The crossed light-by-light box — an exact word list on a K3

The one member of the three-loop light-by-light family whose finite piece was proved beyond every closed-form attack — no basis of known constants can recognise it, and the obstruction is a genuine K3 period — now has an exact answer anyway. The finite piece is an explicit thirteen-letter iterated-integral word list whose letters are built from the K3 period itself, with zero fitted word coefficients. It reproduces a reserved oracle to the oracle's own full precision, and it staked a falsifiable claim: a 122-digit value recorded at a point where no independent evaluation then existed; one made since agrees with it on 72 digits, the full depth of that evaluation.

Family context: this page closes the crossed member of the three-loop light-by-light family; the family page carries the diagram census, the parent topology, and the no-go theorem this result answers.

What stood in the way

The finite piece $m_{18}^{(0)}$ of the top master carried the family's sharpest negative result. Value-fitting is structurally unreachable, not precision-limited: every polylogarithm basis tried, with and without K3 kernel columns, floors at 1.08 to 6.6 digits while its inputs carry 48 to 90, and the exact boundary value at $s=t=0$,

$$m_{18}^{(0)}(0,0)=\bar V_{4N}-6\gamma_E\zeta_3,\qquad \bar V_{4N}=6\zeta_3-22\zeta_4-8\zeta_2\log^2 2+\tfrac{4}{3}\log^4 2+32\,\mathrm{Li}_4(\tfrac12),$$

is verified to 159 digits and carries the irreducible weight-four alternating Euler sum $U_{3,1}$, written here through $\mathrm{Li}_4(\tfrac12)$. The functions the polylogarithm fits miss are iterated integrals over the K3 period. The obstruction dictated the cure: a genuine closure would have to be an iterated-integral expression over the K3 itself; this page delivers that construction.

The result

On the transport curve $1/s+1/t=-\tfrac38$, the finite piece closes as an exact word list over thirteen quadrature letters. The data files served with the evaluator below write it out in three sectors:

$$m_{18}^{(0)}(s)\;=\;\underbrace{x_{17}(\varpi,\varpi';s)\;+\;\int_{-4}^{s}r_{17}(x)\,\mathrm dN_5(x)}_{\text{K3 sector}}\;+\;\underbrace{W_{\rm rat}(s)}_{\text{rational sector}},$$

with a pole tower $(0,\,0,\,2\zeta_3)$ above it — the $\varepsilon^{-3}$ and $\varepsilon^{-2}$ coefficients vanish and $\varepsilon^{-1}$ is exactly $2\zeta_3$ — and every boundary constant in exact closed form.

The K3 sector. $x_{17}$ is an explicit ring word in the K3 period $\varpi$ and its derivative, given exactly; $r_{17}$ is an exact rational function (degree-30 numerator over degree-26 denominator, provided in machine-readable form); and $N_5$ is a genuinely new quadrature letter built from the period,

$$N_5(s)=\int_{-4}^{s}\Bigl[A(x)\,\varpi(x)+B(x)\,\frac{\varpi'(x)^2}{\varpi(x)}\Bigr]\mathrm dx,$$

$$A=\frac{640}{13\cdot 3^{17}}\,\frac{(x+\tfrac{16}{3})(x^{2}+8x+\tfrac{64}{3})}{x\,(x+\tfrac83)^{2}},\qquad B=\frac{80}{13\cdot 3^{17}}\,\frac{x\,(x^{4}+20x^{3}+\tfrac{352}{3}x^{2}+\tfrac{1024}{3}x+\tfrac{4096}{9})}{(x+\tfrac83)(x+\tfrac{16}{3})}.$$

The rational sector. $W_{\rm rat}$ is the canonical all-ones particular solution — twelve words over the seven rational letters the family's alphabet leaves on the curve,

$$\{\,s,\;3s+8,\;3s^{2}+12s+32,\;3s^{2}+48s+128,\;s-4,\;5s+8,\;3s+4\,\},$$

every coefficient exact. The full alphabet has thirteen quadrature letters.

The sector that is provably absent. A twist sector could in principle couple into $m_{18}$ at a branching locus, through a coefficient called $c_q$ that was measured rather than assumed: $|c_q|\le 7.2\times10^{-146}$, with planted-twist controls proving the measurement would detect a real coupling at $10^{-30}$. The physical solution is unbranched there.

Zero fitted word coefficients. Every coefficient in the word list is derived exactly from the differential equation and exact boundary input. The constants that enter the construction as pinned inputs — the transport register $\mu$, the register constants $c_9$ and $c_{10}$, and $c_q$ — are each fixed by transport or a boundary condition, none fit to oracle data.

One qualifier the record itself carries: the symbolic word tower for row 17 — the canonical quadratures of the twelve block words through the 23-core coupling — is recorded as a stop-class rather than open as words: the 23-core coupling is rational in $\varepsilon$ (all 529 entries carry $\varepsilon$ in the denominator) and its Moser reduction stalls in the rational-gauge class at excess 5 (a modular pre-pass, four of four prime–$\varepsilon$ combinations, reproduced 2026-09-06 — a stop of the search, not a proof that no $\varepsilon$-form exists); the evaluator of that row is closed and certified, and every number on this page comes from it.

Verified: the assembled word list reproduces a reserved oracle evaluation — a point that entered no step of the construction — to 49 digits at both working precisions, the oracle's own full precision at its first run, and to 72 digits against its re-evaluation at 60 requested digits (lbl3x-evaluate.py --m18-eps0 checks the 72-digit comparison first and keeps the earlier one as a second line); the word side carries at least 122 digits of internal headroom. Further corroboration: a closed-loop transport consistency check, a non-circular corner-anchor comparison, the exact Euler identity $(s\partial_s+t\partial_t)m_{18}^{(0)}=-2\Delta$ verified at function level, and the $\varepsilon^{-1}$ coefficient agreeing with the exact $2\zeta_3$ to 82 digits at the reserved point.

The prediction, tested. At $s=-7$ on the curve, where $t=-56/13$, the word list gave $m_{18}^{(0)}=-10.41519682\ldots$ in the normalization used throughout this page — recorded to one hundred and twenty-two digits, in lbl3x-prediction.json, with no independent evaluation of that point anywhere. One has now been made: an AMFlow run at that point (60 requested digits, by the same route as the reserved-point check, from a fresh reduction) agrees with the recorded value on 72 digits — the run's own depth, not the prediction's: its 40-digit twin agrees with it to 49.8 digits, and both agree with the recorded value to the full length of their own precision (49.787 and 72.644 digits). The prediction stands as the seventh held-out point; lbl3x-evaluate.py --m18-eps0 --point -7 recomputes it from the downloaded data in about two minutes and checks it against the run.

Run it yourself

The downloadable evaluator (lbl3x-evaluate.py) evaluates the family from analytic definitions — period series, exact kernels, exact boundary closed forms; no numeric node cache anywhere on the chain. The default run recomputes the reference values and their cross-checks — the exact base-point values, the Domb identity, the K3 subsector at three points, one tail integral, the pole coefficient at five points, the Euler identity, the lower block, and the precision-doubling ladder — in about a minute and a half on one core, printing each part as it finishes; --full adds the two deeper subsector points, a second tail point, the second integration engine, and the contour deformation, and takes about six minutes on a laptop; --derive-constants re-derives the four transported boundary constants on the bundle itself, from the exact banana-tower data now included beside the evaluator and with no auxiliary-mass-flow input, in about five minutes, agreeing with the served values to 78 digits or more before the full checks run on them. --point s t [dps] evaluates the closed pieces of the family at any Euclidean point with $s<0$, $t<0$ and $u=-s-t$ in $(2,50]$, at any requested precision; the finite piece $m_{18}^{(0)}$ of the top master is served on the transport curve $1/s+1/t=-\tfrac38$ only (--m18-eps0), and no off-curve evaluator of the amplitude is claimed. On the single excluded line $t=-2$ — where a kernel pole sits exactly on the integration contour and the divergence is genuine — it refuses by name rather than extrapolate. The word data are included beside it, integrity-pinned: lbl3x-k3words.json (the word list and letters), lbl3x-k3seed.json (the K3 subsector's exact seed), lbl3x-legb-x17.json (the $x_{17}$ ring word — the evaluator refuses, naming the file, if any of these have been altered) · lbl3x-expression.md.

References

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