# gg → H elliptic masters G1, G2 (families 2 and 5 of arXiv:2501.14435) — the source's defining integrals, re-evaluated

<!-- Target paper: arXiv:2501.14435 (Marzucca, McLeod, Nega, PRD 2025).
     Written 2026-06-30 for the BootLoops blog download pack. -->

## The result

The two transcendental masters of the elliptic sectors of families 2 and 5
(the source's own definitions, its Eqs. 3.17/3.18, re-evaluated here; the eMPL form of the canonical master integrals themselves, which the source leaves to future work, is not attempted),
in units mt = 1, second internal mass M, virtuality s:

    G1(s) = ∫₀ˢ K1(s′) ϖ₀(s′) ds′ ,   G1(0) = 0  (exact cusp boundary)
    G2(s) = ∫₀ˢ K2(s′) ϖ₀(s′) ds′ ,   G2(0) = 0  (exact cusp boundary)

with purely algebraic kernels over Q[√] (r1 = √(−s), r2 = √(4−s),
r3 = √(M²), r4 = √(4−M²)):

    K1(s) = 2 (M²−1)(M²−s−4) / (M²+3s−4)²

    K2(s) = 2 r3 r4 r1 r2 (M⁴ + 3M²s² − 13M²s − 4M² − 3s³ + 16s)
            ─────────────────────────────────────────────────────
                        (s−4)² s (M²+3s−4)²

In the language of elliptic multiple polylogarithms each is a single iterated integral: the kernel is the
Kronecker first-kind one-form ω₁(z_p(s), τ_C) evaluated at the moving
puncture z_p(s) (Abel–Jacobi image of the additional root s_p1 = (4−M²)/3),
in the LS-period normalization where it pulls back to the elliptic period
ϖ₀(s). All transcendental period content sits in ϖ₀; K1, K2 carry none.

## The period ϖ₀ (paper Eq 3.15, exact transcription)

              2i   √(−M²(M²−4)) · K(zarg)
    ϖ₀(s)  =  ── · ─────────────────────────────────────────────
              π    π √(s−M²) · ⁴√[(s−(M+2)²)(s−(M−2)²)]

    zarg = 1/2 + [M⁴ − 2(s+2)M² + (s−4)s] / [2(M²−s)√((s−(M+2)²)(s−(M−2)²))]

K = complete elliptic integral of the first kind, parameter (m = k²)
convention, i.e. mpmath.ellipk. The square root under K's argument is the
principal root of the PRODUCT (s−(M+2)²)(s−(M−2)²), positive on s < 0: zarg → 0
and ϖ₀ → 1/π as s → 0⁻, the period regular at the cusp (the boundary G_i(0) = 0
rests on it). Taking the roots of the two factors separately gives the other,
logarithmically divergent period of the same curve; the AMFlow towers of the
family through the canonical rotation decide between them (2026-09-09).

## Equivalent split form (paper Eqs 3.21/3.22, the independent assembly route)

    G1 = (8/3)(M²−4)(M²−1) G(k1E) − (2/9)(M²−1) G(k3E)
    G2 = 2 r3 r4 G(k8E)

with elliptic kernels k1E = ϖ₀/(M²+3s−4)², k3E = 3ϖ₀/(M²+3s−4), k8E (Eq 3.42).

## Why DE-transport was mandatory (no compact value-fit exists)

The seed dG/dτ_C carries the period ratio ϖ₀/ψ₁, non-constant in s
(0.082, 0.067, 0.055 at s = −1/2, −1, −2; dps-independent), so no finite
weight-graded dictionary over Q[√] can represent the value directly. At the
connection level the period rides inside the kernel definition and the
matching coefficients K1, K2 stay algebraic (confirmed to 72 digits; the
connection-level match was verified to 60 digits).

## Verification

- The independent check: at (s, M²) = (−7/4, 3/5) the function G₁ is fixed by
  the auxiliary-mass-flow values of the family's thirty master integrals
  through the canonical rotation of arXiv:2501.14435 (its ancillary files,
  DOI 10.5281/zenodo.14843619; the weight-one layer of the canonical basis),
  and the closed form reproduces that value to 57 digits (the two-goal floor;
  receipt 8a8e85a051c2ec9e). One layer deeper (the weight-two identity, the
  deeper towers) the same rotation fixes G₂ to 48 digits (receipt
  ddfde205c92c6628); it enters G₂ as one quarter of the printed Eq 3.18 (the
  normalisation solved from the towers is 1/4 at the 30-digit print floor; the
  printed normalisation fails at 3 digits), and G₂ here is Eq 3.18 itself.
- The evaluator's own consistency: nine Euclidean points (the M²=9/10 grid
  s=−1/2,−1,−2,−3; the paper's kinematics M²=9/10, s=−1/10 and M²=1/3,
  s=−1/75, −1/3; the interior points M²=9/10, s=−137/100 and M²=1/3,
  s=−53/70) at dps 60 and 90, two-depth agreement floor 81 digits; the paper's
  defining integrals (Eqs 3.17/3.18, a separate transcription and quadrature)
  agree with the closed form there to 61 digits. These are consistency checks
  of the closed form with its own definition, not independent oracles.

A self-contained numerical implementation (period, kernels, quadrature, and
the stored reference strings of 2026-09-09) accompanies this file as
ggH-evaluate.py; the values of record with both depths are in the P1-CURE
data file named in CHANGES.md.