Volume 45 certifies a finite compressed canonical shell predicate theorem for the KS→NS Fourier-shell admissibility object inherited from Volume 44 under the Sigma₄1 instrument lineage. The study object is not a new physical transport computation and not a new substrate result. It is a finite grammar-relative compression theorem over the declared ΩV44 transition-cell family emitted by the production notebook. The full declared family contains 1, 695 transition cells across 66 declared (width, halfspan) pairs. The inherited Volume 44 oracle partitions the domain as EXCLUDE = 901, Branch 1 = 712, Branch 2 = 6, Branch 3 = 76, with PV44 = 794 admissible cells. Volume 45 proves that the inherited three-branch reference oracle is exactly compressed on ΩV44 by the canonical predicate: P* = (¬includeqcrit ∧ includeqsup) ∨ includeb3b ∨ (includeqbranchₕi ∧ ¬includeqbranchₗo) Under GAMMASHELLDNFV1 and LEXICOGRAPHICCOSTORDERV1, P* matches the inherited Volume 44 admissibility oracle exactly on ΩV44, with zero mismatches, zero false positives, and zero false negatives. The certified P* cost tuple is (3, 5, 5, 2, 1, 28, 167), compared with the inherited Volume 44 reference cost tuple (3, 13, 7, 8, 3, 64, 437). A lower-cost exclusion search over 123, 954 candidates finds zero lower-cost exact predicates. At the P* cost level, the exact representative count is one and the equivalence-class count is one. The study also certifies that P* is not a disjoint branch assignment. Its three clauses form an overlap-aware clause-incidence cover. The clauses overlap on 580 rows, all inside Branch 1. Clause cover counts are |C0| = 712, |C1| = 448, and |C2| = 526; pair intersections are |C0 ∩ C1| = 372, |C0 ∩ C2| = 520, and |C1 ∩ C2| = 312; the triple intersection is |C0 ∩ C1 ∩ C2| = 312. Inclusion-exclusion recovers the admissible union count exactly: unioncountᵢnclusionₑxclusion = 794 and unioncountdirect = 794. The compressed predicate must therefore be interpreted as an overlap-aware finite-domain cover, not as a replacement disjoint branch law. A final independent re-derivation and full-cube scope audit was added as the sixth notebook cell. This verifier independently reconstructs the domain, partition, exactness totals, grammar counts, minimality search, uniqueness certificate, overlap inclusion-exclusion audit, and P* vector SHA-256. It also tests the full seven-atom Boolean cube. The audit finds that P* disagrees with the inherited oracle on 51 of 128 possible Boolean signatures, while 0 of those disagreements are realized inside ΩV44. This confirms that P* is an exact finite-domain extensional compression on ΩV44, not a global shell-logic replacement for the oracle. Certified Results: Finite compressed canonical shell predicate theorem certified on ΩV44Production certification sweep covers 1, 695 transition cellsExact agreement between P* and the inherited Volume 44 admissibility oracle on ΩV44Zero mismatchesZero false positivesZero false negativesFull certified partition: EXCLUDE = 901, Branch 1 = 712, Branch 2 = 6, Branch 3 = 76, PV44 = 794P* cost tuple: (3, 5, 5, 2, 1, 28, 167) Inherited reference cost tuple: (3, 13, 7, 8, 3, 64, 437) Lower-cost search space: 123, 954 candidatesLower-cost exact predicates: 0Raw exact representatives at P* cost: 1Equivalence classes at P* cost: 1Overlap row count: 580Inclusion-exclusion union count: 794Direct union count: 794Full Boolean cube signatures: 128Realized ΩV44 signatures: 20Full-cube P* versus oracle disagreements: 51Realized ΩV44 disagreements: 0 Certified Predicate: P* = (¬includeqcrit ∧ includeqsup) ∨ includeb3b ∨ (includeqbranchₕi ∧ ¬includeqbranchₗo) Clause Structure: C0 — ¬includeqcrit ∧ includeqsupC1 — includeb3bC2 — includeqbranchₕi ∧ ¬includeqbranchₗo Canonical Constructs: Finite compressed canonical shell predicate, ΩV44 extensional compression, GAMMASHELLDNFV1 grammar, LEXICOGRAPHICCOSTORDERV1, lower-cost exclusion certificate, unique exact normal form at P* cost, overlap-aware clause-incidence semantics, inclusion-exclusion support accounting, full-cube scope audit, Sigma₄1 inherited instrument lineage. Continuity: Volume 45 continues the RSI shell-law program by compressing the closed three-branch oracle certified in Volume 44. Volume 44 supplies the exact branch-resolved admissibility law on the declared ΩV44 family. Volume 45 shows that, under a locked grammar and locked cost order, the same finite oracle has a strictly lower-cost exact predicate representation. The inherited branch interpretation remains valid for explanatory purposes, while Volume 45 supplies the canonical finite-domain compression and proves that the compressed predicate is exact, minimal, and unique at its cost on ΩV44. Reproducibility: Evidence for the certified claim set consists of the locked production notebook, the regenerated ΩV44 transition-cell domain, the inherited Volume 44 admissibility oracle, the declared finite DNF grammar, the locked lexicographic cost order, exact rowwise comparison of P* against the oracle, lower-cost exclusion search, uniqueness enumeration, overlap-aware semantic audit, independent verifier cell, full-cube scope audit, reader-facing article PDF, Colab export, machine-readable summaries, and the final artifact bundle under inherited RSI manifest discipline. Scope Bound: This volume does not certify physical Navier-Stokes transport, cross-substrate transport, PoEM, asymptotic invariance, extension beyond ΩV44, global shell-logic equivalence, global shell-7 sufficiency, Branch-3 global uniqueness, or V46 validity. The independent full-cube audit shows why this scope bound is necessary: P* is exact on ΩV44 but not equivalent to the inherited oracle on the full 128-signature Boolean cube. The certified object is therefore an exact finite-domain compression theorem on ΩV44 only.
Camaron Foster (Tue,) studied this question.