Taylor–BMW Transition Rule

This design note specifies a Taylor-inspired recurrence for forecasting a matrix-valued pre-write state in recurrent associative memory. It distinguishes this proposed mechanism from scalar key-value-cache eviction forecasting. It defines the exponential reference update, a damped first-order trend predictor with a normalized error-correcting write, and the quadratic Newton backward-difference identity. The update variant is not algebraically identical to the baseline and remains a proposal rather than an established or uniquely defined architecture. The package reports controlled synthetic checks: a scalar diagnostic pilot and a matched 16 x 16 synthetic retrieval and stability sweep across dense-random, correlated, and orthogonal keys, multiple loads, delays, fixed trend-gate settings, and random seeds. Results are task-specific and mixed: at delay 64, no tested nonzero-trend setting has lower mean MSE than the exponential reference for dense or correlated keys; nine of twelve do for each tested orthogonal-key load. Some gate settings produce very large memory norms. These experiments do not include learned projections, language-model training, natural-language evaluation, throughput measurement, or evidence of improved model performance. The stated hypothesis that trend may help when latent memory trajectories are locally predictable remains experimentally unverified at model scale. An earlier scalar cache-eviction note is excluded from the evidence because its source scripts and raw data were unavailable; its inconsistent reported values are not relied on here. The ZIP contains the typeset note, editable source, reference implementation, tests, experiment scripts and results, and checksums.