Modular Flatness and Gauge Reduction in Relational Operator Algebras

We isolate the local operator-algebraic part of relational spacetime reconstruction and determine exactly which assumptions turn modular transport into a flat physical translation system. A pair of positive null generators scaled inversely by one modular group yields a positive-energy \(1+1\)-dimensional Lorentz seed, but inverse scaling alone does not imply commutativity: an explicit opposite half-sided modular inclusion closes to \(\mathfrak{sl}(2,\mathbb R)\). We prove two sufficient flatness mechanisms. The first requires agreement of the two characteristic paths on a generating family containing both characteristic sectors and the complete corner sector. The second uses localization of the canonical diamond implementer in shrinking corner algebras with scalar intersection. We then construct a nonempty flat type-\(\mathrm{III}_1\) branch from two chiral conformal components. Finally, without assuming a bulk-edge tensor product, we show that a pure-gauge characteristic diamond vanishes on the fixed-point algebra, while a nontrivial boundary quotient class survives as physical edge holonomy; the BFV analogue states that an exact commutator induces commuting derivations on cohomology. The results provide a local flatness and gauge-reduction theorem, not a derivation of a \(3+1\)-dimensional gravitational dynamics.