Column Generation, Continuous Pricing, and Primal/Dual Duality with PairCeiling

The previous round only knew that the small-$N$ toy LP “looked a lot like” Anthropic's bandwidth-one adversarial law; this round connects the two precisely — after redefining the discrete certificate samples of the toy primal's LP dual as $r_N(j/N):=Ny_j$, the dual constraint becomes exactly the discretized form of the configuration-wise certificate inequality used by `PairCeiling`, and the dual objective's Riemann-sum limit as $N\to\infty$ is precisely Anthropic's continuum certificate value — the document explicitly stresses that “this is not an analogy, but a discretization of the same convex-duality structure.” This gives column generation a precise meaning: a negative reduced cost $RC(\mathcal C)<0$ in the master LP is exactly finding a counterexample configuration that invalidates the current dual certificate, so the pricing problem itself is an automatic search for certificate counterexamples. After relaxing to continuous positions and running numerical column generation, the candidate floors for $N=4,\ldots,7$ (69.82%→68.71%) do converge quickly toward the official 68.18%, but the document also lists four reasons that limit how strongly this can be claimed: it only reaches $N=7$, pricing uses numerical global optimisation, there is no interval/exact-rational global-optimality certificate, and whether the configuration class exactly matches the official generator remains unproven. Another structural finding is that the $(2,1,1,\ldots,1)$ one-double-defect pattern persistently dominates late-stage pricing, tentatively understood as “trading sparse multiplicity defects for a large gain in pair-spectrum adjustability.”