SFT Leaves a Reproducible Signature in Activation Geometry: Calibrating Post-Training Interpretability Against Run-to-Run Noise

1 INTRODUCTION

• A base-versus-descendant diff presumes a rerun would land nearby, although two Qwen3-1.7B SFT runs sit over half as far apart as either sits from the base.Across five SFT runs, the 10,000 strongest edges shift by edge-Jaccard distance 0.057–0.061.Source: §1 · p. 1
• Matched-base and same-recipe rerun comparisons separate training displacement from base properties and run noise, using the same estimator.The audit also includes positive-control, random-dictionary, and precision checks.Source: §1 · p. 2
• 17.8 pairwise-distance SDs: LoRA-SFT’s graph displacement exceeds its own four-run band, replicates on a second backbone, and is disjoint from DPO at the same step count.The z-score is a descriptive scale; the band-free SFT–DPO comparison uses four runs per arm.Source: §1 · p. 2
• DPO’s band counts show no dose ordering, light GRPO is unresolved rather than equivalent to no change, and graph concentration read as an RL signature is already in the zero-RL base.The authors do not infer a changed mechanism from displacement; the registered stage-specificity interaction reversed at two of three budgets.Source: §1 · p. 2

What post-training changes inside a model

• Mechanistic studies describe fine-tuning as reusing or strengthening existing circuitry, while RL-era accounts of internal changes remain less settled.The paper asks how large an internal difference must be to exceed what another run of the same recipe produces.Source: §2 · p. 3

Model diffing with sparse dictionaries

• Sparse dictionaries and related methods support checkpoint comparisons, including studies of chat-only features, fine-tuning changes, and RL-induced tool use.The paper uses a base-trained SAE and a co-activation graph over its codes, with a random-dictionary control.Source: §2 · p. 3

Run-to-run variability as the reference

• Prior work documents seed variation in learned units, fine-tuning outcomes, and SAE features; this paper uses variation in the post-training run itself as the reference for a diff.The same diffing estimator reads both the run-to-run variation and the base-to-post difference.Source: §2 · pp. 3–4

Behavioral and weight-space accounts of RL post-training

• Existing behavioral, weight-space, and state-space accounts do not measure sparse features or specify the internal footprint RL should leave.The paper supplies a seed-calibrated reference for light GRPO on a 1.7B model.Source: §2 · p. 4

3 PROBLEM SETUP

• The roster includes four LoRA-SFT runs, LoRA-DPO runs at 12, 29, and 60 steps, and six full-parameter GRPO-family runs from Qwen3-1.7B.LoRA runs are reseeded before model construction; the GRPO-family band mixes seed, step, and reward variance.Source: §3 · p. 4
• The co-activation graph uses last-position residual states from 500 Pile prompts and SAE encoders trained on the base model.The same basis is applied to each checkpoint; the graph is a co-activation proxy, not a causal attribution.Source: §3 · p. 4
• Single-checkpoint statistics are the inbound-weight Gini or top-1% share, with post-training credited only for the change from the base measured using the same SAE and probes.A degree-preserving permutation addresses concentration versus chance, a different question.Source: §3 · p. 4
• Edge-Jaccard displacement compares base-to-run contrasts with seed-pair distances, and contrasts are outside the band only when they exceed every seed-pair distance.The reported z is a distance scale, not a tail probability; a precision rule determines whether a statistic is scored.Source: §3 · pp. 4–5
• DPO’s displacement stays flat across 12, 29, and 60 steps, while its outside-band counts are 3/4, 1/4, and 3/4 with no consistent dose ordering.Archived cross-GPU graphs gave different counts, so the dose-dependent reading was withdrawn.Source: §3 · p. 5
• The exchangeability bound assumes the base can be treated as one more run, although all runs share it as their common origin.A within-band contrast is only a non-detection and cannot rule out a change shared by all runs.Source: §3 · p. 5
• Under run exchangeability, k runs certify no better than p ≤ 1/(k + 1) when all k contrasts exceed the band, regardless of the number of seed pairs or effect size.With four SFT runs, the stated bound is p ≤ 1/5; the z-score is reported descriptively.Source: §3 · p. 5

4.1 SINGLE-CHECKPOINT LEVELS ARE SET BY THE BASE MODEL

• The zero-RL Qwen3-1.7B base already has 42.7% of inbound weight on its top 1% of active destination features, and light Qwen recipes remain close to it.The base also has nearly all the same top hubs; light recipes stay within a narrow Gini band of the base.Source: §4.1 · pp. 5–6

4.2 THE SEED-NOISE-NULL AUDIT

• At 29 steps, four SFT and four DPO runs occupy disjoint displacement ranges, with the gap several times either arm’s spread.The comparison uses matched adapter geometry and step count, but the recipes also differ in learning rate and dropout.Source: §4.2 · p. 7
• DPO’s contrasts outside their own band count 3/4, 1/4, and 3/4 at 12, 29, and 60 steps, while displacement stays flat and counts show no consistent dose ordering.The dose counts are reported from graphs built on one GPU architecture.Source: §4.2 · p. 7
• All five rebuildable light-GRPO contrasts sit inside the mixed run-to-run band, so the result is a non-detection rather than evidence of absence.The band mixes seeds, steps, and a reward variant; no equivalence margin is defined.Source: §4.2 · pp. 7–8
• At 10,000 edges, ablating sources unique to the SFT graph moves SFT output 12× more in mean and 3.5× more in median than frequency-matched random sources, but the stage-specificity interaction reverses at two budgets and the third fails its gate.The authors therefore describe calibrated structural displacement, not a changed mechanism.Source: §4.2 · p. 8

4.3 PRODUCTION SCALE: THE TULU-3 STAIRCASE

• The Tulu-3 staircase shows large cumulative displacement, with SFT and DPO writing most of it, but its one checkpoint per stage provides no seed band for calibration.Its RLVR increments are compared with one public RLVR run pair, not a seed distribution.Source: §4.3 · p. 8, Table 2 · p. 8

5 DISCUSSION

• Under run exchangeability, a recipe’s band certifies p ≤0.05 only with at least 19 reruns all outside, whereas four-run recipe comparisons with disjoint ranges reach p ≤1/35 for a different question.The latter compares recipes, not base versus post-training.Source: §5 · pp. 8–9
• A within-band GRPO contrast cannot exclude a change shared by all runs, and the light-GRPO non-detection neither measures its footprint nor confirms proposed weight-space or state-distribution accounts.The contrasts sit at seed-pair distance.Source: §5 · p. 9
• Displacement beyond the band indicates changed co-activation structure, not changed computation; the causal sweep’s stage-specificity interaction reversed at two of three budgets.The audit tests only light, accuracy-only GRPO at 1.7B on general text; long-horizon math RLVR at scale is outside its scope.Source: §5 · p. 9

6 LIMITATIONS

• The full detection protocol covers one 1.7B backbone, one SFT recipe, and general-text probe states; the cross-family light-recipe level control uses a heavy instruction-tuned Gemma checkpoint.The top-32 hub set is selected by the inbound weight whose concentration it summarizes, rather than at random.Source: §6 · p. 9
• The GRPO band mixes seed, step, and reward variance, making detection conservative but absence claims anti-conservative, and its deleted baseline leaves five rebuildable runs.The DPO evidence is one LoRA recipe at 1.7B with different learning rate and dropout from SFT, and its band counts have no dose ordering.Source: §6 · p. 9
• The on-task BFCL arm is uninformative under its threshold and failed positive control, while the behavioral check uses n=100 and does not resolve the SFT gain from zero.The two generated-text probes also re-tokenize checkpoint outputs for an external reader with its own noise floor.Source: §6 · p. 9

7 CONCLUSION

• LoRA-SFT displaces the co-activation graph by 17.8 pairwise-distance SDs beyond its own four runs’ mean, a descriptive z-score.The displacement is reproduced on a second backbone and SAE family.Source: §7 · p. 9
• At the identical step count, four SFT and four DPO runs occupy disjoint displacement ranges without a band.The conclusion characterizes DPO displacement as flat over three doses and light GRPO as unresolved from run noise.Source: §7 · p. 9
• The causal sweep finds displaced sources load-bearing, but its stage-specificity interaction reverses at two of three budgets, supporting calibrated structural change rather than a mechanism claim.Source: §7 · p. 9
• Under run exchangeability, k runs certify no better than p ≤1/(k + 1), regardless of effect size.The conclusion recommends scoring each base-to-post contrast beside the same recipe’s rerun band, per dose and with graphs built on one GPU architecture.Source: §7 · p. 9

• The work uses public language models and public corpora and involves neither human subjects nor personal data.Source: ETHICS STATEMENT · p. 15
• Its ethical concern is that post-training differences may be reported as properties of a method without accounting for run-to-run variation.Source: ETHICS STATEMENT · p. 15
• The authors calibrate claims against repeated runs and limit any training-history audit using the signature to the paper’s stated scope.Source: ETHICS STATEMENT · p. 15

• The supplement provides co-activation graphs, receipt JSONs, and readers and checkers that recompute headline statistics without a GPU.Source: REPRODUCIBILITY STATEMENT · p. 15
• The release includes graph-building and figure-generation code, trainers, small BFCL-derived inputs, job specifications, DPO-loss tests, and preregistrations.Source: REPRODUCIBILITY STATEMENT · p. 15
• The release omits model checkpoints, the GRPO training pipeline, and software environments, so it supports statistic recomputation rather than bit-for-bit end-to-end retraining.Source: REPRODUCIBILITY STATEMENT · p. 15
• The GRPO baseline checkpoint and full-parameter SFT checkpoints from discussion-period runs are no longer available for rebuilding graphs.Source: REPRODUCIBILITY STATEMENT · p. 15

USE OF AI ASSISTANTS

• Claude and OpenAI Codex agents were used to implement and check experiment code, collect public datasets, check references, and polish the writing.Source: USE OF AI ASSISTANTS · p. 15

A PROOF OF PROPOSITION 1

• Under run exchangeability, disjoint displacement ranges for two groups of four have probability at most 1/35, while the proposition’s scope does not imply power or make descriptive z a tail probability.Source: A PROOF OF PROPOSITION 1 · p. 16

B.1 INDEPENDENT CONVERGENCE ON THE SFT-OVER-RL ORDERING

• A separate line of work reports that SFT disrupts circuits more than RL in its setting, and this study finds the same direction for SFT versus DPO using a different backbone and mechanistic unit.Source: §B.1 · p. 16

C BEHAVIORAL SANITY-CHECK VISUALIZATION

• On the held-out BFCL validation split, GRPO improved accuracy by +0.36 over base, while SFT’s +0.11 was not resolved from zero by this harness.Source: Figure 3 · p. 17

B.2 THE NULL-CALIBRATION GAP

• Prior mechanistic studies generally report single-run differences without calibrating them against the training algorithm’s run-to-run variability.Source: §B.2 · p. 17
• Nasermoghadasi’s strongest threshold-clearing feature was the top-|d| feature in only 1 of 5 seeded runs, while Paulo and Belrose measured about 30% cross-seed feature overlap.Source: §B.2 · p. 17

• The Qwen3-1.7B setup includes LoRA-SFT and DPO arms, GRPO-family runs, and a Gemma comparison, with model and SAE configurations recorded.The four seeded SFT runs use the same recipe and step count; the four DPO runs use the same trainable parameter count but a distinct training set and schedule.Source: D SETUP AND REPRODUCIBILITY · pp. 17–18
• The graph protocol builds 10,000-edge co-activation graphs from 500 Pile-10k prompts and evaluates concentration, activation metrics, and displacement.Displacement uses edge-set Jaccard or weighted edge cosine; the seed-noise null uses pairwise distances among GRPO-family runs.Source: D SETUP AND REPRODUCIBILITY · p. 18
• The random-dictionary control matches encoder width and sparsity to the trained SAE and recomputes the graph and seed-null pipeline.The control uses one random draw, so it tests whether detection needs the trained dictionary, not invariance across random dictionaries.Source: D SETUP AND REPRODUCIBILITY · p. 18, Table 3 · p. 19
• The H2 cross-tokenizer pipeline re-encodes generated text with a fixed Pythia reader, rather than encoding the audited model’s activations.The H3 protocol compares per-feature entropy across pre- and post-GRPO generated-text sequences using the same fixed reader.Source: D SETUP AND REPRODUCIBILITY · pp. 18–19
• The trained-SAE cosine null was too low-resolution to report a z-score, whereas the random-dictionary null passed the precision check.The trained-SAE null had SD 0.000046, below the 0.0001 reporting step.Source: D SETUP AND REPRODUCIBILITY · p. 19
• The supplement contains headline graphs, metric outputs, control receipts, analysis scripts, and preregistrations, but rerunning training requires external model weights, checkpoints, and environments.The original audited SFT checkpoint cannot be reproduced bit-for-bit; the headline displacement and range separation can be recomputed from committed graph arrays.Source: D SETUP AND REPRODUCIBILITY · pp. 19–20
• The cross-tokenizer H3 run found a mean per-feature entropy delta of −0.004 nats and Cohen’s d of −0.005 over 32 features, below the entropy-collapse gate.The previously reported hub-widening claim did not survive and was withdrawn.Source: D SETUP AND REPRODUCIBILITY · p. 20
• The entropy result is a secondary non-detection: the reported delta and effect size do not pass the entropy-collapse gate.Source: D SETUP AND REPRODUCIBILITY · p. 20

H1 graph statistics

• The top 1% of active features capture 42.7% of inbound edge weight, while a degree-preserving null gives a 33.0%±0.5% top-1% share (z = 20.0); GRPO captures 42.9%.The concentration is therefore above chance but not evidence that RL caused it: the observed concentration is already present in the base graph, and the GRPO checkpoint is indistinguishable on this statistic.Source: E EXTENDED RESULTS: STATISTICS,… · p. 21

Which statistic we score on, and why the other one cannot carry a z

• Edge Jaccard has null SD 0.0016 across 14 distinct values, whereas trained-SAE weighted edge cosine has SD 0.000046 below its reporting precision.The trained-SAE cosine z = −30.7 is withdrawn as a rounding artifact; edge Jaccard is the scored statistic. The SFT detection also survives a single matched random dictionary (z = −16.4 versus −15.9 trained), so detection does not require trained features, but projection invariance is untested.Source: E EXTENDED RESULTS: STATISTICS,… · pp. 21, 22, 25, 26

SFT against the GRPO-family band, and GRPO across steps

• GRPO contrasts fall within run-to-run noise, with z ∈[−2.0, +0.8]; within one run, displacement stays flat at z = +0.3/+0.1/+0.8 across steps 100/200/300.This is a non-detection, not an equivalence result: the matched band does not resolve the change, nor exclude one shared across runs. The SFT-versus-GRPO contrast also differs in objective, data, and duration, so it does not isolate algorithm alone.Source: E EXTENDED RESULTS: STATISTICS,… · pp. 21–22

A per-algorithm null for the SFT stage

• On one architecture, base→SFT displacement is z = −17.8 against its own six-pair SFT seed null, with all five contrasts outside the band.The z-score is a descriptive distance scale because the six null distances share run endpoints. On Llama-3.1-8B with Llama-Scope, all four rebuilt base→SFT contrasts were 0.0442–0.0452, outside the 6-pair seed band; the smallest was 1.72× the band maximum.Source: E EXTENDED RESULTS: STATISTICS,… · pp. 20–22

A per-algorithm null for DPO

• At 12 steps, all four archived base→DPO contrasts fell inside the DPO band despite displacements of 0.0352/0.0354/0.0352/0.0327.On the same-architecture rebuild, three of four contrasts lay just outside the band, but the registered verdict remained D2; across 12/29/60 steps, outside-band counts were 3/4, 1/4, and 3/4, with no consistent dose ordering. The fixed-order DPO null omits data-order variance, making it narrower than SFT’s and potentially anti-conservative for detection.Source: E EXTENDED RESULTS: STATISTICS,… · pp. 23–24

The dose-matched ladder

• At 29 steps, SFT displacement 0.0571–0.0615 and DPO displacement 0.0317–0.0356 form disjoint ranges, separated by a 0.0216 gap.The gap is 5.0× and 5.6× the two ranges’ widths, and mean SFT displacement is 1.80× mean DPO displacement. DPO mean displacement remains 0.0327/0.0329/0.0342 at 12/29/60 steps; a reshuffled 29-step DPO control also remained disjoint from SFT.Source: E EXTENDED RESULTS: STATISTICS,… · pp. 23–25

Behavioral sanity check and on-task arm

• On held-out BFCL validation, GRPO scores 0.8875–0.900 with ∆ = +0.356 (95% CI [+0.21, +0.50]), while SFT’s ∆ = +0.106 (95% CI [−0.03, +0.24]) includes zero.The on-task audit’s 2σ minimum detectable effect is 0.0806, versus SFT’s pile-arm effect of 0.0254, and its SFT positive control does not separate (z = −1.06). The paper therefore draws no behavioral/internal dissociation or conclusion from the on-task arm.Source: E EXTENDED RESULTS: STATISTICS,… · p. 26

Localizing the SFT signature

• 63% of the edge weight SFT drops is inbound to hub #30329, but the ablation supports a correlational co-activation change, not a pruned causal coupling.GRPO’s activation delta is represented by the SAE rather than missed (∥∆ˆh∥2/∥∆h∥2 = 3.0; cosine 0.26), though the moderate cosine leaves difference-specific blind spots possible. Base-trained SAE FVU is 0.0592/0.0588/0.0593 for base/SFT/GRPO, and the random-dictionary control cannot certify the trained-SAE hub naming.Source: E EXTENDED RESULTS: STATISTICS,… · pp. 26–27

Operating-point robustness

• Across both operating points, post-SFT stage increments agree within 0.001, while the dominant base→SFT increment shifts from 0.669 to 0.697.DPO→PPO straddles a rounding boundary (v1 0.958; v2 0.959). Absolute L0 is not compared across SAE families, and Llama-Scope FVU is excluded because of a verified decode-path loader inconsistency.Source: E EXTENDED RESULTS: STATISTICS,… · p. 27

Graph-build environment

• Archived graphs built on different GPU architectures could differ by about one seed-pair distance, prompting same-architecture recounts.Graphs rebuilt bit-for-bit on their original architecture but differed across architectures, despite unchanged code, libraries, probe data, SAE, and weights.Source: F GRAPH-BUILD ENVIRONMENT, ARCHIVED… · p. 27

Discussion-period analyses

• All five SFT contrasts remain outside the SFT band, and SFT and DPO remain disjoint at 29 steps; the Llama replication also holds.The 29-step gap is 5.0× and 5.6× the arms’ spreads.Source: F GRAPH-BUILD ENVIRONMENT, ARCHIVED… · p. 28
• DPO’s same-architecture counts at 12, 29, and 60 steps are 3/4, 1/4, and 3/4, showing no consistent dose ordering.The earlier 0/4, 2/4, 4/4 curve was within the build difference, so its dose-dependent reading was withdrawn.Source: F GRAPH-BUILD ENVIRONMENT, ARCHIVED… · p. 28
• The GRPO family reads 0/5 outside-band contrasts after recount, while its six-run band cannot be rebuilt because the baseline checkpoint was deleted.The step-300 triplet also crosses builds and was not rebuilt.Source: F GRAPH-BUILD ENVIRONMENT, ARCHIVED… · p. 28

Archived readings

• The archived Table 4 readings are superseded because most rows pair base and run graphs built on different GPU architectures.The submitted version’s four-decimal similarities changed some printed z values, though counts were unchanged by that rounding.Source: Table 4 · p. 28

Registered analyses

• With 20 SFT reruns, all 20 base-to-SFT contrasts remain outside the band; full-parameter SFT also exceeds its own four-run band in 4/4 contrasts.The full-parameter result uses the same data, steps, and learning rate, so the SFT verdict is not specific to LoRA.Source: F GRAPH-BUILD ENVIRONMENT, ARCHIVED… · p. 28
• Full-precision cosine and three other readouts also place every LoRA-SFT and full-parameter SFT contrast outside its band.For the five remaining GRPO runs, edge Jaccard and cosine each put 0/5 contrasts outside, while the other three readouts put 1/5 each; these are descriptive because the baseline is deleted.Source: F GRAPH-BUILD ENVIRONMENT, ARCHIVED… · p. 29

Discussion-period analyses

• Table 5 reports discussion-period counts of contrasts outside each readout’s own band, with one GPU architecture and same-job bases.Its SFT rows have registered branches; DPO at 29 steps and GRPO are descriptive, and GRPO has five runs because the baseline checkpoint was deleted.Source: Table 5 · p. 29

G FULL LIMITATIONS AND UNABRIDGED BODY PARAGRAPHS

• The light-RL conclusion is bounded by the 1.7B control and available transfer evidence; the paper does not claim it covers long-horizon, larger-scale math-RLVR.The accessible Gemma post-checkpoint is heavy instruct-tuning, not a matched light-GRPO run, and its raw-text probe may introduce chat-template mismatch.Source: G FULL LIMITATIONS AND UNABRIDGED BODY… · pp. 29, 33
• At matched 29-step doses, four SFT and four DPO runs occupy disjoint displacement ranges, but differences in data, learning rate, and dropout limit attribution to the objectives alone.The comparison covers one LoRA recipe at one scale and does not establish a general preference-learning result.Source: G FULL LIMITATIONS AND UNABRIDGED BODY… · p. 30
• On GSM8K question-and-solution chains, every base-to-GRPO metric has |rel ∆| < 1%, extending the narrowing-metric null to a second off-training-distribution probe.The on-task function-calling audit is reported separately.Source: G FULL LIMITATIONS AND UNABRIDGED BODY… · p. 30
• The GRPO base-to-post contrasts do not escape the run band, but its mixed seeds, steps, and reward recipes make this descriptive consistency—not an equivalence test or certification of absence.The matched-step comparison has only three runs, and its observations share a base checkpoint.Source: G FULL LIMITATIONS AND UNABRIDGED BODY… · pp. 29, 31
• At production scale, the base Gini is 0.780 and every Tulu-3 post-training stage stays within +0.009; the activation battery also changes little.Cumulative displacement is large, but stage analysis resolves no RLVR increment above the available run-pair reference.Source: G FULL LIMITATIONS AND UNABRIDGED BODY… · pp. 31–32
• The generated-text probes corroborate the main null, but their fixed external reader and pilot feature panel make them supporting evidence rather than internal-activation nulls.Activation deltas are polarity-preserving (weighted cosine 0.9998), and entropy effects remain below the pre-registered |d|=0.5 gate.Source: G FULL LIMITATIONS AND UNABRIDGED BODY… · p. 34

G.1 BASE CONTROL FOR THE ACTIVATION-SIDE “NARROWING” METRICS

• On metrics that can vary, light-GRPO activation statistics remain base-dominated, while heavy-instruct Gemma shifts in directions opposite to the predicted narrowing.Qwen’s fixed-k TopK dictionary pins mean L0, so that metric is not informative there.Source: §G.1 · p. 34

H EXTENDED RELATED WORK

• Independent studies report an SFT-over-RL circuit-disruption ordering, but use different backbones, mechanistic units, and training designs; their directional agreement does not supply a run-noise width.Zhang et al. (2025a) study production-scale RL with activation summaries, unlike this paper’s light-GRPO band-calibrated comparison.Source: H EXTENDED RELATED WORK · pp. 34–35

I ADDITIONAL H1 MATERIAL

• The zero-RL Qwen base already shows the full concentration pattern, with 31–32 of 32 top hubs shared across training stages (Jaccard 0.94–1.00).Across five Qwen recipes and seeds, |∆Gini| ≤0.005; Gemma’s zero-RL base also carries the concentration, with Gini 0.708.Source: Table 6 · p. 36, Table 7 · p. 36
• With the denominator fixed at top-1%-of-dSAE, concentration rises from 0.7110 to 0.8395 between 1.7B and 8B, but that rise is mechanical because the 8B dictionary is twice as wide.Width-independent Gini values are flat to slightly decreasing across a 23× parameter range in both SAE families.Source: I ADDITIONAL H1 MATERIAL · pp. 37–38, Table 9 · p. 38
• The observed concentration increase in the dictionary-relative top-1% statistic is not evidence of stronger concentration with scale; the width-independent measures show no monotonic increase.Figure 4’s available layer-pair scan also has unbalanced source/destination coverage, so its rise from about 22% to 49% is descriptive, not an isolated destination-depth effect.Source: Figure 4 · p. 37, I ADDITIONAL H1 MATERIAL · p. 38
• On activation-side metrics, light-GRPO values are base-dominated with |rel ∆| < 1%, while heavy-instruct Gemma’s mean L0 rises and firing-frequency Gini falls, opposite to narrowing.Qwen mean L0 is pinned at 50 by its fixed-k TopK dictionary and is not a model measurement.Source: Table 10 · p. 38
• On a single-architecture rebuild, none of the five recomputable GRPO contrasts lies outside its band; the archived cross-architecture table is descriptive only.The archived table reports SFT at z = −15.9 and five of six RL contrasts within run noise, but its graphs mix GPU architectures.Source: I ADDITIONAL H1 MATERIAL · p. 39, Table 11 · p. 39
• The narrowing-metric null persists on GSM8K question-and-solution chains, where the base firing-frequency Gini is 0.840 versus 0.683 on pile.GSM8K is an off-training-distribution probe; the on-task function-calling audit is reported separately.Source: I ADDITIONAL H1 MATERIAL · p. 39

J ON-TASK (BFCL) ARM AND ITS MINIMUM DETECTABLE EFFECT

• The BFCL arm draws no conclusion: its 2σ minimum detectable effect is 0.0806, about 3× SFT’s pile-arm effect, and the positive control does not separate.The on-task band is about 25× wider than pile’s; the null band is also contaminated and the hub-Jaccard statistic is too quantized to score.Source: J ON-TASK (BFCL) ARM AND ITS MINIMUM… · pp. 40–41, Table 12 · p. 41

K LOCALIZATION OF THE SFT SIGNATURE

• Of 138 edges retained by base and every GRPO run but dropped by SFT, 86 (63% of dropped weight) target the dominant hub #30329.SFT-added edges are diffuse, and “dropped” means below the top-10,000 cutoff, not necessarily zero; the hub is a generic numeric, whitespace, and formatting feature fed by boilerplate/function-word sources.Source: K LOCALIZATION OF THE SFT SIGNATURE · pp. 41–42
• Ablating displaced sources shifts the hub 3.8–15.8× more than a count-matched random control, but the hub rise is not weaker in SFT than in base.The probe therefore supports a change in co-activation statistics, not a pruned causal coupling; the random control is not magnitude-matched.Source: K LOCALIZATION OF THE SFT SIGNATURE · p. 42

L THE REGISTERED FUNCTIONAL-TRANSFER SWEEP

• The registered test compares differential source sets across models and asks whether SFT-specific sources matter more in SFT than in base, after a faithfulness gate.The corrected sweep uses the audited graph with its own adapter and 4,816 held-out prompts; the verdict rule requires one branch across all three edge budgets.Source: L THE REGISTERED FUNCTIONAL-TRANSFER… · pp. 42–43
• At 10,000 edges, SFT-specific sources are 12–37× their random control in SFT, but the sweep returns MIXED and supports no stage-specificity claim.The interaction is negative at 5,000 and 10,000 edges, while the base-specific sources fail the gate at 2,000 edges.Source: L THE REGISTERED FUNCTIONAL-TRANSFER… · p. 43, Table 14 · p. 43

M TULU OPERATING-POINT ROBUSTNESS AND SAE-FAMILY CAVEATS

• The RLVR increments are unresolved against a single public PPO/GRPO reference of 0.097, not a same-recipe seed-null distribution.Figure 6 distinguishes checkpoint levels from stage displacement, defined as 1 minus edge Jaccard; unresolved increments are not certified nulls.Source: Figure 6 · p. 44
• Base→SFT remains the dominant stage displacement after normalization folding, moving from 0.669 to 0.697; post-SFT increments agree within 0.001 across operating points.The Llama-3.1-8B LoRA-SFT replication used the normalization-folded operating point.Source: M TULU OPERATING-POINT ROBUSTNESS AND… · pp. 43–44
• Absolute L0 does not compare across SAE families, and Llama-Scope reconstruction-error numbers are excluded because its decode path returns implausible error.Llama-Scope runs at measured L0 ≈1.5k of 131k versus 50 for qwen-scope; the paper uses encode only and reports within-family deltas.Source: M TULU OPERATING-POINT ROBUSTNESS AND… · p. 44

N H2 / H3 AUXILIARY-NULL DETAIL (V0 PYTHIA-70M PANEL)

• Across accuracy-only GRPO runs, entropy changes remain far below the preregistered |d| = 0.5 collapse gate; the reported outcome is binary non-collapse.The cross-tokenizer run gives ∆H = −0.004, d = −0.005; probe-budget measurements are small and sign-unstable, while the shaped-reward run gives d = −0.29.Source: N H2 / H3 AUXILIARY-NULL DETAIL (V0… · p. 44
• Base concentration is stable across the cited reading, but the proposed pretraining-crystallized interpretation remains compatible rather than tested beyond the matched base.The cited 8B graphs have no SFT or GRPO checkpoint, and identical integer indices across layers are not aligned features.Source: N H2 / H3 AUXILIARY-NULL DETAIL (V0… · p. 45
• H2 and H3 encode generated checkpoint text with a fixed Pythia-70m reader and SAE, so their panel indices are not the Qwen internal hub indices.The 32-feature panel was preselected by association with function-word outcomes; these are auxiliary generation-side probes, not internal-activation tests.Source: N H2 / H3 AUXILIARY-NULL DETAIL (V0… · p. 45
• In the shaped-λ0.1 arm, top-32 activation deltas lie in [−0.008, +0.006], with weighted cosine 0.9998 and all 16 of 16 active counts unchanged.The four top inbound hubs all move negatively, consistent with the arm’s reward penalizing the panel’s activations; no feature switches off.Source: N H2 / H3 AUXILIARY-NULL DETAIL (V0… · p. 45, Table 15 · p. 46
• The earlier “entropywidening” and 4/4 steering-gate claims are withdrawn; corrected steering clears 0/19 at σKL ≥2.The committed cross-tokenizer run does not reproduce the earlier entropy result.Source: N H2 / H3 AUXILIARY-NULL DETAIL (V0… · pp. 45–46
• On the committed entropy run, two of four top-attribution features gain ∆ = +0.11 and a third ties at +0.110, despite aggregate d = −0.005.The aggregate does not pass the gate, and the per-feature entropy changes are heavy-tailed.Source: N H2 / H3 AUXILIARY-NULL DETAIL (V0… · p. 45, Table 16 · p. 46
• Across the 8B layer scan, 13 of 15 audited slots appear in at least one top-30 set, but only three integer ids recur across destination layers.Independently trained layer dictionaries mean index matches are not feature identity; layer 35 is unembedding-dominated.Source: N H2 / H3 AUXILIARY-NULL DETAIL (V0… · p. 46, Figure 7 · p. 47