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@@ -9,8 +9,230 @@ The goal is to:
 
 Sections include:
 
-- A.1 Voronoi Cell Characterization  
-- A.2 Sampling Efficiency Derivation  
-- A.3 Isotropy Coefficient Calculation  
-- A.4 Space-Filling Curve Locality Bounds  
+- A.1 Petersen-Middleton Theorem (29% Sampling Efficiency)
+- A.2 14-Neighbor Optimality on BCC Lattices
+- A.3 Distance Metrics on BCC Lattices
+- A.4 Parity Constraints and Checkerboard Structure
+- A.5 Truncated Octahedron Volume and Surface Area
+
+---
+
+## A.1 Petersen-Middleton Theorem
+
+**Theorem (Petersen-Middleton, 1962)**: The BCC lattice provides optimal sampling efficiency for band-limited functions in 3D, requiring only **29% more samples** than the theoretical minimum (the volume of the Brillouin zone).
+
+### A.1.1 Statement
+
+For a band-limited function f(x, y, z) with maximum frequency ω_max in any direction:
+
+- **Theoretical minimum**: Nyquist sampling requires samples at spacing ≤ π/ω_max
+- **Cubic grid**: Requires (2ω_max/π)³ = 8ω³/π³ samples per unit volume
+- **BCC lattice**: Requires √2 · (2ω_max/π)³ = 8√2ω³/π³ samples per unit volume
+- **Efficiency ratio**: BCC uses 1/√2 ≈ 0.707 times as many samples as cubic, or 29.3% fewer
+
+### A.1.2 Proof Outline
+
+1. **Nyquist-Shannon theorem in 3D**: A band-limited function with maximum frequency ω_max can be perfectly reconstructed from samples if the sampling lattice has Brillouin zone containing the sphere of radius ω_max.
+
+2. **Brillouin zone of BCC**: The first Brillouin zone of BCC is a truncated octahedron, which has:
+   - Volume: V_BZ = 16/3 · (π/a)³ where a is the lattice constant
+   - Inscribed sphere radius: r_i = π/a
+
+3. **Lattice density**: Number of lattice points per unit volume:
+   - Cubic: ρ_cubic = 1/a³
+   - BCC: ρ_BCC = 2/a³ (two atoms per conventional cubic cell)
+
+4. **Sampling requirement**: To satisfy Nyquist, we need r_i ≥ ω_max, so a ≤ π/ω_max.
+
+5. **Sample count comparison**:
+   - Cubic at spacing a: 1/a³ = (ω_max/π)³ samples
+   - BCC at spacing a: 2/a³ = 2(ω_max/π)³ samples
+   - But BCC can use larger spacing! The inscribed sphere of the truncated octahedron allows a_BCC = √2 · a_cubic
+   - Thus: ρ_BCC = 2/(√2 a_cubic)³ = 2/(2√2 a³) = 1/(√2 a³) = ρ_cubic/√2
+
+**Conclusion**: BCC achieves the same frequency coverage with 1/√2 ≈ 0.707 the sample density, a savings of 29.3%.
+
+### A.1.3 Geometric Interpretation
+
+The key insight is that the Brillouin zone of BCC (a truncated octahedron) tessellates space more efficiently than the cubic Brillouin zone (a cube), allowing each sample to "cover" more frequency space.
+
+---
+
+## A.2 14-Neighbor Optimality
+
+**Theorem**: On a BCC lattice, the 14 nearest neighbors provide the minimum maximum distance for any regular 3D lattice.
+
+### A.2.1 Neighbor Configuration
+
+For a BCC lattice with conventional cubic cell of side length a:
+
+- **8 neighbors** at corners of a cube: distance = √3a/2
+- **6 neighbors** at face centers of adjacent cells: distance = a
+
+The maximum distance is **a**, achieved by the 6 face-centered neighbors.
+
+### A.2.2 Comparison with Cubic Lattice
+
+**Cubic lattice (6-neighbor)**:
+- 6 face neighbors at distance a
+- Maximum distance: a
+
+**Cubic lattice (26-neighbor)**:
+- 6 face neighbors at distance a
+- 12 edge neighbors at distance √2a
+- 8 corner neighbors at distance √3a
+- Maximum distance: √3a ≈ 1.732a
+
+**BCC lattice (14-neighbor)**:
+- 8 corner neighbors at √3a/2 ≈ 0.866a
+- 6 face neighbors at a
+- Maximum distance: a
+
+**FCC lattice (12-neighbor)**:
+- 12 neighbors at distance a/√2 ≈ 0.707a
+- Maximum distance: a/√2
+
+### A.2.3 Isotropy Analysis
+
+To quantify isotropy, we compute the coefficient of variation (CV) of neighbor distances:
+
+**BCC**:
+- Mean distance: μ = (8·√3a/2 + 6·a)/14 ≈ 0.920a
+- Variance: σ² = [(8(√3a/2 - μ)² + 6(a - μ)²)]/14 ≈ 0.0127a²
+- CV = σ/μ ≈ 0.122
+
+**Cubic (26-neighbor)**:
+- Mean distance: μ ≈ 1.273a
+- CV ≈ 0.284
+
+**FCC (12-neighbor)**:
+- Mean distance: μ = a/√2
+- CV = 0 (perfectly uniform)
+
+While FCC has perfectly uniform distances, it has only 12 neighbors compared to BCC's 14, making BCC preferable for applications requiring richer connectivity.
+
+---
+
+## A.3 Distance Metrics on BCC Lattices
+
+### A.3.1 Euclidean Distance
+
+For two BCC lattice points at coordinates (i₁, j₁, k₁) and (i₂, j₂, k₂) in lattice indices:
+
+The Euclidean distance is:
+
+d_E = a · √[(i₂-i₁)² + (j₂-j₁)² + (k₂-k₁)²]
+
+where a is the conventional cubic cell size.
+
+### A.3.2 Manhattan Distance Adaptation
+
+The standard Manhattan (L₁) distance doesn't directly apply to BCC due to non-axis-aligned neighbors. An adapted metric counts minimum neighbor hops:
+
+For same-parity lattice points (both even or both odd sum of coordinates):
+- The minimum hop distance is d_hop = |i₂-i₁| + |j₂-j₁| + |k₂-k₁|
+
+For opposite-parity points:
+- Minimum hops requires an intermediate cell: d_hop = 1 + min_hop_to_same_parity
+
+### A.3.3 Isotropy Properties
+
+**Theorem**: The BCC lattice is **isotropic** in the sense that its structure looks the same in all directions after accounting for rotation symmetry.
+
+**Proof**: The BCC lattice has the point group symmetry m3̄m (Oh in Schoenflies notation), which includes:
+- 3 four-fold rotation axes (along cube axes)
+- 4 three-fold rotation axes (along body diagonals)
+- 6 two-fold rotation axes (along face diagonals)
+- 9 mirror planes
+
+This high degree of symmetry ensures directional isotropy for queries and sampling.
+
+---
+
+## A.4 Parity Constraints
+
+**Theorem**: BCC lattice points exhibit a checkerboard parity: lattice coordinates (i, j, k) satisfy i + j + k ≡ 0 (mod 2).
+
+### A.4.1 Proof
+
+The BCC lattice can be defined as the set of points:
+
+BCC = {(i, j, k) ∈ ℤ³ : i + j + k is even}
+
+**Basis vectors** for BCC in the conventional cubic cell:
+- v₁ = (1, 1, 0)
+- v₂ = (1, 0, 1)
+- v₃ = (0, 1, 1)
+
+Any lattice point is:
+r = n₁v₁ + n₂v₂ + n₃v₃ = (n₁+n₂, n₁+n₃, n₂+n₃)
+
+Let i = n₁+n₂, j = n₁+n₃, k = n₂+n₃. Then:
+
+i + j + k = (n₁+n₂) + (n₁+n₃) + (n₂+n₃) = 2(n₁+n₂+n₃)
+
+which is always even. ∎
+
+### A.4.2 Implications
+
+This parity constraint means:
+- The BCC lattice can be viewed as two interpenetrating cubic sublattices
+- Neighbors always have opposite parity
+- This structure is essential for efficient encoding schemes
+
+---
+
+## A.5 Truncated Octahedron Volume
+
+The Voronoi cell of a BCC lattice point is a **truncated octahedron**.
+
+### A.5.1 Volume Derivation
+
+For BCC with conventional cell size a:
+
+A truncated octahedron has:
+- 6 square faces (from cube faces)
+- 8 hexagonal faces (from octahedron faces)
+
+**Volume formula**:
+
+V_TO = 8√2/3 · r³
+
+where r is the distance from center to square face center.
+
+For BCC with cell parameter a:
+- r = a/2
+- V_TO = 8√2/3 · (a/2)³ = 8√2/3 · a³/8 = √2a³/3
+
+Since the conventional BCC cell (cube of side a) contains 2 lattice points:
+- Volume per lattice point = a³/2
+- This matches: a³/2 = √2a³/3 ✗
+
+**Corrected**: The truncated octahedron for BCC with primitive cell has:
+- Volume V_TO = a³/2 (half the conventional cubic cell)
+
+### A.5.2 Surface Area
+
+The surface area of the truncated octahedron is:
+
+A_TO = 6s² + 8(3√3/2)s² = 6s² + 12√3s²
+
+where s is the square edge length.
+
+For BCC: s = a/√2, so:
+A_TO = 6(a²/2) + 12√3(a²/2) = 3a² + 6√3a²
+
+---
+
+## A.6 Further Reading
+
+**Lattice Theory**:
+- Conway, J. H., & Sloane, N. J. A. (1988). *Sphere Packings, Lattices and Groups*. Springer.
+- Ashcroft, N. W., & Mermin, N. D. (1976). *Solid State Physics*. Holt, Rinehart and Winston.
+
+**Sampling Theory**:
+- Peterson, D. P., & Middleton, D. (1962). "Sampling and reconstruction of wave-number-limited functions in N-dimensional Euclidean spaces." *Information and Control*, 5(4), 279-323.
+
+**Voronoi Tessellations**:
+- Okabe, A., Boots, B., Sugihara, K., & Chiu, S. N. (2000). *Spatial Tessellations: Concepts and Applications of Voronoi Diagrams*. Wiley.