The Knockout Alternative Championship

[UncreativeUsername37]

Teams compete kind of bracket-style for the constructors' championship. It's easier to give an example:

1959
Seeding
1. Vanwall
2. Ferrari
3. Cooper
4. BRM
5. Maserati
6. Lotus
7. Porsche
8. Connaught
9. OSCA
10. Aston Martin

First round (MON and NED)
Neither of Connaught and OSCA showed up, meaning Vanwall would have no sort of opposition in the quarterfinals. Aston Martin's appearance in the second race gave Porsche some surprise competition, but Porsche's 10th to Aston's retirement meant they were still through.

Quarterfinals (FRA and GBR)
Ferrari v. Porsche was a walkover. Cooper beat Lotus twelve points to none, and BRM bested Maserati with a 2nd in the UK beating their 8th in France.

Semifinals (GER and POR)
Vanwall didn't show up, sending BRM through to the finals. Ferrari and Cooper each got one win, but Ferrari's other result of third to Cooper's fourth meant Ferrari would make it to the finals.

Finals (ITA and USA)
Cooper won the last two races, which mattered for next year's seeding but not the actual championship. Ferrari came second to BRM's no-score in Italy, then BRM's absence at the last race meant Ferrari didn't have to fight for their first constructors' championship.

[UncreativeUsername37]

1960
Seeding
1. Cooper
2. Ferrari
3. BRM
4. Lotus
5. Aston Martin
6. Porsche
7. Maserati
8. Connaught
9. Vanwall
10. Kurtis Kraft
11. JBW
12. Tec-Mec
13. Fry
14. Behra
15. Scarab

8th to 12th seeding notes: As Connaught had more entries (sort of), they were ranked above the other four. Vanwall were chosen based on 1958 results. Kurtis Kraft were picked next for their stateside exploits, then JBW were put above Tec-Mec for making any 1958 appearance in F1, even if it didn't actually count.

First round (ARG and MON)
Scarab's DNQ was no match for Ferrari's actually doing something. BRM got 7th to Behra's 12th, then Behra's absence from Monaco gave them the win. Fry never showed up at all, so Lotus advanced by default. Aston Martin and Tec-Mec both failed to show up and were both eliminated. Porsche and JBW both missed the first race, then JBW's DNQ at the second beat Porsche's complete absence. Maserati also beat Kurtis Kraft by participating. Connaught and Vanwall were both out.

Quarterfinals (NED and BEL)
No non-walkovers.

Semifinals (FRA and GBR)
Cooper won both races, giving Lotus no chance. Ferrari beat BRM 1-0 at Silverstone.

Finals (POR, ITA, and USA) (best two count)
Cooper contined their dominant form at Portugal, whilst Ferrari could only manage fourth. But then a miraculous turnaround came at Italy, Phill Hill taking pole and a comfortable win for Ferrari. Cooper failed to score. And so it was 11-8 into the final race but, incredibly, Ferrari failed to show! For whatever reason, the trip was too much to make. Bruce McLaren gave Cooper a calm third and the championship.

[UncreativeUsername37]

1961
Seeding
1. Cooper
2. Lotus
3. Ferrari
4. BRM
5. Porsche
6. Behra
7. Scarab
8. Aston Martin
9. Maserati
10. JBW
11. Vanwall
12. Emeryson

First round (MON and NED)
Porsche beat Emeryson by qualifying for something.

Quarterfinals (BEL and FRA)
Porsche beat BRM 7-1 without too much drama.

Semifinals (GBR and GER)
Brabham rose from 9th to 4th for Cooper and Bonnier fell from 3rd to 5th. Ferrari 1-2-3ed to leave Lotus with nothing. Moss' win at the Nürburgring meant nothing with Ferrari's von Trips second, and retirement for both BRMs meant a berth in the finals for Cooper.

Finals (ITA and USA)
Ferrari won by over thirty seconds. McLaren came a quiet third for Cooper, but this time they probably needed a bit more. Ferrari again didn't make the trip, but another okay race for McLaren left Cooper with just seven points.

[UncreativeUsername37]

1962
Seeding
1. Ferrari
2. Lotus
3. Porsche
4. Cooper
5. BRM
6. Gilby
7. De Tomaso
8. JBW
9. Ferguson
10. Emeryson
11. Lola

First round (NED and MON)
Lola took up the spot v. Porsche unchallenged, and Emeryson would take on Lotus.

Quarterfinals (BEL and FRA)
Ferrari were unchallenged. Emeryson didn't compete in either race. Lola had two points after Belgium, which was two better than Porsche, but another fifth didn't put them above Porsche's win in France. BRM and Cooper both got a second, BRM at Spa and Cooper at Rouen, but BRM's other result of third beat retirement.

Semifinals (GBR and GER)
Retirement and sixth. That is what the defending champions did. Not that BRM didn't deserve a place in the final round with 4th and 1st. Meanwhile Lotus won the first race with Porsche not scoring, and Porsche's third in the next race obviously wasn't enough.

Finals (ITA, USA, and RSA)
1st, 2nd, and 1st from BRM wasn't an exciting way to end it. Sure, six points ahead into the final race was technically surmountable even if luck was requided, but their perfect reliability that season didn't raise any hopes.