RH-W-15: Interval–Taylor Parameter Tube Expansion

Switching to a convex combination of corner matrices plus a pure second-order Taylor remainder (replacing the global first-order absolute value envelope of W-14), expanding the radius of the two-dimensional near-zero parameter tube in each direction from 4×10⁻¹² to 10⁻⁷ — a full 25,000 times larger, expanding the rectangular area by 6.25×10⁸ times, while the entire rectangle still strictly satisfies 10⁻⁸<λ_min<5×10⁻⁸.

RH_CLAIM = False — The declared status output by the verifier itself in the package, reproduced here as-is. Simultaneously discovered and corrected the gap in the W-14 archimedean first derivative bound that missed the constant tail outside the support: the original W-14 derivation is marked as SUPERSEDED, and the conclusion is marked as RECERTIFIED—the original conclusion still holds, but the constants (175/215/253→179/218/255) must be replaced by the corrected version.

Connections

Regarding its relationship with other packages, I try to use the words from its own documents, not my interpretation.

Belongs toResearch Starting Point v1.8 · IntervalTaylor Parameter Tube

Previous PackageRH-W-14: Strict Two-Dimensional Parameter Tube

Next PackageRH-W-16: Three-Parameter Near-Zero Spectral Tube

Batch 0115 / 20

"The support function being zero does not mean that the normalized integrand containing the f(0) subtraction term is also zero. This rule applies to any integral certificate with local cancellation, global compensation, or a renormalization constant. If one only checks the function support without checking the complete integrand, it is easy to miss the out-of-support tail." — Excerpted from Document 02 of this package, "Why This Correction Must Be Preserved."

GAP Ledger

From RH-W-15_subgaps_v0.1.csv within the package; the status and summaries are recorded by the package itself, not my retrospective judgments.

Reproduce

Purely rational exact verifier (optional high-precision floating-point cross-check):

python verify_interval_taylor_tube.py
python crosscheck_w15_mpmath.py

01 · Interval–Taylor Parameter Tube Expansion

Main Document — Four-corner linear interpolation replacing the global absolute value bound, corrected second-order global bound, four-corner exact positive definiteness and convexity arguments, full parameter tube upper bound, and methodological significance (first-order variations are no longer treated as adversarial perturbations; only true curvature is converted into absolute error).

02 · Archimedean Outer Support Tail Correction and W-14 Retrospective Re-certification

Discovered gap (constant tail artanh(e⁻ᴿ) outside the support), exact tail integral, corrected first-order bound of W-14, re-execution of purely rational LDL^T confirming the preservation of the original conclusion, and why this correction must be preserved.

03 · GAP Updates and Batch 01 Progress

Closed/still open sub-nodes, Batch 01 progress 15/20, remaining plans (W-16 adding h, W-17 chamber-aware automatic segmentation, W-18 certificate backend unification, W-19 adversarial audit, W-20 Batch 01 integration and handoff package).

Verification Output

From ALL_VERIFY.txt within the package (merged output of cross-regularity verification + interval-Taylor parameter tube verification, including W-14 retrospective re-certification results), reproduced line by line.

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Files

The machine-readable certificate for the W-14 retroactive recertification (RH-W-14_corrected_lipschitz_reaudit.json) is listed below and can be downloaded directly for verification.

Download Full Package (253.6 KB)

sha256 74755c497650d207840ef0169b41b4be35031f7d5961380209871d09f7d36540