Alec Kriebel · Independent Researcher
Exact AI-Assisted Research in Quantum Structures and Mathematical BiologyQuantum structures and mathematical biology
Independent research on quantum correlations, quantum graph theory, stochastic reaction networks, evolutionary dynamics, and exact mathematical models. Manuscripts are released with source code, proof maps, exact certificates, and explicit scope statements to support specialist scrutiny and collaboration.
Research status. Manuscripts are unrefereed unless otherwise stated. The research was developed with extensive generative-AI assistance, including systems from OpenAI and Anthropic, in exploration, proof development, software generation, verification design, literature organization, and drafting. Exact source, certificates, and provenance records are provided for independent scrutiny.
Passing the supplied checks verifies encoded identities and computations; it does not replace specialist review.
Six completed manuscripts with focused theorem statements, explicit boundaries, and reproducible supporting materials.
Research program
Finite-dimensional quantum graph theory, Bell inequalities, and exact operator bounds.
Quantum graph theory · Finite-dimensional projector colorings
Resolves Lalonde’s proposed finite-witness family in the standard finite-dimensional model, allowing arbitrary finite projector rank. The first new unrestricted case is the one-apex graph \(G_{19}\vee K_1\); no commuting-operator extension is claimed.
Quantum information · Fixed-qubit Bell behaviors
Proves equality of the shared-randomness-convexified fixed-qubit behavior sets when each party has two inputs, then gives an exact rational \(3\times2\) separation and minimum-setting classification. The qualitative \(3\times2\) separation is prior art; no raw-set, same-state, or operator-level POVM/PVM equality is claimed.
Quantum information · Cyclic Bell inequalities
Determines exact all-dimensional values, including the commuting-operator strengthening, and identifies phase-permutation freedom producing nonuniform exact maximizers. The result limits what the maximum scalar Bell value alone certifies; it does not contradict randomness conclusions conditioned on the full canonical behavior.
Research program
Stochastic reaction networks, evolutionary dynamics, and exact mass-action geometry.
Stochastic reaction networks · Positive recurrence
Proves that weak reversibility makes every population reachability set a closed communicating class. Under the additional one-linkage and bimolecular hypotheses, each nonabsorbing reachable class is positive recurrent for every positive rate vector. This removes the prior pure-species-complex condition; multiple linkage classes and higher molecularity remain open for recurrence.
Evolutionary graph theory · Simultaneous amplification
Constructs one fitness-independent graph family that simultaneously amplifies Bd and dB fixation for every fixed fitness below an exact algebraic threshold \(1.5028569\ldots>3/2\). This is a strict lower-bound improvement; the unrestricted exact value of \(R_{\mathrm{sim}}\) remains open.
Reaction-network dynamics · Reversible mass action
Determines a four-parameter positive rate family on one fixed reversible graph with a common compact positive equilibrium ellipse, generic geometric coprimality, and exact rate-cone and stability information. It is a fine-tuned family, not a universal realization theorem.
Further completed manuscripts in topology, group theory, recreational mathematics, quantum structures, and evolutionary dynamics.
Linear dynamics · Oberwolfach Report 19/2024
An affirmative answer to the arbitrary-Γ question in the original separable setting, made explicit from earlier criteria of Abbar and Abbar–Kuznetsova. The five-page note gives a direct proof and credits the prior scalar criterion and amplification mechanism.
Homogeneous forms · OWR-17293-016
An affirmative answer to Brandes's 2019 question: a real basis makes the diagonal ordered coefficients one and every mixed entry strictly between zero and one. The three-page proof is accompanied by an independent Cauchy–Schwarz argument and exact rational certificates.
Convex geometry · OWR-4136-011
The general convex-body equality case is exactly an inscribed simplex. A countable extreme-point partition and an exact deficit identity extend the known polytope classification.
Statistical geometry · AMR-059-0011
The round cylinder answers Kurose’s printed Question 3(e) negatively. The four-page note proves radial integrability and gives a two-equation curvature obstruction, including a non-self-dual extension.
Computational topology · Discrete Morse theory
A triangulated interval has a nonincident persistence pair with zero gradient paths in the fixed incident-pair matching. Three boundary reductions disprove Conjecture 2 in Knudson’s 2008 Oberwolfach report.
Finite group theory · Kourovka Notebook 16.45
An explicit group of order 100920 has maximal inclusion-minimal base size three and independence number four. The structural proof includes every nonfaithful and intransitive action.
Recreational mathematics · Sum–product matrices
Proves the DeVincentis conjecture: a matrix of globally distinct positive integers with every row sum and column product equal cannot have three or more rows. An elementary inequality covers every dimension and common value.
Recreational mathematics · Chess attack configurations
Proves that no finite nonempty placement has every knight attack four rooks while every rook sees two knights, with no same-type attacks. A stronger geometric theorem permits knight-to-knight attacks and is accompanied by a separately checkable local impossibility certificate.
Finite group theory · Kourovka Notebook 16.63
Constructs a finite group with \(|G|=|\operatorname{Aut}(G)|=1009^{52359}\), answering MacHale’s existence question through an explicit Lie lattice and an exact count of the full automorphism group.
Yang–Baxter operators · Exceptional Hecke class
Gives a sparse exact \(16\times16\) exceptional Hecke operator, faithful minimal Jones–Wenzl localization, and all-strand equivalence transporting known Family III finite-image and Clifford properties; its direct enhancement and skein calculation yield twice HOMFLYPT at \((i,i)\).
Evolutionary graph theory · Fixed finite structures
Proves a fixed-finite-graph impossibility theorem, distinct from the growing-family simultaneous-amplification result, and supplies exact complete-support and low-dimensional classifications.
Focused entry points for checking load-bearing claims and reproducing exact artifacts.
Independent reproduction, corrections, and substantive scientific collaboration are welcome. Selected manuscripts provide concise overviews, proof maps, exact source, and focused reviewer questions so that specialists can inspect the load-bearing claims without navigating the full research archive.
Technical correspondence: me@aleckriebel.com
A human-directed workflow with extensive model assistance, explicit evidence boundaries, and open artifacts.
Alec Kriebel selects problems, defines scope, directs research iterations, determines which claims are released, and is responsible for public presentation.
Generative-AI systems, including models from OpenAI and Anthropic, have been used extensively in mathematical exploration, proof and counterexample search, software development, exact computation, adversarial review, literature organization, and drafting. Manuscript-specific disclosures identify the tools used.
Model output is not evidence by itself. Releases provide, where possible, exact symbolic checks, rational or algebraic certificates, independent implementations, proof dependency maps, finite exhaustive certificates, and immutable source records.
Internal audits and computational verification do not replace subject-matter review, peer review, or independent reproduction. Not every proof is formally verified.