hbar-perfect graphs: exact certificates for simultaneous Pauli-variance violations, and three errata

the anti-heptagon C̄₇ is ħ-imperfect: seven explicit 3-qubit Pauli strings realise it as a frustration graph, α = 2, and a Gaussian-integer state gives Σᵢ ⟨Sᵢ⟩² ≥ 327090970/156225001 > 2

the anti-nonagon C̄₉ is ħ-imperfect, kernel-certified: α = 2 and β ≥ 2641054/1283689 = 2.0573939…

the 9-vertex graph of arXiv:2511.13531v1 Fig. fig:otherghs(b) is ħ-imperfect: α = 2 and β ≥ 45101177/22227852 = 2.0290389…

the 9-vertex graph of arXiv:2511.13531v1 Fig. fig:otherghs(c) is ħ-imperfect: α = 2 and β ≥ 1044367973/514699969 = 2.0290810…

negative controls: the C̄₇ witness proves β > 209/100 but not β > 21/10; the C̄₉ witness does not reach 2.06; toggling one edge of C̄₇ makes the frustration check fail; α(C̄₇) ∉ 1, 3

boundary control on G₇, the one 7-vertex graph whose ħ-perfectness the source proves by hand (β(G₇) = 2): the same search reaches Σ⟨Sᵢ⟩² > 199/100 and does not cross 2

erratum: the second graph of Table tab:g9 of arXiv:2511.13531v1, as drawn, has α(G,ω) = 3 for the printed ω = (1,…,1,2), not the printed 2

erratum: the third and fourth graphs of Table tab:g9 are drawn identically but are given different SDP upper bounds, 2.00175 and 2.00118

erratum: C̄₉ is ħ-imperfect and contains neither C̄₇ nor G₈ as an induced subgraph, so the sentence at arXiv:2511.13531v1 appendix.tex line 690 ('all ħ-imperfect graphs with no more than 9 vertices have either C̄₇ or G₈ as an induced subgraph') is false as written

G₈ = arXiv:2511.13531v1 Fig. fig:otherghs(a) is ħ-imperfect at the weighted facet ω = (1,1,1,1,2,1,2,1) with α = 3 and β ≥ 167391146/54982225 = 3.0444592…, while the same witness gives nothing at the uniform facet

arXiv:2511.13531v1 Fig. fig:otherghs(e) is ħ-imperfect at the weighted facet ω = (1,1,1,2,1,1,1,1,1) with α = 3 and β ≥ 315096058/103489929 = 3.0447026…, and the same witness gives nothing at the uniform facet

uniform-facet sweep: among all 1897 nine-vertex graphs with α(G) = 2, exactly 69 have β(G,1) > 2 — 66 contain an induced C̄₇ and the other three are C̄₉ and Fig. fig:otherghs(b),(c) — so no new ħ-imperfect graph exists in that class at the uniform weight vector, and in particular no member of the source's 78 undetermined graphs with α(G) = 2 is caught there

a new sufficient criterion for ħ-perfectness at the uniform facet: if Ḡ is triangle-free and every two disjoint Ḡ-edges span an odd number of Ḡ-edges, then β(G,1) = α(G,1) = 2

the source's undetermined cell is not Bernstein-shaped: deciding one of the 78 is block-positivity of an explicit 72×72 or 144×144 Hermitian ℤ[i] matrix at the boundary of the cone, and the box route needs 33⁹ ≈ 4.6e13 root coefficients and cannot work at any budget because the bound is attained

each of the 11 distinct largest-gap graphs of Table tab:g9 has exactly ONE facet normal of STAB(G) with full support, and β = α there to 1e-14 — so deciding each of them is a single tight inequality, not a facet enumeration

First exact upper bounds on a beta number via the shared SOS/PSD checker: beta(C5-bar,1) <= 103/50 (below the source's theta(C5-bar) = 2.2360679...) and beta(C7-bar,1) <= 14/5 on the family's own obsH7, with psiH7 as the exact refutation point (ratio 447155038123/408831003403)

the first exact upper bounds on an anticycle beta number that are below the source's own theta bound: β(C̄₇,1) ≤ 421/200 = 2.105 < ϑ(C̄₇) = 1+1/cos(π/7) = 2.1099162642…, and β(C̄₅,1) ≤ 1029/500 = 2.058; with the family's own HP1 witness this pins 327090970/156225001 = 2.0937171893… ≤ β(C̄₇,1) ≤ 2.105. The certificate is the level-2 SOS relaxation of (†) made affordable by an exact symmetry reduction: the 448 × 448 Gram matrix over the monomials cₐ c_b yₖ is shown to be Ô (Z ⊗ 1) Ô for an explicit orthogonal signed-permutation Ô and a 28 × 28 Hermitian Z, so only 28 × 28 is ever factored

the SOS hierarchy for β(C̄ₙ,1) mapped, level by level, with the level that the theta bound sits at identified: the block-matrix test ρ·1 − C ⪰ 0 is worth B = 3 at every n; SOS over the monomials cᵢ yₖ (equivalently, "ρ‖c‖²·1 − 2Σ cᵢ cᵢ₊₁ Rᵢ is an SOS matrix") has threshold bracketed at (2.23605957, 2.23606873) for n = 5 and (2.1099121, 2.1100586) for n = 7, each containing ϑ(C̄ₙ) and nothing else — so the theta bound is the level-1 SOS-matrix bound here; and level 2, over cₐ c_b yₖ, has threshold (2.05474396, 2.05476379) for n = 5 and (2.10097656, 2.10126953) for n = 7. Also measured: a naive alternating-projection solve in the full 120-monomial basis over-states the n = 5 level-2 threshold by 0.098 (it reports 2.1524 where 2.058 is certified exactly), so a large-basis alternating-projection threshold is an upper bound on the threshold and never the threshold — which is what the n = 5 "on the boundary at B = 2.1" reading of journal/2026-09-02-sos-checker.md §5 was