Company claiming ownership of AI-assisted code I wrote on personal time with personal subscriptions

I'm a senior software developer at a mid-size SaaS company in San Francisco. Over the past 6 months I used GitHub Copilot and Claude Code extensively — both for my work projects and for a personal side project I've been building on evenings and weekends.

Here's where it gets messy: my company recently updated their IP policy and is now claiming that ALL code I wrote with AI assistance — including my personal side project — is company intellectual property. Their argument is:

1. The company's codebase was used as "training context" by the AI tools (since Copilot and Claude see open files)
2. Therefore any AI output is a "derivative work" of company IP
3. Since I used the same AI tools for both work and personal projects, there's "commingling"

My counterargument:

• I used my personal GitHub Copilot subscription ($19/mo) and my personal Claude Pro subscription
• I worked on my side project exclusively on my personal laptop at home, after hours
• My side project is in a completely different domain (fitness tracking app) than my employer's product (B2B analytics)
• I never shared company code with the AI tools while working on my personal project

They've sent me a formal letter demanding I assign all rights to the side project to the company and cease development. Before I respond, I need to understand my legal position. This is in California.