Toward a Geometric-Dynamical Reformulation of the Goldbach Partition Function

This technical report develops a computational-geometric certificate framework for Goldbach-type incidence. It does not claim a proof of the strong Goldbach conjecture. Starting from finite Goldbach maps and billiard-inspired toy models, the project identifies limitations of simulation-first hitting approaches and moves toward certificate-first constructions. The main developed direction is a holonomy-based framework in which geometric objects are generated first, holonomy vectors are measured, and integer labels are decoded by a fixed additive functional. A certified holonomy composition then gives a sound implication A=p+q at the decoded-label level. Computational experiments show that the flat-torus toy holonomy route is positive under restricted targets, robustness tests, and simple null controls. More saddle-like slit-torus models are parameter-sensitive: a constraint-ladder audit and reconciliation tests show Level-3 endpoint-clear plus slit-clear certificates can survive under matched parameter regimes, but broad robustness and a covering theorem remain unresolved. The report frames these results as a research program toward non-circular geometric certificates, not as a proof of Goldbach's conjecture.