\documentclass[11pt]{article}
\usepackage{amsmath,amssymb}
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\title{Time, Frequency, Energy, and Bits}
\author{Hutter RNN Project}
\date{January 31, 2026}

\begin{document}
\maketitle

\section{Setup: Carrier Frequency}

Assume a carrier frequency $f_p$ that defines a fundamental period:
\[
T_p = \frac{1}{f_p}
\]

For concreteness, let $f_p = 1$ Hz, so $T_p = 1$ second.

Within one period, phase $\phi \in [0, 2\pi)$ maps to the unit interval:
\[
t = \frac{\phi}{2\pi} \in [0, 1)
\]

Everything in the system operates on some frequency. The spectrum decomposes signals into frequency components.

\section{Fundamental Relations}

\subsection{Energy and Frequency}
Planck's relation:
\[
E = h f
\]
where $h$ is Planck's constant. Energy is proportional to frequency.

\subsection{Frequency and Time}
Frequency is the reciprocal of period:
\[
f = \frac{1}{T}
\]

\subsection{Energy and Time}
Combining:
\[
E = \frac{h}{T}
\]
Energy and time are reciprocals (up to $h$).

The energy-time uncertainty principle:
\[
\Delta E \cdot \Delta t \geq \frac{\hbar}{2}
\]

\subsection{Power}
Power is energy per unit time:
\[
P = \frac{E}{T} = E \cdot f = h f^2
\]

Integrating power over the period:
\[
\int_0^{T_p} P \, dt = E_{\text{total}}
\]

\section{Information and Energy}

\subsection{Landauer's Principle}
Erasing one bit of information requires minimum energy:
\[
E_{\text{bit}} = k_B T \ln 2
\]
where $k_B$ is Boltzmann's constant and $T$ is temperature.

At fixed temperature, \textbf{bits are proportional to energy}:
\[
\text{bits} = \frac{E}{k_B T \ln 2}
\]

\subsection{Entropy Rate}
If events arrive at rate $f$ (events per second), and each event carries $H$ bits of entropy:
\[
\dot{H} = H \cdot f \quad \text{[bits/second]}
\]

Integrating over the period:
\[
\int_0^{T_p} \dot{H} \, dt = H \cdot f \cdot T_p = H \cdot \frac{f}{f_p} \quad \text{[bits per period]}
\]

\section{The Depth Limit Reinterpreted}

\subsection{Original Form}
The depth limit from float32 precision:
\[
d_{\max} = \frac{24}{-\log_2 p_{\text{avg}}} = \frac{24}{H_{\text{avg}}}
\]

This says: we can process at most $d_{\max}$ events before precision runs out.

\subsection{Energy Interpretation}
Rewrite as:
\[
d_{\max} \cdot H_{\text{avg}} = 24 \text{ bits}
\]

The left side is total entropy processed. The right side is the ``bit budget.''

Via Landauer:
\[
E_{\text{budget}} = 24 \cdot k_B T \ln 2
\]

\textbf{The depth limit is an energy budget.}

Each event ``spends'' $H_{\text{avg}} \cdot k_B T \ln 2$ energy. After $d_{\max}$ events, the budget is exhausted.

\subsection{Time Interpretation}
If events arrive at rate $f$:
\[
T_{\max} = \frac{d_{\max}}{f} = \frac{24}{H_{\text{avg}} \cdot f}
\]

This is the maximum temporal horizon---how far back we can ``remember.''

Higher entropy per event ($H_{\text{avg}}$) or higher event rate ($f$) reduces the horizon.

\section{Unit Analysis Summary}

\begin{center}
\begin{tabular}{lll}
\toprule
\textbf{Quantity} & \textbf{Symbol} & \textbf{Units} \\
\midrule
Frequency & $f$ & $[\text{time}^{-1}]$ \\
Period & $T = 1/f$ & $[\text{time}]$ \\
Energy & $E = hf$ & $[\text{energy}]$ \\
Power & $P = E/T = hf^2$ & $[\text{energy} \cdot \text{time}^{-1}]$ \\
\midrule
Entropy & $H$ & $[\text{bits}]$ \\
Entropy rate & $\dot{H} = Hf$ & $[\text{bits} \cdot \text{time}^{-1}]$ \\
\midrule
Bit-energy & $E_{\text{bit}} = k_B T \ln 2$ & $[\text{energy}]$ \\
Bits from energy & $E / E_{\text{bit}}$ & $[\text{bits}]$ \\
\bottomrule
\end{tabular}
\end{center}

\section{The Chain of Proportionalities}

\begin{align}
\text{bits} &\propto \text{energy} && \text{(Landauer)} \\
\text{energy} &= h \times \text{frequency} && \text{(Planck)} \\
\text{frequency} &= 1/\text{time} && \text{(definition)}
\end{align}

Therefore:
\[
\boxed{\text{bits} \propto \frac{h}{\text{time}}}
\]

More time (lower frequency) means fewer bits per unit action.
More bits requires more energy requires higher frequency requires finer time resolution.

\section{Implications}

\subsection{For RNNs}
The hidden state $h_t$ carries information through time. Each timestep ``costs'' entropy. The precision limit (24 bits for float32) sets the total budget.

W$_{hh}$ injection fails because temporal patterns require carrying bits across multiple steps, exceeding the budget.

\subsection{For Arithmetic Coding}
The interval $[L, H) \subset [0, 1)$ narrows with each symbol. The quotient $Q = 1/(H-L)$ grows. When $Q > 2^{24}$, precision is exhausted.

\subsection{For the Brain}
If neural computation operates on a carrier frequency $f_p$, the energy budget per period constrains how much can be computed.

Higher-frequency oscillations (gamma, ~40 Hz) enable more computation per second but cost more energy.

Lower-frequency oscillations (theta, ~6 Hz) are more efficient but slower.

\section{Connection to Zeta}

The Riemann zeta function:
\[
\zeta(s) = \sum_{n=1}^\infty n^{-s} = \prod_p \frac{1}{1 - p^{-s}}
\]

The sum over integers equals the product over primes---additive structure equals multiplicative structure.

In our framework:
\begin{itemize}
\item Integers $n$ = events encoded as natural numbers (E $\to$ N)
\item Primes $p$ = irreducible factors (atomic event spaces)
\item The duality: time-domain (sum) $\leftrightarrow$ frequency-domain (product)
\end{itemize}

The $\pi$ in $\zeta(2) = \pi^2/6$ connects to:
\begin{itemize}
\item The unit circle (phase $\in [0, 2\pi)$)
\item Fourier transform (integration over $e^{2\pi i f t}$)
\item Gaussian distributions (which arise from maximum entropy)
\end{itemize}

\section{Summary}

\begin{center}
\fbox{\parbox{0.8\textwidth}{
\textbf{Time, frequency, energy, and bits are all aspects of the same thing.}

The depth limit $d_{\max} = 24/H$ is:
\begin{itemize}
\item A bit budget (information)
\item An energy budget (thermodynamics)
\item A temporal horizon (time)
\item A precision limit (computation)
\end{itemize}

Fourier duality (time $\leftrightarrow$ frequency) underlies the Bayes/Thermo duality (probability $\leftrightarrow$ microstates).
}}
\end{center}

\end{document}
