The Mathematician's Ghost is the third episode of Foundation's first season on Apple TV+. It aired on October 1, 2021. The episode was written by Olivia Purnell based on the novels by Isaac Asimov. Alex Graves directed the episode.
Four hundred years earlier, Cleon I prepares for his death and the start of Imperial cloning that will begin the Genetic Destiny. Demerzel assures him his legacy will live on, but he is regretful that he will not see the completion of the Star Bridge.
Nineteen years after the bombing of the Star Bridge, Brother Dusk prepares for his final day before his ascension. Dusk wonders about his legacy and what has become of Cleon I's dream now that the Star Bridge orbits Trantor in a decaying orbit. Reminded by his brothers that they are the true dream and legacy of Cleon I, Trantor Station is destroyed to prevent it from threatening the planet. At his ascension, Brother Dusk proclaims Brother Dawn to be Brother Day, with Brother Day becoming Brother Dusk. The newly born Brother Dawn is present to Dusk, now termed as Brother Darkness. As the ceremony takes place, Darkness hears Dawn's crying and believes something is wrong with the child. Despite his apprehension, Darkness ascends, leaving the Genetic Dynasty with three members once more.
The slow ship eventually arrives at Terminus. The colonists begin to settle the planet and discover the Vault. When they try to approach it, the null field incapacitates them. As Terminus began to grow, the settlers avoided the Vault and myths grew around it: an alien artifact, or a surveillance post established by the Empire. Eventually it became mundane to the growing colony. As the decades pass, Mari and Abbas have a child; Salvor Hardin, who grows to fulfil the role of Warden of Terminus, protector of the colony.
Salvor discovers that the null field projected by the Vault is expanding. Taking her findings to her parents, Salvor is worried that if the field keeps expanding, it will endanger the colony. Guided by visions of a mysterious boy running around, Salvor discovers that they are not alone on Terminus -- a band of Anacreons have also made planetfall.
- Lee Pace as Brother Day (Cleon XIII)
- Lou Llobell as Gaal Dornick
- Leah Harvey as Salvor Hardin
- Laura Birn as Demerzel
- Terrence Mann as Brother Dusk (Cleon XI & Cleon XII)/Cleon I
- Cassian Bilton as Brother Dawn (Cleon XIV)
- Elliot Cowan as Lewis Pirenne
- Daniel MacPherson as Hugo
- Sasha Behar as Mari Hardin
- Kubbra Sait as Phara Keaen
- Pravessh Rana as Rowan
- Jairaj Varsani as Poly
- Buddy Skelton as Keir
- Chloe Lea as Gia
- Joni Morris as Laylo Fulham
- Clarke Peters as Abbas
- Kim Adis as Lowre
- Johanna O'Brien as Shivaughn
- Nikol Kollars as Jacenta
- Ben Bazell as Dusk Double
- Matt Gavan as Jord Hasp
- Stuart McQuarrie as Lunden
- Catriona Loughlin as Vreede
- Garry Mountaine as Master Ilan
- Pius McGrath as Encyclopedist
- Slaney Power as Daia
- Isaiah Chambers as Mystery Boy
- Nikhil Parmer as Freestone