SŌRĪLŪN: Exact Inverse Updates, Sharp Coefficient-Dependency Bounds, and Certified Recursive Self-Improvement - Theory, Compiler Optimization, and Reproducible Experiments

SŌRĪLŪN presents a self-contained mathematical and experimental framework for recursive self-improvement (RSI) in a restricted algebraic setting. It investigates a concrete question: can a persistent learner preserve certified prior structure while installing equivalent implementations that reduce both task-update work and portions of the work required to validate its own successors? The framework combines exact formal-power-series inverse updates, support-sensitive correctness certificates, a sharp theorem on parent-coefficient dependence, and autonomous compiler updates with explicit resource accounting. The installed implementation participates in later successor validation, allowing improvements to affect part of the self-improvement process itself. The release provides a 70-page manuscript, mathematical proofs, executable Python/C++ implementations, six verification suites, and original successful and unsuccessful experimental records. Mathematical contributions 1. Exact inverse inheritance under changing responses.For a formal response (F(x)=x+O(x^2)), its compositional inverse (S), and a perturbation (δ) whose first nonzero term has degree (r≥2), the inverse update is [(F+δ)⁻¹=S-S'δ(S){x^N}] whenever (2r-1≥ N). Under this contract, the first-variation expression is the complete finite-precision correction. Coefficients below degree (r) remain unchanged. Additional Hasse-support conditions exploit the binomial terms that survive in prime characteristic to certify larger exactness windows for suitable response and perturbation supports. 2. Sharp coefficient-dependency bounds.For a specified boundary correction coefficient over a prime field, the manuscript identifies a sufficient ordinary coefficient prefix of the parent inverse and proves that, in the stated nontrivial cases, no shorter prefix works uniformly over admissible parents. For example, over the field with five elements, at precision (N=75) and perturbation (δ=x⁴⁹), the coefficient of (x⁷⁴) in the correction is determined solely by the quadratic parent coefficient (s_2) and equals (-3s_2). This is a sharp dependency result in an explicitly defined prefix-access model, rather than a lower bound on arbitrary compressed encodings. 3. Output-preserving Cartier compilation.Classical Cartier naturality is specialized to inverse-update transport and used as an exact compiler rule. Moving coefficient extraction ahead of expensive composition yields successive implementations with identical input/output contracts and smaller intermediate computations. The contribution is the integration of this established identity with exact inheritance, dependency certificates, and a persistent resource-accounted service. 4. Recursive validation and resource accounting.A bounded autonomous implementation proposes, checks, and installs successors from a supplied program grammar. Same-candidate interventions hold the proposed child fixed while changing the parent evaluator, isolating whether the current implementation reduces subsequent validation work. The release separates operation counts, measured execution time, promotion overhead, checkpoint costs, context-specific timing evidence, and pre-execution resource reservation. Recorded experimental evidence The original native verification run passed 5,811 exact assertions across six suites. A changing-response stream completed 24 consecutive inverse updates, with each updated state matching a fresh inversion and preserving its certified prefix. The arithmetic lineage accepted seven promotions; a separate timing-gated lineage accepted two. A persistent resonant compiler service installed three successors over 18 requests. In the resonant service, the recorded ratio of frozen-reference solver time to adaptive-service solver time was approximately 14.25. The adaptive total includes paired promotion validation, program construction, and checkpoint writing. The comparator is a frozen uncompiled implementation. A static implementation preloaded with the final rewrite reaches the same endpoint without exploration overhead. Research-only baseline reruns, input generation, and development effort are outside the declared solver timing boundary. Stronger controls are retained: under the reported ten-million-unit budget, conventional number-theoretic-transform arithmetic completed 58 tasks, compared with 37 for the arithmetic lineage. The evidence therefore supports restricted recursive implementation improvement rather than universal computational superiority. Reproducibility and data The research release includes complete LaTeX source, Python modules, optional locally compiled C++ kernels, six test suites, experiment configurations, raw JSON/CSV results, checkpoints, figures, integrity manifests, and documentation of failed development runs and corrections. New reproduction runs preserve the original evidence snapshot. A portable execution path is provided. The original 5,811-assertion total includes native-backend checks, so portable totals can differ. The accompanying Hugging Face dataset preparation adds directly loadable, provenance-linked tables containing 729 experiment records, six verification summaries, and 15 claim/evidence/boundary entries. Original records remain available alongside these derived views. Dataset split labels are storage conventions and do not establish an independent machine-learning training/test partition. Scope and research relevance The work is intended for researchers in recursive self-improvement, autonomous learning systems, program synthesis and optimization, symbolic computation, formal methods, and resource-aware AI. Its central distinction is between improving a task answer, adapting a reusable inverse operator, and improving an implementation that also participates in future improvement work. The mathematics, candidate grammar, verifier, and scheduler are supplied. The executed system does not autonomously discover its theorem families. The release provides hand-written proofs and internally executed exact tests; it does not establish proof-assistant verification, independent replication, historical firstness, or general-purpose RSI. The proposed contribution is the combined exact-inheritance and resource-recursive framework, with its mathematical and experimental boundaries available for independent assessment. Research release: 1.0, 8 November 2026.