u/leonardvnhemert

Recent discussions about GPT-5.6 in mathematics have mostly focused on whether a model can produce one successful proof. I used (ChatGPT Work) GPT-5.6 (5.6 Sol Ultra) research agents—in a different way: as a controlled multi-agent research and audit environment for a six-repository program around Weil kernels, explicit formulas, and semilocal zeta spectral operators. The models were required to reconstruct primary sources, freeze conventions before parallel work, derive proofs or counterexamples, run independent numerical implementations, use interval arithmetic where feasible, and preserve failed routes instead of silently discarding them. I provided the research direction, constraints, repeated adversarial prompts, repository curation, and final human responsibility. This is not a proof of the Riemann Hypothesis. The work contains new theorem claims, exact reductions, and certified finite obstructions, but the main analytic and operator-theoretic arguments have not yet completed external human peer review. OpenAI has not endorsed or independently verified these results. All six repositories are collected here: https://github.com/stars/LeonardSEO/lists/riemann Each repository contains its own statement of scope, proofs, sources, tests, certificates, and limitations.

The main analytic result

For one fixed, explicit, real-even, compactly supported smooth packet h, we derived an exact zero-only reflected-packet interaction Q_h, retaining every prime power and auditing the pole, archimedean, and trivial-zero cancellations. For every κ ≥ 0, the resulting growth theorem is: Q_h(y) = O(e^(κy)) if and only if Re(ρ) ≤ 1/2 + κ/2 for every nontrivial zero ρ. Consequently, for this fixed packet, RH is equivalent to each of the following:

• (Q_h) is bounded;
• (Q_h) is polynomially bounded;
• (Q_h) has quantified subexponential growth;
• (\log^+|Q_h(y)|=o(y));
• (Q_h) is continuous and positive definite. These are criteria equivalent to RH, not a verification of the criteria and therefore not a proof of RH. The same fixed-packet analysis also gives an exact zero expansion and unconditional positive and negative values arbitrarily far to the right. That is an oscillation theorem for one explicit scalar interaction; it does not determine the parity of an actual semilocal ground state.

Operator-theoretic results

The separate operator-theory repository records:

• a fixed-λ Fourier form-core theorem;
• conditional Ritz eigenvalue, spectral-projector, admissibility, and projective-determinant convergence;
• fixed-λ smoothed spectral convergence under a simple, isolated, inversion-even ground-state hypothesis;
• exact finite relative-resolvent, trace, determinant, and Stieltjes formulas;
• exact free Poisson-alias and omitted-prime-power terms;
• a neutral spectral–arithmetic defect that remains uncontrolled across (\lambda);
• and an interval-certified counterexample to universal one-step active/free interlacing. The fixed-λ results do not imply the required cross-λ identification.

What did not work

Several routes were stopped rather than presented as evidence:

• finite spectral agreement did not yield infinite-dimensional or cross-parameter convergence;
• the active-ground-state construction repeatedly reduced to the same unresolved spectral–arithmetic identification;
• proxy/Feshbach reductions exposed a precise missing bottom-cluster estimate but did not close it;
• a finite positivity band was certified, while an analytic obstruction showed why the corresponding fixed cross-endpoint polynomial method cannot extend indefinitely;
• a short pilot study of Suzuki's screw-function framework found a genuinely ground-state-free finite construction, but did not establish shift-independent zero divisors, a canonical extension parameter, a canonical normalization, or cross-parameter normality. Accordingly, none of the repositories claims convergence to (\Xi), completeness of zeta zeros, Weil positivity, or RH.

What GPT-5.6 (ChatGPT Work) contributed

The workflow was designed to make failure visible:

• freeze conventions before parallel work;
• separate analytic proofs from numerical diagnostics;
• require independent reconstructions and adversarial audits;
• use two implementations for load-bearing finite computations;
• use interval arithmetic where feasible;
• retain exact unresolved estimates and negative certificates;
• and stop branches whose missing hypothesis already contains the desired conclusion. Codex was useful not only for proposing arguments, but also for organizing independent proof reconstructions, finding circular dependencies, generating counterexample searches, maintaining exact normalizations across branches, and turning negative results into reproducible stopping criteria. That is still not a substitute for expert review. Multiple AI audits are correlated evidence, not independent human verification. The purpose of publishing the complete record is to make the results easier to inspect, reproduce, criticize, and falsify.

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Repositories

Growth criteria and analytic number theory https://github.com/LeonardSEO/reflected-packet-growth-criteria 

Semilocal operator theory and certified obstruction results https://github.com/LeonardSEO/semilocal-zeta-operator-theory 

Exact reflected-packet oscillation theorem https://github.com/LeonardSEO/semilocal-reflected-packet-oscillation 

Smoothed positive spectral kernels and the scalar RH criterion https://github.com/LeonardSEO/smoothed-zeta-spectral-kernels 

Finite positivity certificates and their analytic limitation https://github.com/LeonardSEO/certified-riemann-xi-positivity 

Exact finite proxy/Feshbach reduction and the documented stopping point https://github.com/LeonardSEO/semilocal-weil-proxy-bridge

Feedback requested

I would especially value technically specific criticism of: the Laplace-transform pole argument behind the growth criterion; the cancellation and convergence conventions in the zero-only expansion; the domains, quotient operators, and determinant normalizations in the operator-theory paper; whether the cross-λ spectral–arithmetic defect has been isolated correctly; and whether this audit-and-stop workflow is a useful standard for LLM-assisted mathematics. For readers coming from mathematical physics: the connection is through selfadjoint spectral realizations, finite-rank perturbations, functional calculus, positive-definite kernels, spectral-shift formulas, and the Hilbert–Pólya motivation. No physical model or experimental claim is being made.

https://preview.redd.it/lc7tc6be7kfh1.png?width=1800&format=png&auto=webp&s=5113f1cf1cdaac486eaa13082474c287683b6afa