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That's the shortest explanation I can come up with.
As for if cells replenish, I have no idea. Never stayed in a area very long to know for sure, but I'd imagine it either doesn't refresh, or if it does it follows the same refresh rate that you set for item spawns in houses. Again, not 100% sure on that. Just going by my own experience.
edit: found a link but it's from years ago, so no longer applicable.
https://pzwiki.net/wiki/Weapons
The axe you're trying to use is very flimsy. Think of it like 5 health points on the axe, and every time you use it you have a 1 in 5 chance to damage it. That means, on average (and base stats, with 0 Maintenance skill) you have on average 25 uses of the axe before it breaks. It also only does 15 damage to a tree.
Compare that to a wood axe, which is the best tool in the game for chopping trees, and it has 15 hit points with a 1 in 25 chance to degrade the quality of the axe on a use (with 0 maintenance skill) and does 55 damage to a tree. That means you have to swing it less because each swing will do more damage to the tree, plus you have on average 375 swings on top of that per repair interval. AND the cherry on top of the cake is that the wood axe (as well as fire axe and hand axe and pick axe) can all be repaired, which is unlike stone axe that cannot.
I used to use stone axes all the time too, but then I realized how bad they are and now I can't be bothered with them anymore. You can't repair them either. So short of stashing a couple spares in your pack, it's not worth it. The only nice thing about stone axes is you can have a practically unlimited supply. But as weapons, they're even outclasses by spears. So they're really only good as makeshift axes if your base is too far away to grab a good axe, or in your case, where you haven't found one yet.
That said, if you're so early into the game where you haven't found an axe yet, then you probably don't have a need to start chopping down trees. You won't have the carpentry skill to really use the wood yet, and probably won't have nails yet to make anything if you did. Just start tearing down furniture to level up carpentry, which will also give you some nails and wood.
Also, this is precisely why every character I create in this game, at minimum, will always have the Handyman perk. It gives you 75% faster growth of your maintenance skill. The higher your maintenance skill, the less likely you are to reduce the health of the item you're using.
There are a few perks that also increase the radius of foraging, such as cat eyes and hiker.
Then you also have to consider the time of day. At night, when it's darker, you naturally can't see as much so your foraging radius shrinks drastically. Same goes for fog.
Last thing is the location you're foraging. You were saying you're foraging in streets, which is not a good place to forage for chipped stones. I have maxed out my foraging, and I can tell you that some good stuff is foragable in towns and on streets (I have, on a few rare occasions, found boxes of ammo. But I've also found canned goods, jarred vegetables, etc), but it's less frequent and you need a high level to find some of that loot anyway. For chipped stones, you want vegetation, forest, or deep forest to really make the most of your foraging time (things also more likely to spawn out here are rabbits, squirrels, frogs, rats, birds). Like I said, you can find chipped stones in towns and on streets, and in fact on my level 10 foraging character I find them all the time, I just think your time is better spent in other biomes. Also, keep in mind chipped stones are less frequent than just stone. Probably half as frequent by my guess.
Also, I don't have any hard metrics about it, but it does seem that foraging 'replenishes'. I generally leave foraging on whenever it's daytime and I'm just walking around (you can't find anything if it's not on, and you never know when something great will pop up, so why not leave it on?). Through high traffic areas sometimes I will only find one thing every few days. But I've noticed if I leave town for a couple days to go somewhere else, and return and turn foraging back on, I'll find huge plethora of stuff (sometimes it's just a bunch of twigs and grasshoppers and crickets, but still). It's hard to call it a replenish, but probably closer to a 'spawn chance'. In other words, the items aren't on the ground, it seems as though the game will 'spawn' something in your foraging radius when it's on sometimes. And the chance for that spawn decreases I think the more you forage an area, and goes back up over time (I have loot respawning off on my server, so I can confirm it's infinite).
As for cell reloading, even with a maxed Forage the RNG clumps like crazy. Sometimes you will get three items at once, and other times you will walk for 10 minutes (in game time) without anything, with a Forage of 10. The old system did have a cell exhaust/refresh, but this new system may not. If it does, then it does refresh every day or two in Build 41. My impression is every day but I have not specifically tested for it.
I have never ever, 0x, found chipped stones in vegatation or forest. That is where I started looking just from logic. But then I saw a guide and the person mentioned chipped stones on the street. And that is the only place I found it so far, and extremely rare.
Veg/forest gives me a lot of different stuff, but not stones.
If i go out into the woods searching for stones, i might find some IF humans brought them there for example by creating woodroads or paths.
But deeper into the woods i would only find much larger stones or rocks, barely any usable smaller stones.
Thats what riverbeds are for and how nature works.
You know, water flushing away most of the dirt until only the heavy objects, aka stone, are left over.
And in the forest you typically have vegetation growing over rocks and there is nothing that destroys rocks to create stone fragments. Thats what water does.
So looking close to the water and streets/roads is where you will find stone, not only ingame but also in real life, typically.
At least in central europe.
But to me its not surprising that foraging in forests for stones doesnt yield much reward.
Only 1 rank in foraging so far, and have Outdoorsman, but I think that only increases the visual radius slightly.