Why Mathematics Even Works

In plain words

Imaginary numbers are not real. Infinitesimals are not real scalars. Roots of a polynomial need not lie in the base field. Yet all of them solve problems stated entirely in the lower layer.

The paper separates the lower layer, where the problem lives, from a promoted layer with extra objects. Strict usefulness, it proves, forces two things at once: a hidden object that does not descend, and an audited boundary expression that does.

Four witnesses are worked in full: imaginary numbers, dual numbers, Galois roots and graph cohomology. A transfer theorem says a validated profile is portable without pretending the objects are the same.

What it shows

• The Strict Audited Utility certificate, extracted rather than stipulated.
• An essential boundary necessity theorem.
• Four canonical witnesses with distinct profiles.

What it does not claim

It is about strict useful promoted traces, not mathematical practice in general.

Cite

Tsiokos, I. (2026). Why Mathematics Even Works. Zenodo. https://doi.org/10.5281/zenodo.20712761