2.1 The square bracketed expression [x] will mean the INTEGER portion of x.

For example, [3.1416] = 3, [.25] = 0, 2.718 - [2.718] = .718, etc...

2.2 - Definition of R(a, K)

Given any integers a, A, and K

a + A * K = R(a, K) whenever 0 <= a + A * K <= K – 1.

Algebraically,

R(a, K) = a - [a / K] * K whenever 0 <= a - [a / K] * K <= K - 1.

2.3 The braced expression {a, b, ...,z} will be used to indicate the set of all of the possible values that may be selected for a particular variable or expression.

For example, R(D, 7d) = {0, 2, 3, 5} implies that the week day D may be either
Saturday (0), Monday (2), Tuesday (3), or Thursday (5).

2.4 Common elements within braces may be reduced as follows {a, ..., a, ...} = {a, ..., ...}

For example, {1, 2, 3, 1, 5, 3} = {1, 2, 3, 5}

2.5 The expression {a, b, ...,z} + {x , y, ... , } evaluates to

{a+x, a+y, ...
 b+x, b+y, ...
 ...
 z+x, z+y, ...
}

Hence, the expression {a,b} + {c,d} + {e,f} evaluates to

{a+c, a+d, b+c, b+d} + {e,f}

which results in

{a+c+e, a+c+f, a+d+e, a+d+f, b+c+e, b+c+f, b+d+e, b+d+f}

2.6

Let X = { x(1), x(2), x(3), .... }
Let Y = { y(1), y(2), y(3), .... }
Let Z = { z(1), z(2), z(3), .... }

Then for some x in X,
         some y in Y, and
         some z in Z,

the equation X + Y = Z will have as its solution
all of the pairs (x , y) for which x + y = z.

As an example, suppose that Y - X = 3, and that Y = {6, 7, 8, 9}, then the pairs (x, y) that could be solutions to the the equation are (3, 6), (4, 7), (5, 8) and (6, 9).

Other considerations would have to be made in order to determine whether or not any of {3, 4, 5, 6} are to be found in the set X.

I am leaving alone any more general definitions or discussions of the ideas presented by
Arithmetical Conventions 3, 4, 5, and 6, since I believe that the use of these conventions in the following text is reasonably easy to understand.

T' represents the molad of Tishrei for year H'.
T" represents the molad of Tishrei for year H".

T" - T' = the period of time between two known moladot of Tishrei.

Let T" - T' = M + m

where

M = the integer portion of the time period, ie, [M+m]
m = the fractional portion of that period.

Hence, m = M+m - [M+m]

Thus for 12 Hebrew months, M =  354d and m =  8h 876p
     for 13                M =  383d and m = 21h 589p
     for 19 Hebrew years   M = 6939d and m = 16h 595p

For convenience, the following symbols represent these values

d  = 24h   0p; d' = 23h 1079p; d" = 24h   1p
c  =  8h 876p; c' =  8h  875p; c" =  8h 877p
g  = 15h 204p; g' = 15h  203p; g" = 15h 205p
b  = 21h 589p; b' = 21h  588p; b" = 21h 590p
f  = any fractional part of a day such that 0 <= f <= d'
p' = the number of days that Tishrei 1 is postponed at the start of the year
p" = the number of days that Tishrei 1 is postponed at the end   of the year
T' = the molad of Tishrei of some year H'
T" = the molad of Tishrei of some subsequent year H"
D' = the day of 1 Tishrei H'
D" = the day of 1 Tishrei H"

It is to be noted that

i) d - g = 8h 876p and d - b = 2h 491p
ii) p' and p can be {0, 1, 2} days.

Computer analysis demonstrates that, over the full Hebrew calendar cycle of 689,472 years, each of the possible 181,440 values for the Tishrei Moladot, when reduced to the form R(M, 7d) + m, is repeated exactly 3 or exactly 4 times.

The reduced form of the Tishrei Moladot for the 11th, 13th, and 15th years of the mahzor katan (19 year cycle) known as GUChADZaT are the only ones that are repeated exactly 3 times over the full Hebrew calendar cycle of 689,472 years.

It is therefore reasonable to assume, that for any given M + m, there exists at least one period of Hebrew years whose lunar length in reduced form is R(M, 7d) + m.

As an example, given the value 2d 5h 204p, we will assume that there exists at least one period of Hebrew years, no matter how long, whose lunar length M + m, when reduced to the form
R(M,7d) + m, will be 2d 5h 204p.

The time of the molad T' for year H' is given by
4.1 T' = [T'] + f.

If some postponement rule applies to [T'], then the day for 1 Tishrei H' is 

4.2 D' = [T'] + p'

The molad of Tishrei for year H" is given by
     
     T"  =  T'  +     M + m
         = [T'] + f + M + m (substituting equation 4.1 for T')

Hence 
4.3 [T"] = [T'] + M + [f+m]

If some postponement rule applies to [T"], then the day for 1 Tishrei H" is 

4.4  D"  = [T'] + M + [f+m] + p"

The length L between years H" and H' is given by
subtracting equation 4.2 from equation 4.4, yielding

4.5 L  = D" - D' = M + [f+m] + p" - p'

Absent the Dehiyyot, (ie, the postponement rules) the length L of a single Hebrew year is

L = M + [f+m]
  = {354d, 383d} + {0d, 1d}     (Since M = {354d, 383d} and 5.1 above)
  = {354d, 355d, 383d, 384d}

Lo ADU Rosh requires that the first day of Tishrei be postponed to the next day whenever
R([T],7d) = {1d, 4d, 6d}.

Consequently, both D' and D" = {0d, 2d, 3d, 5d}, and both p'and p" = {0d, 1d}.

Under this rule, the length of one Hebrew year can become

L = M + [f+m] + p" - p'  (By equation 4.5)
  = {354d, 383d} + {0d, 1d} + {0d, 1d} - {0d, 1d}
  = {354d, 383d} + {-1d, 0d, 1d, 2d} (by equation 5.2 above)
  = {353d, 354d, 355d, 356d, 382d, 383d, 384d, 385d}

The actual distribution of these lengths, over the complete Hebrew calendar cycle of 689,472 years, is shown in the following table.

The 356 day and 382 day years are produced whenever a single day is removed from the days of the week which are permitted for 1 Tishrei. It does not matter which day is chosen. This effect is predicted by Equations 4.5 and 5.1.

Now, let w be the day not allowed for 1 Tishrei.

In the following example, Tuesday is removed from the allowable weekdays, while the remaining 6 days are allowed. The actual distribution of the single years lengths, over the complete Hebrew calendar cycle of 689,472 years, is shown in the following table.

Therefore, given the single non-allowable weekday w, the 356 day year will begin on D' = w + 2d and end on D" = w + 1d.

In the above example, with Tuesday removed, the 356 day year is seen to begin on Tuesday + 2d = Thursday. It will end on Tuesday + 1d = Wednesday since R(Thursday + 356d, 7d) = Wednesday.

Therefore, given the single non-allowable weekday w, the 382 day year will begin on D' = w + 1d and end on D" = w - 2d.

In the above example, with Tuesday removed, the 382 day year is seen to begin on Tuesday + 1d = Wednesday. It will end on Tuesday - 2d = Sunday since R(Wednesday + 382d, 7d) = Sunday.

Since R(D" - D', 7d) = R(356d, 7d) = 6d, 
             R(D', 7d) = R(D" - 6d, 7d)
                       = R({0d, 2d, 3d, 5d} - 6d, 7d)
                       =  {1d, 3d, 4d, 6d} 
                       =  {3d} due to Dehiyyah Lo ADU Rosh

Of the 4 values possible for R(D', 7d), only 3d (Tuesday) is allowed.

Therefore, the 356 day year begins on Tuesday, and ends on R(3d+356d, 7d) = 2d (Monday).

Let the molad period of 12 Hebrew months be expressed as 354d + m.

Since the length of any given period of Hebrew years is given by

L = D" - D' = M + [f+m] + p" - p' (Equation 4.5)

and there is yet no 2-day postponement, the 356 day year occurs whenever
f + m > d' , p" = 1, and p' = 0.

So that [f+m] = 1d, it is necessary that f+m > d' (or f > d' - m).

When the molad of Tishrei for year H' is on Tuesday past d'-m,
postponing 1 Tishrei H' to Thursday forces p' = 2d, thus making the length of the 12 month Hebrew year H' 354 days because

                     L = D" - D' 
                       = 354d + [f+m] + p"  - p'   (By equation 4.5)
                       = 354d + 1d    + 1d  - 2d
                       = 354d

Therefore, whenever the molad of Tishrei for a 12 month year occurs past 15h 203p (ie, d' - m) on a Tuesday, postponing the start of Tishrei to Thursday eliminates the 356 day year.

Let the molad period of 13 Hebrew months be expressed as 383d + m, where m = 21h 589p.

Since the length of any given period of Hebrew years is given by

the 382 day year occurs whenever f + m < d , p" = 0, and p' = 1.

Since Dehiyyah Molad Zakein has not yet been introduced, and the year is leap, p' cannot be 2d.

 Now, R(D" - D', 7d) = R(382d, 7d) = 4d, 
            R(D', 7d) = R(D" - 4d, 7d)
                       = R({0d, 2d, 3d, 5d} - 4d, 7d)
                       =  {3d, 5d, 6d, 1d} 
                       =  {3d, 5d} due to Dehiyyah Lo ADU Rosh

Since p' = 1, and Lo ADU Rosh does not postpone 1 Tishrei from
Monday (2d) to Tuesday (3d), but does postpone 1 Tishrei from
Wednesday (4d) to Thursday (5d), D' = {5d}.

Consequently, R(D", 7d) = R(D' + 382d, 7d)
                        = R(5d + 4d, 7d)
                        =  2d

the molad of Tishrei H' is on Wednesday (4d),
and M = 383d,
and f+m < d, implying f < d - m

then p' = 1,
year H" begins on Monday (2d),
making p" = 0 (due to Lo ADU Rosh),
and the length of year H = M + [f+m] + p" - p'   (Equation 4.5)
                         = 383d + 0d + 0d - 1d
                         = 382d 

Since, the earliest time f, for any molad of Tishrei year H, is 0h 0p, the smallest value of f+m for year H+1, following a leap year, is 21h 589p.

That time on Monday (2d) would cause p" to remain zero, and year H to be 382 days unless
1 Tishrei H+1 is postponed to Tuesday (3d) whenever a post-leap year molad of Tishrei is on Monday (2d) past 21h 588p.

In so doing, p" becomes 1 thereby making equation 4.5

D" - D' = 383d + [f + 21h 588p] + 1d - 1d
        = 383d + 0  (since it is given that f + 21h 588p < 1d)