Pivot Postulates
Theorem 3.2.1
A Perfect Palindrome has a Delimiter Pivot Character if and only if a Perfect Palindrome has Odd Parity.
\[\forall \zeta \in K: (\omega_{\zeta} = \sigma) \equiv \zeta \in P_{-}\]
Proof Let \(\zeta \in K\) . Note
\[K \subseteq C \subseteq D\]
Then by definition of Perfect Palindromes \(\zeta = \zeta^{-1}\) .
(\(\rightarrow\) ) Assume \(\omega_{\zeta} = \sigma\) . Assume, for the sake of contradiction, \(\zeta \notin P_{-}\) . Then, since,
\[P_{+} \cup P_{-} = P\]
And,
\[P_{+} \cap P_{-} = \varnothing\]
It follows that it must be the case, \(\zeta \in P_{+}\) . Therefore, \(\exists i \in mathbb{N}: l(\zeta) = 2i\) . Then, by definition of Pivot Characters
\[\overrightarrow{\omega_s} = \overleftarrow{\omega_s} = \sigma\]
Where the last equality follows by assumption. Then,
\[\zeta[\frac{l(s) + 2}{2}] = \zeta[\frac{l(2)}{2}] = \sigma\]
In other words, two consecutive Characters are Delimiter. But this is impossible if \(\zeta \in D\) . Therefore, it must be case,
\[\zeta \in P_{-}\]
(\(\leftarrow\) ) Assume \(\zeta \in P_{-}\) . By Theorem 2.1.5
\[\omega_{\zeta} = \sigma\]
Thus the equivalence is established. Summarizing and generalizing,
\[\forall \zeta \in K: (\omega_{\zeta} = \sigma) \equiv \zeta \in P_{-}\]
∎
Theorem 3.2.2
The Pivot Character of a Perfect Palindrome is not a Delimtier if and only if its Pivot Word is Reflective.
\[\forall \zeta \in K: (\omega_{\zeta} \neq \sigma) \equiv (\Omega_{\zeta} \in R)\]
Proof Let \(\zeta \in K\) .
Theorem 3.2.3
The Pivot Character of a Perfect Palindrome is not a Delimiter if and only if its Pivot Character is the Pivot Character of its Pivot Word.
\[\forall \zeta \in K: (\omega_{\zeta} \neq \sigma) \equiv (\omega_{\Omega_{\zeta}} = \omega_{\zeta} )\]
Inverse Postulates
Theorem 3.2.x
Either the first Word of a Palindrome is contained in the first word of its Inverse, or the first Word of its Inverse is contained in its First Word.
\[\forall \zeta \in P: (\zeta[[1]] \subset_s \zeta^{-1}[[1]]) \lor (\zeta^{-1}[[1]] \subset_s \zeta[[1]])\]
Theorem 3.2.x+1
Either the last Word of a Palindrome is contained in the last word of its Inverse, or the last word of its Inverse is contained in the last word of the Palindrome.
\[\forall \zeta \in P: (\zeta[[\Lambda(\zeta)]] \subset_s \zeta^{-1}[[\Lambda(\zeta)]]) \lor (\zeta^{-1}[[\Lambda(\zeta)]] \subset_s \zeta[[\Lambda(\zeta)]])\]