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[Tables]

With Echo, I think it is possible to win within a turn.
0: Meld Dice
1: Dogma Dice, draw Soap and meld Watermill; Dogma Watermill, tuck Soap-Crossbow-Almanac-Clock-Lightning Rod-Hot Air Balloon-Ice Cream-Air Conditioner-Wristwatch-Artificial Heart-Game Over

Having thought about it, this doesn't work. The second draw from Dice must be a base card, as you have Soap in hand. So you draw and meld a base 2, not an echoes 2.

[GenericKen]

Solo base game (7 achievement win in 8 turns):

0 Clothing over Metalworking. (Green)

1.1 Clothing dogma metalworking (Red), score Code of laws, Pottery.
1.2 Metalworking dogma 8 castle age 1 techs, claim Monument, draw agriculture.
(score 10)

2.1 Clothing dogma agriculture (Yellow), score Sailing, Writing(1), Calendar(2).
2.2 Metalworking dogma 5 castle age 2 techs, draw math.
(score 23)

3.1 Clothing dogma Math (Blue), score canal building, currency, philosophy(2), compass(3).
3.2 Metalworking dogma 4 castle age 3 techs, draw education.
(score 44)

4.1 Clothing dogma education (Purple), score last 4 age 3 cards, Any age 4 card
4.2 Education dogma age 4 card from score, draw any age 5 card.
(score 56)

5.1 Dogma Math age 5 card to meld any age 6 card.
5.2 Claim Achievement 1

6. Achievement 2/3
7. Achievement 4/5
8. Achievement 6...

[GenericKen]

4: Dogma Astronomy, draw and meld Atomic Theory, draw any 6; Dogma Math, draw and meld Electricity

Pretty sure you can't dogma Math while it's underneath Atomic Theory. You can dogma Atomic Theory though.

Starting meld: Writing
1: Dogma Writing, draw Math; meld Math
2: Draw any 2; Dogma Math. draw and meld Paper
3: Dogma Paper, splay blue, draw any 4; Dogma Math, draw and meld Astronomy

4: Dogma Astronomy, draw and meld Atomic Theory, draw any 6 (may also draw and meld any green to claim Universe achivement); Dogma Atomic Theory, draw and meld Electricity
5: Dogma Electricity, return Atomic theory, draw any 8; Dogma math, draw and meld Computers
6: Dogma Computers, draw-meld-dogma-win with any 10 (I believe Bioengineering, Globalization, Self-Service, and Robotics->Software->AI all win in solitaire games).