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 nonexplosion on every closed communicating class and positive recurrence on every nonabsorbing closed class, for one-linkage weakly reversible networks of molecularity at most two and arbitrary positive rates. This removes the prior pure-species-complex condition; multiple linkage classes and higher molecularity remain open.
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.
Two further completed manuscripts with narrower roles in the current portfolio.
Yang–Baxter operators · Exceptional Hecke class
Gives an explicit \(16\times16\) unitary Yang–Baxter operator in the smallest unresolved local dimension of the stated exceptional family, together with a claimed Hecke/Jones–Wenzl localization consequence.
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.