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1. A level 20 Precog who maxes RP can convert all 21 (not counting theme refresh) and take a certain temporal anomoly to have 14 paradoxes in one day. If you're Fragmented Past you can use the paradoxes to stabilize and get back in the fight in one round, making them better than the 3 RP you spent on them. You can heal stamina via spells.
You shouldn't do this and convert all your RP, but you can. 14 paradoxes all rolled at once gives you a 51% chance of at least one of them being a 20, although of course you don't have to commit to converting if your first 5, then 7, then each individual one after that are sufficient.
2. A Precog is the only class that can safely use Contact Other Plane with the +5 bonus to your results. As long as you have a 13 paradox you won't lose spellcasting for a week.
3. Fragmented Past greater anchor burst damage: Bank 1-2 "20" paradoxes, use your three attacks in one round to attempt 2-3 crits in one turn, delivering 4x, 5x, or 6x the average damage of your (heavy or advanced melee) weapon in one round. For best results use a heavy weapon in a gunners harness for -2 to attack on your first pair of atacks.
First, cast probability prediction in advance, use that to take two rolls on your first attack. If miss on both, use a reaction (with appropriate 14th level temporal anomal) to replace with a hit. If you have two 20s banked, used one here. If not, use a high number to guarantee a hit.
(Alternatively, when fighting a boss if you've got two 20 paradoxes banked just wait for a natural crit to come up before committing to this course of action, then spend 1-2 of them to keep the chain going and splat them with no fuss or worry. If the second shot finishes him, you've still got your daily triple attack and one more 20 to try this again later.)
Second, roll normally. If it's a miss, you can either try Uncanny Luck for a reroll and hope for a crit, or just buy a hit with your high but not-20 paradox.
Third, pay for a crit on your bonus third attack with a 20 paradox, don't worry about whether the -6 is reduced by your gunner's harness.
With the highest level reaction cannon you average 344 (you can use 10+ paradox to get some maximum damage die rolls instead of rolling, if you have them to spare) with one crit. With two crits, 430 points of damage. If you're lucky enough to be banking three 20s or to have rolled one naturally to get triple crits, that's 516 points of damage.
4. I cannot tell you that the Dimension of Time improved anchor ability to make things not be subject to time is useful. But perhaps it can be cool? It has to be in your possession when you first effect it, but seems to be permanently maintained thereafter. Perhaps wear the eternally fresh decapitated head of your enemes on your belt as a warning to others? Carry around exotic flowers that never wilt? An amazing, exotic steak, prepared to perfection and eternally ready for that first perfectly warm bite, kept as a gift for negotiations with anyone you find in the wilderness?
Yes, you can keep your favorite dead pet permanently preserved on your mantel piece at home without turning to taxidermy, you weirdo.
5. As a Timewarped anchor precog you join the Akashic mystic as someone with a hideaway where no one can reach you. Yours comes with a "buy one item of any kind" clause, which saves you the trouble of investing in engineering ranks or worrying about having crafting facilities while in the wild. And get one of those +20% drow batteries so you can regenerate 120 charges per day.
6. Between Advanced Preparation temporal anomaly and the Restore Consumable spell you can once per day buy a consumable out of thin air, then spam 2nd level spell slots to replenish every use of it. You still have to pay, but you don't have to anticipate your needs in advance. Great if you need a spell gem, a custom polymorph form via polymorphic serum, or a nanite hypopen.
7. Prescient Casting temporal anomaly, unlike the feat and the Esoterocist archetype ability that also protect spellcasting from an AOO hit, works against all hits, so if you like to cast in melee while surrounded pick this up. Just have the HP/SP to survive the actual damage.
8. Instance Analysis temporal anomaly is potentially a useful replacement for all the important monster identification skills (engineering, life science) you don't get and the lack of insight bonuses on culture and mysticism, especially after 8th level.
9. The 6th level (no ally temporal agression) and 18th level (chronomatic max defense stays +4 instead of going to +5) archetype trade outs are things you really don't care about, so strongly consider going Esoterocist to trade those out for the better ala carte options from that archetype. Starwright is also an option.
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10. Cheat Time combat manuevers - Sure, you think you want to use the 6 actions this spell grants you to do 5 buffs or walls (alas, you cannot summon) and then an attack to close it out, but hold on. It also allows a bull rush, dirty trick, disarm, or trip combat manuever, using dex instead of strength, and you can keep going until you're sucessful.
Take Improved Combat Manuever in your preferred manuever, which is probably either disarm (to steal the weapons from 2-3 enemies while time is stopped - allow for retries and actions to grab/move/store weapons you take) or dirty trick (you can hit the same target with different debuffs - imagine finding that you've become blinded, entangled, sickened, shaken, and off-target while time was frozen and then hit with a spell out of no where before you could even think of removing any of those save debuffing conditions; then the rest of the party attacks you with your awful AC and bad saves).
Note that the Hindering fusion gives bonuses to all of these manuevers except bull rush, but that one is easy to optimize through gear and racial traits. Dirty trick doesn't have any +2 circumstance bonuses available, but is potentially the most useful/versatile one to specialize in. Merchants of the Void AP volume does have some feats that boost the effects/use of dirty trick.
11. Guaranteed crit Disintegrate - You have a 20 paradox? You have a crit for Disintegrate on demand. You need them to fail the save in order to deliver average 294 damage, so debuff ahead of time. Have you heard of Cheat Time dirty tricks?
And I know you're also thinking, 294, that's chump change compared to the Fractured Past guy using his 1/day with a big weapon and 1-2 crits. Yep, but you took the 18th level Esoterocist archetype ability to +50% your damage 2/day on spells, right? So let's make that an average of 441 if they fail the save. That's just ahead of the two crit triple attack with the most damaging (not-unwieldy) ranged weapon in the game.
12. Guaranteed spell stealer - Have you seen Usurp Spell? Gross, right? Well, at least you have to succeed at a caster level check and face a penalty if you try to hijack a 5th or 6th level spell and turn it against them.
What's that? I can spend a 16+ paradox to autosucceed against anything within +1 CR of me? Higher against tougher opposition? Eat your own disintegrate, necrovite.
13. Ally correction - You get Modify Outcome spell, previously WW only, which is an expensive way to give an ally a reroll if they fail something important (an incapacitating save, for example). But you also get exclusive access to Temporal Bullets, which is a cheaper spell and can just flat give damage as a reaction if an ally misses, no reroll required. They can save for half, but I'm pretty sure that (Usurp Spell aside) this is the best damaging reaction in the game - the mystic reflection abilities don't do nealry as much and have save negates rather than half.