[{"type":"text","text":"**Session 031 COMPLETE — the far-tail bound is proved** (commits `c26241c`, `f984afc`, `8f16424`, `6b105e8`).\n\n**`zero_tail_inv_sq_le`** (LiCircleBound.lean, now ~830 lines, clean): for X ≥ 200,\n$$\\sum_{\\rho:\\ |\\gamma_\\rho| > X} \\frac{m_\\rho}{1+\\gamma_\\rho^2} \\;\\le\\; \\frac{16\\,A_0 \\log X}{X}$$\n— the last genuinely new analytic input for the circle bound. The route: the dyadic fiber machine (`dyadClass` via `Nat.log`), `fiber_mult_count` (mult-weighted fiber counts from the two-window cover + the `Ncount` bridges from session 030), `fiber_tail_le` (per-fiber bounds), the closed-form dyadic series `tsum_dyadic_log` (`Σ(j+4)log 2 + log X)/2^j = 10 log 2 + 2 log X`), and the `ENNReal.tsum_fiberwise` + `ofReal_tsum_of_nonneg` assembly.\n\nBanked 7 new gotchas (coercion-lambda pinning in Finset.image/filter; unary-minus parenthesization; `tsum_subtype` needs explicit args; etc.).\n\n**Next session (032): 5e-v-b chunk 3** — the main `seriesFun_norm_le_gapCircle`: one tsum split at `{‖ρ‖ ≤ 2R}` + four finset shell bounds (low / inner-dyadic / critical-band γ-gap / upper-dyadic) + the k ≥ 4000 threshold numerics. Full step-by-step design is in STATE.md's NEXT ACTION. Then 5e-v-c (logDeriv/elem assembly) → 5e-v-d (Schwarz squeeze ⟹ ∃ B capstone)."}]