Purity (assay) of a chemical is determined by quantifying impurities using analytical methods like HPLC, GC-MS, or elemental analysis, then calculating purity as 100% minus total impurities. This numerical purity (e.g., 99.9%) is derived from precise measurement and comparison to standards.

Abdelhak Maghchiche has described the requirements well. I might add that you might need to distinguish between the "purity" and "concentration" of the substance.

For example, hydrogen peroxide is commonly sold as a "30% concentration solution" but has very little contamination from metals or insoluble fractions present (the rest is water).

You may also need to investigate whether the concentration is suitably expressed with a measurement of uncertainty. Assaying the material on different days by different operators against different batches of reagents may be required.

For example, production of sulfuric acid may be tested against known sodium hydroxide standards, to ensure consistency, or at least knowledge of how variable your product is. Is it 98% ±2%, or 95% ±5%?

Finally, expressing concentrations as metal % and metal oxide % might need clarification, as it is often confused or unclear which unit the product is sold as.

@Matthew Wheal and @abdelhak Maghiche is metallurgical OES results reliable for purity claims?

From the figures you give, I guess you're in a critical area as far as numerous quality criteria are concerned. We chemists sometimes receive requests from medical doctors and others who don't understand that we too had to fall into a cauldron of magic potion when we were little. There are very many professions, not just in science in medicine, where one can't possibly gain a working knowledge by reading books - much less from the internet!

One example from almost any analytical lab: if you need to know the purity of a substance to 0.1%, very many substances will gain or lose that much water or residual solvents while you're weighing out the test portions. Usually you have to confront an assay result with the sum of detected impurities, but don't expect the figures ever to match precisely.

All chemists know some analytical chemistry, but some branches of that have surprisingly difficult learning and experience curves that may be particularly invisible to members of other professions. Chemical theory is "informed" everywhere by quantum mechanics, but we need to know also how best to clean the glassware, purify reagents... and not set off explosions.

It seems to me that you will need to engage one or more contract labs with proven experience that work to standards that meet your requirements, without using more resources than are needed.

The learned chemical and industrial societies should be good sources of information.

You can find a third-party testing agency and entrust them to test the product ingredients to obtain COA.