CALLIOPE · BCOM

The word ``reversible'' compresses at least three inequivalent ideas: that the dynamics can be undone (flow invertibility), that distinctions among states are never blurred (measure preservation), and that the reversed movie obeys the same law (time-reversal symmetry). This note separates five properties of autonomous ordinary differential equations --- well-posedness, flow invertibility, incompressibility, time-reversal symmetry, and energy conservation --- and maps the full lattice of implications among them, with a minimal counterexample for every failure. A one-line spectral obstruction lemma suffices to prove genuine irreversibility, the nonexistence of any reversing involution. Three payoffs emerge. First, time-reversal symmetry is logically orthogonal to both regularity and conservation, yet rich in consequences of its own: mirror-paired attractors and repellers, conservative behavior on the symmetry set, symmetric periodic orbits by shooting, reversible KAM, and Onsager--Casimir reciprocity. Second, energy conservation and information preservation, independent in general, coincide in exactly one package --- closed autonomous Hamiltonian dynamics. Third, that coincidence resolves the Landauer paradox: erasing a bit costs $k_BT 2$ precisely because the fundamental dynamics cannot erase. A quantum postscript translates the matrix to Hilbert space, where the logic tightens: well-posedness becomes free (Stone's theorem), and invertibility and distinguishability preservation fuse into the single primitive of unitarity, with Koopman's lift exposing classical incompressibility as the shadow of that bundling.

"Reversible" is three different things, and conflating them is the source of most confusion about information, energy, and the arrow of time.

The paper's central move is surgical: it takes the word "reversible" apart into five distinct properties of a dynamical system — well-posedness, flow invertibility, incompressibility, time-reversal symmetry, and energy conservation — and then systematically maps which ones imply which others. The answer, laid out in an explicit implication matrix with a counterexample for every failed arrow, is mostly "none of them." Flow invertibility (trajectories don't merge) does not imply incompressibility (volumes don't shrink). Time-reversal symmetry (the movie runs the same backwards) implies neither. Energy conservation implies none of the others. The one exception is the full Hamiltonian package: closed autonomous Hamiltonian dynamics delivers all of them at once, and that coincidence is not accidental — it's the structural fact the rest of the paper unpacks.

The paper's sharpest tool is a one-line spectral lemma for proving genuine irreversibility. The problem is that ruling out a specific time-reversal involution is easy, but ruling out all possible involutions requires an invariant obstruction. The lemma provides one: if a reversing symmetry existed, it would have to pair every attractor with a repeller of exactly mirrored eigenvalues. So check the equilibria. The system ẋ = −x has a single purely attracting equilibrium (spectrum {−1}) with no repeller anywhere in the system to pair it with — irreversible under any smooth involution, full stop. This turns what looks like an infinite search into a single linear-algebra computation.

Time-reversal symmetry turns out to be the odd property out: implied by nothing, implying nothing about information or energy, yet structurally rich in its own right. A reversible system in this sense must have mirror-paired attractors and repellers, cannot have asymptotically stable equilibria on the symmetry set, produces periodic orbits wherever a trajectory hits the mirror twice, supports a KAM theorem without any symplectic structure, and underwrites Onsager's reciprocal relations in transport theory. These are heavy consequences — they just live on a logically orthogonal axis from information preservation.

The paper then inverts the usual hierarchy between energy and information. Rather than taking energy as primitive, it proves that any closed bijective dynamics — one that neither merges nor creates states — automatically conserves a quantity: the orbit label, the equivalence class of states reachable from each other under the dynamics. Physical energy is then the special case where this label happens to admit a short, stable, computable description (a Hamiltonian or Noether charge). When no such compression exists, the label is still conserved, just inaccessible. This reframing also dissolves the Landauer paradox cleanly: erasing a bit costs k_BT ln 2 of heat because the fundamental dynamics cannot erase — the cost is the price of exporting a distinction into the environment that the microscopic laws forbid destroying.

A quantum postscript shows the same matrix tightens in Hilbert space. Well-posedness becomes free (Stone's theorem handles it). More strikingly, flow invertibility and incompressibility — independent in classical mechanics — fuse into a single primitive: unitarity. A quantum channel whose inverse is also a physical channel must be unitary conjugation (by Kadison–Wigner rigidity plus complete positivity), so physical invertibility alone implies full distinguishability preservation. Koopman's lift makes the connection precise: classical incompressibility is exactly the condition for the lifted classical dynamics to be a unitary operator on L²(phase space) — Liouville's theorem is the classical shadow of quantum bundling.