A preregistered bench for the two-defect heat-kernel cross term on the square lattice: effective exponents, observable dependence and resolution limits

Technical report, code and data for bench D4 of Volume IV.2: a preregistered measurement of the two-defect heat-kernel cross term I_K = K12 − K1 − K2 + K0, with K_X(t) = exp(−t L_X), for pairs of finite surgeries of the square-lattice torus at separation R. Design. Two surgery families (twisted plaquette, TP; perpendicular two-swap control, SWAP_PERP_MIN), R ∈ {4, 6, 8, 12, 16, 24, 32, 64}, fixed times t ∈ {1, 4, 16} and diffusive times t = λR² with λ ∈ {1/4, 1, 4}, two torus sizes per separation (L, L′ = 2L). The protocol (PREREG_D4.md, Spanish, normative; SHA-256 fdb983750efd7c82098fce99b2491837c73e252c4ccd385e3efd62f928dad104) was frozen, with one dated amendment, before any bench instance was computed. Method. Kernels are evaluated with a Chebyshev–Bessel expansion with a certified truncation bound (2 Σ_{k>n} e^{−4t} I_k(4t) ≤ 1e−24), validated before the bench against expm_multiply and against exact component-wise arbiters (Bessel closed form; positive Poisson uniformization series), up to t = 16384 (engine_validation_hash a2a659c6fa9da2ac). The provenance of every error component is documented; the rounding part is an empirical envelope, not a proved bound. Results. On the points admitted by the preregistered filters, a model with power-law dependence on R and λ-dependent intercepts is preferred to exponential and constant models for TP. The joint effective exponents are 7.22 (midpoint entry Q_mid, ΔAICc = 36.2) and 7.17 (probe-block norm F_RMS, ΔAICc = 11.6). They stay preferred, at 7.13 and 7.04, under a sensitivity analysis declared before the results. Exponents fitted per λ differ (6.57–7.75) and local slopes drift over R ≤ 24, so the joint values are effective exponents over a finite range. For TP at λ = 1/4 the selected model depends on the observable. At R = 64 both observables fall below the operational resolution threshold; no rigorous upper bound on the true value is obtained. No contradiction with the fixed-time Poisson bound of the companion release is found at resolved points. The bench does not establish a universal asymptotic power. No physical interpretation is claimed. Files. Guerrero-Diaz_2026_D4_..._d4-v1.0.pdf: technical report (7 pages). D4_Core_Replication.zip: code (src/d4.py, scripts 20–24), configuration, preregistration, gate records, logs, SHA-256 manifests, derived data, reports (English and Spanish), LaTeX source and figure. D4_Raw_Data.zip: the 120 raw heat-kernel blocks (.npz). From the two archives, scripts/20_run_d4.py assemble, 21_analyze_d4.py and 22_sensitivity_d4.py regenerate the derived data byte for byte (see README.md). Git tag d4-v1.0. Continues the companion release d1-v1.1 (doi:10.5281/zenodo.23326548).