GC3 influence was a mystery to me. So I'm delving into the numbers. This is my attempt at a "Culture 101" thread. I will break up my findings into (many) replies, and edit as I improve the upper and lower bounds.
My goal is to reverse-engineer the cultural influence equations. We did that for economy, so our collective players' knowledge on this forum exceeds the "Econ 101" thread. As a by-product, I plan to influence-flip many AI planets. I don't care about actually winning the game; I probably won't even get that far. eviator's Not Culture Flipping thread made me curious. Since then, I have been enthralled by the beauty in numbers 
I debated whether to put this in the Early Feedback sub-forum, but I think this topic is more general than a Beta 1 issue. We regular users cannot post in Journals, which I guess is restricted to Stardock dev staff.
A. Test game parameters & methodology
Altarian, Large galaxy, Scattered, vs. 3 Godlike. Considerations:
- Racial traits are the first choice point.
- Drengin have =0% Influence. This is the only way to plot the hex radii for influence values < 37.5)
- Terran, Altarian, Iridium have +25% Influence. They're all equally good for this test.
- Iridium's -25% Growth make them very intriguing, since they probably won't win a straight conquest race. But ...
- [0.42 opt-in bug] Iridium Constructor crashes to desktop on Build Starbase command.
I micro-manage (all queues, every turn). For Influence regression, I also jot down:
- for every turn
- for every colony
- all of its Influence-related parameters (as described below)
For my 1st test game, I also wrote simple Excel macros to double-check for consistency, and through 38 81 turns my predictions fit the data perfectly. I hope to use this data to tighten the bounds on the influence-to-radius function's staircase-like thresholds.