In Week 11 Tenacious-Bench, we trained a LoRA adapter on Tenacious-style B2B sales emails using Supervised Fine-Tuning (SFT).

We got a real performance lift: Delta A = +0.263 (p < 0.0001).

But that result exposed a harder question :

Did the adapter learn how Tenacious writes, or just what repeated Tenacious-like samples looked like?

This post answers that at the mechanism level: cross-entropy token-by-token, LoRA gradient flow, and why low-diversity augmentation can make convergence look better than generalization.

1) What SFT cross-entropy actually optimizes

In autoregressive SFT, the model predicts the next token at each step.

Cross-entropy loss measures how much probability mass the model gave the correct next token.

So the objective is:

not “be honest,”

not “be cautious,”

not “be Tenacious,”

but: assign high probability to target tokens in the training distribution.

If your targets consistently reflect Tenacious behavior, style improves indirectly.

But the optimization target is still token…