riemann

NS-5: complete lambda=4 spectral ordering and the ground transform

This is a certified computation using previously archived complete certificates, not a finite-compression diagnostic. No RH hypothesis or zero list is used. The new gates take a few seconds because the old infinite-tail residual enclosures are reused.

Requirements: Python 3, python-flint 0.9.0 (Arb); the complete repository checkout including evidence/v126/. No NumPy, midpoint solve or numerical eigensolver is required by the verifier.

From the repository root:

python3 evidence/v140/certify_ground4.py --bits 1024
python3 evidence/v140/certify_ground4.py --bits 1280
python3 evidence/v140/check_independent.py
python3 tools/verify_manifest.py v140
./manuscript/build.sh

Both reports must say PASS_COMPLETE_SIMPLE_EVEN_GROUND. At the exact threshold 1e-73 the 17-dimensional even lower Schur form must have one negative and 16 positive pivots; the 16-dimensional odd lower form must have 16 positive pivots. Every pivot excludes zero. The exact finite even trial in zero-based column16 has complete Rayleigh quotient <2.454e-75.

Fixed parameters and changes from v1.17

• Window lambda=4; L=2log4; outer head ends at16; normalized parity bases.
• Existing complete tail floor delta=1.7940e-8; no lambda=3 tail constant.
• Existing finite trials supported through4096; no new solve, CG depth, preconditioner, support or normalization is introduced.
• Complete relative Schur constants: even62629/100000; odd428/1000.
• Threshold sigma=1e-73; ordinary head/tail Grams computed from exact frozen dyadic columns, retaining the head congruence.
• Archived complete ingredients include rows through65536 and the order64 infinite remote majorant. These cutoff values belong to the existing lambda4 certificates; the new calculation introduces no additional tail truncation.
• v1.17 recomputed shifted residuals and inverse weights. Here the exact resolvent identity bounds the complete shift directly. Merely subtracting sigma times the identity from an unshifted lower matrix would be wrong.

Inherited inputs and replay scope

certify_ground4.py records the SHA256 of the exact inputs used in each run. They remain in their original evidence/v126/ locations rather than being duplicated here. The new verifier reconstructs both exact frozen trial matrices, checks their heads against saved enclosures, and computes the ordinary Grams afresh. It replays the even relative-margin gate, the odd K-orthogonal complement gate, the exact rational shared-head check, and the odd repaired-direction scalar gate. It independently reevaluates the elementary archimedean tail-floor constant.

The full residual assemblies themselves are inherited theorem inputs: even at768/896bits; odd complement ingredients at768bits; odd repaired direction at1024/1280bits. The two new working precisions do not turn the old768bit odd enclosure into a fresh higher-precision assembly. Full original assembly scripts and witnesses are in g2_simultaneous/, g2_odd_complement/ and g2_weighted_signed/ under evidence/v126/.

Conclusions and limits

The complete selfadjoint W4 has 0<mu0<2.454e-75 and mu1>1e-73, so its ground is simple, even and isolated with gap>9.7546e-74. The hypotheses of Connes–van Suijlekom Theorem6.1 are checked in new_section.tex, including their real distribution on[0,L] and essential selfadjointness on the trigonometric core. Consequently the entire Fourier transform of the complete ground has only real zeros.

This is a fixed-window application of an existing external theorem. It does not assert simple transform zeros, equality or convergence to Riemann Xi, a cofinal estimate, G2 or RH. No PDF is delivered.

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