Core Concepts

Universal invariant formulation

• Central form: Z = A(B / c)
• Interpretation:
  • c is the universal invariant; choose the domain-appropriate constant (e.g., c ≈ 299792458 m/s for relativistic physics, e² for some discrete normalizations).
  • A encodes frame-specific scaling/transformations.
  • B is the dynamic rate/shift input.

Domain-specific forms

• Physical domain: Z = T(v / c)
  • Use for relativistic-like transforms, with explicit causality checks.
  • Guard: raise ValueError if |v| ≥ c.
• Discrete domain: Z = n(Δ_n / Δ_max)
  • For integer sequences and prime-density mapping.
  • Curvature: κ(n) = d(n) · ln(n+1) / e², with guards against zero division.

Geometric resolution

• θ'(n, k) = φ · ((n mod φ) / φ)^k, k ≈ 0.3 recommended for prime-density mapping.
• Use as a resolution / embedding technique for discrete geodesics.

Links

• Axioms and Mathematical Foundations
• Implementations and Code
• Datasets and Resources

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