írj egy appot python es html The backend must always be impl...

Prompt

írj egy appot python es html The backend must always be implemented in a single file, and the frontend must always be contained in a single index.html file, including all subpages. azt akarom hogy ilyen es ehhez hasonlo kepleteket: $$ \begin{aligned} \mathbf{F} = (\mathbf{P}, \mathbf{Q}) \quad : \quad & \begin{cases} \sigma \in \operatorname{Aut}(\mathbb{C}^2) : \begin{cases} u(x, y) = x - 2y(y^2 + x) - (y^2 + x)^2 \\ v(x, y) = y + y^2 + x \end{cases} \implies \det J_{(x, y)}(u, v) \equiv 1 \\[12pt] \mathbf{P}(x, y) = -\dfrac{v(x, y)}{1 + u(x, y)v(x, y)}, \qquad \mathbf{Q}(x, y) = u(x, y)\Big(1 + u(x, y)v(x, y)\Big) \\[12pt] \forall p \ge 5 : \begin{cases} \mathbf{P}^{(p)}(x, y) = -v(x, y) \sum_{k=0}^{p-2} \Big(-u(x, y)v(x, y)\Big)^k \in \mathbb{F}_p[x, y] \\ \mathbf{Q}^{(p)}(x, y) = u(x, y)\Big(1 + u(x, y)v(x, y)\Big) \in \mathbb{F}_p[x, y] \end{cases} \\[14pt] \mathbf{F}_{\mathcal{U}} = \Big[ \big(\mathbf{P}^{(p)}, \mathbf{Q}^{(p)}\big) \Big]_{p \in \mathcal{U}} : \prod_{p \in \mathcal{U}} \overline{\mathbb{F}}_p^2 \to \prod_{p \in \mathcal{U}} \overline{\mathbb{F}}_p^2 \quad \left(\prod_{p \in \mathcal{U}} \overline{\mathbb{F}}_p \cong \mathbb{C}\right) \\[12pt] \mathcal{S} = \left\{ (x, y, w) \in \mathbb{C}^3 \;\middle|\; y = w^2 + \dfrac{32}{9} x^{15} w^9 \right\} \cong \mathbb{A}^2_{(x, w)} : \begin{cases} \mathbf{P}_{\mathcal{S}}(x, w) = x + x^{16} w^7 \\ \mathbf{Q}_{\mathcal{S}}(x, w) = w^2 \end{cases} \end{cases} \\[16pt] \wedge\quad & \det J_{(x, y)}(\mathbf{P}, \mathbf{Q}) = \det J_{(u, v)}(\mathbf{P}, \mathbf{Q}) \cdot \det J_{(x, y)}(u, v) \equiv 1 \cdot 1 = \mathbf{1} \quad \text{mindenütt a reguláris szimplektikus fázistéren } \mathbb{C}^2 \setminus \{1 + uv = 0\} \\[10pt] \wedge\quad & \forall p \ge 5 : \quad \det J_{(x, y)}\big(\mathbf{P}^{(p)}, \mathbf{Q}^{(p)}\big) = 1 + (-1)^{p-2} p \cdot (uv)^{p-1} \equiv \mathbf{1} \pmod p \quad \text{az affin polinomgyűrűben } \mathbb{F}_p[x, y] \\[10pt] \wedge\quad & \det J\big(\mathbf{F}_{\mathcal{U}}\big) = \Big[ \det J\big(\mathbf{P}^{(p)}, \mathbf{Q}^{(p)}\big) \Big]_{\mathcal{U}} \equiv \mathbf{1} \quad \text{a karakterisztika-0 algebrailag zárt test felett } \prod_{p \in \mathcal{U}} \overline{\mathbb{F}}_p \cong \mathbb{C} \\[10pt] \wedge\quad & \forall p \ge 5 : \begin{cases} a_p = \sigma^{-1}\left(\frac{1}{2}, 2\right) \in \mathbb{F}_p^2 \\ b_p = \sigma^{-1}\left(\frac{1}{3}, 6\right) \in \mathbb{F}_p^2 \end{cases} \implies a_p \ne b_p \;\wedge\; \mathbf{F}^{(p)}(a_p) = \mathbf{F}^{(p)}(b_p) = (0, 1) \implies \neg\operatorname{Inj}\big(\mathbf{F}^{(p)}\big) \\[12pt] \wedge\quad & \mathbf{F}_{\mathcal{S}}(0, 1) = \mathbf{F}_{\mathcal{S}}(0, -1) = (0, 1) \implies \neg\operatorname{Inj}(\mathbf{F}_{\mathcal{S}}) \\[10pt] \wedge\quad & \mathrm{d}\mathbf{P}_{\mathcal{S}} \wedge \mathrm{d}\mathbf{Q}_{\mathcal{S}} \equiv \mathrm{d}x \wedge \mathrm{d}y = \left(2w + 32x^{15}w^8\right)\mathrm{d}x \wedge \mathrm{d}w \implies \left(\frac{\mathrm{d}\mathbf{P}_{\mathcal{S}} \wedge \mathrm{d}\mathbf{Q}_{\mathcal{S}}}{\mathrm{d}x \wedge \mathrm{d}y}\right)_{\mathrm{rel}} \equiv 1 \\[10pt] \wedge\quad & [\mathbb{C}(\mathcal{S}) : \mathbb{C}(x, y)] = 9 \;\wedge\; [\mathbb{C}(\mathcal{S}) : \mathbb{C}(\mathbf{P}_{\mathcal{S}}, \mathbf{Q}_{\mathcal{S}})] = \deg(\mathbf{F}_{\mathcal{S}}) = 32 \ge 5 \\[10pt] \wedge\quad & \mathcal{A}_{\mathbf{F}} = \{v = 0\} \;\wedge\; \mathcal{B}_{\mathbf{F}} = \{v = 0\} \cup \left\{ 256\left(\frac{16u}{15}\right)^{30}v^7 = 1 \right\} \\[10pt] \wedge\quad & \pi_1(\mathbb{C}^2 \setminus \mathcal{B}_{\mathbf{F}}) \curvearrowright S_{32} : \begin{cases} \sigma_0 = (1\,3)(2\,4\,5\,\dots\,32) \sim (2)(30) \\ \sigma_c = (1\,2) \sim (2) \end{cases} \implies [\sigma_0, \sigma_c] = (1\,2)(3\,4) \ne \operatorname{id} \implies \text{Nem-abeli étale monodrómia} \\[12pt] \wedge\quad & \widetilde{\mathbf{F}} \in \operatorname{Mor}\Big(\operatorname{Bl}_{\{[1:0:0], [0:1:0]\}}(\mathbb{P}^2) \setminus \widetilde{D}, \, \mathbb{P}^2\Big) \implies \widetilde{\mathbf{F}}^*\left(\frac{\mathrm{d}\mathbf{P} \wedge \mathrm{d}\mathbf{Q}}{1 - \mathbf{P}\mathbf{Q}}\right) = \frac{\mathrm{d}x \wedge \mathrm{d}y}{1 + u(x, y)v(x, y)} = \Omega_{\log} \\[12pt] \wedge\quad & \neg\operatorname{Proper}(\mathbf{F}) \;\wedge\; \neg\operatorname{Bir}(\mathbf{P}_{\mathcal{S}}, \mathbf{Q}_{\mathcal{S}}) \;\wedge\; \neg\operatorname{Inv}(\mathbf{F}_{\mathcal{S}}) \end{aligned} $$ frontenden be illesztek DE BARMILYEN KEPLETTEL MUKODJON es a backend a TELJES KEPLETET az utolso betu/szamig mindent le futtar es vegul ki irja h ervesnyes counterexample e a jacobi sejtesre 2d ben. de ne ugy csinald hogy SZANDEKOSAN csak azt irja ji hogy nem csak azert mert te egy papagaj vagy es csak annyit tudsz szajkozni hogy remains open remain open es MINDEN FELE KEPLETTEL MULODJON Send back the complete code files with all fixes applied. Fix all of the listed errors one by one, actually fixing them. Keep the original imports, because the files exist. Write out every single character, do not shorten anything. Fix every error. Do not write anything else, only the complete code files, and there must not be any comments in them. I never allow any simplified mock, placeholder, dummy, simulated, or fake crap, and write the complete files as unshortened production-ready code in a code block. They must be 100% error-free, and send them as downloadable files. frontend az ultra modetn letisztult fekete apple liquid glass designes pwa spa The provided specification defines a self-contained React toolkit (`window.GlassEngine`)—comprising custom reactive signals, cubic-bezier tweening, spring-wobble physics, optical geometry helpers, procedural canvas displacement-map generation, and both WebGL2 and SVG-filter rendering pipelines to power refractive, liquid glass components alongside a companion script that observes the DOM to automatically mount `GlassMaterial` layers into matching UI elements.

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