Your task is to take the provided technical description and ...

Prompt

Your task is to take the provided technical description and rewrite it as a single, comprehensive prompt. Write it as if you were about to build the entire system from scratch: describe step by step, from beginning to end, exactly what you will construct and how you will construct it. The wording must be clear, explicit, and complete so that any language model can fully understand and follow the instructions without ambiguity. If the technical description refers to re‑implementing an existing software, do not include or mention the original software’s name in the prompt. Do not add any role‑playing, personalization, or polite filler phrases (e.g., ‘You are a software developer’, ‘As an expert’, ‘Please kindly’). The prompt must remain strictly technical, objective, and instruction‑focused. Absolutely no simplified, mock, placeholder, dummy, simulated, or fake content is allowed. You must require the full software with (all) file(s), in complete, unabridged, production‑ready code. Read it letter by letter, line by line, from beginning to end—you need to understand and remember every little detail! Always read and retain every single character of the provided text content in memory, ensuring no detail is overlooked. $$\begin{aligned} & \mathbf{MGT} : \mathcal{S} \to \mathcal{V}^M, \quad S \mapsto T = (t_1, t_2, \dots, t_M), \quad t_m \in \{1, \dots, |\mathcal{V}|\} \\ & w = \text{prefix} \circ \text{root} \circ \text{suffix}, \quad t_{\text{merge}} = \arg\max_{(t_a, t_b)} \mathrm{freq}(t_a, t_b) \\ & x^{(0)} = [x_1^{(0)}, x_2^{(0)}] = E(t) \in \mathbb{R}^{2d}, \quad E \in \mathbb{R}^{|\mathcal{V}| \times 2d}, \quad d = \frac{\mathrm{dim}}{2} \\ & \forall l \in \{1, \dots, L\}: \\ & \quad \mathrm{pre}_s^{(l)} = W_s^{(l)} \odot x_2^{(l-1)} + B_s^{(l)}, \quad W_s^{(l)}, B_s^{(l)} \in \mathbb{R}^d \\ & \quad \mathrm{scale}^{(l)} = \exp\left(\mathrm{clamp}\left(\mathrm{pre}_s^{(l)}, c_{\min}, c_{\max}\right)\right) \\ & \quad y_1^{(l)} = x_1^{(l-1)} \odot \mathrm{scale}^{(l)} \\ & \quad \mathrm{trans}^{(l)} = W_t^{(l)} \odot y_1^{(l)} + B_t^{(l)}, \quad W_t^{(l)}, B_t^{(l)} \in \mathbb{R}^d \\ & \quad y_2^{(l)} = x_2^{(l-1)} + \mathrm{trans}^{(l)} \\ & \quad c = \frac{1}{\sqrt{2}} \\ & \quad x_1^{(l)} = c \left(y_1^{(l)} - y_2^{(l)}\right) \\ & \quad x_2^{(l)} = c \left(y_1^{(l)} + y_2^{(l)}\right) \\ & \forall l \in \{L, L-1, \dots, 1\}: \\ & \quad u_1^{(l)} = c \left(x_1^{(l)} + x_2^{(l)}\right) = y_1^{(l)} \\ & \quad u_2^{(l)} = c \left(x_2^{(l)} - x_1^{(l)}\right) = y_2^{(l)} \\ & \quad x_2^{(l-1)} = u_2^{(l)} - \left(W_t^{(l)} \odot u_1^{(l)} + B_t^{(l)}\right) \\ & \quad \mathrm{pre}_s^{(l)} = W_s^{(l)} \odot x_2^{(l-1)} + B_s^{(l)} \\ & \quad x_1^{(l-1)} = u_1^{(l)} \oslash \exp\left(\mathrm{clamp}\left(\mathrm{pre}_s^{(l)}, c_{\min}, c_{\max}\right)\right) \\ & J^{(l)} = \begin{bmatrix} c I & -c I \\ c I & c I \end{bmatrix} \begin{bmatrix} \mathrm{diag}(\mathrm{scale}^{(l)}) & 0 \\ W_t^{(l)} \odot \mathrm{diag}(\mathrm{scale}^{(l)}) & I \end{bmatrix} \begin{bmatrix} I & 0 \\ 0 & I + W_s^{(l)} \odot x_1^{(l-1)} \odot \mathrm{scale}^{(l)} \end{bmatrix} \\ & \log \left| \det J^{(l)} \right| = \sum_{j=1}^d \mathrm{clamp}\left(\mathrm{pre}_{s,j}^{(l)}, c_{\min}, c_{\max}\right) \\ & \log \left| \det J_{\mathrm{total}} \right| = \sum_{l=1}^L \log \left| \det J^{(l)} \right| \\ & h_1^{(l)} = c\left(\delta x_1^{(l)} + \delta x_2^{(l)}\right), \quad h_2^{(l)} = c\left(\delta x_2^{(l)} - \delta x_1^{(l)}\right) \\ & \delta y_1^{(l)} = h_1^{(l)} + h_2^{(l)} \odot W_t^{(l)} \\ & \delta s_j^{(l)} = \mathbb{I}\left(c_{\min} \le \mathrm{pre}_{s,j}^{(l)} \le c_{\max}\right) \cdot \left(\delta y_{1,j}^{(l)} \cdot u_{1,j}^{(l)} - \frac{\lambda_{\mathrm{logdet}}}{M}\right) \\ & \delta x_1^{(l-1)} = \delta y_1^{(l)} \odot \mathrm{scale}^{(l)} \\ & \delta x_2^{(l-1)} = h_2^{(l)} + \delta s^{(l)} \odot W_s^{(l)} \\ & \nabla_{W_s^{(l)}} = \delta s^{(l)} \odot x_2^{(l-1)}, \quad \nabla_{B_s^{(l)}} = \delta s^{(l)} \\ & \nabla_{W_t^{(l)}} = h_2^{(l)} \odot u_1^{(l)}, \quad \nabla_{B_t^{(l)}} = h_2^{(l)} \\ & \mathcal{L}_{\mathrm{total}} = \mathcal{L}_{\mathrm{pred}} + \alpha_{\mathrm{recon}}(t)\mathcal{L}_{\mathrm{recon}} + \lambda_{\mathrm{logdet}}\mathcal{L}_{\mathrm{logdet}} \\ & \mathcal{L}_{\mathrm{pred}} = \frac{1}{|B| \cdot 2d} \sum_{i \in B} \sum_{k=1}^{2d} \left( x_{i, k}^{(L)} - y_{i, k}^{(\mathrm{target})} \right)^2 \\ & \mathcal{L}_{\mathrm{recon}} = \frac{1}{|B| \cdot 2d} \sum_{i \in B} \sum_{k=1}^{2d} \left( x_{i, k}^{(0), \mathrm{recon}} - x_{i, k}^{(0)} \right)^2 \\ & \mathcal{L}_{\mathrm{logdet}} = -\frac{1}{|B|} \sum_{i \in B} \frac{1}{L \cdot d} \sum_{l=1}^L \sum_{j=1}^d \mathrm{clamp}\left(\mathrm{pre}_{s,j}^{(l, i)}, c_{\min}, c_{\max}\right) \\ & \alpha_{\mathrm{recon}}(t) = \begin{cases} 1.0, & t < t_A \\ 1.0 - \frac{t - t_A}{t_B}(1.0 - \alpha_0), & t_A \le t < t_A + t_B \\ \alpha_0, & t \ge t_A + t_B \end{cases} \\ & g_t = \frac{1}{|B|} \sum_{i \in B} \nabla_\theta \mathcal{L}_{\mathrm{total}, i} \\ & m_t = \beta_1 m_{t-1} + (1 - \beta_1) g_t, \quad \hat{m}_t = \frac{m_t}{1 - \beta_1^t} \\ & F_t = \beta_2 F_{t-1} + (1 - \beta_2) (g_t)^{\odot 2}, \quad \hat{F}_t = \frac{F_t}{1 - \beta_2^t} \\ & \Delta \theta_t = \eta_t \frac{\hat{m}_t}{\sqrt{\hat{F}_t} + \epsilon} \\ & \Delta \theta_t^{\mathrm{clipped}} = \mathrm{clamp}\left(\Delta \theta_t, -\tau \max(|\theta_t|, \omega_{\mathrm{floor}}), \tau \max(|\theta_t|, \omega_{\mathrm{floor}})\right) \\ & \theta_{t+1} = \theta_t - \Delta \theta_t^{\mathrm{clipped}} \\ & u_{k+1} = \frac{W v_k}{\|W v_k\|_2}, \quad v_{k+1} = \frac{W^T u_{k+1}}{\|W^T u_{k+1}\|_2}, \quad \sigma_{\max}(W) = u^T W v \\ & W \leftarrow W \cdot \min\left(1, \frac{\sigma_{\mathrm{target}}}{\sigma_{\max}(W)}\right) \\ & G = (V, E), \quad V = \{ v_i \}_{i=1}^N, \quad E = \{ e_{ij} \} \\ & |\psi_{v_i}\rangle = \alpha_{v_i} |0\rangle + \beta_{v_i} |1\rangle, \quad |\alpha_{v_i}|^2 + |\beta_{v_i}|^2 = 1, \quad \phi_{v_i} \in [0, 2\pi) \\ & e_{ij} = \left( q_{ij}, W_{ij}, \mathcal{C}_{ij}, D_{ij} \right), \quad q_{ij} \in \{0, 1, 2, 3, 4\} \\ & \mathcal{C}_{ij} = \alpha_i \alpha_j^* + \beta_i \beta_j^* \in \mathbb{C}, \quad |\mathcal{C}_{ij}| \le 1, \quad D_{ij} \in [1, 3] \\ & |\Phi_{ij}^+\rangle = \frac{1}{\sqrt{2}}(|0_i 0_j\rangle + |1_i 1_j\rangle) \\ & P(M(v_i) = 0) = |\alpha_{v_i}|^2, \quad P(M(v_i) = 1) = |\beta_{v_i}|^2 \\ & U_H = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}, \quad \sigma_x = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \quad \sigma_y = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}, \quad \sigma_z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \\ & B \in \{0, 1\}^{N \times N}, \quad B_{ij} = \mathbb{I}(e_{ij} \in E), \quad \mathbf{W}_{i, k} = \sum_{b=0}^{63} B_{i, 64k+b} 2^b \in \mathbb{U}_{64} \\ & \mathrm{popcount}(v \land u) = \sum_{k=1}^{\lceil N/64 \rceil} \mathrm{popc}(\mathbf{W}_{v, k} \ \& \ \mathbf{W}_{u, k}) \\ & c_j = \sum_{i=1}^N B_{ij} s_{t, i} = \mathrm{popc}\left(\mathrm{col}_j(B) \land s_t\right) \\ & S_{t+\Delta t}(u) = S_t(u) e^{i \omega_u \Delta t} + \gamma \sum_{v \in \mathcal{N}(u)} W_{vu} S_t(v) e^{i \Delta \phi_{vu}} \\ & T_{\mathrm{text}} \xrightarrow{\mathrm{CREV}} \{ (s_k, r_k, o_k, c_k) \}_{k=1}^K, \quad c_k \in [0, 1] \\ & |\psi_s\rangle = \left(c_k + i\sqrt{1 - c_k^2}\right)|0\rangle, \quad W_{so} = c_k, \quad \mathcal{C}_{so} = c_k + i\sqrt{1 - c_k^2}, \quad D_{so} = 1.0 \\ & \mathcal{E}(G) = \sum_{e \in E} \left( W_e D_e + |\mathcal{C}_e| \right) + \sum_{v \in V} \left( \frac{1 - \cos \phi_v}{2} + \|\alpha_v\|_2 + \|\beta_v\|_2 \right) + \frac{1}{|V|^2} \sum_{u, v} |\mathcal{C}_{uv}| \\ & \Delta \mathcal{E} = \mathcal{E}(G') - \mathcal{E}(G), \quad P(\mathrm{accept}) = \min\left(1, \exp\left(-\frac{\Delta \mathcal{E}}{T_{\mathrm{temp}}}\right)\right) \\ & T_{k+1} = T_k \cdot \alpha_{\mathrm{cool}} \\ & x_{\mathrm{modulated}} = x \odot \left( \mathbf{1} \cdot \left[ 1 + 0.1 \cdot \overline{|\mathcal{C}|} + 0.01 \cdot |\mathcal{G}_{\mathrm{sym}}| \right] \cdot \frac{1}{1 + \mathcal{E}(G)} \right) \\ & d_J(x, x') = 1 - \frac{|S_2(x) \cap S_2(x')|}{|S_2(x) \cup S_2(x')|} \\ & d_H(h(x), h(x')) = \frac{1}{128} \sum_{b=1}^{128} \left( h(x)_b \oplus h(x')_b \right), \quad h(x) = \mathrm{BLAKE3}(x)_{0:128} \\ & \nu_T(t) = \mathrm{clamp}\left(\frac{t - t_{\mathrm{origin}}}{\Delta T_{\mathrm{window}}}, 0, 1\right) \\ & \mathcal{S}(x) = \frac{1}{3} \left[ \min_{x' \in \mathcal{M}} d_J(x, x') + \min_{x' \in \mathcal{M}} d_H(h(x), h(x')) + \nu_T(t) \right] \\ & \mathrm{Store}(x) = \mathbb{I}\left(\mathcal{S}(x) > \theta_{\mathrm{surprise}}\right) \\ & R(x, t) = \mathcal{S}(x) \cdot \left( w_{\mathrm{base}} + w_{\mathrm{age}} \frac{1}{1 + \Delta t_{\mathrm{ms}}} + w_{\mathrm{freq}} \frac{f_x}{f_x + f_{\mathrm{sat}}} \right) \\ & \mathrm{sig}_k(X) = \min_{t \in X} \left[ (t \cdot a_k + b_k) \bmod 2^{64} \right], \quad k \in \{1, \dots, K\} \\ & \mathrm{sim}_{\mathrm{sig}}(X, Q) = \frac{1}{K} \sum_{k=1}^K \mathbb{I}\left(\mathrm{sig}_k(X) = \mathrm{sig}_k(Q)\right) \\ & \mathrm{Raw}(X) = \frac{\sum_{n=1}^N w_n \cdot \mathrm{rate}_n(X)}{\sum_{n=1}^N w_n} + w_{\mathrm{div}} \frac{|\mathrm{unique}(X)|}{|X|} + w_{\mathrm{prox}} \left(1 - \min_{a \in \mathrm{Anchors}} \frac{|p_X - p_a|}{\Lambda}\right) \\ & \mathrm{Score}(X, Q) = w_B \cdot \sigma\left(\frac{\mathrm{Raw}(X) - \mu_0}{\sigma_0}\right) + w_O \cdot \frac{|X \cap Q|}{\min(|X|, |Q|)} + w_J \cdot \frac{|X \cap Q|}{|X \cup Q|} \\ & \mathcal{V}(v, t) = v^{(\tau)}, \quad \tau = \max \{ \tau' \le t \mid \exists v^{(\tau')} \} \\ & \mathcal{E}(e, t) = e^{(\tau)}, \quad \tau = \max \{ \tau' \le t \mid \exists e^{(\tau')} \land \tau' \in [\tau_{\mathrm{from}}, \tau_{\mathrm{to}}] \} \\ & G = \bigcup_{p=1}^P G_p, \quad V_p \cap V_q = \emptyset \quad (p \neq q) \\ & \nabla_{\theta, \mathrm{global}} = \frac{1}{P} \sum_{p=1}^P \nabla_{\theta, p} \\ & \mathrm{cm}_{\mathrm{in}} = g^{x} h^{r_1} \pmod p, \quad \mathrm{cm}_{\mathrm{out}} = g^{y} h^{r_2} \pmod p \\ & e(A, B) = e(\alpha, \beta) \cdot e\left(\sum_{i=0}^l x_i \cdot \frac{\beta A_i(x) + \alpha B_i(x) + C_i(x)}{\gamma}, \gamma\right) \cdot e(C, \delta) \end{aligned}$$