# Unequal-mass kite — final result (closed by DE-transport)

Two-loop self-energy, propagators
`D1=k1²−1, D2=k2², D3=(k1−k2)²−1, D4=(k1−p)², D5=(k2−p)²−2`, `p²=s`.
Top master `J(1,1,1,1,1)`, ε⁰ coefficient. The sunrise sub-block has squared
masses (1,1,2); its maximal cut is a **non-modular** (non-congruence) elliptic
curve, so — by the kernel-modularity probe of this work — no finite
constant-coefficient eMPL basis in the fixed three-puncture word bases reproduces
this master (an eMPL form with s-dependent punctures for three distinct masses is in
Broedel–Duhr–Dulat–Penante–Tancredi, arXiv:1902.09971 Sec. 5). The terminal
form here is the DE-transport representation below.

## The curve and its periods

BMSW Feynman curve of the (1,1,2) sunrise,

    E_(1,1,2):  y² = 4(x−e₁)(x−e₂)(x−e₃),   k² = (e₃−e₂)/(e₁−e₂),  Z₃ = e₁−e₂,

roots e_i(s) fixed by the sunrise thresholds M_a = (±1±1±√2)². Period pair and
Wronskian:

    ψ₁(s) = 2 K(k²)/√Z₃,   ψ₂(s) = 2i K(1−k²)/√Z₃,   W = ψ₁ψ₂′ − ψ₂ψ₁′,

with W fixed exactly by the Legendre relation. At s = −2: k² = (2+√2)/4 and
W(−2) = iπ/8 exactly in this ψ₂ convention (the real-representation records in
CLOSE_kite_Aminus.json quote the magnitude π/8); the fiber at s = −2 is
y² = x³ − 756x + 7344 (conductor 128, Cremona 128a2, j = 10976, non-CM, analytic
rank 1), a short Weierstrass model that is non-minimal at 3: the u = 3 scaling of the
minimal model y² = x³ + x² − 9x + 7, whose real period equals ψ₁(−2) exactly.

## The transport representation (the deliverable)

    J⁰_top(s) = J⁰_top(−2) + ∫_{−2}^{s} ψ₁(s′)³/W(s′) · S(s′) ds′ ,

where S(s) is rational in s, assembled from the lower polylogarithmic masters
by the 14×14 IBP connection A(d,s) (obtained with Kira). Every denominator of A
factors over the 4-letter alphabet {s, s−1, s−2, s²−12s+4}; singularities at
s ∈ {0, 1, 2, 6±4√2}; the negative real axis is pole-free.

Boundary (s = −2, 130 digits; master 13, ε⁰ — DERIVED AMFlow-free by
kite-boundary.py: closed-form two-loop vacuum ring at s = 0, every constant
classical (γ_E, ζ₂, ln 2, Catalan; ξ(1,1,2) = −8·Catalan), plus bounded-branch
regularity at s = 0 and exact-DE transport to s = −2; the held-out AMFlow
evaluation at s = −2 is now a gate only, matched to all 130 stored digits):

    J⁰_top(−2) = −0.888742166608516105643132240826652053999779043266323101677986012109273906989625731092455908650989763261260001376255176083646439855

PSLQ-exact sub-sector facts: M₁₀^(ε⁻²) = (s+2)(s+6), M₉^(ε⁻²) = 3. The
remaining boundary entries are 130-digit transcendentals (irreducible on this
curve; see the PSLQ statement below).

## The named boundary integral

The single new boundary transcendental N_kite := 4s·J⁰_top(s)|_{s=−2} = −8·M13[ε⁰](−2)
= 7.109937332868128845145057926613216431998232346130584813423888096874191255917005848739647269207918106090080011010041408669171518840
(130 d; a theorem of the kite-boundary.py construction — classical constants
plus convergent series — confirmed by the held-out AMFlow value to all 130
stored digits) is delivered as an explicit period-pair variation-of-parameters
(Eichler) integral:

    N_kite = c₁ ψ₁(−2) + c₂ ψ₂(−2)
             + ψ₂(−2)∫_{t⋆}^{−2} ψ₁(t)R(t)/W(t) dt − ψ₁(−2)∫_{t⋆}^{−2} ψ₂(t)R(t)/W(t) dt,

t⋆ = −4, R(t) the explicit rational source from the top row of the 14×14
connection, and (c₁, c₂) DERIVED analytically (2026-07-03, zero fits): s = 0
cusp regularity — the residue spectrum {0¹⁵ (semisimple), −1²⁴, −2³} admits a
unique bounded solution — seeded by the closed-form three-mass vacuum-sunset
ring (γ_E, ζ₂, ln 2, Catalan; in particular ξ(1,1,2) = −8·Catalan). Numerically
(c₁, c₂) = (26.359412432411461897087360205708380411206096239458,
−35.499895175164543101380809177424949770627230513028) — now OUTPUTS of the
derivation; every AMFlow value (s ∈ {−2, −3, −6} and the 15-point oracle set) is a
held-out gate. The deep-digit reproductions — 125.0–125.2 digits at 100 requested
(stable at 160) rising to 130.0–131.1 at 300, where every comparison exhausts the
130-digit reference strings (reference-exhausted) — hold at s ∈ {−5/2, −3, −15}
plus N_kite at s = −2; the remaining gates run 88.9–130.2 digits (s = −6 at 99.6 d,
the 15-point oracle set at 88.9–112.0 d), as the Gate section below records. PSLQ against every furnished named ring
(period-lattice bilinears, conductor-2-power Dirichlet L, BSD/Beilinson
regulator, Mahler measure, Type-III lemniscatic) is honestly NEGATIVE at sane height:
full closure to a named constant is arithmetically blocked, not basis-blocked.

## Gate (held-out verification)

DE-transport from the s = −2 boundary (Taylor integrator, all 42 ε-graded
components carried unconditionally) reproduces an independent 130-digit
AMFlow oracle runs at all 14 held-out Euclidean points:

- audited run (order 120, dps 160):
  min 63.5 d (s=−15), mean 75.8 d, max 97.7 d (s=−5/2);
- higher-precision rerun (order 170, dps 200):
  88.9–130.2 d at the same points.