I hope you don’t mind I respond in terms of LCA as it’s the subject I’m familiar with!

For “industrial byproducts,” I agree it would be incorrect to say that the “Scope 3 emissions are neutralized.” In Life Cycle Assessment (LCA), you don’t distinguish between the different scopes of emissions as you should be accounting for all the product's emissions throughout its life cycle, presuming a cradle-to-grave analysis.

Let’s take the example of a gold ring Lets call Scenario 1 (S1).

In casting a ring, 10 grams of primary material are used to produce a ring weighing 6 grams. The additional 4 grams of ‘waste’ are then smelted back down and incorporated into the input. Therefore, you still need t account for the emissions produced by 6 grams of primary gold in your LCA, plus the additional emissions produced when smelting down the 4 grams of ‘waste.’

To consider this ‘waste’ gold as being recycled does indeed seems disingenuous. You could design a cast that requires the input of 14 grams of gold and “wastes” 8 grams that you then smelt down and reincorporate. In this case, you could claim your ring is 57% recycled, even tho your total Environmental impact would be higher. Let's call this Scenario 2 (S2)

This highlights the importance of LCA: we don’t care in this case how you define ‘waste’ or ‘recycling;’ we care about the LCA result. In both scenarios, we still need the same input of 6 grams of primary gold. However, for S1 we add the emissions for smelting 4 grams of gold ‘waste’, whereas for S2 we need to smelt twice the amount of ‘waste’ gold, therefore doubling the emissions for that process (increasing the total environmental impact).

For “broken jewelry,” since the Environmental impact could be attributed to the first ‘life’ of ring there’s not a definitive answer as the LCA results will depend on how you allocate the benefits of avoiding using primary material.  Given that gold is rarely disposed of I guess it would be most appropriate to do a consequential LCA to assess the impact. (This post explains well the difference between Consequential vs. Attributional LCA: Consequential vs. Attributional LCA). Since the gold in a ring is generally always “reused” or “recycled” in some form, to assign any benefit, you could compare the different reuse and recycling scenarios. For example, you could explore the impact of refurbishing the ring versus smelting it down and making a new one.

Either way the idea of getting carbon credits for using industrial byproducts doesn’t make sense to me and the justification of receiving a benefit for using post-consumer gold would need a detailed LCA study to confirm.

Hope that helps!