Documentation for a Hamiltonian proof that the net of spatially cutoff P(φ)₂ ground states converges in norm on every local algebra. This is a research record, not a published paper.
Read this first. Theorem 12.2 is conditional on the imported interface U_GJS. Nothing here has been refereed; audit item C5 is open; the work is AI-assisted and human verification of it is ongoing. Treat every statement as a claim under active verification. See status.
For the spatially cutoff P(φ)₂ Hamiltonian H(g) with ground state Ω_g, and ω_g the induced state on the time-zero quasi-local algebra A, in the Glimm–Jaffe–Spencer weak-coupling region 0 ≤ λ/m₀² < ε_P the full net (ω_g) — no subnets, no space averaging — converges in norm on every local algebra A(O),
\lim_{g\in\mathcal G}\ \bigl\|(\omega_g-\omega_\infty)|_{\mathfrak A(O)}\bigr\| = 0 ,
with an explicit super-polynomial rate in dist(O, {g < 1}). The limit is locally normal, translation invariant, ℤ₂ invariant for even P, and a ground state of the Glimm–Jaffe dynamics in the spectral sense.
| Status | what is proved, what is conditional, what is open, and the gate record |
| Results | every numbered statement, with dependencies — generated from the manuscript |
| Imports | A1–A8, their sources, and exactly what U_GJS is and is not |
| Definitions | the model, the three cutoff classes, the plateau net |
| Notation | the frozen notation table |
| Provenance | gates, corrections history, the citation dossiers |
| Figures | eight interactive illustrations of the main results |
The manuscript itself is part_i/main.pdf (40 pp, document of
record).
The route is Hamiltonian rather than Euclidean. Two uniform inputs are established over the cutoff class — a local interaction moment (M₂, Theorem 5.1) and a gap (G, Theorem 7.8) — and then a local perturbations perturb locally estimate is run along an affine path of cutoffs. The LPPL step rests on two exact ingredients rather than approximate ones: an exact inverse-energy filter (Lemma 9.1) that converts the gap into a super-polynomially decaying time kernel, and the exact light cone of the cutoff dynamics (Proposition 2.7), which confines a time-evolved local observable. Their combination is the two-regime bound (Lemma 10.3), and summing it over cells gives the rate function Ψ_γ (Lemma 10.5), which decays faster than every polynomial.