# Equal-mass sunrise (two-loop, d = 2−2ε) — final result

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Equal-mass sunrise S₁₁₁(2−2ε, t), t = p²/m², m² = μ² = 1. Engine convention
J₁₁₁ = −S₁₁₁ (AW sign). Status: **validation** — known result (arXiv:1704.08895)
reproduced blind, compared only after the result was committed.

## Closed form at order ε⁰

$$
J^{(0)}(t) = -\frac{\psi_1(t)}{\pi}\Bigl(\tfrac{3}{2}\sqrt{3}\,L(\chi_{-3},2) + I(1,f_3;\,q_C)\Bigr)
$$

with (Γ₁(6) modular frame, q_C the nome on the cusp-connected branch, negative
real for Euclidean t < 0):

- ψ₁/π = (4/π) [(t−1)³(t−9)]^(−1/4) K(k²), K the complete elliptic integral of
  the first kind, k² built from the roots of the sunrise Baikov quartic
  P = y(y−4)(y²−2y(t+1)+(t−1)²). Equivalently as a q-series
  ψ₁/π = 2√3 (e₁(q)+e₁(q²)), e₁ = 1/6 + Σ_{m≥1}(Σ_{d|m}χ₋₃(d)) qᵐ.
- q_C = −e^{iπψ₂/ψ₁}; hauptmodul t(q) = 9 η(6τ)⁸η(τ)⁴ / (η(2τ)⁸η(3τ)⁴).
- I(1, f₃; q_C) = Σ_{n≥1} (aₙ/n²) q_Cⁿ, the depth-two Eichler integral of the
  weight-3 form f₃ = 36√3 (e₁³ − e₁²e₂ − 4e₁e₂² + 4e₂³) = Σ aₙqⁿ on Γ₁(6)
  (e₂(q) = e₁(q²)); in the bootstrap's kernel basis f₃ = −3√3·k3.
- Boundary constant (PSLQ-recovered blind, height 3, 498 digits, residual
  < 10⁻³⁰⁷): (3/2)√3 L(χ₋₃,2) = 2.02988321281930725004…, with
  L(χ₋₃,2) = 0.7813024128964862968… .

## All orders through ε⁴ (the full blind result, weight 6)

$$
e^{2\gamma\varepsilon} J_{111}(2-2\varepsilon, t) = \tfrac{2}{\sqrt3}\, u(q)\, e^{-\varepsilon \tilde w(q)} \sum_{j\ge0} \varepsilon^j H_j(q)
$$

- u = (√3/2)ψ̂₁, the normalized holomorphic period (u(0) = 1); w̃ the exact
  exponential rotation that ε-factorizes the Griffiths–Dwork operator.
- H_j are exact ℚ(√3)-word combinations of iterated integrals over the derived
  alphabet {one, k2h, k3, k4E, c4}, c4 = η(τ)²η(2τ)²η(3τ)²η(6τ)² the weight-4
  cusp form 6.4.a.a (fitted coefficient c = 0). Word counts: 2, 4, 12, 25, 55.
- Eleven numerical conditions total through ε⁴; five came out exactly zero
  (ρ̂₁ = ρ̂₂ = ρ̂₃ = ρ̂₄ = 0 and c = 0); ρ̂₀ = 3√3.
- Boundary constants: B₀ = −(3/2)√3·L(χ₋₃,2);
  B₁ = −6·ImLi₃(1−r₃) − (1/2)π·log²3 − (5/54)π³;
  B₂ = 12·ImLi₄(1−r₃) + (3/2)√3·L(χ₋₃,4) − (1/4)√3·π²L(χ₋₃,2)
  + (5/54)π³log3 + (1/6)πlog³3;
  B₃, B₄ remain 500-digit protocol numerics (the committed PSLQ constant
  dictionary was shown to be insufficient at those weights).

## Gate

Held-out point t = −3 (never used in any fit, committed before the gate):
**304 digits** against the certified literature representation. All 50
ball-overlap tests (10 points × 5 ε-orders, incl. 3 on the physical cut with
+i0) agree at 304–305 digits. Independent AMFlow engine points at t = −3, 7
(held out) agree to their full certified precision (129–131 digits).