Infotropy: a candidate structural unification of information and entropy

Information and entropy are two readings of one thing. Read one irreversible event from the cost position and you get entropy — what was spent, spread, lost, the part physics already names. Read the same event from the record position and you get information — the durable structure that survives and goes on to shape what comes next. The thesis of this paper is that these are not opposites and not two substances but two faces of one structure, the way electricity and magnetism turned out to be two aspects of one electromagnetic field. Infotropy is the name for that candidate unification, and it is the subject of the paper. The mathematics is inherited, not new — the Shannon–Boltzmann identity, Jaynes, Landauer already give the two one shared form; what the account adds is a reading of that shared form as one process seen from two positions over two different sample spaces. Record formation is the frame the reading is tested through (a record carries the costly history that formed it and steers what comes next), but the frame is the instrument, not the thesis. A unification with universal reach is exactly where a theory is most tempted to mean nothing, so the claim is stated with the knife that kills it. If a record's formation history could be rebuilt from the cost ledger alone, the two readings are one wearing two names and the account is false. That falsifier was run. In a driven-dissipative φ⁴ regime a reconstructor given only the cost ledger recovers about 0.001 bits of the real record's formation history but about 1.69 bits on a fixture where the cost ledger genuinely carries it — a firewall holding the two readings apart. In that same regime the record carries recoverable formation history, clearing an end-state-matched self-organization rival by at least +1.03 bits (reproduced on a fresh engine), and steers its downstream future by +0.162 bits at horizon 80 with content held apart from cost. This is a candidate: a strong, live thesis under test — earned on both faces in one regime, with the rest of the perimeter open and stated at full strength; unproven is not disproven. Whether the two-sided structure recurs as a regime-general law across physically unrelated regimes is the open frontier, named as conjecture, not shown. A pre-registered test in a discrete substrate confirmed the asymmetry the cross-regime claim rests on — supply the record-forming condition and recoverable formation history survives a live eraser, remove it and the history is destroyed — but the record there was long-lived and eroding rather than permanent, so the run's sealed grade, scored against a stricter flat-permanence bar, came back indeterminate. The deposit includes the release manuscript (PDF and Markdown source) with its ten figures, and a verified cold-run reproducibility package that recomputes, from a hash-locked engine, the four numbers the keystone rides on (the cost-firewall leak fixture, the real-record cost-only recovery, the REAL−SELF_ORG carries margin, and the steers effect), with the estimator, seed policy, the 0.10-bit decision threshold, the bootstrap 95% confidence intervals, the SHA-256 pre-registration code-hash lock, and a transparent grader-logic correction (a superseded inline REFUTE flip on a noise-floor permutation tick, adjudicated to PASS against the locked threshold, with no locked value altered). The thesis is held speculative: no unification is asserted as proven. The keystone is one regime, one earned point, independently reproduced and held provisional; the cross-regime conjecture remains open. "Value" throughout means physically recoverable information about a costly cause, never worth or meaning to an observer; the account makes no normative claim.