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Notes -
You're assuming the grid of grocery stores is rotated 45 degrees relative to the grid of city blocks, when it would make more sense to for them to be aligned.
I am assuming, not a grid of grocery stores, but a uniform distribution of grocery stores (or, alternatively, of all the myriad establishments that count toward the house's ASTM E2843 walkability score). The average distance (of approximately 72 % of 1/2 mile) is measured to all points (on the road network) that are within 1/2 mile of the house (as measured on the road network), not just the ones that I explicitly marked with big black dots on the edge of the 1/2-mile catchment area.
If you assume the grocery on your map is at (0,0), then your diamond shape only makes sense if there are grocery stores at (1/2, 1/2, (1/2, -1/2), (-1/2, 1/2), and (-1/2, 1/2) with the unit distance being a mile. I take back what I said about this not making sense. It is the distribution of grocery stores with the lowest density of grocery stores that are still reachable from anywhere without travelling more than 1/2 mile.
But I was assuming that the grocery stores would be at (1/2, 0), (1/2, 1/2), (0, 1/2), etc. So, instead of taking the average distance to the points within your diamond shape, you would take the average distance to the points within a 1/2 mile x 1/2 mile square. The average distance to a grocery store is less because this gives you 4 grocery stores per square mile, whereas your arrangement gives you 2 grocery stores per square mile.
I will explicitly abandon the focus on grocery stores. Rather, I will state that I am thinking of all 48 of the different establishments of which ASTM E2843's walkability score requires four to six examples within the 1/2-mile catchment area. Those 48 establishments need not be arranged in evenly spaced grids. Rather, even if they are arranged haphazardly, in total they still can result in the entire city's being considered walkable.
One house may be within range of a supermarket, a movie theater, a public park, and a bank.
A different house may be within range of a post office, a library, a hardware store, a liquor store, a farmers' market, and a zoo.
A third house may be within range of a dentist's office, a day care, a pharmacy, a book store, a bar, and a church.
All three of those houses count as being "in walkable proximity to neighborhood assets" under the requirements of ASTM E2843. Maybe you disagree, and think that: (1) only grocery stores (or perhaps even only supermarkets, not just smaller grocery stores that some would consider convenience stores) should count toward walkability; and therefore (2) a city interested in being walkable (a) would subsidize at least one grocery store within a 1/2-mile distance of every point in the city (if the interest is felt politically by the government), or (b) would result in the free market's doing the same thing (if the interest is felt culturally by the people); (3) causing grocery stores to be present in a grid pattern. The activists who decry "food deserts" (1 2) might agree with you. But ASTM disagrees with you. And I think that, as long as all 48 establishments taken together are uniformly distributed across the network of roads (or walking paths), then my rough sketch is an accurate representation of the situation.
On what point are you saing that ASTM disagrees with me?
You said:
If the city is using some kind of framework to incentivize the building of walkable neighborhoods—and ASTM E2843 is one such framework, as it is incorporated by reference in the International Green Construction Code, which the city might incorporate into its own zoning laws—then grocery stores will constitute only part of the walkability score (though admittedly a very major part, since they have triple weight in the minimum score of six points to be considered walkable), and there is no need to construct them in a perfectly regular grid. My assumption that grocery stores and other amenities that contribute toward walkability will be distributed uniformly but randomly, rather than in a rigid grid, is valid.
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