Half Ounce Research

Daniel Kirtchakov

Independent researcher · daniel@halfounce.io · github.com/05oz

I work on polynomial maps, computational algebra, and the verification of algebraic claims by machine-checkable certificates. Recent work concerns the structure of the counterexample to the Jacobian Conjecture announced in July 2026 — in particular degree minimality within its equivariant class — and the certification of parameters of quantum error-correcting codes.

The organising principle of this work is that a computational claim should travel with a witness that a reader can check independently, without repeating the search and without trusting the software that found it.

Preprints

D. Kirtchakov, Degree minimality in the equivariant class of the Alpöge Keller map, and the moment-map structure of its cotangent lift, August 2026, 20 pp. Two new results about the ℂ*-equivariant class of the degree-7 Keller map announced by L. Alpöge in July 2026. First, a degree-minimality theorem: in that class no Keller map of degree ≤ 6 exists at all in the sector containing Alpöge's map, and every degree-≤6 map elsewhere in the class is an automorphism — established by eight Gröbner-basis unit-ideal certificates over ℚ, reproduced mod 32003. Second, a moment-map identity: the ℂ*-action lifts Hamiltonianly to the cotangent bundle and the cotangent lift preserves the moment map exactly. The cotangent lift itself, its exact preservation of the symplectic form, and the associated Weyl-algebra endomorphism are W. G. P. Mayner's (July 2026), independently verified here; the note adds their identification as a Poisson-conjecture witness and the quantization identity relating them. The equivariant reduction, master equation, S₃ cover and image theorem were obtained independently but are anticipated by T. Shaska (arXiv:2607.20210), by the anonymous ulam.ai note, and by Mayner; priority is credited in the paper. [ PDF · certificates · scripts — links on release ]

Work in progress

Certified parameters for quantum error-correcting codes. The minimum distance of a stabiliser code admits a two-sided certificate: an upper bound is an explicit logical operator, verifiable by two matrix–vector products; a lower bound is a proof that no lighter logical operator exists, expressible as a refutation certificate. Neither is currently supplied with published code parameters. The aim is a checker and a small corpus of certified parameters for codes in common use.

Degree minimality across weight systems. The equivariant reduction extends to the weight systems (1, −1, −k). Certificates for k = 1 and k = 3 are in preparation.

Software and certificates

Every published claim is accompanied by the script that establishes it. The scripts assert their results and fail loudly; the certificates are short enough to check by hand-written code with no solver in the loop.

git clone https://github.com/05oz/certify
cd certify
python3 -m venv venv && venv/bin/pip install sympy
venv/bin/python scripts/core_verify.py    # the map, its Jacobian, the collisions
venv/bin/python scripts/cover_verify.py   # the cover and the image theorem
venv/bin/python scripts/weyl_verify.py    # the Weyl-algebra endomorphism

Reproducing the Gröbner-basis certificates additionally requires msolve. Instructions are in the repository.

Correspondence

Corrections, prior references, and counterexamples are welcome and will be credited. If a result here is already known, a pointer to the source is the most useful thing you can send.