Computational Verification of the Connes-Consani Adele Class Space Framework for the Riemann Hypothesis: Weil Positivity, Toeplitz Factorization, and Gap Mapping

This preprint is withdrawn in full and should not be cited except as a record of retraction. A formal closure notice documenting the errors, their detection, and the reasoning is included as the primary file of this version. The withdrawal is voluntary and author-initiated. The errors were identified through internal adversarial review under the PROS Multi-Domain Adversarial Attack Framework v1. 0. No external party raised them. The notice is published in full so that the reasoning is part of the permanent record rather than a silent deletion. **Why it was withdrawn** The paper's abstract and conclusion stated that it did not claim a proof of the Riemann Hypothesis. That disclaimer stands and was honoured. The withdrawal concerns the weaker claims the paper did make — that it computationally verified Weil positivity, and that it identified a structural bridge to the open p-adic case. Neither claim survives. These are errors of mathematical substance, not of citation. Correcting every reference in the bibliography would leave every one of them intact. 1. **The paper's W (h) is not the Weil functional. ** Section 2. 2 defines W (h) as the prime sum of the explicit formula, taken in isolation and with a positive sign. In Weil's formula that sum enters negatively; Weil positivity is the assertion that the archimedean contribution dominates it. The paper computed the adversary in the inequality, found it positive, and reported that as verification. Under its own definition, a larger W (h) makes RH harder. The reported positivity is a property of the chosen test function — a positive-lobed ground-state PSWF transform against positive, rapidly decaying weights — and tests no property of zeta or of the primes. 2. **The declared primary novel result is a definitional identity, is not Toeplitz, and is verified by a test that cannot fail. ** Mₚ = AₚT Dₚ Aₚ is matrix multiplication restated, not a factorization discovered. Mₚ is a weighted Gram matrix, not a Toeplitz matrix. Positive semidefiniteness of AT D A with non-negative D holds unconditionally for any A, so the Cholesky check across all primes and bandwidths carries no information (FM8, circular validation). Section 4. 3's stated criterion is also false: standard Cholesky exists iff a matrix is positive definite, not positive semidefinite. 3. **The claimed structural parallel to Connes–Consani inverts the logic of its source. ** In Tq = λ (Id − Σⱼ d (j) e (zⱼ) ), positivity follows from the bound Σⱼ d (j) 0 are the obstruction, and the bound is the theorem. The paper equates identical sign with identical role. The bound is never checked and fails at face value for p = 2 (Σₙ (log 2) ·2^ (−n/2) = 1. 673 > 1). Connes and Consani use hermitian Toeplitz matrices to control the difference between the Weil distribution and the Sonin trace — an error-control device on a hard remainder, not positivity by construction. 4. **The Frobenius eigenvalue check verifies its own input. ** |p^ρ| = p^ (1/2) holds for any complex number with real part 1/2; the check is equivalent to Re (ρ) = 1/2, which was supplied as input. It is the statement |e^ (ix) | = 1 and carries no information about zeta. 5. **Cross-model agreement was treated as independent verification. ** Language models with overlapping training corpora share failure modes; their agreement establishes internal plausibility, not truth. Section 5. 5 reports that two engines produced the gap statement "in near-identical language" and adopts it as authoritative on that basis — near-identical language between two models is evidence of a shared corpus, not of independence. Additional defects are catalogued in the closure notice, including a printed Slepian–Pollak matrix in §3. 1 that cannot reproduce the paper's own χ₀ table (the implementation was correct; the printed method omits the c² diagonal term), a direct contradiction between §3. 2 and §5. 4 on the direction of the reported bug, two mutually distinct "precise gaps" each identified as the gap, a dangling "Connes (2020) " citation that duplicates the listed Connes–Consani (2021) with a co-author dropped, Connes–Consani–Moscovici (2024) miscited as an unpublished preprint (it is Ann. Funct. Anal. 15, 87, doi 10. 1007/s43034-024-00388-z), and the unmentioned Λ = 1 restriction on the archimedean result. **What survives** Section 6 — the five-lens zero ordinate screening — survives intact and is the only section of the paper that performs a genuine test. It specifies a null model, states what would constitute a signal, and reports the signal's absence. Replacing the flat prior with a 2, 000-draw Monte Carlo null drawn from the empirical quantile function of the zero distribution, and reporting the apparent perfect-fifth signal collapsing to Z = 0. 14, p = 0. 89, is methodologically correct. It will be reissued as a standalone negative-results note, stripped of all reference to Weil positivity, Toeplitz structure, the Connes–Consani framework, and proof strategy, and re-run at N = 500 and N = 1000 under full Monte Carlo. That work will be linked from this record when posted. **Correct references, for anyone arriving here from the withdrawn text** - Connes, A. & Consani, C. (2021). Weil positivity and trace formula, the archimedean place. *Selecta Mathematica* (N. S. ) 27, 77. doi: 10. 1007/s00029-021-00689-4 (arXiv: 2006. 13771). Proven for cutoff parameter Λ = 1. - Connes, A. , Consani, C. & Moscovici, H. (2024). Zeta zeros and prolate wave operators. *Annals of Functional Analysis* 15, 87. doi: 10. 1007/s43034-024-00388-z (arXiv: 2310. 18423). - Connes, A. (1999). Trace formula in noncommutative geometry and the zeros of the Riemann zeta function. *Selecta Mathematica* (N. S. ) 5 (1), 29–106.