Axioms and Mathematical Foundations

Core principle

• Normalize observations via Z = A(B / c); A = frame-dependent, B = rate/shift, c = invariant.

Axiom summary

1. Empirical Validation First
   • Reproducible tests required; use mpmath with precision target < 1e-16.
   • Explicitly label hypotheses UNVERIFIED until validated.
2. Domain-Specific Forms
   • Physical: Z = T(v / c) with causality checks (ValueError for |v| ≥ c).
   • Discrete: Z = n(Δ_n / Δ_max), κ(n)=d(n)·ln(n+1)/e²; avoid zero-division.
3. Geometric Resolution
   • Use θ′(n,k)=φ·((n mod φ)/φ)^k with k ≈ 0.3 for prime-density mapping.
4. Style and Tools
   • Prefer simple, precise solutions. Use mpmath, numpy, sympy. Cross-check predictions with datasets (e.g., zeta_zeros.csv).

Empirical validation guidelines

• Create unit and integration tests that reproduce numerical results.
• Set mp.dps and document the target precision.
• Record RNG seeds or deterministic steps for reproducibility.

References

• Link to Z5D_Reference_Impl-2.ipynb
• Link to code modules: axioms.py, domain.py

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