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This doesn¹t make any scientific sense. Neutral solutions do not neutralize acids. Bases do. Most acids I use in solid form in the laboratory are solubilized in water before use. This does not neutralize them. If it did, they'd be useless. If I need to neutralize them, I need to add equinormal amounts of a basic solution. The pH of an acid does increase somewhat upon addition of more water but this is strictly an effect of dilution of hydrogen ions and not an effect of neutralization and to increase the pH only one unit would require a 10-fold volume increase with the neutral solution because pH is logarithmic.
i.e. To neutralize a single teaspoon of solution at pH 2 in the absence of an added base would require the addition of 99,999 teaspoons of neutral water (an increase from pH 2 to pH 7, or 5 pH units, requires a 10^5 or 100,000-fold dilution in hydrogen ions). Also, since most tap water is slightly acidic, you'd never really neutralize it no matter how much water were added.
That said:
A buffer is capable of stabilizing a range of pH for a particular range of volume. It is possible that buffered aspirin is too neutral for an aspirin mask but that the buffer capacity is exceeded once the saliva/aspirin mixture is diluted with larger volumes of stomach acid. In this case, I'd assume that if the pH of an aspirin mask is critical then an unbuffered aspirin would be more effective and that the buffered form, may remain neutral as a low volume paste.
The obvious question is what the pH of aspirin actually is in either cheaper, non-buffered or buffered forms when crushed into a paste with water and I don't know the answer to this. I would be more than happy to figure it out empirically in the lab on Monday. However, someone will need to remind me to do it if I forget!
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