# LBL3X (crossed light-by-light box, sector 938) — the served result **Status (record of 2026-07-08, served 2026-09-03): the finite piece m₁₈⁽⁰⁾ on the transport curve 1/s+1/t=−3/8 is an exact iterated-integral word list in three sectors — a K3 sector (the ring word x₁₇ plus one quadrature of the exact rational r₁₇ against the new period letter N₅), a rational sector (twelve words, closing coefficients exactly 1, certified at three primes), and a twist sector measured to be absent (c_q = 0, bounded below 10⁻¹⁴⁵ with planted-twist controls). The pole tower above it is (0, 0, 2ζ₃) exactly. The assembled value reproduces an independent oracle evaluation at (s,t)=(−8,−4) — a point never used in any fit — to 49 digits at both working precisions, the oracle's own full precision, and the value at s=−7, where no independent evaluation exists anywhere, is on record to 122 certified digits as a standing falsifiable prediction (the served chain, re-pinned 2026-09-06 to the ~200-digit constants, re-derives it at two precisions agreeing to 222 digits).** The 39 master integrals of the three-loop crossed-box γγ→γγ family (sector 938 of the ten-propagator parent, internal photons crossing, electron mass m=1) are determined at any Euclidean point (s,t) by DE transport. The function space is mixed: a pure-polylog top sector (8-letter alphabet {s, t, s+t, s+t+4, s+t+16, s−4, t−4, 2s+t}) over the equal-mass three-loop banana K3 subsector at virtuality u = −s−t. A symbolic closed form in known constants is provably out of reach of value-fitting (class C by the A/B/C value-fittability criterion): the obstruction is a genuine K3 period, and the cure is the word list over that period served here. ## The word list (the deliverable) On the transport curve, m18^(0)(s) = x17(ϖ, ϖ'; s) + ∫_{−4}^{s} r17(x) dN5(x) + W_rat(s) - **x₁₇** — the K3 ring word: a 2484-coordinate h⁷ ring particular with 502 nonzero exact rational coordinates, unique given r₁₇ (kernel rank zero), certified by multi-prime CRT with zero remainder at both confirmation primes. Machine-readable: `lbl3x-legb-x17.json` (sha256 1f0e96a4…, verified on every read). - **r₁₇** — an exact rational function, degree-30 numerator over degree-26 denominator, quoted in full in the wordlist record. - **N₅** — the new quadrature letter built from the K3 period ϖ: N₅(s) = ∫_{−4}^s [A(x)ϖ(x) + B(x)ϖ'(x)²/ϖ(x)] dx with the exact rational A, B given in the record. - **W_rat** — the rational sector: twelve words v_j·Qint_j (j = 0–4, 6–10) and h_i·Qtw_i (i = 1, 2), closing coefficients exactly 1, certified at the three primes 2147483629 / 2147483587 / 2147483647 with a drop-one minimality sweep (all twelve required; the thirteenth candidate word vanishes identically). - **The quadrature alphabet** — thirteen letters: N₅, the ten second-kind letters Qint_j, and the two twisted Eichler-class letters Qtw_i, over the rational letters s and 3s+8. - **The absent sector** — the lower tower's twist curve (j-invariant 3796416/1225) could couple in through one scalar c_q; it was measured, not assumed: c_q = 0 with |c_q| ≤ 7.2×10⁻¹⁴⁶ (main engine) at the measured root, planted 10⁻³⁰ twists recovered at 10⁻¹⁵⁸ absolute error. The physical solution is unbranched there. Every word coefficient is derived exactly from the differential equation and exact boundary input; zero coefficients are fit to oracle data. Record of record: `WORDLIST_FINAL5.json` (2026-07-08 final closure, source sha256 8cc9fe13…), served beside this file as the scrubbed copy `lbl3x-k3words.json` (sha256 c98285fc…, integrity-pinned by the evaluator; `--k3-words` quotes letters, words and certificates from it at runtime). ## Exact base-point values (closed form) Euler identity (exact, from IBP): (s∂_s + t∂_t) m18^(0) = −2Δ, Δ(0,0) = 0 Boundary value at s=t=0 (pySecDec normalization): m18^(0)(0,0) = Vbar_4N − 6·γ_E·ζ₃ Vbar_4N = 6ζ₃ − 22ζ₄ − 8ζ₂·log²2 + (4/3)·log⁴2 + 32·Li₄(1/2) [EXACT] The 32·Li₄(1/2) term is PSLQ-proven NOT producible by any polylog-ansatz column at (0,0) — it comes entirely from the K3 tail. K3-tail kernel (exact and rational, verified at 6 rational points): A_s[m18→m13] = 9743/(660·s) − 8/(15·u) + 21/(u−4) + 35/(2·(u−4)²) − 87/(22·(u−16)) ## The values, and how to check them - **Reference point (−8,−4)** (raw normalization, the page's): m₁₈⁽⁰⁾(−8,−4) = −10.42013800… — agrees with the independent oracle record (never used in any fit) to 49.79/49.80 digits at the two working precisions, the oracle's own full precision. - **Standing prediction s=−7** (on-curve t=−56/13): m₁₈⁽⁰⁾(−7) = −10.41519682300476029923264613844543736155834406676181… — certified 122 digits as recorded 2026-07-08 (two-precision agreement 122.5 d; path diversity 156.8 d); the chain re-pinned 2026-09-06 to the ~200-digit constants agrees with itself at its two precisions to 222.6 d (raw; stripped 222.4 d) at this point. An independent goal-60 AMFlow run (2026-09-06) agrees to 72.644 digits at this point (goal-40 twin 49.787); the full record, both conventions, and the source-file pins are in `lbl3x-prediction.json`. - **Run it live:** `lbl3x-evaluate.py --m18-eps0 --point -7` (or `-8`) re-derives the seed from the pin chain and transports through the exact 41-row system at serve time (vendored `row21_chain/` package, every input sha-pinned; substrate mpmath + python-flint + gmpy2). The default gate suite and `--full` check every other layer of the construction from analytic definitions. ## What remains unfinished (quoted from the record) The record's own remaining item, verbatim from `WORDLIST_FINAL5.json`: "row-17 SYMBOLIC word tower (canonical quadratures through the 23-core coupling): still OPEN as words; the EVALUATOR is closed". The value-serving path is closed and certified; the canonical-quadrature symbolic form of the core response tower is the one construction still outstanding. The Euler function-level gate fires with zero freedom once those words arrive. Status 2026-09-06 (cure-queue item 66): the canonical-quadrature symbolic form of the core response tower is recorded as a stop-class: the 23-core coupling is rational in eps and its Moser reduction stalls in the rational-gauge class (excess floor 5 at both the full-eps and the eps=0 layers, mod-p pre-pass at 4 of 4 and 2 of 2 prime-eps combinations); the words would be written over an eps-form that the reduction does not reach. The evaluator stands. ## History (dated; superseded statements live here) **2026-07-03 — the named form and the no-go statements.** Before the final closure, this artifact stated: "A GPL word list for m18 ε⁰ does not exist in any basis; a genuine word-list closure would have to be an eMPL/Eichler expression over the sec-15 K3, which has not been constructed", and "full m₁₈⁽⁰⁾(s,t) is not evaluable from shipped parts". Both were true when written; the 2026-07-08 closure delivered exactly the predicted eMPL/Eichler-class construction (the word list above), and the 2026-08-07 vendoring of the certified evaluator made the on-curve value servable live. The 2026-07-03 pass also recorded an honest correction: the earlier claim that the named variation-of- parameters form terminates as a finite-depth iterated integral was refuted by direct measurement (the 24-row top block is irreducibly coupled at ε⁰ — 553 off-diagonal ε⁰ entries), so in the unrotated frame the representation is a coupled Volterra system, and it becomes a genuine Chen iterated integral only in an ε-factorized frame — the construction the word list now supplies on the curve. No dispersive representation of m18 exists or is claimed. **2026-07-03 — the transport gate (still genuine).** ε-graded Taylor transport on the completed 41-row on-curve connection, gated at the independent oracle point (−8,−4) (oracle = the prior AMFlow ray-scan evaluation of arXiv:2201.11669, never used in any fit): | master | agreement (dps300 / dps115) | |---|---| | m18 ε⁰ | 49.79 / 49.80 d | | m34 ε⁰ | 49.61 / 49.67 d | | m35 ε⁰ | 49.59 / 49.57 d | | m17 ε⁻¹ = 2ζ₃ | 56.51 / 51.12 d | Two-precision-stable; the ~49 d floor is the oracle string's own precision. Corroboration: closed-loop monodromy residual 2.7e−52; 689/689 kernel entries matched mod 2⁶¹−1 against an independent symbolic reduction. **Earlier still — the m²-DE transport route (superseded, still genuine).** Boundary = massless m²→∞ vacuum vector; ε-graded transport of the electron-mass DE; checked at five independent points vs AMFlow (none used in any fit): | point (s,t) | u | gate_d_min | |---|---|---| | (−25,−25) | 50 | 36.27 | | (−11,−2) | 13 | 34.41 | | (−14,−3) | 17 | 33.95 | | (−23,−10) | 33 | 38.51 | | (−25,−11) | 36 | 38.43 | gate_d_min = minimum over the three top-sector masters (indices 17, 34, 35; index 17 = m18 = I(1,1,1,1,1,1,0,…)). **2026-07-08 — final closure.** `WORDLIST_FINAL5.json`: rational-sector words complete (12-word variation-of-parameters, three-prime certificate), homogeneous pinning done to the corner-data limit, twist sector measured zero, value bank at (−8,−4) and the s=−7 prediction written down. **2026-08-07 — the certified value chain vendored.** The phase3 evaluator (seed re-derived at runtime from the pin chain, 41-row transport, both conventions) was packaged as `row21_chain/` with a single MANIFEST root pin. **2026-09-03 — this refresh.** The served wordlist was upgraded from the phase-1 draft (sha256 75358ee1…, whose interim classifications the final closure superseded) to the scrubbed FINAL5 copy; the `--m18-eps0` arm was ported into `lbl3x-evaluate.py`; `lbl3x-prediction.json` now ships the s=−7 record beside the code.