As far as the box search goes since they were reporting being on the 157/337 line an hour after they had arrived they would have had to been flying a modified search pattern as Gary had previously suggested versus an expanding square search pattern.
Your logic escapes me. How does the fact that Earhart reported being on the 157/337 line an hour after they seem to have arrived at the advanced LOP mean anything but that they were on the line at that time? Why could they not simply have been doing what Earhart said they were doing - "running on line north and south."? On what basis do you (and Gary) inject an entirely imagined "box search" or "modified search pattern" into the scenario? If you're going to make up stuff for which there is no evidence you can put the plane anywhere you want.
As I recall Gary suggested the search pattern initially and I agreed with the theory and I still do. While Gary and I disagree in many areas, I think that he brings a lot of good ideas to the table and this would be one of them. He seems to have quite a background in the area of navigation so I would be cautious of kicking all of his ideas to the curb and labeling contrary theories as "making stuff up". There are certain areas that are not knowable, despite the claims, so we theorize as to what might have happened using rational deduction. This is one of those areas of discussion.
As to the idea of the origin of the search pattern theory, Gary had suggested that conducting a search pattern upon arrival would be the correct thing to do given the circumstances (lack of communications, etc). That is probably obvious to any modern aviator, I am not sure if FN would have had this knowledge or not, perhaps Gary can chime in on that since he is the expert in the history of navigation.
From 19:12 GMT to 20:13 GMT they must have been up to something other than just circling around an empty spot in the ocean right? What would you be doing? I know what I would be doing, attempting to find the island in a methodical approach keeping track of time and distances traveled, in other words performing an improvised search pattern. In the meantime I would be on the radio trying to get help from my destination. FN would be busy attempting to get a new fix one way or another by any means at his disposal while simultaneously trying to spot the island himself.
The search pattern theory also explains why they reported being on "the line" 157/337 an hour after arriving to where they thought Howland should be. If you do not know where you are, "the line" offset by 40 miles is still "the line" because you have no idea where the real advanced line of position that passes through Howland really is. If they were on a line of position that passed through Howland, they probably would have found Howland. This seems self evident that the projected advanced line of position did not match reality.
Another reason that I believe that this concept has merit is that it is my understanding that FN could also be taking observations of the Sun during this time which would be 90 degrees to your direction of travel if you were on a 157/337 heading. If successful, he would be able to determine where they were longitudinally on the Earth. In the meantime, AE could be traveling long N/S legs in a search, then performing a 2 times the visibility range offset. The could cover quite a large area in a very short amount of time using this method.
Is this suggestion of a search pattern far fetched? I do not think so. I believe to that to go back and forth on the same advanced line of position (a calculated projection, a completely abstract construct without a real physical reference), expecting to find Howland on each pass would be the very definition of insanity, doing the same thing over and over expecting different results. They must have been in the area at 20:13 GMT as their signal strength was 5.
If you strongly disagree with this line of reasoning I would like to hear the alternative. What is the rational for staying on the line going back and forth (North and South as stated by AE)?
I think the only point of contention here is whether they would have used an offset. Is this really a radical departure from your line of reasoning? I think not.