JHĀNEIRŪ-40 V: Eïmāra Residual Relations - Fixed-Parameter Capability Synthesis, Oracle-Learned Structural Certificates, and the Residual-Horizon Contraction Law

JHĀNEIRŪ-40 V investigates whether developmental evidence can change the mathematical structure of future algorithm discovery. It presents a finite, executable setting in which verified observation capabilities determine which structural distinctions a subsequent discovery procedure must retain. The release combines a self-contained manuscript, definitions and proofs, exact synthesis implementations, synthetic benchmark records, emitted executable procedures, and a Hugging Face-ready dataset with an uploader. Its demonstrated scope is finite-field linear capability synthesis; autonomous recursive self-improvement remains the motivating research objective. The central construction is an anchored residual scope. For a linear transition system over a finite prime field \( F_p\), let \(W\) be the observable row space generated by the existing action alphabet, and let \(H\) be the observable closure under a complete finite catalog of existing and independently installable actions. The residual dimension \[ d= H- W \] measures how many independent observation directions remain outside the current anchor. A candidate action \(B_i\) induces a total linear relation on \(H/W\): \[ R_i(W)= \{(π_W z,π_W(zB_i)):z∈ H\}. \] This relation can be multivalued because an action need not preserve \(W\). Replacing it with an ordinary quotient function would discard the old-to-new leakage that can create additional observable directions. The first mathematical result establishes contextual equivalence for eventual target-row observability. Equal residual relations impose identical invariance constraints in every installed-action context. With independent installations, query-specific permissions and nonnegative costs, minimum-cost selection can therefore retain the cheapest allowed representative of each relation type. The manuscript proves that every inclusion-minimal feasible installed set contains at most \(d\) actions. Combined with at most \(p2d^2\) relation types, this yields an exact \[ pO(d^3)poly(input) \] algorithm for fixed \(p\), or with \((p,d)\) as the parameter pair. The construction permits higher-rank changes, overlapping innovations and arbitrary target rows, extending the earlier rank-one/private-target special case. This is a parameterized tractability result: the worst-case bound can be impractical for large \(d\). A polynomial bit-size kernel is not claimed, and unrestricted selection retains a Set Cover hardness boundary. A complementary countermodel construction characterizes successful selection as covering all old-invariant residual subspaces that omit the target. It connects capability synthesis to explicit and implicit weighted hitting-set methods and provides surviving invariant spaces as negative witnesses. The analysis identifies limits of compressed specifications: hyperplane-only checks can miss lower-dimensional countermodels, while compressed violation barriers require their target, anchor and catalog provenance. The second central result is the Eïmāra Residual-Horizon Contraction Law, the substantive Eve-Maciej anchored-horizon contribution. If verified development enlarges \(W\) to \(W'\) within the same complete horizon \(H\), and \(q:H/W→ H/W'\) is the quotient projection, then \[ R_i(W')=(q× q) R_i(W). \] On the same remaining catalog, previously equivalent relation classes can merge but cannot split. Productive anchor growth decreases the residual dimension governing the search bound. A complete learned response certificate can be transported without further source-oracle queries, while coordinate conversion, grouping, validation and current-state acquisition remain computational costs. The manuscript separately derives the exact additional response-query debt \[ R-(R∩ P) \] when an action has already been queried on span \(P\) and its full response is required on span \(R\). This result assumes an exact linear row-image oracle. A particular capability decision may require fewer probes. The implementation learns its declared scope through charged row-image/adjoint-style queries. Its definitive study uses 24 distinct GF(2) systems, eight fixed seeds and three designed geometries: shared innovation types, cascaded interactions, and mostly distinct noncommuting effects. Sixteen queries per system produce 384 common selection queries and 3,456 records across nine methods. All 432 candidate changes relative to identity have global rank greater than one. These are three geometries of one finite linear-systems domain. All primary queries are feasible; separate small exhaustive cases and unit tests cover infeasibility. All nine methods agree on the optimum installation cost for every primary query. Including declared acquisition and preparation costs, the relation kernel uses 581,537 work units, compared with 532,445 for the strongest measured conventional implicit-countermodel control: 9.22% more work, with an exploratory paired seed-block 95% excess interval of 7.19–11.05%. Cached residual search is also cheaper overall, and an equally informed classical implementation using the same kernel ties exactly. The data support the finite construction and its scope-dependent compression effects without establishing overall empirical superiority. Operation counters are declared implementation units; allocation, serialization and several language-level costs remain outside them. Executable validation includes 24 standalone selector modules and 384 task-program instances representing 379 distinct source hashes. All 3,456 execution traces and 9,216 output/capability checkpoints agree with full-system execution. A separately executed observation-only initialization supplement verifies 1,152 initializations, using 6,510 reset rollouts and 9,018 object-action steps, with zero output mismatches and zero direct-state projection calls. Resettable measurement access is explicit. The supplement does not establish optimal measurement scheduling or an end-to-end speedup. The fixed-horizon history assay records 81 anchor stages, 57 productive installations and 2,788 checks preserving previously equal remaining-action relations. Transport within the learned scope requires no new source queries. Equally informed conventional transcript reuse ties, and outside-catalog actions can invalidate the contraction premise. The archive contains a 41-page self-contained manuscript, complete proofs, source code, fixed-seed generators, original operator/query transcripts, cost records, emitted programs, negative cases, internal audit material and six dataset configurations totaling 5,217 rows. The retained scientific project contains 97 unit methods, including 31 new Stage V methods. Dataset packaging adds twelve separate offline uploader tests. The experimental core uses standard-library Python and was tested on Python 3.12.14. Normalized tables preserve the original records and aggregate costs; their “test” storage split does not imply independent machine-learning generalization evidence. Checksums and reconstruction instructions are included. Exactly three of ten architectural responsibilities are dedicated to the Eve–Maciej principle: horizon genesis, singular developmental continuity, and residual-horizon contraction. This is a mathematical responsibility allocation rather than a measured percentage of compute, novelty or causal effect. The wider forty-structure reservoir is accompanied by explicit finite bridges and deferrals; infinitary forcing, ultrafilters and transfinite recursion are not presented as directly executable learning mechanisms. The proposed significance is a constructive connection between observable-space expansion, contextual operator equivalence, short installation certificates and historical semantic contraction. Prior-art comparisons identify substantial antecedents in weighted automata, observability, operator learning, invariant-subspace semantics, abstract interpretation, structural control and countermodel abduction. Priority for the combined formulation remains unestablished. Proofs and computational records have undergone internal model-assisted checking; external independent reproduction and expert review remain pending. The outer discovery backend and proof rules are supplied, so the release does not demonstrate autonomous invention of that backend, sustained improvement of the improver, or general-intelligence acceleration.