In reciprocity-based over-the-air split learning, channel aging rotates and attenuates the derivative signal returned to the transmitter. Seven case studies combine numerical forecasts, mechanism checks, and competing predictions. Near the correlation zero, an archived SGD comparison finds truncated inversion 0.000813 lower in cross-entropy than polar correction, with a paired 95% interval for the difference from -0.001853 to 0.000060, and a retained-mode quadratic proxy slightly favors truncation. Selecting the return port with the largest received power falls short of the predicted five-point alignment advantage over random selection at both tested spacings, gaining at most 0.97 points, while the local inverse relation between amplitude and phase gradient is supported. Seven specified interference-induced accuracy-collapse predictions are rejected, and independent zero-mean interference preserves the conditional mean under a fixed linear chain. Per-subcarrier correction and spherical-wavefront compensation do not deliver their predicted accuracy gains. Focusing leaves both methods near chance in a single-mode model and adds no observed gain in a model with a line-of-sight component plus scattering. A successful port assignment under nonsymmetric motion refutes an exact-symmetry requirement. The competing positioning-error prediction is supported. Gaussian rail jitter with a standard deviation of 0.383 wavelengths yields high terminal test accuracy, whereas a fixed bias of the same magnitude yields near-chance accuracy. These cases distinguish valid local physical relations from unsupported training forecasts through task support, noise injection, and update moments.
A Gaussian random wave field is sampled on a lattice within a bounded aperture. After displacement, re-indexing leaves a residual in the Voronoi cell. We study whether the empirical lag correlation has positive real part throughout the cell. For each realization in a single-output reciprocal link with noiseless over-the-air return, this is equivalent to the negative returned gradient being a descent direction for all residuals and differentiable losses with nonzero error signals. Return noise is treated exactly. For isotropic fields, ball-shaped apertures, unshifted lattice points, and dilations approaching the critical covering radius, every minimizer is a deep hole except with exponentially small probability. A Gaussian vector on the deep holes determines the limiting failure probability, with explicit covariance and lattice-independent limiting normalization. Square, hexagonal, simple cubic, and body-centered cubic lattices have two, three, four, and twelve sign classes. At fixed target failure probabilities below the critical limiting value, this covariance determines the exact leading aperture constant. When the margin and target failure probability vanish jointly, that constant is common to lattice shapes within each dimension. A cubic correction yields vanishing relative tail error in a moderate-deviation regime. A non-asymptotic two-sided aperture law has explicit constants. With a Rayleigh-faded specular path and positive diffuse margin, a failure floor occurs at every positive path power exactly when the cosine of its phase change is negative somewhere in the cell. The floor vanishes for all directions exactly when the covering radius is at most a quarter wavelength.
In over-the-air split learning over an aging single-antenna Rayleigh channel, the mean returned gradient equals the antenna-domain gradient scaled by the temporal correlation. Uncorrected training collapses in negative-correlation lobes. Near a correlation zero, no nonzero scalar calibration improves the ratio of mean power to innovation power for an individual transmission, and, on the tested grid, using sixteen times as many pilots recovers only 7 percent of the training gap. Returning at the first correlation extremum with phase correction, 1.3 ms later at 3.5 GHz and 15 m/s, achieves 90.2 percent accuracy, compared with 91.8 percent with CSI. In the tested line-of-sight setting, the near-zero gap becomes negligible.
Over-the-air split learning returns the receiver's loss gradient through the reciprocal channel, so terminal motion between the forward and backward transmissions corrupts the backward pass. We repair the aged return at a fixed receive array using only second-order statistics. Two numbers taken from the forward-backward cross-covariance, the alignment reachable by the best common phase and by the best unitary correction, bound the main term of every repair. In the 3GPP tapped-delay-line channels TDL-A/B/C, where every antenna ages by the same scalar, the two numbers coincide, and at a correlation zero all tested fixed corrections stay between 58% and 62% accuracy, whereas a return delayed to the next correlation lobe reaches 98.4%. In 3GPP clustered-delay-line channels a statistical unitary correction stays within 0.02 in mean cross-entropy of ideal channel knowledge in all six tested profile and speed combinations, while common-phase correction falls to 69% for CDL-A at 3 km/h and the tested delay loses up to 10 points. In a two-cluster model spatial correction is decisive only where the common-phase alignment nearly vanishes (64% to 95%). With a ResNet-18 on CIFAR-10, only the delayed return trains at a separable correlation zero (86% against at most 16%), and at 30 km/h in a clustered-delay-line channel the unitary correction reaches 80% where the common phase reaches 60%. Given geometry, displacement and angular statistics, a port permutation chosen by linear assignment reaches 99.5-99.7% accuracy for hexagonal and square arrays in oblique motion without pilots or digital feedback during training, where nearest-position matching drops to 89%.