D-047 — E-026 analysis: inverse-spectral reconstruction — machinery validated, signal diffuse

Date: 2026-08-19 | Session: 008 | Follow-up to the domain-sweep opening Status: complete (machinery built + validated; avenue parked with revival notes). Experiment: E-026 (moment round-trip 3e-14).

1. What was built

E-026: Stieltjes/Lanczos reconstruction (dps 50, full reorthogonalization) of the Jacobi recurrence coefficients {b_k} of the symmetric Clark measures μ_a (from E-022 atoms+masses), a ∈ {4, 8, 12}, 240 coefficients per side. The {b_k} are the discrete shadow of the canonical-system Hamiltonian (the inverse-spectral invariant of the model spaces).

2. Validation

• Moment round-trip: (J^j)_{00}·mass vs input moments, j ≤ 20: rel err ≤ 3e-14. (The script's V1 atom-matching check was misdesigned — a K-coefficient Jacobi matrix matches MOMENTS, not individual atoms; the correct check passes. Script comment fixed below.)
• Symmetry: |a_k| < 1e-30 (exact 0 by the symmetric measure).

3. Findings (the honest negative)

• The b_k are dominated by the window truncation: tail b_k → Z/2 (the band edge), period-2 oscillation from the gap at z = 0.
• The γ-structure moves the coefficients only diffusely: ratio to the equal-mass reference has std ~2%, max 14% (early coefficients), no sharp spectral feature. Narrow-band (B = 60) rerun: no clean γ-beat modes (top FFT modes do not match the predicted beat frequencies 2π/arccos(γ/B)). The inverse-spectral data smooths the localized valley structure over all coefficients.
• Interpretation: the Clark measure is "nearly free + small atomic corrections" in the Jacobi representation; the consolidation signal (mass migration) is a ~few-percent effect there, versus order-unity in the K-diagonal picture (D-040/D-045).

4. Status of the avenue

Parked. The machinery is validated and archived (b_k series for a = 4/8/12 in E-026-jacobi.csv); the sharp observable for the spectral-purity question remains the K-diagonal (valley depth, residual). Revival options if the avenue is reopened: (i) full canonical-Hamiltonian decode (Krein-string conventions; needs the string literature, not just the Jacobi shadow); (ii) longer coefficient sequences + band-matched truncations; (iii) use the b_k as input to the de Branges inverse algorithm proper (subspace chain) rather than FFT-style feature hunts. Note: E-026 script's V1 check should read "moment match" (the shipped check compares atoms and reports 1e2 — cosmetic; moments verified post-hoc to 3e-14).