Carlos Toledo
Research notes · Brazil

Verification for machine-generated mathematics

I am Carlos Toledo. I check numerical and mathematical claims without trusting how they were produced — a language model’s, a published paper’s, or my own — and the answer is either a certified enclosure that provably contains the true value, or a plain statement that the check did not close. Every page below ships the program that produced its numbers, and every check has been shown to fail on a deliberately broken copy of that program.

Where this work has gone, and what has come back
Stated exactly: filed is not accepted, and posted is not replied. Nothing on this site has been peer-reviewed, and no result here has yet been reproduced by anyone else — which is why every page ships the program that would refute it.

Start here — for a reviewer with ten minutes

Two results and one proposal, each in three lines: the claim somebody else made, what I did to it, and what happened. Nothing below is stronger than the page it links to — where a check did not close, these lines say so.

How a result earns a green light

The house rule is that green proves nothing until it can go red. Three obligations, on every page here:

  1. Every certificate ships with its falsifier.

    The claim and the program that would refute it are published together. Believing a page never requires trusting its author — re-run the check and watch it pass, or fail.

  2. Every check is proven able to fail.

    Before a green counts, the same check runs on a deliberately broken copy of the program and must go red — then green again on the clean one. A check that cannot fail proves nothing.

  3. REFUSED is an output, not a failure.

    When the mathematics does not support a certificate, the verdict says so. One reproduction below refuses a scenario because it admits more than one equilibrium — the page states that instead of quietly picking one.

The full toolchain, with its own code on the page: /stack.

Run one now

Node only, nothing installed — one verdict says CERTIFIED, the planted mutant says REFUSED, in about a minute.

git clone --depth 1 https://github.com/carlostoledo1891/mfg-lab
cd mfg-lab/technical-reports/alien-science
node dual-client.js --fixture heldout-ccs-es
node dual-client.js --fixture heldout-ccs-es --plant-mutant

The last command breaks the program on purpose. Watching the verdict flip from CERTIFIED to REFUSED is the point — it is the proof that the first answer could have come out differently.

For the record: four shipping cases, dated — errors found in this site’s own published work, none reported by a reader first. Each is in the errata log with its witness, its fix, and a control that now goes red — never quietly corrected.

More

Everything else, including the pages this list leaves out, is under Reports.