Δ𝑚
2
21
and 𝜌
Λ
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
2
31
and the Planck 𝜌
Λ
:
Δ𝑚
2
21
= (7.56 ± 0.11) × 10
−5
eV
2
. (13)
The measured value is (7.530 ± 0 .097) × 10
−5
eV
2
[4, 3]. The difference is +0.2𝜎 (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.
Any of the chance relations of Section 6 would satisfy a similar constraint about as well. Most of
the uncertainty in Eq. (13) comes from Δ𝑚
2
31
(0.093) and from ℎ (0.049). The prediction moves by
0.047 × 10
−5
eV
2
for each 0.01 × 10
−3
eV
2
change in Δ𝑚
2
31
. The NuFIT 6.1 value that includes the
atmospheric data, 2.511 × 10
−3
eV
2
[4], gives 7.51.
The constraint becomes a test as its inputs improve. It has four consequences.
It depends on the Hubble constant. At fixed Ω
𝑚
ℎ
2
, 𝜌
Λ
grows with ℎ, and the implied splitting
falls (Fig. 2a). It is 7.56 for the Planck ℎ and 7.32 for ℎ = 0.70. For the distance-ladder value
ℎ = 0.730 ± 0.010 [18] it is 7.08 ± 0.12, which is 3.0𝜎 below the measurement. In the other direction,
the neutrino data and the relation give 𝜌
1/4
Λ
= 2.245 ± 0.036 meV. With the Planck Ω
𝑚
ℎ
2
this gives
𝐻
0
= 67.6 ± 1.5 km s
−1
Mpc
−1
. This agrees with the Planck value and lies 3.0𝜎 below the distance-
ladder value. The result is conditional, not independent. If the relation holds, a high 𝐻
0
is disfavored.
The distance-ladder case also pairs ℎ with the Planck Ω
𝑚
ℎ
2
, which is not a self-consistent cosmology.
It needs better inputs. After six years, JUNO projects 0.3% on Δ𝑚
2
21
and 0.2% on Δ𝑚
2
31
[5]. 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
2
31
and a resolved Hubble constant.
It excludes the inverted ordering. With 𝐾
𝜈
= 1/2, the inverted ordering has a lightest mass near
10
−8
meV (Section 2). Eq. (6) would then put 𝜌
1/4
Λ
about eight orders of magnitude below its value. A
confirmed inverted ordering rules the relation out.
8
65 70 75
H
0
(km s
−
1
Mpc
−
1
)
7.0
7.2
7.4
7.6
7.8
Δ
m
2
21
(10
−
5
eV
2
)
Planck SH0ES
(a) the solar splitting predicted from
ρ
Λ
predicted,
ρ
Λ
= 64
m
4
1
measured (global
fi
t with JUNO)
JUNO 6-year precision (illustrative)
0.8 1 1.5 2 3 4
ρ
1/4
Λ
implied (meV)
channel: cell corner bonds (Eq. 10)
channel: frame bonds
D4 tick, whole tick per cell
FCC tick, whole tick per cell
D4 tick, cell time extent
D4 tick, cell mean extent
channel: all 24 D4 bonds
FCC tick, cell mean extent
FCC tick, cell time extent / FCC site
octahedron volume
tetrahedron volume / bond
conventional FCC cube
⟨
100
⟩
cell, FCC tick
⟨
100
⟩
network, own tick
null shadow, own tick
(b) every cell choice tried
Planck
ρ
1/4
Λ
Figure 2: (a) The solar splitting implied by Eq. (6) with 𝐾
𝜈
= 1/2 and the measured Δ𝑚
2
31
, as a function
of the Hubble constant that sets 𝜌
Λ
at fixed Ω
𝑚
ℎ
2
. The blue band is the 1𝜎 range from Δ𝑚
2
31
; the
gray bands are the current measurement and, for illustration, the 0.3% precision JUNO projects for six
years [5]. (b) The value of 𝜌
1/4
Λ
implied by each cell choice of Table 3, with the Planck value.
It needs a constant dark energy. The DESI data mildly prefer evolving dark energy [8]. If dark energy
evolves, a fixed relation to 𝑚
1
would require the neutrino mass to evolve with it, as in mass-varying
models [6]. 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
Í
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
2
𝜃
12
= 0.3096 and sin
2
𝜃
13
= 0.0220 [4, 2]. They lie far
below the present limits [19, 20]. 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 [21]. The relation predicts no deviation there.
Table 2 collects these consequences. Its first three rows state one constraint in three 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
Í
𝑚
𝜈
well above 60 meV, or a double-beta signal above
about 5 meV.
6 The search, and the probability of a chance match
A numerical relation must be judged against the alternatives that could also have matched. We report
every choice we evaluated.
Cell choices. Table 3 and Fig. 2b list the fifteen bond, cell and clock choices we evaluated. Two give
64. One is the hypercube reading of Section 4.7. The other charges a whole tick to each cell (scripts
9
Table 2: What the relation and 𝐾
𝜈
= 1/2 imply, and what is measured. The first three rows are equivalent;
the last three follow from 𝐾
𝜈
= 1/2 alone.
Quantity Implied Measured or limit Status
𝜌
1/4
Λ
from neutrino data 2.245 ± 0.036 meV 2.239 ± 0.013 meV [1] +0.2𝜎
Δ𝑚
2
21
(10
−5
eV
2
) 7.56 ± 0.11 7.530 ± 0.097 [4] +0.2𝜎
𝐻
0
(km s
−1
Mpc
−1
) 67.6 ± 1.5 67.4 ± 0.5 [1]; 73.0 ± 1.0 [18] agrees; −3.0𝜎
𝑚
1
from 𝜌
Λ
alone 0.792 ± 0.005 meV not measurable at present untested
mass ordering normal not yet determined pending
dark energy constant, 𝑤 = −1 evolution mildly preferred [8] open
inverse-square law at 62 𝜇m no deviation none down to 52 𝜇m [21] consistent
Í
𝑚
𝜈
59.7 meV < 64.2 meV [8] consistent
𝑚
𝛽
8.9 meV < 0.45 eV [19] below reach
𝑚
𝛽𝛽
(Majorana) 1.0–4.3 meV < 28–122 meV [20] below reach
Table 3: Every bond, cell and clock choice tried for the energy density of one bond quantum 𝑚
1
𝑐
2
. The
coefficient is that of 𝑚
4
1
in natural units; the last column compares the implied 𝜌
1/4
Λ
with 2.239 meV.
Cell Coefficient 𝜌
1/4
Λ
(meV) vs Planck
Vacuum channel on the cell’s corner bonds (16/4) (Eq. 10) 64 2.245 +0.3%
Vacuum channel on the frame bonds (8/4) 8 1.34 −40%
FCC tick, energy per mid-time section of that cell,
√
2𝐿
3
45.3 2.059 −8.0%
FCC tick, per average section, 𝐿
3
/
√
2 (= FCC volume per site) 90.5 2.448 +9.3%
FCC tick, per octahedron volume 135.8 2.710 +21.0%
FCC tick, per tetrahedron volume or per bond 543 3.83 +71%
FCC tick, per conventional FCC cube 22.6 1.73 −23%
FCC tick, per ⟨100⟩ cubic cell or tetrahedral-void cube 181 2.91 +30%
⟨100⟩ cubic network with its own tick (𝐾/𝑑 = 2) 8 1.34 −40%
𝐷
4
null-bond spacing with its own tick 1 0.79 −65%
FCC tick, one whole tick per four-cell 256 3.18 +42%
𝐷
4
tick, one whole tick per four-cell 64 2.245 +0.3%
𝐷
4
tick, per four-cell over its time extent 32 1.89 −16%
𝐷
4
tick, per four-cell over its mean extent 16 1.59 −29%
Vacuum channel on all 24 𝐷
4
bonds (24/4) 216 3.04 +36%
derive64.py, cubes.py).
Mass combinations. We compared 𝜌
1/4
Λ
with nine combinations of the masses. Five lie within 0.5%
of some simple constant. The closest is not 𝑚
1
. It is (𝑚
1
𝑚
2
𝑚
3
)
1/3
/𝜋, within 0.08% (script explore.py).
Constants. We formed 206 distinct constants (𝑎/𝑏)𝑠 between 1 and 20. Here 𝑎 and 𝑏 come from
{1, 2, 3, 4, 5, 6, 8, 9, 12, 16, 24}, and 𝑠 comes from {1,
√
2,
√
3,
√
6, 𝜋, 1/𝜋,
√
𝜋}. Four of these constants lie
within 1𝜎 of the ratio in Eq. (5). Place a ratio at random within 2𝜎 of its value. The probability that some
constant lies within 0 .3% of it is 0.56. Within 0.7% the probability is 0.89 (script lookelsewhere.py).
A second reading of 2
√
2. The factor also equals the four-dimensional diagonal of the four-cell, 2𝐿,
divided by the cubic spacing 𝐿/
√
2. For the lattices 𝐷
𝑛
, this reading gives
2(𝑛 + 1). The bond rule
gives (𝐾/𝑛)
𝑛/(𝑛+1)
with 𝐾 = 2𝑛(𝑛 − 1). The two agree only at 𝑛 = 3, where three different quantities
equal 4 (script disguise.py). This is a coincidence of small numbers. It is not a second derivation.
The closeness of the match is therefore not evidence. A relation this close would arise by chance
more often than not, and the same search produced several others.
10
7 Discussion
What would make it a derivation. Four steps are needed. The first is a reason for the model to have a
lattice at the neutrino scale 𝐿
𝜈
. The second is a derivation of the vacuum’s stiffness in the time channel
and of the matching condition, from the code’s boundary dynamics. The third is a derivation of the two
choices of Section 4.7. It must show why the vacuum inside the neutrino’s cell transmits along the corner
bonds, and why the neutrino’s quantum is held for one Euclidean step. The fourth is a Hamiltonian for
the time channel that gives the step amplitude and the Lorentzian dispersion. The count per four-cell
needs no separate reason, because it follows from the neutrino being a worldline (Section 4). If all four
steps follow from the model, every factor of 𝜌
Λ
= (𝐾/𝑑)
3
𝑚
4
1
comes from the lattice. The relation then
becomes a falsifiable Koide-type law. Ref. [15] proposes, as a program and not a result, that the time-axis
condensate sets the neutrino mass scale. A mechanism there that set both 𝑚
1
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
1
. In the matching reading, the lightest time-only
worldline sets the charging energy of the time channel. The lightest charged defect plays the same
role for the photon channel. The relation then 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 [16]. A residual energy tied to the lowest excitation is therefore at least of a consistent kind.
Holographic arguments give the same scale from 𝑀
2
Pl
𝐻
2
0
[25]. The relation here ties it to 𝑚
1
instead.
This is why it constrains the neutrino sector.
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 hypotheses H
𝜈
1–H
𝜈
4 of Section 4. The present agreement of Eq. (13)
is not independent of it.
8 Conclusion
For the 𝐾
𝜈
= 1/2 neutrino spectrum, the dark-energy density and the lightest neutrino mass satisfy
𝜌
Λ
= 64 𝑚
4
1
to 0.2𝜎. In the close-packed lattice of the Selection–Stitch Model, the factor can be read as
(𝐾/𝑑)
3
. This reading has three parts. The bond rule applies at a neutrino-scale spacing. The neutrino is
a time-only worldline in an orthogonal-frame four-cell of 𝐷
4
. 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. 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, not derived. Lorentzian counting with null steps
gives 16 to 90, and the cell’s other bonds give 8 or 216. By look-elsewhere standards the match itself is
unremarkable. The content of the relation is a constraint. With 𝐾
𝜈
= 1/2 it fixes the solar splitting from
the atmospheric splitting and the dark-energy density, Δ𝑚
2
21
= (7.56 ±0.11) ×10
−5
eV
2
. It also requires
the normal ordering and a constant dark energy. It becomes a sharp test once JUNO’s six-year data and
a resolved Hubble constant bring the inputs below one percent.
11
Data availability
The scripts that reproduce every number are archived at https://github.com/raghu91302/ss
mtheory/blob/main/neutrino_dark_energy_scripts.zip. The script nde_numbers.py
covers Sections 2–5, including the Hubble dependence, the error budget and the sensitivity to
Δ𝑚
2
31
. The script koide_zero.py checks Eq. (3). Section 4 and Table 3 use derive64.py,
cubes.py, lorentz_cell.py, frame_cube.py, nu_impedance.py, clocks.py, hypercube.py
and nu_operator.py. Section 6 uses explore.py, lookelsewhere.py and disguise.py. The
script predict_dm21.py is an independent check of Eq. (13), and make_figs.py draws Fig. 2, and
make_fig_ssm.py draws Fig. 1. All scripts use numpy. Most also use scipy, and the figure script uses
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.
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