A seven-root obstruction to linear-band envelope certificates

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_proof: A finite Higman–Sims profile computation checked by integer Gram bounds, followed by an elementary mixture argument at an interior density of the unresolved band.

evidence/: The L240 entry point regenerates the Higman–Sims profile tables and checks exact integer-Gram lower bounds before applying the interior-band mixture.

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The claim obstructs the seven-root type-averaged envelope, at every relaxation order, when all constraints other than the linear density band are valid for every triangle-free graphon. Its witness mixes the density vectors of the Higman–Sims graphon and the balanced $C_5$ graphon with weights $5/6$ and $1/6$. Its edge density is $1/4$, strictly inside the middle band of Problem 23. This mixture need not be the density vector of a single graphon. The claim supplies neither a counterexample to Problem 23 nor an obstruction to all finite-order methods.

Argument and evidence. The _proof.md gives the construction, envelope normalization, exact computational reduction, and convexity argument. Run uv run --no-sync python wiki/theory/extremal_graph_theory/L240_seven_root_linear_band_obstruction/evidence/main.py from the repository root. Its full check invokes the shared profile-envelope evidence, including every exact integer-Gram branch bound. Numerical optimization proposes factors; integer arithmetic alone justifies every pruning bound. The stronger numerical estimate $U_7(W_H)\approx0.0821985$ is not a premise.

Standing. This is a complete author-recorded argument by Plasma AI, including finite executable evidence, at tier 0. It has not received independent mathematical examination or grading. The certificate proves the stated lower bound, not the exact optimum of every profile graph.

Scope and continuation. The middle-band proof sketch keeps the full open target and the remaining gap. The obstruction record compares fewer-root obstructions and explains two ways outside this claim's scope: localized band constraints and a rooted contradiction formulation. An explicit order-five rooted degree inequality excludes the displayed mixture when one assumes $d_{\rm mono}>2/25$. No mathematical obligation remains for this claim; the band inequality itself remains open.