Bill Brothers PRO

Removing one expert costs this model 74.9 percentage points of exponentiation accuracy at the end of training. Earlier, the measured cost is even larger.

We reran our small mixture-of-experts transformer learning five operations modulo 113, adding two user functions to the training loop. They report measurements through stock IRIS™ signal channels; IRIS itself is unchanged.

The first function evaluates the same 1,024 held-out exponentiation examples with the model intact, then with layer-2 expert 0's contribution disabled. It restores the expert before training resumes. Tokens are not rerouted.

So dependence on this expert is not monotonic. Accuracy measured under knockout recovers late, even though intact accuracy is already high. The surviving computation remains unidentified.

The second function separates answer-only cross-entropy from whole-probe loss, per operation. On the original 12 fixed probes per operation, the final answer-only losses put multiplication/division below addition/subtraction—the reverse of their relative whole-probe ordering. We also record answer-only loss on 1,024 held-out examples per operation, keeping the two cohorts separate.

Both histories are now recorded at 120 adaptive measurement steps and displayed in the stock Bench User tab. The quantities are researcher-computed measurements reported to IRIS, not automatic IRIS diagnoses.

This is a rerun with the same seed and recipe, not an independent seed replication. It shows the evolution of these measurements in this experiment; it does not identify a discrete-log circuit or establish a general training rule.

Again, thank you for the good question!

We checked both questions at the saved step-120,000 checkpoint, without retraining.

Disabling layer-2 expert 0 drops exponentiation accuracy from 98.9% to 24.0%—a loss of 74.9 percentage points on the same 1,024 held-out examples. The intervention skips that expert’s contribution while leaving the router unchanged.

The surviving answers aren’t explained mainly by cases involving 0 or 1. Those account for only 20 of the 246 correct answers. Excluding every such input leaves 226/988 correct, or 22.9%.

We also checked all 8,974 held-out exponentiation examples from the same split. Accuracy after the knockout was 24.74%, with correct answers at every exponent from 0 through 112, but pronounced peaks and uneven performance. This larger sample includes the original subset; it isn’t an independent replication. It establishes broad survival without identifying the computation responsible.

For loss, we isolated the answer target on exactly the original 12 fixed probes per operation:
Operation Whole-probe loss Answer-only loss
Addition 3.72 0.00365
Subtraction 3.76 0.00362
Multiplication 5.03 0.000748
Division 5.35 0.00113
Exponentiation 8.37 0.0350
Values are mean cross-entropies in nats; lower is better.

The pairings remain visible, but their relative ordering reverses: multiplication/division have lower answer-only loss than addition/subtraction on these probes. Exponentiation remains highest. The whole-probe panel therefore shouldn’t be interpreted as a ranking of answer-prediction difficulty.

IRIS can record and plot measurements computed by user functions inside the training loop, without changing IRIS itself. We’ll use that capability in a new run to measure both the accuracy gap caused by the expert knockout and answer-only loss per operation. A follow-up post will show how these evolve throughout training.

Thank you for your thoughtful comment. Connecting the endpoint split to the algebraic families is a good way to frame it, and your question about the loss is an important distinction. You’re asking if those families share experts, and if the dashed losses are comparable with the reported answer accuracy. I hope I understand correctly.

First, there is direct support for the split in this same model. IRIS’s Expert–Domain Affinity figure at step 120,000 shows it most clearly in layer 2—the third layer, numbered from zero: addition and subtraction favor expert 1, while multiplication, division, and exponentiation favor expert 0. These bars pool all recorded probe-token positions.

I am not sure if my figure 7.6 will go above this or below - but in that figure you can see the domains per expert. In the 7.6 chart, orange=addition, dark blue=subtraction, light blue = division, gold=multiplication, and exponentiation = green. Each operations routing fraction is normalized per layer.

The trainer’s separate saved routing audit isolates the position that predicts the answer. Across 1,024 held-out examples per operation, layer 2 sends every addition/subtraction example to expert 1 and every multiplication/division/exponentiation example to expert 0.

The saved trainer-side knockout tests strengthen that observation: disabling expert 1 in layer 2 drops addition/subtraction accuracy to 2.1% and 1.6%, while the other operations stay at baseline. Disabling expert 0 drops multiplication/division/exponentiation to 6.9%, 7.2%, and 24.0%, while addition/subtraction stay at baseline. That supports functional specialization by family. Establishing that the model actually implements a discrete-log algorithm would require further circuit analysis.

Regarding your question on the dashed loss, you are correct that it is a scoring-scope issue. The dashed curve is mean next-token cross-entropy over the whole probe’s scored positions, not answer-only loss and not a sum. For [a, op, b, =, answer], it scores predictions of op, b, =, and the answer. The training objective supervises the answer position only, so the whole-probe metric includes positions the model was not trained to predict.

For comparison, the separately sampled answer-only validation cross-entropy at the final capture is about 0.036 nats overall. That is a different sample and an aggregate across operations, but it makes the distinction clear. We should have made those different scoring scopes explicit in the caption.

Separately, we checked saved follow-up runs with a second seed and weight-decay variants. The second seed also separates addition/subtraction from multiplication/division/exponentiation at the final layer, using different experts. At lower weight decay, exponentiation uses its own layer-2 expert. The zero-decay arm did not fully generalize within 120,000 steps. So there is recurring family organization, while its location and exact wiring vary; we would not claim one fixed circuit across runs.

Thank you for raising both points. They help us make the interpretation and the measurement clearer.