RH-W-12: Mixed-Order Dictionary and Cross-Cancellation Mode

Establish the first 10-dimensional real Weil interval matrix mixing m=1 (linear) and m=3 (cubic) B-splines — a purely rational certificate exactly brackets the mixed spectral bottom at 1/2000 < λ < 1/1000, strictly lower than the respective lower bounds of the two isolated channels (λ⁽¹⁾>1/20, λ⁽³⁾>1/250). The new low mode arises from a cross-block cancellation of approximately 93.4% of the self-energy, rather than being the direct product of any single kernel, and is named the "cross-regularity cancellation mode".

RH_CLAIM = False — The declared status output by the verifier itself in the package, reproduced here as-is. CERTIFIED_POSITIVE_ON_THIS_MIXED_SUBSPACE does not equal RH; no true Weil negative witness was found.

Connections

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

Belongs toResearch Starting Point v1.5 · Mixed-Order Cross Cancellation

Previous PackageRH-W-11: Kernel Sensitivity and Regularity Duality

Next PackageRH-W-13: Cross-Regularity Continuation and Canonical Parameters

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"The important phenomenon is not that lower-order channels independently produce lower spectral values. Quite the opposite: λ⁽¹⁾>λ⁽³⁾>λ_mixed. That is to say, the new low modes arise from the cross-coupling of m=1 and m=3, rather than from any isolated kernel." — Excerpt from Section 5 of Document 01 in this package. Mixed regularity is not channel splicing; rather, it generates new spectral modes that do not exist in a single channel.

GAP Ledger

From RH-W-12_subgaps_v0.1.csv in the package; the status, statements, and evidence are all recorded by the package itself, not my retrospective judgments.

Reproduce

Mixed dictionary construction → Purely rational exact verifier → 80-digit mpmath independent cross-check:

python build_mixed_dictionary.py
python verify_mixed_dictionary.py
python crosscheck_mixed_mpmath.py

01 · Mixed-Order Dictionary and Cross-Cancellation Mode

Main Document — The first strict chamber (2N=10), multi-distance arithmetic sensing map (three prime power layers: 2, 3, 4), three Archimedean tail bounds working simultaneously, exact spectral sandwich certificate, cross-cancellation mode (93.4% cancellation), and the effect of prime-power on the witness.

02 · Certificate Architecture and Trust Boundaries

Three-layer architecture of exploration layer / interval construction layer / exact verifier, interval perturbation principles, mixed witness upper bounds, exact sign determination of cross-cancellation, and independent mpmath cross-checks.

Verification Output

From ALL_VERIFY.txt in the package (merged output of exact verifier + mpmath cross-check, including the interval width for each block-lag), reproduced line by line.

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