Isoperimetric stability uniformly in codimension

How close must a nearly isoperimetric boundary be to a round sphere when the number of ambient directions is allowed to grow? This article establishes a quantitative guarantee that remains uniform from finite-dimensional Euclidean spaces to arbitrary real Hilbert spaces. It lets researchers pass to increasing-dimensional approximations without losing the strength of the deficit-to-distance estimate. The geometric quantities. Let \(T\) be a nonzero finite-mass closed integral \((n-1)\)-current in a real Hilbert space \(H\), with fixed \(n≥2\). Write \( M\) for current mass and \(ω_n\) for the volume of the Euclidean unit \(n\)-ball. The true integral filling mass and its associated radius are \[ F_H(T)=inf∂ Q=T M(Q), r_T=(F_H(T)/ω_n)1/n. \] The infimum runs over integral \(n\)-currents in the same space. Define the normalized deficit and filling asymmetry by \[ D_H(T)={ M(T)}{nω_n r_Tⁿ⁻¹}-1, A_H(T)=r_T⁻ⁿinf_B F_H(T-∂ B), \] where \(B\) ranges over oriented affine flat \(n\)-disks of radius \(r_T\). Thus the comparison disk has the same mass as the least filling of the original boundary. A global estimate uniform in codimension. For every fixed \(n≥2\), there is \(c_n>0\) such that \[ { D_H(T)≥ c_n A_H(T)^2} \] for every such cycle in every real Hilbert space. The constant depends on the intrinsic filling dimension, but not on the ambient dimension. Equivalently, the normalized filling distance is at most \(c_n-1/2√ D_H(T)\). The theorem requires no bounded support, first moment, or attainment of the filling infimum. The distance is an integral filling distance between boundaries, rather than a Hausdorff distance between their supports. Why equality rigidity controls uniform stability. For \(n≥3\), the paper proves an equivalence between a single stability constant valid in all finite ambient dimensions and round-boundary rigidity for normalized compact Hilbert equality pairs. It establishes this rigidity directly: an equality boundary is the consistently oriented boundary of a flat unit disk. Selected near-extremizers can then enter the local quadratic comparison with constants that retain their intrinsic dependence. Reusable analytic and geometric tools. Finite-net alignment through Gram matrices produces strong norm compactness under uniform covering bounds, including rank loss. A whole-current Rellich argument retains integer multiplicity and explicitly controls cancellation from other projected sheets. Finally, cutting and capping an unbounded cycle before applying an integrated projection homotopy extends the estimate to all finite-mass Hilbert data without a first-moment assumption. Relation to prior work. Bögelein, Duzaar and Fusco (2015) established the quadratic filling-asymmetry estimate in every fixed Euclidean codimension; their local spherical-graph comparison already has intrinsic-dimensional constants. The present contribution is global codimension uniformity, its equality-rigidity mechanism and the full Hilbert-space conclusion. The manuscript explicitly credits the classical current, regularity, Sobolev and radial-comparison inputs used in its proof. The integrated 24-page article contains the complete proofs, two geometric figure groups and references.