Theorem 12.2 is an implication from these eight assumptions (§2.4, pp. 7–9). Their exact statements are in the result inventory; this page records what each one is for, and what its status is.
| assumption | what it supplies | |
|---|---|---|
| A1 | cutoff construction | existence and normalization of the ground state Ω_g of H(g) |
| A2 | Feynman–Kac–Nelson transfer | the slab objects h_{T,g}, ψ_s, c_g and the Euclidean ↔ Hamiltonian passage |
| A3 | finite-volume stability | U_h in every L^p and e^{−ϱU_h} integrable, so 0 < Z(h) < ∞ and the interacting measures dq_h are well defined |
| A4 | free hypercontractivity | the L^p smoothing used in the moment estimates |
| A5 | domain smoothing; common core | Ω_g ∈ D(V_j), and a core on which the algebra acts |
| A6 | free propagation; patching; the dynamics α | the exact light cone and the patched limit dynamics |
| A7 | covariance data | translation covariance and parity |
| A8 | the GJS package | the cluster-expansion estimates: CE1, CE2, the printed norms, and U_GJS |
A1–A7 are literal quotations of printed theorems, cited to the page. A8 is a package of several printed items, and one of them is not literal.
It is A8(vi): the cutoff-uniform form of the Glimm–Jaffe–Spencer moment estimate (Ann. of Math. 100 (1974), Thm. 1.1.8), on the slab family used here. In the manuscript’s own display,
\text{one constant for the whole slab family }\mathcal C^{\rm slab},
\qquad\text{uniformly in the cutoff }h .
It is not a quantifier strengthened beyond the source. Since the scope amendment of 5 September 2026 the reading is this: the quantifier is the one GJS themselves state on p. 629 and consume in their printed proofs of Theorems 1.1.7 and 1.1.1 on pp. 594–597. A8(vii′) quotes those passages verbatim, in three steps:
It is still a hypothesis. The manuscript does not re-derive the cluster expansion. Appendix C is a source-mechanism audit plus proofs of two analytic prerequisites — it is not a reconstruction. Theorem 12.2 therefore remains conditional, and any citation of it should say so.
The only proportionate route to an unconditional Theorem 12.2 is a display-by-display trace of the h-dependence through the printed expansion on GJS pp. 598–629, completing Proposition C.1, so that U_GJS becomes derived rather than imported. This is recorded as an open problem and is not scheduled: it is judged disproportionate for a record whose imports are named and whose scope is stated.
Separately, a Hamiltonian bypass was investigated and found closed — g-uniform local moments were explicitly open in the 1970–71 literature, and the available substitutes cover only translation-averaged states or carry constants that are not uniform in the needed sense. That is a verdict on the printed record, not a no-go theorem.
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