Boundary Certificates and an Information Barrier for a Six-Direction Affine Plank Problem on a Triangle

Abstract

Bang's affine plank conjecture asserts that any collection of planks covering a convex body K in R^d has total relative width at least 1. The conjecture remains open, including in the plane, where the best known dimension-wide lower bound is 2/(1+sqrt(d)). This paper does not address the unrestricted conjecture. Instead it studies a prescribed six-direction affine plank covering problem for a triangle, consisting of the three facet directions together with three specific nonconcurrent affine directions. We develop a cevian boundary-certificate framework, prove a general three-arm support theorem, and obtain a closed-form evaluation for the cyclic three-cevian family relevant to the six-direction seed. Combining these certificates with a location-sensitive linear relaxation, we prove the certified rational lower bound C ≥ 13505875189/14039754651 ≈ 0.9619737327844277 for the six-direction covering constant C of the triangle. We then show that the self-overlap inequality responsible for the final sharpening of this bound is itself sharp: a one-step boundary self-overlap coefficient equal to 1, and an induced rescue coefficient equal to 576, both admit exact rational witnesses attaining equality. The same collision mechanism saturates at every 144-adic scale without generating an independent second charge, and this saturation can be embedded arbitrarily close to (though never exactly coincident with) a genuine boundary equality configuration. Together these results identify a precise structural barrier: within the compressed boundary framework developed here, several natural strengthenings are simultaneously exhausted, and any improvement must exploit information absent from this framework's marginal/setwise transfer data. This is stated as a boundary information barrier theorem, with its exact scope and explicit exclusions. All numerical constants, linear-programming certificates and extremal witnesses are exact rational objects; no floating-point optimization enters any proof. Eleven deterministic exact-rational checker scripts are included as ancillary files.