Paper 62BD: Effective Field Equations of Holosphere Gravity From the Conventional Einstein Tensor and Bianchi Target to a Support-Source Tensor, Correction Ledger, and Readability-Gated GR-Limit Field Equation

Paper 62BD formulates the first effective field-equation bridge for Holosphere gravity. It follows Paper 62BC, which assembled the branch-bounded GR-limit recovery stack. The purpose of Paper 62BD is to organize that stack into a single effective tensor equation with a divergence-safe source side, not to claim a completed all-order derivation of general relativity.

The conventional target is the Einstein field equation, where the Einstein tensor is proportional to the stress-energy tensor, together with the Bianchi identity and source conservation. Paper 62BD translates this target into Holosphere language by defining an effective geometry, a support-source tensor, an effective coupling, and a correction tensor. The proposed effective Holosphere field equation is G_eff = kappa_H T_support + C_H. Here G_eff is the Einstein-tensor-like curvature of the support-loaded effective metric, T_support is the support-source tensor, kappa_H is the effective coupling, and C_H is the correction tensor.

The correction tensor is treated as a source ledger, not as a free fitting term. It accounts for off-corridor burden, delayed support response, macroscopic redistribution, unresolved burden, and branch-transition burden. If correction channels are active, they must be named and included in the conservation balance. They cannot be hidden inside the field equation.

The central consistency condition is that the total source side must be divergence-safe. In normalized form, the support-source tensor plus the correction tensor divided by kappa_H must have zero Holosphere covariant divergence. This is the Holosphere analogue of the Bianchi/source-conservation condition in general relativity. Without this condition, the equation would only be a formal metric expression rather than a viable field-equation bridge.

The clean readable branch is defined by open readability, controlled correction burden, a support-source tensor that maps to ordinary matter, and an effective coupling that approaches the GR coupling. Under those assumptions, the effective Holosphere field equation approaches the Einstein form to the declared order. The paper also checks that the field-equation bridge projects correctly to the weak scalar source law, the compact spherical exterior burden, the Schwarzschild-compatible second-order loading branch, the perturbation branch, and the weak rotating-source branch.

The result is bounded. Paper 62BD establishes organization and consistency, not microscopic origin. It does not derive Newton’s constant from first principles, does not derive the full Einstein field equations all-order, does not solve full Kerr, black-hole thermodynamics, Hawking radiation, singularity resolution, numerical relativity, or full gravitational-wave phenomenology. Its contribution is to put the Holosphere gravity stack into one divergence-safe, readability-gated effective field-equation form and to identify the next open tasks: deriving the coupling, deriving the correction dynamics, and finding a possible Holosphere action principle.