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# CY3 threshold banana (1,1,1,1,16) — closed forms for c_{5/4} and c_{7/4}

**Object.** Four-loop equal-vertex banana at masses-squared (1,1,1,1,16),
i.e. the threshold coalescence sqrt(M5) = 4 = 1+1+1+1. The order-6
Picard–Fuchs operator has threshold indicial polynomial (exact, residual 0):

    3057647616 * (rho-2)^2 (rho-1)^2 (4 rho-7)(4 rho-5)
    => exponents {1,1,2,2, 5/4, 7/4},  monodromy eigenvalues {1^4, +i, -i},
       semisimple block of ORDER 4.

**Result (closed, function-level).** The connection coefficients of
the two quarter-integer Frobenius branches onto the transported holomorphic
period are

    c_{5/4} = -1 / ( sqrt(2*pi) * Gamma(1/4)^2 )
            = -Gamma(3/4)^2 / ( 2*sqrt(2) * pi^(5/2) )

    c_{7/4} = -5 * Gamma(1/4)^2 / ( 384*sqrt(2) * pi^(5/2) )

    c_{5/4} * c_{7/4} = 5 / (768 * pi^3)        (exact; Gamma(1/4)Gamma(3/4)=pi*sqrt(2);
                                                 768 = 12*(sum sqrt(M_i))^2)

First appearance of the lemniscatic constant Gamma(1/4) in the banana ladder.

**Reference numerics** (Arb-ball-certified values from this work):

    c_{5/4} = -0.03034924668822965573017531637062298790051610508293110352324
               3534531847089485880107793205779575282757646610878370867091580
               8100848637145028098185258535596997066664622708287607349490105
               14066624930272879007...          (198 d ball-certified)

    c_{7/4} = -0.00691848893175028418738600387437833392618847603076836389479
               8534336415339266092596593361322224742229001215813306151159038
               0274835819153479063682706611409319805744979771549613839702428
               9252721055999309161...           (197 d ball-certified)

The high-precision Route-B rerun (1280-bit working precision, 1044/1041
accurate ball bits) matches the closed forms at 295.47 d / 295.13 d.

**How it was extracted (spectral projector, no 6x6 solve).** With threshold
monodromy eigenvalues {1,1,1,1,+i,-i},

    P_{+i} =  (i/8) (M0 - I)^4 (M0 + i I)
    P_{-i} = -(i/8) (M0 - I)^4 (M0 - i I)

so P_{+-i} . varpi0 = c_{5/4,7/4} * Phi_{5/4,7/4}, with six-row state
consistency as the runtime check.

**Certification (no external Feynman oracle — AMFlow hardware-blocked):**
- Two independent routes sharing only the exact-Q PF operator:
  Route A (mpmath, s-chart, BFKNS multinomial seed, small CCW circle) vs
  Route B (Julia/Arb, z-chart, operator-recursion seed, large CW circle at
  z=infinity). Cross-route gate: 72.53 d (c_{5/4}), 71.02 d (c_{7/4}) —
  Route-A dps-80-limited; Route-B internal s_dec-independence 260.4 d / 258.7 d.
- Seed cross-check: 0/301 coefficient mismatches (multinomial vs op-recursion).
- PSLQ over {1, log pi, log 2, log 3, log 5, log 7, log Gamma(1/4)}:
  height 4 (c_{5/4}: [2,0,1,1,0,0,0,4]) and height 15
  (c_{7/4}: [2,0,5,15,2,-2,0,-4]); two-precision-stable at 120 d & 200 d
  (an earlier 896-bit run used 118 d & 197 d).
- K3 positive control on the same code path: c_{3/2} = -sqrt(3)/(36*pi)
  reproduced at 74.8 d (Route A) / 223.3 d (Route B mid).
- Convention note: under the order-4 monodromy each c_alpha picks up +-i per
  circuit; |c_alpha| is the convention-robust quantity (lower-branch
  arg t = -pi convention here).

**Derivation tiers of cy3-banana-evaluate.py** (added 2026-09-11; the default evaluation above is unchanged).
- `--derive [--bits B]` (B >= 1400, default 1400): the DIRECT LINEAR decomposition of the transported
  holomorphic period onto the full local Frobenius basis at s = 0 -- the exact multinomial-squared
  series annihilated exactly by the Picard–Fuchs operator of record, an exact seed jet above the
  threshold with a rigorous tail bound, a ball-arithmetic Taylor transport at B and B/2+100 bits,
  the exact-Q Frobenius basis, one 6x6 solve (the units of direct_linear_extract.py, served beside).
  All six coordinates are printed. At 1400 bits (pair 800): in the canonical basis
  (Phi_1 = t + O(t^3), Phi_2 = t^2(1+O(t)), t = e^{-i pi} s)
      alpha_1^(s) = -0.112055098597095358029096362504371312392...   (two-precision 231.6 d; alpha_1^(t) = -alpha_1^(s))
      alpha_2^(s) = -0.001640947945459396244597465618520841162...   (two-precision 230.2 d)
  the logarithmic coefficients ell_1 (on s log s) and ell_2 (on s^2 log s) vanish to 234.0 / 234.1
  digits; c_{5/4}, c_{7/4} agree with the closed forms above to 412.0 / 411.7 d (two-precision
  232.1 / 231.3 d) and with the Arb literals to their 295.5 / 295.1 digits. K3 (1,1,1,9)
  control, same code path: A_0 = sqrt(3) log(24)/(12 pi) to 413.2 d, A_1 = -sqrt(3)/(24 pi) to 412.7 d,
  A_2 = 0 to 237.7 d, c_{3/2} = -sqrt(3)/(36 pi) to 412.6 d. Wall 0:36.20 (m:ss, one core-pair under a CPU quota).
- `--hankel [--dps N]` (default 60): the analytic derivation of c_{5/4}, c_{7/4} from the non-oscillatory
  tail of J_0(y)^4 J_0(4y) after the rotation of the Bessel representation to the Hankel kernel and the
  Mellin formula (threshold_hankel_tail.py --check --family CY3, served beside: STRUCTURE EQUAL to the
  closed forms above, the tail ratios c_{9/4}/c_{5/4} = -149/2560 and c_{11/4}/c_{7/4} = -5185/75264 equal
  to the operator's exact Frobenius coefficients), then the Bessel moments at two (Y, K, dps) settings:
      int_0^oo y J_0(y)^4 J_0(4y) dy = 0                        to 37.3 / 45.8 digits (absolute)
      int_0^oo y ln(y) J_0(y)^4 J_0(4y) dy = alpha_1^(s)         to 35.8 / 44.3 digits
  (the first is ell_1 = 0; the second identifies the analytic coefficient alpha_1 with the log-moment of
  the Bessel kernel), with the (1,1,1,9) control int y J_0(y)^3 J_0(3y) dy = sqrt(3)/(12 pi) to 34.6 / 43.4 d
  and int y ln(y) J_0(y)^3 J_0(3y) dy = -sqrt(3)(ln 12 + gamma_E)/(12 pi) to 34.5 / 43.3 d. Wall 2:42.43 (m:ss).
- `--planted`: the operator of record with its theta^2 z^1 coefficient shifted by one no longer
  annihilates the period series -- refused by name (exit 3) on (1,1,1,1,16) and on (1,1,1,9).
The five files the tiers read (direct_linear_extract.py, threshold_hankel_tail.py, calpha_rings.py,
fixtures/) are pinned in the evaluator by sha256 and listed in MANIFEST.sha256; they are the same bytes
the threshold-banana bundle serves.