[under construction - preliminary draft needs typography]
We take the infinitesmally closed interval in deference to finitesmality, and to a possibly finer ruler.
Let a, a+1, ... be numbers adjacent by smallest measurable quanta=1, one finitesmal; and let x: a+1>x>a be a measurable value on the same domain. If (and we're looking for the contradiction to if) x leans distinguishably more toward a than toward a+1, which we denote, a+1 m> x, meaning x measures less than a+1, then in fact we could have resolved further the distinguishable adjacency of a, a+1,..., and we may presume then we already did so [in eliminating the contradiction] - to wit: if x : a+1 m> xo > y > a (validly: a+1 > xo) then xo, 2a+1-xo subdividee the interval (a,a+1) into three distinguishable pieces: lesser quanta, around and between a,a+1, that is, around a,a+.5,a+1, where "a+.5" is short notation for, between a,a+1. Obviously a contradiction: x measured cannot be more accurate/precise than the ruler, else we'd have merely defined a way of obtaining a+.5: after finishing that definition at a,a+1,.... This may change our skill at taking measurements, by observing around-ness and between-ness; but it cannot improve on the best possible measure of x, and therefor neither a+1>x nor x>a - x is not distinguishably toward, around, nor between a,a+1.
Whence x is indistinguishable on [x-1,x+1], while it is distinguishable outside. Thus the a's themselves are vestigially indistinguishable, as adjacent, while distinguishable further away than immediate. Just possibly a m= a+1.
Any x then, must measure indistinguishably as either of the nearest two a's, or in the extreme case of x=a, as among {a-1,a,a+1}. [Possibly xm=x-1,x+1]
[x : ... , is read, "x for/ [be] such-that/where [it[is]] ..." . . . . . .
However, the precise nature of the semi-natural normal distributed uniform finite covering is such that, although the representative selection of an exact sample as itself, is just 50%, the misselection by some near-representation (of an exact sample) is minutely greater at 0.500006975 - a curiosity that infinite sums of equally spaced consecutive normal distributions converge rapidly to uniform on the whole, for smaller meshes, but not identically: the half-way between numbers are representatively selected slightly less than 100% (for all possible) and thereby compensates mostly. To characterize this more reasonably, the normal distribution is not perfectly ideal. *
* [sampling on some uniform detectors yields a triangular distribution]
Thus by notion our ruler line is graduated with near-triangular half-height distributions overlapping on half-widths (still each a full unit-for-unit likelihood, but spread over two), whereín each exactly half-way number is representatively selected as (equally) either nearest neighbor, 82% total occasionally or 41% each nearest neighbor (better than strictly triangular 75%, or 37.5% each). . . . .
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