HASH-EMBEDDING MEMORY INJECTION: CROSS SCALE TRANSFERABLE DIFFERENTIAL MEMORY FOR LARGE LANGUAGE MODELS

Fine-tuning is expensive and adapters such as LoRA are tied to one base model, so domain knowledge is voided at every backbone upgrade. We propose Hash-Embedding Memory Injection (HEMI), which stores knowledge in a hash $n$-gram table trained once on a source backbone and reused across backbones of the same family and pre-training lineage through each backbone's anchor basis. Three contributions: (1) a critical-frequency theory giving a closed-form $f^{\star }$ and the table-size bound $V^{\star }=T\cdot L/f^{\star }$; pruning at $f^{\star }$ costs under $1\%$ held-out loss over $48$ configurations in $3$ domains; (2) differential injection of only the residual between stored memory and the hidden state, under evidential gating and RMS balancing—ablations show the gate and the adapter decide whether a transferred table helps or hurts (removing the adapter hurts in all three settings; removing the gate does so in two and leaves $\sim 25\%$ of the gain in the third), while dropping differencing or RMS costs $0.005$–$0.039$ nats; (3) anchor rendering via thin QR factorization, plus a migration stability theorem bounding direction error by three additive sources: finite-frequency noise, Stage-2 drift, and anchor-subspace mismatch. On DeepSeek-R1-Distill-Qwen, a table trained on $1.5$B improves the $7$B target in four domains by $+0.150$ to $+0.284$ nats with $65,811$ target-side parameters ($1/307$ of LoRA at $r=8$), on top of a $7.68$M-parameter table trained once on the source; the gain does not survive on the $14$B target ($-0.003$ on PubMed, $-0.149$ on Math; $+0.071\pm 0.043$ against $+0.238\pm 0.006$ on $7$B under a second protocol), where a zero-table control gains more than the trained table ($+0.127$ on PubMed, $+0.045$ on Math), so the content of the table contributes nothing there.