Brief: kernel/, gap (a), the archimedean Sonin projection S_∞ on windows

You own hunts/weil_propagation/c4_s2/kernel/ and write nowhere else. Time box: 3 hours of work from your start. At the box, write RESULTS.md with what stands and stop.

The gap

S_∞, the archimedean Sonin projection, has no closed kernel on these windows. Connes-Consani arXiv:2006.13771 s6 proves the S = {∞} case with numerically identified prolate data. Your job is to make S_∞ computable on window functions in the shared CCM basis of the mission, so that two_adic/ can compose it with P_2 and checker/ can measure R_S.

Inputs

• Connes-Consani arXiv:2006.13771: Intro, s3 (Cor 3.8, Rem 3.9), s4, s6, Thm 6.11, the trace remainder δ(ρ) (their eqs. (8), (10)).
• Connes arXiv:math/9811068 s VII; CCM arXiv:2511.22755 (prolate functions, s7-8); theory RESULTS §7.3 item 2 (the factorization Π_S = S_∞ ⊗ P_2 + 1 ⊗ (1 − P_2); graded there as a derivation, unreviewed).
• The rejected prior attempt, read and do not trust: remainder.py and the review, at the path in your task spec. Its split Q = Q_∞ − √2 log 2 · g(log 2) is sound in substance. Its two defects, not to be repeated: a claimed 1e-25 tolerance that was never achieved (tolerances come from a measured deviation), and the x → 0 limit of A(f) taken by a hard-coded 1e-6 finite difference (do it analytically or by a stated series).
• hunts/rogue_frontier/weil_trunc/galerkin.py and SOURCE.md for the basis and the archimedean entries.

Tasks, in order

1. Fix the S = {∞} objects exactly from the source, with section and equation numbers: the Sonin space and S_∞, the representation ϑ(f), the trace term Tr(ϑ(f) S_∞ ϑ(f)*), the function class of Thm 6.11 (which vanishing conditions on the transform are imposed), and δ(ρ). Translate into the mission's notation (our f is their g). Milestone 1: status message.
2. Make S_∞ Galerkin-computable on windows: a closed kernel if one exists, otherwise numerically identified prolate data as in their s6, with a convergence study (basis size and precision response). State which, and grade it.
3. T_inf_matrix(c, N, dps): the S = {∞} trace term as a Hermitian matrix on the shared basis. Calibrate at c ∈ {1.5, 1.9} against the statement of Thm 6.11 (the inertia and lowest eigenvalues of Q_∞ − T_∞ at N = 8, 16, 32, on the full space and on the function class), and report the precision response (dps 40 against 60). Milestone 2: status message with the calibration numbers.
4. If feasible within the box, support the Γ_ℂ archimedean type (Γ_ℂ(s) = Γ_ℝ(s)Γ_ℝ(s+1), the archimedean data of Dedekind ζ_{Q(√−23)}) so the positive control can run. If not, say so plainly in INTERFACE.md and RESULTS.md.
5. INTERFACE.md: the functions, their conventions, the JSON keys, and the exact representation of S_∞ that two_adic/ consumes (for example a matrix on a stated basis of L²(ℝ), or a kernel function with its quadrature rule). The coordinator routes it to two_adic/ and checker/.

Values for the mission cells c ∈ {2.2, 2.5, 2.9} go to a JSON in your folder.

Standing rules (identical in all four briefs)

• Read first: hunts/weil_propagation/c4_s2/MISSION.md (scope, shared basis and cells, interface contract, kill-controls, done), hunts/weil_propagation/MISSION.md, AGENTS.md (hard rules, certainty ladder, original vs novel), ALIGNMENT.md sections 4 and 5, and theory hunts/weil_propagation/theory/RESULTS.md sections 0, 7, 7.1, 7.3, 8.
• Python, from the worktree root: PYTHONPATH=$PWD .venv/bin/python. The .venv/bin/python named in AGENTS.md does not exist in this worktree. Tests: PYTHONPATH=$PWD .venv/bin/python -m pytest -q -n 2 <your folder> tests/test_hunt_probe_discipline.py tests/test_docs_numbering.py (testpaths is tests, so name your folder explicitly).
• Compute: four workers share a 16 GB laptop. Every local run under 10 minutes and a few GB; N ≤ 32 and dps ≤ 60 unless you state why; pytest -n 2 at most. Anything larger goes in your RESULTS.md as a CI proposal with a measured per-unit estimate. No lake build.
• Git (shared worktree, shared branch teal-sea/weil-c4-s2): commit only your own folder: git add hunts/weil_propagation/c4_s2/<folder>/ then git commit -m "<msg>" -- hunts/weil_propagation/c4_s2/<folder>/. Never git commit -a, git add -A or ., git stash, reset, rebase, checkout of files you do not own, or push. Files outside your folder in git status belong to peers: leave them. On an index.lock error wait a few seconds and retry. Small commits; end each message with Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>.
• Read-only inputs from the sibling branch: git show teal-sea/weil-propagation:hunts/weil_propagation/numerics/<file>. Source PDFs go in your session scratchpad, never in the repository.
• Lexical ban: tests/test_hunt_probe_discipline.py fails if the past participle of "certify" appears anywhere under hunts/ in a .py, .md or .json file, case-insensitive, including inside a longer word and inside a sentence disclaiming it. Do not write it. Never dump a raw zeta.rigor result dict to JSON (it carries that key); rename keys. House style: no em dashes in prose you write.
• Provers: AXLE only (axle on PATH, runs remotely: axle check, verify-proof, disprove, repair-proofs). No Leanstral, no Aristotle, and no escalation about their keys (operator's decision). Prover output is input, not evidence: it counts only after zero-sorry checking, and is graded kernel-checked for exactly that lemma and nothing wider.
• Grades: the AGENTS.md ladder; a composite claim takes the grade of its weakest step. Every number in your RESULTS.md is pinned by a test in your folder. Tolerances come from a measured deviation at a stated dps, never asserted. A search you did not run is "not searched".
• Orca: every --from and --terminal you pass is your own handle from your preamble, never the coordinator's. Check your inbox at each checkpoint (before starting a new file, after a test run) and immediately before worker_done. Blocking questions go through your preamble's ask command. Send a status message to the coordinator at each milestone named below. Anything outside the mission, irreversible, or costing money: ask, do not do.
• Finish: RESULTS.md in your folder whose first five lines say what was built, the key numbers with their grades, and what is open; an ALIGNMENT s5 status (refuted, obstructed in a restricted class, unresolved, or paused by allocation). An exact statement of where the construction is stuck is a complete result; a fabricated number is not. Then worker_done once, with --outcome succeeded or --outcome failed, --files-modified and --report-path.