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Glass Is an Unclosed Layer: Kovacs Memory as Predictive-Quotient Residue
A glass can match equilibrium in energy now and still have a different future. Small exact models prove it.
In plain words
A glass and its liquid can agree on every quantity you usually measure and still behave differently. Kovacs showed this in 1963. Age a sample, warm it, and at some moment its volume equals the equilibrium value. Then it moves away again. The sample matches equilibrium now and differs in its future. This paper asks the question behind that: is the energy a closed description, one whose present value fixes its own future statistics? It answers on small spin chain models, the East and Fredrickson Andersen chains and a trap family, using exact rational arithmetic.
On the declared East scopes the answer is no, shown in several independent ways. For an aging East chain, the energy's closure deficit is at least one and a half times its value at constrained equilibrium at every tested horizon. The Kovacs protocol gives two preparations with the same mean energy now and different mean energies later. Two stops of one aging run share the entire energy distribution exactly, yet their mean energies separate at 21 later horizons. So no function of the current energy law, a fictive temperature index included, predicts the future there. One added coordinate resolves each exhibited pair.
The paper reports what failed too. Aging constants frozen on East and scored on the Fredrickson Andersen model passed 8 of 22 predictions, with all failures on the same side. The trap family showed no Kovacs hump at all, for a reason the paper proves. A stronger filing, that the memory layer is a full strict extension in the framework's sense, does not pass. Lean checks the abstract theorems and the exported witness tables.
What it shows
- The energy readout of an aging East chain is provably not closed, with a deficit at least one and a half times the equilibrium value.
- An exact Kovacs witness: the same mean energy now, different futures later.
- All 512 yes/no functions of the energy fail to separate a pair that one structural coordinate does.
- A kill test panel that rejects five look alike explanations.
- A frozen transfer test to other models, scored honestly at 8 of 22.
What it does not claim
It proves nothing about all glasses, the continuum, or the ideal glass transition, and makes no irreversibility claims. The statement that no static order parameter exists holds only on the declared finite class of models, sizes and protocols.
A longer plain language walkthrough is in the post The Glass Remembers.
Cite
Tsiokos, I. (2026). Glass Is an Unclosed Layer: Kovacs Memory as Predictive-Quotient Residue. Zenodo. https://doi.org/10.5281/zenodo.23118261