A theory and a proof do different work. Theory proposes a structure capable of organizing what is known and revealing what should be sought next. Proof belongs to the measurements, derivations, and discriminating experiments that can decide among rival accounts.
The distinction matters because a cross-disciplinary theory can easily become greedy. Every spiral begins to look like confirmation; every resonance becomes the same resonance; every recurrence is treated as identity. That would destroy the very insight the work is trying to preserve.
The admission rule does not claim that a nautilus shell, a cortical wave, a galaxy, and a chemical spiral share one local mechanism. They do not. It claims that local mechanisms execute forms within a constrained geometric field of possibility. The recurrence of related forms across unrelated media is a reason to investigate that field, not permission to erase the differences among the media.
For that reason, the findings gathered here are witnesses rather than trophies. The folded cortex carries traveling spiral and concentric waves. A growing layered gel folds into sulci and gyri. Membrane mechanics can produce pearling and division. Topological braids can protect information through global relation. Each result stands first on its own measurement. Together they indicate where the geometry-first theory may have explanatory reach.
The hand-off is therefore precise. The theory proposes that geometry is not the residue left after substance has done the real work. Geometry establishes capacity. Form and function arise together. Physical systems select among admitted states. Living systems add enclosure, memory, copying, and lineage. Evolution runs upon the resulting field of possibility.
The experiments should ask where that account predicts more than the material-first alternative. If cortical folding sets the permitted large-scale modes, then individual differences in folding should be connected to measurable differences in those modes. If membrane geometry contributes causally to function, controlled changes in curvature should alter the relevant field, transport, or signaling behavior in reproducible ways. If the origin-of-life threshold depends upon physically generated enclosure and division, then inheritance should strengthen when those processes arise from the membrane’s own dynamics rather than from an imposed split.
The theory is offered in that spirit. It does not ask to be protected. It asks to be used.
The shape came first. The language followed. The proof belongs to whoever can make the shape answer back.