# LBL3SE — box ⊗ kite dressing: final form (one-fold over closed kernels)

The light-by-light three-loop self-energy-insertion master: one-loop massive box
(four on-shell massless legs) with one rung replaced by the two-loop equal-mass
kite self-energy Σ(ℓ²), mass pattern (0,0,m,m,m). Eight propagators, UV-finite,
two-particle-reducible at the insertion; the elliptic content (Γ₁(6) sunrise
curve on the three-massive cut w = 9m²) is inherited, not new.

## Final form (the deliverable)

The 2PR structure gives an *exact* one-fold over the internal mass of the
dressed rung — this is the result, not an approximation:

    I(s,t,m²) = (1/π) ∫_{m²}^{∞} dw  ρ(w) · Box₁(s,t; M² = w)

with every factor an analytic object evaluable to arbitrary precision:

- **Box₁(s,t;w)** — the one-loop massive box in 1-dimensional closed
  dilogarithmic form (4-root partial fractions + one fixed Gauss–Legendre
  rule; uniformly convergent in w).

- **ρ(w) = −Im J_top(w+i0)** — the discontinuity of the ε⁰ kite-family top
  master J[1,1,1,1,1]. **No stored spectral table**: ρ is a Picard-Fuchs /
  IBP-connection series solution, computed at runtime by adaptive local Taylor
  series of the exact rational 8×8 system dJ/dw = A(w,d)·J (eps-graded to ε²)
  along (m²,∞), seeded once by the derived (AMFlow-free) boundary Laurent
  vector at w = 5m².
  The elliptic content is the equal-mass sunrise 2×2 block of A — the Γ₁(6)
  curve, cusps {0, 1, 9, ∞} in w/m². The cusp data appear as the closed-form
  turn-on word ρ ≃ −2π·(w−1)·log(w−1) at w = m² (subtracted and added back as
  exact power-log moments) and the finite Γ₁(6) jump at w = 9m² (crossed on
  the Im w > 0 arc, the Feynman +i0 sheet).

- **IBP onto kite-family masters, exhibited**: the non-elliptic kite masters
  have Γ/₂F₁ closed forms in the same normalization —
  J[1,1,0,0,0] = TAD², J[1,1,0,1,0] = TAD·FMIX(w), J[1,1,0,1,1] = FMIX(w)²
  with TAD = −e^{εγ}Γ(ε−1), FMIX = e^{εγ}Γ(ε)·₂F₁(ε,1;2−ε;w+i0)/(1−ε) —
  and the evaluator checks the transported masters against them live.
  The remaining functions are the Γ₁(6) sunrise block and the m00 sunset;
  the top master is their VoP one-fold through the rational connection.

- **Quadrature** — singularity-subtracted tanh–sinh level ladder; truncation
  guards at the two cusps and the tail are *derived from the requested
  precision* and printed with their bounds; raising `QUAD_DPS`/`LEVEL`/`DPS`/
  `NORD` tightens everything. There is no numeric node cache anywhere;
  a fresh run regenerates every number.

Boundary seed — **derived, AMFlow-free**: the w=5 eps-Laurent vector of the
kite masters is computed by `lbl3se-w5-seed.py` (p²=0 vacuum closed forms —
tadpoles, the equal-mass vacuum sunset in its ₂F₁(…;1/4) form, vacuum IBP —
then Frobenius at the regular-singular point w=0 and an exact-DE Taylor march
to w=5 through the upper half plane), shipped as `lbl3se-w5-derived.json`
(emitted at dps 80; rerunnable at any precision). The retired AMFlow w=5
vector `lbl3se-kite-boundary-w5.json` is kept as a held-out cross-check; the
evaluator prints the recomputed seed agreement each run. The exact rational
connection is `lbl3se-kite-de.json` (Kira IBP, reconstructed from 43 samples,
validation diff 0).

## Gate (measured live by lbl3se-evaluate.py, default settings)

At the held-out Euclidean point (s,t,m²) = (−1, −1/3, 1), ε⁰, this run:

    I (tanh–sinh L5) = 0.52495777678114463233299641528264604205679097
    GT_AMF   (independent 8-prop AMFlow ε-grid, ≥40 d)   agreement 36.1 d
    GT_PARENT (independent 9-prop parent-family Neville)  agreement 36.1 d

The dispersive route shares no reduction, quadrature, or boundary data with
either oracle. Supporting live checks from the same run:

    rho(12) PF-series vs held-out AMFlow solve_integrals   51.1 d
      (archived dps=130 record: 99.95 d; also 99.90/99.89 d at w=50/100)
    kernel-swap (1/π)∫ρ/(w+1)dw vs −Σ_kite(−1) (≥60 d)     41.8 d
    non-elliptic masters vs Γ/₂F₁ closed forms             17.9 d
      (ceiling set by the ε-probe, not the transport)

Archived full-precision original record (<archive>/phys_lbl3se, not recomputed by
the default run): gate 41.70 d at L5/L6, ρ checks 99.89–99.95 d.

## Verification summary

Two independent held-out oracles for the full integral (the 8-propagator
AMFlow black-box ε-grid and the 9-propagator LBL3KP parent family's ν₃=0
member), an independent solve_integrals oracle for the spectral function at
w=12, an independent oracle for the kernel-swap self-test, and live
closed-form checks of every factorizable kite master. All agreements above
are recomputed at runtime as −log₁₀|f−oracle|/|oracle|.