<!--
C3 central-rung-mass planar double box: final closed result through weight 4.
All coefficients copied verbatim from the closure records of this work
(explicit w<=2 2D form, 2026-06-24; s=-1 slice w3/w4 explicit GPL words,
2026-06-26; DE-transport gate vs held-out AMFlow, 2026-06-19; explicit
2-variable w3/w4 closed forms, closed 2026-07-03, vendored
as c3-dbox-2var.json).
-->

# C3 central-rung-mass planar double box — final result (w <= 4)

Two-loop planar double box, four massless legs, one internal mass m on the central
rung; m^2 = 1, scales (s, t). Leading-singularity prefactor R = s^2 (t - 1).
Pure function: g(eps) = R eps^4 e^{2 eps gamma_E} J = sum_w g^(w) eps^w, where J is
the top-sector master. g^(0) = 1 exactly (|g0 - 1| < 1e-129 on every sampled point).

## Alphabet (7 rational letters, m^2 = 1)

  { s,  t,  s+t,  s+1,  1-s,  1-t,  s+t-s*t }

First-entry set {s, 1-s, 1-t}. No square root and no further letter enters the
top master through weight 4 (an early maximal-cut scan proposed s^2-t as an eighth
letter; no column of the weight-3/4 tables uses it). s+t enters only at weight >= 3.
The family is topology C of Schwanemann-Weinzierl, arXiv:2412.07522 (SciPost Phys. 18
(2025) 172): their canonical master J^C_17 equals (s/m^2)^{2 eps} g(eps) on the s + i0
sheet (mu^2 = s), checked against their published program at Euclidean points.
c3-dbox-identities.py in this directory also records, in exact arithmetic, a one-parameter
rational parametrisation of the two square roots sqrt(s(s+4)), sqrt(s(s-8)) that the early
maximal-cut scan returned; they belong to no master of the family and enter nothing below.

## Explicit 2D closed form, weights 0-2 (closed; 99 digits on 40 fresh points)

  g^(0) = 1
  g^(1) = -2 log(s) - 4 log(1-t)
  g^(2) =  2 log(s)^2 - 2 log(s) log(s+1) + 8 log(s) log(1-t) + 8 log(1-t)^2
          - 2 Li2(-s) + 10 Li2(t) + 2 Li2(s+t-s*t) - (1/6) pi^2

All coefficients PSLQ-clean rationals, max height 16. Weight-2 symbol (indices into
the alphabet above):
  S[g2] = 4(s,s) + 8(s,1-t) + 8(1-t,s) + 16(1-t,1-t) - 10(1-t,t)
          - 2(s,s+1) - 2(1-t,s+t-st) - 2(1-s,s+t-st)

## Explicit s = -1 slice, weights 3-4 (closed; 108.7 / 108.8 held-out digits)

On the slice s = -1 the t-alphabet is {t, 1-t, 2t-1}; G[...] = G(a_vec; -t) with
letters {t, 1-t, 2t-1} <-> {0, -1, -1/2} in x = -t. Constants live in the
MZV (x) log2 ring (log2 = the s=-1 image of the first-entry letter 1-s).

Weight 3 (15 words):

  g^(3) =  -8 G[t,t,1-t]   + 40 G[t,1-t,1-t]   - 12 G[t,2t-1,1-t]
          + 40 G[1-t,t,1-t] - 64 G[1-t,1-t,1-t] +  4 G[1-t,2t-1,1-t]
          - 16 G[2t-1,t,1-t] + 8 G[2t-1,1-t,1-t] + 8 G[2t-1,2t-1,1-t]
          + log2 ( -12 G[t,2t-1] + 4 G[1-t,2t-1] + 8 G[2t-1,2t-1] )
          + (5/3) pi^2 G[1-t] + ( -(2/3) pi^2 + 4 log2^2 ) G[2t-1]
          - (50/3) zeta3

Weight 4 (44 words): 26 integer-coefficient weight-4 words (|c| <= 256, last letter
1-t), 9 weight-3 words x log2, 6 weight-2 words x {pi^2, log2^2}, 2 weight-1 words
x {zeta3, log2^3}, and the constant
  -pi^4/5 - 14 zeta3 log2 + (2/3) pi^2 log2^2 - (2/3) log2^4 - 16 Li4(1/2).

Full list (coefficients verbatim from the closure record of this work):

  g^(4) =   32 G[t,t,1-t,1-t]        - 20 G[t,t,2t-1,1-t]
          + 164 G[t,1-t,t,1-t]       - 160 G[t,1-t,1-t,1-t]
          -  28 G[t,1-t,2t-1,1-t]    -  96 G[t,2t-1,t,1-t]
          +  48 G[t,2t-1,1-t,1-t]    +  48 G[t,2t-1,2t-1,1-t]
          +  52 G[1-t,t,t,1-t]       - 160 G[1-t,t,1-t,1-t]
          +  24 G[1-t,t,2t-1,1-t]    - 152 G[1-t,1-t,t,1-t]
          + 256 G[1-t,1-t,1-t,1-t]   -   8 G[1-t,1-t,2t-1,1-t]
          +  32 G[1-t,2t-1,t,1-t]    -  16 G[1-t,2t-1,1-t,1-t]
          -  16 G[1-t,2t-1,2t-1,1-t] -  36 G[2t-1,t,t,1-t]
          +  64 G[2t-1,t,1-t,1-t]    +  12 G[2t-1,t,2t-1,1-t]
          -  28 G[2t-1,1-t,t,1-t]    -  32 G[2t-1,1-t,1-t,1-t]
          +  20 G[2t-1,1-t,2t-1,1-t] +  64 G[2t-1,2t-1,t,1-t]
          -  32 G[2t-1,2t-1,1-t,1-t] -  32 G[2t-1,2t-1,2t-1,1-t]
          + log2 ( -20 G[t,t,2t-1]  - 28 G[t,1-t,2t-1]  + 48 G[t,2t-1,2t-1]
                  + 24 G[1-t,t,2t-1] -  8 G[1-t,1-t,2t-1] - 16 G[1-t,2t-1,2t-1]
                  + 12 G[2t-1,t,2t-1] + 20 G[2t-1,1-t,2t-1] - 32 G[2t-1,2t-1,2t-1] )
          + (16/3) pi^2 G[t,1-t]
          + ( -4 pi^2 + 24 log2^2 ) G[t,2t-1]
          - (10/3) pi^2 G[1-t,1-t]
          + ( (4/3) pi^2 - 8 log2^2 ) G[1-t,2t-1]
          - (8/3) pi^2 G[2t-1,1-t]
          + ( (8/3) pi^2 - 16 log2^2 ) G[2t-1,2t-1]
          + (140/3) zeta3 G[1-t]
          + ( -8 zeta3 - (16/3) log2^3 ) G[2t-1]
          - pi^4/5 - 14 zeta3 log2 + (2/3) pi^2 log2^2 - (2/3) log2^4 - 16 Li4(1/2)

## Full 2D w3/w4: explicit 2-variable closed forms (closed 2026-07-03)

Off the slice, weights 3-4 are explicit closed forms (closed 2026-07-03;
exact-rational tables vendored in c3-dbox-2var.json,
evaluated live by the full-2D section of c3-dbox-evaluate.py). Basis:
products of GPL words

  G(a_vec; x = -t), x-letters {0, -1, s, s/(1-s)}
    (dlog images of the alphabet letters t, 1-t, s+t, s+t-s*t; the tables
    carry a fifth x-letter, -s^2, that no column uses),
  G(b_vec; y = -s), y-letters {0, 1, -1}  (letters s, s+1, 1-s),

times powers of log x, log y and a constant.

  w3: 39 exact-rational columns; constants ONLY the classical zeta2, zeta3.
      The interim "genuinely-new weight-3 transcendental" of the earlier
      audit is RETIRED: it was the numeric transport boundary at t = -1/3,
      not a new period. The constant ring is classical.
  w4: 123 exact-rational t-sector columns (constants 1, zeta2, zeta3;
      certified upstream by a t-independent remainder at ~1e-79 across 45
      s-values) + the pure-y block C4(y) as 29 EXACT terms (constants
      1, zeta2, zeta3, zeta4, max coefficient height 38; DUAL-DECODE
      confirmed -- two independent lattice decodes, projection-CVP and
      gauge-invariant quotient, agree on all 29 terms). No fitted numeric
      constants remain: all five weights are closed forms over
      Q[zeta2, zeta3, zeta4].

For s < -1 the y-transport continues along a fixed upper-half-plane detour
around the y = 1 letter; the assembled weights are real (|Im| <= 8e-129
live branch-cancellation certificates at dps 130).

The earlier certified DE-transport representation -- the 16-master
eps-graded system dM/dt = A(eps, s=-1, t) M, built from Kira IBP plus the
external-invariant D_t operator with the s+t+u=0 constraint D_t u = -1,
transported from the single AMFlow boundary at t = -1/3 by a high-order
Taylor stepper (t-poles at {0, 1/2, 1}, Euclidean path pole-free) -- stands
as independent validation of the same function.

## Gate summary (held-out AMFlow oracle, dps 55)

  TOP master, s = -1:        eps^-2 (w2)   eps^-1 (w3)   eps^0 (w4)
    t = -1/2                  54.94 d       54.75 d       54.89 d
    t = -1                    48.47 d       47.73 d       46.83 d
    t = -2                    40.37 d       39.06 d       39.38 d
  Non-circular (t=-2 -> -3, boundary never touched): 54.96-55.92 d (eps^-2..0).
  Explicit slice formula vs 4 fresh AMFlow points: w3 >= 108.7 d, w4 >= 108.8 d.
  Explicit 2D w2 form: 99 digits on 40 fresh points never used in the fit.

  Explicit 2-var w3/w4 forms vs 20 held-out AMFlow points (110-digit strings,
  slices s in {-7/2, -3, -5/2, -3/2} NEVER used in any fit; fit points
  excluded; measured 2026-07-04 by the shipped evaluator, --gate-full, dps 130):
    w3: 109.51-112.90 d      w4 (full, all exact): 109.73-111.13 d
    |Im| branch-cancellation certificates <= 8.04e-129 on every point.