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

Worker session of 2026-09-23 on branch teal-sea/weil-c4-s2, folder hunts/weil_propagation/c4_s2/kernel/. Nothing here is a claim about RH. Grades follow the AGENTS.md ladder. Every number below is asserted by test_sonin.py in this folder (the pinning test is named in brackets, or in the table of s7). Code: sonin.py (construction), calibrate.py (class analysis), run_cells.py (writes cells_dps40.json, cells_dps60.json). Interface for two_adic/ and checker/: INTERFACE.md.

0. Status (ALIGNMENT s5)

• Gap (a): closed as a construction. S_∞ is computable on window functions, both as the trace functional that T_∞ needs and as a projection on L²(ℝ*_+, d*u).
• C4 (theory §7.2): unresolved. Nothing in this folder composes S_∞ with the place 2. That is two_adic/'s gap (b).
• The S = {∞} calibration agrees with every statement of the source it was compared with (s3). Nothing was refuted.

1. The objects, from the source

All from Connes-Consani arXiv:2006.13771 ("CC"), read in full for s1-s6 and Thm 6.11; PDF in the session scratchpad, not committed.

Translation to the mission notation (our f is CC's g). A window function F on [0, L], L = log c, sits in L²(ℝ*_+, d*u) as u ↦ F(log u) (any translate gives the same traces). Its autocorrelation g is CC's f = g∗g*, and d*ρ = dx. Then:

• A(f) of theory §0 is CC's W_∞. It is the digamma form (ψ(1/4) − log π)g(0) + ∫_0^∞ 2[g(0) − g(x)] e^{−x/2}/(1−e^{−2x}) dx, which equals the §0 formula through ψ(1) − ψ(1/4) and the tail −log(1 − e^{−2L}) (pinned in [test_arch_block_matches_theory_s0_formula_verbatim]).
• T_∞(f) := Tr(ϑ(f)S_∞ϑ(f)*) = A(f) + E(f), where E(f) = ∫_{−L}^{L} g(x) ε(e^{|x|}) dx.
• R_∞ := Q_∞ − T_∞ = (P + A) − (A + E) = P − E. The pole block P vanishes on the class C1 (ĝ(−i/2) = 0 means ∫F e^{−y/2} = 0). So on C1, R_∞ = −E, and Thm 6.11 reads E ≤ c₀|∫F|² there.
• ĝ(0) = ∫_0^L F dy.

2. The construction (gap (a))

The kernel of E in closed form plus a fast series (derivation, this session, unreviewed). With η_n = Fξ_n = λ_n φ_n (entire, CC after Prop 5.3), CC's ε is Σ_n λ_n²/(1−λ_n²) ρ^{1/2}∫_{1/ρ}^{1} φ_n(x)φ_n(ρx) dx (their proof of Lemma 5.4). Splitting ∫_{1/ρ}^1 = ∫_0^1 − ∫_0^{1/ρ}:

ε(ρ) = ρ^{1/2}A(ρ) − ρ^{−1/2}A(1/ρ), A(y) = Σ_n (1−λ_n²)^{−1} ∫_0^1 η_n(t)η_n(yt) dt.

Write 1/(1−λ²) = 1 + λ²/(1−λ²). The "1" part sums by completeness of {ξ_n} in L²[0,1], because Σ_n η_n(t)η_n(s) = 4∫_0^1 cos(2πxt)cos(2πxs) dx:

A0(y) = 2[Si(2π(1−y))/(2π(1−y)) + Si(2π(1+y))/(2π(1+y))], ρ^{1/2}A0(ρ) = δ(ρ) exactly, A1(y) = Σ_n v_n ∫_0^1 η_n(t)η_n(yt) dt, v_n = λ_n²/(1−λ_n²).

The weights v_n decay super-exponentially, with 20 terms at dps 40 and 23 at dps 60 [prolate block in the JSON]. A1 is stored as an even polynomial from the Taylor moments of η_n. So the only non-elementary input to S_∞ is the standard prolate eigenproblem (a tridiagonal Legendre-Galerkin matrix), solved to the working precision. No "numerically identified" data enter: CC s6 identifies data only for the constant c, not for S_∞.

Checks of the split (independent routes, hardened):

• Σ_n of the prolate Taylor data reproduces A0 to 10^{−30}.
• ρ^{1/2}A0 equals CC eq. (49) to 10^{−35} [test_completeness_part_is_delta_closed_form].
• ε from the split equals CC's own series ε = Σ v_n ρ^{1/2}∫_{1/ρ}^1 φ̃φ̃(ρ·), by direct quadrature at ρ = 1.6, to 10^{−30} [test_eps_basic_properties].
• ε'(1+) by CC Lemma 5.4 (Σ v_n ξ_n(1)²) and by differentiating the split agree to 10^{−30} [test_edge_terms_and_eps_derivative_match_cc].

The Galerkin blocks. A form ∫ g(x) K(|x|) dx has matrix M(n,m) = ∫_0^L K q_nm (CCM Lemma 2.3). It is built from two moment sequences s_k, d_k (form_from_moments), computed by Gauss-Legendre with 192 or 384 nodes. Doubling the rule changes no moment by more than 10^{−40} at dps 40 [test_quadrature_rule_by_doubling].

• A is built from the digamma form at interior nodes with expm1. The x → 0 limit is removable, so no finite difference is taken anywhere; this is the defect of the rejected prior attempt, not repeated. A agrees with the CCM closed forms of hunts/rogue_frontier/weil_trunc/galerkin.py to below 10^{−36} at c = 1.5 and 2.5, N = 8, dps 40 [test_arch_block_matches_ccm_closed_forms].
• P matches CCM's W02 to below 10^{−36} [test_pole_block_matches_w02].

S_∞ as a projection (for two_adic/, INTERFACE §3). ran P equals Sonin ⊕ span{ζ_n}: a vector of ran P orthogonal to both would satisfy ‖P̂v‖² = ⟨v, PP̂Pv⟩ = 0, and ran P ∩ ker P̂ = 0 by uncertainty. Hence

S_∞ = 1_{|x|≥1} − Σ_n |ζ_n⟩⟨ζ_n| (ordinary argument from CC eq. (81), unreviewed).

The sum converges super-exponentially on vectors supported in |x| ≤ X. η_n is computed for every s by a spherical-Bessel sum. It agrees with the Taylor route to below 10^{−34} for s ≤ 2.5 [test_eta_bessel_route_matches_taylor_route]. For v = 1_{[1,1.5]}, the Fourier transform of S_∞v vanishes on [−1,1] to below 10^{−35} with 40 terms [test_projection_maps_into_sonin_space]. The compressed matrix ⟨V_m|S_∞|V_n⟩ on the CCM basis at [c^{−1/2}, c^{1/2}] has its spectrum in [0, 1], with truncation tail below 10^{−45} at K = 20 (c = 2.5) [test_window_projection_matrix].

For two_adic/'s composition through the Mellin variable, the transforms M_n(s) = ∫_1^∞ ζ_n(v) v^{−1/2−is} dv are given in closed form by Tate's local functional equation (zeta_mellin_all). They are checked three ways [test_zeta_mellin_closed_form]: the moment recurrence against quadrature to 10^{−38}, the Tate constant on 1_{[0,1]} to 10^{−38}, and a direct integral over [1, V], whose error falls like V^{−2} and is below 10^{−4} at V = 60. Any number of modes is available (prolate_vectors, orthonormal to 10^{−40} for 150 modes [test_many_modes_orthonormal]). λ_n is resolved only for n below n_max; beyond it the interface uses η_n directly.

3. Calibration against CC (S = {∞}), measured

Prolate data (CC s4, Rem 4.6, Lemma 5.4, footnote 7), dps 40:

• λ_0 … λ_5 reproduce CC's printed 0.999971, −0.979485, 0.524086, −0.0589766, 0.00273233, −0.0000762914 within half a unit of the last printed digit [test_prolate_eigenvalues_match_cc_list].
• Σλ_n² = 2.237484835 = 2(Si(4π)/(4π) + 1) to 10^{−35} [test_sum_lam2_equals_delta_at_1_closed_form].
• Even plus odd sums to 4 (CC footnote 7) [test_even_plus_odd_sum_is_4].
• t(n) = v_n ξ_n(1)² reproduce CC's 11.9719, 8.77574, 2.20528, 0.0433983, 0.000125459, and ε'(1+) = 22.99648 (CC: 22.9965).

Positivity statements (N = 16 at every cell, both parities; N = 8, 16, 32 in the JSON):

• T_∞ = A + E ⪰ 0 (Thm 4.7).
• A + D ⪰ 0 (Cor 2.3).
• D − E ⪰ 0 (eq. (90)) [test_T_inf_psd_and_D_minus_E_psd].

The Thm 6.11 window, c = 2 [test_calibration_statements]:

• λ_max(K_I) is a Rayleigh-Ritz value on {h: h/2 − h' ∈ C1 ∩ band}, via E(F) = 2ε'(1+)(⟨h|K_I h⟩ − ‖h‖²) with F = h/2 − h' (CC Prop 5.5, Lemma 3.3), so it increases in N.
• The first-order (1/N) extrapolation 2·(N=32) − (N=16) gives 1.05160 and 15.236. This is a measured heuristic, not a bound.
• R_∞ + c₀|∫F|² ⪰ 0 on C1 and R_∞ ⪰ 0 on C2 hold at every N, as Thm 6.11 and Thm 1 require.

The calibration cells c = 1.5 and 1.9 (N = 8, 16, 32, same counts at every N):

At c = 1.5, R_∞ ⪰ 0 already on the full space. At c = 1.9, λ_max(K_I) > 1, so the one remainder direction of Thm 6.11 is present below log 2. This agrees with CC's Figure 11 (PDF page 36, read off the plot, no printed number): their largest eigenvalue crosses 1 between a = 0.5 and 0.6, below log 1.9 = 0.642, and stays below 1 at log 1.5 = 0.405. The positive part of R_∞'s spectrum accumulates at 0 like N^{−2} (for example 1.871e−4 at c = 1.5, N = 32). That is the essential spectrum of −E at high frequency (ε has the kink ε'(1+) > 0), not a remainder direction. What is N-stable is the inertia and the negative eigenvalues.

4. The Γ_ℂ type (task 4)

Γ_ℂ(s) = Γ_ℝ(s)Γ_ℝ(s+1), so the archimedean block is A_{a=1/4} + A_{a=3/4}. The Sonin term is taken as S_ev ⊕ S_odd, the Sonin projection of all of L²(ℝ) split by parity. The odd sector uses the sine transform, the odd prolates (μ_0 … = 0.99878, −0.84956, 0.20740) and A0_odd with the sign of the second Si term flipped.

Grade: derivation by analogy with CC s1-s4 (cos → sin), unreviewed. The checks are:

• A_odd − A_even = ∫ g/(2cosh(x/2)) to below 10^{−36}. This is CC's τ with the sine kernel.
• A_odd equals weil_trunc's DH archimedean block minus log 5 to below 10^{−36} [test_odd_arch_block_matches_dh_route_and_sech_identity].
• T_odd ⪰ 0, A_odd + D_odd ⪰ 0 and D_odd − E_odd ⪰ 0 at all six c (N = 16).
• Σμ² = 1.762515165 = 2(1 − Si(4π)/(4π)), and ε'_odd(1+) = 16.73365.

T_inf_matrix(c, N, dps, arch_type="C") returns T_ev + T_odd.

Not included, deliberately: the conductor term of Dedekind ζ_{Q(√−23)} (log 23 · g(0)). It belongs to the ramified place 23, which is not in S = {∞, 2}. Whether the complex-place (disk) Sonin space should replace S_ev ⊕ S_odd for GL₁ over Q(√−23) is not decided here.

5. The mission cells c ∈ {2.2, 2.5, 2.9} (archimedean part only)

These are the S = {∞} objects that two_adic/ composes. They are not R_S.

The counts are identical at N = 8, 16, 32 and at dps 40 and 60. Thm 6.11's inequality with c₀ fails at all three windows, as expected: they lie outside its support [2^{−1/2}, 2^{1/2}]. For S = {∞} alone, the remainder of bounded rank grows from one direction (c ≤ 2.5) to two (c = 2.9); a second vanishing condition would be needed there. The generating moments of A, E and P (both parities) for k ≤ 32 are in moments[c] of both JSON files (INTERFACE §5).

6. Numerics

• Precision response. Over every stored eigenvalue, c*, Thm 6.11 value and K value, dps 40 against dps 60 differ by at most 5.1e−39 (at c = 2.2, N = 32, c*). All inertias are identical [test_precision_response_40_vs_60].
• Recompute. analyze_cell at N = 8 reproduces the stored cells to 10^{−30} [test_json_recompute_N8]. The N = 32 check at c = 2 is marked slow.
• Truncations (JSON prolate): 20 weighted prolates, 74 Taylor terms and a 76-function Legendre basis at dps 40; 23 prolates at dps 60.
• Runtime. One run each, single process on a laptop: run_cells.py --dps 40 about 216 s, --dps 60 about 365 s, both under the 600 s limit (the stored seconds_total is checked against that limit). The kernel test tier runs under pytest -n 2 (s9).

7. Grades

No prover was used; no statement is kernel-checked.

8. Threads (observations, not pursued)

• The window where each remainder direction appears. For S = {∞}, the first negative of R_∞ on C1 appears between c = 1.5 and 1.9 (CC's Figure 11 puts it between a = 0.5 and 0.6, that is c ≈ 1.65 to 1.82). The second, on C2, appears between 2.5 and 2.9. A bisection at N = 32 would locate both. This is the S = {∞} version of the mission's "bounded rank" count, and it gives two_adic/ a baseline.
• Shifted W_∞ matrix. Needed only if the semilocal Γ-sum produces translated autocorrelations h(x) = g(x − k log 2) inside the trace against S_∞. It would be the §0 formula applied to the even part of h.
• Complex-place Sonin (disk prolates) as the alternative Γ_ℂ model.
• N-convergence is first order in 1/N for λ_max(K_I), c* and the C1 negative eigenvalue. This is consistent with the jump of the periodic Fourier basis at the window edge. A basis with edge functions would converge faster.

9. Reproduction

From the worktree root, with PY="PYTHONPATH=$PWD .venv/bin/python":

$PY hunts/weil_propagation/c4_s2/kernel/run_cells.py --dps 40 $PY hunts/weil_propagation/c4_s2/kernel/run_cells.py --dps 60 $PY -m pytest -q -n 2 hunts/weil_propagation/c4_s2/kernel tests/test_hunt_probe_discipline.py tests/test_docs_numbering.py