In this appendix it is shown that if the length of the dial is

$$\begin{aligned} 2\ell = \frac{1}{2} c\tau (1 + \beta ) \end{aligned}$$

(46)

or shorter, then the triad partner of a quantum

\(\widehat{2}\) is the first quantum

\(\widehat{3}\) following that

\(\widehat{2}\) on the way toward the recorder. In this case the identification of the triad partners

\((\widehat{2}, \widehat{3})\) can be done without the knowledge of the serial number

k . However, if the dial is longer than the value in (

46 ), namely if

$$\begin{aligned} 2\ell = \frac{m}{2} c\tau (1 + \beta ), \quad m = 2,3, \dots , \end{aligned}$$

(47)

then the pairing depends on the value of

k .

The proof of the above statements relies on the comparison of the distance \(x_{\widehat{3}_k} - x_{\widehat{3}_{k - 1}}\) between the consecutive quanta \(\widehat{3}\) with the distances \(x_{\widehat{3}_k} - x_{\widehat{2}_k}\) between the triad partners \(\widehat{2}_k\) , \(\widehat{3}_k\) .

It is easy to see that

$$\begin{aligned} x_{\widehat{3}_k} - x_{\widehat{3}_{k - 1}} = c\tau (1 + \beta ), \quad k = 1,2,\dots , n. \end{aligned}$$

(48)

Indeed, as told before Eq. (

30 ),

\(c\tau (1 + \beta )\) is the distance between the consecutive triads impinging on the clock, in particular between the consecutive quanta

\(\widetilde{3}\) scattered back by dial body

\(\mathbf 3\) at rest. The scattered

\(\widetilde{3}\) ’s are the

\(\widehat{3}\) ’s, therefore the distance between the consecutive

\(\widehat{3}\) ’s is equal to the same distance

\(c\tau (1 + \beta )\) .

Let us now look at the distances

\(x_{\widehat{3}_k} - x_{\widehat{2}_k}\) . According to (

22 )

$$\begin{aligned} x_{\widehat{3}_k} - x_{\widehat{2}_k} = 2\left( \ell - x_h^{(k)}\right) , \quad k = 1,2,\dots , n, \end{aligned}$$

(49)

where

\(x_h^{(k)}\) is the coordinate of the hand at the moment of its scattering with the quantum

\(\widetilde{2}_k\) which becomes

\(\widehat{2}_k\) at that very moment. As told before Eq. (

30 ) such scatterings follow each other at time intervals

\(\tau \) . During one interval the hand covers the distance

$$\begin{aligned} u\tau = \frac{2\ell }{T} \tau = \frac{2\ell }{n} \end{aligned}$$

(50)

in the negative

X direction, therefore the relation between

\(x_h^{(k)}\) and

\(x_h^{(1)}\) is

$$\begin{aligned} x_h^{(k)} = x_h^{(1)} - (k - 1) \frac{2\ell }{n}, \quad k = 1,2,\dots , n. \end{aligned}$$

(51)

Since the hand starts from the point

\(x_{d_3} = \ell \) , its scattering with the first quantum

\(\widetilde{2}_1\) occurs in the interval

\(\left[ \ell , \ell - \frac{2\ell }{n}\right] \) . Let us consider the case when this scattering happens at the middle point

$$\begin{aligned} x_h^{(1)} = \ell - \frac{\ell }{n} \end{aligned}$$

(52)

of the interval. The results obtained in this special case concerning the time-reading process are valid also for other points of the interval. The mathematical derivation involves cumbersome notation in the general case.

From (

49 ), (

51 ) and (

52 ) it follows that

$$\begin{aligned} x_{\widehat{3}_k} - x_{\widehat{2}_k} = 4\ell \frac{k - \frac{1}{2}}{n}, \quad k = 1,2, \dots , n. \end{aligned}$$

(53)

Let us now look at the case when the length of the dial is given by (

46 ):

$$\begin{aligned} 2\ell = \frac{1}{2} c\tau (1 + \beta ). \end{aligned}$$

(54)

Then according to (

48 )

$$\begin{aligned} x_{\widehat{3}_k} - x_{\widehat{3}_{k - 1}} = 4\ell , \end{aligned}$$

(55)

and (

53 ) leads to

$$\begin{aligned} x_{\widehat{3}_k} - x_{\widehat{2}_k} = \left( x_{\widehat{3}_k} - x_{\widehat{3}_{k - 1}}\right) \frac{k - \frac{1}{2}}{n} < x_{\widehat{3}_k} - x_{\widehat{3}_{k - 1}} \end{aligned}$$

(56)

for all values of

k . (

56 ) says that

\(\widehat{2}_k\) travels toward the recorder between

\(\widehat{3}_{k - 1}\) and

\(\widehat{3}_k\) (Fig.

2 ), so the triad partner of

\(\widehat{2}_k\) is indeed the first quantum

\(\widehat{3}\) following

\(\widehat{2}_k\) .

Fig. 2 The case \(2\ell = \frac{1}{2} c\tau (1 + \beta )\)

Let us now consider the cases when the length of the dial is given by relation (47 ). The general result for \(m \ge 2\) described in Sect. 4 can be inferred from the special cases \(m = 2\) and \(m = 3\) discussed below.

For

\(m = 2\) (

47 ) and (

48 ) give

$$\begin{aligned} x_{\widehat{3}_k} - x_{\widehat{3}_{k - 1}} = 2\ell , \end{aligned}$$

(57)

so (

53 ) becomes

$$\begin{aligned} x_{\widehat{3}_k} - x_{\widehat{2}_k} = \left( x_{\widehat{3}_k} - x_{\widehat{3}_{k - 1}}\right) \frac{2k - 1}{n}. \end{aligned}$$

(58)

According to this relation if

$$\begin{aligned} k < \frac{n + 1}{2} \end{aligned}$$

(59)

then

$$\begin{aligned} x_{\widehat{3}_k} - x_{\widehat{2}_k} < x_{\widehat{3}_k} - x_{\widehat{3}_{k - 1}}, \end{aligned}$$

(60)

\(\widehat{2}_k\) travels between

\(\widehat{3}_{k - 1}\) and

\(\widehat{3}_k\) , thus the triad partner of

\(\widehat{2}_k\) is again the first

\(\widehat{3}\) following

\(\widehat{2}_k\) (Fig.

3 a).

Fig. 3 The cases \(m = 2\)

If

$$\begin{aligned} k = \frac{n + 1}{2} \end{aligned}$$

(61)

then

\(\widehat{2}_k\) and

\(\widehat{3}_{k - 1}\) travel together, and the triad partner of

\(\widehat{2}_k\) is still the first

\(\widehat{3}\) following

\(\widehat{2}_k\) . However, if

$$\begin{aligned} k > \frac{n + 1}{2} \end{aligned}$$

(62)

then

$$\begin{aligned} x_{\widehat{3}_k} - x_{\widehat{2}_k} > x_{\widehat{3}_k} - x_{\widehat{3}_{k - 1}}, \end{aligned}$$

(63)

\(\widehat{2}_k\) travels between

\(\widehat{3}_{k - 2}\) and

\(\widehat{3}_{k - 1}\) , and the triad partner of

\(\widehat{2}_k\) is the second

\(\widehat{3}\) following

\(\widehat{2}_k\) .

Dealing with relations involving k and n one should take into account that both k and n are integers. Thus for even values of n the case (61 ) does not occur, and in (59 ) the highest occurring value of k is n / 2.

If the recorder is turned on at a moment during the arrival of the quanta, the serial number

k of a registered quantum

\(\widehat{2}\) is not known, and it is ambiguous whether the first or the second quantum

\(\widehat{3}\) is its triad partner. According to (

57 ) the distance between these

\(\widehat{3}\) ’s is

\(2\ell \) , and from (

32 ) it follows that the ambiguity in

\(t_c\) is

$$\begin{aligned} \Delta t_c = \frac{2\ell }{4\ell } T = \frac{T}{2}, \end{aligned}$$

(64)

much larger than the accuracy

\(\tau \) of the clock.

Let us now look at the case

\(m = 3\) . Then according to (

47 ) and (

48 )

$$\begin{aligned} x_{\widehat{3}_k} - x_{\widehat{3}_{k - 1}} = \frac{4\ell }{3}, \quad k = 1,2,\dots , n, \end{aligned}$$

(65)

and with (

53 )

$$\begin{aligned} x_{\widehat{3}_k} - x_{\widehat{2}_k} = \left( x_{\widehat{3}_k} - x_{\widehat{3}_{k - 1}}\right) 3 \frac{k - \frac{1}{2}}{n}. \end{aligned}$$

(66)

The triad partner of

\(\widehat{2}_k\) is now the first

\(\widehat{3}\) following

\(\widehat{2}_k\) if

$$\begin{aligned} x_{\widehat{3}_k} - x_{\widehat{2}_k} \le x_{\widehat{3}_k} - x_{\widehat{3}_{k - 1}}, \end{aligned}$$

(67)

i.e. according to (

66 ) if

$$\begin{aligned} k \le \frac{n}{3} + \frac{1}{2}. \end{aligned}$$

(68)

The triad partner of

\(\widehat{2}_k\) is the second

\(\widehat{3}\) following

\(\widehat{2}_k\) if

$$\begin{aligned} x_{\widehat{3}_k} - x_{\widehat{3}_{k - 1}} < x_{\widehat{3}_k} - x_{\widehat{2}_k} \le x_{\widehat{3}_k} - x_{\widehat{3}_{k - 2}} = 2 \left( x_{\widehat{3}_k} - x_{\widehat{3}_{k - 1}}\right) , \end{aligned}$$

(69)

i.e. if

$$\begin{aligned} \frac{n}{3} + \frac{1}{2} < k \le \frac{2n}{3} + \frac{1}{2}, \end{aligned}$$

(70)

and the triad partner is the third

\(\widehat{3}\) if

$$\begin{aligned} \frac{2n}{3} + \frac{1}{2} < k \le n + \frac{1}{2}. \end{aligned}$$

(71)

If the value of

k is not registered by the recorder the ambiguity in the choice of the triad partner

\(\widehat{3}\) of a

\(\widehat{2}\) is now threefold. According to (

65 ) the distance between

\(\widehat{3}_k\) and

\(\widehat{3}_{k - 1}\) , as well as between

\(\widehat{3}_{k - 1}\) and

\(\widehat{3}_{k - 2}\) is

\(4\ell / 3\) , between

\(\widehat{3}_k\) and

\(\widehat{3}_{k - 2}\) it is

\(8\ell / 3\) . Therefore from (

32 ) it follows that the ambiguities in the value of

\(t_c\) are

T / 3 and 2

T / 3.