Variational Backbone and Regime Closures I: Energy as a Variational Language A Density-Fixed Reading Protocol and a Scoped First-Derivative Backbone Representative

Energy is treated as a variational language: a local C2scalar functional E on configurations (g, IR, II ) whose local density is held fixed across regimes/readings. Assumingfirst-derivative locality, a coercive quadratic backbone, adjointable DI , and admissible localmetric dependence, we isolate—within a controlled equivalence class (boundary terms, rescalings/time reparametrizations, lower-order stabilizations, and linear reparametrizations)—ascoped convenient representative whose derivative spine is |∇RIR|2 + |DI II |2 with a minimal bilinear interface term ⟨IR, DI II ⟩. The representability claim is explicitly restricted tothe first-derivative coercive backbone within the stated equivalence moves. From the samefixed E we define three density-fixed readings: (R1) Euler–Lagrange stationarity; (R2) anL2 gradient flow defined via a Riesz representative; and (R3) a minimal time-completiontemplate preserving the fixed spatial density e(·). Novelty is not that GR or wave equationsadmit variational formulations, but that one fixed primitive E, under the declared readingprotocol and admissible eliminations, supports both the geometric metric closure (Part II)and the KG→Sch→HJ conservative corridor (Part III) as internal consequences, withoutadopting their standard axioms as additional axioms. Part II and Part III then provide thecorresponding functional-analytic formulations (including gauge fixing/normalizations whenneeded) and close the dissipative and conservative readings, respectively; interpretation isdeferred to Part IV.