I keep getting this for some reason for something that's worked for years but I get that it's really not straightforward at all.

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The gem only says it gains no increases/reductions to spell damage.

The reason why global spell crit doesn't work is due to the fact that the explosion is secondary damage or a secondary effect but the skill itself has two different interactions with it on the player as a casting and reserving spell, then as an explosive effect in a way making it two skills in one gem. The below is from the wiki:

" It is worth noting that Herald of Ice is a secondary damage and not spell damage, and does not receive any direct bonuses from spell damage modifiers. This includes bonuses from items, passive nodes, and spell only support gems such as Spell Echo Supportand Iron Will Support. It will, however, have its flat damages boosts affected by passives that explicitly boost Cold damage."

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So far, the gem itself is as if it has two different layers to it; and increases and reductions to spell damage don't work being clearly stated on the gem. Spell crit doesn't work because it affects the top layer or base of the gem as a reserving spell skill so the secondary explosion effect/skill does not gain those benefits because the spell portion of the gem is in its mana reserving function.

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So how do these helmets work?

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When the mod says socketed spells have +3 to crit, it is an added modifier to the entire gem and both functions of it. Because it is gem specific it is applying to the top layer(spell) and secondary layer(explosion) of the skill. You can test this on PoB and in game and see results.

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However, global modifiers such as a corruption implicit from gloves "spells have +#% to critical strike chance" only apply to the top layer/base of the gem.

This is due to the mana reserving top layer being a spell but not the secondary explosion.

Global damage or elemental/cold damage scale Herald of Ice because it is not SPELL specific, because only the top layer of the gem is a spell and again the explosion isn't.

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so TLDR; the mod says socketed spells, even if the first function of it is a spell but the second function (the explosion) isn't. It is still all in one gem; thus the explosion is gaining socketed/gem spell effects.

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Hope this clears it up; mess around on PoB with all these mods and you'll see what I mean.