Can positive geometry fix the Standard Model? Evidence from a Lean-verified agentic research programme (Paper II: the evidence)

Positive geometries, such as the amplituhedron and the associahedron, encode scattering amplitudes without reference to spacetime or a Lagrangian. We ask whether a single such structure can fix a dimensionless quantity of the Standard Model with no fitted parameters, and we report what a human-directed, AI-agentic research programme has established so far. Every claim carries a status tag: proved in Lean 4, proved on paper, computed in a preregistered simulation with locked checkers and planted controls, cited, or stated as a conjecture, a speculation, a record or a reviewer judgement; the method is the subject of the companion Paper I. The evidence spans physics and mathematics. In physics, a blind preregistered test of Lovelace-Shapiro coupling ratios against measured resonance widths passed within its bands. A one-loop colour-factor no-go shows that, in the cubic Tr φ³ colour model, no colour group SU(N) with N ≥ 3 makes the triangle-type and crossed-box colour factors sign-coherent. Renormalisation-group consistency narrows the top Yukawa coupling to a window but does not select a value, and one electroweak coupling constraint is reproduced, conditionally, from amplitudes. In mathematics, we give exact closed forms for the Ollivier-Ricci curvature of associahedron flip graphs, whose mean changes sign at the decagon. We also state as a conjecture an equality between Ollivier and capped Forman curvature, which holds on every one of the 140,107 edges of the 513 simple polytopes we tested. Together these form a map of where positive geometry reaches the Standard Model and where it does not yet: a Lean library of 1,378 theorems with no unproved steps or extra axioms, 77 mapped routes, and an exclusion ledger of proved no-gos. Two patterns run across the routes, and we offer them as conjectures: the geometry tends not to see quantum numbers, and consistency yields windows rather than points. Deriving a quantum number is the open target.