Validated Intersection Certificate

Validated Separation–Positivity Intersection Certificate v0.2 — Reconstructs the floating-point candidates of v0.1 into explicit piecewise linear functions, establishing interval certificates for both the regional negative direction and arithmetic positivity separately; all 480 sub-rectangles are certified, with 0 pending.

RH_CLAIM = False — The certificate_summary.txt within the package self-reports certificate_passed=True, and the scope_warning field defined by CERTIFICATE_SCHEMA.md explicitly states the non-RH status, reproduced exactly as is. This is a replayable validated-numerics certificate, not a verified formal proof; it neither proves nor disproves the Riemann Hypothesis.

Connections

The technical notes of this package self-report right at the beginning which version it succeeds and what loopholes it corrects, quoted as is.

"v0.1 has found the same set of floating-point coefficients... but this still has two loopholes: 1. Negative values on a grid do not imply that the continuous rectangular region is negative everywhere; 2. Positive floating-point values cannot be directly treated as strict arithmetic lower bounds. v0.2 no longer uses black-box arrays, but reconstructs the candidate into an explicit piecewise linear function, and then establishes interval certificates for both signs separately." — Excerpt from §1 of the technical notes in this package.

Previous Package · v0.1Separation-Positivity Intersection Solver

Next Package · v0.1Zero-Side Leakage Budget

Corresponding TheorySequence ④ · Off-Axis Positive Obstructions in Explicit Formulas §7

Technical Notes

Main document — explicit test function, closed form of the Fourier transform, interval subdivision of the continuous rectangle negative certificate, interval bounds for finite position terms and Archimedean terms, and intersection conclusions.

Result Data

Continuous region upper bounds, arithmetic intervals, and certified sub-rectangle statistics, from RESULTS.md in the package.

Trust Boundaries

Which components are trusted, which trust dependencies were reduced compared to v0.1, and what is still lacking to reach the proof assistant level, from TRUST_BOUNDARY.md within the package.

Figures

Three images from outputs/ within the package, reproduced as is, without post-processing.

Validated piecewise-linear test function — Graph of the explicitly defined piecewise-linear test function psi(t) — Explicit test function ψ(t): a superposition of 601 node hat functions, not a black-box array.

Adaptive continuous-region certificate cover — Adaptive interval subdivision cover heatmap of the target rectangle — Adaptive continuous region covering: all 480 sub-rectangles certified, with a maximum subdivision depth of 3 layers.

Arithmetic certificate intervals — Interval error bar chart of finite position terms, Archimedean terms, and the sum, with the sum maintained on the right side of zero. — Arithmetic certificate intervals: finite, Archimedean terms, and the sum; the sum interval remains on the right side of zero.

Files

The raw node data (corrected_nodes.csv, certified_region_cells.csv) and the complete certificate.json are not listed individually; they are all in the full zip file.

Download Complete Package (195.0 KB)

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