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Given the very close results of the 4 techniques used to measure transmissivity, one must conclude the filter is best described as a ND0.4 or ND.4 filter. Perhaps the filter was made by Fujimi to target the Tiffen market which uses this designation. Or some error in their translation or even manufacturing.

Canon EOS 1100D manual mode metering: without filter I get the meter at the center with exposure of 0"5, and with filter it goes 4 steps lower, so that to compensate I have to click the wheel 4 times, getting exposure "5" (i.e. 1/5 s). The ratio (1/5)/0.5 is thus also 40%.

Thorlabs

There are several ways neutral filters are designated. The most common is in multiples of half stops. Thus an ND2 filter is 1 f-stop, ND4 is 2 f-stops and so on. This yields transmissivity of .50, and .25 which are not even close. Another convention not in play here is to label the filter with a number followed by "x" which shows the transmissivity decrease factor. Thus one stop of attenuation would be labeling the filter 2x, 2 stops, 4x, 3 stops 8x, etc.

Filters specified as ND .4 have a transmissivity of 10^(1/0.4) which is .398. Looking at the 4 measurements the OP made, the reflectance at 520nm (approx. peak of visual response) is .40. The other 3, lux, exposure steps of 1/3 f-stop, and raw histogram range from .38 to .41. This corresponds very closely to an ND.4 filter.

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As I understand, ND4 filter should transmit about 25% of the incident light. I have measured transmittance spectrum of my Fujimi ND4 58mm filter, and got the following result:

Canon EOS 1100D raw photo: raw histograms of the photos with the same exposure have corresponding peaks at values having the ratios 38% to 41%, depending on color channel (corroborating the tint visible in the transmittance spectrum).

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So, transmittance is about 40%-45% in the visible range of wavelengths. That's quite far from the expected 25%. Is my filter not really ND4 (although I bought it as such, and it has the marking on its rim)? Or is my expectation wrong, and this filter is normal?

e.g. Let’s say that in step 4 you got a shutter speed 1/1000”, since your ND reduces the light by 4 f-stops, step 5 should give you around 1/62”. Halving the original speed 4 times or mathematically: shutter_speed/2^f-stops in our example (1/1000)/16 = 1/62,5”