What ingredients can I add to food or gum recipes to add grittiness without any carbs?

I found a patent application for Chewing gum possessing tooth cleaning effect and a teeth cleaning method. Paragraph 130 addresses polishing agents:

Consequently, a polishing material can be any material that does not abrade dental enamel and dentine. Typical materials include silica gels and precipitates, aluminas, phosphates, and mixtures thereof. Specific examples include dicalcium orthophosphate dihydrate, calcium pyrophosphate, Bamboo, tricalcium phosphate, hydrated alumina, beta calcium pyrophosphate, calcium carbonate, sodium polymetaphosphate, sodium hexametaphosphate, Calgen, Giltex, Quadrafos, Hagan phosphate, micromet, calcium phosphate dibasic, calcium monohydrogen phosphate, dicalcium orthophosphate secondary calcium phosphate, carbonic acid calcium salt, cacti, calcichew, calcidia, citrical, aragonite, calcite, valerite, aluminum oxide, alumina, silicon dioxide, silica, silicic anhydride, and resinous abrasive materials such as particulate condensation products of urea and formaldehyde and others such as disclosed in U.S. Pat. No. 3,070,510. Mixtures of polishing agents can also be used.

I don't know all those ingredients. The ones that looked interesting are food grade silica gel and bamboo (powdered form).

I was surpised to see calcichew and citracal in there, I'm not familiar with their abrasiveness.

Also, whole (not ground) Chia seeds.

Baking soda is commonly used as a mild abrasive in toothpaste, and is a food ingredient.

Liquorice root is like a tasty chewy twig, and is chewed to clean teeth in some countries. Maybe you could chop some of that up if you can get it. Assuming the flavour is acceptable of course.

Otherwise you might be looking for seed husks of some sort, dried and ground. Cardamom perhaps? It's cooked in pilau rice and while you might choose to eat round it, accidentally biting into/swallowing a whole pod is normal. You'd have to experiment with how much of the flavour you can handle, and indeed with his much is in the husk.

Have you looked for traditional toothpaste recipes for ideas?

Food grade Diatomaceous earth is available. It is used for brushing teeth. For something a bit softer than silica, you might try a pumice powder. Looks like it'll take a bit of digging to find genuine food grade stuff.

I would use poppy seeds, or some other seed of the same size. Many other things fit your criteria - for example the ground hulls of stone fruit kernels (not the soft nut within), but I would be afraid that they could be too abrasive.

If you could grind fennel or cardamom seeds to the right consistency to get the right abrasive action and "bite" (please excuse the pun!), this would also have dental benefits.

See: http://ijpsr.com/bft-article/effect-of-chewing-fennel-and-cardamom-seeds-on-dental-plaque-and-salivary-ph-a-randomized-controlled-trial/?view=fulltext

You might look into bamboo charcoal, or something like that.

I know it's sold as a tooth cleaner, so it should work in that regard. I also know it's edible (well, if food grade), there are recipes for bamboo charcoal bread, and other culinary uses. And the grittishness of the powder depends on the grinding, I know commercial powders are very smooth, homemade was quite a bit crunchier when I was trying a charcoal bread recipe - depends on what level of grit you're looking for (I imagine charcoal is soft enough that even coarse grit is not going to be too, too abrasive for one's teeth).

I don't know how the flavor will stack up in your gums, etc - in charcoal bread it's likely to be far more diluted than it will be in a gum, especially in high enough quantities to be an effective abrasive. But it might well be worth trying.