[{"type":"thinking","thinking":"psi1/psi2 are nan — the digamma of a BOX input returns nan (python-flint's acb.digamma probably doesn't support wide ball inputs, or the box construction is wrong). Let me test acb.digamma on a box directly. Maybe acb.digamma only accepts point balls, or the issue is `arb(\"0\", \"5.125\")` — an arb centered at 0 with radius 5.125 crosses zero; the box2's imaginary part [−5.125, 5.125] and real [−1.375, −1.125] — that's fine as a complex box. But digamma box evaluation may not be implemented (returns indeterminate).\n\nTest: acb.digamma(acb(arb('1.625','0.125'), arb('0','5.125'))) — see what happens. If box digamma unsupported, alternatives:\n(a) Cover the box with small point balls? No — need certified bound over the whole box.\n(b) Bound |ψ(w)| on the box analytically: use ψ(w) = log w − 1/(2w) − 2∫... or the series ψ(w) = −γ + Σ_{k≥0}(1/(k+1) − 1/(k+w)) — hmm, the Weierstrass series ψ(w) = −γ + Σ_{k=0}^∞ (1/(k+1) − 1/(k+w)) converges for w not a negative integer; termwise bound on the box: |1/(k+1) − 1/(k+w)| = |(w−1)/((k+1)(k+w))| ≤ (|w|+1)/((k+1)|k+w|)... for the box2 (Re w ∈ [−1.375,−1.125]): k+w for k=0: w ∈ box, |w| ≤ 1.375+5.125... |1/(k+w)| with k=0: |w|.lower could be small? |w| ≥ 1.125 (Re part), so |w| ∈ [1.125, ~5.3]: 1/|w| ≤ 0.889. For k ≥ 1: |k + w| ≥ k − 1.375 ≥ ... for k=1: |1+w| ≥ |Re(1+w)| = |1−1.25±0.125| ≥ 0.125 — that's the near-pole term (ψ's pole at −1 lives in this box's neighborhood... wait NO: pole at w = −1: box2 is Re ∈ [−1.375,−1.125] — does it contain −1? No! −1 ∉ [−1.375,−1.125]. Distance from box to −1 is 0.125. So |1 + w| ≥ 0.125 for w in box2 ✓ (the earlier estimate). Termwise: |1/(k+1) − 1/(k+w)| = |(w−1)/((k+1)(k+w))|. |w−1| ≤ |w|+1 ≤ 5.3+1 = 6.3. k=0 term: |1/1 − 1/w| = |(w−1)/w| ≤ 6.3/1.125 = 5.6. k=1: |1/2 − 1/(1+w)| = |(w−1)/(2(1+w))| ≤ 6.3/(2·0.125) = 25.2?! Hmm — that's the dominant term. k=2: |1/3 − 1/(2+w)| ≤ 6.3/(3·|2+w|), |2+w| ≥ 0.625 → ≤ 3.36. k=3: |3+w| ≥ 1.625 → 1.29. k≥4: |k+w| ≥ k−1.375: term ≤ 6.3/((k+1)(k−1.375)) ~ 6.3/k². Sum ≈ 5.6 + 25.2 + 3.36 + 1.29 + Σ_{k≥4} ≈ 35.5 + 6.3·(π²/6 − 1.36) ≈ 35.5 + 1.8 ≈ 37. So |ψ(box2)| ≤ ~37 certified. Hmm — that's much bigger than the ~10 I guessed (the pole proximity). Then M ≈ 1.14 + 1 + 18.5 + 0.13 ≈ 20.8, L ≈ 83, allowance L·5e-5 ≈ 0.0042 > 0.00283. FAILS.\n\nImprove: the Weierstrass bound is loose at k=1 (using |1/(1+w)| ≤ 8 and |w−1| ≤ 6.3 with full triangle). Tighter: |1/2 − 1/(1+w)| exactly for the box: = |(w−1)/(2(1+w))|. With w ∈ box2: |w−1| ≤ 2.375 + ... hmm |w−1|: w = x+iy, x∈[−1.375,−1.125], |y|≤5.125: |w−1| = sqrt((x−1)²+y²) ≤ sqrt(2.375² + 5.125²) = 5.65. And |1+w| ≥ 0.125 → ≤ 5.65/(0.25) = 22.6. No better.\n\nBetter idea: evaluate ψ on box2 via the RECURRENCE to shift away from the pole: ψ(w) = ψ(w+2) − 1/w − 1/(w+1). Then bound |ψ(w+2)| where w+2 has Re ∈ [0.625, 0.875] — far from poles — the Weierstrass series there is tame: |ψ(w+2)| ≤ γ + Σ|(w+1)/((k+1)(k+w+2))| ≈ log-ish ~ 2. And 1/|w| ≤ 1/1.125 = 0.889, 1/|w+1| ≤ 8. So |ψ(box2)| ≤ 2 + 0.889 + 8 ≈ 10.9. That's the ~10.5 estimate. But implementing the Weierstrass bound by hand = more code, more proof burden.\n\nALTERNATIVE: does Arb's digamma support ball input? The nan suggests python-flint 0.9.0's acb.digamma doesn't propagate balls for wide inputs. Maybe it works for NARROW balls? Test quickly.\n\nALTERNATIVE 2: use acb.polygamma(0, w)? polygamma exists in dir(acb)! acb.polygamma(0, box) = ψ(box) — maybe polygamma supports box input even if digamma doesn't.\n\nLet me test both.","signature":""},{"type":"text","text":"`digamma` returns nan on box inputs. Testing whether `polygamma(0, ·)` handles boxes."},{"type":"tool_use","id":"Bash_153","name":"Bash","input":{"command":"./.venv/Scripts/python.exe -c \"\nfrom flint import acb, arb, ctx\nctx.prec = 128\nb1 = acb(arb('1.625','0.125'), arb('0','5.125'))\nb2 = acb(arb('-1.25','0.125'), arb('0','5.125'))\nprint('digamma point:', acb.digamma(acb('1.7','2.0')))\ntry:\n    print('polygamma box1:', acb.polygamma(0, b1))\nexcept Exception as e:\n    print('polygamma box1 err:', e)\ntry:\n    print('polygamma box2:', acb.polygamma(0, b2))\nexcept Exception as e:\n    print('polygamma box2 err:', e)\ntry:\n    print('digamma box1:', acb.digamma(b1))\nexcept Exception as e:\n    print('digamma box1 err:', e)\n\" 2>&1 | head -10","description":"Test polygamma/digamma on box inputs","timeout":120000},"extra_content":null}]