Main-content extraction commonly precedes language identification (LID) and determines its input in web-corpus pipelines. We measure this dependency on 7,808 WebMainBench pages using three extractors and three LID systems. We compare human-selected main text with annotated-region and full-page extractor outputs. A context flip occurs when annotated-region extraction preserves the route on human-selected text but full-page extraction changes it. For pages with nonempty text and matching human-selected and annotated-region routes, context-flip rates are 1.10–3.20%. On the common eligible set, Magic-HTML has lower paired rates than Trafilatura for all three LID systems, with each corresponding 95% interval excluding zero. All-page route loss counts empty outputs or routes that differ from the route on human-selected text. Magic-HTML’s empty full-page outputs add 5.84 percentage points. The fixed snapshot’s coarse non_en–en difference is positive in all nine configurations and remains positive after standardization to the page-complexity field (meta.level). A 100-page audit finds 35 unsupported, 29 supported, and 36 indeterminate routes. Of 35 unsupported flips, 26 move from predicted English to another route, and 17 involve changed content selection alone or in mixed cases. Among supported flips, 21 move to English. Extraction context can change text availability and downstream routing.

Systems that return several valid derivations require explicit choices about proof identity and proof-set correspondence. This study examines both choices on ProofWriter. An application-aware representation distinguishes grounded uses of the same source rule through canonical premise derivations. Source-ID projection collapses distinct reference proofs in 28 of 400 stratified derivable questions (7.0%). The comparison includes exact proof-set F₁, flattened-atom F₁, Optimal Subpattern Assignment with Jaccard distance (Jaccard-OSPA), and MCT-J. The label MCT-J denotes a balanced optimal-transport distance with Jaccard distance as the ground metric. On 160 fixed outputs from 32 derivable questions, source-ID projection lowers MCT-J distance by 0.0606 on average. A cluster bootstrap over the underlying ProofWriter worlds gives a 95% interval of 0.0316–0.0955. MCT-J equals Jaccard-OSPA on all 117 equal-cardinality outputs, while Jaccard-OSPA has the larger distance on 38 of 43 unequal-cardinality outputs. Of 433 generated proof members, 430 pass independent rule-level replay. Excluding the three replay failures changes the mean representation effect from 0.0606 to 0.0589. Proof identity and proof-set correspondence therefore change structural evaluation on fixed outputs.