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The Square Root Has Accomplices

Written 2026-07-26. Claims re-read against their sources on 2026-09-27: 7 checked, 5 confirmed, 2 wrong, 0 unverifiable. By claims-audit controls 2026-09-27 (audit agent + fix agent).

Both audit findings were re-read against gitcoin.co/mechanisms/quadratic-funding by the fix agent before editing. Verifier research/quadratic-funding/verify-quadratic-funding.mjs: 206 passed before, 206 passed after (the change is wording; no number moved).

Claims

• WRONG the finite-pool attack payout pool × 2W/(2W+H) follows the normalization "Gitcoin describes"
  https://gitcoin.co/mechanisms/quadratic-funding : Gitcoin squares each project's sum of square roots and says "The resulting values represent relative allocation weights and are normalized across all projects so that the total matching distributed equals the size of the matching pool": it normalizes the funding weight, the page normalizes the subsidy
• MINOR "Current Gitcoin rounds use Connection-Oriented Cluster Matching with Passport-based detection"
  https://gitcoin.co/mechanisms/quadratic-funding : the page says Passport scoring, COCM and pairwise analysis "may reduce, reallocate, or otherwise adjust final allocations"; it does not say every current round uses COCM
• CONFIRMED Buterin, Hitzig & Weyl, "A Flexible Design for Funding Public Goods", Management Science 65(11) 5171-5187 (2019)
  https://doi.org/10.1287/mnsc.2019.3337 : Crossref: vol 65, issue 11, pp. 5171-5187
• CONFIRMED Buterin, "Pairwise coordination subsidies", Ethereum Research, 4 June 2019; per-pair subsidy 2MW/(M+W) < 2M
  https://ethresear.ch/t/pairwise-coordination-subsidies-a-new-quadratic-funding-design/5553 : post by vbuterin created 2019-06-04; text gives exactly 2MW/(M+W) < 2M and k(k-1)/2 pairs
• CONFIRMED Jiajia, Bernardineli & Emmett, "How to Attack and Defend Quadratic Funding", BlockScience, 10 March 2021
  https://blog.block.science/how-to-attack-and-defend-quadratic-funding/ : article:published_time 2021-03-10; all three authors named
• CONFIRMED Gitcoin says pairwise adjustments and COCM mitigate but do not eliminate the risk
  https://gitcoin.co/mechanisms/quadratic-funding : "Pairwise matching adjustments and COCM mitigate but do not eliminate this risk"
• CONFIRMED funding weight (Σ√c)² and subsidy (Σ√c)² − Σc; n identities splitting C give nC weight and (n−1)C subsidy
  research/quadratic-funding/verify-quadratic-funding.mjs : algebra re-derived; the verifier builds the arrays directly

What was done

• [fixed] the two-colluders card no longer says Gitcoin describes this normalization: it says scaling the subsidy is the model's choice and that Gitcoin describes normalizing the funding weight, a different rule; README scope paragraph reworded to match; dated correction line added to the page's source list and README
• [fixed] "Current Gitcoin rounds use COCM with Passport-based detection" (limits list) and "Gitcoin's recent rounds use..." (production details) now report Gitcoin's own "may reduce, reallocate or otherwise adjust" wording; the src/content summary and README caveat softened the same way