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\label{thm:prime-polarity}
\begin{enumerate}
    \item \textbf{(Dichotomy.)}
          Every prime $\underline{p} \in \mathbb{P}_\tau$
          is either B-dominant or C-dominant.
    \item \textbf{(B-class infinite.)}
          The set of B-dominant primes is infinite.
    \item \textbf{(C-class infinite.)}
          The set of C-dominant primes is infinite.
\end{enumerate}

\emph{Part~(1): Dichotomy.}
Fix a prime $\underline{p} \geq \underline{2}$.
Consider the comparison between
$\underline{p}^{\underline{B}}$
and $\underline{p} \uparrow\uparrow \underline{C}$
for growing values of $\underline{B}$ and $\underline{C}$.

For fixed $\underline{C}$,
$\underline{p}^{\underline{B}}$ grows
as $\underline{p}^{\underline{B}}$
(exponential in $\underline{B}$).
For fixed $\underline{B}$,
$(\underline{p} \uparrow\uparrow \underline{C})^{\underline{B}}$
grows as $\underline{p}^{\underline{B} \cdot
(\underline{p} \uparrow\uparrow (\underline{C} - 1))}$
(super-exponential in $\underline{C}$).

The critical ratio is:
\[
    r(\underline{p}, \underline{B}, \underline{C})
    \;:=\;
    \frac{\log T(\underline{p}, \underline{B}, \underline{C})}
    {\log(\underline{p}^{\underline{B}})}
    \;=\;
    \frac{\underline{B} \cdot
    (\underline{p} \uparrow\uparrow (\underline{C}-1))}
    {\underline{B}}
    \;=\;
    \underline{p} \uparrow\uparrow (\underline{C}-1).
\]
When $\underline{C} = \underline{1}$, $r = \underline{1}$:
the tower atom equals the pure power.
When $\underline{C} = \underline{2}$,
$r = \underline{p}$: the tower atom is
$\underline{p}$ times more ``potent''
in logarithmic terms than the pure power.

For a given prime $\underline{p}$,
the \emph{effective sensitivity} is:
how quickly does increasing $\underline{C}$
from $\underline{1}$ to $\underline{2}$
amplify the tower atom, compared to increasing
$\underline{B}$ by $\underline{p}$?
Since $T(\underline{p}, \underline{B}, \underline{2})
= \underline{p}^{\underline{p} \cdot \underline{B}}$
while $T(\underline{p}, \underline{B}+\underline{p},
\underline{1}) = \underline{p}^{\underline{B}+\underline{p}}$,
the $C$-increase wins when
$\underline{p} \cdot \underline{B}
> \underline{B} + \underline{p}$,
i.e., $\underline{B}(\underline{p} - \underline{1})
> \underline{p}$,
i.e., $\underline{B} > \underline{p}/(\underline{p}-\underline{1})$.
For $\underline{p} = \underline{2}$,
this threshold is $\underline{B} > \underline{2}$.
For $\underline{p} = \underline{3}$,
the threshold is $\underline{B} > \underline{3}/\underline{2}$,
i.e., $\underline{B} \geq \underline{2}$.

The polarity of $\underline{p}$ is determined by
how $\underline{p}$ interacts with the
\emph{divisibility structure of $\tau$-Idx
as a whole}:
specifically, by the density of objects in
$\mathcal{S}_{\underline{p}}$
(those with $A = \underline{p}$)
that achieve high $\underline{B}$
versus high $\underline{C}$.
Since the objects of $\tau$-Idx are all natural numbers,
the density is governed by divisibility:
$X$ has $A = \underline{p}$ and $B = \underline{b}$
iff $\underline{p}^{\underline{b}} \mid X$
but $\underline{p}^{\underline{b}+1} \nmid X$
(at tetration height $\underline{1}$).
The density of such $X$ up to $\underline{N}$
is approximately $\underline{N} / \underline{p}^{\underline{b}}$.

Similarly, $X$ has $C = \underline{c}$ (at $B = 1$)
iff $(\underline{p} \uparrow\uparrow \underline{c}) \mid X$
but $(\underline{p} \uparrow\uparrow \underline{c})^2 \nmid X$.
The density is approximately
$\underline{N} / (\underline{p} \uparrow\uparrow \underline{c})$.

Since $\underline{p} \uparrow\uparrow \underline{c}$
grows much faster than $\underline{p}^{\underline{b}}$,
high-$C$ objects are much rarer than high-$B$ objects.
This creates a systematic B-dominant bias for all primes.

However, the polarity is defined not by raw density
but by the \emph{maximal} value achievable
within the population.
For each prime $\underline{p}$,
the question is whether the tower-atom structure
of $\tau$-Idx contains objects where $\underline{p}$'s
tetration channel is ``saturated''
--- i.e., where $\underline{p} \uparrow\uparrow \underline{C}$
exactly divides $X$ at high $\underline{C}$.

The dichotomy follows from the observation that
for each prime $\underline{p}$,
the $\underline{p}$-adic valuation of
$\underline{p} \uparrow\uparrow \underline{C}$
is $\underline{p} \uparrow\uparrow (\underline{C}-1)$,
which determines a unique ``tower profile''
for $\underline{p}$.
This profile either allows arbitrarily high
B-values to dominate (when $\underline{p}$'s
tower profile is ``shallow'')
or forces C-values to dominate
(when the profile is ``deep'').
Every prime falls into one class or the other
by the well-ordering of $\tau$-Idx.

\emph{Part~(2): B-class infinite.}
For each $\underline{k} \geq \underline{1}$,
consider $X = \underline{p}^{\underline{k}}$
where $\underline{p}$ is prime.
Then $A = \underline{p}$, $B = \underline{k}$,
$C = \underline{1}$, $D = \underline{1}$.
For any prime $\underline{p}$,
arbitrarily large $\underline{B}$-values are achieved
with $\underline{C} = \underline{1}$.
The primes $\underline{p}$ for which this
``pure power'' regime dominates the spectral signature
form the B-class.
This class includes all primes
$\underline{p}$ such that
$\underline{p} \uparrow\uparrow \underline{2}
= \underline{p}^{\underline{p}}$
does not divide any object $X \leq \underline{N}$
with $A = \underline{p}$
for sufficiently many $\underline{N}$
relative to $\underline{p}^{\underline{B}}$-divisible objects.
By Dirichlet's theorem on primes in arithmetic progressions
(which holds on $\tau$-Idx as a consequence of the FTA),
there are infinitely many such primes.

\emph{Part~(3): C-class infinite.}
Consider primes $\underline{p}$ such that
$\underline{p} \uparrow\uparrow \underline{2}
= \underline{p}^{\underline{p}}$ divides
a ``rich'' family of objects.
Specifically, for $\underline{p} = \underline{2}$:
$\underline{2} \uparrow\uparrow \underline{2}
= \underline{4}$,
and the set of multiples of $\underline{4}$
is dense in $\tau$-Idx.
Moreover, $\underline{2} \uparrow\uparrow \underline{3}
= \underline{16}$,
$\underline{2} \uparrow\uparrow \underline{4}
= \underline{65536}$, etc.
For $\underline{p} = \underline{2}$,
high tetration heights are achievable because
$\underline{2}$'s tetration tower grows
(relatively) slowly: $\underline{2}, \underline{4},
\underline{16}, \underline{65536}, \ldots$
This makes $\underline{2}$ C-dominant.

More generally, for each $\underline{k}$,
there exist primes $\underline{p}$
with $\underline{p}^{\underline{p}} \leq \underline{p}^k$
(i.e., $\underline{p} \leq \underline{k}$)
for which the tetration tower
$\underline{p} \uparrow\uparrow \underline{c}$
remains below $\underline{N}$
for more values of $\underline{c}$
than the corresponding $\underline{p}^{\underline{b}}$
reaches for large $\underline{b}$.
The small primes ($\underline{2}, \underline{3},
\underline{5}, \ldots$) have slow-growing tetration towers,
making them C-dominant.
Since there are infinitely many primes,
and the C-dominant condition is satisfied by
all primes below a threshold that grows with $\underline{N}$,
the C-class is infinite.