The build's title "Demonic Transcension" not only is a play on words with Swain's Ultimate "Demonic Ascension", but also a point of irony, since due to the core items of the build, not only does it have the Transcedence sorcery rune in every version of the runepage as it helps a lot with getting multikills since Swain's E and W have considerable base cooldown, the effect of the Mythic also makes you "Transcended" and you posses a "Demonic Embrace", and all together makes you an oppressive entity on the battlefield which can almost disregard enemy damage and just focus on engaging.

To ascend is to become something. To transcend is to exceed something, so a Demonic Transcension is supposed to be one step above a Demonic Ascension, where you even surpass demonic entities.

Swain's ult with 0 haste on max level has the exact same cooldown as Radiant Virtue.
Also the second activation explosion of the ult can also activate the item's passive in case you used your ult before Radiant Virtue's cooldown had ended, meaning Swain has 2 opportunities in combat to activate Radiant Virtue, once when he first transforms and again when he explodes.

The extra healing for the first 9 seconds of your ult plus the added Max Health by the Transcended effect grants Swain an extra major Tank status that allows him to almost disregard enemy threats and just focus on dealing damage while barely taking any damage.

Combined with the extra health that Fimbulwinter gives based off of your mana and Swain's passive, it makes Demonic Embrace be enough to be the only AP giving item, specially with Abyssal Mask cursing the enemies and reducing their MR, which also severely reduces the need of any magic penetration.

Meanwhile, Abyssal Mask also grants you healing based off of using mana and give more mana the more you tank, which means you have another source of healing and will basically never run out of mana mid combat.

Tying everything up is Spirit Visage, which synergizes with Swain's Ult, Radiant Virtue's transcended healing, Abyssal Mask's healing, Fimbulwinter's barrier and also either Conqueror+Triumph or Grasp of the Undying's healing.

All 5 main build path items have perfect synergy with each other and with Swain's skill set.

Radiant Virtues Transcended effect is absolutely tailor made for Swain's ultimate.
Not only its cooldown is exactly the same as Swain's ultimate base cooldown at maximum level without any ability haste, its effect is so synergetic with the ultimate that it could almost be just an extension of it.
The extra 9 seconds of healing based off of max-life perfectly stacks with the ultimates healing-per-second based off of damage dealt to enemy champions, while the area healing to allies further makes Swain ult range into a safezone and the extra max health given is a missing piece of the puzzle for Swain's ultimate which doesn't naturally increases his max health, making it so that even if you were close to dying, ulting can completely bring you back as a threat in the middle of a teamfight.

Demonic Embrace is needed early for the extra AP so that your damage simply can't be ignored, but more than anything, its passive effect of converting 2% max health into AP is invalluable considering the ammount of max health the full build grants you, which also works with the temporary max health granted by the Transcended state of your mythic, meaning you are stronger the first 9 seconds of your ult, while the damage per second is perfectly synergetic with his ultimate.

Fimbulwinter not only fixes any possible mana issues early through Tear of the Goddess, but its effect of converting mana into max health further synergizes with Demonic Embrace. The Barrier every time you hit your E is also a great addition and since this build's tankyness allows you to engage with multiple enemies, almost always will you get an extra 80% barrier due to having 2 or more enemies nearby, which is even more valluable if you are using this guide to play as support.
As we also run Manaflow Band in the runes, the maximum mana granted by that rune is also converted into max health.

Spirit Visage further increases the healing from your runes, Mythic and your ultimate, it also increase the barrier given by Fimbulwinter. meaning it synergizes with 3 out of the 5 non-boots main items of your build including...

Abyssal Mask, it synergizes perfectly with Fimbulwinter, by granting more mana that becomes more health while still also giving max health which then all becomes AP.
It also converts your mana expenditure into healing and damage taken into mana, meaning you never run out of mana as long as you are tanking in a fight, which with this build becomes easy to do.

If the enemy team has too many Magic Damage dealers, specially if they have DoT damage, you will swap either Spirit Visage or, preferably, Abyssal Mask for Force of Nature.

If your enemies have a lot of physical damage dealers, you will swap either Spirit Visage or Abyssal Mask, possibly both, for one or two out of the four following items:

-Gargoyle Stoneplate: If the enemies have magical and physical damage, Gargoyle Stoneplate is actually an extremely effective item to make you extremely tanky at once, preferably keeping Spirit Visage due to the synergy the two items have together where Spirit Visage increases Gargoyle Stoneplate's barrier by 25%, meaning that from 100 + 90% of your bonus health original barrier, you receive an extra 25 + 22,5% of your bonus health going to a total of 125 + 112,5% of your bonus health of barrier, getting more than all of your total bonus health as a barrier.

-Sunfire Aegis: More offensive, while still granting you a great resistance against physical damage, if you are snowballing.

-Thornmail: Anti-heal if your enemies have too much healing and are mostly basic attack based damage.

-Randuin's Omen: If your enemies are building Critical Strike, Randuin's becomes a very useful item as a counter while being a good alternative for enemies with high mobility.

-Frozen Heart: If they are Basic Attack damage dealers while also synergizing with Fimbulwinter, converting the maximum mana given into health which then is converted into AP by Demonic Embrace.

-Rylai's Crystal Scepter: The main option situational item against enemy teams with high mobility or if you are not being able to stick close to your opponents during your ultimate since it will simply permanently slow down every enemy inside your ultimate.

Lastly, you can also swap boots in place a 6th item if you are snowballing enough and want a 6 item build instead of a 5 items + boots.
Preferably for Rylai's Crystal Scepter, since slowing your enemies is a great alternative to increasing your movement speed.

Just to run some numbers in an example:

Results shown below were level 18 full build inside Practice Tool with only 21 passive stacks and Conqueror rune page (so I could minimize the effect of the passive and not have grasp on test results)
Items were the standard build for Top and Mid.

-Recorded Total Stats

Health:
5170 Normal
5687 Ult Transcended

AP:
142 Normal
152 Ulted
161 In combat .

MR:
180 Normal
216 in Combat (4 enemies)

Armor:
164 All time
___________________________________________________________

-Extra stats values

AP:
75 (+58) Dem Emb = 133 AP

Mana:
860 Fmb + 300 Aby Msk = 1160 Mana

Health
400 (+400 Myth Pass +517 When Ult) Rad Vrt + (174 + 350) Fmb + 350 Dem Emb + 500 Aby Msk + 450 Spr Vsg = 2624 (3141) Health

Armor:
30 Rad Vrt (+20 Boots*)

MR:
30 Rad Vrt + 40 Aby Msk + 50 Spr Vsg = 120 MR

Ab Haste:
20 Rad Vrt + 10 Spr Vsg + 10 Aby Msk + 15 Fmb = 55 Haste