# LBL3KP — kite parent: QED box × crossed-photon self-energy (closed)

The genuine all-eeγ QED 3-loop γγ→γγ diagram (V=8, E=10, planar, one closed
fermion loop): the 1-loop electron LbL box with one rung dressed by the 2-loop
crossed-photon ("kite") electron self-energy, two ℓ²−m² stubs flanking the
insertion. Scalar top-sector integral ν=[1,1,1,2,1,1,1,1,1] (stub doubled),
UV-finite.

## Final result — IBP-reduction onto kite-family masters + one-fold with closed-form kernels

$$
I_{\rm KP} \;=\; \frac{1}{\pi}\int_{m^2}^{\infty} \rho(w)\,K_P(w)\,\mathrm{d}w,
\qquad
K_P(w) \;=\; \frac{K(w)-K(m^2)}{(w-m^2)^2}\;-\;\frac{K'(m^2)}{w-m^2},
$$

where **every ingredient is an analytic object evaluated at runtime to the
requested precision — there is no numeric node table anywhere**:

- $K(w) = \mathrm{Box}_1(s,t;\,m^2,m^2,m^2,\,M^2{=}w)$ is the 1-loop
  3-massive + 1-massive-corner box in its **closed dilogarithmic** 1-dim form
  (4-root partial fractions + one fixed Gauss–Legendre rule, uniform in $w$);
- $\rho(w) = -\,\mathrm{Im}\,J_{\rm top}(w+i0)$ is the discontinuity of the
  kite-family TOP master $J[1,1,1,1,1]$ — the IBP content of the reduction.
  It is computed as a **Picard–Fuchs / IBP-connection series solution**: the
  exact rational 9×9 kite system $\mathrm{d}J/\mathrm{d}w = A(w,d)\,J$
  (`lbl3kp-kite-de.json`, Kira) is ε-graded and solved by adaptive local
  Taylor series along $(m^2,\infty)$ from a single seed at $w=5m^2$, itself
  **derived AMFlow-free** (p²=0 vacuum closed forms + Frobenius at $w=0$ +
  exact-DE march, `lbl3kp-w5-seed.py`; shipped as `lbl3kp-w5-derived.json`,
  with the retired AMFlow vector `lbl3kp-kite-boundary-w5.json` kept as a
  held-out cross-check printed each run).  The elliptic content
  is the equal-mass sunrise block of $A$ — the $\Gamma_1(6)$ curve with cusps
  $\{0,1,9,\infty\}$; the turn-on at $w=m^2$ is the closed-form word
  $-2\pi\,(w{-}m^2)\log(w{-}m^2)$ and the $w=9m^2$ finite jump is the
  $\Gamma_1(6)$ 3-particle cut.  The non-elliptic kite masters have
  $\Gamma$/${}_2F_1$ closed forms, verified live against the transported
  Laurent blocks;
- the partial-fraction kernel $K_P$ is exact (the two stubs share momentum ℓ
  with the dispersive $(\ell^2-w)^{-1}$) and regular at $w=m^2$ (Taylor route
  there, live seam checks); $K(m^2)$, $K'(m^2)$ are computed from $(s,t)$ by
  two independent routes each (direct / Chebyshev fit / Richardson FD);
- the quadrature is singularity-subtracted tanh–sinh (Dispersify): the
  threshold word is subtracted and added back in closed form (exact power-log
  moments × kernel Taylor coefficients).

Single-stub ($\nu_3{=}1$) and no-stub ($\nu_3{=}0$ = LBL3SE) members of the
same family use $K_1(w) = [K(w)-K(m^2)]/(w-m^2)$ and $K(w)$ with the same
runtime $\rho$.

Kernel constants at $(s,t,m^2)=(-1,-\tfrac13,1)$, computed live and
cross-validated against archived AMFlow box values (63.8/68.8 d this run):

```
K(m^2)  =  0.17805022679323064388457145940884606798964278254996
K'(m^2) = -0.081823711317293982171269102248590831053460209669384
```

## Gated values at (s,t,m²) = (−1, −1/3, 1)  (live, default settings, this run)

| integral | this work (runtime) | AMFlow Neville (held-out oracle) | gate |
|---|---|---|---|
| ν₃=2 (top, $I_{\rm KP}$) | 0.11206248143395833056839569474909314353199754 | 0.11206248143395833056839569474909314335268853120486 | 35.8 d |
| ν₃=1 (single stub, $I_{K1}$) | −0.21628648795256062927405122224776344544791881 | −0.21628648795256062927405122224776344616801354202879 | 35.5 d |
| ν₃=0 (= LBL3SE cross-check) | 0.52495777678114463233299641528264604205679097 | 0.524957776781144632332996415282646042436171 | 36.1 d |

(Gate digits are relative-error digits, $d=-\log_{10}|a-b|/|b|$, recomputed at
runtime. Oracle values are Neville ε→0 extractions from an entirely
independent 9-prop AMFlow black-box solve, x_order=200, 714 s — themselves
~40 d artifacts; archived full-precision campaign records 40.04/40.22/43.32 d.
The transport seed is itself gated each run: the derived AMFlow-free w=5
vector agrees with the retired AMFlow vector to ≥79.9 d. The runtime side
keeps growing with `--dps`, exhibited by the held-out 163-digit $\rho(12)$
check (51.1 d this run) and the 138-digit kernel-constant checks (63.8/68.8 d
this run).)

## Method note

The partial-fraction kernel identity is the method deliverable: any 2PR
self-energy-on-a-rung graph reduces to divided differences of the bare-rung
kernel with the sub-integral discontinuity ρ unchanged — LBL3KP is a
kernel-swap extension of LBL3SE, not a new computation. Round 1 of this page
shipped ρ as a 1669-node cache (precision-capped, rejected form); the cache is
deleted — ρ is now solved from its exact rational IBP connection at runtime,
so precision is limited only by requested series order / quadrature level.

**Remaining gap (stated):** a fully kinematic-space iterated-integral form
(the family's m²-DE transported from the m²→∞ vacuum, VoP/Eichler words over
the kite/sunrise Γ₁(6) kernels in m²) is derivable — the symbolic 9-prop
family connection in m² is costed but not yet built; the one-fold above is in
the insertion-mass variable $w$, with all $s,t$ dependence in the closed-form
kernel.