# Modular graph function C_{2,2,2,2} — the weight-8 dihedral (banana) Laurent polynomial ## The result C_{2,2,2,2}(tau) is the weight-8 four-edge dihedral (banana) modular graph function, |A|+|B|=16. Near the cusp it collapses to a Laurent polynomial ``` C_{2,2,2,2}|_Laurent = sum_k c_k * y^k, y = pi*tau_2, k in {+8, +1, -1, -2, -3, -4, -5, -6, -7} (all other c_k = 0). ``` **All nine nonzero coefficients are closed and verified.** 8/8 products two-precision-PSLQ PASS (150↔200 on 280d input); c₋₅ CLOSED as svMZV, 94d beyond the fit precision (350d fresh reverify). Independent per-coefficient check: **PASS — all 9 at 41.5–60.5 digits beyond the fit** (τ₂=21; independent-oracle run record). | k | c_k | wt | 2-prec | |---|---|---|---| | +8 | 229 / 3 322 873 125 | 0 | PASS | | +1 | ζ₇/540 | 7 | PASS | | −1 | −7ζ₉/180 | 9 | PASS | | −2 | ζ₅²/12 | 10 | PASS | | −3 | 63ζ₁₁/80 | 11 | PASS | | −4 | −21ζ₅ζ₇/4 | 12 | PASS | | **−5** | **13803/800·ζ₁₃ + 153/10·ζ₃ζ₅² + 51/400·ζ_sv(5,3,5)** | 13 | PASS | | −6 | −(810ζ₅ζ₉ + 567ζ₇²)/64 | 14 | PASS | | −7 | 6825ζ₁₅/512 | 15 | PASS | with ζ_sv(5,3,5) = 2ζ(5,3,5) − 22ζ₅ζ(3,5) − 120ζ₃ζ₅² − 10ζ₅ζ₈ [Brown 1309.5309 eq.7.4, convention ζ(n₁..n_r)=Σ_{k₁<…12 constants known only in scattered n≤3-vertex instances]). ## Method (how the closed forms were derived) Sector expansion (generalises D'Hoker–Green 1904.06603 from a_r=1 to arbitrary a_r): winding split + Poisson resummation → K_{a-½} kernel Φ_a is polynomial; u-integral localises; every sector = rational × signed Mordell–Tornheim sum. |S|≤2 → single ζ (exact); |S|=3 → 2-fold Tornheim → depth-2 MZV; |S|=4 → 3-fold MT → depth-3 MZV. All 9 c_k at 280d in 96.6s (350d in 197s). PSLQ then closes each coefficient on a small odd-zeta pool (plus, for c₋₅, the svMZV₁₃ basis). ## Verification record - **Two-precision PSLQ:** 8 products at dps 150 & 200 (both < 280d input): identical sign+gcd-canonical vectors, resid ≤10⁻¹⁴⁷/10⁻¹⁹⁷. c₋₅ at dps 200 & 250: identical vector on 13-elt ℚ-indep pool + on 5-elt svMZV₁₃ basis, positive control PASS. - **Beyond the fit:** c₋₅ closed form vs FRESH 350d sector-sum → 343.97d = 93.97d beyond the 250d fit leg. sv-form vs same 350d → 309.3d = 59.3d beyond it. - **sv-purity:** In sv-aligned MZV₁₃ spanning set, every ζ₂-containing and every non-sv-irreducible element has coefficient 0. Weight-11 control: Brown's ζ_sv(3,5,3) formula reproduced from independent mzv3 numerics. - **Positive controls:** D₄=C_{1,1,1,1} (6/6 vs DGV), C_{2,2,2}, C_{1,1,2,2}, C_{2,2,2,1} — every coeff closes on odd-zeta products. C_{1,1,1}=E₃+ζ₃ hand-checked. W₄((1,1,1,1);1)=30ζ₅−12ζ₂ζ₃ analytic identity. - **Independent oracle: PASS.** Batch-evaluated outer tanh-sinh torus-quadrature evaluator (no shared code with the sector-expansion evaluator beyond mpmath) at τ₂∈{12,15,18,21}, dps=80 md=8, selfcons ≥95.8d each point. Per-coefficient exact-subtraction check at τ₂=21: c₊₈ 60.5d, c₊₁ 52.2d, c₋₁ 49.9d, c₋₂ 48.5d, c₋₃ 47.6d, c₋₄ 46.6d, **c₋₅ 45.2d**, c₋₆ 43.6d, c₋₇ 41.5d. Residual 3.0e-30→7.4e-54 across τ₂ 12→21, fitted slope 0.961×(−2π) — tail model confirmed, no missing Laurent term. Controls: C₂₂ vs exact E₄/π⁴ 73–90d at the same settings; D₄ through the same subtraction check vs DGV: 41.0–49.9d all 5 coeffs, slope 0.981×(−2π). - **Leave-one-out:** N/A (analytic derivation, not multi-point fit). ## Reproducing the numbers The companion script `mgf-evaluate.py` (same download directory, Python 3 + mpmath only) recomputes every coefficient at runtime from the closed forms above; nothing is embedded as a floating-point literal. The single non-classical constant, ζ_sv(5,3,5), is computed live from Brown's explicit formula [arXiv:1309.5309 eq.(7.4)] with the underlying MZVs evaluated by Richardson–Hurwitz nsum (dps-keyed caches, with the MZV layer re-verified at dps+30 as a two-precision raise). The script checks all nine coefficients against the stored 280-digit sector-sum strings on EVERY run — unconditionally, exiting 1 on any drift (measured 121.1 digits of agreement at --dps 120, mutation control rc=1); the strings come from the sector expansion that produced the PSLQ *input*, and the closed forms were fit with legs at 200/250 dps, so agreement beyond 250 digits is beyond the precision of the fit, and any agreement is a computation of the closed form, never a copy of the reference string. Usage: `python3 mgf-evaluate.py --dps 120` (optionally `--tau2 T` to evaluate the assembled Laurent polynomial).