Five entropy-like or entropy-adjacent objects do load-bearing work in this sequence — three genuine entropy-valued state functionals (the entanglement entropy of a region; the coarse-grained state entropy of a declared record-bearing support; an access-weighted accessible record entropy), one warrant-side public-standing profile routinely conflated with entropy, and one induced-gravity counterterm routinely reached through horizon-entropy arguments. The last two are not entropies, and they are kept in the ledger deliberately: they are where the conflations land. Paper 36 crossed the dynamical-coupling gate by reading the Clausius entropy as the second of these — importing, as a live defect, the which-entropy circularity: the entropy whose standardness underwrites the Clausius relation depends on a declared coarse-graining that the relation was supposed to legitimate. Paper 42 left two recovery rows conditional on it (A1, local equilibrium — exhibition deferred; A3, the Clausius defect DeltaCl <= epsilonCl — defined, never evaluated) ; Paper 47 built a four-column entropy ledger to bridge-audit depth only and named this paper its sole successor candidate. Whoever would settle the matter by announcing which entropy is "the right one, " context-free, is a Global Thermometer — and this paper, opening the fifteenth grouping (Papers 51-53, the Entropy and Thermodynamic Standing Program), retires that seat by construction. Delivered: the five-column type ledger — each object typed by definition, owner, codomain (R-valued state functional / profile / coefficient motion), dependencies, divergence structure, and world/warrant side, with an annex of typed non-members and a hard completeness sweep: every entropy-like symbol in the thermodynamic debt chain (Papers 3, 6, 9, 29, 33, 36, 38, 39, 42, 47, plus Paper 44's A1/A3 instrument symbols) lands in exactly one column or the annex, displayed. The non-substitutability matrix: all ten pairwise conflations refuted at the right type — numeric witnesses where codomains match (a declared finite model family computing three pairwise-distinct values, 0. 5731, 0. 4982, and 0. 4515 nats, at one displayed parameter point), type witnesses where they do not (Paper 29's audited-state underdetermination theorem, unconditional part, for the standing profile; Paper 47's kill, at its exact scope, for the counterterm) — each cell carrying a named breakage the corpus already paid for. The licensed-move graph: every legitimate transition between columns as a gated edge; the single world-to-warrant crossing sourced by the capacity-attribution-provenance package, never by a scalar entropy; nothing reaching the counterterm except through Paper 47's closed gate. On that graph the which-entropy circularity is located: it is a mixed cycle — one transition edge (the record-bearing entropy's occupancy of the Clausius slot) and one justification meta-edge (the admissibility gates that fix the coarse-graining were built to make the Clausius reading standard). Locating it types its exit: a pair of independent re-derivations — a sector-internal qualification of the graining that does not consume Clausius standardness, and an accounting boundary for exported heat that does not presuppose the audited ledger. Both halves are checked syntactically against the gate definitions and both are typed; the realized verdict is TaxStatus = typed-scoped, with the qualification half handed to Paper 52 (local equilibrium; the A1 exhibition) and the accounting half to Paper 53 (Clausius balance; the A3 evaluation) as written consumption specifications. The defect is located and its exit typed under discharge — not resolved: nothing here exhibits an equilibrated sector or evaluates a Clausius defect. No datum is scored; no band, floor, or feasibility verdict moves; the execution layer is untouched; the substantive hope that a final theory will unify these entropies is declined, not refuted. What retires is the seat: which entropy occupies a thermodynamic role is fixed by a declared ledger and its licensed moves — not by a thermometer held from outside.
Tomoyuki Uchida (Sat,) studied this question.