The τ = 2 Conjecture, the Replication Conjecture and the f-Flowing Conjecture, and the classification of binary matroids with the sums of circuits property are foundational to Clutter Theory and have far-reaching consequences in Combinatorial Optimization, Matroid Theory and Graph Theory. We prove that these conjectures and result can equivalently be formulated in terms of cuboids, which form a special class of clutters. Cuboids are used as means to (a) manifest the geometry behind primal integrality and dual integrality of set covering linear programs, and (b) reveal a geometric rift between these two properties, in turn explaining why primal integrality does not imply dual integrality for set covering linear programs. Along the way, we see that the geometry supports the τ = 2 Conjecture. Studying the geometry also leads to over 700 new ideal minimally non-packing clutters over at most 14 elements, a surprising revelation as there was once thought to be only one such clutter. CONTENTS AHMAD ABDI, GÉRARD CORNUÉJOLS, NATÁLIA GURICANOVÁ, AND DABEEN LEE 5.1. Products and coproducts of clutters 31 5.2. Products and coproducts of sets 32 5.3. Reflective products of sets 33 5.4. Strict connectivity and the R k,1 's 36 6. The spectrum of strictly non-polar sets of constant degree 37 6.1. Proof of Theorem 1.20 37 6.2. Generating strictly non-polar sets of degree at most 4 43 7. Concluding remarks and open questions 46 Acknowledgements 48 THE STRICTLY NON-POLAR SETS OF DEGREE AT MOST 4 AND DIMENSION AT MOST 7 The strictly non-polar sets of degree at most 4 and dimension at most 7, ordered according to (degree, dimension) and by size (0,3) ("100" "010" "001" "111")