# Threshold banana (1,1,1,9) — non-unipotent connection coefficient, closed form Three-loop banana K3 at masses-squared (1,1,1,9), i.e. on the additive normal-threshold coalescence sqrt(M4) = sqrt(M1)+sqrt(M2)+sqrt(M3). The order-4 Picard–Fuchs operator has threshold indicial polynomial 18432 (rho-1)^3 (2 rho - 3) => exponents {1, 1, 1, 3/2} so the local monodromy at t = p^2 -> 0 is NON-unipotent (eigenvalue e^{3 pi i} = -1 on the t^{3/2} branch; order 2). The single new function-level datum is the connection coefficient c_{3/2} of the holomorphic period onto that branch, varpi_0(t) = A0 Phi^(0)(t) + A1 Phi^(1)(t) + A2 Phi^(2)(t) + c_{3/2} Phi_{3/2}(t). ## Closed form (function-level, closed) c_{3/2} = - sqrt(3) / (36 pi) equivalently 432 pi^2 c_{3/2}^2 = 1 PSLQ/LLL relations (primary relation height 36; squared relation coefficient 432): - 36 c_{3/2} pi + sqrt(3) = 0 residual 194.2 d - -2 log|c_{3/2}| - 2 log pi - 4 log 2 - 3 log 3 = 0 residual 196.1 d - 432 pi^2 c_{3/2}^2 = 1 residual 192.8 d Where the constants come from: in the Hankel-tail derivation (FeynmanGeometry Sec. 2.4; this bundle's threshold_hankel_tail.py) the non-oscillatory tail of J0(y)^3 J0(3y) is -sqrt(3)/(18 pi^2) y^-3 + ..., where 1/sqrt(3) = (m1 m2 m3 m4)^(-1/2) and 1/pi^2 comes from the four Bessel asymptotics; the Mellin transform of H0^(1) at lambda = -1 supplies 2^-3 Gamma(-1/2)^2 = pi/2, giving -sqrt(3)/(36 pi). One may also read 36 = (sum_i sqrt(M_i))^2, the normal threshold; at this mass point the two readings coincide. The Chowla–Selberg Gamma(a/15) CM-period ring of the level-15 K3 motive is PSLQ-NEGATIVE (zero coefficients in every relation). ## Certified value (Route B, Julia/Arb, 768-bit balls, 195 digits) c_{3/2} = -0.01531469153949422359753684717536026226103851334351779591109350056363973065192416096778833413397696914465827970348884947342805812313051692629590881161944422277729854364955194835822349292058384942464 Im part rigorously |Im| < 3.3e-198 (Arb ball). Ball-certified accuracy 649-650 bits. Sign convention: the sign is branch-dependent — the order-2 monodromy negates the t^{3/2} branch, so |c_{3/2}| = sqrt(3)/(36 pi) is the convention-robust statement. (Route A's mpmath e^{-i pi} lands on arg = +pi via roundoff, so Route A reports +|c_{3/2}|.) ## How it was extracted (the projector trick) Exact spectral projector onto the -1 eigenspace of the threshold monodromy M_0: P_{-1} = -(M_0 - 1)^3 / 8 , P_{-1} . varpi_0_state = c_{3/2} . Phi_{3/2}_state (exactly) (M_0 - 1)^3 annihilates the rank-3 unipotent log tower (nilpotent index 3) and acts as -8 on Phi_{3/2}, killing algebraically the ~1e4 unipotent contamination that had walled the earlier 4x4 lu_solve extraction at 3.3-3.7 digits. Two independent transport routes, sharing ONLY the exact-rational PF operator: - Route A: Python/mpmath, s-chart, holomorphic-period multinomial seed (Boenisch-Fischbach-Klemm-Nega-Safari, arXiv:2008.10574), small CCW circle |s|=s_dec around s=0. s_dec-independence 102.5 d, two-base-point agreement 95.9 d (110 working digits); re-run at 210 and 225 working digits the two runs agree to 204 d (internal floor 204 d). - Route B: Julia/Arb ball-arithmetic theta-companion transport, z=-1/s chart, operator-recursion seed (0/155 mismatches vs the multinomial seed), large CW circle around all six finite singularities. s_dec in {1/4, 7/20} agree to all 195 ball-certified digits; row consistency 231.2 d. Two-method agreement A vs B: 195 d, every ball-certified digit of Route B (the mpmath route re-run at 210/225 working digits; the earlier 80.7 d figure was the record's 80-digit printout cap, not the route's accuracy). The non-unipotency itself was certified four independent ways (indicial polynomial, Frobenius monodromy, parity, and series-contrast checks), all from the same exact-rational Picard–Fuchs operator. ## Honest caveats - No external Feynman oracle: a distinct-mass AMFlow evaluation was not available for this configuration. Cross-validation = two-independent-method gate + s_dec-independence + 4-row consistency (>= 100 d) + height-36 PSLQ closure at 195 d. - Both routes share the exact-rational PF operator (built once via CRT + rational reconstruction, annihilation residual exactly 0).