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Well, this sounds like hundred percent black hat - basically presenting some content to the search engine while not showing it to the visitors. You are risking a penalty, yes.

But I'm not really "hiding" the content. I mean they are right there on the page. In fact the additional words are all wrapped inside the same H1 tag. It's just that because they are white and only 1px in size, you can't see them - but they aren't actually hidden. If you mouse over the area where they are and drag>select, you can see them.
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Could you please PM me the site in question? I'd like to take a look, maybe something can be done without hiding the words.

I'm working locally at the moment so there is no public facing site yet. I'm uploading the work to a staging site soon for the client to review (maybe tomorrow) so I can send you a link then.
In order for Google to penalize the site, it would have to know that the content wasn't visible to the user. How would G know this? You would only be able to know this if you actually looked at the page and then compared it with the content in the source. I'm curious, how would Google know to penalize me??
Thanks for taking the time to reply...