Brief: theory worker

Read first: hunts/weil_propagation/MISSION.md, AGENTS.md (certainty ladder, original vs novel), ALIGNMENT.md, then hunts/rogue_frontier/weil_trunc/RESULTS.md and THEOREM_FEASIBILITY.md (the Davenport-Heilbronn negativity at c = 31), and docs/24, docs/35.

Question

Is there any known or constructible mechanism by which positivity of the Weil form on window L, together with the Euler product, forces positivity on window L + delta? What does the literature already know about how positivity or the ground state changes with the window?

Tasks, in order

1. Primary sources, read and cite with arXiv IDs and section numbers: Connes-Consani semilocal trace formula and "Weil positivity and trace formula, the archimedean place"; Connes-Consani-Moscovici (arXiv:2511.22755); Connes-van Suijlekom (arXiv:2511.23257); Connes (arXiv:2602.04022) s6; arXiv:2608.24827 (support-1.6 positivity and the doubly exponential decay law); Yoshida's original result. Summarize only what each proves about dependence on the window, the role of each prime entering, and where each program says it is stuck.
2. Write down the exact derivative or difference of the form in L: the archimedean change, the new prime-power atoms entering at n = e^L, and the effect on the admissible function space. Derive, don't recall; check any constant numerically with the lab's zeta/weil.py where cheap.
3. Propose at most three candidate propagation lemmas. For each: exact statement, what it would imply, where the Euler product enters, and the DH test (DH must violate the lemma's hypothesis or conclusion at the c = 30 -> 31 crossing, otherwise the lemma is refuted as stated).
4. For each candidate, give the smallest check the numerics worker could run, and say whether it is a restatement of RH with no new estimate.

Output

Write only in hunts/weil_propagation/theory/: RESULTS.md with sources, derivations, candidates and their status on the certainty ladder. Commit to this branch in small commits. Do not push. No heavy computation; nothing over a few minutes locally. No lake build.

Finish with a five-line summary at the top of RESULTS.md: strongest candidate, what it would give, what refutes it, whether it is new or known (state what was searched), next step.