Ilyrion-P / Néyara: Counterfactual Parry Learning - Exact Causal Improvement of Fresh Child Training, Cubic Inverse Metric Refinement, Structured Nonlinear Transfer, and Adaptive Step Repair

Ilyrion-P / Néyara is a self-contained mathematical and computational study of a precise question in recursive learning: can successive versions of a learning mechanism become better at training the same freshly initialized child, rather than merely inheriting a better starting point? The deposit provides complete proofs in a restricted symmetric positive-definite (SPD) matrix model, two executed synthetic experiments, raw observations, learned matrix states, verification scripts, and a reproducible research package. The central construction is a two-sided residual teacher. Given an SPD training geometry A, a parent inverse approximation P, and a fixed child learning metric C, the teacher is T_P(C) = 2P − PAP + (I − PA) C (I − AP). Writing E_A(X) = I − A^(1/2) X A^(1/2), its exact normalized error satisfies E_A(T_P(C)) = E_A(P) E_A(C) E_A(P). The resulting spectral-norm bound is e_A(T_P(C)) ≤ e_A(P)^2 e_A(C). It applies without assuming that the child commutes with the parent or training matrix. The bound alone does not impose a universal ordering on arbitrary teachers. A stronger causal result is established along the specified self-applied refinement sequence: when P_(g+1) = T_(P_g)(P_g), the parent error cubes, and the error obtained by teaching the same fixed child at the next generation is at most e_A(P_g)^4 times its preceding value. Under the stated contraction condition, this gives a rigorous same-child improvement law. Self-application expands to the classical cubic hyperpower inverse iteration, P_(g+1) = 3P_g − 3P_g A P_g + P_g A P_g A P_g. The study explicitly identifies this equivalence rather than claiming invention of hyperpower inversion. Its contribution is the complete causal-teaching formulation, the associated analysis and controlled experiments, and a documented deployment failure and repair. Cubic contraction concerns inverse metric error under explicit assumptions; it is not presented as general intelligence compounding. A finite coefficient search over 343 candidates uses a human-supplied cancellation criterion and is described as bounded synthesis, not unrestricted autonomous algorithm discovery. The main experiment uses 24 independently seeded 32-dimensional geometries, nine refinement generations, 384 final tasks, 6,144 solver trajectories, and 240 saved parent states. Four nonlinear scalar loss families-log-cosh, pseudo-Huber, oscillatory convex, and shifted logistic-are excluded from metric refinement. They share a declared curvature relationship with the training geometry, so transfer means new objective forms under a common structural assumption, not generalization to unrelated domains. The full training matrix is also available to the informed conventional controls. At a horizon of 32 joint objective-and-gradient observations, mean bounded log-error utility rises from 0.057373 for the initial parent to 1.000000 for generations six and nine. A utility of one means that the best observed objective reaches the specified 10^−12 relative-objective threshold; it does not mean zero numerical error or a measured increase in general intelligence. The matched classical hyperpower implementation and the equally informed exact-metric Cholesky controls also reach this threshold. The maximum relative difference between the self-refined and classical cubic matrices is approximately 1.24 × 10^−14. Superiority to these strong matched controls is not established. The experiment also exposes a consequential failure: a more accurate inverse metric can produce worse short-horizon optimization when the deployment step remains fixed. A separately sealed follow-up tests a repair on 24 fresh geometries and 384 fresh tasks, retaining 2,304 additional solver trajectories. The repair is a clipped preconditioned Barzilai-Borwein scalar step, using already observed parameter and gradient differences without additional gradient observations. This established step-size method is credited as such, and its arithmetic is not treated as free. In that follow-up, the final fixed-step metric has mean utility 0.891975 after 16 observations. With adaptive steps, its mean utility reaches 1.000000, and all 384 tasks attain the relative-objective threshold by that horizon. The largest recorded relative objective at that 16-observation horizon is approximately 4.04 × 10^−16. The paired lineage-level utility gain is 0.108025, with an exploratory percentile 95% interval of [0.107044, 0.108968]. An exact-metric Cholesky control given the identical adaptive rule ties the repaired method. The outcome therefore demonstrates a reproducible repair, not a new generally dominant optimizer. Robustness limits are retained. All 24 unannounced-rotation stress cases fail the contraction guard for the untransported final metric. A transported control receives the coordinate transformation explicitly and is not credited with discovering it. Forty-eight bounded-drift cases check the conditional geometry-transfer bound as post-study diagnostics; they are distinguished from the sealed experiments and are not used to select a favorable method. Verification includes exact rational identities with noncommuting matrices, 160 directional derivative checks, reconstruction of all reported main and follow-up scores, checks of the 240 saved matrix states, and a complete one-lineage replay of 256 solver trajectories whose non-timing records match exactly in the recorded environment. These are numerical and algebraic checks with within-environment replay, not independent external replication. Source seals are local records, not external preregistration. The retained development pilot is explicitly separated from the final evidence because its exact historical source snapshot was not preserved. Resource accounting distinguishes metric preparation, dense matrix operations, solver observations, and measured CPU time. The recorded main solver histories contain 393,216 observations; rotation stress and Adam tuning are separately counted. The follow-up contains 147,456 solver observations. Each refinement generation uses three dense matrix multiplications in the compiled implementation. The matched classical implementation can use the same allowance. CPU measurements are descriptive results from one host, and lifecycle projections are labeled calculations rather than independently timed deployment cohorts. The deposit includes the 16-page manuscript and editable source, Python implementations, original protocols and results, raw trajectories, saved matrices, figures, retained pilot and replay material, and reproduction instructions. The accompanying Hugging Face preparation adds a research card, citation metadata, three explicitly configured viewer tables, integrity checks, and a dataset-only uploader with Windows launchers. The exported horizon tables are correlated views of the original runs; independent geometry lineage remains the statistical unit. Uploading the package does not rerun the experiments. Stellar Blade supplies the creative engineering imagery: Eve’s paired parry motivates correction from both sides; Lily represents repair after an observed regression; Tachy represents the fixed-child comparison; and Raven represents an adversarial geometry change. These are research metaphors, not evidence for the equations or claims about game algorithms. The Maciej-Eve dedication expresses a commitment to let methods change while preserving the meaning of explicit promises. No proprietary game artwork or code is included, and no publisher affiliation is implied. The supported result is a bounded, analytically transparent example in which successive refined parents improve their ability to teach an identical fresh child, with structured nonlinear transfer and a fresh-seed adaptive-step repair. The experiments do not establish unrestricted source-code self-rewriting, large-language-model improvement, unrelated-domain transfer, intelligence explosion, or a computational advantage over equally informed classical methods. The release is intended for reproducibility research, numerical analysis, and the careful evaluation of causal claims in recursive learning.