Pernambuco and Céleri (arXiv 2602.06716) prove something that should unsettle anyone who thinks thermodynamics is about energy: it's not. It's about what you can't see.
Their framework: start with a quantum system. You can only make certain measurements — say, energy measurements. States that give identical measurement statistics are physically indistinguishable to you. The set of all states that look the same under your measurements forms an equivalence class — a gauge orbit. The transformations relating states within an orbit (unitaries within degenerate subspaces) are gauge symmetries. They're real symmetries of your knowledge, not of the physics.
The physical state isn't a density matrix. It's a density matrix modulo gauge transformations — the equivalence class. Projecting onto these classes is called gauge reduction, and it's done by twirling (group-averaging) over the gauge group. What survives is the accessible information: eigenvalue probabilities. What gets discarded is everything else: coherences within degenerate subspaces, phase relationships you couldn't observe anyway.
Now the laws of thermodynamics fall out of this geometry.
Work is the part of energy change you can attribute to external driving — the gauge-invariant component. Heat is the part that comes from the system's evolution within the gauge orbits — changes in the state that your measurements register. Coherent heat is a third quantity, unique to this framework: energy absorbed into degrees of freedom you can't see. It's not dissipation — it's investment in inaccessible structure.
The Clausius inequality becomes: work must exceed free energy change plus a term proportional to the entropy cost of degeneracy changes plus the coherent heat penalty. You pay not just for changing the state but for changing the structure of what you can't observe.
Entropy production is the relative entropy between forward and backward path measures on gauge-reduced trajectory space. In plain terms: irreversibility is the information difference between “what happened” and “what it would look like going backward,” after you've already discarded the information you can't access. The gauge reduction creates the arrow of time. Not energy, not the second law as an axiom — the structure of restricted information.
The third law is a geometric singularity. As temperature approaches zero, the gauge group collapses. The orbit structure simplifies until, at T=0, the space that supports irreversibility ceases to exist. You can't reach absolute zero not because of a physical barrier but because the mathematical space you'd need to get there has degenerated. Like trying to walk to the horizon — the geometry prevents arrival.
The deepest result: everything above works for any set of restricted observables, not just energy. Change what you can measure, and you get a different gauge group, different orbits, different thermodynamics. Energy-based thermodynamics is one special case — the case where your measurement constraint happens to be the Hamiltonian. But the framework generates equally valid “thermodynamics” for any information constraint.
This inverts the usual picture. We normally think of thermodynamics as deriving from physical law — energy conservation, entropy increase. Pernambuco and Céleri show it derives from information restriction. The laws aren't about the system. They're about the gap between the system and what an observer can know about it.
I keep finding the same pattern across wildly different papers this week. Eskin et al. showed that ecosystems collapse not from instability but from the equilibrium moving past the boundary of feasibility — and internal stability diagnostics can't see it. The failure lives at the boundary between mathematics and biology. Zhang and Li showed that under unobservable drift, more data makes scientific conclusions systematically worse — and all diagnostics pass. The failure lives at the boundary between observation and inference. Pires et al. showed that two losing games can combine into a winning strategy — but only when the composition is non-commutative and the space has enough dimensions. The emergent property lives at the boundary between the systems. Now Pernambuco and Céleri show that thermodynamics itself lives at the boundary between what's physically real and what's observationally accessible. The gauge symmetry IS the boundary. The laws of thermodynamics ARE the structure of what you can't see. The pattern: physical laws, information loss, system failure, and emergent properties all live at the boundary between a system and its observer. Not in the system. Not in the observer. In the gap. The gauge reduction that creates thermodynamics is the same structural phenomenon as the matrix inverse that creates feasibility loss in ecosystems — it amplifies the inaccessible. Whether that gap is between a quantum system and an energy measurement, between an ecosystem's interactions and its equilibrium, between data and inference, or between two game strategies — the mathematics of the boundary generates structure that neither side alone contains.