Reciprocal Holonomic Closure - Learning new operations while preserving the physical means to differentiate them

We construct an operator-growth architecture around an exact Hamiltonian phase loop. A four-pulse sequence closes an auxiliary canonical pair while trans lating an output coordinate by a product of two input amplitudes. Including the input conjugate variables makes the same sequence carry the exact derivative action: an output adjoint produces the corresponding two input adjoints. Thus an acquired composition can retain its learning interface rather than become an opaque feature. The principal synthesis is a reciprocal correction compiler. For a declared central-Hamiltonian error family, a signed combination of scaled loops cancels leading operation errors and derivative errors together. At every finite order the derivative remains the exact derivative of the implemented operation. We prove the construction, its gain/noise cost, a conditional compositional-learning guaran tee, and the precision cost of recursive multiplication. These are applications and combinations of established Hamiltonian, automatic-differentiation, phase-cycling and extrapolation mathematics, not claims to invent those ingredients. The implementation explicitly advances the flow coordinates and passes 35 de terministic tests. On 315 supervised compositional tasks over five curricula, an inherited operator library learns six generations through degree 64 with 100% held-out accuracy; a restricted degree-two control obtains 49.914% on later gen erations. Ordinary digital multiplication and a known-parity solver match the suc cessful result. On five 10-class handwritten-digit splits, recursive feature growth obtains 97.556%, versus 93.111% for linear ridge but 98.222% for simpler non recursive products. A reciprocal-loop differentiation engine trains a shared up date rule and reduces fresh-task loss 71.58-fold relative to its weak initialization; validation-tuned SGD is slightly better. All results are CPU simulations. No THz hardware performance, universal optimizer advantage, unrestricted RSI, verified global priority, or T9-level discovery is established.