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<title>Prime Grokking: Scratchpad & Filler Token Experiment</title>
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<body>
<div class="container">
<header>
<div class="badge">Research Addendum 8</div>
<h1>Prime Grokking: Scratchpads & Fillers</h1>
<p class="subtitle">A visual breakdown of why standard neural networks fail on prime numbers, and how token-space computation tests the boundaries of algorithmic emergence.</p>
</header>
<!-- SECTION 1: THE PUZZLE -->
<section>
<h2>1. The Core Puzzle: Why Primes Break Standard Grokking</h2>
<div class="card">
<p>When researchers train neural networks on <strong>modular arithmetic</strong> (like <code>(a + b) mod 97</code>), the network suddenly experiences <strong>grokking</strong>: after long overfitting, it abruptly discovers the generalized circle algorithm (Fourier transform) because it is simpler and lower-norm than memorization.</p>
<p><strong>Next-Prime is completely different.</strong> Finding the next prime after <em>n</em> requires a <em>two-level nested algorithm</em>:</p>
<div class="diagram-container">
<svg viewBox="0 0 800 220" xmlns="http://www.w3.org/2000/svg">
<!-- Candidate search -->
<rect x="50" y="30" width="200" height="60" rx="8" fill="#1e3a8a" stroke="#3b82f6" stroke-width="2"/>
<text x="150" y="65" fill="#ffffff" text-anchor="middle" font-size="14" font-weight="bold">Outer Loop: Candidates</text>
<text x="150" y="80" fill="#93c5fd" text-anchor="middle" font-size="12">c = n+1, n+2, n+3, ...</text>
<!-- Arrow -->
<path d="M 250 60 L 330 60" stroke="#94a3b8" stroke-width="2" marker-end="url(#arrow)"/>
<!-- Divisibility test -->
<rect x="330" y="20" width="240" height="80" rx="8" fill="#064e3b" stroke="#10b981" stroke-width="2"/>
<text x="450" y="50" fill="#ffffff" text-anchor="middle" font-size="14" font-weight="bold">Inner Loop: Trial Division</text>
<text x="450" y="70" fill="#6ee7b7" text-anchor="middle" font-size="12">Check if c is divisible by</text>
<text x="450" y="88" fill="#6ee7b7" text-anchor="middle" font-size="12">p ∈ {2, 3, 5, 7, ..., √c}</text>
<!-- Output decision -->
<path d="M 570 60 L 630 60" stroke="#94a3b8" stroke-width="2"/>
<rect x="630" y="30" width="130" height="60" rx="8" fill="#831843" stroke="#ec4899" stroke-width="2"/>
<text x="695" y="58" fill="#ffffff" text-anchor="middle" font-size="13" font-weight="bold">If no divisors:</text>
<text x="695" y="76" fill="#fbcfe8" text-anchor="middle" font-size="12">Return c</text>
<!-- Rejection loop -->
<path d="M 450 100 L 450 160 L 150 160 L 150 90" fill="none" stroke="#fb7185" stroke-width="2" stroke-dasharray="5,5"/>
<text x="300" y="150" fill="#fb7185" font-size="12">If divisible: reject & next candidate</text>
<defs>
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<p>A standard feedforward Transformer has a <strong>fixed number of layers</strong>. It cannot run an unbounded while-loop. In experiments E1–E7, we proved that even training for <strong>4,000,000 steps</strong> never produces grokking on next-prime: the model just memorizes a lookup table for numbers in range, and scores <strong>0%–3% out-of-range</strong>.</p>
</div>
</section>
<!-- SECTION 2: THE IDEA -->
<section>
<h2>2. The Proposed Solution: Computation in Token Space</h2>
<div class="card">
<p>If the model doesn't have enough internal depth to perform multiple serial operations, we can give it <strong>intermediate tokens</strong> to think before answering.</p>
<p>Recent theoretical work by <em>Pfau et al. (2024, "Let's Think Dot by Dot")</em> made a surprising prediction: <strong>filler tokens (e.g. dots or pauses) only help if the hidden subcomputations are parallelizable</strong>. Checking divisors {2, 3, 5, 7} are independent parallel operations! Therefore, next-prime is the ideal testbed for this hypothesis.</p>
</div>
</section>
<!-- SECTION 3: THE 4-ARM EXPERIMENT -->
<section>
<h2>3. The 4-Arm Experimental Design</h2>
<p>To scientifically isolate <em>why</em> intermediate tokens help (or don't), we test four distinct arms:</p>
<div class="grid-2">
<!-- ARM A -->
<div class="arm-card arm-a">
<div class="arm-header">
<span class="arm-tag">Arm A: Direct Baseline</span>
<span class="badge" style="background:#334155;color:#f8fafc;border:none;">Control</span>
</div>
<p>Input goes directly to output. No intermediate steps.</p>
<div class="token-sequence">
<span class="token tok-input">4</span>
<span class="token tok-input">2</span>
<span class="token tok-sep">→</span>
<span class="token tok-ans">4</span>
<span class="token tok-ans">3</span>
<span class="token tok-sep"><eos></span>
</div>
<p style="font-size:0.9rem;"><em>Tests: Can pure fixed-depth attention solve next-prime? (We know from E7 this fails out-of-range).</em></p>
</div>
<!-- ARM B -->
<div class="arm-card arm-b">
<div class="arm-header">
<span class="arm-tag highlight-green">Arm B: Structured Scratchpad</span>
<span class="badge" style="background:rgba(16,185,129,0.2);color:#34d399;border:none;">Algorithm Supervision</span>
</div>
<p>Ground-truth trial division steps are spelled out in tokens.</p>
<div class="token-sequence">
<span class="token tok-input">4</span><span class="token tok-input">2</span>
<span class="token tok-sep">→</span>
<span class="token tok-trace">c=43</span>
<span class="token tok-trace">d2:0</span>
<span class="token tok-trace">d3:0</span>
<span class="token tok-trace">d5:0</span>
<span class="token tok-sep">#</span>
<span class="token tok-ans">4</span><span class="token tok-ans">3</span>
</div>
<p style="font-size:0.9rem;"><em>Tests: If the algorithm is explicitly supervised, does the model grok and generalize to unseen ranges?</em></p>
</div>
<!-- ARM C -->
<div class="arm-card arm-c">
<div class="arm-header">
<span class="arm-tag highlight-blue">Arm C: Pause / Filler Tokens</span>
<span class="badge" style="background:rgba(56,189,248,0.2);color:#38bdf8;border:none;">Pfau Hypothesis</span>
</div>
<p>Fixed 16 identical pause tokens (no intermediate labels).</p>
<div class="token-sequence">
<span class="token tok-input">4</span><span class="token tok-input">2</span>
<span class="token tok-sep">→</span>
<span class="token tok-pause"><p></span>
<span class="token tok-pause"><p></span>
<span class="token tok-pause"><p></span>
<span class="token tok-pause">...</span>
<span class="token tok-sep">#</span>
<span class="token tok-ans">4</span><span class="token tok-ans">3</span>
</div>
<p style="font-size:0.9rem;"><em>Tests: Does extra compute depth alone allow the transformer to execute parallel divisor checks silently?</em></p>
</div>
<!-- ARM D -->
<div class="arm-card arm-d">
<div class="arm-header">
<span class="arm-tag highlight-amber">Arm D: Random Noise Tokens</span>
<span class="badge" style="background:rgba(251,191,36,0.2);color:#fbbf24;border:none;">Length / Bias Control</span>
</div>
<p>Fixed 16 random letters from a disjoint alphabet (<code>[a-p]</code>).</p>
<div class="token-sequence">
<span class="token tok-input">4</span><span class="token tok-input">2</span>
<span class="token tok-sep">→</span>
<span class="token tok-rand">x</span>
<span class="token tok-rand">m</span>
<span class="token tok-rand">k</span>
<span class="token tok-rand">...</span>
<span class="token tok-sep">#</span>
<span class="token tok-ans">4</span><span class="token tok-ans">3</span>
</div>
<p style="font-size:0.9rem;"><em>Tests: Does any sequence expansion help, or is static pause token embedding special?</em></p>
</div>
</div>
</section>
<!-- SECTION 4: WHAT EACH OUTCOME PROVES -->
<section>
<h2>4. What Each Comparison Proves</h2>
<div class="card">
<table class="comparison-table">
<thead>
<tr>
<th>Observed Pattern</th>
<th>Scientific Interpretation</th>
<th>Significance</th>
</tr>
</thead>
<tbody>
<tr>
<td><strong class="highlight-green">Arm B >> Arm A, C, D</strong></td>
<td><strong>Algorithm Decomposition Required:</strong> Extra compute alone does nothing; explicit intermediate supervision is required to guide SGD.</td>
<td>Confirms Nye et al. Scratchpad mechanism.</td>
</tr>
<tr>
<td><strong class="highlight-blue">Arm C >> Arm A</strong></td>
<td><strong>Pfau et al. Confirmed:</strong> Unsupervised filler tokens provide enough hidden attention routing to compute parallel divisibility tests.</td>
<td><strong>Major theoretical finding!</strong> First clean toy demonstration on non-group tasks.</td>
</tr>
<tr>
<td><strong class="highlight-amber">Arm D ≈ Arm C > Arm A</strong></td>
<td><strong>Pure Depth Invariance:</strong> Expanding sequence length improves representational capacity regardless of token semantics.</td>
<td>Points to transformer attention capacity dynamics.</td>
</tr>
<tr>
<td><strong class="highlight-rose">All Arms Fail ($P4$) Out-of-Range</strong></td>
<td><strong>Absolute Algorithmic Wall:</strong> Neither internal recurrence nor token-space scratchpads allow next-token models to generalize prime discovery.</td>
<td>Falsifies scratchpad extrapolation for unbounded search.</td>
</tr>
</tbody>
</table>
</div>
<div class="takeaway-box">
<h3>Summary of the Proposed Experiment</h3>
<p>We will train a 2-layer GPT Transformer across all four arms in the $[2, 1000]$ regime for 200k steps with weight decay $0.1$. We evaluate exact-match accuracy both <strong>in-range ($[2, 1000]$)</strong> and on the unseen <strong>probe range ($[1001, 2000]$)</strong> to test true algorithmic generalization.</p>
</div>
</section>
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