FluctLight Equation: basically all geometric fields.

By Ethan G Appleby

The FluctLight Equations

1. Matrix-Valued Helmholtz Equation

The multi-component wavefield is

ψ(x,t) ∈ ℂᴺ

and obeys

∇²ψ + Q(x,t)² ψ = 0

where Q(x,t) ∈ ℂᴺˣᴺ is the local matrix wavenumber field.

If Λ(x,t) is the matrix wavelength field, define

Q(x,t) = 2π Λ(x,t)⁻¹

so that

K(x,t) = Q(x,t)²

or, for a positive semidefinite stiffness operator,

K(x,t) = Q(x,t)† Q(x,t)

The Hermitian part of Q controls conservative phase propagation, the anti-Hermitian part controls absorption or gain, and non-commuting components encode gyrotropic, Berry, or topological coupling.

2. Covariant First-Order Form

The first-order matrix wave equation is

i Γ̃ᵘ Dᵤ ψ − M(x,t) ψ = 0

where

Dᵤ = ∂ᵤ + Aᵤ

is the covariant derivative, Aᵤ is the internal gauge/connection field, Γ̃ᵘ are generalized Dirac matrices encoding anisotropy, and M(x,t) is the matrix mass/absorption potential.

The second-order reduction has the form

Dᵤ(Λᵘᵛ(x,t) Dᵥ ψ) + K(x,t) ψ + curvature terms = 0

with curvature

Fᵤᵥ = ∂ᵤAᵥ − ∂ᵥAᵤ + [Aᵤ, Aᵥ]

The curvature terms represent the non-abelian geometric/topological coupling.

3. Non-Abelian Wilson Transport

For a ray γ, the internal phase transport is

U_γ = 𝒫 exp(−i ∫_γ Qᵤ(x,t) dxᵘ)

where 𝒫 denotes path ordering.

This is necessary because, in general,

[Qᵤ(x), Qᵥ(y)] ≠ 0

so the simple exponential of an integral is not valid unless the matrices commute.

4. Matrix Hyperoctant Kernel

The spatial hyperoctant routing remains abelian, while the internal transport remains matrix-valued.

For commuting Cartesian spatial axes, the scalar hyperoctant expansion is

∏ᵢ₌₁ᵈ [cos((k/√d)xᵢ) + i nᵢ sin((k/√d)xᵢ)]

∑_{σ ∈ {±1}ᵈ} P(σ|n) exp(i(k/√d) σ·x)

with

P(σ|n) = ∏ᵢ₌₁ᵈ (1 + σᵢ nᵢ)/2

The non-abelian internal part is carried by the Wilson transport U_γ or by matrix amplitudes.

Thus the total structure is:

spatial abelian routing × internal non-abelian transport

5. Forward Measurement Channels

The raw hyperoctant measurements are matrix-valued:

Yσ(k) = ∫{𝕊ᵈ⁻¹} U(n;k) A(n;k) P(σ|n) dμ(n) + η_σ

where

A(n;k) ∈ ℂᴺˣᴺ is the directional matrix amplitude,

U(n;k) = 𝒫 exp(−i ∫_{ray(n)} Qᵤ dxᵘ)

is the non-abelian transport operator,

P(σ|n) is the scalar hyperoctant routing probability,

η_σ is matrix-valued noise.

The multiplication order must be fixed. Common choices are

U A, A U, or U A U†

depending on whether transport acts on the left, right, or by conjugation.

6. Walsh-Hadamard Decoupling

Because the spatial parity group (ℤ₂)ᵈ is abelian, the Walsh characters remain scalar:

χS(σ) = ∏{i∈S} σᵢ

The Walsh moment channels are

CS(k) = ∑{σ ∈ {±1}ᵈ} χ_S(σ) Y_σ(k)

Each C_S(k) ∈ ℂᴺˣᴺ.

Using the identity

∑{σ ∈ {±1}ᵈ} χ_S(σ) P(σ|n) = ∏{i∈S} nᵢ

we obtain

C_S(k) = ∫{𝕊ᵈ⁻¹} U(n;k) A(n;k) ∏{i∈S} nᵢ dμ(n) + η_S

Thus Walsh-Hadamard decoupling extracts matrix-valued angular moments while preserving the internal non-abelian structure.

7. Matrix-Valued Denoising

For each Walsh moment matrix, take the SVD:

C_S(k) = U_S Σ_S V_S†

with singular values

Σ_S = diag(sᵣ)

Apply singular-value shrinkage:

Γ_S(k) = diag(sᵣ² / (sᵣ² + λ_S(k)))

Then the filtered moment is

C̃_S(k) = U_S Γ_S(k) Σ_S V_S†

In the scalar-norm limit,

C̃_S(k) = ( ‖C_S(k)‖_F² / (‖C_S(k)‖_F² + λ_S(k)) ) C_S(k)

8. Inverse Walsh-Hadamard Synthesis

The filtered hyperoctant channels are recovered by

B̃σ(k) = 2⁻ᵈ ∑{S ⊆ {1,…,d}} χ_S(σ) C̃_S(k)

This reconstructs denoised matrix-valued hyperoctant data.

9. Matrix Geometry Transfer Function

The sensor aperture 𝓚 ⊂ ℝᵈ has matrix-valued Fourier signature

H_𝓚(ξ) = ∫_𝓚 exp(−i ξ·x) W(x) dx

where W(x) ∈ ℂᴺˣᴺ is the local sensor weight.

The regularized matrix pseudoinverse is

H_𝓚⁺(ξ) = (H_𝓚(ξ)† H_𝓚(ξ) + λI)⁻¹ H_𝓚(ξ)†

or, for the opposite operator ordering,

H_𝓚⁺(ξ) = H_𝓚(ξ)† (H_𝓚(ξ) H_𝓚(ξ)† + λI)⁻¹

The correct side depends on whether the aperture operator acts on the left or right.

10. Matrix Reconstruction

The reconstructed matrix field is

ρ_final(x) = 𝟙𝓚(x) Re[ ∑k ∑{σ′ ∈ {±1}ᵈ} Y{σ′}(k) ∑_{σ ∈ {±1}ᵈ} exp(i(k/√d) σ·x) H_𝓚⁺((k/√d)σ) G(σ ⊙ σ′;k) ]

where

G(σ ⊙ σ′;k) = 2⁻ᵈ ∑_{S ⊆ {1,…,d}} χ_S(σ ⊙ σ′) Γ_S(k)

If a scalar image is required, apply a readout trace:

ρ_scalar(x) = 𝟙_𝓚(x) Re Tr[R† ρ_final(x)]

where R is a detector/readout matrix.

11. Decoupled Walsh-Space Reconstruction

Equivalently, reconstruction may be performed directly in Walsh space:

ρ_final(x) = 𝟙_𝓚(x) Re[ ∑k H_𝓚⁺(k) ∑{S ⊆ {1,…,d}} Φ_S(x;k) C̃_S(k) ]

with parity basis

Φ_S(x;k) = i^|S| ∏{j∈S} sin((k/√d)xⱼ) ∏{j∉S} cos((k/√d)xⱼ)

This form separates spatial parity modes from internal matrix dynamics.

12. Algebraic Closure

In the flat, factorized case, the symmetry algebra is

𝔤 = 𝔱ᵈ ⊕ 𝔤_internal

corresponding to

spatial translations ⊕ internal symmetry

In the fully non-abelian spatially varying case, the covariant derivatives obey

[Dᵢ, Dⱼ] = Fᵢⱼ

so the closure is more generally

𝔤 ≈ 𝔱ᵈ ⋉ 𝔤_internal

with curvature obstruction

Fᵢⱼ = ∂ᵢAⱼ − ∂ⱼAᵢ + [Aᵢ, Aⱼ]

The direct product is recovered only when the internal connection is flat or commutes with spatial transport.

Final Compact Statement

The corrected framework is:

Dᵤ = ∂ᵤ + Aᵤ

U_γ = 𝒫 exp(−i ∫_γ Qᵤ dxᵘ)

Dᵤ(Λᵘᵛ Dᵥ ψ) + Kψ = 0

Yσ(k) = ∫{𝕊ᵈ⁻¹} U(n;k) A(n;k) P(σ|n) dμ(n) + η_σ

C_S(k) = ∑_σ χ_S(σ) Y_σ(k)

C̃_S(k) = U_S Γ_S Σ_S V_S†

B̃_σ(k) = 2⁻ᵈ ∑_S χ_S(σ) C̃_S(k)

ρ_final(x) = 𝟙_𝓚(x) Re[ ∑_k H_𝓚⁺(k) ∑_S Φ_S(x;k) C̃_S(k) ]