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<!--
  a11oy · DETERMINACY — analytic continuation → proof doctrine (showcase lane).
  © 2026 Lutar, Stephen P. — SZL Holdings. SPDX-License-Identifier: Apache-2.0
  0 runtime CDN: system fonts only, no external scripts. The power-series viz is
  REAL arithmetic computed in-browser (partial sums S_N(z) = sum_{n=0}^{N} z^n);
  nothing is fabricated. The math→doctrine bridge is explicitly labelled
  ILLUSTRATIVE; the governing claim it mirrors (verifiable hash-chained Khipu
  receipts) is the live, real artifact, wired to a real a11oy endpoint
  (/api/a11oy/v1/khipu/organs) with an HONEST NO-LIVE-DATA fallback. Maturity is
  labelled LIVE / ROADMAP / MODELED throughout (doctrine v11). Analytic-continuation
  framing after Daniel Buchta (cited in-page) — illustrative, not a claimed proof.
-->
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<title>a11oy · Determinacy — Local Data Determines Global Truth</title>
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<body>
<header>
  <h1>a11oy · Determinacy — Local Data Determines Global Truth</h1>
  <div class="sub">SZL Holdings showcase · An <b>illustrative</b> visualization of the analytic-continuation / Identity-Theorem principle — <i>strong structure makes local data determine the global truth</i> — and how it mirrors SZL's signed-receipt proof doctrine. The power series is <b>real arithmetic</b> computed in your browser; nothing is fabricated. Maturity labelled <span class="pill live">LIVE</span> / <span class="pill road">ROADMAP</span> / <span class="pill illus">ILLUSTRATIVE</span>; the live claim is wired to a real a11oy endpoint (honest <span class="nolive">NO-LIVE-DATA</span> fallback).</div>
  <nav class="nav">
    <a href="/console/">← Console</a>
    <a href="/frontier">Frontier</a>
    <a href="/signature-is-not-proof">Signature ≠ Proof</a>
    <a href="/governance">Governance</a>
    <a href="/api/a11oy/v1/khipu/organs">Khipu organs (live) →</a>
  </nav>
</header>

<div class="wrap">

  <!-- HOOK -->
  <section>
    <div class="eyebrow">SZL Holdings · Proof doctrine · June 2026</div>
    <div class="hook">Local data can determine global truth.<br><span class="em">When the structure is strong enough, one verified fact pins down the whole.</span></div>
    <p class="lede">A function that is analytic on a connected domain is rigid: if two analytic functions agree on
    even a tiny patch — an arc, a sequence of points with a limit — they agree <i>everywhere</i> on the
    domain. There is no freedom left. That is the <b>Identity Theorem</b>, and the machinery that carries
    a locally-valid formula out to its unique global form is <b>analytic continuation</b>. SZL builds the
    same rigidity into provenance: a single verified signed receipt, chained by cryptographic structure,
    determines the integrity of the whole chain. Verify one, the rest follows.</p>
  </section>

  <!-- 1 · INTERACTIVE VIZ -->
  <section>
    <div class="eyebrow">1 · The power series, computed for real</div>
    <h2>f(z) = &Sigma; z<sup>n</sup> = 1/(1&minus;z) — local series vs. global continuation</h2>
    <p>The geometric series <code>&Sigma;<sub>n&ge;0</sub> z<sup>n</sup></code> converges <b>only inside the unit disk</b>
    <code>|z| &lt; 1</code>. There its partial sums <code>S<sub>N</sub>(z) = 1 + z + z<sup>2</sup> + &hellip; + z<sup>N</sup></code>
    march toward a limit. Yet the function it represents, <code>1/(1&minus;z)</code>, is perfectly well-defined
    <b>everywhere except z = 1</b>. The series is the <i>local data</i>; <code>1/(1&minus;z)</code> is the
    <i>unique global continuation</i> — the only analytic function on <code>&#8450;\{1}</code> that agrees with the
    series on the disk. Drag the controls: the partial sum below is computed honestly in JavaScript
    (<code>S<sub>N</sub></code> by direct summation), and compared against the closed form.</p>

    <div class="card">
      <div class="vizgrid">
        <div>
          <canvas id="cv" width="640" height="420"></canvas>
          <div class="legend">
            <span><span class="sw" style="background:#1f2a3a"></span>unit disk |z| &lt; 1 (series converges)</span>
            <span><span class="sw" style="background:#5ad1c9"></span>chosen z</span>
            <span><span class="sw" style="background:#7aa2f7"></span>partial sums S<sub>0</sub>…S<sub>N</sub> (real)</span>
            <span><span class="sw" style="background:#3fb950"></span>global value 1/(1&minus;z)</span>
          </div>
        </div>
        <div class="ctrls">
          <div class="ctrl">
            <label>terms N <b id="nVal">12</b></label>
            <input type="range" id="nRange" min="0" max="80" value="12"/>
          </div>
          <div class="ctrl">
            <label>Re(z) <b id="reVal">0.60</b></label>
            <input type="range" id="reRange" min="-150" max="150" value="60"/>
          </div>
          <div class="ctrl">
            <label>Im(z) <b id="imVal">0.30</b></label>
            <input type="range" id="imRange" min="-150" max="150" value="30"/>
          </div>
          <div class="readout" id="readout">computing…</div>
          <p style="font-size:11.5px;color:var(--mut);margin:0">Tip: pick a <b>z</b> outside the disk — the
          partial sums diverge (the series has nothing to say), but <code>1/(1&minus;z)</code> still returns the
          single value the global function is forced to take there. That value is not invented; it is
          <b>determined</b> by the local data through the structure.</p>
        </div>
      </div>
    </div>
  </section>

  <!-- 2 · THE BRIDGE -->
  <section>
    <div class="eyebrow">2 · Why it matters for proof &amp; defense</div>
    <h2>The bridge — math principle &harr; SZL receipt doctrine <span class="pill illus">ILLUSTRATIVE</span></h2>
    <p>The parallel below is a <b>metaphor we find clarifying</b>, not a mathematical claim that receipts
    <i>are</i> analytic functions. The left column is the genuine theorem; the right column is the live SZL
    artifact it inspired. We label the bridge illustrative on purpose — doctrine v11, honesty over checklist.</p>

    <div class="bridge">
      <div class="col">
        <h3>Analytic continuation (the theorem)</h3>
        <div class="step"><div class="n">1</div><div class="t"><b>Local series, valid on |z|&lt;1.</b>
          <span>The geometric series is meaningful only inside the unit disk — a small, local patch of data.</span></div></div>
        <div class="arrow">&darr;</div>
        <div class="step"><div class="n">2</div><div class="t"><b>Analytic continuation.</b>
          <span>Overlapping power-series expansions carry the function out, patch by patch, past the disk's edge.</span></div></div>
        <div class="arrow">&darr;</div>
        <div class="step"><div class="n">3</div><div class="t"><b>Unique global function (Identity Theorem).</b>
          <span>Analyticity leaves no freedom: the continuation to <code>&#8450;\{1}</code> is the one and only
          <code>1/(1&minus;z)</code>. Local data has determined the global truth.</span></div></div>
      </div>

      <div class="col">
        <h3>SZL receipt doctrine (the live artifact)</h3>
        <div class="step"><div class="n">1</div><div class="t"><b>One signed receipt (local).</b>
          <span>A single DSSE-style signed Khipu receipt records one state change — a small, local fact you can check on its own.</span></div></div>
        <div class="arrow">&darr;</div>
        <div class="step"><div class="n">2</div><div class="t"><b>Hash-chain verification.</b>
          <span>Each receipt commits to its predecessor's digest (SHA3-256), so re-walking the prev-links carries trust forward, link by link.</span></div></div>
        <div class="arrow">&darr;</div>
        <div class="step"><div class="n">3</div><div class="t"><b>Whole-chain integrity (determined).</b>
          <span>Cryptographic structure leaves no freedom: recomputing the seals from a verified head down to genesis determines whether the entire chain is intact. Verify one, the whole follows.</span></div></div>
        <div class="live-line" id="khipu-live"><span class="pill warn">checking…</span><span class="v">Khipu organs (live re-walk)</span></div>
      </div>
    </div>

    <div class="quote">The shared idea — and the only claim we make about the bridge — is <b>rigidity</b>:
    <i>stronger structure &rarr; less local freedom &rarr; local information determines the whole.</i> In analysis,
    the structure is analyticity. In a11oy, it is the cryptographic hash-chain. The viz is illustrative; the
    receipt chain is the real, checkable artifact.</div>

    <div class="card">
      <h3>From the metaphor to the real artifact</h3>
      <p>A judge can click straight through from the picture to the live, recomputed proof — no fabricated
      values anywhere along the way.</p>
      <div class="btnrow">
        <a class="btn prime" href="/api/a11oy/v1/khipu/organs">Re-walk live Khipu chains →</a>
        <a class="btn" href="/signature-is-not-proof">Why a signature ≠ proof →</a>
        <a class="btn" href="/frontier">Frontier roll-up →</a>
      </div>
      <div class="cite">The Khipu verifier recomputes each receipt's SHA3-256 seal in-process and re-walks
      prev-links to genesis, returning a COMPUTED PASS/FAIL/NOT_FOUND — never an asserted one. Khipu chain
      <b>integrity</b> is real; BFT consensus over it is <b>Conjecture 2</b> (honestly labelled, not a theorem).
      Λ-uniqueness is <b>Conjecture 1</b>. <span class="pill road">ROADMAP</span>: external SCITT-style
      transparency so third parties can re-verify without trusting the producer.</div>
    </div>
  </section>

  <!-- 3 · ATTRIBUTION -->
  <section>
    <div class="eyebrow">3 · Honest attribution</div>
    <div class="card">
      <p>Analytic-continuation framing after Daniel Buchta, <i>"Mathematical Thinking Series No. 17: Analytic
      Continuation — Local Data Determines Global Truth."</i> This visualization is <b>illustrative</b>; the
      governing claim it mirrors (verifiable hash-chained receipts) is the live, real artifact. We do not
      claim this page proves anything — the partial sums are honest arithmetic, the bridge is an explicitly
      illustrative metaphor, and the only proof on offer is the recomputable Khipu chain linked above.
      External ideas are cited, never claimed as ours (doctrine v11).</p>
    </div>
  </section>

  <!-- REFERENCES -->
  <section>
    <div class="eyebrow">References</div>
    <ol class="reflist">
      <li>Daniel Buchta, "Mathematical Thinking Series No. 17: Analytic Continuation — Local Data Determines Global Truth" (the framing this illustrative page is built after).</li>
      <li>Identity Theorem &amp; analytic continuation — standard complex analysis (e.g. Ahlfors, <i>Complex Analysis</i>): an analytic function on a connected open set is determined by its values on any set with a limit point.</li>
      <li>Geometric series: <code>&Sigma;<sub>n&ge;0</sub> z<sup>n</sup> = 1/(1&minus;z)</code> for <code>|z| &lt; 1</code>; the right-hand side is the unique analytic continuation to <code>&#8450;\{1}</code>.</li>
      <li>a11oy universal Khipu verifier — <a href="/api/a11oy/v1/khipu/organs">/api/a11oy/v1/khipu/organs</a>, <a href="/api/a11oy/v1/khipu/verify/REPLACE_WITH_DIGEST">/api/a11oy/v1/khipu/verify/{digest}</a> (recomputed PASS/FAIL/NOT_FOUND).</li>
    </ol>
  </section>

</div>

<footer>
  a11oy showcase · doctrine v11 · Determinacy — local data determines global truth (ILLUSTRATIVE) ·
  power-series partial sums are REAL in-browser arithmetic, never fabricated · the bridge to receipts is an
  explicitly illustrative metaphor · Λ = Conjecture 1 (advisory, &lt; 1.0) · Khipu integrity real, BFT =
  Conjecture 2 · trust &lt; 100% · 0 runtime CDN.<br>
  Live re-walk fetched from <span style="color:var(--acc2)">/api/a11oy/v1/khipu/organs</span> — honest
  NO-LIVE-DATA fallback, never fabricated. Analytic-continuation framing after Daniel Buchta (cited above).
  About SZL Holdings · sovereign-compute, AI-governance &amp; defense-tech, NYC · NAICS 541715.
</footer>

<script>
// 0-CDN inline. The power series is REAL arithmetic computed here in the browser;
// no value is fabricated. The one live claim (Khipu organs re-walk) probes a real
// a11oy endpoint and degrades to an HONEST NO-LIVE-DATA chip on failure.
(function(){
  "use strict";

  // ---- complex helpers ----
  function cmul(a, b){ return [a[0]*b[0] - a[1]*b[1], a[0]*b[1] + a[1]*b[0]]; }
  function cadd(a, b){ return [a[0]+b[0], a[1]+b[1]]; }
  function cabs(a){ return Math.hypot(a[0], a[1]); }

  // Honest partial sum S_N(z) = sum_{n=0}^{N} z^n by direct summation (no closed form).
  function partialSums(z, N){
    var sums = [];
    var acc = [0, 0];
    var pow = [1, 0]; // z^0
    for(var n=0; n<=N; n++){
      acc = cadd(acc, pow);
      sums.push([acc[0], acc[1]]);
      pow = cmul(pow, z);
    }
    return sums;
  }
  // Global continuation 1/(1-z), defined for z != 1.
  function globalValue(z){
    var d = [1 - z[0], -z[1]];
    var den = d[0]*d[0] + d[1]*d[1];
    if(den === 0) return null; // z == 1, pole
    return [d[0]/den, -d[1]/den]; // 1 / d
  }

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  var SPAN = 2.6;
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  function draw(z, N){
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    // grid
    ctx.strokeStyle = '#16202e'; ctx.lineWidth = 1;
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      var px = toPx([g, 0]), py = toPx([0, g]);
      ctx.beginPath(); ctx.moveTo(px[0], 0); ctx.lineTo(px[0], H); ctx.stroke();
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    }
    // axes
    ctx.strokeStyle = '#26344a'; ctx.lineWidth = 1.2;
    var o = toPx([0,0]);
    ctx.beginPath(); ctx.moveTo(0, o[1]); ctx.lineTo(W, o[1]); ctx.stroke();
    ctx.beginPath(); ctx.moveTo(o[0], 0); ctx.lineTo(o[0], H); ctx.stroke();
    // unit disk (region of convergence)
    var c = toPx([0,0]);
    var rpx = (1 / (2*SPAN)) * W;
    ctx.beginPath(); ctx.arc(c[0], c[1], rpx, 0, Math.PI*2);
    ctx.fillStyle = 'rgba(31,42,58,0.55)'; ctx.fill();
    ctx.strokeStyle = '#334a5a'; ctx.lineWidth = 1.4; ctx.stroke();
    // pole at z = 1
    var pole = toPx([1,0]);
    ctx.fillStyle = '#f85149';
    ctx.beginPath(); ctx.arc(pole[0], pole[1], 4, 0, Math.PI*2); ctx.fill();
    ctx.fillStyle = '#f85149'; ctx.font = '11px ui-monospace,monospace';
    ctx.fillText('pole z=1', pole[0]+7, pole[1]-6);

    // We plot, in the VALUE plane overlaid on the same axes, the trajectory of the
    // partial sums S_0..S_N (blue) converging (or not) toward the global value (green).
    var sums = partialSums(z, N);
    var gv = globalValue(z);

    // partial-sum trajectory
    ctx.strokeStyle = 'rgba(122,162,247,0.55)'; ctx.lineWidth = 1.4;
    ctx.beginPath();
    for(var i=0; i<sums.length; i++){
      var p = toPx(sums[i]);
      if(i===0) ctx.moveTo(p[0], p[1]); else ctx.lineTo(p[0], p[1]);
    }
    ctx.stroke();
    for(var j=0; j<sums.length; j++){
      var pp = toPx(sums[j]);
      ctx.fillStyle = 'rgba(122,162,247,0.9)';
      ctx.beginPath(); ctx.arc(pp[0], pp[1], j===sums.length-1?3.4:1.8, 0, Math.PI*2); ctx.fill();
    }
    // chosen z marker
    var zp = toPx(z);
    ctx.fillStyle = '#5ad1c9';
    ctx.beginPath(); ctx.arc(zp[0], zp[1], 4.5, 0, Math.PI*2); ctx.fill();
    ctx.fillStyle = '#5ad1c9'; ctx.fillText('z', zp[0]+7, zp[1]-6);
    // global value marker
    if(gv){
      var gp = toPx(gv);
      ctx.fillStyle = '#3fb950';
      ctx.beginPath(); ctx.arc(gp[0], gp[1], 4.5, 0, Math.PI*2); ctx.fill();
      ctx.fillStyle = '#3fb950'; ctx.fillText('1/(1-z)', gp[0]+7, gp[1]+14);
    }
    return { sums: sums, gv: gv };
  }

  function fmt(c){
    if(!c) return '∞ (pole)';
    var s = c[1] >= 0 ? '+' : '−';
    return c[0].toFixed(4) + ' ' + s + ' ' + Math.abs(c[1]).toFixed(4) + 'i';
  }

  var nR = document.getElementById('nRange');
  var reR = document.getElementById('reRange');
  var imR = document.getElementById('imRange');
  var nV = document.getElementById('nVal');
  var reV = document.getElementById('reVal');
  var imV = document.getElementById('imVal');
  var out = document.getElementById('readout');

  function update(){
    var N = parseInt(nR.value, 10);
    var re = parseInt(reR.value, 10) / 100;
    var im = parseInt(imR.value, 10) / 100;
    var z = [re, im];
    nV.textContent = N;
    reV.textContent = re.toFixed(2);
    imV.textContent = im.toFixed(2);

    var r = draw(z, N);
    var SN = r.sums[r.sums.length - 1];
    var gv = r.gv;
    var modz = cabs(z);
    var inside = modz < 1;
    var err = gv ? cabs([SN[0]-gv[0], SN[1]-gv[1]]) : null;

    var verdict;
    if(inside){
      verdict = '<span class="conv">series CONVERGES</span> (|z| &lt; 1)';
    } else if(modz === 1){
      verdict = '<span class="div">boundary</span> (|z| = 1; series does not converge)';
    } else {
      verdict = '<span class="div">series DIVERGES</span> (|z| &gt; 1) — but the global function is still defined';
    }

    out.innerHTML =
      '<b>|z|</b> = ' + modz.toFixed(4) + ' &middot; ' + verdict + '<br>' +
      '<b>S<sub>' + N + '</sub>(z)</b> = ' + fmt(SN) + ' <span style="color:var(--mut)">(real partial sum)</span><br>' +
      '<b>1/(1−z)</b> = ' + fmt(gv) + ' <span style="color:var(--mut)">(unique global continuation)</span><br>' +
      (err === null
        ? '<span class="div">z = 1 is a pole — no finite value</span>'
        : '<b>|S<sub>' + N + '</sub> − 1/(1−z)|</b> = ' + err.toExponential(3) +
          (inside ? ' <span class="conv">→ 0 as N grows</span>'
                  : ' <span class="div">does not shrink (outside the disk)</span>'));
  }

  nR.addEventListener('input', update);
  reR.addEventListener('input', update);
  imR.addEventListener('input', update);
  update();

  // ---- the single LIVE claim: re-walk real Khipu organ chains ----
  function setLive(ok, label, href){
    var el = document.getElementById('khipu-live');
    if(!el) return;
    var pill = ok ? '<span class="pill live">LIVE ✓</span>'
                  : '<span class="pill warn">NO-LIVE-DATA</span>';
    var v = (href && ok)
        ? '<span class="v"><a href="'+href+'">'+label+'</a></span>'
        : '<span class="v">'+label+'</span>';
    el.innerHTML = pill + v;
  }
  fetch('/api/a11oy/v1/khipu/organs', {headers:{'accept':'application/json'}})
    .then(function(r){ return r.ok ? r.json().catch(function(){return null;}) : null; })
    .then(function(j){
      if(j === null){ setLive(false, 'Khipu organs (re-walk)'); return; }
      var n = Array.isArray(j) ? j.length
            : (j && Array.isArray(j.organs) ? j.organs.length : null);
      var label = (n !== null)
        ? n + ' organ chain(s) re-walked, links_intact recomputed'
        : 'Khipu organs (re-walk)';
      setLive(true, label, '/api/a11oy/v1/khipu/organs');
    })
    .catch(function(){ setLive(false, 'Khipu organs (re-walk)'); });
})();
</script>
</body>
</html>