Marginal Saturation as a Variational Principle: The Gravitational Sector of the Record Geometry

Abstract

The companion papers leave this framework’s internal geometry without a stabilising sector. This note opens that sector under a registration whose criteria pre-date every candidate, and reports one positive result, four closures, and a boundary. The positive result. The capacity rule itself selects the internal shape. Because the K = 5 shell’s charge second moment is exactly ⟨q²⟩ = 8, the self-consistent budget becomes geometric, B(ζ) = 16 + 8/ζ, and carrier admission confines the shape to ζ ≤ 7/6. Minimising capacity subject to admission is a constrained optimisation with a strictly positive Karush–Kuhn–Tucker multiplier ν = 8/7, so the constraint is active and the solution globally unique: ζ* = 7/6 exactly, B* = 160/7, s* = 21/668 — a global constrained variational principle, not a local field equation, containing no fitted quantity. It scores 0.197 points against a criterion of 0.2 committed beforehand and survives; a covariance-controlled discriminator, now complete, separates it from the round configuration and currently favours the latter at about 7.7:1 once each geometry is allowed to locate its own crossing scale — so the paper’s own candidate is disfavoured though not excluded, but an evolution is being draft and yet to be presented. The closures. The installed Einstein–Maxwell branch has no stable stationary point (exact mixed Hessian signature on the whole positive branch); its configuration destroys the truncation producing the carrier; its calibrated radius is refuted by the gauge concordance under a zero-parameter test; and the capacity sector contains no running coupling. The boundary. The record enters through a dimensionless heat weight, so its predictive content occupies exactly the dimensionless a₄ slot of the induced expansion, while Newton’s constant and the vacuum energy sit in the dimensionful a₂ and a₀ slots. An explicit audit confirms the internal radius cancels from every prediction. Gravity is therefore not missing from this framework: it is the curvature of the background on which the record sits, and was assumed at the outset. What survives as an empirical claim is dimensionless — the gauge-matching and quartic-zero scales must coincide, predicting mt(mt) = 162.1 ± 0.4 GeV.