Geometry from Brackets Consistency

Abstract

We argue for Brackets Consistency to be a `Pillar of Geometry', i.e. a foundational approach, other Pillars being 1) Euclid's constructive approach, 2) the algebraic approach, 3) the projective approach, and 4) the geometrical automorphism groups `Erlangen' approach. We proceed via a distinct Pillar 5) - Killing's computational route to Erlangen's groups - firstly by picking out the ensuing geometrically significant brackets subalgebras as consistent possibilities. Secondly, by considering which generators, arising piecemeal from different generalized Killing equations, combine brackets-consistently, recovering the affine-conformal and projective-conformal alternatives. Thirdly, we cease to rest on Killing's Pillar altogether, replacing it by mere polynomial ans\"{a}taze as source of generators. Even in this case - demanding (Lie) Brackets Consistency - suffices to recover notions of geometry, as indicated by the below alternative just dropping out of the ensuing brackets algebra, and is thus declared to be Pillar 6. The current article is further motivated, firstly, by the gradually accumulating successes in reformulating Theoretical Physics to run on the Dirac Algorithm for classical constraint algebraic structure consistency. Secondly, this is sufficiently driven by Brackets Consistency alone to transcend to whichever context possesses such brackets. Thirdly, we derive the conformal-or-projective alternative in top-automorphism-group $d \geq 3$ Flat Geometry. In a manner by now familiar in Dirac Algorithm applications, a) these two top geometries moreover arise side by side as roots of a merely algebraic quadratic equation. b) This equation arises from requiring that the self-bracket of a general ansatz for generators be strongly zero. c) For, if not, an infinite cascade of generators would follow, by which the ansatz would not support a finite Shape Theory.