I started doing some freelancing for a young (22yo) startup founder around four weeks ago

They contracted me as a senior architect/system design person - I said I could cut code, but that wasn't likely where they were going to get value from me. (I'm currently the only person they've hired - they are hoping to add one frontend person).

Their product is a marketplace app, but they're looking to integrate a bunch of AI/ML features (mostly computer vision or NLP related), and make heavy use of video content (both ingest/capture and realtime streaming), and a bunch of social graph features.

Since then I've been meeting up roughly 2-3 hours per week (in person), and 3-4 hours per week (video calls).

* Most of that was project scoping calls with the founder - to try to get an idea of their vision, and what they're trying to accomplish - bounce ideas, and give them a sense for what's achievable, what's not, and what the prior art might be. Each week we'd get some large A3 sheets of paper out, post-its some marker pens, and start helping them brainstorm and capture all their ideas.
* I've also jumped on various calls each week with ML/AI software vendors, and walked the founder through pros/cons of each vendor, versus also trying to implement ourselves (i.e. build vs buy). I've helped review product specs for each solution, read through their API documentation, and taken apart their various demos and spec sheets, so that I could give the founder an honest assessment of each vendor. As well as a crash-course in ML.
* I've helped walk them through the gotcha of their video streaming problem space, and what tricks they might use to optimise the user experience - and started planning out how the backend (RTMP ingest, WebRTC for streaming would work).
* I've also helped interview various frontend people for them, or done background research on their portfolios/prior work etc.
* I've given them guidance on things they need to consider - e.g. spam/fraud detection, GDPR compliance, CSAI detection etc.
* I've also started teasing out a schema design from them, based on our weekly meetings, and their various "I want app do to this".

The problem now is - I've just submitted my first invoice for the last 4 weeks (I averaged it out to 20 hours a week), and they're balking at paying me.

I feel completely sick inside - it's both incredibly frustrating that I've spent all this spare time helping them - and also that they don't seem to see the value of any of my time.

They've said "I can pay you for the time we spend meeting up face to face, or when we were on video calls" - but that completely discounts all the time I spent afterwards collating my handwritten notes or our project boards, reviewing academic papers or finding the current state-of-art in various topics for them, reviewing documentation for them, or coming up with various ideas to help their project along.

Also - I did a small task for them - and I just put 15 hours on the invoice as a bucket, to include all the time for debugging, scaling it, and dealing with some idiosyncrasies of the endpoints and not tripping up any WAF (web application firewalls) - and they've said, my "tech consultant" (I'm assuming their other 22 year old programmer friend, who I've met once online) said he could do it in 2 hours.

They asked for some artefacts to justify the invoice - and I've send them the current schema (which I assume they'll look at and be like, pftt, anybody could type that text out in 2 hours), as well as my feature planning summary notes (around 15 pages, handwritten shorthand, which I'm going to convert into the ticket system I set up for them...assuming I get paid).

**Does anybody have any ideas on how to salvage this situation?
Or try to explain the value I'm hoping I can bring to the project?
What would you do in this situation?**