# The nonplanar $\mathcal T_4$ line form, written out

**What this file is.** The symbolic form of record of the two-loop nonplanar
$\mathcal T_4$ integrals for $q\bar q\to W^+W^-$ (exact top mass) on the
kinematic line $(x,z)=(17/2,\,5/3)$, $y$ free — the structures the
[results page](../../diagrams/qqww-t4.html) displays, each next to its
receipt. Everything here is machine-checkable: `python3 eval_row35.py
--check-expression` (in this directory) recomputes the sector-487
zero-remainder substitution, the letter alphabet, $r_\pm$, and the exact
$j$-invariant chain live, in exact arithmetic, and exits nonzero on any
mismatch. Full construction: Sec. 2.18 of
[the portfolio paper](../../SmatrixBootLoops.pdf).

Generated 2026-09-03 05:00:28Z by script; every digest and list below is
script-emitted from the named source file, never typed.

## The family and the line

$\mathcal T_4[\nu_1,\nu_2,\nu_3,\nu_6,\nu_7,\nu_8,\nu_9]$, family
`wpairT4`, $d=4-2\varepsilon$: massless quark legs, on-shell $W$ legs,
exact $m_t$ in the crossed chain (propagators on the results page). The
form is constructed on the line $(x,z)=(17/2,5/3)$ with
$y=-t/m_t^2$ running (Mandelstam $t$; below, the symbol $t$ is reused for the sector-487 line variable and for the $X_0(2)$ Hauptmodul, each defined where used); base point $y=1/3$.

The 79 components (76 master integrals + 3 sector-487 auxiliaries carried
by the numeric basis) satisfy the first-order system
$\partial_y M = A(\varepsilon,y)\,M$. The connection $A$ is known exactly:
1240 nonzero entries, each an exact rational function
$N(d,y)/Q(d,y)$ (`row35_data.json` in this directory, key `A_entries`;
entry `"i,j"` holds the $y$-coefficient lists of $N$ and $Q$, each
coefficient itself an exact rational function of $d$).

## The 79 rows

| # | master |
|---|---|
| 0 | `wpairT4[0,1,1,0,0,0,0,0,1]` |
| 1 | `wpairT4[0,1,1,0,0,1,0,0,1]` |
| 2 | `wpairT4[0,1,0,0,0,1,1,0,0]` |
| 3 | `wpairT4[1,0,0,0,0,0,1,0,1]` |
| 4 | `wpairT4[1,-1,0,0,0,0,1,0,1]` |
| 5 | `wpairT4[0,1,0,0,0,0,1,0,1]` |
| 6 | `wpairT4[-1,1,0,0,0,0,1,0,1]` |
| 7 | `wpairT4[0,1,1,0,0,0,1,0,1]` |
| 8 | `wpairT4[-1,1,1,0,0,0,1,0,1]` |
| 9 | `wpairT4[0,1,1,0,0,-1,1,0,1]` |
| 10 | `wpairT4[1,1,1,0,0,0,1,0,1]` |
| 11 | `wpairT4[1,1,1,0,0,-1,1,0,1]` |
| 12 | `wpairT4[1,0,0,0,0,1,1,0,1]` |
| 13 | `wpairT4[0,1,0,0,0,1,1,0,1]` |
| 14 | `wpairT4[-1,1,0,0,0,1,1,0,1]` |
| 15 | `wpairT4[1,1,0,0,0,1,1,0,1]` |
| 16 | `wpairT4[1,1,-1,0,0,1,1,0,1]` |
| 17 | `wpairT4[0,1,1,0,0,1,1,0,1]` |
| 18 | `wpairT4[1,1,1,0,0,1,1,0,1]` |
| 19 | `wpairT4[1,0,0,0,0,1,0,1,0]` |
| 20 | `wpairT4[1,-1,0,0,0,1,0,1,0]` |
| 21 | `wpairT4[1,1,1,0,0,1,0,1,0]` |
| 22 | `wpairT4[1,1,1,0,0,1,0,1,-1]` |
| 23 | `wpairT4[0,0,0,0,0,0,0,1,1]` |
| 24 | `wpairT4[0,0,1,0,0,0,0,1,1]` |
| 25 | `wpairT4[-1,0,1,0,0,0,0,1,1]` |
| 26 | `wpairT4[0,0,0,0,0,1,0,1,1]` |
| 27 | `wpairT4[1,0,0,0,0,1,0,1,1]` |
| 28 | `wpairT4[1,-1,0,0,0,1,0,1,1]` |
| 29 | `wpairT4[1,0,-1,0,0,1,0,1,1]` |
| 30 | `wpairT4[0,0,1,0,0,1,0,1,1]` |
| 31 | `wpairT4[-1,0,1,0,0,1,0,1,1]` |
| 32 | `wpairT4[0,-1,1,0,0,1,0,1,1]` |
| 33 | `wpairT4[1,0,1,0,0,1,0,1,1]` |
| 34 | `wpairT4[1,-1,1,0,0,1,0,1,1]` |
| 35 | `wpairT4[1,0,1,0,0,1,-1,1,1]` |
| 36 | `wpairT4[0,1,1,0,0,1,0,1,1]` |
| 37 | `wpairT4[-1,1,1,0,0,1,0,1,1]` |
| 38 | `wpairT4[0,1,1,0,0,1,-1,1,1]` |
| 39 | `wpairT4[1,1,1,0,0,1,0,1,1]` |
| 40 | `wpairT4[1,1,1,0,0,1,-1,1,1]` |
| 41 | `wpairT4[1,0,0,0,0,1,1,1,0]` |
| 42 | `wpairT4[1,1,0,0,0,1,1,1,0]` |
| 43 | `wpairT4[1,1,-1,0,0,1,1,1,0]` |
| 44 | `wpairT4[1,1,1,0,0,1,1,1,0]` |
| 45 | `wpairT4[1,0,0,0,0,0,1,1,1]` |
| 46 | `wpairT4[1,-1,0,0,0,0,1,1,1]` |
| 47 | `wpairT4[1,0,-1,0,0,0,1,1,1]` |
| 48 | `wpairT4[1,0,1,0,0,0,1,1,1]` |
| 49 | `wpairT4[1,-1,1,0,0,0,1,1,1]` |
| 50 | `wpairT4[1,0,1,0,0,-1,1,1,1]` |
| 51 | `wpairT4[1,1,1,0,0,0,1,1,1]` |
| 52 | `wpairT4[1,1,1,0,0,-1,1,1,1]` |
| 53 | `wpairT4[0,0,0,0,0,1,1,1,1]` |
| 54 | `wpairT4[1,0,0,0,0,1,1,1,1]` |
| 55 | `wpairT4[1,-1,0,0,0,1,1,1,1]` |
| 56 | `wpairT4[1,0,-1,0,0,1,1,1,1]` |
| 57 | `wpairT4[1,0,0,-1,0,1,1,1,1]` |
| 58 | `wpairT4[1,0,0,0,-1,1,1,1,1]` |
| 59 | `wpairT4[1,-2,0,0,0,1,1,1,1]` |
| 60 | `wpairT4[0,1,0,0,0,1,1,1,1]` |
| 61 | `wpairT4[-1,1,0,0,0,1,1,1,1]` |
| 62 | `wpairT4[-2,1,0,0,0,1,1,1,1]` |
| 63 | `wpairT4[1,1,0,0,0,1,1,1,1]` |
| 64 | `wpairT4[1,1,-1,0,0,1,1,1,1]` |
| 65 | `wpairT4[1,1,0,-1,0,1,1,1,1]` |
| 66 | `wpairT4[1,1,0,0,-1,1,1,1,1]` |
| 67 | `wpairT4[1,1,-2,0,0,1,1,1,1]` |
| 68 | `wpairT4[1,1,-1,-1,0,1,1,1,1]` |
| 69 | `wpairT4[0,0,1,0,0,1,1,1,1]` |
| 70 | `wpairT4[1,0,1,0,0,1,1,1,1]` |
| 71 | `wpairT4[1,-1,1,0,0,1,1,1,1]` |
| 72 | `wpairT4[1,0,1,-1,0,1,1,1,1]` |
| 73 | `wpairT4[1,0,1,0,-1,1,1,1,1]` |
| 74 | `wpairT4[0,1,1,0,0,1,1,1,1]` |
| 75 | `wpairT4[1,1,1,0,0,1,1,1,1]` |
| 76 | `wpairT4[1,1,1,-1,0,1,1,1,1]` |
| 77 | `wpairT4[1,1,1,-2,0,1,1,1,1]` |
| 78 | `wpairT4[1,1,1,0,-1,1,1,1,1]` |

## The 36 diagonal blocks (29 dlog / 6 radical / 1 elliptic)

Block-lower-triangular structure of $A$, read off
`finalgate/GATE_TABLE.json` (sector → rows). The class column: `dlog` =
$\varepsilon$-factorizes to exact d-log form (GPL words), `radical` =
square-root letters (sectors 417, 421, 449, 453, 482, 483), `elliptic
maximal-cut curve` = sector 481 (its fibrewise curve has non-constant $j$,
while its $\varepsilon^0$ differential-equation block is reducible to first
order: six exact hyperexponential factors on both lines, the `sector_481`
entry of `row35_L1/blocks_epsform.json` and `row35_L2/blocks_epsform.json`,
re-run by `qqww-t4-sec481-chain.py --line L1|L2`; the full $\varepsilon$-dependent block is regular singular with the local exponent $-3/2$ at the chain's two conic letters, so it admits no $\varepsilon$-factorised form with rational letters on the line, an $\varepsilon$-form over the two conic square roots not excluded); sector 487 is `dlog` in the basis with $[1,1,1,-1,-1,1,1,1,1]$
in place of $[1,1,1,-2,0,1,1,1,1]$ and Moser-irreducible only in the
Laporta basis carrying $[1,1,1,-2,0,1,1,1,1]$ (its radicals cancel at
$d=4$, where its basis is pure rational).

| sector | rows | row ids | class |
|---|---|---|---|
| 98 | 1 | 2 | dlog |
| 161 | 2 | 19–20 | dlog |
| 167 | 2 | 21–22 | dlog |
| 225 | 1 | 41 | dlog |
| 227 | 2 | 42–43 | dlog |
| 231 | 1 | 44 | dlog |
| 262 | 1 | 0 | dlog |
| 294 | 1 | 1 | dlog |
| 321 | 2 | 3–4 | dlog |
| 322 | 2 | 5–6 | dlog |
| 326 | 3 | 7–9 | dlog |
| 327 | 2 | 10–11 | dlog |
| 353 | 1 | 12 | dlog |
| 354 | 2 | 13–14 | dlog |
| 355 | 2 | 15–16 | dlog |
| 358 | 1 | 17 | dlog |
| 359 | 1 | 18 | dlog |
| 384 | 1 | 23 | dlog |
| 388 | 2 | 24–25 | dlog |
| 416 | 1 | 26 | dlog |
| 417 | 3 | 27–29 | radical |
| 420 | 3 | 30–32 | dlog |
| 421 | 3 | 33–35 | radical |
| 422 | 3 | 36–38 | dlog |
| 423 | 2 | 39–40 | dlog |
| 449 | 3 | 45–47 | radical |
| 453 | 3 | 48–50 | radical |
| 455 | 2 | 51–52 | dlog |
| 480 | 1 | 53 | dlog |
| 481 | 6 | 54–59 | elliptic maximal-cut curve (the $\varepsilon^0$ differential-equation block is reducible to first order: six exact hyperexponential factors on both lines; the full block is regular singular with exponent $-3/2$ at the two conic letters, so no rational-letter $\varepsilon$-form) |
| 482 | 3 | 60–62 | radical |
| 483 | 6 | 63–68 | radical |
| 484 | 1 | 69 | dlog |
| 485 | 4 | 70–73 | dlog |
| 486 | 1 | 74 | dlog |
| 487 | 4 | 75–78 | dlog (in the basis with $[1,1,1,-1,-1,1,1,1,1]$ in place of $[1,1,1,-2,0,1,1,1,1]$; Moser-irreducible only in the Laporta basis carrying $[1,1,1,-2,0,1,1,1,1]$) |

## The 15-letter alphabet

The polylogarithmic cone's GPL words are written over fifteen letters:

$$a \in \{0,\ -1,\ \pm\tfrac53,\ -\tfrac{31}{12},\ -\tfrac{31}6,\ -\tfrac{41}6,\ -\tfrac{25}6,\ -\tfrac72,\ -\tfrac{73}{48},\ -\tfrac{175}{48},\ \tfrac{91}{24},\ -\tfrac{215}{24},\ r_+,\ r_-\},$$

with $r_\pm = \tfrac1{12}(-31\pm\sqrt{561})$ the roots of
$18y^2+93y+50$. Each letter is certified against the connection itself:
the check mode factors all 1240 denominators $Q(4,y)$ exactly and confirms
every rational letter as a root and the quadratic $18y^2+93y+50$ as an
irreducible factor. The denominator locus also carries eight additional
apparent singularities ($y=1,\ -21,\ -\tfrac{97}3,\ -\tfrac{47}6,\ -\tfrac{37}6,\ \tfrac83,\ \tfrac{95}6,\ \tfrac{163}6$)
and further quadratics (among them $18y^2+93y+67$, the sector-483 radical
kernel) that appear in no GPL word of the cone — apparent poles of the
connection, not letters of the solution. In the basis with
$[1,1,1,-1,-1,1,1,1,1]$ in place of $[1,1,1,-2,0,1,1,1,1]$ the sector-487
block is an $\varepsilon$-factorised dlog form, regular at $y=\infty$, with
the two letters $18y^2+93y+50$ and $18y^2+93y+203$ on this line (the
sector-487 entry of `row35_L1/blocks_epsform.json`; Sec. 2.18 of the
portfolio paper).

## The polylogarithmic tower (first two $\varepsilon$-layers)

The 40-row polylogarithmic cone carries explicit GPL words over the
alphabet. The first two $\varepsilon$-layers of the main 12-row tower are
shipped exactly in
[`qqww-t4-tower12-layers.json`](qqww-t4-tower12-layers.json)
(exact rational and $\mathbb Q(\sqrt{561})$ coefficients; deeper layers
reach depth six in the same letters). The results page's sample entry is
layer-1 entry $(11,10)$ of that file:

$$-\tfrac23\,G(0;y)\;+\;\tfrac{187}{18}\,G(-1;y).$$

## Sector 487: the exact rational basis

Rows 75–78. With $y = \tfrac13 + \tfrac4{15}t$ and
$D(t) = t^2 + \tfrac{175}8 t + \tfrac{1475}8$, the four solution
columns $Y_1..Y_4$ displayed on the results page are shipped exactly in
[`qqww-t4-sector487.json`](qqww-t4-sector487.json). The certificate is an
exact zero-remainder substitution,

$$\frac{d}{dt}Y_c \;-\; \tfrac4{15}\,A(t)\,Y_c \;=\; 0
\quad\text{exactly, for all four columns,}$$

with $A(t)$ the rows-75–78 block of the exact connection at $d=4$ under
the same substitution — recomputed live by `--check-expression` (exact
`Fraction` arithmetic; a $10^{-30}$ perturbation of one basis
coefficient makes the check fail, and that control ships as
`--check-expression --mutate`).

## Sector 483: the radical basis

Rows 63–68: three exact rational columns plus a nested-twist radical pair
over the line letters, radical kernel
$1/\sqrt{t^2 + \tfrac{175}8 t + \tfrac{625}8}$ (the $t$-image of
the $18y^2+93y+67$ kernel), certified by the same exact zero-remainder
container identities (source `sec483x/B483_BASIS_FULL.json`, digest
below).

## Radical sectors 417 / 449 / 482 (and 421 / 453)

Masters 27–29, 45–47, 60–62 close as layered variation-of-parameters
containers with radical letters; the certified containers ship in
`row35_radical_feed.json` (this directory) and are evaluated directly by
the evaluator (exact zero-remainder certificates: 35/35, 35/35, 52/52
container identities). Sectors 421 and 453 (masters 33–35, 48–50) have
the same class of certified containers, about a gigabyte together —
not downloadable; the shipped evaluator computes those six masters by
transport instead, with the same certificates as every other row.

## Sector 481: the block with an elliptic maximal-cut curve and its $X_0(2)$ anchor

Rows 54–59. The maximal cut localizes onto $w^2 = g_1(z_4)\,G_2(z_4)$,
a cubic in $z_4$ fibered over $z_5$ ($g_1$, $G_2$ written out on the
results page). The block is solved by an order-by-order variation of
parameters on a certified rank-six basis (`sec481/HOMOG_BASIS.json`,
`RATSOL.json`, `LOGSOL.json`), not by any named class of special
functions: the curve is non-CM and non-congruence. The sector has an elliptic maximal-cut curve: its fibrewise curve has non-constant $j$ (an order-two Picard--Fuchs operator in the fibre variable $z_5$; $j$ non-constant along the mass-scaling ray as well), while its $\varepsilon^0$ differential-equation block is reducible to first order.
On both served lines and along two mass-scaling rays the $\varepsilon^0$
block factors into a chain of six first-order operators with exponents in
$\tfrac12\mathbb{Z}$ at the sector's letters (two conic quadratics carrying
exponent $-3/2$, the rest integer), remainder rank zero -- two exact rational
solutions and four solutions that are iterated integrals of radicals; the
certified thimble column lies in the chain's solution space (the `sector_481`
entry of `row35_L1/blocks_epsform.json` / `row35_L2/blocks_epsform.json`;
the receipts under `vendor_row35_sec481/`; `qqww-t4-sec481-chain.py`
re-runs the factorisation on the served connection). The full $\varepsilon$-dependent block is regular singular at every singular point on the first line (at generic fixed dimension an exact rational gauge transformation brings it to Fuchsian form), and its local exponents lie in $\mathbb{Z}+\mathbb{Z}\varepsilon$ at every letter except the chain's two conic letters, $Q_1 = 225y^2+5370y+19801$ and $Q_2 = 900y^2-12180y-7751$, where the exponent is $-3/2$, independent of the dimension: no $\varepsilon$-factorised form with rational letters exists on the line, and an $\varepsilon$-form over the function field extended by $\sqrt{Q_1}$, $\sqrt{Q_2}$ is not excluded.

**The $j$-invariant derivation, end to end.** At the base fiber
($y=\tfrac13$, $z_5=\tfrac73$ — one of four rational fibers where the
exact-$j$ reconstruction from the cubic's invariants was gated
exact-match in `sec481/HAUPTMODUL_PROBE.json`):

1. the cubic $g_1G_2$ in $z_4$ has exact rational coefficients; its
   invariants give
   $$j \;=\; \frac{c_4^3}{\Delta} \;=\; \frac{232620187166690272611921}{9846172967579601725} \;\approx\; 23625.44188;$$
2. the curve carries a rational 2-torsion point, so $j$ factors through
   $X_0(2)$: $j = (t+256)^3/t^2$ with $t$ the $X_0(2)$ Hauptmodul;
3. solving $(t+256)^3 = j\,t^2$ exactly gives the unique rational root
   $$t \;=\; \frac{6787441}{213725},$$
   and the identity $(t+256)^3 = j\,t^2$ holds exactly at that $t$ —
   this is the anchor value the results page displays.

`--check-expression` recomputes this whole chain in exact rational
arithmetic.

## The boundary constants

The 395 boundary constants (the first five $\varepsilon$-Laurent
coefficients of each of the 79 components at the base point, real and
imaginary parts) are digit strings in
[`qqww-t4-boundary.json`](qqww-t4-boundary.json), each with its source
ball radius and a cross-source agreement column; at every evaluator run
the boundary value path is re-certified against the two-precision node
bank (certificate min 124.2 digits). Boundary provenance across the 79
components (paper, Sec. 2.18): 23 import-GPL limits, 11 derived closed
forms, 5 derived written integrals, and 40 components defined by the
$\eta$-transport written integral; the radical closed forms consume the
boundary constants of 32 components (120 $(i,k)$ constants, refreshed at
load from the certified bank — the evaluator's `BND_VAL REFRESH` line).

## Verification numbers on the results page, traced

* **37 digits** — worst row of the consolidated table at 40-digit working
  precision: row 76 (sector 487) at 37.0 digits,
  `finalgate/GATE_TABLE.json` (quoted per-row minimum over both fresh
  points and orders $\varepsilon^{-3}..\varepsilon^2$ vs independent
  reference evaluations never used in any fit; rises to 57–58 digits at
  dps 60 — pure precision-floor scaling).
* **47 digits** — overlap with the published planar results: the paper's
  own functions reproduced at 47–49 digits on four masters (paper,
  Sec. 2.18, ancillary-value comparison).
* **76 masters, 23 new / 53 mapped** — family reduction count and the
  symbolically verified dictionary onto the planar branches (paper,
  Sec. 2.18).
* **36 blocks = 29 + 6 + 1** — the table above.
* **395 constants** — counted in `qqww-t4-boundary.json` (script-emitted).

## Source digests (form of record)

The archive paths are relative to the `solve_row35/qqww_2026-07-03`
form-of-record store; the first 16 hex of each SHA-256:

| file | sha256 (16) | role |
|---|---|---|
| `finalgate/GATE_TABLE.json` | `6f96b0ed99daef55…` | the consolidated 79-row verification table (36 sectors, per-row digits) |
| `finalgate/RESULT.md` | `f17fdd370090e3b5…` | the per-class verification record |
| `sec483x/B487_BASIS_FULL.json` | `994954da7a1c54a3…` | sector-487 exact rational fundamental system (shipped, scrubbed, as qqww-t4-sector487.json) |
| `sec483x/B483_BASIS_FULL.json` | `afbeb1ebd46b931f…` | sector-483 basis: three rational columns + a nested-twist radical pair |
| `sec481/HOMOG_BASIS.json` | `cee4f08d305336fd…` | sector-481 certified rank-six homogeneous basis |
| `sec481/RATSOL.json` | `03759cf54600af85…` | sector-481 rational solution data |
| `sec481/LOGSOL.json` | `dda766e6bd98b4cf…` | sector-481 logarithmic solution data |
| `sec481/HAUPTMODUL_PROBE.json` | `de119fac692ea3fc…` | the X0(2) Hauptmodul factorization probe (exact j at four rational fibers) |
| `radical/RADICAL_CLOSURE.json` | `05d9e9c3d0d169d7…` | index of the radical closed-form containers |
| `radical/RADICAL_CLOSURE_b417.json` | `64b4bc4e46f73d39…` | sector-417 cluster closed forms (shipped in row35_radical_feed.json) |
| `radical/RADICAL_CLOSURE_b449.json` | `4214aaecb0d8960f…` | sector-449 cluster closed forms (shipped in row35_radical_feed.json) |
| `radical/RADICAL_CLOSURE_b482.json` | `cfa5ba620e7dfc7f…` | sector-482 cluster closed forms (shipped in row35_radical_feed.json) |
| `radical/RADICAL_CLOSURE_b421_k2.json` | `01ae2bb66c0661c3…` | sector-421 certified containers (~0.5 GB; not downloadable) |
| `radical/RADICAL_CLOSURE_b453_k2.json` | `80f69bcb7a54c557…` | sector-453 certified containers (~0.5 GB; not downloadable) |
| `amf_probe/out_T4_anchor79deep.json` | `c0aa080d5487855c…` | the deep base-point run behind the boundary-constant table (qqww-t4-boundary.json) |
| `assembly/CONE_WORDS2_tower12_K2.json` | `4f5ead1a911fd753…` | the main-tower GPL words, first two eps-layers (shipped, scrubbed, as qqww-t4-tower12-layers.json) |

Shipped-bundle digests are in [`MANIFEST.sha256`](MANIFEST.sha256).