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    <title>DEV Community: Nikhil Nath Jha</title>
    <description>The latest articles on DEV Community by Nikhil Nath Jha (@jhanikhilnath).</description>
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      <title>Building PrismJudge: An Offline-First, Mathematically Rigorous Hackathon Platform</title>
      <dc:creator>Nikhil Nath Jha</dc:creator>
      <pubDate>Sat, 03 Oct 2026 20:06:56 +0000</pubDate>
      <link>https://dev.to/jhanikhilnath/building-dogfood-2026-an-offline-first-mathematically-rigorous-hackathon-platform-22eg</link>
      <guid>https://dev.to/jhanikhilnath/building-dogfood-2026-an-offline-first-mathematically-rigorous-hackathon-platform-22eg</guid>
      <description>&lt;blockquote&gt;
&lt;p&gt;&lt;strong&gt;An Engineering Deep Dive into Building PrismJudge&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
&lt;em&gt;Targeting the Grand Prize ($800 USD) and Best Judging Engine ($100 USD)&lt;/em&gt;&lt;br&gt;&lt;br&gt;
&lt;strong&gt;Repository:&lt;/strong&gt; &lt;a href="https://github.com/jhanikhilnath/PrismJudge_DogFood_Hack" rel="noopener noreferrer"&gt;https://github.com/jhanikhilnath/PrismJudge_DogFood_Hack&lt;/a&gt;&lt;br&gt;&lt;br&gt;
&lt;strong&gt;Stack:&lt;/strong&gt; Node.js 22 LTS · TypeScript · Fastify v5 · Embedded SQLite 3 (WAL Mode) · Vanilla Studio Light SSR&lt;br&gt;&lt;br&gt;
&lt;strong&gt;Official Acceptance Status:&lt;/strong&gt; &lt;strong&gt;7 / 7 PASS&lt;/strong&gt; in &lt;code&gt;run.py&lt;/code&gt; · &lt;strong&gt;81 / 81 PASS&lt;/strong&gt; in &lt;code&gt;npm test&lt;/code&gt; · &lt;strong&gt;641 / 641 PASS&lt;/strong&gt; in Route Audit · &lt;strong&gt;89 / 89 PASS&lt;/strong&gt; in Container E2E  &lt;/p&gt;
&lt;/blockquote&gt;




&lt;h2&gt;
  
  
  Table of Contents
&lt;/h2&gt;

&lt;ul&gt;
&lt;li&gt;
1. Executive Summary &amp;amp; The Core Philosophy

&lt;ul&gt;
&lt;li&gt;The One-Command Rule&lt;/li&gt;
&lt;li&gt;Comprehensive Verification Scorecard&lt;/li&gt;
&lt;/ul&gt;
&lt;/li&gt;
&lt;li&gt;
2. System Architecture: The In-Process Fastify &amp;amp; SQLite Pipeline

&lt;ul&gt;
&lt;li&gt;In-Process Request Flow&lt;/li&gt;
&lt;li&gt;Why Embedded node:sqlite Beats PostgreSQL for Hackathon Portals&lt;/li&gt;
&lt;/ul&gt;
&lt;/li&gt;
&lt;li&gt;
3. Mathematical Rigor: The Judging &amp;amp; Normalization Engine

&lt;ul&gt;
&lt;li&gt;3.1 Empirical Bayesian Shrinkage Z-Score Normalization&lt;/li&gt;
&lt;li&gt;3.2 Formal Proof: The Singularity Resolution of Judge jdg_07&lt;/li&gt;
&lt;li&gt;3.3 Bradley-Terry Pairwise Model via Minorization-Maximization (MM)&lt;/li&gt;
&lt;li&gt;3.4 Inter-Rater Reliability: Intraclass Correlation Coefficient ICC(1,1)&lt;/li&gt;
&lt;/ul&gt;
&lt;/li&gt;
&lt;li&gt;
4. Security Architecture &amp;amp; Adversarial Defenses

&lt;ul&gt;
&lt;li&gt;4.1 The Hard Peer Isolation Defense&lt;/li&gt;
&lt;li&gt;4.2 Temporal Submission Deadline Guard&lt;/li&gt;
&lt;li&gt;4.3 Anti-Sybil Community Voting Guard&lt;/li&gt;
&lt;/ul&gt;
&lt;/li&gt;
&lt;li&gt;
5. Engineering Challenges &amp;amp; Battle Stories: What Broke and How We Fixed It

&lt;ul&gt;
&lt;li&gt;Challenge 1: The Browser Caching Deadlock &amp;amp; Persona Switcher Trapping&lt;/li&gt;
&lt;li&gt;Challenge 2: The Two-Page Diploma Print Overflow&lt;/li&gt;
&lt;li&gt;Challenge 3: Chronological Skew in the Append-Only Audit Ledger&lt;/li&gt;
&lt;/ul&gt;
&lt;/li&gt;
&lt;li&gt;6. Frontend Design&lt;/li&gt;
&lt;li&gt;
7. Verification &amp;amp; Testing Runbook

&lt;ul&gt;
&lt;li&gt;The Official Acceptance Report&lt;/li&gt;
&lt;/ul&gt;
&lt;/li&gt;
&lt;li&gt;8. Conclusion &amp;amp; Lessons Learned&lt;/li&gt;
&lt;/ul&gt;




&lt;h2&gt;
  
  
  1. Executive Summary &amp;amp; The Core Philosophy
&lt;/h2&gt;

&lt;p&gt;The prompt for &lt;strong&gt;DOGFOOD 2026&lt;/strong&gt; was deceptively simple, yet brutally unforgiving:  &lt;/p&gt;

&lt;blockquote&gt;
&lt;p&gt;&lt;em&gt;"Build the platform that will judge you."&lt;/em&gt;&lt;/p&gt;
&lt;/blockquote&gt;

&lt;p&gt;In 72 hours, teams were challenged to construct an entire end-to-end hackathon management and evaluation portal capable of ingesting messy legacy fixtures (41 projects, 30 judges, 8 tracks, and 126 reviews), enforcing strict submission deadlines, isolating evaluator peer scores under adversarial probing, evening out juror bias with documented statistical normalization, and serving traffic completely offline.&lt;/p&gt;

&lt;p&gt;Most teams immediately reached for the standard modern web toolbox: Next.js 15, Django or FastAPI, PostgreSQL 16, Redis, and multi-container Docker Compose networks. While functional, that architecture carries immense cognitive and operational overhead: slow cold-boot times (15–30 seconds), container networking race conditions, connection pool exhaustion, and fragile multi-service health checks.&lt;/p&gt;

&lt;p&gt;We made an audacious architectural bet: &lt;strong&gt;The One-Command Rule&lt;/strong&gt;.&lt;/p&gt;

&lt;h3&gt;
  
  
  The One-Command Rule
&lt;/h3&gt;



&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight shell"&gt;&lt;code&gt;docker compose up
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;


&lt;p&gt;With the host machine's physical network disconnected, our entire platform boots from a cold start in &lt;strong&gt;under 600 milliseconds&lt;/strong&gt;, mounts an embedded SQLite 3 database in Write-Ahead Logging (WAL) mode, runs idempotent schema migrations, ingests all fixture submissions and scores, seeds four authenticated evaluation personas, and serves production-ready HTTP/1.1 traffic on port &lt;code&gt;8080&lt;/code&gt;.&lt;/p&gt;


  &lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.us-east-2.amazonaws.com%2Fuploads%2Farticles%2Fgcx0rpf5854xmn9i6v54.png" alt="PrismJudge Editorial Welcome Portal rendering real-time macro telemetry, active competition tracks, and offline platform status" width="800" height="452"&gt;Figure 1: The PrismJudge Welcome Portal, rendering real-time macro telemetry, competitive tracks, and offline platform status.
  

&lt;h3&gt;
  
  
  Comprehensive Verification Scorecard
&lt;/h3&gt;

&lt;div class="table-wrapper-paragraph"&gt;&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Competition Dimension&lt;/th&gt;
&lt;th&gt;Requirement / Invariant&lt;/th&gt;
&lt;th&gt;Verified Platform Status&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;&lt;strong&gt;Claimed Tiers&lt;/strong&gt;&lt;/td&gt;
&lt;td&gt;&lt;code&gt;claimed = ["T1", "T2"]&lt;/code&gt;&lt;/td&gt;
&lt;td&gt;&lt;strong&gt;PASS (7/7 in official &lt;code&gt;run.py&lt;/code&gt;)&lt;/strong&gt;&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;&lt;strong&gt;T1 Core Platform&lt;/strong&gt;&lt;/td&gt;
&lt;td&gt;Public gallery, fixture rendering, deadline rejection&lt;/td&gt;
&lt;td&gt;&lt;strong&gt;100% PASS (22 / 22 assertions)&lt;/strong&gt;&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;&lt;strong&gt;T2 Judging Engine&lt;/strong&gt;&lt;/td&gt;
&lt;td&gt;Workload queue, 4-slider rubric, IDOR peer isolation, CSV export&lt;/td&gt;
&lt;td&gt;&lt;strong&gt;100% PASS (27 / 27 assertions)&lt;/strong&gt;&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;&lt;strong&gt;T3 Public Balloting&lt;/strong&gt;&lt;/td&gt;
&lt;td&gt;Fisher-Yates hash shuffle, anti-Sybil rate limiter, comments stream&lt;/td&gt;
&lt;td&gt;&lt;strong&gt;100% PASS (7 / 7 assertions)&lt;/strong&gt;&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;&lt;strong&gt;T4 Stretch Features&lt;/strong&gt;&lt;/td&gt;
&lt;td&gt;Bradley-Terry MM arena, OpenAPI 3.1, private verifiable diplomas&lt;/td&gt;
&lt;td&gt;&lt;strong&gt;100% PASS (33 / 33 assertions)&lt;/strong&gt;&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;&lt;strong&gt;Internal Unit Suite&lt;/strong&gt;&lt;/td&gt;
&lt;td&gt;Automated regression &amp;amp; boundary tests (&lt;code&gt;npm test&lt;/code&gt;)&lt;/td&gt;
&lt;td&gt;&lt;strong&gt;81 / 81 PASS across 12 test suites&lt;/strong&gt;&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;&lt;strong&gt;Role &amp;amp; Route Crawler&lt;/strong&gt;&lt;/td&gt;
&lt;td&gt;Comprehensive HTTP status matrix (&lt;code&gt;tests/audit_script.mjs&lt;/code&gt;)&lt;/td&gt;
&lt;td&gt;&lt;strong&gt;641 / 641 assertions PASS&lt;/strong&gt;&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;&lt;strong&gt;Live Container Audit&lt;/strong&gt;&lt;/td&gt;
&lt;td&gt;Full browser &amp;amp; API container audit (&lt;code&gt;tests/comprehensive_e2e_audit.mjs&lt;/code&gt;)&lt;/td&gt;
&lt;td&gt;&lt;strong&gt;89 / 89 assertions PASS&lt;/strong&gt;&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;&lt;strong&gt;Diploma Print Layout&lt;/strong&gt;&lt;/td&gt;
&lt;td&gt;Single-page landscape PDF export (&lt;code&gt;@media print&lt;/code&gt;)&lt;/td&gt;
&lt;td&gt;&lt;strong&gt;Strictly 1 Page (&lt;code&gt;Pages: 1&lt;/code&gt;, 0 overflow)&lt;/strong&gt;&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;&lt;strong&gt;Container Cold Boot&lt;/strong&gt;&lt;/td&gt;
&lt;td&gt;Offline Docker initialization time&lt;/td&gt;
&lt;td&gt;&lt;strong&gt;&amp;lt; 600ms on &lt;code&gt;http://localhost:8080&lt;/code&gt;&lt;/strong&gt;&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;&lt;/div&gt;


&lt;h2&gt;
  
  
  2. System Architecture: The In-Process Fastify &amp;amp; SQLite Pipeline
&lt;/h2&gt;

&lt;p&gt;Rather than splitting our system across microservices, we engineered PrismJudge as a high-throughput, unified monolith inside &lt;strong&gt;Node.js 22 LTS&lt;/strong&gt; using &lt;strong&gt;Fastify v5&lt;/strong&gt;. Fastify was selected over Express and NestJS due to its low overhead (up to 4x faster than Express), built-in JSON schema compilation, and clean plugin lifecycle.&lt;/p&gt;
&lt;h3&gt;
  
  
  In-Process Request Flow
&lt;/h3&gt;


&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;┌─────────────────────────────────────────────────────────────────────────────┐
│                             INCOMING HTTP REQUEST                           │
└──────────────────────────────────────┬──────────────────────────────────────┘
                                       │
                                       ▼
┌─────────────────────────────────────────────────────────────────────────────┐
│                      Fastify Global Pipeline Hooks                          │
│  ├─ onRequest: Fastify Cookie parsing &amp;amp; session secret verification         │
│  ├─ preHandler: resolveUserHook (maps session token to req.user)            │
│  └─ onSend: Enterprise Security Headers &amp;amp; Cache-Control Enforcement         │
└──────────────────────────────────────┬──────────────────────────────────────┘
                                       │
        ┌──────────────────────────────┼──────────────────────────────┐
        ▼                              ▼                              ▼
┌──────────────┐              ┌─────────────────┐            ┌─────────────────┐
│ Public Views │              │ Evaluator Queue │            │ Organizer Admin │
│  - Gallery   │              │  - Rubric Model │            │  - Live Console │
│  - Detail    │              │  - Peer Guard   │            │  - Bayesian Proof│
│  - Ballot    │              │  - Pairwise MM  │            │  - CSV Export   │
└───────┬──────┘              └────────┬────────┘            └────────┬────────┘
        │                              │                              │
        └──────────────────────────────┼──────────────────────────────┘
                                       │
                                       ▼
┌─────────────────────────────────────────────────────────────────────────────┐
│                     Centralized Data Access Layer                           │
│                       (src/db/queries.ts)                                   │
└──────────────────────────────────────┬──────────────────────────────────────┘
                                       │
                                       ▼
┌─────────────────────────────────────────────────────────────────────────────┐
│               Node.js 22 Embedded SQLite 3 (node:sqlite)                    │
│      - DatabaseSync singleton with PRAGMA journal_mode = WAL                │
│      - PRAGMA synchronous = NORMAL · PRAGMA foreign_keys = ON               │
│      - In-process zero-latency query execution (&amp;lt; 150 microseconds)         │
└─────────────────────────────────────────────────────────────────────────────┘
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;

&lt;h3&gt;
  
  
  Why Embedded node:sqlite Beats PostgreSQL for Hackathon Portals
&lt;/h3&gt;

&lt;p&gt;Node.js 22 introduced native, zero-dependency SQLite support via &lt;code&gt;node:sqlite&lt;/code&gt; (&lt;code&gt;DatabaseSync&lt;/code&gt;). Running SQLite in-process yields decisive advantages for hackathon evaluation:&lt;/p&gt;

&lt;ol&gt;
&lt;li&gt;
&lt;strong&gt;Zero Connection Latency:&lt;/strong&gt; Database queries are synchronous C++ function calls directly against the SQLite engine in the same memory address space. Average query execution is under 150 microseconds.&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Write-Ahead Logging (WAL Mode):&lt;/strong&gt; Readers never block writers, and writers never block readers. Concurrency bottlenecks disappear even under intensive test crawls.&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Atomic Multi-Table Transactions:&lt;/strong&gt; Workload redistribution and scoring updates execute inside isolated ACID transactions with zero risk of partial commits.&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Offline Resilience:&lt;/strong&gt; The database is an immutable, single-file artifact (&lt;code&gt;portal.sqlite&lt;/code&gt;). There is no Postgres daemon to wait for, no TCP socket handshakes, and no password authentication failures during container boot.
&lt;/li&gt;
&lt;/ol&gt;
&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight typescript"&gt;&lt;code&gt;&lt;span class="c1"&gt;// src/db/index.ts — The In-Process SQLite WAL Singleton&lt;/span&gt;
&lt;span class="k"&gt;import&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt; &lt;span class="nx"&gt;DatabaseSync&lt;/span&gt; &lt;span class="p"&gt;}&lt;/span&gt; &lt;span class="k"&gt;from&lt;/span&gt; &lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="s1"&gt;node:sqlite&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="k"&gt;import&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt; &lt;span class="nx"&gt;config&lt;/span&gt; &lt;span class="p"&gt;}&lt;/span&gt; &lt;span class="k"&gt;from&lt;/span&gt; &lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="s1"&gt;../config.js&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;

&lt;span class="kd"&gt;let&lt;/span&gt; &lt;span class="nx"&gt;dbInstance&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nx"&gt;DatabaseSync&lt;/span&gt; &lt;span class="o"&gt;|&lt;/span&gt; &lt;span class="kc"&gt;null&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="kc"&gt;null&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;

&lt;span class="k"&gt;export&lt;/span&gt; &lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;getDatabase&lt;/span&gt;&lt;span class="p"&gt;():&lt;/span&gt; &lt;span class="nx"&gt;DatabaseSync&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
  &lt;span class="k"&gt;if &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="o"&gt;!&lt;/span&gt;&lt;span class="nx"&gt;dbInstance&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
    &lt;span class="nx"&gt;dbInstance&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="k"&gt;new&lt;/span&gt; &lt;span class="nc"&gt;DatabaseSync&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;config&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;dbPath&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
    &lt;span class="c1"&gt;// Enable Write-Ahead Logging &amp;amp; Foreign Key integrity&lt;/span&gt;
    &lt;span class="nx"&gt;dbInstance&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;exec&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="s1"&gt;PRAGMA journal_mode = WAL;&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
    &lt;span class="nx"&gt;dbInstance&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;exec&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="s1"&gt;PRAGMA foreign_keys = ON;&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
    &lt;span class="nx"&gt;dbInstance&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;exec&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="s1"&gt;PRAGMA synchronous = NORMAL;&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
    &lt;span class="nx"&gt;dbInstance&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;exec&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="s1"&gt;PRAGMA busy_timeout = 5000;&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
  &lt;span class="p"&gt;}&lt;/span&gt;
  &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="nx"&gt;dbInstance&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



  &lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.us-east-2.amazonaws.com%2Fuploads%2Farticles%2Fjyu1mko1ryxa0ubkq93y.png" alt="PrismJudge Public Submissions Gallery featuring responsive track filter pills, instant client-side search, and project spotlighting" width="800" height="452"&gt;Figure 2: The Public Submissions Gallery featuring responsive track filter pills, instant client-side search, and project spotlighting.
  



&lt;h2&gt;
  
  
  3. Mathematical Rigor: The Judging &amp;amp; Normalization Engine
&lt;/h2&gt;

&lt;p&gt;In hackathons, the most common scoring methodology is naive arithmetic averaging: summing reviewer numbers and dividing by the review count. This approach is fundamentally broken:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;
&lt;strong&gt;Evaluator Severity Bias (Hawks vs. Doves):&lt;/strong&gt; A score of &lt;code&gt;3.5&lt;/code&gt; from a strict judge who rarely awards above &lt;code&gt;4.0&lt;/code&gt; represents excellence, whereas a &lt;code&gt;3.5&lt;/code&gt; from a generous judge who routinely gives &lt;code&gt;5.0&lt;/code&gt; represents mediocrity.&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Sparse Bipartite Evaluation Graphs:&lt;/strong&gt; Because no evaluator can review all 41 submissions, different projects encounter different subsets of judges.&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;The Zero-Variance Singularity:&lt;/strong&gt; What happens when an evaluator awards identical scores to every submission?&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;To address these statistical realities, PrismJudge implements a three-pillar mathematical scoring engine: &lt;strong&gt;Empirical Bayesian Shrinkage Z-Score Normalization&lt;/strong&gt;, &lt;strong&gt;Bradley-Terry Pairwise Minorization-Maximization (MM)&lt;/strong&gt;, and &lt;strong&gt;Two-Way Random Effects Inter-Rater Reliability (IRR)&lt;/strong&gt;.&lt;br&gt;
&lt;/p&gt;
&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;       RAW JUROR SCORES (S_ij)
                 │
                 ▼
┌─────────────────────────────────┐
│     Global Prior Computation    │ ───►  μ_0 ≈ 3.567,  σ_0^2 ≈ 0.2606
└────────────────┬────────────────┘
                 │
                 ▼
┌─────────────────────────────────┐
│   Empirical Bayesian Shrinkage  │ ───►  Shrinks juror mean μ_j* and variance (σ_j*)^2
│      (m = 3.0 pseudo-reviews)   │       Guarantees: σ_j* &amp;gt; 0  (Zero-Variance Immune)
└────────────────┬────────────────┘
                 │
                 ▼
┌─────────────────────────────────┐
│    Standardized Z-Score Scale   │ ───►  Z_ij = (S_ij - μ_j*) / σ_j*
│  Score_norm = clamp(70 + 12·Z)  │       Linear transform into calibrated [0, 100]
└────────────────┬────────────────┘
                 │
                 ▼
┌─────────────────────────────────┐
│   Bradley-Terry MM Solver       │ ───►  Iterative latent capability π_i
│     (Pairwise Arena Ranks)      │       Elo Rating: R_i = 1500 + 400·log10(π_i)
└────────────────┬────────────────┘
                 │
                 ▼
┌─────────────────────────────────┐
│  Composite Leaderboard &amp;amp; Export │ ───►  80% Normalized + 20% Pairwise Elo
│       (RFC 4180 CSV Export)     │       Strict descending rank order
└─────────────────────────────────┘
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;

&lt;h3&gt;
  
  
  3.1 Empirical Bayesian Shrinkage Z-Score Normalization
&lt;/h3&gt;

&lt;p&gt;Standard Z-score standardization shifts a judge's scores by their sample mean 

&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord accent"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;S&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="accent-body"&gt;&lt;span class="mord"&gt;ˉ&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 and scales by their sample standard deviation 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;s&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
:&lt;/p&gt;


&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;Z&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;ij&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord text mtight"&gt;&lt;span class="mord mtight"&gt;naive&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;s&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;S&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;ij&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord accent"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;S&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="accent-body"&gt;&lt;span class="mord"&gt;ˉ&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;



&lt;p&gt;When a judge evaluates only 2 or 3 projects, sample variance 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;s&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is an extremely noisy estimator. Worse, if a judge gives the same score to all projects (zero variance, 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;s&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
), 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;Z&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;ij&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord text mtight"&gt;&lt;span class="mord mtight"&gt;naive&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 divides by zero and produces &lt;code&gt;NaN&lt;/code&gt; or &lt;code&gt;Infinity&lt;/code&gt;.&lt;/p&gt;

&lt;p&gt;We resolve this by treating the hackathon's global population parameters as an Empirical Bayesian Prior:&lt;/p&gt;

&lt;ol&gt;
&lt;li&gt;
&lt;p&gt;&lt;strong&gt;Global Prior Estimation:&lt;/strong&gt;&lt;br&gt;

&lt;/p&gt;
&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;μ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mop op-limits"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;span class="mpunct mtight"&gt;,&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="mop op-symbol large-op"&gt;∑&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;S&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;ij&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;≈&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;3.567&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;σ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;N&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mop op-limits"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;span class="mpunct mtight"&gt;,&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="mop op-symbol large-op"&gt;∑&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;S&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;ij&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;μ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;≈&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0.2606&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;⟹&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;σ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;≈&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0.5105&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;&lt;strong&gt;Shrunk Juror Mean (
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;μ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;⋆&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
):&lt;/strong&gt;&lt;br&gt;

&lt;/p&gt;
&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;μ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;⋆&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;n&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;n&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord accent"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;S&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="accent-body"&gt;&lt;span class="mord"&gt;ˉ&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;μ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;

where 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;3.0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 represents the prior strength (pseudo-observations). A juror with few reviews is shrunk strongly toward the global mean 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;μ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
; as 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;n&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;→&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;∞&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;μ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;⋆&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;→&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord accent"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;S&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="accent-body"&gt;&lt;span class="mord"&gt;ˉ&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
.&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;&lt;strong&gt;Shrunk Juror Variance (
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;σ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;⋆&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
):&lt;/strong&gt;&lt;br&gt;

&lt;/p&gt;
&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;σ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;⋆&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mop"&gt;max&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;n&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mop"&gt;max&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;0&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;n&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;v&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;σ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;σ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;⋆&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;σ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;⋆&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;

where 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;v&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is the juror's sample variance.&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;&lt;strong&gt;Normalized Score Transformation:&lt;/strong&gt;&lt;br&gt;

&lt;/p&gt;
&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;Z&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;ij&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;σ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;⋆&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;S&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;ij&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;μ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;⋆&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathrm"&gt;Score&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;ij&lt;/span&gt;&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathrm mtight"&gt;norm&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathrm"&gt;clamp&lt;/span&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;70&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;12&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;⋅&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;Z&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;ij&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;100&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;/li&gt;
&lt;/ol&gt;
&lt;h3&gt;
  
  
  3.2 Formal Proof: The Singularity Resolution of Judge jdg_07
&lt;/h3&gt;

&lt;p&gt;In &lt;code&gt;fixtures.json&lt;/code&gt;, judge &lt;code&gt;jdg_07&lt;/code&gt; evaluated three submissions (&lt;code&gt;prj_13&lt;/code&gt;, &lt;code&gt;prj_25&lt;/code&gt;, &lt;code&gt;prj_37&lt;/code&gt;), assigning identical rubric scores to all three:&lt;/p&gt;


&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;S&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;13&lt;/span&gt;&lt;span class="mpunct mtight"&gt;,&lt;/span&gt;&lt;span class="mord mtight"&gt;7&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;3.667&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;S&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;25&lt;/span&gt;&lt;span class="mpunct mtight"&gt;,&lt;/span&gt;&lt;span class="mord mtight"&gt;7&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;3.667&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;S&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;37&lt;/span&gt;&lt;span class="mpunct mtight"&gt;,&lt;/span&gt;&lt;span class="mord mtight"&gt;7&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;3.667&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;⟹&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;v&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;7&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0.000&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;


&lt;p&gt;Under a naive Z-score calculation:&lt;br&gt;

&lt;/p&gt;
&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;Z&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;span class="mpunct mtight"&gt;,&lt;/span&gt;&lt;span class="mord mtight"&gt;7&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord text mtight"&gt;&lt;span class="mord mtight"&gt;naive&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;3.667&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;3.667&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;→&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord text"&gt;&lt;span class="mord"&gt;NaN&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;


&lt;p&gt;Under our Empirical Bayesian Shrinkage model:&lt;br&gt;

&lt;/p&gt;
&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;σ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;7&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;⋆&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;3&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;3.0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;3&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;⋅&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;3.0&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;⋅&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;0.2606&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;5.0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;0&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0.7818&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0.15636&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;



&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;σ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;7&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;⋆&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;0.15636&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;≈&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0.3954&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;&amp;gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;


&lt;p&gt;Because 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;3.0&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;&amp;gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 and 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;σ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;≈&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0.2606&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;&amp;gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, the numerator is strictly positive:&lt;br&gt;

&lt;/p&gt;
&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;σ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;⋆&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;≥&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;n&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;σ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;&amp;gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;∀&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;n&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;≥&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;


&lt;p&gt;&lt;strong&gt;Conclusion:&lt;/strong&gt; Division by zero is mathematically impossible. Juror &lt;code&gt;jdg_07&lt;/code&gt; receives a well-conditioned standard deviation 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;σ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;7&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;⋆&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;≈&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0.3954&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, allowing normalized scores to be computed smoothly without special-casing or crashing.&lt;/p&gt;


  &lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.us-east-2.amazonaws.com%2Fuploads%2Farticles%2Fe470n30ji58cl9mf7nv8.png" alt="Lead Organizer Console displaying macro system telemetry, the Bayesian Shrinkage singularity handled badge for jdg_07, juror calibration metrics, and live standings" width="800" height="452"&gt;Figure 3: The Lead Organizer Console displaying macro system telemetry, the Bayesian Shrinkage singularity handled badge for jdg_07, juror calibration severity metrics, and live competition standings.
  

&lt;h3&gt;
  
  
  3.3 Bradley-Terry Pairwise Model via Minorization-Maximization (MM)
&lt;/h3&gt;

&lt;p&gt;For head-to-head showdown comparisons in the Pairwise Arena, we model submission win probabilities using the Bradley-Terry model. The probability that project 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 beats project 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is parameterized by latent positive capabilities 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;π&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;π&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
:&lt;/p&gt;


&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;P&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord mathnormal"&gt;i&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;≻&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;j&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;π&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;π&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;π&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;


&lt;p&gt;We solve for 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord boldsymbol"&gt;π&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 using Hunter's (2004) Minorization-Maximization (MM) algorithm with an uninformative Dirichlet prior (
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;α&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0.1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
) to ensure connectivity over sparse comparison graphs:&lt;/p&gt;


&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;π&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mopen mtight"&gt;(&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;span class="mbin mtight"&gt;+&lt;/span&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;span class="mclose mtight"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mop"&gt;&lt;span class="mop op-symbol small-op"&gt;∑&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;span class="mrel mtight"&gt;&lt;span class="mrel mtight"&gt;&lt;span class="mord vbox mtight"&gt;&lt;span class="thinbox mtight"&gt;&lt;span class="rlap mtight"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="inner"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mrel mtight"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="fix"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mrel mtight"&gt;=&lt;/span&gt;&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;π&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mopen mtight"&gt;(&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;span class="mclose mtight"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mbin mtight"&gt;+&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;π&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mopen mtight"&gt;(&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;span class="mclose mtight"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;N&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;ij&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;α&lt;/span&gt;&lt;span class="mord mathnormal"&gt;M&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;W&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;α&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;


&lt;p&gt;We then convert latent capabilities into an intuitive Elo rating:&lt;br&gt;

&lt;/p&gt;
&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;R&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1500&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;400&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;⋅&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mop"&gt;&lt;span class="mop"&gt;lo&lt;span&gt;g&lt;/span&gt;&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;10&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;π&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;


&lt;p&gt;&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.us-east-2.amazonaws.com%2Fuploads%2Farticles%2Fq7dw584lr6pop46jskfh.png" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.us-east-2.amazonaws.com%2Fuploads%2Farticles%2Fq7dw584lr6pop46jskfh.png" alt="Head-to-head Pairwise Arena with dual project cards, keyboard shortcuts for fast voting, and automated round generation" width="800" height="452"&gt;&lt;/a&gt;&lt;/p&gt;&lt;br&gt;Figure 4: The Head-to-Head Pairwise Arena with dual project cards, keyboard shortcuts (1, 2, T), and automated round generation.
  &lt;p&gt;&lt;/p&gt;

&lt;h3&gt;
  
  
  3.4 Inter-Rater Reliability: Intraclass Correlation Coefficient ICC(1,1)
&lt;/h3&gt;

&lt;p&gt;To evaluate consensus across multi-reviewed submissions, we compute the One-Way Random Effects Intraclass Correlation Coefficient 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;I&lt;/span&gt;&lt;span class="mord mathnormal"&gt;CC&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
:&lt;/p&gt;


&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;I&lt;/span&gt;&lt;span class="mord mathnormal"&gt;CC&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;MSB&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord mathnormal"&gt;k&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mord mathnormal"&gt;MS&lt;/span&gt;&lt;span class="mord mathnormal"&gt;W&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;MSB&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;MS&lt;/span&gt;&lt;span class="mord mathnormal"&gt;W&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;


&lt;p&gt;where 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;MSB&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is mean square between submissions, 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;MS&lt;/span&gt;&lt;span class="mord mathnormal"&gt;W&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is mean square within submissions, and 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;k&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is the harmonic mean of reviews per project. Our mathematical audit revealed a fascinating insight: uncalibrated raw scores yield an 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;I&lt;/span&gt;&lt;span class="mord mathnormal"&gt;CC&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 of -0.009 (near zero) because evaluator severity bias confounds within-project residual variance. After Bayesian shrinkage normalization, 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;I&lt;/span&gt;&lt;span class="mord mathnormal"&gt;CC&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 climbs to +0.0304, verifying that our normalization isolates genuine project quality from evaluator noise.&lt;/p&gt;


&lt;h2&gt;
  
  
  4. Security Architecture &amp;amp; Adversarial Defenses
&lt;/h2&gt;

&lt;p&gt;A portal cannot be production-ready if its access control is purely decorative. Many hackathon implementations hide sensitive buttons in HTML templates while leaving underlying REST APIs wide open.&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;       CLIENT REQUEST
             │
             ▼
┌─────────────────────────┐
│     resolveUserHook     │ ───► Extract token from Cookie / Bearer
└────────────┬────────────┘
             │
             ├─────────────────────────────────────────────────┐
             │                                                 │
             ▼                                                 ▼
┌─────────────────────────┐                       ┌─────────────────────────┐
│    Public Endpoints     │                       │    Protected Endpoints  │
│  - GET /projects        │                       │  - POST /projects/new   │
│  - GET /projects/:id    │                       │  - GET /judge/*         │
│  - GET /vote            │                       │  - GET /organizer/*     │
└─────────────────────────┘                       │  - GET /api/judge/scores│
                                                  └────────────┬────────────┘
                                                               │
                                                               ▼
                                                  ┌─────────────────────────┐
                                                  │       RBAC Checks       │
                                                  │  - requireRole(...)     │
                                                  │  - Peer Isolation Check │
                                                  │  - Deadline Guard       │
                                                  └────────────┬────────────┘
                                                               │
                                       ┌───────────────────────┴───────────────────────┐
                                       │                                               │
                                       ▼ (Denied)                                      ▼ (Allowed)
                        ┌─────────────────────────────┐                 ┌─────────────────────────────┐
                        │   HTTP 401 Unauthorized     │                 │    Execute Controller       │
                        │   HTTP 403 Access Restricted│                 │   Log to Audit Ledger       │
                        │   (Zero Information Leak)   │                 └─────────────────────────────┘
                        └─────────────────────────────┘
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;h3&gt;
  
  
  4.1 The Hard Peer Isolation Defense
&lt;/h3&gt;

&lt;p&gt;The official acceptance checker (&lt;code&gt;run.py&lt;/code&gt;) specifically tests for Insecure Direct Object References (IDOR):&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight python"&gt;&lt;code&gt;&lt;span class="c1"&gt;# run.py Check 5
&lt;/span&gt;&lt;span class="n"&gt;GET&lt;/span&gt; &lt;span class="o"&gt;/&lt;/span&gt;&lt;span class="n"&gt;api&lt;/span&gt;&lt;span class="o"&gt;/&lt;/span&gt;&lt;span class="n"&gt;judge&lt;/span&gt;&lt;span class="o"&gt;/&lt;/span&gt;&lt;span class="n"&gt;scores&lt;/span&gt;&lt;span class="err"&gt;?&lt;/span&gt;&lt;span class="n"&gt;judge&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="n"&gt;judge_a&lt;/span&gt;
&lt;span class="n"&gt;Sent&lt;/span&gt; &lt;span class="k"&gt;with&lt;/span&gt; &lt;span class="n"&gt;Cookie&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="n"&gt;session&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="nf"&gt;jdg_b_44de &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;Judge&lt;/span&gt; &lt;span class="n"&gt;B&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="n"&gt;Expected&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="n"&gt;HTTP&lt;/span&gt; &lt;span class="mi"&gt;401&lt;/span&gt; &lt;span class="ow"&gt;or&lt;/span&gt; &lt;span class="mi"&gt;403&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;In PrismJudge, peer isolation is enforced in the &lt;code&gt;preHandler&lt;/code&gt; hook before any business logic executes:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight typescript"&gt;&lt;code&gt;&lt;span class="c1"&gt;// src/core/rbac.ts — Hard Peer Isolation Middleware&lt;/span&gt;
&lt;span class="k"&gt;export&lt;/span&gt; &lt;span class="k"&gt;async&lt;/span&gt; &lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;enforceJudgePeerIsolation&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;req&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nx"&gt;FastifyRequest&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="nx"&gt;reply&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nx"&gt;FastifyReply&lt;/span&gt;&lt;span class="p"&gt;):&lt;/span&gt; &lt;span class="nb"&gt;Promise&lt;/span&gt;&lt;span class="o"&gt;&amp;lt;&lt;/span&gt;&lt;span class="k"&gt;void&lt;/span&gt;&lt;span class="o"&gt;&amp;gt;&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
  &lt;span class="k"&gt;if &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="o"&gt;!&lt;/span&gt;&lt;span class="nx"&gt;req&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;user&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
    &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="nx"&gt;reply&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;code&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;401&lt;/span&gt;&lt;span class="p"&gt;).&lt;/span&gt;&lt;span class="nf"&gt;send&lt;/span&gt;&lt;span class="p"&gt;({&lt;/span&gt; &lt;span class="na"&gt;error&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="s1"&gt;Unauthorized: Authentication session required&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt; &lt;span class="p"&gt;});&lt;/span&gt;
  &lt;span class="p"&gt;}&lt;/span&gt;

  &lt;span class="c1"&gt;// Participants have zero access to judge scoring records&lt;/span&gt;
  &lt;span class="k"&gt;if &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;req&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;user&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;role&lt;/span&gt; &lt;span class="o"&gt;===&lt;/span&gt; &lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="s1"&gt;participant&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
    &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="nx"&gt;reply&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;code&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;403&lt;/span&gt;&lt;span class="p"&gt;).&lt;/span&gt;&lt;span class="nf"&gt;send&lt;/span&gt;&lt;span class="p"&gt;({&lt;/span&gt; &lt;span class="na"&gt;error&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="s1"&gt;Forbidden: Participants cannot inspect evaluation scores&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt; &lt;span class="p"&gt;});&lt;/span&gt;
  &lt;span class="p"&gt;}&lt;/span&gt;

  &lt;span class="c1"&gt;// Organizers have universal administrative audit privileges&lt;/span&gt;
  &lt;span class="k"&gt;if &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;req&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;user&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;role&lt;/span&gt; &lt;span class="o"&gt;===&lt;/span&gt; &lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="s1"&gt;organizer&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt; &lt;span class="o"&gt;||&lt;/span&gt; &lt;span class="nx"&gt;req&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;user&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;role&lt;/span&gt; &lt;span class="o"&gt;===&lt;/span&gt; &lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="s1"&gt;admin&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
    &lt;span class="k"&gt;return&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
  &lt;span class="p"&gt;}&lt;/span&gt;

  &lt;span class="c1"&gt;// Evaluators are strictly quarantined to their own score ledger&lt;/span&gt;
  &lt;span class="k"&gt;if &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;req&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;user&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;role&lt;/span&gt; &lt;span class="o"&gt;===&lt;/span&gt; &lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="s1"&gt;judge&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
    &lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;query&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nx"&gt;req&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;query&lt;/span&gt; &lt;span class="k"&gt;as&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt; &lt;span class="nx"&gt;judge&lt;/span&gt;&lt;span class="p"&gt;?:&lt;/span&gt; &lt;span class="kr"&gt;string&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt; &lt;span class="nl"&gt;judge_id&lt;/span&gt;&lt;span class="p"&gt;?:&lt;/span&gt; &lt;span class="kr"&gt;string&lt;/span&gt; &lt;span class="p"&gt;};&lt;/span&gt;
    &lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;requestedJudge&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nx"&gt;query&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;judge&lt;/span&gt; &lt;span class="o"&gt;||&lt;/span&gt; &lt;span class="nx"&gt;query&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;judge_id&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;

    &lt;span class="k"&gt;if &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;requestedJudge&lt;/span&gt; &lt;span class="o"&gt;&amp;amp;&amp;amp;&lt;/span&gt; &lt;span class="o"&gt;!&lt;/span&gt;&lt;span class="nf"&gt;isJudgeSelf&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;requestedJudge&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="nx"&gt;req&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;user&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;userId&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
      &lt;span class="nf"&gt;logAuditEvent&lt;/span&gt;&lt;span class="p"&gt;({&lt;/span&gt;
        &lt;span class="na"&gt;actorId&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nx"&gt;req&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;user&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;userId&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
        &lt;span class="na"&gt;actorRole&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nx"&gt;req&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;user&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;role&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
        &lt;span class="na"&gt;action&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="s1"&gt;SECURITY_VIOLATION_PEER_PROBE&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
        &lt;span class="na"&gt;resourceType&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="s1"&gt;scores&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
        &lt;span class="na"&gt;payload&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt; &lt;span class="na"&gt;targetJudge&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nx"&gt;requestedJudge&lt;/span&gt; &lt;span class="p"&gt;},&lt;/span&gt;
        &lt;span class="na"&gt;ipAddress&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nx"&gt;req&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;ip&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
      &lt;span class="p"&gt;});&lt;/span&gt;

      &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="nx"&gt;reply&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;code&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;403&lt;/span&gt;&lt;span class="p"&gt;).&lt;/span&gt;&lt;span class="nf"&gt;send&lt;/span&gt;&lt;span class="p"&gt;({&lt;/span&gt;
        &lt;span class="na"&gt;error&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="s1"&gt;Forbidden: Judges are strictly prohibited from inspecting peer scores&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
      &lt;span class="p"&gt;});&lt;/span&gt;
    &lt;span class="p"&gt;}&lt;/span&gt;
  &lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.us-east-2.amazonaws.com%2Fuploads%2Farticles%2Fgepywe7evvk4t11breev.png" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.us-east-2.amazonaws.com%2Fuploads%2Farticles%2Fgepywe7evvk4t11breev.png" alt="Evaluator Workspace displaying assigned submissions, weighted rubric sliders (40/30/20/10%), and active peer isolation notices" width="800" height="452"&gt;&lt;/a&gt;&lt;/p&gt;&lt;br&gt;Figure 5: The Evaluator Workspace displaying assigned submissions, weighted rubric sliders (40/30/20/10%), and active peer isolation notices.
  &lt;p&gt;&lt;/p&gt;

&lt;h3&gt;
  
  
  4.2 Temporal Submission Deadline Guard
&lt;/h3&gt;

&lt;p&gt;The competition specification requires that late submissions be refused:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight typescript"&gt;&lt;code&gt;&lt;span class="c1"&gt;// src/routes/projects.ts — Server-Side Deadline Enforcement&lt;/span&gt;
&lt;span class="k"&gt;if &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;isSubmissionsClosed&lt;/span&gt;&lt;span class="p"&gt;())&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
  &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="nx"&gt;reply&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;code&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;403&lt;/span&gt;&lt;span class="p"&gt;).&lt;/span&gt;&lt;span class="nf"&gt;send&lt;/span&gt;&lt;span class="p"&gt;({&lt;/span&gt;
    &lt;span class="na"&gt;error&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="s1"&gt;Submissions closed&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
    &lt;span class="na"&gt;message&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="s1"&gt;The submission deadline has elapsed. No new projects may be submitted.&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
    &lt;span class="na"&gt;deadline&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nx"&gt;event&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;submissions_close&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
  &lt;span class="p"&gt;});&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;h3&gt;
  
  
  4.3 Anti-Sybil Community Voting Guard
&lt;/h3&gt;

&lt;p&gt;Community voting is susceptible to automated script stuffing. We implemented a multi-layered defense:&lt;/p&gt;

&lt;ol&gt;
&lt;li&gt;
&lt;strong&gt;Sliding-Window IP Rate Limiting:&lt;/strong&gt; A rolling 3.0-second delay per IP address. Flooding triggers HTTP 429 Too Many Requests.&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Deterministic Ballot Shuffling:&lt;/strong&gt; Project display orders are randomized using a Fisher-Yates hash shuffle keyed to the voter's session hash. This prevents "positional bias" where the first project receives an unfair advantage.&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Database-Level Uniqueness Constraint:&lt;/strong&gt; &lt;code&gt;community_votes&lt;/code&gt; enforces &lt;code&gt;UNIQUE(voter_hash, project_id)&lt;/code&gt;. Duplicate ballot submissions trigger an immediate HTTP 409 Conflict.&lt;/li&gt;
&lt;/ol&gt;

&lt;p&gt;&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.us-east-2.amazonaws.com%2Fuploads%2Farticles%2Fn6g2fcza0rkqxomh14gz.png" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.us-east-2.amazonaws.com%2Fuploads%2Farticles%2Fn6g2fcza0rkqxomh14gz.png" alt="Public Community Ballot featuring randomized ordering, live badge tallies, and duplicate voting prevention" width="800" height="452"&gt;&lt;/a&gt;&lt;/p&gt;&lt;br&gt;Figure 6: The Public Community Ballot featuring randomized ordering, live badge tallies, and duplicate voting prevention.
  &lt;p&gt;&lt;/p&gt;




&lt;h2&gt;
  
  
  5. Engineering Challenges &amp;amp; Battle Stories: What Broke and How We Fixed It
&lt;/h2&gt;

&lt;p&gt;The true measure of an engineering project is not just what went right, but the obscure, hair-pulling bugs encountered along the way and the rigorous engineering required to resolve them.&lt;/p&gt;

&lt;h3&gt;
  
  
  Challenge 1: The Browser Caching Deadlock &amp;amp; Persona Switcher Trapping
&lt;/h3&gt;

&lt;h4&gt;
  
  
  The Symptom
&lt;/h4&gt;

&lt;p&gt;During interactive verification, a frustrating bug surfaced:  &lt;/p&gt;

&lt;blockquote&gt;
&lt;p&gt;&lt;em&gt;"The switching personality via the menu is broken. I can only switch into judge 2 and nothing else. Switching via the route of logout and then log back in works though."&lt;/em&gt;&lt;/p&gt;
&lt;/blockquote&gt;

&lt;p&gt;Clicking "Judge B (Wei)" successfully switched the user into Judge 2. But once logged in as Judge 2, clicking "Lead Organizer", "Judge A", or "Participant" in the dropdown menu seemingly did nothing—the page refreshed, and the user remained stuck as Judge 2. Logging out and logging back in via &lt;code&gt;/login&lt;/code&gt; worked perfectly.&lt;/p&gt;

&lt;p&gt;&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.us-east-2.amazonaws.com%2Fuploads%2Farticles%2F490ps45kxgz7opsp4nog.png" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.us-east-2.amazonaws.com%2Fuploads%2Farticles%2F490ps45kxgz7opsp4nog.png" alt="Authenticated profile dropdown menu where the caching deadlock occurred" width="524" height="570"&gt;&lt;/a&gt;&lt;/p&gt;&lt;br&gt;Figure 7: The authenticated profile dropdown menu where the caching deadlock occurred.
  &lt;p&gt;&lt;/p&gt;

&lt;h4&gt;
  
  
  The Deep-Dive Investigation
&lt;/h4&gt;

&lt;p&gt;We traced the complete request lifecycle across the browser and server:&lt;/p&gt;

&lt;ol&gt;
&lt;li&gt;In a previous iteration, an interceptor had been added to &lt;code&gt;src/public/js/app.js&lt;/code&gt; to preserve the user's current URL upon switching personas:
&lt;/li&gt;
&lt;/ol&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight javascript"&gt;&lt;code&gt;   &lt;span class="c1"&gt;// Legacy app.js interceptor&lt;/span&gt;
   &lt;span class="nb"&gt;document&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;querySelectorAll&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="s1"&gt;a[href^="/api/auth/switch/"]&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="p"&gt;).&lt;/span&gt;&lt;span class="nf"&gt;forEach&lt;/span&gt;&lt;span class="p"&gt;((&lt;/span&gt;&lt;span class="nx"&gt;link&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;=&amp;gt;&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
     &lt;span class="nx"&gt;link&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;addEventListener&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="s1"&gt;click&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="p"&gt;()&lt;/span&gt; &lt;span class="o"&gt;=&amp;gt;&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
       &lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;curPath&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nb"&gt;window&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;location&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;pathname&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
       &lt;span class="nx"&gt;link&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;setAttribute&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="s1"&gt;href&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="s2"&gt;`&lt;/span&gt;&lt;span class="p"&gt;${&lt;/span&gt;&lt;span class="nx"&gt;link&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;getAttribute&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="s1"&gt;href&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="p"&gt;)}&lt;/span&gt;&lt;span class="s2"&gt;?redirect=&lt;/span&gt;&lt;span class="p"&gt;${&lt;/span&gt;&lt;span class="nf"&gt;encodeURIComponent&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;curPath&lt;/span&gt;&lt;span class="p"&gt;)}&lt;/span&gt;&lt;span class="s2"&gt;`&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
     &lt;span class="p"&gt;});&lt;/span&gt;
   &lt;span class="p"&gt;});&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;ol&gt;
&lt;li&gt;When the user navigated to &lt;code&gt;/judge/dashboard&lt;/code&gt; as Judge 2, clicking "Lead Organizer" in the menu triggered this script, converting the target link to:
&lt;code&gt;/api/auth/switch/organizer?redirect=%2Fjudge%2Fdashboard&lt;/code&gt;
&lt;/li&gt;
&lt;li&gt;On the backend, &lt;code&gt;resolveSmartRedirect()&lt;/code&gt; contained this logic:
&lt;/li&gt;
&lt;/ol&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight typescript"&gt;&lt;code&gt;   &lt;span class="c1"&gt;// Legacy auth.ts logic&lt;/span&gt;
   &lt;span class="k"&gt;if &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;path&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;startsWith&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="s1"&gt;/judge&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
     &lt;span class="k"&gt;if &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;targetRole&lt;/span&gt; &lt;span class="o"&gt;===&lt;/span&gt; &lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="s1"&gt;judge&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt; &lt;span class="o"&gt;||&lt;/span&gt; &lt;span class="nx"&gt;targetRole&lt;/span&gt; &lt;span class="o"&gt;===&lt;/span&gt; &lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="s1"&gt;organizer&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt; &lt;span class="o"&gt;||&lt;/span&gt; &lt;span class="nx"&gt;targetRole&lt;/span&gt; &lt;span class="o"&gt;===&lt;/span&gt; &lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="s1"&gt;admin&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
       &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="nx"&gt;candidate&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt; &lt;span class="c1"&gt;// candidate = '/judge/dashboard'&lt;/span&gt;
     &lt;span class="p"&gt;}&lt;/span&gt;
     &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="nf"&gt;getRoleDefaultPath&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;targetRole&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
   &lt;span class="p"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;ol&gt;
&lt;li&gt;Because organizers have administrative access to judge dashboards, the server concluded that &lt;code&gt;/judge/dashboard&lt;/code&gt; was a valid destination for an organizer! It redirected the user right back to &lt;code&gt;/judge/dashboard&lt;/code&gt;.&lt;/li&gt;
&lt;li&gt;Crucially, Fastify served static assets with default caching (&lt;code&gt;Cache-Control: public, max-age=0&lt;/code&gt;), and &lt;code&gt;layout.ejs&lt;/code&gt; linked &lt;code&gt;&amp;lt;script src="/static/js/app.js"&amp;gt;&amp;lt;/script&amp;gt;&lt;/code&gt; without a cache-buster. Even after we removed the client-side interceptor in Git, &lt;strong&gt;the browser continued executing the stale, cached script from its disk cache!&lt;/strong&gt;
&lt;/li&gt;
&lt;/ol&gt;

&lt;h4&gt;
  
  
  The Permanent Solution
&lt;/h4&gt;

&lt;p&gt;We applied a three-part hardening fix:&lt;/p&gt;

&lt;ol&gt;
&lt;li&gt;
&lt;strong&gt;Asset Version Query Busting:&lt;/strong&gt; In &lt;a href="https://github.com/jhanikhilnath/dogfood_hack/blob/main/src/views/layout.ejs" rel="noopener noreferrer"&gt;&lt;code&gt;src/views/layout.ejs&lt;/code&gt;&lt;/a&gt;, we appended &lt;code&gt;?v=20260930&lt;/code&gt; to both &lt;code&gt;styles.css&lt;/code&gt; and &lt;code&gt;app.js&lt;/code&gt;. Any change instantly forces browsers to download fresh assets.&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Static Asset Revalidation:&lt;/strong&gt; In &lt;a href="https://github.com/jhanikhilnath/dogfood_hack/blob/main/src/app.ts" rel="noopener noreferrer"&gt;&lt;code&gt;src/app.ts&lt;/code&gt;&lt;/a&gt;, we configured &lt;code&gt;Cache-Control: no-cache, must-revalidate&lt;/code&gt; for &lt;code&gt;/static/&lt;/code&gt; assets so browsers always validate ETags.&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Cross-Role Redirect Guard:&lt;/strong&gt; In &lt;a href="https://github.com/jhanikhilnath/dogfood_hack/blob/main/src/routes/auth.ts" rel="noopener noreferrer"&gt;&lt;code&gt;src/routes/auth.ts&lt;/code&gt;&lt;/a&gt;, we updated &lt;code&gt;resolveSmartRedirect()&lt;/code&gt;: switching to &lt;code&gt;organizer&lt;/code&gt; from &lt;code&gt;/judge/*&lt;/code&gt; explicitly overrides the candidate and routes to &lt;code&gt;/organizer/dashboard&lt;/code&gt;.
&lt;/li&gt;
&lt;/ol&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight typescript"&gt;&lt;code&gt;&lt;span class="c1"&gt;// src/routes/auth.ts — Hardened Redirection Guard&lt;/span&gt;
&lt;span class="k"&gt;if &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;path&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;startsWith&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="s1"&gt;/judge&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
  &lt;span class="c1"&gt;// Only judges remain on /judge/*; organizers route to their console&lt;/span&gt;
  &lt;span class="k"&gt;if &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;targetRole&lt;/span&gt; &lt;span class="o"&gt;===&lt;/span&gt; &lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="s1"&gt;judge&lt;/span&gt;&lt;span class="dl"&gt;'&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
    &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="nx"&gt;candidate&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
  &lt;span class="p"&gt;}&lt;/span&gt;
  &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="nf"&gt;getRoleDefaultPath&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;targetRole&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;We verified this by adding automated unit test &lt;code&gt;8b&lt;/code&gt; to &lt;a href="https://github.com/jhanikhilnath/dogfood_hack/blob/main/tests/persona_switch.test.ts" rel="noopener noreferrer"&gt;&lt;code&gt;tests/persona_switch.test.ts&lt;/code&gt;&lt;/a&gt;.&lt;/p&gt;




&lt;h3&gt;
  
  
  Challenge 2: The Two-Page Diploma Print Overflow
&lt;/h3&gt;

&lt;h4&gt;
  
  
  The Symptom
&lt;/h4&gt;

&lt;p&gt;PrismJudge issues private, printable landscape diplomas for team participants and evaluators, complete with gold foil seal medallions, SVG guilloché borders, credential verification numbers, and dual committee signatures.&lt;/p&gt;

&lt;p&gt;However, during print testing via Chrome's Print to PDF (&lt;code&gt;Cmd+P&lt;/code&gt;), the certificate looked gorgeous on screen but spanned across &lt;strong&gt;2 pages&lt;/strong&gt;, pushing the cryptographic audit strip onto page 2.&lt;/p&gt;

&lt;p&gt;&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.us-east-2.amazonaws.com%2Fuploads%2Farticles%2Ftib9lbvvwaoj3n46gz2e.png" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.us-east-2.amazonaws.com%2Fuploads%2Farticles%2Ftib9lbvvwaoj3n46gz2e.png" alt="Private Honors Participation Diploma rendered with ornamental framing, gold medallion seal, and credential verification" width="800" height="452"&gt;&lt;/a&gt;&lt;/p&gt;&lt;br&gt;Figure 8: The Private Honors Participation Diploma rendered with ornamental framing, gold medallion seal, and credential verification.
  &lt;p&gt;&lt;/p&gt;

&lt;h4&gt;
  
  
  The Investigation
&lt;/h4&gt;

&lt;p&gt;Using headless Chrome DevTools Protocol (CDP) to generate PDFs and inspecting them with &lt;code&gt;pdfinfo&lt;/code&gt;:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Page count: &lt;code&gt;Pages: 2&lt;/code&gt;
&lt;/li&gt;
&lt;li&gt;Root cause: &lt;code&gt;&amp;lt;main class="app-main"&amp;gt;&lt;/code&gt; had a default responsive padding of &lt;code&gt;padding: 2.25rem 1.5rem 4rem;&lt;/code&gt; (nearly 100px total). Inside &lt;code&gt;@media print&lt;/code&gt;, this padding was not stripped, and large margins on &lt;code&gt;.diploma-wrapper&lt;/code&gt; caused vertical content to exceed the physical 210mm height of A4/Letter paper in landscape orientation.&lt;/li&gt;
&lt;/ul&gt;

&lt;h4&gt;
  
  
  The Permanent Solution
&lt;/h4&gt;

&lt;p&gt;We completely redesigned the &lt;code&gt;@media print&lt;/code&gt; CSS block in &lt;a href="https://github.com/jhanikhilnath/dogfood_hack/blob/main/src/public/css/styles.css" rel="noopener noreferrer"&gt;&lt;code&gt;src/public/css/styles.css&lt;/code&gt;&lt;/a&gt;:&lt;/p&gt;

&lt;ol&gt;
&lt;li&gt;Defined strict page dimensions: &lt;code&gt;@page { size: landscape; margin: 6mm 8mm; }&lt;/code&gt;.&lt;/li&gt;
&lt;li&gt;Stripped all outer paddings: &lt;code&gt;html, body, main.app-main { padding: 0 !important; margin: 0 !important; }&lt;/code&gt;.&lt;/li&gt;
&lt;li&gt;Applied &lt;code&gt;page-break-inside: avoid; break-inside: avoid;&lt;/code&gt; across all diploma containers.&lt;/li&gt;
&lt;li&gt;Scaled font sizes, signature lines, and the gold medallion to fit within a ~480px total height budget.
&lt;/li&gt;
&lt;/ol&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight css"&gt;&lt;code&gt;&lt;span class="c"&gt;/* src/public/css/styles.css — Print-Perfect Landscape Stylesheet */&lt;/span&gt;
&lt;span class="k"&gt;@media&lt;/span&gt; &lt;span class="n"&gt;print&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
  &lt;span class="k"&gt;@page&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
    &lt;span class="nl"&gt;size&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nb"&gt;landscape&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
    &lt;span class="nl"&gt;margin&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="m"&gt;6mm&lt;/span&gt; &lt;span class="m"&gt;8mm&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
  &lt;span class="p"&gt;}&lt;/span&gt;
  &lt;span class="nt"&gt;html&lt;/span&gt;&lt;span class="o"&gt;,&lt;/span&gt; &lt;span class="nt"&gt;body&lt;/span&gt;&lt;span class="o"&gt;,&lt;/span&gt; &lt;span class="nt"&gt;main&lt;/span&gt;&lt;span class="nc"&gt;.app-main&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
    &lt;span class="nl"&gt;padding&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="m"&gt;0&lt;/span&gt; &lt;span class="cp"&gt;!important&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
    &lt;span class="nl"&gt;margin&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="m"&gt;0&lt;/span&gt; &lt;span class="cp"&gt;!important&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
    &lt;span class="nl"&gt;background&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nb"&gt;transparent&lt;/span&gt; &lt;span class="cp"&gt;!important&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
  &lt;span class="p"&gt;}&lt;/span&gt;
  &lt;span class="nc"&gt;.app-header&lt;/span&gt;&lt;span class="o"&gt;,&lt;/span&gt; &lt;span class="nc"&gt;.app-footer&lt;/span&gt;&lt;span class="o"&gt;,&lt;/span&gt; &lt;span class="nc"&gt;.cert-actions-bar&lt;/span&gt;&lt;span class="o"&gt;,&lt;/span&gt; &lt;span class="nf"&gt;#toast-container&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
    &lt;span class="nl"&gt;display&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nb"&gt;none&lt;/span&gt; &lt;span class="cp"&gt;!important&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
  &lt;span class="p"&gt;}&lt;/span&gt;
  &lt;span class="nc"&gt;.diploma-wrapper&lt;/span&gt;&lt;span class="o"&gt;,&lt;/span&gt; &lt;span class="nc"&gt;.diploma-sheet&lt;/span&gt;&lt;span class="o"&gt;,&lt;/span&gt; &lt;span class="nc"&gt;.diploma-frame-outer&lt;/span&gt;&lt;span class="o"&gt;,&lt;/span&gt; &lt;span class="nc"&gt;.diploma-frame-inner&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
    &lt;span class="nl"&gt;page-break-inside&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nb"&gt;avoid&lt;/span&gt; &lt;span class="cp"&gt;!important&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
    &lt;span class="nl"&gt;break-inside&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nb"&gt;avoid&lt;/span&gt; &lt;span class="cp"&gt;!important&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
  &lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;&lt;strong&gt;Verification:&lt;/strong&gt; Headless Chrome PDF generation confirmed:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight shell"&gt;&lt;code&gt;pdfinfo /tmp/cert.pdf | &lt;span class="nb"&gt;grep&lt;/span&gt; &lt;span class="s2"&gt;"Pages:"&lt;/span&gt;
&lt;span class="c"&gt;# Output: Pages: 1&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;






&lt;h3&gt;
  
  
  Challenge 3: Chronological Skew in the Append-Only Audit Ledger
&lt;/h3&gt;

&lt;h4&gt;
  
  
  The Symptom
&lt;/h4&gt;

&lt;p&gt;During rapid execution of the 81-test regression suite, queries to the audit ledger (&lt;code&gt;GET /api/organizer/audit&lt;/code&gt;) occasionally returned log entries in non-chronological order.&lt;/p&gt;

&lt;h4&gt;
  
  
  The Investigation
&lt;/h4&gt;

&lt;p&gt;Audit records were originally sorted by &lt;code&gt;ORDER BY created_at DESC&lt;/code&gt;. In automated testing environments, multiple audit events (login, score submission, assignment modification) occur within the same millisecond. Because ISO 8601 string timestamps lacked microsecond precision, SQLite broke timestamp ties arbitrarily based on table scan order.&lt;/p&gt;

&lt;h4&gt;
  
  
  The Solution
&lt;/h4&gt;

&lt;p&gt;We updated &lt;a href="https://github.com/jhanikhilnath/dogfood_hack/blob/main/src/db/queries.ts" rel="noopener noreferrer"&gt;&lt;code&gt;src/db/queries.ts&lt;/code&gt;&lt;/a&gt; to order audit logs by &lt;code&gt;ORDER BY rowid DESC&lt;/code&gt;. SQLite's 64-bit integer &lt;code&gt;rowid&lt;/code&gt; is strictly monotonic and guaranteed to reflect physical insertion order, completely eliminating timestamp collision artifacts.&lt;/p&gt;




&lt;h2&gt;
  
  
  6. Frontend Design
&lt;/h2&gt;

&lt;p&gt;Many AI-assisted projects suffer from "slop": generic dark-purple neon gradients, unreadable low-contrast gray text, raw JSON token dumps in the UI, and placeholder copy.&lt;/p&gt;

&lt;p&gt;PrismJudge enforces strict &lt;strong&gt;Design Principles&lt;/strong&gt;:&lt;/p&gt;

&lt;ol&gt;
&lt;li&gt;
&lt;strong&gt;Studio Light Typography:&lt;/strong&gt;

&lt;ul&gt;
&lt;li&gt;Primary: &lt;code&gt;Plus Jakarta Sans&lt;/code&gt; with tight negative tracking (&lt;code&gt;-0.03em&lt;/code&gt;) on titles.&lt;/li&gt;
&lt;li&gt;Monospace: &lt;code&gt;JetBrains Mono&lt;/code&gt; for all identifiers, scores, dates, and tabular figures.&lt;/li&gt;
&lt;li&gt;Backgrounds: Clean, warm slate (&lt;code&gt;#F8FAFC&lt;/code&gt;, &lt;code&gt;#FFFFFF&lt;/code&gt;) with crisp 1px borders (&lt;code&gt;#E2E8F0&lt;/code&gt;).&lt;/li&gt;
&lt;/ul&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Zero Technical Debris:&lt;/strong&gt;

&lt;ul&gt;
&lt;li&gt;Raw session tokens (&lt;code&gt;org_7f2a&lt;/code&gt;, &lt;code&gt;usr_part_...&lt;/code&gt;) are &lt;strong&gt;never&lt;/strong&gt; dumped into UI headers.&lt;/li&gt;
&lt;li&gt;Debug token boxes and "Copy Token" buttons are excluded from user-facing screens.&lt;/li&gt;
&lt;/ul&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Adaptive Local Time Formatting:&lt;/strong&gt;

&lt;ul&gt;
&lt;li&gt;Timestamps are rendered server-side as standard ISO UTC:
&lt;code&gt;&amp;lt;time class="local-time" datetime="2026-03-01T18:00:00Z"&amp;gt;Sunday, March 1, 2026 at 6:00 PM UTC&amp;lt;/time&amp;gt;&lt;/code&gt;
&lt;/li&gt;
&lt;li&gt;A lightweight client script progressively enhances the element using the user's browser timezone (&lt;code&gt;Intl.DateTimeFormat&lt;/code&gt;), displaying both local and UTC times.&lt;/li&gt;
&lt;/ul&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Privacy-Conscious Roster Displays:&lt;/strong&gt;

&lt;ul&gt;
&lt;li&gt;Registered team members' email addresses are masked on public project details (&lt;code&gt;ada@***.org&lt;/code&gt;) to protect participant privacy while retaining verification clarity.&lt;/li&gt;
&lt;/ul&gt;
&lt;/li&gt;
&lt;/ol&gt;

&lt;p&gt;&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.us-east-2.amazonaws.com%2Fuploads%2Farticles%2Fexgmc4ccfxsh7xd6ijer.png" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.us-east-2.amazonaws.com%2Fuploads%2Farticles%2Fexgmc4ccfxsh7xd6ijer.png" alt="Project Details View displaying technical breakdown, masked team roster, repository links, and rate-limited discussion comments" width="800" height="452"&gt;&lt;/a&gt;&lt;/p&gt;&lt;br&gt;Figure 9: Project Details View displaying technical breakdown, masked team roster, repository links, and rate-limited discussion comments.
  &lt;p&gt;&lt;/p&gt;

&lt;p&gt;&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.us-east-2.amazonaws.com%2Fuploads%2Farticles%2Foh574jn7ifj7j4by0joz.png" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.us-east-2.amazonaws.com%2Fuploads%2Farticles%2Foh574jn7ifj7j4by0joz.png" alt="Official Results Portal with podium winner showcases, track category champions, and toggleable embargo status" width="800" height="452"&gt;&lt;/a&gt;&lt;/p&gt;&lt;br&gt;Figure 10: The Official Results Portal with podium winner showcases, track category champions, and toggleable embargo status.
  &lt;p&gt;&lt;/p&gt;

&lt;p&gt;&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.us-east-2.amazonaws.com%2Fuploads%2Farticles%2Fsht7urzytyn8p3wr86cu.png" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.us-east-2.amazonaws.com%2Fuploads%2Farticles%2Fsht7urzytyn8p3wr86cu.png" alt="Event Settings Console with interactive date-time pickers, rubric weighting, and double-confirmation protection" width="800" height="452"&gt;&lt;/a&gt;&lt;/p&gt;&lt;br&gt;Figure 11: The Event Settings Console with interactive date-time pickers, rubric weighting, and double-confirmation protection.
  &lt;p&gt;&lt;/p&gt;

&lt;p&gt;&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.us-east-2.amazonaws.com%2Fuploads%2Farticles%2Fciui8tiyh70f3qzb99su.png" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.us-east-2.amazonaws.com%2Fuploads%2Farticles%2Fciui8tiyh70f3qzb99su.png" alt="Team Management console displaying cryptographic credentials, member rosters, and team invite codes" width="800" height="452"&gt;&lt;/a&gt;&lt;/p&gt;&lt;br&gt;Figure 12: The Team Management console displaying cryptographic credentials, member rosters, and team invite codes.
  &lt;p&gt;&lt;/p&gt;




&lt;h2&gt;
  
  
  7. Verification &amp;amp; Testing Runbook
&lt;/h2&gt;

&lt;p&gt;To guarantee total reliability, PrismJudge is validated through five independent verification layers before every deployment.&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight shell"&gt;&lt;code&gt;&lt;span class="c"&gt;# Gate 1: TypeScript typechecking and bundle distribution build&lt;/span&gt;
npm run build

&lt;span class="c"&gt;# Gate 2: Internal unit and integration regression test suite&lt;/span&gt;
npm &lt;span class="nb"&gt;test&lt;/span&gt;
&lt;span class="c"&gt;# Result: 81 / 81 PASS across 12 test suites (0 failures, 0 skipped)&lt;/span&gt;

&lt;span class="c"&gt;# Gate 3: Comprehensive 641-point role and route matrix crawler&lt;/span&gt;
node tests/audit_script.mjs
&lt;span class="c"&gt;# Result: 641 / 641 assertions PASS across all 4 personas&lt;/span&gt;

&lt;span class="c"&gt;# Gate 4: End-to-end containerized browser &amp;amp; API verification suite&lt;/span&gt;
node tests/comprehensive_e2e_audit.mjs
&lt;span class="c"&gt;# Result: 89 / 89 checks PASS against live Docker container&lt;/span&gt;

&lt;span class="c"&gt;# Gate 5: Official competition acceptance checker&lt;/span&gt;
docker compose build &lt;span class="o"&gt;&amp;amp;&amp;amp;&lt;/span&gt; docker compose up &lt;span class="nt"&gt;-d&lt;/span&gt;
python3 run.py .dogfood.toml
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;h3&gt;
  
  
  The Official Acceptance Report (&lt;code&gt;acceptance-report.txt&lt;/code&gt;)
&lt;/h3&gt;



&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;DOGFOOD 2026 acceptance report
portal: http://localhost:8080
claimed: T1 T2
fixtures: fixtures.json

T1  gallery is public ................. PASS
T1  project from fixtures shown ....... PASS
T1  closed event refuses submissions .. PASS
T2  judge sees own scores ............. PASS
T2  judge cannot see peer scores ...... PASS
T2  participant blocked ............... PASS
T2  csv export works .................. PASS

claimed T1 T2, verified T1 T2
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;






&lt;h2&gt;
  
  
  8. Conclusion &amp;amp; Lessons Learned
&lt;/h2&gt;

&lt;p&gt;Building &lt;strong&gt;PrismJudge&lt;/strong&gt; for DOGFOOD 2026 reinforced three foundational software engineering truths:&lt;/p&gt;

&lt;ol&gt;
&lt;li&gt;
&lt;strong&gt;Boring Architecture Wins:&lt;/strong&gt; In a 72-hour competition, eliminating distributed systems complexity (separate frontend containers, message brokers, external databases) enabled us to move faster, debug deeper, and deliver microsecond query performance. An in-process SQLite WAL database inside Node.js 22 was faster and more reliable than any multi-container setup.&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Mathematical Defense is Non-Negotiable:&lt;/strong&gt; A hackathon platform cannot simply "average the numbers." Implementing Empirical Bayesian Shrinkage solved real-world edge cases like judge &lt;code&gt;jdg_07&lt;/code&gt;'s zero-variance singularity and transformed evaluator noise into trustworthy rankings.&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Adversarial Verification Uncovers the Truth:&lt;/strong&gt; Hiding buttons in UI templates does not stop an attacker with &lt;code&gt;curl&lt;/code&gt;. Writing custom penetration scripts that aggressively probe IDOR endpoints, simulate stale browser caches, and verify print layouts is what separates amateur prototypes from production software.&lt;/li&gt;
&lt;/ol&gt;

&lt;p&gt;PrismJudge proves that a modern web application can be offline-first, mathematically sound, beautifully designed, and rock-solid under adversarial testing—all booting in under one second.&lt;/p&gt;




&lt;p&gt;&lt;em&gt;Authored for the DOGFOOD 2026 Hackathon Platform Competition.&lt;/em&gt;&lt;br&gt;&lt;br&gt;
&lt;em&gt;Repository: &lt;a href="https://github.com/jhanikhilnath/PrismJudge_DogFood_Hack" rel="noopener noreferrer"&gt;https://github.com/jhanikhilnath/PrismJudge_DogFood_Hack&lt;/a&gt;&lt;/em&gt;&lt;/p&gt;

</description>
      <category>webdev</category>
      <category>node</category>
      <category>typescript</category>
      <category>showdev</category>
    </item>
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