The Sign Premise Is a State Clause: Pumped Baths, the de Sitter Thermostat, and the Limits of Scale Without Law

Abstract

A double-edged sequel to 'Scale Without Law' (10.5281/zenodo.21016309), adversarially verified. (1) POSITIVE EDGE: the anti-MOND sign theorem's operative premise is the bath STATE, not ghost-freedom -- an explicitly ghost-free two-level bath in a population-inverted (non-KMS) steady state flips the sign of the worldline mass shift (delta_m proportional to p_g - p_e), with unitarity intact; lab-real physics (gain-assisted anomalous dispersion, Wang-Kuzmich-Dogariu 2000); the spectral density generalizes in a stationary NESS to rho_NESS(omega) = sum |B_nm|^2 (p_n - p_m) delta(omega - omega_mn). Ghost-freedom and unitarity do NOT force the anti-MOND sign; passivity does. (2) A NEW RIGIDITY WALL: for linear coupling to a FREE bosonic field the commutator is a c-number, so the response is STATE-INDEPENDENT -- no thermal, squeezed, or pumped Gaussian state of a free field can flip the sign; pumping free fields only adds noise. The escape requires an ANHARMONIC medium; the de Sitter-Unruh free-field bath -- this framework's own premise -- cannot carry the MOND sign at ANY occupation. (3) THE THERMOSTAT THEOREM: 'the de Sitter horizon is the pump' fails four independent ways -- the Gibbons-Hawking state is KMS at T_dS (= the passivity clause; zero extractable work, Pusz-Woronowicz 1978); horizon occupation in the galactic band (~10^2-10^3 H0) is < 10^-900; the real universe's departure from de Sitter supplies no in-band inversion (short of the minimum pumping rate by ~10^10 under maximally generous assumptions); and counterfactually, gain at the horizon's own frequency produces the ANTI-MOND dispersive sign in the band above it. The clean statement: the very thermality that lets T_dS set the SCALE a0 forbids the horizon from supplying the SIGN. (4) Physical gain saturation cannot produce mu(a/a0): wrong asymptotic exponents at both ends, an unenforced c=1 equivalence-principle fine-tuning, and a free medium scale -- a0 = cH_Lambda/Z is not delivered. NET: the sign-theorem escape and the published galactic-pump specification MERGE into one precisely-specified, currently unoccupied requirement (universal anharmonic cosmologically-regulated medium, band-limited T-odd coupling, maintained inversion, hand-shaped saturation, exact EP condition, external a0) -- with a scheduled class falsifier: wide-binary gamma -> 1 at Gaia DR4. Scale Without Law's trichotomy stands unweakened on its own turf; its nonlocal horn is now stated with edges instead of hope. Claims explicitly NOT made: no pump, no derived mu or a0, no weakening of the passive theorem, nothing beyond gravity. All results backed by 7 committed scripts (exit 0), independently re-derived by an adversarial verifier. The author retracted all earlier 'theory of everything' claims (2026-06-23); this note makes none.

The authors' abstract, as published at the source. Zenodo (CERN European Organization for Nuclear Research), 2026 · DOI ↗