berkovich-padic-logic

name	berkovich-padic-logic
description	Guidelines and reference for Berkovich spaces and p-adic logic in the project, ensuring agents read the underlying mathematical LaTeX definitions prior to editing or analyzing the implementation.

Berkovich & p-adic Logic Skill

This skill governs the mathematical and implementation rules for Berkovich spaces and $p$-adic numbers in the animated-transformer repository.

1. Prerequisites and Mathematical Grounding

Before reading, editing, or making decisions about the code in berkovich.ts (or any file importing or interacting with it), you MUST read and understand the mathematical formulation documented in the LaTeX file:

• Mathematical Reference: berkovich.tex

Do not attempt to modify the implementation of $p$-adic arithmetic, valuation, distance, Berkovich points, tree navigation, or gradient descent steps without verifying the definitions and notation in the LaTeX file.

2. Core Concepts

Refer to berkovich.tex for detailed descriptions of:

• $p$-adic Numbers & Valuations ($|x|_p = p^{-\nu_p(x)}$)
• The $\mathbb{Q}_p$-tree ($\Gamma_p$) representing points by center-radius tuples $(c, \rho)$ representing closed rational disks $\bar{D}(c, p^\rho)$.
• Equivalence Relation: Two tuples $(c, \rho)$ and $(c', \rho')$ are identical iff $\rho = \rho'$ and $|c - c'|_p \le p^\rho$.
• Point Classification:
  • Type I (Leaves, $\rho \to -\infty$): Classical $p$-adic points.
  • Type II (Vertices, $\rho \in \mathbb{Z}$): Branching points.
  • Type III (Edges, $\rho \notin \mathbb{Z}$): Points on the segments between vertices.
• Hsia Kernel Distance: $\delta(x, y) = \max(p^{\rho_x}, p^{\rho_y}, |c_x - c_y|_p)$.
• Gradient Descent Steps:
  • Type III Edges: Continuous gradient flow (updating $\rho$ coordinate).
  • Type II Vertices: Branch transitions (evaluating candidates in $\mathbb{F}_p \cup {\infty}$ and taking the argmin of the loss).