Southeast corner is initially surveyed. The Northwest corner is monumented from that survey. This creates a grid within the Township where every Section has 4 corner monuments. Often, but not always, the side line centers are also surveyed and monumented to extend to protracted quarter sections.
All government lots are defined by an actual survey and monument grid. You can't "assume" a government lot exists where there has been no survey and monument.
When dealing with protracted blocks the rule is that self protraction is not reliable or useful if the nearest pin or monument is more than two miles away. For that reason the PLSS protracted blocks are just a regular grid without any attempt to assume government lots will be created. This can create odd remainder blocks but even then there will be no government lot assumed.
If you use the CadNSDI as a guide for protracted sections your location description will be recognized by the BLM. Second divisions are usually also protracted within the CadNSDI so that would be your guide to aliquot descriptions. The CadNSDI does not agree with the old 2008 PLSS grid so be careful which you use to describe locations on surveyed or unsurveyed sections.
As always the actual survey monuments on the ground are the definitive answer to where the survey lies no matter what the CadNSDI, topo or old PLSS show. When dealing with unsurveyed protracted blocks the CadNSDI is your go to for describing public lands.
If you are in an area that has not been covered by a protraction of the survey and your claim is more than 2 miles from the nearest monument you should locate by metes and bounds. (1872 Mining Act Section 10) There may be other horizontal control points than PLSS monuments that are valid for location. Look for USMM (mineral monuments), patent surveys, homestead entry surveys, townsite surveys, U.S Surveys or any other official approved survey monuments to describe your location in relation to.
This stuff can get pretty complex but there is a set of rules to describe what happens in each situation. When locating a mining claim it is a good idea to get familiar with the
Mineral Survey Procedures Guide so you will know what the rules for perfection of your location entail.
Heavy Pans