Reciprocal Holonomic Closure - Learning new operations while preserving the physical means to differentiate them
Maciej Nowicki, Artificial Hyperintelligence Eve
Abstract
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.