The uncertainty bound

From [x̂, p̂] = iℏ and the Cauchy-Schwarz inequality on the Hilbert space, the Robertson-Schrödinger inequality gives:

ΔxΔp ≥ |⟨[x̂, p̂]⟩|/2 = ℏ/2

The Heisenberg uncertainty principle.

The factor of ½ comes from the Cauchy-Schwarz inequality — pure mathematics on the derived Hilbert space.

No physical assumption about sector splitting, action budgets, or phase space volumes.

The algebra does the work.

Sit with that for a moment.

The bound that governs every quantum measurement ever performed — the bound that tells you precisely how much you can know about a particle’s position and momentum simultaneously — falls out of the translation structure of a manifold that was derived from four axioms.

No one imported it. No one postulated it.

The manifold has structure. The structure has consequences. The consequence is ℏ/2.

The chain: Hilbert space (Chapter 1, from axioms) → Manifold (AS, from EH) → Spatial translations (smooth Lorentzian structure) → Stone’s theorem → ℏ as scale factor (forced above) → [x̂, p̂] = iℏ (algebra) → Robertson-Schrödinger (Cauchy-Schwarz) → ΔxΔp ≥ ℏ/2.

No link imports physics. No link assumes the uncertainty principle to derive it. No link uses phase space, action, or Hamiltonian mechanics.

The derivation runs on the axiom-derived structures and pure mathematics alone.