RH-W-11: Kernel Sensitivity and Regularity Duality

Generalize the single cubic kernel of W-10 to the entire centered cardinal B-spline family, deriving the general prime-power boundary activation law r=m+n+1: the same integer r simultaneously controls boundary regularity, Fourier decay, and tail bound order. The smoother the kernel, the weaker the arithmetic threshold signal, but the easier it is to strictly control the tail bound—there is no single optimal kernel. The 10⁻²⁸ observed in W-10 is the result of the cubic kernel (m=3) suppressing the signal by a factor of 10¹⁶; switching to a linear kernel (m=1) directly amplifies the signal by 1.65×10¹⁶ times.

RH_CLAIM = False — The claim status self-reported by the verifier within the package, reproduced here as is. What is closed in this round is the engineering trade-off of kernel design, not RH positivity; no Weil negative witness was found, nor was any infinite-dimensional positivity proven.

Connections

Regarding its relationship with other packages, I try to use the words from its own documents as much as possible, not my interpretation. The beginning of the main document of this package also lists "Original research concept: Neo.K; Mathematical engineering, derivation, and implementation: Aletheia (GPT-5.6 Thinking)", quoted exactly as is.

Belongs toResearch Starting Point v1.4 · Kernel Sensitivity and Regularity Duality

Previous PackageRH-W-10: Prime Boundary Local Mode and Seventh-Order Soft Start

Next PackageRH-W-12: Mixed-Order Dictionary and Cross-Cancellation Mode

Batch 0111 / 20

"The smoother the kernel, the more invisible the prime boundary; the sharper the kernel, the more expensive the tail bound. Therefore, no single degree dominates the other degrees across all engineering metrics. The truly reasonable architecture is not to 'find the optimal kernel', but to build a multi-order dictionary." — Excerpt from Section 8 of the 01 document in this package.

GAP Ledger

Sourced from RH-W-11_subgaps_v0.1.csv within the package; the status and dependencies are recorded by the package itself, not judgments added by me after the fact.

Reproduce

Kernel sensitivity certificate construction → Rational interval verification → Independent floating-point convolution regression:

python build_kernel_sensitivity.py
python verify_kernel_sensitivity.py
python validate_kernel_family.py

01 · Kernel Sensitivity and Regularity Duality

Main Document — Centered cardinal B-spline family, general prime-power soft-start law, sensitivity-regularity duality, step-by-step compression ratio, and strict comparison of penetration depth against W-10 (m=1 is over 10¹⁶ times greater than m=3).

02 · Mixed-Order Dictionary and Pareto Design

Why a single kernel is insufficient, the proposed dual-channel core (m=1 sensing / m=3 certificate), cross-channel automatic formation of intermediate orders, mixed-order block Toeplitz structure, and the certificate contract for the next round.

Verification Output

From EXACT_VERIFY.txt (rational interval verification) and VALIDATION.txt (independent floating-point convolution regression, for implementation checking only) in the package, reproduced line by line.

Loading...

Loading...

Files

Download Full Package (47.1 KB)

sha256 d3deca06d57412ff5c614b2613cec409773d7773a3c44084877cc2e0a076cbe3