Category: certified computation and proved implication. This run executes the requested complete-operator spectral-ordering task. It does not substitute finite eigenvalue diagnostics.
At the exact ordinary spectral threshold sigma=10^-73, outward Arb LDL certifies the following counts at both1024 and1280bits:
| Sector | Head dimension | Negative pivots | Positive pivots |
|---|---|---|---|
| even | 17 | 1 | 16 |
| odd | 16 | 0 | 16 |
The exact saved even finite trial has complete Rayleigh quotient
2.45361754930e-75 < q(f)/||f||^2 < 2.45361754931e-75.
Together with prior complete strict positivity, this proves
0 < mu0 < 2.454e-75, mu1 > 1e-73,
mu1 - mu0 > 9.7546e-74.
Eigenvalues here belong to the complete ordinary Hilbert-space operator and are counted with multiplicity. Thus the ground is simple and even.
The v1.17 inertia logic transfers; its hard-coded lambda3 tail constants do not. Instead of rebuilding every shifted residual, we bound the entire inverse change using the already certified complete lambda4 Schur forms. For the old exact trial G=(V;Z), set K=GWG, Hh=VV, Ht=Z*Z. The existing complete bounds are S^V>=kappa K, with kappa=.62629 even and .428 odd, and T>=delta I, delta=1.7940e-8. Then for 0<sigma<delta:
S_sigma^V >= L_sigma
= kappa K - sigma [Hh + 2/(1-sigma/delta)(Ht + K/delta)].
Proof: write r=BV+TZ, C=r*T^-1r, Y=T^-1BV. The exact identities are S^V=K-C and
S_sigma^V = S^V - sigma Hh
- sigma Y*(I-sigma T^-1)^-1Y.
Since 0<=C<=K and Y=-Z+T^-1r,
Y*Y <= 2Ht + 2C/delta <= 2Ht + 2K/delta.
This gives the displayed lower bound by spectral calculus. It retains the ordinary Gram and the full inverse-shift term. A negative eigenvalue of the lower form alone would not prove a negative shifted Weil direction; the separate exact Rayleigh witness supplies that indispensable direction. Square completion and compact resolvent then give the complete ordering.
Both new runs reconstruct the same exact dyadic trials through4096 and recompute their ordinary Grams. The old complete residual gates are replayed on their saved enclosures: even768/896bits, odd complement768bits, odd repaired direction1024/1280bits. The latter trial has support8192 but exactly the same relevant physical head. The old infinite-tail assemblies are not represented as newly recomputed. They remain inherited inputs from v1.25/v1.26, with hashes recorded by the new verifier.
The earlier eight-order diagnostic concerns the even-sector finite compression and is not a global complete spectral gap. The smaller complete trial values differ from the N64 diagnostic, as expected when the support grows. The log records the initially rejected threshold.
Connes–van Suijlekom, Theorem6.1 requires a real distribution on[0,L] whose form on trigonometric polynomials defines a lower-bounded essentially selfadjoint operator, with simple isolated lowest eigenvalue and reflection-even eigenfunction. It does not require a positive lowest eigenvalue or a nonzero endpoint.
The complete proof supplies the actual one-sided distribution, including the origin counterterm and correct logarithmic diagonal; merely saying “an even kernel” would not address the caveat in their Remark4.3. The manuscript’s prop:v114-weil-core supplies the essential selfadjointness and compact resolvent. Taking L=2log4, the centered unitary Fourier transform differs from their transform by a nonzero scalar and exp(-izL/2), so its zeros are unchanged. Therefore xi-hat_4, the entire Fourier transform of the complete ground eigenfunction, has only real zeros. Simplicity of those zeros is not asserted.
This is a new certified fixed-window application within the project, continuing v1.17’s spectral mechanism and applying the cited existing theorem. Worldwide novelty or publication readiness has not been established. No convergence to Riemann Xi is proved. No G2 gap was closed; RH remains unproved.