A cutoff return is not an exact minimum.

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<html lang="en"><head><meta charset="utf-8"><meta name="viewport" content="width=device-width,initial-scale=1"><title>A cutoff return is not an exact minimum.</title><style>:root{color-scheme:light;--ink:#192e38;--accent:#255e61;--wash:#e6f1ed;--paper:#faf9f4}*{box-sizing:border-box}body{margin:0;background:var(--paper);color:var(--ink);font:17px/1.6 system-ui,sans-serif}main{max-width:1040px;margin:auto;padding:22px 24px 42px}header{border-top:5px solid var(--accent);padding-top:12px;margin-bottom:17px}.eyebrow{font-size:16px;font-weight:750;color:var(--accent);margin:0 0 4px}h1{font-size:clamp(26px,4vw,38px);line-height:1.2;letter-spacing:-.025em;margin:0 0 10px}h2{font-size:23px;line-height:1.3;margin:0 0 12px}h3{font-size:19px;margin:0 0 8px}p{margin:0 0 13px}.lede{max-width:850px}.stage{background:white;border:1px solid #b4c4c8;border-radius:10px;padding:20px;margin:15px 0}.sample{background:var(--wash);border-left:5px solid var(--accent);padding:16px;margin:12px 0}.result{border-top:3px solid var(--accent);padding:16px 0 4px;margin-top:14px;font-weight:700;font-size:19px;white-space:pre-wrap;overflow-wrap:anywhere}.detail{white-space:pre-wrap;overflow-wrap:anywhere}button,input,textarea,select{font:inherit}button{min-height:46px;background:var(--accent);color:white;border:1px solid var(--accent);border-radius:6px;padding:9px 14px;cursor:pointer;font-weight:650}button:hover{filter:brightness(.9)}button[aria-pressed=true]{box-shadow:inset 0 0 0 3px #d5f525}button:focus-visible,input:focus-visible,textarea:focus-visible,summary:focus-visible{outline:3px solid #245bad;outline-offset:3px}button:disabled{opacity:.6;cursor:default}.controls{display:flex;flex-wrap:wrap;gap:8px;margin:12px 0}label{display:block;font-weight:650;margin:11px 0 6px}input,textarea{display:block;width:100%;min-width:0;max-width:100%;background:white;color:var(--ink);border:1px solid #8199a1;border-radius:5px;padding:10px 12px}textarea{min-height:110px;resize:vertical;font:16px/1.5 ui-monospace,Consolas,monospace}.grid{display:grid;grid-template-columns:1fr 1fr;gap:22px}.grid>*{min-width:0}ul,ol{padding-left:23px}li{margin:6px 0}.small,summary,footer{font-size:16px;color:#415964}code,pre{font:16px/1.5 ui-monospace,Consolas,monospace}pre{white-space:pre-wrap;overflow-wrap:anywhere;background:#edf1f1;padding:16px;border-radius:6px}code{overflow-wrap:anywhere}.chip{display:inline-block;border:1px solid #95aeb4;border-radius:4px;padding:2px 8px;font-size:16px;margin:3px}svg{display:block;width:100%;height:auto}svg text{font-family:system-ui,sans-serif}details{margin:20px 0;border-top:1px solid #a7bdc2;padding-top:13px}summary{cursor:pointer;font-weight:750}mark{background:#d5f525;color:#202600}footer{border-top:1px solid #a7bdc2;padding-top:15px;margin-top:22px}table{width:100%;border-collapse:collapse;table-layout:fixed}th,td{padding:10px 8px;text-align:left;vertical-align:top;border-bottom:1px solid #c2cfd2;overflow-wrap:anywhere}th{background:var(--wash)}.number{font-size:32px;font-weight:800}.note{border-left:3px solid #a78e51;padding-left:12px}@media(max-width:650px){main{padding:15px 17px 30px}.grid{grid-template-columns:1fr;gap:14px}.stage{padding:15px}header{margin-bottom:12px}.controls button{flex:1 1 auto}h1{font-size:27px}th,td{padding:8px 5px}}</style></head><body><main style="--accent:#534f86;--wash:#eeecfb"><header><p class="eyebrow">ALGORITHMS / EXACT VS BOUND</p><h1>A cutoff return is not an exact minimum.</h1><p class="lede">Original MAX-over-MIN fixture. JavaScript trace comparison; no Python interpreter runs here.</p></header><section class="stage"><p class="sample">A [4,7] · B [3,-20] · C [6,5]. MAX chooses among MIN branches.</p><div class="controls"><button type="button" id="trace">Trace current order</button><button type="button" id="reverse">Reverse branches</button><button type="button" id="reset">Reset</button></div><p id="result" class="result" role="status" aria-live="polite"></p><pre id="traceResult"></pre><p id="baseline"></p></section><label for="tree">Ordered branch JSON</label><textarea id="tree" spellcheck="false">[[&quot;A&quot;,[4,7]],[&quot;B&quot;,[3,-20]],[&quot;C&quot;,[6,5]]]</textarea><details><summary>Port the bounded loop to Python</summary><p>For each MIN list, beta=min(beta,value). Cut at beta&lt;=alpha only if leaves remain. Record skipped IDs and upper_bound; otherwise exact. Update root choice on strict improvement. Keep full baseline results separate.</p></details><footer>Original fictional practice example. Current-page interaction only; no saved records or external requests.</footer></main>
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