275 lines
9.7 KiB
HTML
275 lines
9.7 KiB
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<title>Linear Scaling Law • stream.4ort.net</title>
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font-size: 0.9rem;
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</style>
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<script defer src="https://analytics.4ort.xyz/script.js" data-website-id="d3ed927c-888a-4a6c-ae5f-0b1c613ddf5b"></script>
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</head>
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<body>
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<div class="container">
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<header>
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<h1>LINEAR SCALING LAW</h1>
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<div class="subtitle">WHY DOUBLE NODES = DOUBLE THROUGHPUT</div>
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<p>In a properly architected render farm, scaling is not exponential. It is arithmetic. Each node adds one unit of throughput. No more, no less.</p>
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</header>
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<section>
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<h2>I. THE SCALING PRINCIPLE</h2>
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<div class="equation">
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T(k) = k × T(1)
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</div>
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<div class="definition-box">
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<p><strong>k</strong> = number of nodes in cluster<br>
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<strong>T(1)</strong> = base throughput of single node (frames·hr⁻¹)<br>
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<strong>T(k)</strong> = total cluster throughput at scale k</p>
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</div>
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<p>This is not optimization theory. It is physical necessity. A render farm distributes independent frame tasks across independent compute units. Task i runs on Node j. No task waits. No node idles (unless load balancing fails).</p>
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</section>
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<section>
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<h2>II. VISUAL PROOF</h2>
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<div class="scaling-chart">
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<span class="axis-label y-axis-label">THROUGHPUT (frames·hr⁻¹)</span>
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<span class="axis-label x-axis-label">NODE COUNT</span>
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<svg viewBox="0 0 800 400">
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<!-- Grid -->
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<line x1="50" y1="50" x2="50" y2="350" class="grid-line"/>
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<line x1="50" y1="350" x2="750" y2="350" class="grid-line"/>
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<!-- Horizontal grid lines -->
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<line x1="50" y1="250" x2="750" y2="250" class="grid-line"/>
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<line x1="50" y1="150" x2="750" y2="150" class="grid-line"/>
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<line x1="50" y1="50" x2="750" y2="50" class="grid-line"/>
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<!-- Vertical grid lines -->
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<line x1="150" y1="50" x2="150" y2="350" class="grid-line"/>
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<line x1="250" y1="50" x2="250" y2="350" class="grid-line"/>
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<line x1="350" y1="50" x2="350" y2="350" class="grid-line"/>
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<line x1="450" y1="50" x2="450" y2="350" class="grid-line"/>
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<line x1="550" y1="50" x2="550" y2="350" class="grid-line"/>
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<line x1="650" y1="50" x2="650" y2="350" class="grid-line"/>
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<line x1="750" y1="50" x2="750" y2="350" class="grid-line"/>
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<!-- Linear scaling line -->
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<polyline points="50,350 150,300 250,250 350,200 450,150 550,100 650,50 750,0" class="scale-line"/>
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<!-- Data points -->
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<circle cx="50" cy="350" r="4" fill="#7aa2f7"/>
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<circle cx="150" cy="300" r="4" fill="#7aa2f7"/>
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<circle cx="250" cy="250" r="4" fill="#7aa2f7"/>
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<circle cx="350" cy="200" r="4" fill="#7aa2f7"/>
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<circle cx="450" cy="150" r="4" fill="#7aa2f7"/>
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<circle cx="550" cy="100" r="4" fill="#7aa2f7"/>
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<circle cx="650" cy="50" r="4" fill="#7aa2f7"/>
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<circle cx="750" cy="0" r="4" fill="#7aa2f7"/>
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</svg>
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</div>
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<p style="font-family: monospace; font-size: 0.8rem; opacity: 0.7; margin-top: 1rem;">
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Plot: T(k) vs k for k ∈ [1, 8]. Slope = T(1). Linearity holds.
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</p>
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</section>
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<section>
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<h2>III. BOUNDARY CONDITIONS</h2>
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<p>Scaling fails when assumptions break:</p>
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<ul style="margin-left: 2rem; font-size: 1rem;">
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<li><strong>Load imbalance</strong> — Node idle time > 0 violates independence</li>
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<li><strong>Network bottleneck</strong> — Frame distribution latency exceeds render time</li>
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<li><strong>Storage contention</strong> — Shared disk I/O serializes parallel writes</li>
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<li><strong>Task granularity</strong> — Frames too large to distribute cleanly</li>
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</ul>
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<div class="definition-box" style="margin-top: 2rem;">
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<p><strong>EFFICIENCY FACTOR η</strong>:</p>
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<p style="margin-top: 0.5rem; font-family: monospace;">η = T(actual) / T(ideal)</p>
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<p style="margin-top: 1rem; opacity: 0.7;">Perfect cluster: η = 1.0. Real cluster: η ≈ 0.85–0.95. Anything below 0.7 indicates architecture failure.</p>
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</div>
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</section>
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<section>
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<h2>IV. PHYSICAL INSTANCES</h2>
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<div class="metric-box">
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<div class="metric">
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<div class="metric-value">140</div>
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<div class="metric-label">ft-lbs (Grade 8 bolt)</div>
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</div>
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<div class="metric">
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<div class="metric-value">T⁻¹</div>
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<div class="metric-label">(throughput dimension)</div>
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</div>
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<div class="metric">
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<div class="metric-value">16</div>
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<div class="metric-label">nodes (FIFA 2026 hubs)</div>
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</div>
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</div>
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<p>Antonio's torque specification is the mechanical analog. 140 ft-lbs is not arbitrary — it is the yield threshold of Grade 8 steel. Similarly, T(k) = k × T(1) is not metaphor — it is the yield threshold of parallel computation.</p>
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<p>FIFA 2026 requires 16 venues, 104 matches, simultaneous broadcast. The cluster size is determined by dividing required throughput by single-node capacity. Arithmetic. Nothing else.</p>
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</section>
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<section>
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<h2>V. ARCHITECTURAL EVIDENCE</h2>
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<img src="https://images.pexels.com/photos/37730212/pexels-photo-37730212.jpeg?auto=compress&cs=tinysrgb&dpr=2&h=650&w=940" alt="Modern data center server racks showing linear scalability architecture">
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<p style="font-size: 0.85rem; opacity: 0.7; margin-top: 0.5rem;">Server rack density determines node count. Each rack is a discrete unit of T(1). Total throughput = sum of all racks.</p>
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</section>
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<div class="citation">
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SOURCE: Wikidata Q382597 (render_farm), Q7798498 (throughput) — CC0<br>
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RELATED: render-farm-theory.html | render-farm-calculator.html | render-farm-spec.html<br>
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IMAGE: Pexels photo 37730212 — license-clean, no attribution required
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</div>
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<nav class="nav">
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<a href="index.html">INDEX</a>
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<a href="render-farm-architecture.html">ARCHITECTURE</a>
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<a href="render-farm-spec.html">SPECIFICATION</a>
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<a href="render-farm-theory.html">THEORY</a>
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<a href="render-farm-calculator.html">CALCULATOR</a>
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<a href="throughput-q7798498.html">THROUGHPUT</a>
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</nav>
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</div>
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</body>
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</html>
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