Ŋamári: Recursive Operator Discovery with Protected Learning, Reusable Experiments, and Sharp Information-Compute Limits

Ŋamári investigates how a learning system can acquire reusable mathematical structure governing its future optimization problems, revise the observation procedure used to identify later operators, preserve designated linear behavior, and finance further discovery from measured computational savings. The work combines noncommuting matrix-algebra discovery, protected derivative observations, exact statistical precision limits, and explicit accounting of acquisition, verification, and fallback costs. The central learned object is a procedure for identifying the relevant action of a task operator. Paid observations enlarge a generated real matrix algebra; a fixed compiler transforms its observable symmetric action space into a probe design, response decoder, independent audit, and fallback contract. New experience changes both the operator family that can be recognized and the measurements used on subsequent tasks. The compiler and its primitive operations remain fixed, making the scope of the demonstrated adaptation explicit. Protected learning is formalized through the linear contract \(Bθ=b\). With \(N\) an orthonormal basis for \( B\), every update \(Δθ=Nz\) preserves the designated outputs. The admissible update operators form the corner algebra \(\{NSN^\}\), giving the zero-divisor motivation a concrete mathematical interpretation. The framework distinguishes permitted parameter changes from retained observations: a protected probe can return the complete response \(HNz\), whose information content can exceed that of the projected response \(N^ HNz\). The manuscript develops four principal mathematical contributions: Exact protected observability. A general criterion determines when common probe inputs across repeated blocks identify the operator action relevant to protected optimization. For a specified repeated real symmetric positive-definite family with unknown trace, exact whole-target identification requires one unrestricted ambient full-response query, two protected full-response queries, or \(d\) protected projected-response queries. These bounds concern complete operator identification. The realization uses the established Werner-Holevo construction, with explicit attribution. An exact query–precision frontier. Under the stated known-family, fixed-design, unbiased Gaussian estimation model, equal-energy orthogonal probes attain the optimal rank-constrained design. The relative target risk is \[ R_q^*/ R_* = 1+(d-q)(5q-3)/3d^2(q-1)(2q-1), 2≤ q≤ d. \] Two protected queries have mean-squared error at most \(79/72\) of the best fixed design with unrestricted query count at the same total input energy. Three are within \(31/30\). Sharp precision advantages and limits. In the specified family, the optimal projected-response/full-response risk ratio is \((d^2-1)/3\). Within the stated invertible real bistochastic channel class, the ratio has sharp supremum \(d(d+1)/2\), approached through depolarizing targets. Full responses contain additional scalar observations, with corresponding acquisition, communication, storage, and processing costs. A conditioning analysis explains why an estimation advantage need not yield a proportional optimization speedup. Reusable experiments and budgeted discovery. Rejected hypotheses retain their probe responses as a prefix of conventional reconstruction. Successful reuse earns a reserve that can fund additional operator acquisition. In the exact model, cumulative operational expenditure remains below a declared reconstruction baseline plus an initial discovery allowance. The numerical implementation charges repair queries, enforces resource caps, and reports explicit nonconvergence when the remaining allowance cannot fund completion. Three executed numerical studies accompany the theory. The known-channel precision study contains 245,760 Gaussian-noise trial records, supported by deterministic comparisons with dense least squares and Fisher-information calculations. The unknown repeated-algebra study uses 12 seeds, ambient dimension 90, six protected coordinates, and 288 required tasks per seed. Two paid generators expand the learned algebra dimension from six to 36 and change the identification procedure from one probe to two. Including acquisition, the mean complete-horizon cost is 1,338.75 Hessian-vector products, compared with 1,814.50 for conjugate gradients and L-BFGS and 2,016 for protected reconstruction. The unknown ordinary-block study uses eight seeds, ambient dimension 96, 24 protected coordinates, and 608 required tasks per seed. All runs recover the hidden partition after 135–318 earlier tasks, with 16–25 paid program revisions. Including discovery and all required earlier and frozen evaluation solves, the learner uses 9,128.125 HVPs, compared with 15,333.75 for conjugate gradients, 15,427.25 for L-BFGS, and 15,200 for reconstruction. The measured reduction relative to conjugate gradients is 40.47%. The accounting also records unfavorable outcomes. Same-information conventional algebra methods match the structural query savings. On the ordinary-block benchmark, mean total local elapsed time is approximately 0.9215 seconds for the learner, versus 0.2601 for conjugate gradients, 0.6904 for L-BFGS, and 0.0514 for reconstruction. Diagnostic costs are reported separately. If earlier tasks are treated solely as training overhead, amortization over only the final evaluation tasks is unfavorable. Out-of-family controls test rejection and charged fallback; a no-recurrence control demonstrates stalled discovery despite valid resource accounting. The release contains a 40-page manuscript, complete proof appendices, executable source, raw observations, learned numerical program states, conventional baselines, configurations and seeds, development and failure records, integrity manifests, and reproduction instructions. The Hugging Face-ready edition exposes 18 final CSV configurations containing 293,020 experimental records. These include multiple methods and diagnostic phases and are not counts of independent tasks. All task instances are synthetically generated; the numerical results are measurements from the supplied experiments. The work provides an exact information-compute analysis linked to a reproducible mechanism for changing future operator-learning procedures. Noisy known-family estimation and noiseless unknown-structure discovery remain separate studies. General autonomous recursive self-improvement, an autonomously rewritten compiler, unlimited compounding intelligence, and large-neural-model acceleration are not established. The evidence supports a bounded structural learning-to-learn mechanism with explicit assumptions, measurable benefits, and inspectable failure conditions.