Topology constrains, under a gap, what geometry and spectrum then determine: a programme for testing where topology drives physics, with a verified history, a measured first node, and the tools to continue

'Physics is driven by topology' is a slogan with three centuries of results behind it and no testable content as written. This programme paper states the programme that tests it: a family of statements in which an integer invariant of a bulk determines a discrete fact at a boundary, robust inside a class of perturbations and failing outside it, each node carrying the theorem (kernel-checked in Lean 4 where reachable), a preregistered measurement, and the measured limit. A verified history (Euler to hyperbolic circuits) and a verified map of what is proved, measured and refuted in each domain lead to the defensible thesis: topology constrains, under a gap, what geometry and spectrum then determine. A counter-programme is already further advanced than the thesis: on the discrete inverse conductance problem geometry and spectrum decide and the topology of the data does not (the author's earlier records). The first thesis node, the finite Su-Schrieffer-Heeger chain, is measured: the invariant fixes the side of the end mode, chiral symmetry fixes its energy, the gap fixes the margin (a third of the gap for on-site terms, a fifth for same-sublattice hopping); the two residue integrals of the chain's symbol are machine-checked against Mathlib, with the remaining links to the winding number listed as not formalised. Research and experimental directions for seven nodes (three of them controls without an invariant), and a verified map of tools with a proposed topology solver and simulator pipeline for classical-wave laboratories, close the paper. Every failed gate and refuted prediction of the base is kept. Nothing here concerns holography.