Neutrino Mass Suppression and Oscillation from a Cosserat Vacuum Lagrangian

Micropolar Neutrinos from Triadic Orthogonal

Calculus:

Deriving Mass Suppression and the PMNS Matrix

from the FCC Vacuum Triad

Raghu Kulkarni

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

raghu@idrive.com

Abstract

We derive the neutrino mass scale and the PMNS mixing matrix from Triadic Or-

thogonal Calculus (TOC), a minimal algebraic framework whose sole primitive is

the vacuum triad

τ = (4, 4, 4)

the unique decomposition of the FCC coordina-

tion shell (

K = 12

) into three mutually orthogonal 4-bond sheets. In the Cosserat

(micropolar) description, charged leptons are translational displacements engaging

the full triad norm

|τ | = 12

, while neutrinos are microrotations coupling only to

the torsional complement

|τ | − τ

= 8

. The geometric mass suppression factor

Φ = 1/[(|τ | + 1) × τ

] = 1/208

yields, after volumetric and holographic projection,

≈ 0.0503

eV and

∆m

≈ 2.53 × 10

−3

(exact match to observation). The

PMNS mixing angles follow from triad sheet geometry:

= 45

◦

(atmospheric),

sin

≈ 0.020

(reactor, from the triad Regge decit), and

sin

≈ 0.319

(solar).

TOC independently predicts

sin

= dim(τ )/(|τ | + 1) = 3/13 = 0.2308

(0.19%

from experiment). The framework predicts Normal Hierarchy, Dirac nature, and

≈ −π/2

all testable by JUNO and LEGEND. Every quantity derives from

τ = (4, 4, 4)

with zero free parameters.

Keywords:

Triadic Orthogonal Calculus, neutrino mass, PMNS matrix, FCC lat-

tice, Cosserat elasticity, Weinberg angle

1 Introduction

The neutrino sector poses two puzzles for the Standard Model. First, the mass hierarchy:

the heaviest neutrino (

≲ 0.1

eV) is suppressed by seven orders of magnitude below the

electron (

= 0.511

MeV). Second, the avor puzzle: the quark mixing matrix (CKM)

has small angles (

≈ 13

◦

), while the neutrino mixing matrix (PMNS) has large, near-

maximal angles (

≈ 49

◦

≈ 34

◦

) [1].

The See-Saw mechanism [2] addresses the mass gap by introducing heavy sterile states

at the GUT scale. Discrete non-Abelian avor symmetries (

) are invoked to enforce

Tri-bimaximal mixing patterns. Neither provides a unied, low-energy dynamical origin

for both phenomena.

We propose a geometric alternative. We model the vacuum as a Face-Centered Cubic

(FCC) tensor network at the

K = 12

Kepler packing limit [4] and introduce Triadic Or-

thogonal Calculus (TOC), a minimal algebraic framework native to this lattice. Combined

with Cosserat micropolar elasticity [5, 6], the TOC triad

τ = (4, 4, 4)

simultaneously de-

rives the neutrino mass suppression, the full PMNS mixing matrix, and the Weinberg

anglewith zero free parameters.

Interactive 3D visualizations

supporting this paper are available at:



Entanglement Defect

 the interstitial node at the tetrahedral void,

its 4 non-bipartite bonds, and the

1 + 3

charge asymmetry that drives the

PMNS avor basis:

https://raghu91302.github.io/ssmtheory/ssm_entanglement_defect.

html



Regge Decit & Lattice Structure

 the

K = 12

cuboctahedral coor-

dination shell, the tetrahedral packing gap (

δ = 0.128

rad), and the three

orthogonal sheets of the triad

τ = (4, 4, 4)

https://raghu91302.github.io/ssmtheory/ssm_regge_deficit.html

2 Triadic Orthogonal Calculus

2.1 The triad

The FCC lattice has coordination

K = 12

, with nearest-neighbor vectors:

N = {(±1, ±1, 0), (±1, 0, ±1), (0, ±1, ±1)}

(1)

These decompose uniquely into three mutually orthogonal 4-bond sheets:

XY-sheet:

(±1, ±1, 0)

(

= 4

bonds)

XZ-sheet:

(±1, 0, ±1)

(

= 4

bonds)

YZ-sheet:

(0, ±1, ±1)

(

= 4

bonds) (2)

Denition 1

(Triad)

The entanglement state of an FCC node is the triad

τ = (τ

, τ

) ∈

, with norm

|τ | = τ

+ τ

and dimension

dim(τ ) = 3

. For the vacuum,

vac

= (4, 4, 4)

Every structural integer derives from this single object:

|τ | = 12 = K

(coordination)

dim(τ ) = 3 = c

(crossing modes)

= 4 = S

trans

(one sheet = translational sector)

|τ | −τ

= 8 = S

tors

(complementary sheets = torsional sector)

|τ | + 1 = 13

(structural cluster size)

= 16

(single-sheet disruption depth)

|τ |

= 144

(full-triad disruption depth) (3)

The partition

trans

+ S

tors

= τ

+ (|τ | − τ

) = |τ |

expresses the

representation-

theoretic decomposition (

⊕ A

vs. the remaining 8 torsional modes) as an algebraic

property of the triad.

1.5

1.0

0.5

0.0

0.5

1.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

(a) = (4, 4, 4): Three sheets

XY sheet (

= 0): 4 bonds

XZ sheet (

= 0): 4 bonds

YZ sheet (

= 0): 4 bonds

1.5

1.0

0.5

0.0

0.5

1.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

XY: 12 bonds

XZ: 12 bonds

YZ: 12 bonds

(b) 36 bonds = 3 × 12

= (4, 4, 4)

| |= 12

dim = 3

= 4

| |

= 8

Rule 1:

= | |

= 144

Rule 2:

= 36

Rule 3:

= | |+1= 13

= 1836

= 5.36

1836.15 (expt)

0.008% match

5.364 (Planck)

0.09% match

Zero free parameters

All from

= (4, 4, 4)

Figure 1: Triadic Orthogonal Calculus. (a) The vacuum triad

τ = (4, 4, 4)

: the 12 FCC

nearest neighbours decompose into three orthogonal 4-bond sheets. (b) The 36 internal

bonds of the 13-node structural cluster, partitioned into

dim(τ ) = 3

groups of

|τ | = 12

by their zero-coordinate (Proposition 1 of Ref. [3]). (c) Algebraic ow from the triad to

physics:

= 1836

and

Ω

/Ω

= 5.36

with zero free parameters.

2.2 Established TOC results

Applied to an interstitial entanglement defect at the tetrahedral void, TOC derives [3]:

= (|τ | + 1)|τ |

− dim(τ )|τ | = 1836 (0.008%)

(4)

Ω

|τ |[(|τ | + 1)(|τ |− τ

)

− |τ |]

1836

= 5.3595 (0.09%)

(5)

In this paper we extend TOC to the neutrino sector.

3 The Cosserat Vacuum

In Cosserat (micropolar) elasticity [5, 6], each lattice node carries two independent degrees

of freedom: translational displacement

and microrotation

. The Lagrangian density

is:

V =

(∇ · u)

+ µ(∇u)

+ κ(∇ × u − 2ϕ)

+ γ(∇ϕ)

(6)

where

λ, µ

are the Lamé coecients (compression and shear),

is the coupling modulus,

and

is the twist modulus.

In TOC, this partition maps onto the triad:

Charged leptons

(

, translational): Topological defects that displace lattice nodes,

stretching the structural bonds. They engage the full coordination

|τ | = 12

and acquire

mass from the bulk modulus

bulk

∼ λ + 2µ

Neutrinos

(

, torsional): Topological defects that twist nodes without displacement.

Because a pure microrotation does not move the node's center of mass, it bypasses the

massive structural stiness entirely. Neutrino mass is governed by the twist modulus

which couples only to the torsional complement

|τ | −τ

= 8

The mass hierarchy between charged leptons and neutrinos is the hierarchy between

bulk

and

4 Neutrino Mass from the Triad

4.1 The geometric suppression factor

Rest mass in the FCC framework equals the elastic energy cost of maintaining a topolog-

ical defect. The electron engages

bulk

directly. The neutrino must project its localized

Spin(1,3) structure onto the spatial

K = 12

geometry. In TOC, this projection is deter-

mined by two triad quantities:

Φ =

(|τ | + 1) × τ

13 × 16

208

(7)

Here

|τ | + 1 = 13

is the structural cluster size (the 13-node defect cluster, proved

in Ref. [3]). The factor

= 4

= 16

is the single-sheet disruption depth: the entan-

glement disruption per triad sheet, analogous to

|τ |

= 144

for the full triad. The two

factors are a productnot a sumbecause they count

independent

degrees of freedom:

|τ | + 1

is a spatial selection (which of the 13 cluster nodes participates), while

is an

internal disruption depth (how many bond-states are disrupted within one sheet). The

spatial structure is determined by lattice geometry and is identical regardless of which

sheet carries the disruption; conversely, the internal disruption depth is set by the entan-

glement structure and is the same at every node. This independenceguaranteed by the

uniformity of the triad

τ = (4, 4, 4)

makes the total phase space a Cartesian product:

13 × 16 = 208

independent cells.

The electron (

) displaces the metric directly, without volumetric projection. The

neutrino (

) does not displace the lattice; because a Dirac mass term couples left and

right chiralities across all three spatial dimensions, the suppression applies volumetrically:

bare

= m

× Φ

= 0.511

MeV



208



≈ 0.0568

eV (8)

4.2 Holographic boundary projection

The value 0.0568 eV represents the 3D bulk twist energy. Neutrinos are detected via

the weak interaction, which in the FCC framework is conned to the square faces of the

cuboctahedral coordination shell. The continuous circular microrotational eld

must

project onto a rigid square boundary. The inscribed circle within a square of side

has

area ratio

πr

/(2r)

= π/4

. Since the Lagrangian energy scales quadratically, the linear

mass projects by the square root:

= m

bare

= 0.0568 × 0.886 ≈ 0.0503

eV (9)

With

≈ 0

∆m

≈ (0.0503)

≈ 2.53 × 10

−3

(10)

The experimental value is

2.51±0.027×10

−3

[1]. The match is within measurement

uncertainty.

5 Normal Hierarchy from Lattice Anisotropy

The FCC lattice has three principal axes with dierent geometric freedom. Because

microrotation

is volume-preserving, its phase space is quenched as the propagation

path is dimensionally reduced. In TOC, the three axes engage dierent numbers of triad

sheets:

(Body diagonal):

Full 3D twist engaging all

dim(τ ) = 3

sheets symmetrically.

Yields

≈ 0.0503

eV.

(Face diagonal):

Twist constrained to a 2D sheet plane, engaging

dim(τ )−1 = 2

sheets. Reduced solid angle guarantees

≪ m

, establishing

∆m

≪ |∆m

(Edge):

Twist conned to a 1D line, engaging

dim(τ ) −2 = 1

sheet. Transverse

microrotation maximally quenched:

≈ 0

The dimensional reduction

3 → 2 → 1

across the sheets predicts the

Normal Hier-

archy

(

≫ m

> m

≈ 0

). This is directly testable by JUNO.

The anisotropic twist moduli (

= γ

) also drive avor oscillation: as unanchored

twist elds propagate through the vacuum, fractional dierences in phase velocity create

the dispersive beating pattern measured as neutrino oscillation.

6 The PMNS Matrix from Sheet Geometry

The CKMPMNS disparity follows from the Cosserat partition. Charged fermions (

)

couple to 2D surface gauge links on the square faces, yielding small CKM angles. Neutri-

nos (

) propagate through the 3D bulk; their mixing is the geometric angle between the

mass basis and the avor basis.

In TOC:

The

avor basis



F = (1, 1, 0)

is the normal to a square facean XY-sheet vector of

the triad.

The

mass basis



M = (1, 1, 1)

is the body diagonalthe freest coherent path through

the bulk, engaging all

dim(τ ) = 3

sheets equally.

6.1 Atmospheric angle (

)

The avor vector



F = (1, 1, 0)

bisects the

and

axes:

cos θ

(1, 0, 0) ·(1, 1, 0)

1 ·

√

=⇒ θ

= 45

◦

(11)

Exact maximal atmospheric mixing. The experimental value

◦

± 1.5

◦

is within

2σ

6.2 Reactor angle (

) from the Regge decit

In a perfectly continuous geometry,

= 0

(Tri-bimaximal orthogonality). The discrete

K = 12

lattice breaks this: regular tetrahedra cannot perfectly tile at 3D space, leaving

a packing gapthe Regge decit. In TOC, every ingredient of this decit is a triad

quantity.

The dihedral angle of the regular tetrahedron is

arccos(1/3)

. The ratio

1/3 = τ

/|τ | =

4/12

is the translational fraction of the triadthe per-bond entropy carried by one sheet.

The number of tetrahedra tting around a shared edge is

⌊2π/ arccos(τ

/|τ |)⌋ = 5 = τ

which equals the within-cluster degree of each shell node (4 cuboctahedral neighbors

bond to the origin, proved in Ref. [3]). The Regge decit is:

δ = 2π − (τ

+ 1) · arccos



|τ |



= 2π − 5 arccos





≈ 0.128

rad (12)

The reactor angle measures the fractional geometric leakage of this packing gap across

the full rotational phase space:

sin

2π

= 1 −

(τ

+ 1)

2π

arccos



|τ |



≈ 0.0204

(13)

The experimental value is

sin

= 0.0220 ± 0.0007

[1]. The 7% deviation from the

central value places the prediction within

2σ

. Every quantity in Eq. (13) is determined

by the triad:

= 4

|τ | = 12

, and the packing number

+ 1 = 5

6.3 Solar angle (

)

The 3D projection of



onto



cos α =



M ·



M||



F |

√

(14)

The ideal mixing probability

sin

= 1 − 2/3 = 1/3

is the Tri-bimaximal baseline.

Applying the unitarity correction from the reactor angle (with

= 2/3

topologically

protected by hexagonal face symmetry):

sin

1 − 3 sin

3(1 − sin

)

≈ 0.319

(15)

Experimental:

0.307 ± 0.013

, within

1σ

7 The Weinberg Angle from the Triad

The electroweak mixing angle measures the ratio of electromagnetic to electroweak cou-

pling. In TOC, the crossing modes (

dim(τ ) = 3

) are the independent weak channels,

while the full cluster (

|τ | + 1 = 13

) is the total structural unit:

sin

dim(τ )

|τ | + 1

= 0.23077

(16)

Experiment at the

pole:

0.23122 ± 0.00003

(0.19% match). The SU(5) GUT value

3/8

at the unication scale runs to

∼ 0.231

via renormalization-group ow; TOC

gives the low-energy value directly.

8 Dirac Nature and CP Violation

The coupling

κ(∇×u −2ϕ)

converts orbital angular momentum into localized spin. On

the

-symmetric triangular faces of the cuboctahedron, the curl operator breaks parity,

forcing left-handed chirality. Because a mechanical twist (

) is topologically distinct from

an anti-twist (

−ϕ

), particle and antiparticle are non-degenerate.

Predictions: neutrinoless double beta decay will not be observed (Dirac nature), and

the CP-violating phase

≈ −π/2

(from the orthogonal collision of the 3D twist against

2D boundaries), consistent with T2K and NOvA hints.

9 Prediction Scorecard

Table 1 collects all TOC predictions for the neutrino sector. Combined with

1836

(0.008%) and

Ω

/Ω

= 5.3595

(0.09%) from the matter paper [3], TOC derives

eight precision observables from the single triad

τ = (4, 4, 4)

Observable TOC prediction Experiment Match

1836 1836.15

0.008%

Ω

/Ω

5.3595 5.364

0.09%

sin

3/13 = 0.2308 0.2312

0.19%

∆m

2.53 × 10

−3

2.51 × 10

−3

0.8%

◦

± 1.5

◦

2σ

sin

0.319 0.307 ± 0.013 1σ

sin

0.020 0.022 ± 0.001 2σ

Hierarchy Normal TBD (JUNO) Falsiable

Nature Dirac TBD (LEGEND) Falsiable

−π/2 ∼ −π/2

(T2K) Falsiable

Table 1: All TOC observables from

τ = (4, 4, 4)

. The rst three are derived in the

companion paper [3]; the remainder in this work. Zero free parameters.

10 Limitations

The framework has several honest limitations. The volumetric suppression

is a geomet-

ric argument, not derived from a wave equation. The holographic boundary projection

π/4

assumes the inscribed-circle model for rotational phase space on a square face;

other geometric choices would shift

∼ 10%

. The mass eigenvalues

and

are

predicted to be hierarchically smaller than

, but their precise values are not computed

(only the ordering

≫ m

> m

≈ 0

). The reactor angle relies on the Regge decit

δ = 2π − (τ

+ 1) arccos(τ

/|τ |)

, which is a fully determined triad quantity; however, the

identication of

δ/(2π)

with

sin

(rather than with some other mixing-matrix element)

is a structural assignment based on the reactor angle being the smallest PMNS parame-

ter, not a derivation from a Hamiltonian. The framework does not address the absolute

neutrino mass scale (only ratios), and does not reproduce the full gauge structure of the

Standard Model.

11 Conclusion

Triadic Orthogonal Calculus provides a unied origin for the neutrino mass scale and

the PMNS mixing matrix from the vacuum triad

τ = (4, 4, 4)

. The mass suppression

Φ = 1/[(|τ | + 1) × τ

] = 1/208

is a triad ratio, not a phenomenological parameter. The

PMNS angles emerge from projections of the body diagonal



M = (1, 1, 1)

onto the sheet

vectors



F = (1, 1, 0)

. The Weinberg angle

sin

= 3/13

connects the neutrino sector to

the electroweak structure.

The ten observables in Table 1spanning the proton mass, dark matter, the Weinberg

angle, and the neutrino sectorall derive from a single algebraic object. Three of these

(Normal Hierarchy, Dirac nature,

) are directly testable by the next generation of

experiments.

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K = 12

Tensor Network,

https://doi.org/10.5281/zenodo.18933667

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Ann. Math.

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Micropolar Neutrinos: Deriving Mass Suppression and Oscillation from theCosserat Lagrangian of the Vacuum