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Mike Tate Mathematics

Galois & Lie Group Action

Galois Group Action

\( f(x) = x^3 – 2 \)

The Galois group \( \mathrm{Gal}(\mathbb{Q}(\sqrt[3]{2}, \omega)/\mathbb{Q}) \cong S_3 \) permutes the three roots of the polynomial in discrete cycles.

Ψ: Discrete Root Permutations · Φ: Path Morphs · ρ: Lie Flow

Spectral Arithmetic Flow

\( \Lambda(n) \cdot T_n(x) \)
on a finite harmonic lattice

A hybrid simulation of Chebyshev oscillations \( T_n(x) \) overlaid with spikes from the Von Mangoldt function \( \Lambda(n) \), producing wave-particle interference rooted in arithmetic structure.

Ψ
Root Permutations
Discrete symmetries of polynomial roots (Sₙ group)
Φ
Path Morphs
Interpolated transitions between permutation states
ρ
Lie Flow
Continuous motion under Lie group evolution (e.g. SU(2))
Λ
Prime Weighting
Von Mangoldt spikes encoding prime power density

Symbolic Explainers

Ψ — Root Permutations (Discrete Symmetry)
Ψ encodes discrete algebraic symmetry — like the Galois group of \( x^3 – 2 \), where roots are permuted without changing the structure. These form groups like \( S_3 \), foundational to modern algebra.
Φ — Path Morphs (Interpolated Deformations)
Φ reflects continuous morphing between discrete outcomes. In our simulations, it corresponds to flowing transitions between permutation states — connecting Ψ via geometry and homotopy.
ρ — Lie Flow (Continuous Symmetry)
ρ expresses smooth symmetries — governed by Lie groups (like SU(2)). These flows act on roots as rotations or evolutions, contrasting with Ψ’s discrete jumps. They’re geometric, topological, and physical.
Λ — Von Mangoldt Function (Prime Encoding)
Λ(n) injects arithmetic into the system — spiking only at prime powers. It’s a core part of the explicit formulas that connect prime distribution to complex function theory, like ζ(s).

🧠 Unified Symbolic Framework

The four symbolic operators form a dynamic grammar of mathematical motion:

  • Ψ — Permutations of algebraic roots (symmetric group structures)
  • Φ — Morphs and deformations between states (interpolated flows)
  • ρ — Continuous Lie group actions (orbiting flows on root space)
  • Λ — Encoded prime density via Von Mangoldt (discrete arithmetic spikes)

Together, they act as operators in a symbolic dynamics: **Ψ** defines state space, **Φ** connects states, **ρ** flows over them, and **Λ** seeds the entire field with number-theoretic resonance.