Great Bustard
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Since I'm the author of the Equivalence Essay (but, obviously, not the first, or even close to the first, to understand the concepts), I think I'm rather well qualified to debunk your statement above.The weakest part of the “theory” is, in order to make the theory stand, we need to assume the fixed 1.6x noise/ISO difference between FF and aps-c. This is hardly true when you look at dslr from different generations or even dslr of the same generation from different vendors. I know the author understand this even though he sometimes try not to mention it, but most people who quote the theory do not. It becomes another source of confusion and reason for the conflict in the cases you mentioned in your op.
First of all, the whole notion that Equivalence is based on equal noise is one of the Ten Myths:
h ttp: www.josephjamesphotography.com/equivalence/index.htm#7
The most controversial visual property of equivalent images is that people incorrectly assume that Equivalence is based on equal noise.
So why would "the author" of Equivalence "sometimes try not to mention it" when "it" gets a special mention in the Essay? But, guess what? It's also mentioned in the very definition of Equivalence:
h ttp: www.josephjamesphotography.com/equivalence/index.htm#equivalence
Equivalent images are not "equal", but instead have five equal attributes which all correspond to the visual properties of the final photo. So, while equivalent images on different formats will usually have the most similar visual properties, they will not be identical, as other visual elements, such as noise, detail, flare, moiré, distortion, bokeh, etc., will not necessarily be the same, and sometimes, radically different.
So, I'd say that what's "weak" is people misrepresenting what "the author" said, and did not say. Whether that misrepresentation (and/or confusion) comes from laziness (criticizing a document they've not read), poor reading comprehension, ignorance, or contrary agendas, well, I'm sure that depends on the individual. Shame, really.