Explicit bounds for the signless Brouwer conjecture at all k: realisability, a k-free certificate, and the anatomy of the obstruction

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We study the signless Brouwer conjecture (Ashraf–Omidi–Tayfeh-Rezaie): the inequality S_k(Q) ≤ m + C(k+1,2) for the sum of the k largest eigenvalues of the signless Laplacian Q = D + A of a graph with m edges. Building on the k=3 realisability framework (Venti, 2026), we carry the programme to general k, where the conjecture is open for 3 ≤ k ≤ n−3. Explicit bounds. We give explicit bounds S_k(Q) ≤ m + C(k+1,2) + ε_k at general k, all verified in exact rational arithmetic, with ε_3 ≤ 0.360043 and ε_4 ≤ 0.465454. Measured against the conjectured threshold, the additive excess is smaller than Lew's previous signless bound C(k,2) + (4k−2)·sqrt(k) by a factor 52 at k=4; to our knowledge these are the sharpest explicit signless bounds of this shape at fixed k ≥ 3. A k-free certificate. The realisability constraint G ≼ I is a rank-k orthogonal projection for every k, and the hard core of the k=3 certificate — the pairwise verifications, the Bernstein subdivision, the corner lemma — is independent of k. The machinery therefore transfers structurally rather than by re-derivation. We show the constraint is complementary to, not subsumed by, the scalar interlacing bound of Sun–Min–Das: it distinguishes graphs that the four-scalar bound cannot, and on the extremal family the scalar bound grows linearly. The certificate understood. The optimal certificate has a rank-one multiplier and six named active constraints, and admits an exact closed form at the extremal profile attaining C(k+1,2) at every k (verified k = 3..10). The per-graph statement sharpens to S_k(Q) ≤ m + C(k+1,2) − Ψ(G) with Ψ ≥ 0 a computable deficit. Anatomy of the obstruction. The plateau obstructing a proof is a one-sided kink, not a smooth barrier; its degeneracy lives entirely on the invisible mass-1 and mass-0 classes. On the extremal family the threshold decomposes exactly by pair type, C(k+1,2) = C(k−1,2) + (k−1) + 1 + (k−1), with the boundary-layer term carrying a full third of the total at k=3; we show this decomposition is a property of the extremal family, not a universal identity. Finally, every two-point realisability constraint — the exact pairwise domain, the Schoenberg hierarchy, the rank-one G ≼ I multiplier, the 2×2 block domain, and three-point moments — is shown either used or measured inert, so closing ε_k to zero requires an ingredient of a different nature. The conjecture at general k remains open; we delimit precisely what a proof along this route cannot come from.

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