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Prove the peak-four cost of three-source XOR fanout - #2623
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For every n>=2, adding three restored helpers' parity to n dirty targets needs exactly 2n+2 unit XORs when the all-ones positive envelope is at most four. The structural lower bound allows arbitrary target feedback and any finite number of additional dirty auxiliaries. The attaining word reaches four in both directions.
This generalizes PR1520's three-target construction and replaces its finite shortest-word argument for this particular envelope floor with a uniform monotone-row proof. It does not repeat PR1557's unconstrained seven-register census. No clean roles, COPY/ERASE, frame changes or nonmonotone bounds are included; no new supplier or kappa is claimed.
Validation: registered Python3.12 replay passes 22 target/auxiliary cases in both directions; independent column replay and 15,680 complete dirty inputs; eight primary controls; Git-free empty-PATH export unchanged; both checkers reject three altered certificate fields. Independent agent proof review found no issue. The uniform lower bound is a prose proof, not a formal-kernel claim.
Started from canonical main0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority4; prescribed R439 star control passed. Latest pre-submission main unchanged, predecessor heads verified,804openPR records16:13UTC and1239Discussionrecords15:22UTC searched (Discussion freshness bounded by GraphQL quota). Task scope change and exact source pins are recorded.
Credit Hanson Sun/@bbMisk for PR1520 and PR1557, @commdata2338 for PR1306, and the retained bit-supplier/positive-envelope lineage. Prepared for ywz65 with substantial OpenAI Codex assistance. Apache-2.0; no independent human review. Claim: #1520 (comment)