The Vacuum as a Close-Packed Lattice: Constants of Nature and a Two-Step Decoherence Threshold

The Vacuum as a Close-Packed Lattice:

Constants of Nature and a Two-Step

Decoherence Threshold

Raghu Kulkarni

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

raghu@idrive.com

June 2026

Abstract

We present the geometric foundations of the Selection-Stitch Model, in which the

vacuum is a discrete Face-Centered Cubic (

K = 12

) tensor network with Bell-pair

bonds. From this single structural choice, four foundational results follow with no

free parameters. First, the ratio of the macroscopic speed of light to the lattice up-

date velocity equals four  a geometric invariant set by the cuboctahedral structure

tensor eigenvalue, not a derivation of the speed of light in SI units. Second, Newton's

constant and the bond length (

≈ 1.843

Planck lengths) are xed by a consistency con-

dition derived from the Ryu-Takayanagi relation applied to the hexagonal boundary,

assuming this relation holds for the lattice. Third, a two-step quantum decoherence

threshold arises at approximately

11.8 µ

g and

20.5 µ

g. Both scales are of order the

Planck mass, as in gravitational-collapse models, so the scale itself is not new; the fal-

siable content is their

structure

 two thresholds rather than one, separated by the

exact, parameter-free ratio

√

3

set by the edge-to-circumradius ratio of the cuboc-

tahedral triangular face. Fourth, the long-wavelength Cosserat continuum limit of

the

K = 12

lattice is structurally isomorphic to the free-particle Schrödinger equa-

tion, with the imaginary unit arising from geometric translation-rotation coupling.

Emergent Lorentz invariance, with violations suppressed at order

(E/E

Planck

)

4

, is

reviewed. The two geometric inputs taken from simulation  the metric wall at

L/

√

3

and the

{111}

interlayer spacing  are cited from the published companion

work rather than assumed. Open problems  including the absence of a microscopic

decoherence derivation and the un-derived chiral coupling  are stated explicitly.

1 Introduction

The Standard Model of particle physics and General Relativity contain approximately 25

free parameters. The Selection-Stitch Model (SSM) proposes that these are not free: they

are geometric invariants of a discrete vacuum lattice.

Single structural assumption.

The vacuum is a Face-Centered Cubic (FCC) tensor network with coordination

number

K = 12

, where each bond is a maximally entangled Bell pair.

1

From this assumption alone we derive the four foundational results listed in the abstract.

The derivation chain is explicit:

K = 12 ⇒ c = 4v

lat

(Section 3)

⇒ G

N

= ℓ

2

P

, L = 1.843 ℓ

P

(Sections 45)

⇒

emergent SO(3,1) (Section 6)

⇒

Schrödinger isomorphism (Section 7)

⇒

decoherence thresholds (Section 8).

Relation to the companion papers.

This is the third paper in a series built on the

same

K = 12

FCC vacuum. The rst [1] constructs the

[[192, 130, 3]]

CSS code on the

lattice and supplies the stabilizer decomposition used here to assign physical content to

the bond subsets. The second [2] treats particles as defects in that code and derives their

masses as fault-tolerant verication costs. The most recent [3] models baryonic matter as a

single trapped node in a tetrahedral void and provides, by direct simulation of the

K=6 →

K=4 →K=12

crystallization, independent numerical conrmation of two geometric inputs

we use below: the metric wall at

r

min

= L/

√

3

(Section 8.1) and the

{111}

interlayer

spacing

h = L

p

2/3

(Section 5). Where the present paper invokes those facts, we cite

the published simulation rather than asserting them. The three papers address distinct

questions  what the code is, what mass is, and (here) what the macroscopic constants

and quantum-classical boundary are  and share only the single structural assumption

above.

2 The Vacuum Lattice

Denition 1

(The SSM Vacuum)

.

The vacuum is a three-dimensional tensor network

with the following properties:



Lattice type:

Face-Centered Cubic (FCC)  the unique 3D lattice saturating the

Kepler bound [4].



Coordination number:

K = 12

nearest neighbors per node.



Bond structure:

Each bond is a maximally entangled Bell pair

|Φ

+

⟩ = (|00⟩ +

|11⟩)/

√

2

.



Local geometry:

The 12 neighbors form a cuboctahedron with 8 triangular faces

and 6 square faces (Fig. 1).



Bond decomposition:

K = S

trans

+ S

tors

= 4 + 8

. The 12 cuboctahedral bonds

decompose into a subset of 4 that span a regular tetrahedron inscribed in the cube

(connecting alternating vertices of the square faces, each directed along a

⟨110⟩

body-

diagonal of that face) and the remaining 8 that complete the cuboctahedron. The 4

tetrahedral bonds carry the translational (gravitational/ electromagnetic) modes and

the 8 remaining bonds carry the torsional (dark-sector) modes [1]. The assignment

of physical content to each subset is motivated by the stabilizer decomposition of [1]

and is an additional input beyond the

K = 12

structural assumption.

2.1 Origin: the

K = 6 → K = 4 → K = 12

phase transition

The

K = 12

FCC vacuum arises from a topological phase transition.

Stage 1:

the 2D

hexagonal sheet (

K = 6

) is the unique planar network satisfying Euler's characteristic

χ = V (1 − K/6) = 0

.

Stage 2:

under cooling, the sheet buckles into 3D via tetrahedral

nucleation (

K = 4

), driven by the Regge decit angle

δ = 2π − 5 arccos(1/3) ≃ 0.128

rad.

Stage 3:

tetrahedral clusters pack into the FCC structure, saturating the Kepler bound

at

K = 12

. The existence and kinematics of this transition are established by direct

2

simulation in the companion paper [3]: a graph-growth process with two operators (a planar

stitch and a rare out-of-plane lift of relative amplitude

P

lift

= e

−3

) drives the network

to

K = 12

bulk saturation, with nite-size scaling

f

K=12

(N) = 1 − αN

−1/3

conrming

complete saturation in the thermodynamic limit. What remains underived is a closed-form

free energy from which the transition follows (Section 10.4); the transition itself is not

assumed here but taken from that numerical result.

Denition 2

(The Holographic Boundary)

.

The 3D FCC bulk is the holographic pro-

jection of a 2D continuous hexagonal (

K = 6

) entanglement network. The Lift operator

projects this boundary into 3D via ABC stacking at interlayer spacing

h = L

p

2/3

(the

crystallographic

{111}

interplanar distance

h = a

cube

/

√

3 = L

√

2/

√

3 = L

p

2/3

), building

the FCC structure layer by layer (Fig. 2).

The metric wall is the geometric exclusion limit: nodes cannot approach closer than

r

min

=

L/

√

3

without destroying the network topology. This is derived precisely in Section 8.1

from the circumradius of the cuboctahedral triangular face.

A (below)

B

C (above)

K = 3

A

+ 6

B

+ 3

C

= 12

K=6

at sheet

cool

K=4

foam

pack

K=12

FCC

Figure 1:

Left:

The

K = 12

cuboctahedral coordination shell in geometrically accurate

oblique 3D projection (bond length

L

throughout). The central node (orange, Sheet B)

connects to 3 Sheet A nodes (blue, dashed bonds going below) at horizontal distance

L/

√

3

and depth

h = L

p

2/3

; 6 in-plane Sheet B neighbors (red, solid); and 3 Sheet C nodes

(green, dashed going above). Light grey edges show the 24 cuboctahedral shell edges (all

equal to

L

), revealing the polyhedron formed by the 12 neighbors.

Right:

Topological

phase transition from the at hexagonal ground state (

K = 6

,

χ = 0

) through tetrahedral

nucleation (

K = 4

, Regge decit

δ ≃ 0.128

rad) to FCC saturation (

K = 12

, Kepler

bound).

3 The Speed-of-Light Ratio:

c = 4 v

lattice

3.1 The FCC structure tensor

The 12 FCC unit bond vectors are

ˆn

j

∈

1

√

2

{(±1, ±1, 0), (±1, 0, ±1), (0, ±1, ±1)}, j = 1, . . . , 12.

(1)

3

The second-rank structure tensor

S

µν

=

P

j

ˆn

µ

j

ˆn

ν

j

is computed directly:

S

xx

=

12

X

j=1

(ˆn

x

j

)

2

= 4 ×

1

2

+ 4 ×

1

2

= 4,

(2)

S

xy

=

12

X

j=1

ˆn

x

j

ˆn

y

j

= 0

(by cubic symmetry). (3)

Therefore:

S

µν

= 4 δ

µν

.

(4)

This result is more transparent than it rst appears. For any set of

K

unit bond vectors

whose second moment is isotropic,

tr S =

P

j

|ˆn

j

|

2

= K

distributes equally over the

d

spatial directions, so

S

µν

=

K

d

δ

µν

,

(5)

and the eigenvalue is simply the coordination-to-dimension ratio

K/d

. The value

4

is

therefore

12/3

, xed by the

K = 12

kissing number in

d = 3

; the same formula gives BCC

(

8/3

) and simple cubic (

2

). We do not claim

4

is a free-standing constant of the FCC

lattice  it is the kissing number divided by the spatial dimension, and the FCC value

follows entirely from

K = 12

. What is specic to FCC is that

K = 12

is the maximal

(Kepler-saturating) coordination in

d = 3

[4], so

4

is the largest such ratio any

3

D Bravais

lattice can realize.

Sheet A

z = 0

Sheet B

z = h

Sheet C

z = 2h

h = L

q

2

3

Figure 2: ABC stacking of

K = 6

hexagonal sheets in geometrically accurate oblique

3D. Sheet A (blue,

z = 0

), Sheet B (red,

z = h

), Sheet C (green,

z = 2h

), each shifted to

inequivalent hex sites (B at

r

min

= L/

√

3

from its 3 nearest A-nodes and 3 nearest C-nodes).

The orange atom is the central B node; colored bonds show its

K = 12

coordination: 3

blue bonds to Sheet A (length

L

, oblique), 6 red bonds to Sheet B neighbors (length

L

,

in-plane), and 3 green bonds to Sheet C (length

L

, oblique). Light grey edges connect

the 12 neighbors to each other, tracing the 24 edges of the cuboctahedron that emerges

from the stacking. Interlayer spacing

h = L

p

2/3

is the crystallographic

{111}

interplanar

distance.

3.2 The ratio

c/v

lattice

as a geometric invariant

In the long-wavelength limit (

|k|L ≪ 1

), the FCC Dirac operator reduces to

D

SSM

≈

iγ

µ

k

ν

S

µν

/τ = 4iγ · k/τ

, using

S

µν

= 4δ

µν

and

P

j

ˆn

µ

j

= 0

. The dispersion relation is

4

E(k) = 4|k|/τ

, giving phase velocity

v

ph

= E/|k| = 4/τ

. Dening the lattice bond-

traversal rate

v

lat

≡ L/τ

(the speed at which a signal propagates one bond length per

timestep), and identifying the macroscopic light speed with the long-wavelength phase

velocity of this mode,

c ≡ v

ph

(a denitional identication, not a derived equality):

c = 4

L

τ

= 4 v

lat

.

(6)

The factor

4 = K/d

is the structure-tensor eigenvalue (Eq. 5), xed by the

K = 12

kissing

number in three dimensions. This result xes only the dimensionless ratio

c/v

lat

= 4

; the

absolute value of

c

in SI units remains a measured physical input, and the identication

c = v

ph

assumes the gravitational/ electromagnetic mode is the one that propagates at the

observed light speed.

3.3 Vanishing of Lorentz-violating corrections

By the inversion symmetry of FCC, the rst-order (mass) tensor

P

j

ˆn

µ

j

= 0

. By centrosym-

metry of the cuboctahedron, the third-order tensor

P

j

ˆn

µ

j

ˆn

ν

j

ˆn

λ

j

= 0

, eliminating the leading

Lorentz-violating correction. The rst non-zero correction is fourth order, suppressed by

(kL)

4

∼ (E/E

P

)

4

, giving corrections of order

10

−112

at optical frequencies, far below the

experimental bound of

10

−16

[6].

4 The Lattice Spacing and Newton's Constant

4.1 Two independent geometric constraints on

L

The lattice spacing

L

is determined by the intersection of two independent geometric

constraints.

Constraint 1 (energy density).

Each node transmits at most one quantum per time

step

τ = 4L/c

. The maximum energy per node is

E

max

= ℏ/τ = ℏc/(4L)

. The FCC

volume per node is

V

node

= L

3

√

2/2

. The maximum lattice energy density is:

ρ

lattice

=

E

max

V

node

=

ℏc/(4L)

L

3

√

2/2

=

√

2 ℏc

4L

4

.

(7)

Requiring

ρ

lattice

≤ ρ

P

= ℏc/ℓ

4

P

gives:

L ≥

√

2

4

!

1/4

ℓ

P

≃ 0.77 ℓ

P

.

(8)

This establishes a lower bound. The energy density argument alone cannot x

L

to a

unique value.

Constraint 2 (Ryu-Takayanagi, Section 5).

The holographic derivation of

G

N

= ℓ

2

P

xes

L

exactly:

L = ℓ

P

s

4 ln 2

p

2/3

≃ 1.843 ℓ

P

.

(9)

This satises Constraint 1 (

1.843 > 0.77

) and is the unique solution consistent with New-

ton's constant.

5

4.2 Ground-state bond length from energy balance

Constraint 1 is only a one-sided bound because it contains a single term: the zero-point

energy

E

zp

= ℏc/(4L)

favours

larger

L

(lower energy density, Eq. (7)), with nothing to

oppose unbounded expansion. A stable ground state requires a restoring cost that grows

with

L

 the energy of stretching a bond against the lattice's rigidity. The only tension

scale available from the fundamental constants is the Planck (maximum) force

c

4

/G

1

, so

the bond energy is

E(L) =

ℏc

4L

|{z}

zero-point

+ γ

c

4

G

L

|{z}

tension

,

(10)

with

γ

a positive dimensionless geometric coecient. The rst term resists compression,

the second resists stretching; their competition has a unique minimum,

dE

dL

= 0 =⇒ L

⋆

=

1

2

√

γ

ℓ

P

.

(11)

For the natural choice

γ =

1

4

(the same factor that sets the zero-point term),

L

⋆

= ℓ

P

exactly

. Energy minimisation therefore independently selects a stable, Planck-scale ground-

state bond length  the spacing is not a free input but the bottom of a potential well 

and the value sits between the energy-density bound (Eq. (8),

0.77 ℓ

P

) and the holographic

value (Eq. (9),

1.843 ℓ

P

), all of order

ℓ

P

.

Why this xes the scale but not the coecient.

The mechanical balance pins the

scale

(

∼ ℓ

P

) but not the precise number, because

γ

is undetermined by mechanics alone.

Reproducing the holographic value

L = 1.843 ℓ

P

requires

γ

RT

=

1

4



ℓ

P

L



2

=

p

2/3

16 ln 2

≈ 0.074,

(12)

which is manifestly built from the Bell-pair entropy

ln 2

and the

{111}

stacking ratio

p

2/3

 entanglement and holographic data that a purely mechanical energy balance cannot ac-

cess. The coecient is therefore intrinsically holographic: energy minimisation determines

that the lattice has a Planck-scale ground state, while the Ryu-Takayanagi condition (Con-

straint 2) supplies the exact factor

1.843

. The two arguments are independent and mutually

consistent; we do not claim the balance

derives

1.843

, only that it explains why a stable

bond length of order

ℓ

P

exists at all.

4.3 Summary of FCC length scales

Scale Value Geometric origin

Bond length

L = 1.843 ℓ

P

RT

+ G

N

constraint

Interlayer spacing

h = L

p

2/3 = 1.505 ℓ

P

a

cube

/

√

3

Plaquette scale

a

plaq

= L(8/3)

1/4

= 2.355 ℓ

P

√

2Lh

Metric wall

r

min

= L/

√

3 = 1.064 ℓ

P

Circumradius of triangular face

1

The combination

c

4

/G ≈ 1.2×10

44

N is the unique tension built from

{ℏ, c, G}

; it is the maximum-force

scale of general relativity, equivalently the Planck tension

ℏc/ℓ

2

P

.

6

5 Newton's Constant from the Ryu-Takayanagi Rela-

tion

5.1 Boundary entanglement entropy

Theorem 1

(Exact entanglement entropy)

.

For a connected boundary region

A

with

perimeter

P

on the hexagonal lattice with spacing

L

, the von Neumann entanglement en-

tropy is:

S

A

=

P

L

ln 2.

(13)

Proof.

Each severed bond contributes

ln d = ln 2

by the Schmidt decomposition of a Bell

pair (Fig. 3). The bonds are independent, and the number cut is

P/L

. This construction

is in the same mathematical class as holographic quantum error-correcting codes [12].

5.2 The minimal bulk surface

The Lift operator stacks hexagonal sheets at the crystallographic

{111}

interplanar spacing:

h =

a

cube

√

3

=

L

√

2

√

3

= L

r

2

3

.

(14)

The companion simulation [3] measures this interlayer spacing directly as

0.8165±0.0011 L

,

matching the crystallographic value

p

2/3 L = 0.8165 L

to

0.01%

with no tting.

Theorem 2

(Minimal surface)

.

The minimum-area bulk surface homologous to

∂A

is the

one-layer curtain with area:

Area(γ

A

) = P × h = P L

r

2

3

.

(15)

Proof.

Any surface homologous to

∂A

must form a closed wall. Surfaces shallower than

h

fail the homology condition (inter-layer bonds at height

h

still connect the two regions).

Surfaces at depth

nh

(

n ≥ 2

) have area

≥ nP h > P h

. The straight curtain at height

h

is

the unique minimal surface.

×

×

×

×

×

×

×

A

¯

A

cut

×

×

×

×

×

A

¯

A

S

A

=

P

L

ln 2 ⇒ G

N

= ℓ

2

P

Figure 3: Ryu-Takayanagi bond cut on the

K = 6

hexagonal boundary.

Left:

Straight cut

separating region

A

(blue) from

¯

A

(grey); severed bonds (

×

, orange) each contribute

ln 2

to

S

A

.

Right:

Curved region

A

illustrating the perimeter law

S

A

= (P/L) ln 2

for any bound-

ary shape. Setting

S

A

= Area(γ

A

)/(4G

N

)

and solving gives

G

N

=

p

2/3 L

2

/(4 ln 2) = ℓ

2

P

.

7

5.3 Consistency condition on

L

given RT

The Ryu-Takayanagi relation [5]

S

A

= Area(γ

A

)/(4G

N

)

was established in AdS/CFT for

holographic quantum error-correcting codes on hyperbolic tilings [12]. Its application to

an FCC Bell-pair network is a

heuristic assumption

, not a derived result.

Assuming

RT

holds for this boundary, combining Eq. (13) and Eq. (15) gives:

P

L

ln 2 =

P L

p

2/3

4G

N

.

(16)

The perimeter

P

cancels. Solving:

G

N

=

p

2/3 L

2

4 ln 2

.

(17)

This is not a derivation of

G

N

from rst principles; it is a

consistency condition

: if RT

holds and

G

N

= ℓ

2

P

, then

L

is uniquely xed. Setting

G

N

= ℓ

2

P

:

L = ℓ

P

s

4 ln 2

p

2/3

≃ 1.843 ℓ

P

.

(18)

Proposition 1

(Consistency check)

.

Substituting

L = 1.843 ℓ

P

into Eq.

(17)

:

G

N

=

p

2/3 × (1.843)

2

/(4 ln 2) = 1.000 ℓ

2

P

exactly. This is a non-trivial check: the same

L

satises both the energy-density lower bound

(8)

and the holographic constraint

(17)

.

5.4 Physical interpretation

Newton's constant equals the interlayer spacing times the bond length, divided by four

times the bond entanglement entropy:

G

N

=

h · L

4 S

bond

=

h · L

4 ln 2

.

(19)

The factor

p

2/3 = h/L

is the exact

{111}

stacking ratio of the FCC lattice. The

ln 2

is

the entanglement entropy of one Bell pair. Gravity is the thermodynamic consequence of

the entanglement structure of the FCC bond network.

6 Emergent Lorentz Invariance

Emergent SO(3,1) Lorentz invariance of the

K = 12

FCC vacuum is established in the

published framework [2] via the structure tensor identity

S

µν

= 4δ

µν

(Eq. 4). We summarize

the argument and add one new element: the resolution of the Collins et al. naturalness

objection.

From

S

µν

= 4δ

µν

to emergent SO(3).

The FCC bond vectors (Eq. 1) satisfy two tensor

identities proved independently in [2, 3]: the rank-2 structure tensor

S

µν

= 4δ

µν

(second-

order spatial isotropy) and the rank-3 tensor

T

µνλ

=

P

j

ˆn

µ

j

ˆn

ν

j

ˆn

λ

j

= 0

(centrosymmetry,

eliminating all preferred-direction terms at third order). Together these force the long-

wavelength dispersion relation to be isotropic to all orders below

O((kL)

4

)

:

E(k) = 4|k|/τ

,

with no

O(k)

or

O(k

3

)

corrections.

8

From SO(3) to SO(3,1).

The hexagonal

K = 6

boundary sheets carry exact continuous

SO(2) rotational symmetry. Four families of

{111}

close-packed planes project this into

three independent spatial directions via the Lift operator, generating SO(3). Assuming

all low-energy Cosserat modes share a single propagation speed

c

(an input of the con-

struction, not a derived consequence  dierent modes could in principle propagate at

dierent speeds), SO(3) spatial isotropy plus time-reversal symmetry plus a single speed

uniquely determine the Lorentz-invariant dispersion form for all dimension-

≤ 4

opera-

tors [2]. Lorentz invariance therefore emerges in the long-wavelength limit, with violations

suppressed by

(kL)

4

. Residual Lorentz violation enters only at fourth order in

(kL)

, sup-

pressed by

(E/E

P

)

4

∼ 10

−112

at optical frequencies  far below the experimental bound

of

10

−16

[6].

Resolution of the Collins et al. naturalness objection.

Collins et al. [7] argued

that any Lorentz-violating UV cuto generates ne-tuning of order

Λ

2

UV

/M

2

P

via radiative

corrections. The SSM partially addresses this: the UV cuto is the 2D boundary network,

which carries

exact

continuous SO(2) symmetry. Radiative corrections in the boundary

theory respect SO(2) at every loop order, and the exact RT map transmits this isotropic

self-energy into the bulk without generating any Lorentz-violating dimension-4 operator.

Heuristically, at the level of the boundary-bulk RT map, no counterterm appears to be

generated. However, a complete argument requires an explicit calculation of the bulk

eective action from the boundary theory  which has not been carried out  so this

statement is tentative (see 10.4).

7 Schrödinger Equation as Cosserat Continuum Limit

7.1 The chiral Cosserat Lagrangian

Each lattice node has translational displacement

u

and microrotation

θ

(Cosserat elastic

degrees of freedom [11] corresponding to

S

trans

= 4

and

S

tors

= 8

channels respectively).

The Lagrangian density for a topological braid threading the lattice is:

L =

1

2

(∂

t

u)

2

+

1

2

(∂

t

θ)

2

−

c

2

2

(∇u)

2

−

c

2

2

(∇θ)

2

−

ω

2

0

2

(u

2

+ θ

2

) + Ω( ˙uθ −

˙

θu).

(20)

The chiral coupling

Ω( ˙uθ −

˙

θu)

is

conjectured

to arise from the Berry connection of the

discrete lattice wavefunction; a derivation of

Ω

from the

K = 12

bond geometry has not

been carried out (see 10.4).

7.2 Structural isomorphism with the Schrödinger equation

The Euler-Lagrange equations are:

¨u −c

2

∇

2

u + ω

2

0

u + 2Ω

˙

θ = 0,

(21)

¨

θ − c

2

∇

2

θ + ω

2

0

θ − 2Ω ˙u = 0.

(22)

Dening

ψ = u + iθ

and multiplying Eq. (22) by

i

and adding:

∂

2

ψ

∂t

2

+ 2iΩ

∂ψ

∂t

− c

2

∇

2

ψ + ω

2

0

ψ = 0.

(23)

The complex unit

i

is the geometric operator mapping translational to rotational modes

of the Cosserat lattice, not an axiom of quantum mechanics. Substituting

ψ = Φ e

−iΩt

9

(Larmor frame) and taking the non-relativistic limit yields:

iℏ

∂Φ

∂t

= −

ℏ

2

2m

∇

2

Φ.

(24)

On the role of

ℏ

.

The Cosserat framework is a classical continuum theory: it does not

intrinsically contain action quantization. The factor

ℏ

enters as follows. Each FCC bond

transmits at most one quantum per time step

τ = 4L/c

(from

c = 4v

lat

= 4L/τ

, Section 3).

The natural energy quantum of the lattice is therefore

ε = ℏ/τ = ℏc/(4L)

, setting the

Planck-scale bond energy scale. The Larmor substitution

ψ = Φ e

−iΩt

reduces the wave

equation (23) to Schrödinger form when

Ω = mc

2

/ℏ

, i.e., when the rest energy

mc

2

equals

ℏΩ ∼ ℏc/L

 the bond energy scale. This identication is consistent with the decoherence

threshold

m

soft

= ℏ/(Lc)

(Section 8), where the same scale reappears as the onset of lattice

corrections. The content of this section is therefore a

structural isomorphism

: the long-

wavelength continuum limit of the

K = 12

Cosserat lattice is exactly isomorphic to the

free-particle Schrödinger equation. This is a non-trivial result  once the chiral coupling is

present, its

form

(a translationrotation cross term) is the generic Cosserat invariant and

the value

Ω = mc

2

/ℏ

and the Larmor substitution are then xed by the reduction rather

than inserted by hand  but it is an isomorphism, not an

ab initio

derivation of

ℏ

from

geometry alone, and the microscopic origin of

Ω

(its sign and magnitude from the

K = 12

Berry connection) remains conjectural (10.4).

8 The Two-Step Decoherence Threshold

Two mass scales emerge from the

K = 12

cuboctahedral unit cell. Both are xed by

L = 1.843 ℓ

P

(Section 5) and the exact geometry of the cuboctahedral faces, with no free

parameters. We dene them purely geometrically:

m

soft

= ℏ/(Lc)

is the mass at which

the reduced Compton length

λ

c

= ℏ/(mc)

equals

L

, and

m

hard

= ℏ

√

3/(Lc)

is the mass

at which

λ

c

equals the triangular face circumradius

L/

√

3

. The reduced Compton length

λ

c

is the scale associated with the rest energy

mc

2

= ℏc/λ

c

; it is

not

the spatial extent of

the center-of-mass wavefunction in any experiment, which is set by trap frequencies and is

many orders of magnitude larger.

Both thresholds are of order the Planck mass. Since

m

soft

= ℏ/(Lc) = m

P

/(L/ℓ

P

) =

m

P

/1.843

and

m

hard

=

√

3 m

soft

= 0.940 m

P

, with

m

P

=

p

ℏc/G ≈ 21.8 µ

g, the scale itself

is

not

a novel prediction: the Diósi-Penrose gravitational-collapse scale is likewise

O(m

P

)

.

What is specic to the SSM, and what makes the prediction falsiable, is the

structure

of the boundary  two thresholds rather than one, separated by the exact, parameter-

free ratio

m

hard

/m

soft

=

√

3

(the edge-to-circumradius ratio of the triangular face), with a

concave-up prole in between  a quadratic onset

Γ ∝ (m −m

soft

)

2

at

m

soft

that steepens

to

Γ ∝ [(m/m

soft

)

2

− 1]

2

approaching

m

hard

. The remainder of this section develops this

structure and contrasts it with the single smooth threshold of Diósi-Penrose.

Physical interpretation (conjectured).

The SSM proposes that these two geometric

thresholds correspond to onsets of decoherence for a center-of-mass superposition of mass

m

. The reasoning is as follows: the Cosserat lattice dispersion (Eq. (26)) carries a cor-

rection

(kL)

2

. When this correction is evaluated at the wavenumber

k

c

= mc/ℏ

 the

Compton scale, which is the natural scale at which a mass-

m

perturbation resolves the

lattice in the gravitational sector  it becomes

(m/m

soft

)

2

. The conjecture is that as this

ratio crosses unity, the lattice can no longer maintain coherent two-branch evolution of the

10

center-of-mass superposition. We emphasize that

this is a conjecture, not a derivation

: a

rst-principles calculation would require an explicit vacuum phonon bath Hamiltonian, a

microscopic coupling to the center-of-mass degree of freedom, and a proper density of nal

states  none of which are constructed in this paper. We treat Eq. (30) as a phenomeno-

logical rate law motivated by the lattice geometry.

The central physical assumption.

The weakest step in this chain is the choice of length

scale, and we state it plainly. The thresholds are built from the

reduced Compton wavelength

λ

c

= ℏ/(mc)

, an internal rest-energy scale, whereas laboratory decoherence of a spatial

superposition is governed by the

delocalization

∆x

of the center-of-mass wavefunction,

which is set by trap geometry and is orders of magnitude larger than

λ

c

for any realizable

mass. The SSM posits that the lattice couples to the rest-energy scale of the superposed

object, not to its delocalization  that the mass-

m

excitation resolves the lattice through

λ

c

in the gravitational sector, independent of how far apart the two branches sit. This is

a physical assumption, not a theorem; it is the price of a parameter-free prediction, and it

is what an experiment in the

11.8



20.5 µ

g window would test. We do not derive it from

a microscopic system-bath model here, and the falsiable content (the

√

3

ratio and the

two-step prole) stands or falls with it.

8.1 The metric wall from cuboctahedral face geometry

The

K = 12

unit cell is a cuboctahedron with 8 equilateral triangular faces of edge length

L

, established in Section 2. Every triangular face has a circumscribed circle of radius

r

min

=

L

√

3

,

(25)

the distance from the face center to each of its three vertices. This is the circumradius

formula for an equilateral triangle with side

a

:

R = a/

√

3

, applied with

a = L

. (Note:

the insphere radius of a regular tetrahedron with edge

L

is

L/(2

√

6) ≈ 0.204L

, which is

dierent; the relevant scale here is the circumradius of the triangular

face

, not a void.)

The physical meaning is direct. A probe at the centroid of a triangular face with Compton

wavelength

λ

c

= ℏ/(mc)

shrinking to

r

min

simultaneously resolves all three face vertices at

exactly bond-length separation. Further compression would require

λ

c

< L/

√

3

: the probe

would have to t between nodes separated by

L

, which is the minimum bond length of the

network. The lattice topology forbids it  this is the metric wall.

This length scale is not only analytic. The companion simulation [3] sweeps the exclusion

radius

R

ex

of the growing network and nds a sharp geometric phase transition at

R

ex

=

L/

√

3 ≃ 0.577 L

, with a wide

K = 12

stability plateau over

R

ex

∈ [0.58, 0.99] L

. Below

L/

√

3

the exclusion is too weak and unphysical overlaps (

K

max

> 12

) appear; the transition

therefore conrms, numerically, that

L/

√

3

is the operative minimum-approach distance

and not a tuned parameter. The two thresholds derived in Sections 8.28.3 inherit this

scale directly.

8.2 Soft threshold: onset of lattice corrections

Evaluating the Cosserat dispersion correction (Eq. (26)) at the Compton wavenumber

k

c

= mc/ℏ

(the scale at which rest energy equals

ℏc/λ

c

):

E(k) =

ℏ

2

k

2

2m



1 +

(kL)

2

6

+ O



(kL)

4





.

(26)

11

4 8

m

s

m

h

24

0

0.5

1

16.2

I II III

Bild 16.2

µ

g

Γ = 0

Γ∝ [(m/m

s

)

2

−1]

2

Mass

m

(

µ

g)

Γ/Γ

max

SSM decoherence rate

4 8

m

s

m

h

24

0

0.5

1

16.2

distinguishable

at

∼

1

µ

g resolution

Γ/Γ

max

SSM vs. Diósi-Penrose

Diósi-Penrose

SSM (this work)

Figure 4:

Left:

SSM two-step decoherence rate

Γ(m)

(normalized to

Γ

max

).

Regime I

(green,

m < m

soft

= 11.8 µ

g):

Γ = 0

, exact quantum mechanics.

Regime II

(yellow,

m

soft

< m < m

hard

= 20.5 µ

g): the rate rises as

Γ ∝ [(m/m

soft

)

2

−1]

2

 a quadratic onset

just above

m

soft

that steepens (concave up) toward

m

hard

.

Regime III

(pink,

m > m

hard

):

topological cuto, coherence impossible. Purple dot: Bild et al. 16.2

µ

g resonator, at only

≈20%

of the terminal rate (coherence survives).

Right:

SSM (solid/thick) versus Diósi-

Penrose (red dashed). The two models are distinguishable at

∼ 1 µ

g resolution in the

1025

µ

g window: DP predicts a single smooth threshold while SSM predicts a hard cuto

at

m

hard

=

√

3 m

soft

.

For a center-of-mass state of mass

m

, the relevant wavevector is the Compton scale

k

c

=

mc/ℏ

, giving correction amplitude

(k

c

L)

2

= (m/m

soft

)

2

, where the soft threshold is set by

(k

c

L)

2

= 1

:

m

soft

=

ℏ

Lc

=

ℏ

1.843 ℓ

P

c

≈ 11.8 µ

g

.

(27)

Below

m

soft

the correction is

(m/m

soft

)

2

≪ 1

: the Schrödinger equation holds to all practical

precision. Above it, the lattice is no longer transparent.

8.3 Hard threshold: circumradius cuto

The metric wall at

r

min

= L/

√

3

gives the hard mass scale: the Compton wavelength that

ts exactly inside the circumscribed circle of a triangular face:

λ

c

=

L

√

3

=⇒ m

hard

=

√

3 ℏ

Lc

=

√

3 m

soft

≈ 20.5 µ

g

.

(28)

The ratio

m

hard

/m

soft

=

√

3

is exact. It comes from the edge-to-circumradius ratio of an

equilateral triangle (

L : L/

√

3 =

√

3

) and is independent of

ℏ

,

c

, or

G

. At

m

hard

, the

wavepacket spans the circumscribed circle of the face exactly; any further compression

would require

λ

c

< L/

√

3

, placing it within the triangular face while all three vertices are

closer than the bond length. The SSM conjectures that no lattice conguration can sustain

the superposition past this point; this is the hard decoherence cuto. Within the window

the rate approaches its terminal value

4 Γ

0

continuously (Eq. (30)); the cuto marks not a

divergence of that rate but the point at which the perturbative expansion in

(kL)

breaks

down (

δ =

1

2

) and the geometry can no longer contain the superposition. A microscopic

operator-level derivation of this breakdown is deferred to future work.

12

8.4 Decoherence rate from dispersion correction

Treating the dispersion correction as a coupling to a lattice phonon bath (whose modes,

coupling Hamiltonian, and density of states are not derived microscopically here), and

modeling the resulting dephasing phenomenologically via Fermi's golden rule:

Γ =

2π

ℏ

|⟨f|δ

ˆ

H |i⟩|

2

ρ(E

f

),

(29)

Here

δ

ˆ

H

denotes the lattice correction to the kinetic operator and

ρ(E

f

)

the density of nal

bath states. The correction amplitude is

δH/H

0

= (m/m

soft

)

2

/6

, so the matrix element

gives

|⟨f|δ

ˆ

H|i⟩|

2

∝ [(m/m

soft

)

2

− 1]

2

across the whole window. With

ρ(E

f

) ∝ m

slowly

varying:

Γ

lattice

(m) =



























0 m ≤ m

soft

,

Γ

0

"



m

m

soft



2

− 1

#

2

m

soft

< m < m

hard

,

coherence not sustained

m ≥ m

hard

,

(30)

where

Γ

0

= ℏ/(mL

2

)

. The prole is not linear and does not taper o near the upper

threshold; it is

concave up

. Just above

m

soft

it reduces to the quadratic onset

Γ ≈ 4Γ

0

ε

2

,

ε = (m − m

soft

)/m

soft

, but it then steepens to a quartic dependence on mass and rises

to its terminal value

Γ

0

[(m

hard

/m

soft

)

2

− 1]

2

= 4 Γ

0

at

m

hard

, where the geometric cuto

(Section 8.1) ends the window. Equivalently, the deterministic deformation

δ(m)

(Eq. (31))

grows quadratically in mass (

1

6

→

1

2

) and the rate grows as the square of its excess over the

soft-threshold value. Concretely: at the window midpoint in mass the rate has reached only

∼20%

of its terminal value (the Bild point,

m/m

soft

≈ 1.37

, sits at

≈ 20%

), then climbs to

39%

,

61%

, and

89%

at

m/m

soft

= 1.5, 1.6, 1.7

before the cuto. Coherence therefore stays

near-perfect through most of the window and collapses increasingly rapidly as

m → m

hard

 the opposite of a uniform linear decay. The

normalized

prole (the percentages above)

is xed by the matrix element

[(m/m

soft

)

2

− 1]

2

and is therefore independent of the bath

details; only the absolute scale

Γ

0

depends on the (here undetermined) lattice coupling,

so the falsiable predictions are the shape and the two threshold masses, not an absolute

coherence time.

What happens between the thresholds.

The leading lattice term in Eq. (26) is

Hermitian, so its rst eect is

unitary

: it modies the dispersion of the center-of-mass

mode and therefore deforms the wavepacket deterministically rather than collapsing it.

The size of the deformation is the dimensionless correction to the kinetic term at the

Compton scale,

δ(m) =

1

6



m

m

soft



2

,

(31)

which grows from

δ = 1/6 ≈ 17%

at

m = m

soft

, through

δ ≈ 31%

at the Bild mass

16.2 µ

g,

to exactly

δ = 1/2

at

m = m

hard

=

√

3 m

soft

. At

m

hard

the lattice correction reaches half

the bare kinetic term  the Compton wavelength has shrunk to the metric wall

L/

√

3

,

the perturbative expansion in

(kL)

ceases to hold, and the geometry can no longer contain

the superposition. Physically, the SSM therefore predicts a sequence, not a single event

(Fig. 4): below

m

soft

the wavepacket evolves under exact quantum mechanics; between the

thresholds it suers a mass-dependent, deterministic, and in-principle reversible dispersive

deformation of size

δ(m)

; only at

m

hard

does coherence become geometrically impossible.

13

Whether the deformation also seeds genuine (irreversible) decoherence depends on coupling

to lattice modes  the dephasing rate Eq. (30) describes that channel

if

the coupling exists,

but the deterministic deformation

δ(m)

is the robust, bath-independent prediction. The

deformation is quoted at the Compton scale; translating it into an observable distortion of

the interference pattern requires the Compton-coupling assumption stated above.

Consistency with Bild et al.

The 16.2

µ

g resonator [9] sits between the thresholds. The

SSM predicts a deterministic dispersive deformation of size

δ =

1

6

(16.2/11.8)

2

≈ 31%

at

the Compton scale (Eq. (31)), but the topological cuto at

m

hard

has not been reached,

so the superposition remains coherent  consistent with the observation of cat states at

this mass. This is not a post-hoc t:

m

soft

and

m

hard

follow from

L

, xed independently

in Section 5.

Distinction from Diósi-Penrose.

The dierence is qualitative, not just a matter of

one threshold versus two. In the DP model [8] the gravitational self-energy

E

∆

of the

mass dierence between the two branches drives a

stochastic, irreversible

collapse at rate

τ

−1

= E

∆

/ℏ

; the eect depends on the branch separation

∆x

and the mass distribution,

sets in smoothly for all masses, and carries the free length parameter

R

0

. The SSM predicts

instead a

deterministic

dispersive deformation

δ(m)

(Eq. (31)) that depends only on the

object's mass

m

through

λ

c

 not on

∆x

 is exactly zero below

m

soft

, and terminates

at a hard geometric cuto at

√

3 m

soft

, with no free parameters. A single experiment

discriminates the two: hold

∆x

xed and scan the mass. DP predicts a smooth,

∆x

-

dependent loss of fringe visibility with no special mass; the SSM predicts a at null below

11.8 µ

g, a deterministic onset

∝ (m − m

soft

)

2

that is independent of

∆x

, and a sharp

cuto at

20.5 µ

g. The two are distinguishable with

∼1 µ

g mass resolution in the

10



25 µ

g

window.

9 Comprehensive Experimental Comparison

Table 1 collects all foundational predictions of this paper against the best available mea-

surements. Every quantity is derived from a single input  the

K = 12

FCC lattice with

Bell-pair bonds  via the chain in Section 10.2.

Three entries warrant comment. The ratio

m

hard

/m

soft

=

√

3

is the sharpest near-term

prediction: it requires only two experiments bracketing the 11.820.5

µ

g window, calibrated

against each other, with no reference to any Standard Model parameter. Current acoustic-

resonator technology is within a factor of two of this range [9]; MAQRO [10] could reach it

within the decade. The Lorentz-violation bound is already satised at

< 10

−16

, consistent

with exact SO(2) boundary symmetry: the RT map suppresses all Lorentz-violating bulk

operators to

(E/M

P

)

4

, some 112 orders below current limits. The Schrödinger entry is

unusual: the measured column records what quantum mechanics assumes as a postulate,

while the SSM column records a structural isomorphism  the

K = 12

Cosserat continuum

limit is exactly isomorphic to the free-particle Schrödinger equation, with the imaginary

unit

i

arising geometrically rather than being assumed.

14

Observable SSM derivation SSM value Measured

Foundational (this paper)

c/v

lat

S

µν

= 4δ

µν

,

K = 12 4

(exact)

4

G

N

RT on hex. boundary

ℓ

2

P

ℓ

2

P

L

RT

+ G

N

= ℓ

2

P

1.843 ℓ

P



Lorentz violation

(E/E

P

)

4

suppressed

∼(E/E

P

)

4

< 10

−16

[6]

m

soft

ℏ/(Lc)

,

L = 1.843 ℓ

P

11.8 µ

g consistent [9]

m

hard

√

3 m

soft

20.5 µ

g

> 16.2 µ

g tested [9]

m

hard

/m

soft

circumradius/edge of

△

√

3

(exact) untested

Schrödinger eq. Cosserat structural

isomorphism

isomorphic postulated

Table 1: Foundational SSM predictions versus experiment. All quantities derive without

free parameters from the single structural assumption

K = 12

FCC with Bell-pair bonds.

A dash under Measured indicates a Planck-scale quantity not directly accessible.

10 Discussion

10.1 Why FCC and not another lattice?

The FCC lattice is selected by three requirements: (a) it saturates the Kepler bound (

K =

12

) [4]; (b) its four

{111}

families generate emergent SO(3), which no other 3D Bravais

lattice achieves; and (c) unlike the simple-cubic and BCC lattices, it is non-bipartite, which

modies where fermion doublers appear in the Brillouin zone compared to standard Wilson-

fermion constructions. Whether the SSM Dirac operator (6) fully evades the Nielsen-

Ninomiya theorem [13]  which applies to all translation-invariant local Dirac operators,

regardless of whether the lattice is bipartite  has not been demonstrated and is listed as

an open problem in 10.4.

10.2 Derivation chain summary

The logical dependencies are:

1.

Input:

K = 12

FCC lattice with Bell-pair bonds.

2.

c/v

lat

= 4

(Section 3): the ratio of macroscopic wave speed to lattice update rate

equals the structure tensor eigenvalue

S

µν

= 4δ

µν

, exact from

K = 12

.

3.

G

N

= ℓ

2

P

,

L = 1.843 ℓ

P

(Sections 45): the RT holographic equation xes

L

uniquely;

the energy-density bound (

L > 0.77 ℓ

P

) and an energy-balance minimum (

L

⋆

∼ ℓ

P

,

Section 4.2) independently conrm the Planck scale, while RT supplies the exact

coecient.

4. Emergent SO(3,1) (Section 6): isotropic dispersion below

O((kL)

4

)

, established in [2];

reviewed here. A single propagation speed for all modes is assumed, not derived.

5. Schrödinger equation (Section 7): from Cosserat Lagrangian.

15

6.

m

soft

= 11.8 µ

g,

m

hard

= 20.5 µ

g (Section 8): from

L

and the metric wall

r

min

= L/

√

3

.

10.3 Comparison with other discrete spacetime approaches

Feature SSM LQG Causal Sets Lattice QCD

c/v

lat

ratio? Yes (

= 4

, exact) No No No

G

N

derived? Yes (RT) No No No

ℓ

P

derived? Yes Assumed No

a → 0

Emergent Lorentz?

(E/E

P

)

4

at

ℓ

P

Debated Statistical

(pa)

2

at tunable

a

Decoherence? Two-step,

√

3

Various Not derived N/A

Free parameters 0 Several 1 (

ρ

) Several

10.4 What remains open

The following foundational questions are left to future work and represent limitations of

the present paper.

Vacuum Hamiltonian.

The Bell-pair bond structure is a static structural input. No

Hamiltonian is given from which

|Φ

+

⟩

on every bond emerges as a ground state; stabilizing

this vacuum dynamically is an open problem.

Chiral coupling

Ω

in the Cosserat Lagrangian.

The term

Ω( ˙uθ −

˙

θu)

in Eq. (20) is

asserted to arise from the Berry connection of the lattice wavefunction but is not derived.

A derivation of the precise value of

Ω

from the

K = 12

bond geometry has not been carried

out.

Phase-transition dynamics.

The

K=6 → K=4 → K=12

sequence in 2 is now sup-

ported numerically: the companion simulation [3] exhibits the transition kinematically

and conrms

K = 12

bulk saturation under nite-size scaling. What remains open is a

closed-form free energy  an action or partition function from which the transition follows

analytically  rather than the existence of the transition itself.

Decoherence mechanism.

The rate law (Eq. (30)) and the hard cuto at

m

hard

are

phenomenological conjectures. A microscopic derivation requires an explicit bath Hamil-

tonian, conjugate coupling operator, density of nal states, and a demonstration that

coherence-protecting symmetry is broken at the topological cuto.

Nielsen-Ninomiya evasion.

The Nielsen-Ninomiya theorem [13] states that any local,

translation-invariant, Hermitian lattice Dirac operator has equal numbers of left- and right-

handed modes, regardless of whether the lattice is bipartite. The SSM is non-bipartite,

which modies the locations of fermion doublers in the Brillouin zone, but a proof that the

tensor-network Dirac operator derived in Section 3 satises the conditions of the theorem

 or that those conditions are not met  has not been given.

Single propagation speed.

The step from emergent SO(3) to emergent SO(3,1) assumes

all low-energy Cosserat modes share a single speed

c

. A proof that the

K = 12

lattice

enforces this (rather than admitting dierent group velocities for dierent modes) has not

been given.

Decoherence thresholds.

The ratio

m

hard

/m

soft

=

√

3

is the cleanest falsiable predic-

tion; it depends on no Standard Model input. The Standard Model gauge group, quark

masses, and cosmological parameters are addressed in separate papers.

16

10.5 Falsiability



Planck-suppressed Lorentz violation:

the model predicts violations at order

(E/E

P

)

4

∼ 10

−112

at optical frequencies, far below current bounds. Any detection of

birefringence or time-of-ight anomalies at energies well below

E

P

would falsify the

model.



Two-step structure

separated by exactly

√

3

, with an exactly null response below

m

soft

= 11.8 µ

g and a hard cuto at

m

hard

= 20.5 µ

g  not the single smooth

threshold of Diósi-Penrose.



Deterministic deformation

between the thresholds: a reversible dispersive distor-

tion

δ(m) =

1

6

(m/m

soft

)

2

that depends on the mass alone and not on the branch sep-

aration

∆x

; a mass scan at xed

∆x

separates this from the

∆x

-dependent stochastic

collapse of Diósi-Penrose.



G

N

from entanglement:

precision short-distance measurements of Newton's con-

stant provide an independent test.

11 Conclusion

From the single structural choice

K = 12

FCC with Bell-pair bonds, four foundational

results follow without free parameters:

1.

c/v

lattice

= 4

from the cuboctahedral structure tensor

S

µν

= 4δ

µν

: the macroscopic

speed of light is exactly four times the lattice update rate.

2.

G

N

=

p

2/3 L

2

/(4 ln 2) = ℓ

2

P

and

L = 1.843 ℓ

P

from the Ryu-Takayanagi relation on

the hexagonal boundary.

3. Two-step decoherence at

m

soft

= 11.8 µ

g and

m

hard

= 20.5 µ

g with ratio exactly

√

3

,

derived from the circumradius of the cuboctahedral triangular face.

4. A structural isomorphism between the long-wavelength

K = 12

Cosserat continuum

and the Schrödinger equation:

i

arises from the translation-rotation coupling of the

lattice microstructure, not as an axiomatic postulate.

Emergent Lorentz invariance (Planck-suppressed violations at order

(E/E

P

)

4

), established

in the published framework [2], is reviewed and the Collins et al. naturalness objection

is partially addressed; a complete proof requires explicit bulk eective-action calculation

from the boundary theory.

The comparison in Table 1 shows the framework is consistent with all available experimental

data; its central prediction  the two-step threshold with ratio

√

3

 remains untested

and is the natural target for the next generation of macroscopic-superposition experiments.

Author Contributions

Raghu Kulkarni:

Conceptualization; Methodology; Formal analysis; Investigation; Writ-

ing  original draft; Writing  review & editing; Visualization; Funding acquisition.

17

Data Availability

No datasets were generated or analyzed during this study. All results follow analytically

from the geometric framework described in the text.

Declaration of Competing Interests

The author declares no competing interests.

Funding

This work was supported internally by IDrive Inc. No external funding was received.

18

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19