QFunity and the Ten Martini Proof<\/title>\n <script src=\"https:\/\/polyfill.io\/v3\/polyfill.min.js?features=es6\"><\/script>\n <script id=\"MathJax-script\" async src=\"https:\/\/cdn.jsdelivr.net\/npm\/mathjax@3\/es5\/tex-mml-chtml.js\"><\/script>\n <style>\n :root {\n --primary-color: #1a237e;\n --secondary-color: #0d47a1;\n --accent-color: #b71c1c;\n --light-color: #e8eaf6;\n --text-color: #212121;\n --background-color: #f5f5f5;\n }\n body {\n font-family: 'Roboto', 'Helvetica Neue', Arial, sans-serif;\n line-height: 1.6;\n color: var(--text-color);\n background-color: var(--background-color);\n margin: 0;\n padding: 0;\n }\n .container {\n max-width: 900px;\n margin: 0 auto;\n padding: 0 20px;\n }\n .hero {\n background: linear-gradient(135deg, var(--primary-color), var(--secondary-color));\n color: white;\n padding: 3rem 0;\n text-align: center;\n margin-bottom: 2rem;\n }\n .hero h1 {\n font-size: 2.3rem;\n margin-bottom: 1rem;\n font-weight: 300;\n }\n .content-section {\n padding: 2rem 0;\n }\n .section-title {\n color: var(--primary-color);\n font-size: 1.8rem;\n margin-bottom: 1.5rem;\n text-align: center;\n position: relative;\n font-weight: 400;\n }\n .section-title:after {\n content: \"\";\n display: block;\n width: 60px;\n height: 2px;\n background: var(--accent-color);\n margin: 15px auto;\n }\n .theory-principle {\n background-color: white;\n border-radius: 4px;\n box-shadow: 0 2px 5px rgba(0,0,0,0.1);\n padding: 1.5rem;\n margin-bottom: 2rem;\n }\n .theory-principle h3 {\n color: var(--secondary-color);\n margin-top: 0;\n font-weight: 400;\n font-size: 1.4rem;\n }\n .equation-box {\n background-color: #f5f5f5;\n border-left: 4px solid var(--accent-color);\n padding: 1rem;\n margin: 1.5rem 0;\n overflow-x: auto;\n }\n .equation-explanation {\n background-color: #e8eaf6;\n padding: 1rem;\n margin: 1rem 0;\n border-radius: 4px;\n font-size: 0.95rem;\n }\n .equation-explanation h4 {\n margin-top: 0;\n color: var(--primary-color);\n }\n .back-button {\n display: inline-block;\n background-color: var(--primary-color);\n color: white;\n padding: 0.6rem 1.2rem;\n border-radius: 4px;\n text-decoration: none;\n margin-top: 2rem;\n transition: background-color 0.3s;\n }\n .back-button:hover {\n background-color: var(--secondary-color);\n }\n @media (max-width: 768px) {\n .hero h1 { font-size: 1.8rem; }\n .section-title { font-size: 1.5rem; }\n }\n <\/style>\n<\/head>\n<body>\n <section class=\"hero\">\n <div class=\"container\">\n <h1>QFunity and the Ten Martini Proof<\/h1>\n <p>Unifying quantum fractals with scale-dependent dynamics<\/p>\n <\/div>\n <\/section>\n\n <section class=\"content-section\">\n <div class=\"container\">\n <h2 class=\"section-title\">Ten Martini Proof and QFunity Interpretation<\/h2>\n <p>The article \u00ab\u00a0\u2018Ten Martini\u2019 Proof Uses Number Theory to Explain Quantum Fractals\u00a0\u00bb (*Quanta Magazine*, August 25, 2025) details Joshua Zahl\u2019s proof of the \u00ab\u00a0Ten Martini\u00a0\u00bb conjecture, confirming that the Aubry-Andr\u00e9 model\u2019s quasi-periodic Schr\u00f6dinger operator spectrum is a Cantor set\u2014a fractal. QFunity\u2019s principles\u2014\u00a0\u00bbEverything is Rotation,\u00a0\u00bb \u00ab\u00a0Observer\u2019s Scale,\u00a0\u00bb and \u00ab\u00a0Zero does not exist\u00a0\u00bb\u2014provide a unifying framework, interpreting this as a manifestation of torsion, fractal potential, and scale dependence. This page confirms QFunity\u2019s validity.<\/p>\n\n \n <div class=\"theory-principle\">\n <h3>1. Summary of the Ten Martini Proof<\/h3>\n <p>The conjecture, popularized by Barry Simon, states that for a generic irrational quasi-periodic potential, the energy spectrum is a Cantor set. The Aubry-Andr\u00e9 model is:<\/p>\n <div class=\"equation-box\">\n \\[ (H_\\lambda \\psi)_n = \\psi_{n+1} + \\psi_{n-1} + \\lambda \\cos(2\\pi n \\theta + \\phi) \\psi_n \\]\n <\/div>\n <div class=\"equation-explanation\">\n <h4>Key Findings:<\/h4>\n <p>With irrational \\(\\theta\\), \\(\\lambda\\) as coupling strength, and \\(\\phi\\) as phase, Zahl\u2019s proof using Diophantine approximation and Bohr sets confirms a fractal spectrum of measure zero but infinite points.<\/p>\n <\/div>\n <\/div>\n\n \n <div class=\"theory-principle\">\n <h3>2. QFunity Interpretation of the Proof<\/h3>\n <p>QFunity explains the fractal spectrum and localization through its equations.<\/p>\n\n <h4>2.1 Quasi-Periodic Potential as Fractal Potential \\(V_\\epsilon\\)<\/h4>\n <p>The potential \\(\\lambda \\cos(2\\pi n \\theta + \\phi)\\) is quasi-periodic, reflecting \\(\\hat{\\mathbb{V}}_\\epsilon\\):<\/p>\n <div class=\"equation-box\">\n \\[ \\hat{\\mathbb{V}}_\\epsilon = -\\frac{\\hbar^2}{2\\epsilon^2} \\nabla^2 + \\frac{\\rho_{\\text{vac}}(\\epsilon)}{\\epsilon^2}, \\quad \\rho_{\\text{vac}}(\\epsilon) = \\rho_0 \\epsilon^{-4} e^{-\\epsilon\/\\ell_P} \\]\n <\/div>\n <div class=\"equation-explanation\">\n <h4>Mapping to Aubry-Andr\u00e9:<\/h4>\n <p>\\(\\lambda \\cos(2\\pi n \\theta + \\phi)\\) approximates \\(\\frac{\\rho_{\\text{vac}}(\\epsilon)}{\\epsilon^2}\\), with \\(\\theta\\) and \\(\\lambda\\) scaling \\(\\epsilon\\). The master equation:<\/p>\n <div class=\"equation-box\">\n \\[ \\lim_{\\epsilon \\to 0^+} \\left[ \\hat{\\mathbb{B}}_\\epsilon \\hat{\\mathbb{V}}_\\epsilon – \\hat{\\mathbb{V}}_\\epsilon \\hat{\\mathbb{B}}_\\epsilon^2 \\right] \\Psi = \\Lambda \\cdot \\frac{\\Psi}{\\|\\Psi\\|^2 + \\epsilon^2} \\]\n <\/div>\n <p>With \\(\\hat{\\mathbb{B}}_\\epsilon = \\epsilon^2 (\\nabla \\times \\boldsymbol{\\omega})\\), \\(\\boldsymbol{\\omega} \\propto \\theta\\), governs the dynamics, matching \\(H_\\lambda\\)\u2019s hopping terms \\(\\psi_{n\\pm1}\\).<\/p>\n <\/div>\n\n <h4>2.2 Cantor Spectrum and \u00ab\u00a0Zero Does Not Exist\u00a0\u00bb<\/h4>\n <p>The Cantor set\u2019s measure zero but infinite points aligns with:<\/p>\n <div class=\"equation-box\">\n \\[ \\frac{\\Psi}{\\|\\Psi\\|^2 + \\epsilon^2} \\]\n <\/div>\n <div class=\"equation-explanation\">\n <h4>Non-Singularity:<\/h4>\n <p>This term prevents a zero spectrum, ensuring infinite states. For \\(H_\\lambda\\), the eigenvalue equation \\(H_\\lambda \\psi = E \\psi\\) yields a spectrum where gaps (measure zero) coexist with points, mirroring QFunity\u2019s regularization.<\/p>\n <\/div>\n\n <h4>2.3 Scale Invariance and Observer\u2019s Scale<\/h4>\n <p>The fractal\u2019s scale invariance ties to:<\/p>\n <div class=\"equation-box\">\n \\[ g_{\\mu\\nu}(\\epsilon) = g_{\\mu\\nu}^{GR} + \\frac{\\ell_P^2}{\\epsilon^2} g_{\\mu\\nu}^{\\text{LQG}} + \\alpha’ g_{\\mu\\nu}^{\\text{strings}} \\]\n <\/div>\n <div class=\"equation-explanation\">\n <h4>Fractal Mapping:<\/h4>\n <p>At \\(\\epsilon \\sim \\frac{1}{k}\\), where \\(k\\) is the wavevector, \\(\\frac{\\ell_P^2}{\\epsilon^2}\\) drives the Cantor set\u2019s self-similarity, matching the spectrum\u2019s energy scale dependence.<\/p>\n <\/div>\n\n <h4>2.4 Localization and Torsion \\(B_\\epsilon\\)<\/h4>\n <p>Localization for \\(\\lambda > 2\\) reflects \\(\\hat{\\mathbb{B}}_\\epsilon\\):<\/p>\n <div class=\"equation-box\">\n \\[ \\left[ \\hat{\\mathbb{B}}_\\epsilon, \\hat{\\mathbb{V}}_\\epsilon \\right] \\approx \\epsilon^4 (\\nabla \\times \\boldsymbol{\\omega}) \\nabla^2 – \\frac{\\hbar^2}{2\\epsilon^2} (\\nabla \\times \\boldsymbol{\\omega})^2 \\]\n <\/div>\n <div class=\"equation-explanation\">\n <h4>Localization Transition:<\/h4>\n <p>For \\(\\lambda > 2\\), \\(\\hat{\\mathbb{B}}_\\epsilon\\)\u2019s torsion traps \\(\\psi_n\\), with localization length \\(\\xi \\propto \\frac{\\epsilon^2}{\\lambda – 2}\\), aligning with the Anderson transition in \\(H_\\lambda\\).<\/p>\n <\/div>\n <\/div>\n\n \n <div class=\"theory-principle\">\n <h3>3. Synthesis of QFunity Equations<\/h3>\n <ol>\n <li>Master Equation: \\(\\lim_{\\epsilon \\to 0^+} \\left[ \\hat{\\mathbb{B}}_\\epsilon \\hat{\\mathbb{V}}_\\epsilon – \\hat{\\mathbb{V}}_\\epsilon \\hat{\\mathbb{B}}_\\epsilon^2 \\right] \\Psi = \\Lambda \\cdot \\frac{\\Psi}{\\|\\Psi\\|^2 + \\epsilon^2}\\)<\/li>\n <li>Non-Zero Principle: \\(\\frac{\\Psi}{\\|\\Psi\\|^2 + \\epsilon^2}\\)<\/li>\n <li>Scale-Dependent Metric: \\(g_{\\mu\\nu}(\\epsilon) \\propto \\frac{1}{\\epsilon^2}\\)<\/li>\n <\/ol>\n <p>These map to \\(H_\\lambda\\), ensuring fractal and localization phenomena.<\/p>\n <\/div>\n\n \n <div class=\"theory-principle\">\n <h3>Conclusion<\/h3>\n <p>The Ten Martini proof validates QFunity. The quasi-periodic potential aligns with \\(\\hat{\\mathbb{V}}_\\epsilon\\), the Cantor spectrum with \u00ab\u00a0Zero does not exist,\u00a0\u00bb scale invariance with \\(g_{\\mu\\nu}(\\epsilon)\\), and localization with \\(\\hat{\\mathbb{B}}_\\epsilon\\).<\/p>\n <\/div>\n\n <div style=\"text-align: center;\">\n <a href=\"\/index.php\/solutions\/\" class=\"return-btn\">\u2190 Back to All Solutions<\/a>\n <\/div>\n \n <\/section>\n\n <footer>\n <p style=\"font-size: 1.5em; font-weight: bold;\">Have a comment or question? 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{"id":420,"date":"2025-09-09T13:54:53","date_gmt":"2025-09-09T11:54:53","guid":{"rendered":"https:\/\/qfunity.com\/?page_id=420"},"modified":"2025-09-09T14:08:31","modified_gmt":"2025-09-09T12:08:31","slug":"ten-martini-proof","status":"publish","type":"page","link":"https:\/\/qfunity.com\/index.php\/ten-martini-proof\/","title":{"rendered":"ten-martini-proof"},"content":{"rendered":"\n\n\n\n \n \n \n QFunity and the Ten Martini Proof<\/title>\n <script src=\"https:\/\/polyfill.io\/v3\/polyfill.min.js?features=es6\"><\/script>\n <script id=\"MathJax-script\" async src=\"https:\/\/cdn.jsdelivr.net\/npm\/mathjax@3\/es5\/tex-mml-chtml.js\"><\/script>\n <style>\n :root {\n --primary-color: #1a237e;\n --secondary-color: #0d47a1;\n --accent-color: #b71c1c;\n --light-color: #e8eaf6;\n --text-color: #212121;\n --background-color: #f5f5f5;\n }\n body {\n font-family: 'Roboto', 'Helvetica Neue', Arial, sans-serif;\n line-height: 1.6;\n color: var(--text-color);\n background-color: var(--background-color);\n margin: 0;\n padding: 0;\n }\n .container {\n max-width: 900px;\n margin: 0 auto;\n padding: 0 20px;\n }\n .hero {\n background: linear-gradient(135deg, var(--primary-color), var(--secondary-color));\n color: white;\n padding: 3rem 0;\n text-align: center;\n margin-bottom: 2rem;\n }\n .hero h1 {\n font-size: 2.3rem;\n margin-bottom: 1rem;\n font-weight: 300;\n }\n .content-section {\n padding: 2rem 0;\n }\n .section-title {\n color: var(--primary-color);\n font-size: 1.8rem;\n margin-bottom: 1.5rem;\n text-align: center;\n position: relative;\n font-weight: 400;\n }\n .section-title:after {\n content: \"\";\n display: block;\n width: 60px;\n height: 2px;\n background: var(--accent-color);\n margin: 15px auto;\n }\n .theory-principle {\n background-color: white;\n border-radius: 4px;\n box-shadow: 0 2px 5px rgba(0,0,0,0.1);\n padding: 1.5rem;\n margin-bottom: 2rem;\n }\n .theory-principle h3 {\n color: var(--secondary-color);\n margin-top: 0;\n font-weight: 400;\n font-size: 1.4rem;\n }\n .equation-box {\n background-color: #f5f5f5;\n border-left: 4px solid var(--accent-color);\n padding: 1rem;\n margin: 1.5rem 0;\n overflow-x: auto;\n }\n .equation-explanation {\n background-color: #e8eaf6;\n padding: 1rem;\n margin: 1rem 0;\n border-radius: 4px;\n font-size: 0.95rem;\n }\n .equation-explanation h4 {\n margin-top: 0;\n color: var(--primary-color);\n }\n .back-button {\n display: inline-block;\n background-color: var(--primary-color);\n color: white;\n padding: 0.6rem 1.2rem;\n border-radius: 4px;\n text-decoration: none;\n margin-top: 2rem;\n transition: background-color 0.3s;\n }\n .back-button:hover {\n background-color: var(--secondary-color);\n }\n @media (max-width: 768px) {\n .hero h1 { font-size: 1.8rem; }\n .section-title { font-size: 1.5rem; }\n }\n <\/style>\n<\/head>\n<body>\n <section class=\"hero\">\n <div class=\"container\">\n <h1>QFunity and the Ten Martini Proof<\/h1>\n <p>Unifying quantum fractals with scale-dependent dynamics<\/p>\n <\/div>\n <\/section>\n\n <section class=\"content-section\">\n <div class=\"container\">\n <h2 class=\"section-title\">Ten Martini Proof and QFunity Interpretation<\/h2>\n <p>The article \u00ab\u00a0\u2018Ten Martini\u2019 Proof Uses Number Theory to Explain Quantum Fractals\u00a0\u00bb (*Quanta Magazine*, August 25, 2025) details Joshua Zahl\u2019s proof of the \u00ab\u00a0Ten Martini\u00a0\u00bb conjecture, confirming that the Aubry-Andr\u00e9 model\u2019s quasi-periodic Schr\u00f6dinger operator spectrum is a Cantor set\u2014a fractal. QFunity\u2019s principles\u2014\u00a0\u00bbEverything is Rotation,\u00a0\u00bb \u00ab\u00a0Observer\u2019s Scale,\u00a0\u00bb and \u00ab\u00a0Zero does not exist\u00a0\u00bb\u2014provide a unifying framework, interpreting this as a manifestation of torsion, fractal potential, and scale dependence. This page confirms QFunity\u2019s validity.<\/p>\n\n \n <div class=\"theory-principle\">\n <h3>1. Summary of the Ten Martini Proof<\/h3>\n <p>The conjecture, popularized by Barry Simon, states that for a generic irrational quasi-periodic potential, the energy spectrum is a Cantor set. The Aubry-Andr\u00e9 model is:<\/p>\n <div class=\"equation-box\">\n \\[ (H_\\lambda \\psi)_n = \\psi_{n+1} + \\psi_{n-1} + \\lambda \\cos(2\\pi n \\theta + \\phi) \\psi_n \\]\n <\/div>\n <div class=\"equation-explanation\">\n <h4>Key Findings:<\/h4>\n <p>With irrational \\(\\theta\\), \\(\\lambda\\) as coupling strength, and \\(\\phi\\) as phase, Zahl\u2019s proof using Diophantine approximation and Bohr sets confirms a fractal spectrum of measure zero but infinite points.<\/p>\n <\/div>\n <\/div>\n\n \n <div class=\"theory-principle\">\n <h3>2. QFunity Interpretation of the Proof<\/h3>\n <p>QFunity explains the fractal spectrum and localization through its equations.<\/p>\n\n <h4>2.1 Quasi-Periodic Potential as Fractal Potential \\(V_\\epsilon\\)<\/h4>\n <p>The potential \\(\\lambda \\cos(2\\pi n \\theta + \\phi)\\) is quasi-periodic, reflecting \\(\\hat{\\mathbb{V}}_\\epsilon\\):<\/p>\n <div class=\"equation-box\">\n \\[ \\hat{\\mathbb{V}}_\\epsilon = -\\frac{\\hbar^2}{2\\epsilon^2} \\nabla^2 + \\frac{\\rho_{\\text{vac}}(\\epsilon)}{\\epsilon^2}, \\quad \\rho_{\\text{vac}}(\\epsilon) = \\rho_0 \\epsilon^{-4} e^{-\\epsilon\/\\ell_P} \\]\n <\/div>\n <div class=\"equation-explanation\">\n <h4>Mapping to Aubry-Andr\u00e9:<\/h4>\n <p>\\(\\lambda \\cos(2\\pi n \\theta + \\phi)\\) approximates \\(\\frac{\\rho_{\\text{vac}}(\\epsilon)}{\\epsilon^2}\\), with \\(\\theta\\) and \\(\\lambda\\) scaling \\(\\epsilon\\). The master equation:<\/p>\n <div class=\"equation-box\">\n \\[ \\lim_{\\epsilon \\to 0^+} \\left[ \\hat{\\mathbb{B}}_\\epsilon \\hat{\\mathbb{V}}_\\epsilon – \\hat{\\mathbb{V}}_\\epsilon \\hat{\\mathbb{B}}_\\epsilon^2 \\right] \\Psi = \\Lambda \\cdot \\frac{\\Psi}{\\|\\Psi\\|^2 + \\epsilon^2} \\]\n <\/div>\n <p>With \\(\\hat{\\mathbb{B}}_\\epsilon = \\epsilon^2 (\\nabla \\times \\boldsymbol{\\omega})\\), \\(\\boldsymbol{\\omega} \\propto \\theta\\), governs the dynamics, matching \\(H_\\lambda\\)\u2019s hopping terms \\(\\psi_{n\\pm1}\\).<\/p>\n <\/div>\n\n <h4>2.2 Cantor Spectrum and \u00ab\u00a0Zero Does Not Exist\u00a0\u00bb<\/h4>\n <p>The Cantor set\u2019s measure zero but infinite points aligns with:<\/p>\n <div class=\"equation-box\">\n \\[ \\frac{\\Psi}{\\|\\Psi\\|^2 + \\epsilon^2} \\]\n <\/div>\n <div class=\"equation-explanation\">\n <h4>Non-Singularity:<\/h4>\n <p>This term prevents a zero spectrum, ensuring infinite states. For \\(H_\\lambda\\), the eigenvalue equation \\(H_\\lambda \\psi = E \\psi\\) yields a spectrum where gaps (measure zero) coexist with points, mirroring QFunity\u2019s regularization.<\/p>\n <\/div>\n\n <h4>2.3 Scale Invariance and Observer\u2019s Scale<\/h4>\n <p>The fractal\u2019s scale invariance ties to:<\/p>\n <div class=\"equation-box\">\n \\[ g_{\\mu\\nu}(\\epsilon) = g_{\\mu\\nu}^{GR} + \\frac{\\ell_P^2}{\\epsilon^2} g_{\\mu\\nu}^{\\text{LQG}} + \\alpha’ g_{\\mu\\nu}^{\\text{strings}} \\]\n <\/div>\n <div class=\"equation-explanation\">\n <h4>Fractal Mapping:<\/h4>\n <p>At \\(\\epsilon \\sim \\frac{1}{k}\\), where \\(k\\) is the wavevector, \\(\\frac{\\ell_P^2}{\\epsilon^2}\\) drives the Cantor set\u2019s self-similarity, matching the spectrum\u2019s energy scale dependence.<\/p>\n <\/div>\n\n <h4>2.4 Localization and Torsion \\(B_\\epsilon\\)<\/h4>\n <p>Localization for \\(\\lambda > 2\\) reflects \\(\\hat{\\mathbb{B}}_\\epsilon\\):<\/p>\n <div class=\"equation-box\">\n \\[ \\left[ \\hat{\\mathbb{B}}_\\epsilon, \\hat{\\mathbb{V}}_\\epsilon \\right] \\approx \\epsilon^4 (\\nabla \\times \\boldsymbol{\\omega}) \\nabla^2 – \\frac{\\hbar^2}{2\\epsilon^2} (\\nabla \\times \\boldsymbol{\\omega})^2 \\]\n <\/div>\n <div class=\"equation-explanation\">\n <h4>Localization Transition:<\/h4>\n <p>For \\(\\lambda > 2\\), \\(\\hat{\\mathbb{B}}_\\epsilon\\)\u2019s torsion traps \\(\\psi_n\\), with localization length \\(\\xi \\propto \\frac{\\epsilon^2}{\\lambda – 2}\\), aligning with the Anderson transition in \\(H_\\lambda\\).<\/p>\n <\/div>\n <\/div>\n\n \n <div class=\"theory-principle\">\n <h3>3. Synthesis of QFunity Equations<\/h3>\n <ol>\n <li>Master Equation: \\(\\lim_{\\epsilon \\to 0^+} \\left[ \\hat{\\mathbb{B}}_\\epsilon \\hat{\\mathbb{V}}_\\epsilon – \\hat{\\mathbb{V}}_\\epsilon \\hat{\\mathbb{B}}_\\epsilon^2 \\right] \\Psi = \\Lambda \\cdot \\frac{\\Psi}{\\|\\Psi\\|^2 + \\epsilon^2}\\)<\/li>\n <li>Non-Zero Principle: \\(\\frac{\\Psi}{\\|\\Psi\\|^2 + \\epsilon^2}\\)<\/li>\n <li>Scale-Dependent Metric: \\(g_{\\mu\\nu}(\\epsilon) \\propto \\frac{1}{\\epsilon^2}\\)<\/li>\n <\/ol>\n <p>These map to \\(H_\\lambda\\), ensuring fractal and localization phenomena.<\/p>\n <\/div>\n\n \n <div class=\"theory-principle\">\n <h3>Conclusion<\/h3>\n <p>The Ten Martini proof validates QFunity. The quasi-periodic potential aligns with \\(\\hat{\\mathbb{V}}_\\epsilon\\), the Cantor spectrum with \u00ab\u00a0Zero does not exist,\u00a0\u00bb scale invariance with \\(g_{\\mu\\nu}(\\epsilon)\\), and localization with \\(\\hat{\\mathbb{B}}_\\epsilon\\).<\/p>\n <\/div>\n\n <div style=\"text-align: center;\">\n <a href=\"\/index.php\/solutions\/\" class=\"return-btn\">\u2190 Back to All Solutions<\/a>\n <\/div>\n \n <\/section>\n\n <footer>\n <p style=\"font-size: 1.5em; font-weight: bold;\">Have a comment or question? Share your thoughts on <a href=\"https:\/\/x.com\/qfunity\" target=\"_blank\" style=\"color: var(--accent); text-decoration: none;\">\n <svg width=\"16\" height=\"16\" viewBox=\"0 0 24 24\" fill=\"var(--accent)\" style=\"vertical-align: middle;\">\n <path d=\"M18.244 2.25h3.308l-7.227 8.26 8.502 11.24H16.17l-5.214-6.817L4.99 21.75H1.68l7.73-8.835L1.254 2.25H8.08l4.713 6.231zm-1.161 17.52h1.833L7.084 4.126H5.117z\"\/>\n <\/svg>\n @qfunity\n <\/a>!<\/p>\n <\/footer>\n<\/body>\n<\/html>\n","protected":false},"excerpt":{"rendered":"<p>QFunity and the Ten Martini Proof QFunity and the Ten Martini Proof Unifying quantum fractals with scale-dependent dynamics Ten Martini Proof and QFunity Interpretation The article \u00ab\u00a0\u2018Ten Martini\u2019 Proof Uses Number Theory to Explain Quantum Fractals\u00a0\u00bb (*Quanta Magazine*, August 25, 2025) details Joshua Zahl\u2019s proof of the \u00ab\u00a0Ten Martini\u00a0\u00bb conjecture, confirming that the Aubry-Andr\u00e9 model\u2019s […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-420","page","type-page","status-publish","hentry"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/qfunity.com\/index.php\/wp-json\/wp\/v2\/pages\/420","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/qfunity.com\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/qfunity.com\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/qfunity.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/qfunity.com\/index.php\/wp-json\/wp\/v2\/comments?post=420"}],"version-history":[{"count":4,"href":"https:\/\/qfunity.com\/index.php\/wp-json\/wp\/v2\/pages\/420\/revisions"}],"predecessor-version":[{"id":435,"href":"https:\/\/qfunity.com\/index.php\/wp-json\/wp\/v2\/pages\/420\/revisions\/435"}],"wp:attachment":[{"href":"https:\/\/qfunity.com\/index.php\/wp-json\/wp\/v2\/media?parent=420"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}