# Deviation from Stock-to-Flow Model:

### 1. **Stock-to-Flow (S2F) Model**:

Given by:
\[ \text{S2F} = \frac{\text{Total stock (current supply)}}{\text{Flow (annual production)}} \]

Bitcoin price, according to the S2F model, can be given as:
\[ P_{S2F} = c \times \text{S2F}^\lambda \]
Where \( c \) and \( \lambda \) are constants.

### 2. **Quantum Representation**:

Let the price state of Bitcoin be represented as:
\[ |\Psi_{BTC}\rangle = \alpha|P_{low}\rangle + \beta|P_{high}\rangle \]

Where \( \alpha \) and \( \beta \) are probability amplitudes such that:
\[ |\alpha|^2 + |\beta|^2 = 1 \]

### 3. **Deviation from S2F Model**:

The expected value (mean) of the Bitcoin price can be represented as:
\[ \langle P_{BTC} \rangle = |\alpha|^2 P_{low} + |\beta|^2 P_{high} \]

The deviation, \( \Delta P \), from the S2F model is:
\[ \Delta P = \langle P_{BTC} \rangle - P_{S2F} \]

### 4. **External Factors**:

Let's represent external factors as a matrix operator, \( H \) (akin to the Hamiltonian in quantum mechanics), such that:
\[ H |\Psi_{BTC}\rangle = \gamma|\Psi_{BTC}\rangle \]
Where \( \gamma \) is a complex number representing the impact of the external factor on the state of Bitcoin's price.

### 5. **Short Run vs. Long Run Deviation**:

For short-term deviation, the wavefunction can be represented as:
\[ |\Psi_{short}\rangle = e^{-iHt/\hbar} |\Psi_{BTC}\rangle \]
For long-term deviation, we might have:
\[ |\Psi_{long}\rangle = e^{-iHt'/\hbar} |\Psi_{BTC}\rangle \]
Where \( t \) and \( t' \) represent short and long time intervals, respectively, and \( \hbar \) is the reduced Planck constant.

### 6. **Infinity**:

Mathematically, the idea of deviation until infinity implies that the time duration extends to infinity:
\[ \lim_{t \to \infty} |\Psi(t)\rangle \]
However, practically, an infinite deviation in financial markets would mean an unpredictable, unbounded behavior.

In summary, using a blend of quantum mechanics and financial modeling, the price behavior of Bitcoin and its deviation from the S2F model can be represented in a structured mathematical format. Still, these representations remain largely metaphorical and are not direct applications of quantum mechanics to finance.