Some notes on TGE -Itu Since the advent of Endwalker and the nonstandard revolution, Black Mages have started to develop a framework to understand how to optimally use transpose to put together strong fight plans. A line is a sequence of ice, fire and paradox casts (including possibly transpose weaves) which starts and ends at the end of a fire phase (on AF3 and enochian assumed to have a few seconds left). Because ogcds, Xenoglossy and Thunder III usages are flexible, the hope is to see which lines are worth using in different contexts and use them as building blocks to form good fight plans. The discussion below assumes familiarity with Black Mage optimization, see here and here for references, the lines comparison sheet is especially important reading. All numbers are calculated at 840 sps with 0.13s caster tax and 0.7s ogcd clip. The strategy is to build a theoretical framework by considering very long time horizons first, and only then thinking about shorter timeframes and applications to actual fight plans. Potency, PPS, PGE The potency of the basic spells at the time of writing is as follows: Spell AF3 AF2 AF1 UI3 UI2 UI1 Base Fire 1 324 288 252 180 Fire 3 468 416 364 182 208 234 260 Fire 4 558 496 434 310 Despair 612 544 476 340 Blizzard 1 126 144 162 180 180 180 180 Blizzard 3 182 208 234 260 260 260 260 Blizzard 4 310 310 310 310 Paradox 500 500 500 500 500 500 500 We also have Xenoglossy at 800, and Thunder III at 50 with a 350 thundercloud (note the DoT scalar is not affected here) and 35 base potency tick for 30 seconds. The two main attributes of a line are Time and Potency. The final goal of optimization is to obtain the most amount of potency in a given time frame determined by the fight, so we could use the total potency of lines and compare them in this way. But this number is hard to compare across lines with different length, so Potency Per Second (PPS) is a first idea to achieve our goal. If we maximize the PPS of our plan, we have also maximized the total potency. This is useful to compare lines that can be repeated indefinitely, but has a number of limitations. ID Casts Potency Time PPS 114 UI1 PD AF1 F3P 2xF4 Desp 2592 13.47 192.43 120 UI1 B4 AF1 F3P 3xF4 PD 3xF4 Desp (F3P) 5321 28.3 188.04 0 B3 B4 PD F3 3xF4 PD 3xF4 Desp (F3P) 5821 33.14 175.67 For example, it is not clear that a higher PPS line is better than a lower PPS line. Assume that we start with a rotation where we only repeat the Standard line (N0), and we are given a chance to include one of the above lines exactly once. PPS might lead us to think N114 is the correct choice, but in fact N120 is stronger in this setting. What this means is: measuring the total potency of both sequences at a fixed but arbitrary point in the future, the sequence N0 N0 ... N0 N114 N0 ... is expected to measure lower than the sequence N0 N0 ... N0 N120 N0 ... What does this mean exactly, and how do we know this? The first line has replaced the standard rotation for 13.47 seconds, replacing a 175.67pps line with 192.43pps, for an expected potency gain of (192.43-175.67)*(13.47) = 225.82, while for the second line the expected gain is (188.04-175.67)*(28.3) = 350.20. The potency gain equivalent (PGE) of a line with time T and pps P, with reference to a baseline pps S, is (P-S)*T. In out setting above, the baseline was the pps of the Standard line, chosen by the assumption that our rotation, aside from the comparison being made, consists on repeating the standard line. This number measures how much potency we expect to gain by replacing a generic part of our baseline rotation with a particular line. PGE has very desirable properties. First, our goal of maximizing the total potency in a given timeframe is equivalent to maximizing the total PGE in the same timeframe, no matter the baseline. Second, given two lines L1 and L2, the total PGE of the sequence L1+L2 is PGE(L1+L2) = PGE(L1) + PGE(L2). Therefore lines can be quickly compared in isolation. We can for example conclude that in the comparison above we expect to gain 350.20-225.85 = 124.35 potency by using N120 over N114. However, PGE comparisons are sensitive to the choice of baseline. If we had instead chosen a hypothetical strong rotation with a baseline pps of 190, then the PGE of N114 and N120 would be 35.73 and -55.47 respectively, telling us that N114 is a better choice under this assumption. What this tells us is that in order to accurately use PGE, we need to choose a baseline pps that will best approximate expected close to optimal performance. OGCDs and TGE The next question is how to evaluate the value of OGCDs, which can let us know if it’s worth it to use them in alternative ways, compared to losing a usage for alignment gains for example. Using a Sharpcast on an F1 or Fire Paradox cast increases the chance of obtaining a Fire III proc (F3p) from 40% to 100%, and using it on a Thunder III cast increases the chance of a Thundercloud proc (T3p) from about 35% to 100%. The value of a T3p is at last 350 potency (since the T3p can be used as a filler) and the value of F3p is at least 182 potency (since AF1 F3p standard is generally guaranteed, and F3p can be used to enable different lines). For a first choice of baseline, we will start by using the Standard line and value obtaining an F3p at 182 potency. We write aPPS for PPS adjusted with this F3p valuation, but will usually just refer to it as PPS when needed. One can interpret this is as the baseline of doing AF1 F3p Standard, though it is not exactly the same thing, and will refer to it as the AF1 F3p baseline. Casts F3p Potency Time aPPS B3 B4 PD F3 3F4 PD 3F4 Desp 0.4 5634 32.47 175.76 In order to evaluate the loss from clipping the Thunder III DoT, we may assume that our DoT will be kept up, since this is the correct thing to do for arbitrary long lengths. Therefore we may ignore the DoT and T3p generation. Losing 30 seconds of clip then forces to use an extra Thunder III cast, which has a 65% chance of having a T3p and casting at 400 potency, and 35% chance of not having a T3p and casting at 50 potency (a weighted T3 cast). The PGE with AF1 F3p baseline of this 30 second clip is -171.26, so clipping 10 seconds expects a loss of 171.26/3 = 57.09 potency, for example.To evaluate instead the loss of 30 seconds of DoT, we again consider a weighted T3, and we include the DoT in the potency, giving a PGE of 185.04. We can use these evaluations somewhat concretely. If there are 60 seconds left of fight, and out DoT is running out with no T3p and no Sharpcasts, do we cast Thunder III? The first cast has 50 potency plus the DoT for -48.76 PGE while the second is a weighted T3 with 185.04 PGE as before, for a total of 142.57 PGE. Enabling Manafont in the usual way gives an inclusion of the line F4 Desp, so we can use PGE to find find how much we potency we expected to gain by doing so: an extra F4 gives 58.27 PGE and an extra Despair gives 77.12 PGE, for a total of 135.38 PGE. Time OGCDs will need a new approach. Ley Lines give about a (30/8.55 – 30) = 5.29s improvement, while getting instant spells from Swiftcast and Triplecast will give us 0.33s from Fire 4, 0.44s from Despair, and 1.10s from slow Fire 3. In practice, we need to compare various potency and time gains and losses from different options. Line Casts F3p Potency Time aPPS PGE TGE(s) Standard B3 B4 PD F3 3F4 PD 3F4 Desp 0.4 5634 32.47 175.76 0 0 4F4 Transpose UI1 PD F3 4F4 Desp 0 3552 20.36 174.46 -26.39 -0.15 Consider the line 4F4 Transpose once again in the context of playing AF1 F3p standard. On its own, it tracks a PGE of -26.39, which makes it a loss. However, in the presence of Swiftcast, the best we can do in the Standard line is Desp* for a gain of 0.44s, while 4F4 transpose enables an F3* for a gain of 1.10s. Under these conditions, by doing 4F4 Transpose we have lost 26.39 potency but gained 0.66s. Is this worth it? To put potency and time analysis in equal footing, we can fix an exchange rate between one or the other. Given a long amount of free time, we can always convert it into potency by doing our baseline rotation (AF1 F3p Standard in this case). The Time Gained Equivalent (TGE) of a line with time T and pps P with reference to a baseline pps S is (P/S -1)*T. We may interpret the ratio P/S as the relative PPS compared to our baseline, then the quantity (P/S-1) calculates how much time of doing our baseline pps we are saving each second by doing P pps instead, and TGE calculates the total amount of time we have saved by using the line. Notice that TGE and PGE are scalar multiples of each other by PGE = TGE*S, so TGE enjoys all the good properties of PGE discussed before. The usage of TGE in general is justified by the flexibility of BLM: given an initial strong plan and a given, maybe small, amount of extra time, one can carefully modify the plan to fit the new timeframe with little change to final total pps. Said differently, aside from cooldown resource timing, we may assume the length of a fight does not significantly modify the average damage expected, so we have use the baseline as an exchange rate between TGE and PGE. Back to our example above, doing 4F4 Transpose when we have a Swiftcast available incurs in -26.39 PGE and +0.66 TGE. By using our exchange rate of 175.76, we can net a total PGE of +89.61, making it a gain. What this means is, compared to doing AF1 F3p standard on a dummy and using Swift on Despair when available, doing AF1 F3p standard on a dummy and using 4F4 Transpose with Swift when available will expect a total potency gain of 89.61 for every time this is done. Line Comparison, Filler Value Line Casts F3p Potency Time aPPS PGE TGE(s) Standard w AF1 F3P B3 B4 PD AF1 F3P 3F4 PD 3F4 Desp 0.4 5816 32.47 181.36 182.00 1.04 3F4 2xTranspose F3P UI1 PD AF1 F3P 3F4 Desp 0 3150 16.41 191.92 265.26 1.51 Using PGE allows us to compare lines in concrete terms. 3F4 Double Transpose F3p (N111) above is a popular strong line enabled to F3p. We see that using the F3p in this way as opposed to AF1 F3p standard nets a PGE of 265.26-182 = 83.26, at the cost of the necessary fillers (2). One could use this to infer that the value of a filler is at least 40 potency. However N111 also enables the used of a triplecast to become UI1 PD AF1 F3P 3F4* F4 Desp (N112), which nets an extra F4 (+58.27 PGE) and the usage of Triplecast on 3F4* (+1.27 TGE) with the opportunity cost of using it on 2F4* Desp* (-1.47 TGE) netting a total of +23.12 PGE from this alternative Triplecast usage. Image: An example sheet with the AF1 F3p baseline discussed above, listing the viable lines. The additive property of TGE and PGE allows us to value variants and inclusions, as well as quickly identify gains on each line obtained by using Swift or Triplecast. In this framework, we may get higher value out of fillers by executing non-f3p transpose lines. Line N70 above (UI1 B4 AF1 PD F1 3F4 Desp) has a PGE of 120.10 for two fillers (when the mana ticks are correctly lined up), while N69 has a PGE of 178.37 for three fillers. This suggests that the optimal playstyle for long periods of time includes these lines. Similarly, one may see that 4F4 Double Paradox has slightly higher value than Standard (67.98-58.27=9.27 PGE). Conclusions The proposal here is about methodololgy, the specific numbers and lines are less relevant, though I have tried to discuss the most important ones. I am not convinced that AF1 F3p standard is the correct baseline. The current meta involves alternating between Standard and N111/N112/I14, so a studying a baseline based on this rotation should give better results. I am convinced the theory can be refined to more coherently include the value of filler generation, ogcd opportunities of usage, and so on. I do believe that non-F3p double transpose lines and double paradox 4F4 are the way to go. Most of these lines are hard to fit on a fight plan because we rely on their natural F3p generation chance to make gains, which means we need to adjust on the fly. Using stronger lines, and as a result misaligning ogcds, clipping and dropping DoTs, losing Swiftcast usages, and using unoptimal triples, will result in losses. Actual fights are tight, and the level of adjustment and mental calculation required to play this style optimally seems daunting. My hopes are that people will at least be motivated to try this approach, and find practical ways of including aspects of it to push the boundaries of BLM gameplay. I want to give special thanks to those involved with the development of the theory of BLM and the Black Mages of The Balance and for teaching it to me. I want to finish with an excerpt of a log of StrikingDummy gameplay, the AI developed by Lady Yunalesca at 840sps with Endwalker 6.3 BiS, pushing a DPS of 10880 before the 5% party buff (for a total of around 11.4k rdps with the 5%). ============= DPS: 10880.28 T3 uptime: 97.99% F4 % damage: 39.19% Desp % damage: 12.60% Xeno % damage: 14.89% T3 % damage: 6.71% Dot % damage: 5.37% ============= [00:00:00.00] 8000 T3 at 0.0s left on dot [00:00:02.53] 7600 TRIPLE** (T3p w/ 39.4s) [00:00:03.25] 7600 F4 (T3p w/ 38.7s) [00:00:03.97] 6000 AMPLIFIER (T3p w/ 37.9s) [00:00:04.88] 6000 LEYLINES (T3p w/ 37.0s) [00:00:05.67] 6000 XENO* [[F4]] (T3p w/ 36.2s) [00:00:06.39] 6000 SHARP* [[HQ_TINCTURE_OF_INTELLIGENCE]] (T3p w/ 35.5s) [00:00:07.73] 6000 F4 (T3p w/ 34.2s) [00:00:08.45] 4400 HQ_TINCTURE_OF_INTELLIGENCE (T3p w/ 33.5s) [00:00:09.79] 4400 F4 (T3p w/ 39.0s) [00:00:10.51] 2800 SWIFT [[TRIPLE*]] (T3p w/ 38.2s) [00:00:11.22] 2800 TRIPLE* [[NONE]] (T3p w/ 37.5s) [00:00:11.94] 2800 DESPAIR (T3p w/ 36.8s) [00:00:12.66] 0 MANAFONT (T3p w/ 36.1s) [00:00:14.00] 3000 F4 (T3p w/ 34.7s) [00:00:16.06] 1400 DESPAIR (T3p w/ 32.7s) [00:00:16.78] 0 TRANSPOSE (T3p w/ 32.0s) [00:00:17.53] 3200 LUCID [[NONE]] (T3p w/ 31.2s) [00:00:18.25] 3200 PARADOX (T3p w/ 30.5s) [00:00:20.31] 8450 F3 (T3p w/ 28.4s) [00:00:22.37] 8450 F4 (T3p w/ 26.4s) [00:00:24.79] 6850 F4 (T3p w/ 24.0s) [00:00:27.20] 5250 F4 (T3p w/ 21.5s) [00:00:29.62] 3650 F4 (T3p w/ 19.1s) [00:00:32.04] 2050 DESPAIR (T3p w/ 16.7s) [00:00:34.62] 0 T3p at 0.0s left on dot (T3p w/ 14.1s) [00:00:35.62] 0 TRANSPOSE (T3p w/ 39.0s) [00:00:36.34] 0 SHARP* [[NONE]] (T3p w/ 38.3s) [00:00:37.06] 0 XENO* [[PARADOX]] (T3p w/ 37.6s) [00:00:39.48] 3750 B4 (T3p w/ 35.1s) [00:00:42.02] 6350 T3p at 23.6s left on dot (T3p w/ 32.6s) [00:00:43.88] 9550 TRANSPOSE (T3p w/ 38.1s) [00:00:44.60] 9550 PARADOX (T3p w/ 37.4s) [00:00:47.14] 7950 F1^ (T3p w/ 34.9s) [00:00:49.68] 7150 F4 (T3p w/ 32.3s) [00:00:52.50] 6350 F4 (T3p w/ 38.2s) [00:00:55.33] 5550 F4 (T3p w/ 35.4s) [00:00:58.16] 3950 F4 (T3p w/ 32.6s) [00:01:00.98] 2350 DESPAIR (T3p w/ 29.8s) [00:01:04.01] 0 XENO* (T3p w/ 26.7s) [00:01:04.73] 0 TRANSPOSE (T3p w/ 26.0s) [00:01:05.44] 3200 SHARP* (T3p w/ 25.3s) [00:01:06.43] 3200 B4 [[PARADOX]] (T3p w/ 24.3s) [00:01:08.97] 5800 T3p at 4.0s left on dot (T3p w/ 21.8s) [00:01:10.88] 9000 TRANSPOSE (T3p w/ 38.1s) [00:01:11.60] 9000 F3p (T3p w/ 37.4s) [00:01:14.02] 9000 F4 (T3p w/ 34.9s) [00:01:16.85] 8200 PARADOX [[F4]] (T3p w/ 32.1s) [00:01:19.39] 6600 F4 (T3p w/ 29.6s) [00:01:22.21] 5800 F4 (T3p w/ 38.5s) [00:01:25.04] 5000 F4 (T3p w/ 35.7s) [00:01:27.87] 3400 F4 (T3p w/ 32.9s) [00:01:30.69] 1800 XENO* (T3p w/ 30.1s) [00:01:31.41] 1800 SWIFT (T3p w/ 29.3s) [00:01:33.11] 1800 DESPAIR (T3p w/ 27.6s) [00:01:33.83] 0 SHARP* [[TRANSPOSE]] (T3p w/ 26.9s) [00:01:34.55] 0 TRANSPOSE (T3p w/ 26.2s) [00:01:35.53] 3200 PARADOX (T3p w/ 25.2s) [00:01:36.25] 3200 LUCID [[TRIPLE*]] (T3p w/ 24.5s) [00:01:36.97] 3200 TRIPLE* (T3p w/ 23.8s) [00:01:37.95] 8450 T3p at 2.0s left on dot (T3p w/ 22.8s) [00:01:39.55] 8450 TRANSPOSE (T3p w/ 38.4s) [00:01:40.37] 8450 F3 (T3p w/ 37.6s) [00:01:42.79] 4450 F4 (T3p w/ 35.2s) [00:01:45.21] 2850 DESPAIR (T3p w/ 39.5s) [00:01:45.93] 0 TRANSPOSE (T3p w/ 38.8s) [00:01:46.65] 550 SHARP* [[NONE]] (T3p w/ 38.1s) [00:01:47.63] 3750 B4 (T3p w/ 37.1s) [00:01:50.17] 6900 T3p at 18.8s left on dot (T3p w/ 34.6s) [00:01:51.17] 6900 TRANSPOSE (T3p w/ 39.0s) [00:01:52.59] 6900 PARADOX (T3p w/ 37.6s) [00:01:55.13] 5300 F1 (T3p w/ 35.0s) [00:01:57.66] 4500 F4 (T3p w/ 39.1s) [00:02:00.49] 3700 F4 (T3p w/ 36.3s) [00:02:03.32] 2900 F4 (T3p w/ 39.4s) [00:02:06.15] 1300 DESPAIR (T3p w/ 36.6s) [00:02:09.17] 0 T3p at 12.0s left on dot [[XENO*]] (T3p w/ 39.6s) [00:02:09.89] 0 LEYLINES [[TRANSPOSE]] [00:02:10.61] 0 TRANSPOSE [00:02:11.59] 3200 B4 [00:02:13.77] 2600 XENO* [00:02:15.00] 5800 AMPLIFIER [00:02:15.83] 5800 XENO* [00:02:16.88] 9000 TRANSPOSE [00:02:17.89] 9000 PARADOX [00:02:20.07] 7400 F1 [00:02:22.24] 6600 F4 [00:02:24.66] 5800 F4 [00:02:27.08] 5000 F4 (T3p w/ 39.7s) [00:02:29.49] 3400 F4 (T3p w/ 37.3s) [00:02:31.91] 1800 F3p [[XENO*]] (T3p w/ 34.8s) [00:02:32.63] 1800 TRIPLE* [[SWIFT]] (T3p w/ 34.1s) [00:02:33.97] 1800 DESPAIR (T3p w/ 38.8s) [00:02:34.69] 0 MANAFONT (T3p w/ 38.1s) [00:02:36.03] 3000 F4 (T3p w/ 36.7s) [00:02:36.75] 1400 SHARP* (T3p w/ 36.0s) [00:02:38.09] 1400 DESPAIR (T3p w/ 34.7s) [00:02:38.81] 0 TRANSPOSE (T3p w/ 33.9s) [00:02:40.15] 0 PARADOX (T3p w/ 32.6s) [00:02:40.87] 0 SWIFT (T3p w/ 31.9s) [00:02:41.59] 4700 LUCID (T3p w/ 31.2s) [00:02:42.57] 4700 T3p at 0.0s left on dot (T3p w/ 30.2s) [00:02:43.88] 9950 TRANSPOSE (T3p w/ 38.7s) [00:02:44.99] 9950 F3 (T3p w/ 37.6s) [00:02:47.41] 5950 F4 (T3p w/ 35.2s) [00:02:50.24] 4350 F4 (T3p w/ 32.3s) [00:02:53.07] 2750 F4 (T3p w/ 29.5s) [00:02:55.89] 1150 XENO* [[DESPAIR]] (T3p w/ 26.7s) [00:02:56.61] 1150 TRANSPOSE (T3p w/ 26.0s) [00:02:57.33] 1150 SHARP* (T3p w/ 25.2s) [00:02:58.31] 1150 B4 (T3p w/ 24.3s) [00:03:00.85] 4300 T3p at 12.7s left on dot (T3p w/ 21.7s) [00:03:02.59] 8050 TRANSPOSE (T3p w/ 38.3s) [00:03:03.30] 8050 PARADOX (T3p w/ 37.5s) [00:03:05.84] 6450 F1 (T3p w/ 35.0s) [00:03:08.38] 5650 F4 (T3p w/ 32.5s) [00:03:11.20] 4850 F4 (T3p w/ 29.6s) [00:03:14.03] 4050 F4 (T3p w/ 26.8s) [00:03:16.86] 2450 F4 (T3p w/ 24.0s) [00:03:19.68] 850 DESPAIR (T3p w/ 21.2s) [00:03:22.71] 0 XENO* (T3p w/ 18.1s) [00:03:23.43] 0 TRANSPOSE (T3p w/ 17.4s) [00:03:24.15] 0 SHARP* [[NONE]] (T3p w/ 16.7s) [00:03:25.13] 0 PARADOX (T3p w/ 15.7s) [00:03:27.55] 4700 T3p at 4.3s left on dot (T3p w/ 13.3s) [00:03:28.88] 9400 TRANSPOSE (T3p w/ 38.7s) [00:03:29.97] 9400 F3p (T3p w/ 37.6s) [00:03:32.39] 9400 F1 [[F4]] (T3p w/ 35.2s) [00:03:34.93] 7800 F4 (T3p w/ 32.6s) [00:03:37.76] 6200 F4 (T3p w/ 29.8s) [00:03:40.58] 4600 F4 (T3p w/ 27.0s) [00:03:43.41] 3000 F4 (T3p w/ 24.1s) [00:03:46.24] 1400 DESPAIR (T3p w/ 38.5s) [00:03:49.26] 0 T3p at 9.3s left on dot [[XENO*]] (T3p w/ 35.5s) [00:03:49.98] 0 TRANSPOSE [00:03:50.70] 0 TRIPLE* [00:03:51.68] 0 PARADOX [00:03:52.41] 0 LUCID [[NONE]] [00:03:54.10] 4700 XENO* [00:03:54.82] 4700 SHARP* [[NONE]] [00:03:55.88] 9950 TRANSPOSE [00:03:56.60] 9950 F3 [00:03:59.02] 5950 F4 [00:04:01.44] 4350 F4 [00:04:02.16] 2750 SWIFT [00:04:03.86] 2750 DESPAIR [00:04:04.58] 0 TRANSPOSE [00:04:05.30] 3200 TRIPLE* [[NONE]] [00:04:06.28] 3200 PARADOX [00:04:08.70] 8450 F3 [[XENO*]] [00:04:10.41] 8450 LEYLINES [00:04:11.13] 8450 F4 [00:04:13.19] 6850 F4 [00:04:15.25] 5250 F4 [00:04:17.67] 3650 XENO* [00:04:18.38] 3650 TRANSPOSE [00:04:19.10] 3650 AMPLIFIER [00:04:19.82] 3650 T3 at 0.4s left on dot [00:04:21.99] 6450 XENO* (T3p w/ 39.4s) [00:04:23.38] 9650 TRANSPOSE (T3p w/ 38.0s) [00:04:24.10] 9650 PARADOX (T3p w/ 37.3s) [00:04:26.28] 8050 F1^ [[F3p]] (T3p w/ 35.1s) [00:04:28.45] 6450 F4 (T3p w/ 32.9s) [00:04:30.87] 4850 F4 (T3p w/ 30.5s) [00:04:33.29] 3250 F4 (T3p w/ 28.1s) [00:04:35.70] 1650 DESPAIR (T3p w/ 25.7s)