From Coefficient Differentials to General Safe Operations
The Post-Trade Elasticity Model was initially developed around a small set of actions: Reserve Asset swaps, single-asset liquidity, proportional liquidity, and LP Token burns. Those actions were proved to satisfy the coefficient order individually and then composed into more elaborate transactions such as Surplus settlement.
The coefficient safety conditions reveal that the known actions are not the whole design space. They are particular paths through a larger region of admissible state transitions.
The differential of each coefficient identifies the local coefficient-order directions. Its exact finite form identifies the complete endpoint envelope induced by that order. The envelope admits direct permanent-reserve allocations, general LP under-minting, more general withdrawals, and multi-asset liquidity operations that need not share one common complement multiplier.
Execution pricing enters at a different layer. Coefficient order determines whether an endpoint is admissible. The execution rule determines whether value-flow balance reaches such an endpoint automatically. For positive scale elasticity, post-trade execution is the unique fixed linear execution rule that makes every finite fixed-coefficient reserve-only swap coefficient-safe without an additional admissibility check or protocol subsidy.
This article develops those results from first principles. It assumes exact arithmetic, positive reserves and scales, and homogeneous elasticities unless a section states otherwise.
The article describes the mathematical design space. The newly identified transitions are candidates for protocol operations, not claims about actions already exposed by the current implementation. A candidate becomes supported only after its consideration, account movements, rounding, and property tests have been specified and implemented.