Directed Computer-Assisted Certification of a Local Order-10 Operator-Chain Corridor on the Real Interval 5 ≤ s ≤ 11.55 via Finite Uniform b₁-Simpson Envelope Sequences and Proof-Preserving Macroparent Compression

Abstract This preprint presents a directed computer-assisted certification of strict positivity and corridor inequalities for a constructed order-10 real-parameter operator chain on the local interval 5 ≤ s ≤ 11.55. The certification begins with a gap-free chain of directed continuum certificates over a historical partition of the real interval [5, 11.00], comprising interval components of widths 0.05, 0.10, and 0.20. It then develops a proof-preserving compression route to degree-10 macroparents of width 0.10. The late-stage continuation is controlled through directed block-remainder estimates, finite uniform b₁-Simpson envelope sequences, and recursive reuse of a fixed conservative coefficient K_unif. Primary and Control certificates use separate numerical accumulation paths while retaining a common analytic anchor and certificate construction. The terminal certified state contains 92 historical directed cells—75 of width 0.05, 11 of width 0.10, and 6 of width 0.20—followed by five degree-10 macroparents of width 0.10. The resulting chain contains 97 proof-authoritative continuum parents with maximum endpoint gap zero. In this record, a proof-authoritative continuum parent is a retained interval component whose directed Primary/Control gates and endpoint compatibility form part of the certificate chain. The fifth-slab stress test identifies the first explicit finite-range validity boundary of the recursive uniform-K strategy. At the inherited Primary panel width h₁ = 0.00065, the direct certificate remains positive, but the conservative uniform-K substitution first fails in the Primary Q⁺ gate. A directed panel-width boundary h₁,max^(P,+) = 0.0006035392387… is obtained. Recomputing the Primary route at h₁ = 0.00060 restores positivity without increasing, refitting, or locally replacing K_unif. Certified scope Within the stated analytic-computational framework, the accompanying directed certificates establish, for every real s in [5, 11.55]: - Q₁₀,₁₀⁺(s) > 0;- Q₁₀,₈⁻(s) > 0;- W₁₀(s) > 0;- 8W₁₀(s) < s b₁₀(s)W₁₁(s) < 10W₁₀(s). The finite slab-envelope ordering K₁ ≥ K₂ ≥ K₃ ≥ K₄ ≥ K₅ is certified only for the five tested macro-slabs [11.05, 11.15], [11.15, 11.25], [11.25, 11.35], [11.35, 11.45], and [11.45, 11.55]. Explicit claim boundary This record does not claim or establish: - a proof of the Riemann Hypothesis;- PF∞;- global q-monotonicity;- global Order-Defect-Cone closure;- pointwise monotonicity K′(s) ≤ 0;- a global coefficient-decay law;- automatic or recursive continuation beyond s = 11.55. The contribution is a local directed operator-chain certificate, not a global zero-geometry theorem. Reproducibility and provenance The accompanying reproducibility package contains the selected authoritative foundation artifacts, directed cell-chain packages, proof-preserving compression and macroparent packages, manifests, ledgers, source scripts, control-path outputs, freeze reports, and an artifact index. Superseded, duplicated, and internal-only transfer artifacts are excluded from the public package. The files attached to this record constitute the public preprint release v1.0.