The Mathematical City · Mini AI Lab
SCALAR
Algebraic Quantum Scalar Fields
What scalar field survives the algebra?
V₂⊗³ → C → K → E₄₇ → P₄₇ → φ
Carrier
125
V₂ ⊗ ³
Invariant
47
E₆ ⊕ E₃₀
Ωc
0.376
dim(E47) / dim(V)
Q[ψ]
0.29089
kernel occupancy
Transverse
0.70911
→ 0 under Γε
‖Kψ‖
177.4868
terminal criterion
Field view
Seed
Contraction
Finite operator: K = (C − 6I)(C − 30I). Contraction: Γε = I − εK². Field convention: L = −Δ, so the terminal field obeys (L − 6)(L − 30)φ = 0.
The plane-wave lift is an explicit visualization map. The finite 125→47 spectral theorem is the audited core; continuum interpretation is a separate layer. Step 0 · Full algebraic scalar field.
Full algebraic scalar field
contraction step 0
Casimir shells
Population by total-spin eigenvalue. Cyan shells are the E47 kernel.
0
d=1
2
d=9
6
d=25
12
d=28
20
d=27
30
d=22
42
d=13
E47Transverse
Scalar potential
Octic well centered at Ωc = 47/125.
f f′ f″