THE ENGINE
Derivation Map
Postulate
S
1
= ∂(Mobius) ↪ S
3
, ∂S
3
= ∅
Boundary Condition
ψ(y + L) = -ψ(y)
Half-integer modes: ν = m + ½
Ground state m = 0:
ψ
0
= sin(πΘ)
120 Domain
|2I| = 120
Native to S
3
≅ SU(2)
Domain: 120 positions
Hurwitz →
Fibonacci wells
{13, 21, 34, 55, 60}/120
Bounded Domain
ℓ
P
→ R
H
Spans 10
122
IR↔UV: x ↦ Ω/x
Fixed point: x = √Ω
√Ω ≈ 10
61
The observer
Bosonic Filter → Phase Operator
C(Θ) = 2 sin
2
(πΘ)
|ψ|
2
projects 2I → I
60R-grid (even only)
Position on the wave
Embedding Hierarchy
S
1
⊂ Mobius ⊂ S
3
Volume dilution: V
n
~ (√Ω)
n
Squared amplitude: |ψ|
2
~ (√Ω)
-n
How far from Planck, on which manifold
Selection Rules
n
Manifold
Scale
Observables
1
Edge S
1
Ω
H
H
0
, a
0
2
Surface
Ω
Λ
Λ
3
Space S
3
Ω
Λ
Null "dark matter"
3/2
Interface
Gravity (Gauss-Codazzi)
If it evolves with epoch → Ω
H
. If it is set by the surface → Ω
Λ
.
The Scaling Law
A / A
P
≈ C(Θ) · (√Ω)
-n
The event occurs at (t, Θ)
a
0
Θ = 13/120 · n = 1 · Ω
H
C = 0.22
→ 2.2 × 10
-62
a
P
H
0
Θ = 34/120 · n = 1 · Ω
H
C = 1.21
→ 1.2 × 10
-61
t
P
-1
Λ
Θ = 60/120 · n = 2 · Ω
Λ
C = 2.00
→ 3.0 × 10
-122
ℓ
P
-2
α
Θ = 13/60 · n = 1/30 · Ω
Λ
C = 0.79
→ 0.00733
Phase Field
Θ
f
=
2
/
120
· 𝟙(𝒯 ≥ 𝒯
c
)
L
f
= v
c
2
/a
0
≈ 13 kpc
H
0
shifts 8.4%
67.4 × 1.084 ≈ 73 km/s/Mpc
Gauss-Codazzi
Λ
obs
=
3
/
2
Λ
top
R
Σ
= Λ
top
(intrinsic surface curvature)
R
spatial
= 3R
Σ
= 2Λ
3D embedding of 2D surface
Topology holds.
Wave is.
Particle samples.