Local Equilibrium Without a Global Bath: Sector Qualification for Record Thermodynamics

When does a physical system have a temperature? Laboratory practice answers by inheritance: the system sits in a cryostat, the cryostat has a dial, and the dial's number is the system's temperature. The presumption behind that answer — temperature as environmentally conferred, universally available, needing a thermometer but never qualification conditions — is what this paper names the Global Bath, and for one concrete family of systems it retires the presumption by construction. The systems are the record-bearing sectors of this sequence's adopted record dynamics: the subsystems that hold certified records and export them at intensity nu. Paper 36's Clausius reading needs such sectors to be locally equilibrated wherever it is applied; Paper 42 fixed the sector-existence criterion (row A1 of its recovery audit) and honestly deferred the exhibition; Paper 44 attached the criterion to its circuit-QED design as a named-audit instrument, specified, never executed; and Paper 51 located the which-entropy circularity as a cycle whose qualification half it assigned to this paper. Nothing in the sequence had yet exhibited one qualifying sector. This paper — the fifty-second, centre panel of the Entropy and Thermodynamic Standing Program (Papers 51-53) — builds the qualification gate and runs it. Delivered: LocalEqProf, a per-sector qualification profile, and the two-tier gate G-LE. The existence tier is Paper 42's A1 criterion upgraded to checkable components — timescale separation sigmaₛep = tauᵣelax * nu <= 1e-1 (a declared decade rule), the stationarity residual against 42's epsilonₑq = 1e-2, production-ledger stationarity, and 42's scale clause completed to the declared inequality ellc / Lₛupp <= 1e-1 — and "local equilibrium" in this paper means exactly this tier, 42's own usage. The temperature tier is what inheritance never asks: the sector's stationary state must be Gibbsian under its own declared relaxation dynamics (the KMS-equivalent detailed-balance check for the discrete-mode model, at declared order), and the inverse temperature read from its stationary occupancy must agree with the one read from its up/down transition rates within a declared tolerance (1e-2). One honesty row governs everything: the candidate's relaxation model does consume a local, calibrated dissipative environment — Paper 44's instrument speaks plainly of the injected-bath intensities and their calibration — and this paper does not pretend otherwise. What the gate never consumes is a universal-bath premise or an unconditioned temperature assignment: the calibration number by itself assigns nothing; assignment waits on the gate, and the same gate refuses the same calibrated bath's number on the same hardware when the sector's profile fails. The strict-gate chain — record-bearing sector existence, local equilibrium, temperature assignment, and flux-accounting standing are four different standings, no two equal — is carried by four displayed instances: a ramped sector failing stationarity; a declared-synthetic driven sector that passes the entire existence tier while its stationary number distribution is visibly non-Gibbsian and its two temperature probes disagree by a factor of roughly 1. 8 of their mean (no temperature licensed — stationarity is not thermality) ; a fully qualified sector with certified export nulled by architecture (Paper 50's unrecorded arm), showing a licensed temperature determines nothing about flux accounting — the link handed to Paper 53, never adjudicated here; and the frozen benchmark's own two regimes under one gate, where Regime M fails the separation component at every declared corner (sigmaₛep = 0. 3 and 7. 0) while Regime V passes. The relaxation time is defined, not chosen — the inverse spectral gap of the sector's declared generator (tauᵣelax = 2/kappa for the damped mode), with the tempting misreading (the qubit's induced-dephasing time, a different subsystem) displayed and foreclosed. The flagship executes the candidate test: the Regime-V benchmark sector, specified as a record-bearing sector (the monitored register whose certified fraction of the output line is amplified into the persistent monitoring record; export intensity nu = etacert * kappa * nᵣes, Paper 42's own audit identity at the idle occupancy), passes every component at both declared corners — sigmaₛep = 1. 2e-3 and 2. 8e-2 against the declared 1e-1; the production ledger exhibited row by row, constant; the stationary number distribution geometric with ratio 0. 0196 displayed; the state-probe and rate-probe temperatures in exact agreement — and the derived Tₗoc = hbar*omegac / kB ln (1 + 1/nᵣes), approximately 73 mK at the declared device-class frequency of 6. 0 GHz, agrees with the injected-bath calibration as a displayed consistency row, never a premise. The verdict pair is (LocalEqStatus, TempStatus) = (exhibited, assigned) — at model grade: a fully specified sector at frozen corpus parameters passing a gate by displayed arithmetic, never a device claim. Consequences, each in its pre-committed wording: Paper 42-A1's deferred exhibition is performed — tied to the existence tier alone, which is all 42's row ever asked, and under this paper's declared discrete-mode completion of the scale clause (the literal hydrodynamic reading being inapplicable to the sector class, and so recorded) — recorded by annotation, with Paper 42 standing unedited; Paper 51's exit-Q moves from typed to discharged-in-one-instance; the first non-empty use-domain statement for Paper 36's Clausius/Jacobson route is issued, narrow and said to be narrow; the qualified domain and the licensed local temperature are handed to Paper 53; and the A1 instrument receives add-only, protocol-grade upgrade material. No datum is scored; no band, floor, or feasibility verdict moves; no eligibility verdict is pronounced; no Einstein equation is derived; the substantive thesis that temperature is ontologically fundamental is declined, not refuted. What retires is temperature-by-inheritance-without-qualification — for record-bearing sectors, a temperature is something a sector qualifies for.