Mathematical Foundations¶

P-adic Numbers and Q_p¶

The p-adic numbers Q_p form a complete metric space extension of the rationals, equipped with a non-archimedean metric. Unlike the reals (which use the standard absolute value), p-adic arithmetic uses a different notion of "distance."

P-adic Valuation¶

For a prime p and integer n ≠ 0, the p-adic valuation v_p(n) is:

v_p(n) = the largest integer k such that p^k divides n

Examples: - v_2(8) = 3 (since 8 = 2³) - v_3(27) = 3 (since 27 = 3³) - v_5(10) = 1 (since 10 = 2 × 5) - v_p(1) = 0 for all primes p - v_p(0) = ∞ (by convention)

P-adic Norm¶

The p-adic norm on Q_p is defined as:

|x|_p = p^{-v_p(x)}  if x ≠ 0
|0|_p = 0

Key properties: 1. Non-negativity: |x|_p ≥ 0 2. Identity: |x|_p = 0 iff x = 0 3. Multiplicativity: |xy|_p = |x|_p · |y|_p 4. Ultrametric inequality: |x + y|_p ≤ max(|x|_p, |y|_p) [stronger than triangle inequality]

The ultrametric property is crucial: in p-adic geometry, "large" numbers (small norms) are closer together.

Example with p = 2: - |8|_2 = 2^{-3} = 1/8 - |1|_2 = 2^0 = 1 - |10|_2 = 2^{-1} = 1/2

P-adic Inner Product¶

The p-adic inner product of vectors v₁, v₂ ∈ Q_p^n is:

⟨v₁, v₂⟩_p = Σᵢ wᵢ · v₁[i] · v₂[i]

where the weight wᵢ depends on the p-adic valuation:

wᵢ = |v₁[i] · v₂[i]|_p = p^{-v_p(v₁[i] · v₂[i])}

This weighting emphasizes components with high p-adic valuation (small norm), which is geometrically meaningful in p-adic spaces.

Example Computation¶

For v₁ = [1, 2] and v₂ = [1, 2] with p = 2:

Component 0: product = 1 × 1 = 1
            v_2(1) = 0
            weight = 2^{-0} = 1
            contribution = 1 × 1 = 1

Component 1: product = 2 × 2 = 4
            v_2(4) = 2 (since 4 = 2²)
            weight = 2^{-2} = 1/4
            contribution = 1/4 × 4 = 1

⟨v₁, v₂⟩_2 = 1 + 1 = 2

Adelic Numbers and Restricted Products¶

The adeles form a topological ring that encodes all completions of Q simultaneously. For the RTSG framework, we use the restricted product.

Archimedean vs. Non-archimedean¶

Archimedean place: The standard absolute value on ℝ (Euclidean metric)
Non-archimedean places: The p-adic norms for primes p

Restricted Adelic Product¶

The restricted adelic inner product is:

⟨v₁, v₂⟩_adelic = ⟨v₁, v₂⟩_∞ × ∏_{p ∈ S} ⟨v₁, v₂⟩_p

where: - ⟨v₁, v₂⟩_∞ = Σᵢ v₁[i] · v₂[i] (Euclidean inner product) - S is a finite set of primes - The product combines archimedean and p-adic contributions multiplicatively

Example: Two-Prime Adelic Product¶

For v₁ = [1, 0] and v₂ = [0, 1] with S = {2, 3}:

Archimedean: ⟨v₁, v₂⟩_∞ = 1×0 + 0×1 = 0
2-adic:      ⟨v₁, v₂⟩_2 = (depends on components)
3-adic:      ⟨v₁, v₂⟩_3 = (depends on components)

⟨v₁, v₂⟩_adelic = 0 × (⟨v₁, v₂⟩_2) × (⟨v₁, v₂⟩_3) = 0

When the archimedean part is zero, the entire product is zero regardless of p-adic contributions.

Gram Matrices¶

The Gram matrix G of a basis {v₁, ..., vₙ} has entries:

G[i,j] = ⟨vᵢ, vⱼ⟩_metric

Properties¶

Symmetry: G[i,j] = G[j,i]
Positive semi-definiteness (for real/complex metrics): All eigenvalues ≥ 0
Full rank iff vectors are linearly independent

Real (Euclidean) Gram Matrix¶

G[i,j] = Σₖ vᵢ[k] · vⱼ[k]

For orthonormal vectors, G = I (identity matrix).

P-adic Gram Matrix¶

G[i,j] = ⟨vᵢ, vⱼ⟩_p = Σₖ |vᵢ[k] · vⱼ[k]|_p

The non-archimedean nature means the Gram matrix structure differs fundamentally from the real case.

Adelic Gram Matrix¶

G[i,j] = ⟨vᵢ, vⱼ⟩_adelic = ⟨vᵢ, vⱼ⟩_∞ × ∏_{p ∈ S} ⟨vᵢ, vⱼ⟩_p

Combines Euclidean and p-adic information in a multiplicative structure.

Eigenvalue Computation via Jacobi Method¶

The Jacobi method is a classical iterative algorithm for computing eigenvalues of symmetric matrices.

Algorithm Overview¶

Initialize: A ← M (copy input matrix)
Iterate until convergence:
Find the largest off-diagonal element A[p,q]
Compute rotation angle θ to zero out A[p,q]
Apply Givens rotation: A ← Rₚ,q(θ)ᵀ · A · Rₚ,q(θ)
Extract eigenvalues from diagonal

Givens Rotation¶

The rotation angle θ is computed to eliminate element (p,q):

tan(2θ) = 2A[p,q] / (A[p,p] - A[q,q])

The rotation matrix Rₚ,q(θ) is applied from both sides to preserve symmetry.

Convergence Properties¶

Quadratic convergence under normal circumstances
Guaranteed convergence for any symmetric matrix
Complexity: O(n³) with typically 5-30 iterations

Implementation Details¶

In our library: - Maximum iterations: 1000 - Convergence tolerance: 1e-14 - Eigenvalues are sorted in descending order - Returns NAN on error (not panic)

Floating-Point P-adic Valuation Estimation¶

Since we work with floating-point numbers rather than exact rationals, we estimate the p-adic valuation through the mantissa and exponent.

Method¶

mantissa, exponent = extract_mantissa_exponent(x)
approx_int = round(mantissa * 2^53)
v_p ≈ count_factors(approx_int, p)

This approach: - Works for typical floating-point values - Provides reasonable estimates for the metric - May lose precision for very large/small numbers - Is deterministic and reproducible

Limitations¶

Loss of information: Float→int conversion loses precision
Boundary cases: Numbers very close to powers of p may have estimation errors
Theoretical gap: True p-adic arithmetic on rationals vs. float approximation

For exact arithmetic on rationals, external libraries (num-rational, rug) would be needed.

Numerical Stability¶

Gram Matrix Computation¶

Double precision (IEEE 754) sufficient for typical use
Direct computation of inner products (no orthogonalization)
Gram matrix inherits numerical properties of inner product

Jacobi Eigenvalue Method¶

Numerically stable for symmetric matrices
Uses only additions, multiplications, trigonometric functions
No explicit inverse or decompositions (higher condition number)
Quadratic convergence means exponential improvement in accuracy

Guard Against Underflow¶

In adelic products, we terminate early if intermediate results become very small (< 1e-300) to prevent numerical underflow.

References and Further Reading¶

Books¶

Cassels, J.W.S. (1986). Local Fields. London Mathematical Society Student Texts. [Definitive reference on p-adic theory]
Serre, J.-P. (1979). Local Fields. Springer-Verlag. [Cleaner presentation]
Weil, A. (1967). Basic Number Theory. Springer-Verlag. [Adeles and algebraic geometry]
Golub, G.H. & Van Loan, C.F. (2013). Matrix Computations. Johns Hopkins University Press. [Jacobi method details]

Papers¶

Jacobi, C.G.J. (1846). "Über ein leichtes Verfahren, die in der Theorie der Säcularstörungen..." [Original Jacobi eigenvalue paper]
Freiling, G., et al. (2000). "Jacobi matrices for measures supported on the unit circle." Linear Algebra and its Applications, 317(1-3), 1-12.