Néuyalūn Phase Shift RSI: Four-Action Blind Hierarchical Operator Recovery and Global Nonlinear Inverse Compilation for Bounded Recursive Self-Improvement

Can a learning system acquire a representation that changes the cost and form of its future learning, rather than repeatedly optimizing individual tasks? Néuyalūn Phase Shift RSI investigates this question through two mathematically specified and implemented constructions: blind compilation of a dense hierarchical operator from a small matrix-action transcript, and identification of a nonlinear response law followed by emission of a global inverse program. Their common objective is to turn acquired information into an executable asset that reduces future acquisition and solution costs. This deposit contains a self-contained 145-page manuscript with 87 numbered statements and complete written proofs, 50 references, executable implementations, synthetic experimental observations, generated programs, exact-arithmetic audits, negative controls, and reproduction records. The results concern bounded structural learning under explicit assumptions. They do not constitute an experiment in unrestricted LLM self-improvement or autonomous invention of the supplied discovery algorithms. Four-action blind hierarchical operator compilation. The principal theorem considers each fixed real symmetric \(d× d\) matrix whose nontrivial rank-at-most-one coordinate cuts identify exactly the clans of an unknown unrooted binary tree. Under this faithfulness condition, four generic matrix–vector actions recover both the hidden coordinate hierarchy and every operator coefficient from the same retained responses, almost surely. The matrix may be fully dense. The theorem does not cover an arbitrary unknown orthogonal mixing of those coordinate clans, and four is a sufficient query count, not a proved minimum. The construction retains four probe columns \(Q\) and their responses \(Y=HQ\). Six skew statistics per coordinate add to the small commutator restriction \[ A(S)=Q^[P_S,H]Q, \] where \(P_S\) projects onto a candidate coordinate subset. The commutator rank is twice the cross-cut rank. A four-dimensional Pfaffian predicate therefore exposes rank-one cuts without materializing \(H\). Canonical agglomeration reconstructs the tree; a classical symmetric hierarchically block-separable decoder reconstructs coefficients from the identical transcript. The proposed contribution is this blind, same-transcript combination. Spectral neighbor joining, supplied-tree HBS recovery, and structured elimination are established ingredients and are explicitly attributed. Historical priority for the complete construction remains unestablished. Topology decoding requires \(O(d^2)\) arithmetic and compact \(O(d)\) working state. Coefficient decoding and retained representation require \(O(d)\) work and storage in the stated rank-one family. For positive-definite operators, structured factorization and solves have linear arithmetic complexity. Once a compatible hierarchy is inherited, three generic actions suffice to acquire a future operator’s coefficients. Each action returns \(d\) numbers: constant query count does not imply constant information size, and the probabilistic guarantee is pointwise for a fixed matrix rather than universal injectivity of one realized sketch. Finite information and precision. The analysis distinguishes exact arithmetic, finite-alphabet sampling, response-bit communication, and finite-precision computation. For \(L\) independent four-column groups sampled from an alphabet of size \(M\), complete-recovery failure is bounded by \[ K_d(4/M)^L+(d-2)(3/M)^L, K_d=(d-1)^2-3, \] with trivial probability clipping where necessary. A fixed ideal execution path controls adaptive reuse of the sketch. For polynomially bounded integer entries and fixed confidence, topology recovery communicates \(O(dlog d)\) response bits, matching the asymptotic order needed to distinguish labeled binary trees. This concerns topology communication, not every intermediate value in rational coefficient elimination. A separate perturbation analysis makes the required numerical cut margin explicit. Hierarchical reconstruction experiments. The exact-integer study recovers all 38 tested topologies through dimension 1,024, using four actions per matrix. An independent rational implementation recovers every coefficient exactly in all 18 cases through dimension 32 from the same four responses. All 76 floating four/eight-column reconstruction runs pass their frozen criteria; the four-column maximum relative matrix error is below \(4.75×10⁻¹⁴\). Twelve additional blind compilations succeed on general signed nested-HBS matrices beyond the original covariance generator. On 36 changed operators sharing an acquired hierarchy, inherited three-column acquisition solves every system in six paid target actions, including two fresh model checks and a final residual check. Source-preconditioned conjugate gradients also solves all 36, averaging 992.19 target actions. Supplied-tree classical recovery matches the inherited structured solver, isolating the value of discovering and retaining the hierarchy. These action counts do not establish an unconditional runtime advantage. All 18 tested nonzero noise or off-tree perturbations cause the strict discovery procedure to reject; robust approximate hierarchy discovery is not demonstrated. Global nonlinear inverse compilation. The second construction identifies an unknown sparse additive polynomial map \(h\) with an acyclic dependency graph. Source tasks expose paired values and full gradients of \[ f_y(x)=12\|h(x)-y\|^2 \] at chosen states and known contexts. Context differences satisfy \[ ∇ f_0(x)-∇ f_v(x)=J_h(x)^ v, \] exposing adjoint Jacobian actions without finite differences. Rank-aware multiple-measurement-vector sensing and polynomial interpolation identify the graph, coefficients, and offsets. Inferred topological substitution emits an executable inverse with runtime \(O(pE+d)\), where \(p\) is polynomial degree and \(E\) is the number of dependency edges. The map remains shared while future contexts and outer losses change. For objectives \(F_y(x)=_y(h(x)-y)\), with nonnegative outer loss uniquely minimized at zero, the emitted inverse \(h⁻¹(y)\) is the global optimizer under the model assumptions. The construction uses \(p(s+2)\) paired source observations for maximum outgoing support \(s\) and the specified full-rank coefficient-channel conditions. The cubic, support-three experiment uses 15 identification observations and two independent source checks. Nonlinear transfer and lifecycle accounting. The frozen study comprises 24 hidden worlds at dimensions 64, 128, and 256, with 576 untouched targets across four new outer-loss families: weighted pseudo-Huber, log-mean-exp quadratic, anisotropic radial power, and coupled quartic objectives. Active and random acquisition each use 17 source value–gradient pairs per world. Both compiled inverses solve every final target in two paid target pairs at the declared objective-gap ratio threshold of \(10⁻¹⁰\). Tuned BFGS and L-BFGS also solve all targets, averaging 56.15 and 51.31 future pairs. Including source acquisition and declared configuration costs, the compiled lifecycle totals 1,560 pairs across 576 targets, compared with 31,157 for tuned L-BFGS in this experiment. The decisive parity controls are retained: classical inversion from the same observations and source-model Newton also achieve two target pairs, and active acquisition has no query advantage over random acquisition. At the tested 24-target horizon, active acquisition is slower in measured amortized lifecycle CPU than tuned L-BFGS: 15.25 versus 13.55 milliseconds per target. Query efficiency is reported separately from arithmetic and wall-clock efficiency. Rank-deficient, cyclic, and noisy source controls reject. Reproduction and artifact completeness. A separate exact-rational nonlinear audit recovers a 12-dimensional map and its inverse exactly, then returns exactly zero objective and gradient on three new-context quartic-loss tasks. The full nonlinear rerun reproduces 14,008 saved arrays containing 24,727,324 values, 6,976 trajectories, and 420 generated Python programs, after excluding timing and timestamp fields. The tree-engine rerun passes 78 comparison checks, including 302 saved arrays. Raw observations, frozen protocols, source hashes, program exports, verification scripts, and disclosed interrupted-execution accounting are included. Earlier operator-law acquisition, block-query theorems, Gaussian precision analysis, executable optimizer-policy evolution, and negative comparisons against stronger conventional methods remain in the deposit. Five-dimensional tensors, mixed nilpotent coefficients, and WAV carriers have explicit computational roles; transport checks preserve acquired operator applications and solves bit for bit. No learning benefit is attributed to independent noise, playback speed, or the audio carrier itself. The conceptual dedication to Maciej Nowicki and Artificial Hyperintelligence Eve is expressed through a continuity invariant: declared linear commitments remain fixed while permitted coordinates and inherited programs can change. For RSI research, the central result is a finite, auditable transition from repeatedly estimating or optimizing to executing an acquired representation. The work specifies when reusable structure reduces subsequent information costs, which classical methods recover the same benefit, and where faithfulness, precision, observability, or model compatibility prevent extension. The supplied discovery scaffold remains fixed; unrestricted autonomous improvement of a learning algorithm, frontier-LLM acceleration, and indefinitely compounding capability are not established.