Shared basis and cells as in ../MISSION.md: U_n(y) = L^{-1/2} exp(2πiny/L) on [0, L], L = log c, n = −N … N, index 0 = n = −N; a quadratic form F is the matrix M with F(f) = v* M v for f = Σ v_n U_n; inner products are conjugate linear in the first slot. Python: PYTHONPATH=<worktree> <venv python>; modules are imported by basename after putting this folder on sys.path (the test files show the pattern).
Delivered
| function | returns | notes |
|---|---|---|
ta_es.shift_corr(c, N, dps, k=1) | C^{(k)}_{mn} = ⟨U_m, D^k U_n⟩, closed form | D = dilation by 2 = shift by log 2 in y; zero when k log 2 ≥ L |
ta_es.theta_gram(c, N, alphas, dps) | Σ_j Gram of Θ_{α_j} = 1 − α_j 2^{−1/2} D on the window basis | 2 < c < 4 only (three-term form) |
ta_es.es_gram(c, N, alphas, dps) | Σ_j Gram of E_S (the class of 1_{Z_2} ⊗ f), unitary α | 2 < c < 4 only |
ta_es.prime_block_from_C(c, N, alphas, dps) | Wp_α = (log 2/√2) Σ_j (α_j C + ᾱ_j C*) | equals the CCM prime block for real α |
ta_es.galerkin_prime_block(c, N, dps) | the prime block read from weil_trunc/galerkin.py | independent route |
ta_data.validate(data, degree=None, tol=None) | LocalData or raises NonUnitaryLocalData | data = ("satake", alphas) or ("tower", {k: s_k}) with degree |
ta_ts.T_S_matrix(c, N, dps, local_data, arch_type="Gamma_R", s_inf=None) | T_S matrix, or raises | see order of checks below |
ta_ts.delta_T_matrix(grid, zetas, alpha, L, N) | (ΔT, M_inf, M_S) for a finite mode family on a grid | float64; exercised on synthetic modes only |
ta_prolate.ProlateModes(nvec, dps), hats_modes, delta_T_cells(pm, cells, N, S) | kernel/'s prolate modes in float64; ζ̂_n, b̂_n; ΔT per cell, and diagnostics gram_z_offI, cond_Gb (formed G_b, float64; saturates near 1e16 above 1/eps), cond_Fz, cond_Fb (the Gram factors, from singular values; cond(G) = cond(F)², reliable while cond(F) is well below 1/eps). Reading rule (RESULTS.md s10.3 A2): ρ measured good to 1.4e−5 relative at cond(F) = 6.7e11 and off by 6.5e−2 at 3.8e14; a build with cond(F) above about 4e14 is not determined by its float64 samples; between the two, not measured | Mellin route, measured grade |
ta_mellin.rho(hat_rows, *, factor) | ρ(s) = x* G⁻¹ x, x = conj(ŵ(s)), for G = F* F, from a Householder QR of F and a triangular solve; G is never formed | since 2026-09-24 (RESULTS.md s10); factor is keyword-only so that a Gram matrix cannot be passed as a factor by mistake; arithmetic error grows like eps·cond(F) |
ta_mellin.rho_inv(hat_rows, G) | the route before 2026-09-24, np.linalg.inv of the formed G | kept as the reference; error grows like eps·cond(G) = eps·cond(F)² |
ta_prolate.gram_factor_s(H, sw, J, S, A, dil), ta_mellin.gram_factor_v(fns, Kmax, per_panel) | F with gram_s(...) = F* F (weighted samples plus the two tail rows), resp. gram_v(fns) = F* F | gram_s, gram_v unchanged |
ta_rho_diag parts, ta_rho_check parts | the diagnosis of cond(G) (s10.1) and the acceptance runs of the QR route (s10.3) | one part per process; JSON below |
ta_ts.KernelProvider(nvec=None, S=None) | T_inf_matrix (kernel/'s moments JSON) and delta_T | the provider T_S_matrix uses by default |
ta_gram_probe.run(nvec, S) | on c = 2.2, N = 8: ΔT, the Gram probe, T_S's lowest eigenvalues, max |G_z^s − I| | measurement for RESULTS.md §7b |
ta_gram_probe.merge_modal(out) | copies modal/out/gram_*.json runs into out unchanged, with source; adds modal_calibration | reads only; idempotent |
ta_gram_probe.band_terms(runs, nvec, S=4800) | the §7b band's three terms at nvec (S, nvec, probe; None where no run forms one), band, binding, complete | band is a lower bound unless complete |
ta_hs.hs2_partial(K, alpha=1, dps=30) | ‖P F_S P‖²_HS over Euler levels j ≤ K | closed form in Si |
ta_local.local_report(K) | exact local statements at 2 | sympy rationals |
T_S_matrix order of checks: (1) local data at 2: NonUnitaryLocalData for any |α| ≠ 1 or a tower that is not a power-sum tower of d unitary parameters (W_a and the Epstein (1,1,6) tower are refused); local_data=None switches the place 2 off and returns kernel/'s T_inf; (2) arch_type: "Gamma_R" accepted, "Gamma_C" raises FrameworkLimit (no idele class character of C_S is odd at ∞ and unramified at 2); (3) builds T_S = T_inf (kernel/) + ΔT and returns a float64 numpy array (ΔT is of measured grade, error band 3.5e−3 to 8.2e−3 by (c, N), RESULTS.md §5b and its correction notice; this line said "about 6e−3" before the Kmax rerun; on c = 2.2, N = 8 the mode truncation is larger, 4.5e−3 in spectral norm from 80 to 100 modes, RESULTS.md §7b); dry_run=True stops after the checks. It never returns a number it did not compute. Defaults: nvec = max(80, 8N/L + 40) prolate modes and S = max(1200, 24πN/L); at N = 32 use at least 200 modes (130 leave spurious residual pairs; rechecked under the Kmax rule, it stands). The default nvec at N = 32 is up to 364 modes (Kmax 15), by extrapolation well past the 10-minute local limit: pass nvec and S to KernelProvider explicitly. The w-range must satisfy 2^Kmax ≳ n_max²/(2π) (ta_prolate.kmax_for, guarded in hats_modes): the asymptotic tail in 1/w has term ratio about (2n)²/(4π(m+1)W) and diverges in practice below that (at W = 2^10 and n = 199 its terms reach about 1 against a first term of 4e−5). Local data at 2 for the controls are in ta_data: ZETA, DEDEKIND_Q_SQRT_M23, W_A_QUARTER, EPSTEIN_116_TOWER.
JSON
ta_es_cells.json: meta.bound_lo, meta.bound_hi (3/2 ∓ 1/√2), and cells[] with keys c, N, dps, theta_gram_eig_min, theta_gram_eig_max, theta_gram_max_imag, prime_block_eig_min, prime_block_eig_max, identity_defect, galerkin_dev, seconds.
ta_ts_cells.json: cells[] with keys c, N, dps and one entry per data set (zeta, dedekind_Q_sqrt_m23, W_a_quarter, epstein_116_tower, place_2_off), each {outcome, message} with outcome one of refused_nonunitary, framework_limit, awaiting_kernel, matrix (ζ and place 2 off are matrix); hs2_P_FS_P with archimedean, partial_sums (K → value), asymptote.
ta_ts_prolate.json: tate_check_max_abs, seconds_total, and rows[] with keys nvec, S, N, c, T_inf_eig_min, T_inf_eig_max, T_S_eig_low3, T_S_eig_max, T_S_n_below_m002, gram_sensitivity, resid_top8 (eigenvalues of ΔT + Wp largest in modulus, signed), resid_n_above_01, resid_n_below_m01, seconds_hats, and Kmax on the rows rerun under kmax_for (configurations 2 to 4; seconds_by_config records those reruns). Since the regeneration of 2026-09-24 under the QR route of rho (RESULTS.md s10.5) every row also carries cond_Fz, cond_Fb and Kmax. Delivered rows: (nvec, S) = (80, 1200) for N = 8, (120, 1600) for N = 16, (200, 2400) for N = 32. Converged to what (checker/'s measurement, RESULTS.md correction notice; this paragraph listed all four as "converged" before): (80, 1200) to about 5.6e−3 at N = 8 and not at N = 16 (80 → 120 modes moves T_S by up to 7.8e−2); (120, 1600) to 1.6e−2 at c = 2.2 and within the probe at 2.5 and 2.9; (200, 2400), under the QR route of rho (checker/ s7.8, 3e36fa4): to 3.0e−2 at c = 2.2 and 1.4e−2 at 2.5 (set by the 240 and the 319-mode refinement respectively; the 319-mode row also changes S, to 2633, and lies beyond A2's last clean case, RESULTS.md s10.3) and within the probe (8.2e−3) at 2.9, where no refinement moves T_S by more than 4.95e−3. On the old route (checker/ s7.7, 3dc0a74) its 240-mode refinement moved T_S by 3.9e−2, 2.4e−2, 3.2e−2, and checker/ read that row as not a clean refinement. For checker/: R_S = Q − T_S with Q = Q_∞ − Wp, so R_S = R_∞ − (ΔT + Wp).
ta_gram_probe.json: c, N, Q_low (checker/'s value), and runs keyed "nvec,S" with nvec, S, Kmax, dT (the matrix), dT_probe_maxentry, dT_probe_norm2, TS_low3, gz_dev, gz_dev_diag_max, seconds, and source (file, platform, python; for modal/'s runs also tree_commit, blas_core, avx512, cpu_seconds, peak_rss_mib). modal_calibration: modal/'s copy of run 80,4800 against the local one (max_abs_diff_dT, max_abs_diff_scalars, file, tree_commit, platform).
Consumed (from kernel/, routed 2026-09-23 22:10)
Used as routed: kernel/INTERFACE.md §3 (prolate_vectors, prolate_data, zeta_mellin_all as the independent check, form_from_moments with the moments of cells_dps40.json). The composition (RESULTS.md s5) needs S_∞ through its complement, 1 − S_∞ = P + Q_∞ with Q_∞ the projection onto span{ζ_n} (Connes-Consani arXiv:2006.13771 Prop 4.5, eq. (81)), in either form:
- ξ_n on [0, 1] (coefficients in a stated basis, or a callable at a stated dps) with λ_n, from which ζ_n = (1 − P) F ξ_n / √(1 − λ_n²) and its Mellin transform follow; or
- ζ_n on u ≥ 1 together with their Mellin transforms on a stated s-grid.
The number of modes must cover the Mellin band of the window basis (about 2πN/L, i.e. of order 100 modes at N = 32), not only the first few prolates: the semilocal time-frequency operator is not Hilbert-Schmidt (RESULTS.md s6). Also T_inf_matrix(c, N, dps) on the shared basis, since T_S = T_inf + ΔT.
JSON of follow-up 3 (RESULTS.md s10)
ta_rho_diag.json: toeplitz_cond_ratio (33.97); scan80, scan200 with nvec, Smax, Kmax, hats, seconds, rows[] of S, S_over_nvec2, and per family z, b: eig_min, eig_max, cond_formed (float64 G), cond_F, cond_F_sq; refine (rows[] with label, s_per_panel, w_per_panel, Kmax, S, z, b); lowdir; hats_accuracy (float64 hats against kernel/'s closed form at dps 30: nodes[], max_abs, max_rel); node_counts[].
ta_rho_check.json: eps; A1 keyed "nvec,S" (cond_Fz, cond_Fb, per family max_dev, max_rho, bound, ratio_to_eps_condG_rho, and dT_change_norm2 keyed "c|N"); A2 keyed by S (cond_Fz, new_max_rel_dev, old_max_rel_dev, sample_max_abs_dev, J_max_abs_dev, A_max_abs_dev, max_rho_ref, n_nodes, n_compared); A3 (old, new keyed by c: spectral change of ΔT under a 2^−52 relative perturbation); A4 (cond_Fz, cond_Fb, band, cells keyed by c with new_low3, old_low3, new_n_below_band, old_n_below_band, new_2400_low3, new_S_response_norm2); gram_probe_dependence keyed "nvec,S".