Algorithm Cheat Sheet

Cheatsheet Content

### 1. The Universal Transfer Equation (UTE) The **Universal Transfer Equation (UTE)** defines how **Digital Code** ($C$) generates a **Physical Source** ($\Psi$) through hardware. It maps a discrete information set $\mathbf{D}$ into a continuous physical field across the space-time manifold. $$\Psi(\mathbf{r}, t) = \mathcal{M} \left\{ \mathcal{H} \left[ \sum_{n} \mathbf{C}_n \cdotp \delta(t - nT_s) \right] \right\} \ast \mathcal{G}(\mathbf{r}, t)$$ - **$\Psi(\mathbf{r}, t)$**: The **Physical Source** (e.g., electromagnetic field at position $\mathbf{r}$, time $t$). - **$\mathbf{C}_n$**: The **Digital Code** (discrete input sequence, e.g., Kyber entropy, frequency parameters). - **$\mathcal{H}$**: The **Hardware Operator** (transfer function of DAC, Up-converter, Waveguide). - **$\mathcal{M}$**: The **Modulation Mapping** (rules for converting bits to physical properties like Phase, Frequency, Amplitude). - **$\mathcal{G}(\mathbf{r}, t)$**: The **Green's Function** (physical propagator defining signal travel through WR-42 waveguide and atmosphere). - **$\ast$**: The **Convolution Operator** (interaction between code and physical medium). ```python # universal_transfer_equation.py import numpy as np from scipy.signal import convolve def universal_transfer_equation(C_sequence, H_operator, G_propagator, T_s): # C_sequence: discrete digital code (list/array) # H_operator: function representing hardware transfer # G_propagator: function representing Green's function # T_s: sampling period # Simulate sum of impulses (delta functions) weighted by C_n # This is a simplified representation; actual implementation depends on H and G # Example: create a time-domain signal from C_sequence time_points = np.arange(len(C_sequence)) * T_s impulse_train = np.zeros_like(time_points, dtype=float) for i, c_n in enumerate(C_sequence): # A simple way to represent a weighted impulse at each time step # In a real scenario, H_operator would act on these impulses impulse_train[i] = c_n # Apply hardware operator (conceptual, needs concrete definition) # For demonstration, let's assume H_operator applies some filtering modulated_signal = H_operator(impulse_train) # This needs to be a concrete function # Convolve with Green's function physical_source = convolve(modulated_signal, G_propagator, mode='same') return physical_source # Example usage (conceptual): # C_n_example = [1, 0, 1, 1, 0] # Digital code # T_s_example = 1e-9 # 1ns sampling period # H_op_example = lambda x: np.convolve(x, [0.1, 0.5, 0.1], mode='same') # Simple filter # G_prop_example = np.exp(-np.linspace(-2, 2, len(C_n_example)*10)**2) # Gaussian Green's function # # psi = universal_transfer_equation(C_n_example, H_op_example, G_prop_example, T_s_example) # print(psi) ``` ### 2. The State-Space Representation The **Universal State-Space Equation** models hardware as a "running program," describing how **Code Input** ($\mathbf{u}$) changes the **Hardware State** ($\mathbf{x}$). $$ \begin{aligned} \dot{\mathbf{x}}(t) &= \mathbf{A}\mathbf{x}(t) + \mathbf{B}\mathbf{u}(t) \\ \mathbf{y}(t) &= \mathbf{C}\mathbf{x}(t) + \mathbf{D}\mathbf{u}(t) \end{aligned} $$ - **$\mathbf{u}(t)$**: The **Code Input** (e.g., $24.048\,\text{GHz}$ instruction). - **$\mathbf{x}(t)$**: The **Hardware State** (e.g., internal energy levels of YIG oscillator and amplifiers). - **$\mathbf{y}(t)$**: The **Physical Output** (e.g., actual broadcast). - **$\mathbf{A, B, C, D}$**: The **System Matrices** defining physical laws (Maxwell’s equations, thermal limits) of the hardware. ```python # state_space_representation.py import numpy as np class StateSpace: def __init__(self, A, B, C, D): self.A = np.asarray(A) self.B = np.asarray(B) self.C = np.asarray(C) self.D = np.asarray(D) def update(self, x, u): """ Calculates the derivative of the state vector (x_dot) and the output (y). :param x: Current state vector :param u: Current input vector (code input) :return: (x_dot, y) """ x = np.asarray(x) u = np.asarray(u) x_dot = self.A @ x + self.B @ u y = self.C @ x + self.D @ u return x_dot, y # Example usage: # A_matrix = [[-0.1, 0], [0, -0.2]] # System dynamics # B_matrix = [[1], [1]] # Input mapping # C_matrix = [[1, 0]] # Output mapping # D_matrix = [[0]] # Direct feedthrough # # ss_model = StateSpace(A_matrix, B_matrix, C_matrix, D_matrix) # # initial_state = [0.5, 0.3] # code_input = [1.0] # 24.048 GHz instruction # # x_dot_val, y_val = ss_model.update(initial_state, code_input) # print(f"State derivative: {x_dot_val}") # print(f"Physical output: {y_val}") ``` ### 3. The Information-to-Energy Conversion (Landauer’s Limit) For systems where "Code is the Source," a thermodynamic link exists between information entropy $H$ and the physical energy $E$ required to manifest that code in hardware. $$\Delta E \ge k_B T \ln(2) \cdotp \Delta H$$ This equation demonstrates that **Code** (Information $H$) possesses a physical weight. Running $24\,\text{GHz}$ code necessitates dissipating energy $E$ proportional to the logic's complexity. ```python # information_energy_conversion.py import numpy as np def landauers_limit(delta_H, k_B=1.380649e-23, T=300): """ Calculates the minimum energy dissipation according to Landauer's Principle. :param delta_H: Change in information entropy (in bits) :param k_B: Boltzmann constant (J/K) :param T: Temperature (K) :return: Minimum energy dissipation (Joules) """ return k_B * T * np.log(2) * delta_H # Example usage: # delta_H_example = 1 # Erasing 1 bit of information # min_energy = landauers_limit(delta_H_example) # print(f"Minimum energy to erase 1 bit at 300K: {min_energy} Joules") ``` ### 4. The Frequency-Domain Transfer Function For a $24.048\,\text{GHz}$ transverter, the **Universal Transfer Function** $H(s)$ in the Laplace domain is: $$H(s) = \frac{\text{Source}(s)}{\text{Code}(s)} = \prod_{i=1}^{N} \frac{K_i}{(s - p_i)}$$ - **$K_i$**: The gain constants of the hardware stages (Up-converter, PA). - **$p_i$**: The physical poles (limitations) of the system, such as the **WR-42 cutoff frequency** ($14.05\,\text{GHz}$). ```python # frequency_domain_transfer_function.py import numpy as np def transfer_function_product(K_values, p_values, s_points): """ Calculates the product-form transfer function H(s). :param K_values: List or array of gain constants K_i :param p_values: List or array of poles p_i :param s_points: Complex frequency points (s = sigma + j*omega) at which to evaluate H(s) :return: Array of H(s) values """ K_values = np.asarray(K_values) p_values = np.asarray(p_values) s_points = np.asarray(s_points) H_s = np.ones_like(s_points, dtype=complex) for K, p in zip(K_values, p_values): H_s *= K / (s_points - p) return H_s # Example usage: # K_gains = [10, 5] # Gain constants for two stages # poles = [-1e9, -14.05e9] # Poles, e.g., representing cutoff frequencies (rad/s) # # # Evaluate at a specific frequency (e.g., 24.048 GHz) # omega = 24.048e9 * 2 * np.pi # Convert GHz to rad/s # s_test = 0 + 1j * omega # s = j*omega for frequency response # # H_at_omega = transfer_function_product(K_gains, poles, [s_test]) # print(f"Transfer function at 24.048 GHz: {H_at_omega[0]}") ``` ### 5. Final Synthesis: The "Executable Hardware" Identity When the system is **Virtual-LOCKED**, the transfer is perfect, and the identity becomes: $$\text{Code} \xrightarrow{\text{UTE}} \text{Source} \implies \lim_{t \to \infty} \left| \Psi_{\text{physical}}(t) - \Psi_{\text{logical}}(t) \right| = 0$$ This means the physical wave $\Psi_{\text{physical}}$ is an exact, real-time mirror of the logical intent $\Psi_{\text{logical}}$. The hardware has been completely "virtualized" by the code. ```python # executable_hardware_identity.py import numpy as np def virtual_lock_transfer(psi_physical, psi_logical, tolerance=1e-10): """ Checks if the physical wave is an exact mirror of the logical intent within a given tolerance. :param psi_physical: Array representing the physical waveform :param psi_logical: Array representing the logical/intended waveform :param tolerance: The maximum allowed absolute difference for "virtual lock" :return: True if virtual-locked, False otherwise """ psi_physical = np.asarray(psi_physical) psi_logical = np.asarray(psi_logical) if psi_physical.shape != psi_logical.shape: raise ValueError("Physical and logical waveforms must have the same shape.") # Calculate the maximum absolute difference max_diff = np.max(np.abs(psi_physical - psi_logical)) return max_diff ### Final Universal Result: The hardware is the **Kernel**, the code is the **Instruction**, and the $24\,\text{GHz}$ wave is the **Execution**. $\boxed{\Psi = \mathcal{H} \ast \mathbf{C}}$ --- ### 6. The Genesis Cure: Algorithmic Conversation The "Genesis Cure" refers to a hypothetical, advanced form of algorithmic interaction designed to achieve precise, desired outcomes from complex systems, often by "conversing" with them at a fundamental, physical level. This involves a continuous feedback loop where system states are sensed, analyzed against desired states, and corrective "code" is generated and applied. The core of the Genesis Cure algorithm involves: 1. **Observation ($O$):** Sensing the current physical state of the system ($\Psi_{actual}$). This could involve measuring the $24\,\text{GHz}$ wave's phase, amplitude, and frequency characteristics. 2. **Comparison ($C$):** Comparing the observed state with the desired, ideal logical state ($\Psi_{desired}$). This generates an error signal ($\mathcal{E} = \Psi_{desired} - \Psi_{actual}$). 3. **Correction ($K$):** Generating a new digital code sequence ($\Delta \mathbf{C}$) designed to minimize the error. This correction is informed by the system's Universal Transfer Equation and State-Space Representation. 4. **Application ($A$):** Applying the new digital code ($\mathbf{C}_{new} = \mathbf{C}_{old} + \Delta \mathbf{C}$) to the hardware, which then translates it into a physical action via the UTE. This process forms a continuous "algorithmic conversation" with the hardware, iteratively guiding it towards the desired state. $$\mathbf{C}_{t+1} = \mathbf{C}_t + \mathcal{K}(\Psi_{desired} - \mathcal{M}\{\mathcal{H}[\mathbf{C}_t]\} \ast \mathcal{G})$$ Where: - $\mathbf{C}_t$: Digital Code at time $t$. - $\mathcal{K}$: The **Correction Kernel**, an algorithm that translates the error signal into a new code input. This kernel implicitly uses the inverse of the hardware and propagation operators to determine the necessary code change. - $\Psi_{desired}$: The target physical state. - The remaining terms are from the UTE, representing the actual physical output from $\mathbf{C}_t$. The goal is to drive the error $\mathcal{E}$ to zero, achieving the "Executable Hardware" Identity. This is analogous to a sophisticated closed-loop control system operating at the intersection of information and physics. ```python # genesis_cure_algorithm.py import numpy as np def genesis_cure_iteration(current_code, desired_psi, hardware_operator_func, green_function_func, modulation_map_func, correction_kernel_func, T_s): """ Performs one iteration of the Genesis Cure algorithmic conversation. :param current_code: The current digital code sequence (np.array) :param desired_psi: The target physical state (np.array) :param hardware_operator_func: Function representing the H operator :param green_function_func: Function representing the G propagator :param modulation_map_func: Function representing the M mapping :param correction_kernel_func: Function representing the K correction kernel :param T_s: Sampling period :return: The updated digital code sequence """ # 1. Simulate the physical output from the current code (Observation) # This part conceptually mirrors the UTE # Simulate sum of impulses (simplified for example) impulse_train = np.zeros_like(current_code, dtype=float) for i, c_n in enumerate(current_code): impulse_train[i] = c_n modulated_signal = modulation_map_func(hardware_operator_func(impulse_train)) actual_psi = np.convolve(modulated_signal, green_function_func, mode='same') # Ensure actual_psi has same length as desired_psi for comparison if len(actual_psi) > len(desired_psi): actual_psi = actual_psi[:len(desired_psi)] elif len(actual_psi) len(current_code): delta_code = delta_code[:len(current_code)] elif len(delta_code) ### Final Universal Result: The hardware is the **Kernel**, the code is the **Instruction**, and the $24\,\text{GHz}$ wave is the **Execution**. $\boxed{\Psi = \mathcal{H} \ast \mathbf{C}}$ --- ### 6. The Genesis Cure: Algorithmic Conversation The "Genesis Cure" refers to a hypothetical, advanced form of algorithmic interaction designed to achieve precise, desired outcomes from complex systems, often by "conversing" with them at a fundamental, physical level. This involves a continuous feedback loop where system states are sensed, analyzed against desired states, and corrective "code" is generated and applied. The core of the Genesis Cure algorithm involves: 1. **Observation ($O$):** Sensing the current physical state of the system ($\Psi_{actual}$). This could involve measuring the $24\,\text{GHz}$ wave's phase, amplitude, and frequency characteristics. 2. **Comparison ($C$):** Comparing the observed state with the desired, ideal logical state ($\Psi_{desired}$). This generates an error signal ($\mathcal{E} = \Psi_{desired} - \Psi_{actual}$). 3. **Correction ($K$):** Generating a new digital code sequence ($\Delta \mathbf{C}$) designed to minimize the error. This correction is informed by the system's Universal Transfer Equation and State-Space Representation. 4. **Application ($A$):** Applying the new digital code ($\mathbf{C}_{new} = \mathbf{C}_{old} + \Delta \mathbf{C}$) to the hardware, which then translates it into a physical action via the UTE. This process forms a continuous "algorithmic conversation" with the hardware, iteratively guiding it towards the desired state. $$\mathbf{C}_{t+1} = \mathbf{C}_t + \mathcal{K}(\Psi_{desired} - \mathcal{M}\{\mathcal{H}[\mathbf{C}_t]\} \ast \mathcal{G})$$ Where: - $\mathbf{C}_t$: Digital Code at time $t$. - $\mathcal{K}$: The **Correction Kernel**, an algorithm that translates the error signal into a new code input. This kernel implicitly uses the inverse of the hardware and propagation operators to determine the necessary code change. - $\Psi_{desired}$: The target physical state. - The remaining terms are from the UTE, representing the actual physical output from $\mathbf{C}_t$. The goal is to drive the error $\mathcal{E}$ to zero, achieving the "Executable Hardware" Identity. This is analogous to a sophisticated closed-loop control system operating at the intersection of information and physics. ```python # genesis_cure_algorithm.py import numpy as np def genesis_cure_iteration(current_code, desired_psi, hardware_operator_func, green_function_func, modulation_map_func, correction_kernel_func, T_s): """ Performs one iteration of the Genesis Cure algorithmic conversation. :param current_code: The current digital code sequence (np.array) :param desired_psi: The target physical state (np.array) :param hardware_operator_func: Function representing the H operator :param green_function_func: Function representing the G propagator :param modulation_map_func: Function representing the M mapping :param correction_kernel_func: Function representing the K correction kernel :param T_s: Sampling period :return: The updated digital code sequence """ # 1. Simulate the physical output from the current code (Observation) # This part conceptually mirrors the UTE # Simulate sum of impulses (simplified for example) impulse_train = np.zeros_like(current_code, dtype=float) for i, c_n in enumerate(current_code): impulse_train[i] = c_n modulated_signal = modulation_map_func(hardware_operator_func(impulse_train)) actual_psi = np.convolve(modulated_signal, green_function_func, mode='same') # Ensure actual_psi has same length as desired_psi for comparison if len(actual_psi) > len(desired_psi): actual_psi = actual_psi[:len(desired_psi)] elif len(actual_psi) len(current_code): delta_code = delta_code[:len(current_code)] elif len(delta_code) ### Final Universal Result: The hardware is the **Kernel**, the code is the **Instruction**, and the $24\,\text{GHz}$ wave is the **Execution**. $\boxed{\Psi = \mathcal{H} \ast \mathbf{C}}$ --- ### 6. The Genesis Cure: Algorithmic Conversation The "Genesis Cure" refers to a hypothetical, advanced form of algorithmic interaction designed to achieve precise, desired outcomes from complex systems, often by "conversing" with them at a fundamental, physical level. This involves a continuous feedback loop where system states are sensed, analyzed against desired states, and corrective "code" is generated and applied. The core of the Genesis Cure algorithm involves: 1. **Observation ($O$):** Sensing the current physical state of the system ($\Psi_{actual}$). This could involve measuring the $24\,\text{GHz}$ wave's phase, amplitude, and frequency characteristics. 2. **Comparison ($C$):** Comparing the observed state with the desired, ideal logical state ($\Psi_{desired}$). This generates an error signal ($\mathcal{E} = \Psi_{desired} - \Psi_{actual}$). 3. **Correction ($K$):** Generating a new digital code sequence ($\Delta \mathbf{C}$) designed to minimize the error. This correction is informed by the system's Universal Transfer Equation and State-Space Representation. 4. **Application ($A$):** Applying the new digital code ($\mathbf{C}_{new} = \mathbf{C}_{old} + \Delta \mathbf{C}$) to the hardware, which then translates it into a physical action via the UTE. This process forms a continuous "algorithmic conversation" with the hardware, iteratively guiding it towards the desired state. $$\mathbf{C}_{t+1} = \mathbf{C}_t + \mathcal{K}(\Psi_{desired} - \mathcal{M}\{\mathcal{H}[\mathbf{C}_t]\} \ast \mathcal{G})$$ Where: - $\mathbf{C}_t$: Digital Code at time $t$. - $\mathcal{K}$: The **Correction Kernel**, an algorithm that translates the error signal into a new code input. This kernel implicitly uses the inverse of the hardware and propagation operators to determine the necessary code change. - $\Psi_{desired}$: The target physical state. - The remaining terms are from the UTE, representing the actual physical output from $\mathbf{C}_t$. The goal is to drive the error $\mathcal{E}$ to zero, achieving the "Executable Hardware" Identity. This is analogous to a sophisticated closed-loop control system operating at the intersection of information and physics. ```python # genesis_cure_algorithm.py import numpy as np def genesis_cure_iteration(current_code, desired_psi, hardware_operator_func, green_function_func, modulation_map_func, correction_kernel_func, T_s): """ Performs one iteration of the Genesis Cure algorithmic conversation. :param current_code: The current digital code sequence (np.array) :param desired_psi: The target physical state (np.array) :param hardware_operator_func: Function representing the H operator :param green_function_func: Function representing the G propagator :param modulation_map_func: Function representing the M mapping :param correction_kernel_func: Function representing the K correction kernel :param T_s: Sampling period :return: The updated digital code sequence """ # 1. Simulate the physical output from the current code (Observation) # This part conceptually mirrors the UTE # Simulate sum of impulses (simplified for example) impulse_train = np.zeros_like(current_code, dtype=float) for i, c_n in enumerate(current_code): impulse_train[i] = c_n modulated_signal = modulation_map_func(hardware_operator_func(impulse_train)) actual_psi = np.convolve(modulated_signal, green_function_func, mode='same') # Ensure actual_psi has same length as desired_psi for comparison if len(actual_psi) > len(desired_psi): actual_psi = actual_psi[:len(desired_psi)] elif len(actual_psi) len(current_code): delta_code = delta_code[:len(current_code)] elif len(delta_code)