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80 Cost
93 Cycles
13 Area
However you could obviously solve the first problem by just taking a "Single arm and gripper" and a "Glyph of Projection"
That would give you:
40 Cost
Huge Cycles.....
6 Area
Since there is no way to solve the problem without either the arm or the Glyph I am confident in saying that those numbers are the absolute minimum.
But how do you compare area to cycles in order to decide better?
A bunch of solutions are here, I have most of the "hard" problems in chapters 1 and 2 optimized for cycle count. for refined gold, here are my best: https://imgur.com/a/s37tv
179 cycles
9 area
These are not for single solutions, but best scores optimizing one category at a time (some solutions have both minimum cost and space, though).
In puzzle order, best scores for cost/cycles/area:
Prologue:
40/15/7
Chapter 1:
40/64/6
40/32/7
40/27/8
40/40/8
40/22/12
60/28/14
40/31/xx
60/43/xx
Chapter 2:
40/50/10
50/39/xx
65/36/12
60/22/14
50/xx/20
40/27/10
xx/32/xx
If you want to look at Refined Gold specifically (I assume this is the gold puzzle you are talking about), my two solutions are
40 g/179 cycles/6 area
175g/64 cycles/15 area
http://steamcommunity.com/app/558990/discussions/0/2381701715719274247/
If you're looking for the current known minimum scores for each category:
https://www.reddit.com/comments/7ip2q3/table_of_minimum_cost_solutions/
https://www.reddit.com/comments/7qmkv6/list_of_current_cycle_optimal_scores/
https://www.reddit.com/comments/7house/list_of_current_area_optimal_scores/