The Hawking spectrum's insensitivity to modified dispersion holds only under conditions such as a free-falling vacuum and adiabatic evolution.

The claim conjoins a sufficiency half (the named conditions suffice for the Hawking spectrum to be recovered) and a necessity half ("only": outside such conditions, deviations occur). Both halves were checked against the primary literature and secondary syntheses.

Sufficiency: Unruh and Schützhold, "Universality of the Hawking effect" (arxiv.org/abs/gr-qc/0408009; Phys. Rev. D 71, 024028, 2005) state general conditions on ultra-high-energy degrees of freedom under which the Hawking effect is reproduced to lowest order: existence of a freely falling frame near the horizon, outgoing modes born in their ground state with respect to freely falling observers, and adiabatic evolution. This generalizes the model calculations recorded in the supporting subclaim that modified-dispersion black-hole models preserve the thermal spectrum↗︎ (Unruh 1995; Brout, Massar, Parentani and Spindel 1995; Corley and Jacobson 1996), which are undisputed as mathematical results.

Necessity: Unruh and Schützhold exhibit counterexamples where their conditions fail and the spectrum deviates. Barceló, Garay and Jannes (arxiv.org/abs/0807.4147) show that for superluminal dispersion the standard result depends on physics inside the horizon and on ground-state conditions in the free-fall frame. Coutant, Parentani and Finazzi's S-matrix treatment (arxiv.org/pdf/1106.1805) quantifies deviations when scale separation is not maintained, and notes that the general effect of dispersion is controlled precisely by such conditions. Later work (e.g. Phys. Rev. D 109, 025001, 2024, journals.aps.org/prd/abstract/10.1103/PhysRevD.109.025001) continues to confirm robustness within, and probe deviations outside, this regime. The philosophy-of-physics analyses of universality arguments (Gryb, Palacios and Thébault, arxiv.org/pdf/1812.07078; philsci-archive.pitt.edu/15900/) take the conditionality as given and debate only its evidential significance.

Weighing: the claim reads "only under conditions such as," asserting sufficiency of the named conditions plus the existence of failures outside them, not that every violation destroys thermality; the evidence supports exactly that reading, and no credible source disputes it. A candidate subclaim making the necessity half explicit resolved, on matching, to the negation of the parent claim that this claim already qualifies via a contradicts edge, so that content is carried here in prose rather than as a node. Status stays supported rather than verified because this pass rested on the papers' stated results and secondary syntheses rather than an independent working-through of the derivation; credence 0.9 reflects that the result is uncontested among specialists. What would change the conclusion: a direct verification of the theorem and counterexamples would raise the status toward verified; a demonstration that thermality is generic even outside the stated conditions would contradict the necessity half and lower it.