View source for UPT Example: 2-State Synchrony Markov Chain

[[Category:Timeline Paradigm]]
[[Category:UPT]]
[[Category:Markovian iteration]]
[[Category:quantum field theory]]
[[Category:Time Matrix Technology]]

= UPT Example: 2-State Synchrony Markov Chain =

This page gives a concrete toy example of Unit Perception Tests (UPTs) and the resulting
2×2 empirical transition matrix.

== 1. Setup ==

We consider a binary synchrony state at each timeline index ''t'':

* State ''S'': synchrony detected at time ''t'' (the UPT returns 1).
* State ''N'': no synchrony detected at time ''t'' (the UPT returns 0).

We write this as a state variable ''X_t'' with two possible values:

* <math>X_t \in \{S, N\}</math>.

The underlying data (streams ''A_t'' and ''B_t'') and the Unit Perception Test ''S(t)''
are abstracted away here; we focus only on the resulting sequence of states.

== 2. Example Sequence of UPT Outcomes ==

Suppose that, after running UPTs over 10 consecutive timeline indices, we observe
the following sequence of synchrony states:

* <math>(X_1, X_2, X_3, X_4, X_5, X_6, X_7, X_8, X_9, X_{10}) = (S, N, N, S, S, N, S, N, N, S)</math>.

For convenience, rewrite this as:

* Time 1: ''S''
* Time 2: ''N''
* Time 3: ''N''
* Time 4: ''S''
* Time 5: ''S''
* Time 6: ''N''
* Time 7: ''S''
* Time 8: ''N''
* Time 9: ''N''
* Time 10: ''S''

We will use this sequence to estimate the transition probabilities between ''S'' and ''N''.

== 3. Transition Counts ==

We look at **one-step transitions** ''X_t → X_{t+1}'' for ''t = 1, …, 9''.
The possible ordered pairs are:

* ''S → S''
* ''S → N''
* ''N → S''
* ''N → N''

From the example sequence:

# Time 1 → 2: ''S → N''
# Time 2 → 3: ''N → N''
# Time 3 → 4: ''N → S''
# Time 4 → 5: ''S → S''
# Time 5 → 6: ''S → N''
# Time 6 → 7: ''N → S''
# Time 7 → 8: ''S → N''
# Time 8 → 9: ''N → N''
# Time 9 → 10: ''N → S''

Counting each transition type:

* ''S → S'': occurs 1 time (at 4 → 5)
* ''S → N'': occurs 3 times (at 1 → 2, 5 → 6, 7 → 8)
* ''N → S'': occurs 3 times (at 3 → 4, 6 → 7, 9 → 10)
* ''N → N'': occurs 2 times (at 2 → 3, 8 → 9)

We can summarize these counts as ''N_{ij}'', where ''i'' and ''j'' are in ''{S, N}'':

* <math>N_{SS} = 1</math>
* <math>N_{SN} = 3</math>
* <math>N_{NS} = 3</math>
* <math>N_{NN} = 2</math>

== 4. Empirical 2×2 Transition Matrix ==

The empirical transition probability from state ''i'' to state ''j'' is defined as:

* <math>\hat{P}_{ij} = \dfrac{N_{ij}}{\sum_k N_{ik}}</math>

For our 2-state case ''{S, N}'':

* Total departures from ''S'':
** <math>\sum_k N_{Sk} = N_{SS} + N_{SN} = 1 + 3 = 4</math>
* Total departures from ''N'':
** <math>\sum_k N_{Nk} = N_{NS} + N_{NN} = 3 + 2 = 5</math>

So the empirical transition probabilities are:

* From ''S'':
** <math>\hat{P}_{SS} = \dfrac{N_{SS}}{N_{SS} + N_{SN}} = \dfrac{1}{4} = 0.25</math>
** <math>\hat{P}_{SN} = \dfrac{N_{SN}}{N_{SS} + N_{SN}} = \dfrac{3}{4} = 0.75</math>

* From ''N'':
** <math>\hat{P}_{NS} = \dfrac{N_{NS}}{N_{NS} + N_{NN}} = \dfrac{3}{5} = 0.6</math>
** <math>\hat{P}_{NN} = \dfrac{N_{NN}}{N_{NS} + N_{NN}} = \dfrac{2}{5} = 0.4</math>

We can write the empirical 2×2 transition matrix ''\hat{P}'' as:

<math>
\hat{P} =
�egin{pmatrix}
\hat{P}_{SS} & \hat{P}_{SN} \
\hat{P}_{NS} & \hat{P}_{NN}
\end{pmatrix}
=
�egin{pmatrix}
0.25 & 0.75 \
0.60 & 0.40
\end{pmatrix}.
</math>

Here the first row/column corresponds to state ''S'', and the second row/column to state ''N''.

== 5. Interpretation ==

From this small example, we can already see qualitative behavior:

* When the system is in synchrony (''S''), it tends to move to no-synchrony (''N'') on the next step with probability 0.75.
* When the system is in no-synchrony (''N''), it tends to move back to synchrony (''S'') with probability 0.6.
* Both states are unstable in the sense that the most likely transition is to the *other* state.

With longer sequences (more UPTs), these empirical probabilities stabilize and define a
'''perceptive envelope''' over synchrony dynamics: a concise description of how likely
synchrony is to persist, dissolve, or re-emerge over time.