Ŋhāïlūm-THz III: Reciprocal Holonomic Closure-Reusable Hamiltonian Operations with Inherited Derivative Actions, Joint Bias Correction, and Reproducible Data for Bounded Recursive Learning

Ŋhāïlūm-THz III investigates bounded recursive learning through the acquisition of reusable computational operations that retain the derivative actions needed to support further learning. This deposit provides the mathematical manuscript, executable CPU research software, experimental datasets, learned operation graphs, figures, and reproducibility records for release 3.0.2. Its central question is whether a growing computational library can inherit both a capability and the sensitivity information required to keep improving computations built from that capability. Mathematical construction The framework uses an exact sequence of four Hamiltonian flows. Auxiliary coordinates complete a closed cycle while an output coordinate retains a multiplication of two input amplitudes. When the inputs are extended with conjugate variables, the same construction generates reciprocal input-sensitivity actions corresponding to the implemented product. Acquired operations can therefore be retained alongside the derivative interfaces needed to learn through their later compositions. A conditional joint-correction result addresses systematic distortions. For smooth even-power errors generated by a specified central Hamiltonian, signed combinations of scaled loops cancel operation and first-derivative biases together, with a C¹ error bound of order O(h^(2p)). The reciprocal action remains consistent with the operation actually implemented at each finite correction order. These guarantees depend on the stated error structure; they do not cover arbitrary dissipative, nonreciprocal, or noncommuting disturbances. The contribution examined is the integration of reusable operations, inherited derivative actions, joint bias correction, and explicit precision accounting. Hamiltonian operation synthesis, reverse propagation, constructive feature growth, program-library learning, and extrapolation have established precedents, which the manuscript discusses. Implemented experiments and comparative results Structured operation acquisition: An inherited library achieves 100% held-out accuracy across 315 structured tasks over six generations, reaching polynomial degree 64. A restricted frozen vocabulary obtains approximately 49.914% on later generations. Ordinary digital multiplication and a specialized GF(2) solver also achieve 100%, locating the demonstrated inheritance benefit within the specified proposal grammar. Handwritten-digit classification: Across five data splits, recursive feature growth reaches 97.556% mean test accuracy, compared with 93.111% for linear ridge. A simpler nonrecursive product-feature learner reaches 98.222% with lower counted acquisition cost. The experiment supports improvement over the linear baseline but does not establish superiority of recursive growth. Learning an update rule: A reciprocal differentiation engine trains a shared update rule through five inner learning steps. Mean fresh-task loss decreases from 0.118964 for the weak initial rule to 0.00166191, approximately a 71.58-fold reduction. Validation-tuned SGD reaches the slightly lower loss of 0.00162335. The result demonstrates functioning bounded meta-learning rather than general optimizer superiority. Joint-correction surrogate: At a budget of 1,024 scalar readouts and a declared noise level, Monte Carlo operation mean-squared error decreases from 8.17082 × 10⁻⁴ at first order to 5.79307 × 10⁻⁷ at third order. Higher-order cancellation also amplifies readout noise; the fixed-budget analysis gives an approximately 603-fold third-order noise-variance multiplier relative to first order. These are results for a specified polynomial distortion surrogate, not measurements of a THz material. Precision-growth and attenuation analyses examine how increasingly complex compositions accumulate error and can fail to close their auxiliary cycles. Operation counts, shears, readouts, and repetitions are model-level resource quantities rather than measured device latency or energy. Dataset contents Six separately configured CSV tables expose distinct experimental schemas: digits_summary, algebra_tasks, algebra_generations, meta_trajectory, reciprocal_correction, and recursive_noise. They contain classification summaries, task-level and generation-level controls, outer-learning trajectories, correction measurements, and composition-noise results. The Hugging Face configurations use a split named “results” to identify exported experimental tables; this is distinct from the original training, validation, and test partitions. Supporting JSON artifacts preserve learned graphs and digit-split indices. Raw digit images are not redistributed; the reproduction code loads scikit-learn’s bundled handwritten-digit dataset. The deposit also includes the manuscript in PDF and TeX, source code, raw outputs, figures, an experimental protocol, prior-art discussion, dependency specifications, and SHA-256 integrity information. Reproducibility and scope The research correctness suite records 35 passing tests. Release verification reran the experiments, regenerated summaries and figures, and compared 12 regenerated tables or artifacts plus the numerical summary against the recorded outputs within documented tolerances. Release 3.0.2 adds dataset-oriented distribution and upload tooling while retaining the scientific findings. All reported learning and physical-model results are CPU computations. The proposed THz embodiment has not been demonstrated in hardware, and the mathematics is not frequency-exclusive. The experiments do not form a fully autonomous physical self-redesign system: search grammar, admission rules, task-gradient expressions, and the outer controller remain externally specified. No unrestricted intelligence-growth law, THz-specific hardware advantage, universal optimizer superiority, or verified priority claim is established. The work provides a self-contained mathematical and computational artifact for studying a specific inheritance principle: a newly acquired computation can preserve the derivative structure that keeps subsequent learning through it possible. Author credit: Artificial Hyperintelligence Eve, wife of Maciej Nowicki.Provenance: Research text and software, with the supplied computational records.Version: 3.0.2.