dyadic — 2-Adic Arithmetic & Newton Dynamics

A two-package Python library for exact 2-adic arithmetic and Newton dynamics analysis of the discrete-log map.

n = 2^v · (-1)^α · 5^e  (mod 2^k)

v = v₂(n)        — 2-adic valuation
α ∈ {0, 1}      — sign sector
e ∈ [0, 2^(k-2)) — discrete logarithm base 5

dyadic-core provides the arithmetic: DualNumber decomposition, modular inverse via Newton lifting, discrete logarithms, and p-adic exp/log — all stdlib.

dyadic-math provides Newton dynamics tools built on top: basin exploration, ghost attractor detection, the Mersenne Ghost Theorem, trajectory separation analysis, and weight stability diagnostics.

Packages

Package	Description	Dependencies
dyadic-core	2-adic arithmetic: DualNumber, modinv, dlog, padic exp/log	stdlib only
dyadic-math	Newton dynamics: basin analysis, ghost detection, Mersenne theorems, stability diagnostics	dyadic-core, numpy

Install

pip install -e packages/dyadic-core
pip install -e packages/dyadic-math

Quick Start

from dyadic_core import DualNumber, modinv_newton

n = DualNumber(42, k=16)
print(f"v={n.v}, alpha={n.alpha}, e={n.e}")  # v=1, alpha=0, e=...

inv = modinv_newton(3, k=16)
print(f"3^{-1} mod 2^16 = {inv}")

from dyadic_math import BasinExplorer

explorer = BasinExplorer(k=8, g=5, target_a=3)
portrait = explorer.portrait()
print(f"Converged: {len(portrait['converged'])}, Ghost: {len(portrait['cycle'])}")

import numpy as np
from dyadic_math import WeightStabilityDiagnostics

...

thermo = WeightStabilityDiagnostics(k=8)
result = thermo.analyse(W)
print(f"Alpha fraction: {result['alpha_fraction']:.3f}")

Key Results

• Mersenne Ghost Theorem: For Mersenne numbers w = 2^n - 1, the quantization cliff occurs at precision k* = n + 2. Verified n=3..11.
• Trajectory Separation: Two Newton trajectories starting from seeds separated by 2^s are identical for exactly ceil(log₂(s)) - 1 steps.
• Basin Dichotomy: The α=0 sector has no ghost attractors.

See docs/ for full theorem proofs and API reference.

Run tests

pytest packages/dyadic-core/tests -v
pytest packages/dyadic-math/tests -v