How votes are counted

What Amendment II says

In every vote of the Union, voting power is shared as follows: the President holds twenty percent (20%) of the total, the Vice President holds five percent (5%), and the remaining seventy-five percent (75%) is divided equally among all citizens at one vote each. The President's and Vice President's vote counts are calculated from the number of citizens voting so as to preserve these shares, rounded up to the nearest whole vote, with a minimum of one (1) vote each. Neither office holds a veto: a united three-quarters of the citizenry can always overrule them both.

How that is worked out

Let N be the number of ordinary citizens who cast a ballot. The President and the Vice President are not counted in N: their ballots carry their weighted share instead of a citizen's single vote, which is what "shared as follows" means.

N  = ordinary citizen ballots cast
T  = N / 0.75                    the total voting power
President  = ceil(0.20 * T) = ceil(4N / 15), minimum 1
Vice Pres. = ceil(0.05 * T) = ceil( N / 15), minimum 1

Every figure is a whole number throughout. Nothing in a final result is ever computed in floating point.

A worked example

Six citizens vote, and so do both officers.

N = 6, so T = 6 / 0.75 = 8
President  = ceil(0.20 * 8) = 2
Vice Pres. = ceil(0.05 * 8) = 1
                              --
total weight                   9

One consequence worth stating plainly

Because both offices round up, the citizen block holds at most three quarters and reaches exactly three quarters only when N is a multiple of 15. Below that the rounding favors the two offices slightly, and the minimum-one-vote floor means that when only one or two citizens vote, the offices together can match or outweigh them. That follows from the text of the Amendment rather than from any choice made here, and it is written down so nobody has to discover it from a surprising result.

Thresholds

An ordinary vote carries on more than half of the weighted votes cast. An Amendment needs at least two thirds, under Amendment VI, and a result landing exactly on two thirds carries. A tie carries nothing.

Secrecy and checking

Your ballot records that you voted, which is what stops anyone voting twice, and what you chose, which no interface anywhere will show to anyone. When you vote you get a receipt token, shown once and never stored; only its SHA-256 is kept. Every hash is published with the result, so you can prove your ballot was counted, and nobody can work backwards from the published list to how any person voted.