| # Engineering Specification for Jacobian-Transported ISS |
|
|
| ## 0. Purpose |
|
|
| This document specifies how to implement the official Internal State Stability metric, abbreviated as ISS, using Jacobian-transported hidden states. |
|
|
| The formal metric remains: |
|
|
| \[ |
| z^\ell_{m,f,q}=J^\ell_m h^\ell_{m,f,q}, |
| \] |
| |
| where no vocabulary unembedding matrix is applied. The implementation may approximate \(J^\ell_m\), but the approximation must be independently validated and its numerical error must be reported. |
|
|
| Raw-ISS is retained only for debugging and ablation. It is not the official metric. |
|
|
| --- |
|
|
| # 1. Official Definition |
|
|
| For model \(m\), fact \(f\), query \(q\), and source layer \(\ell\), extract the query-end residual state: |
|
|
| \[ |
| h^\ell_{m,f,q}\in\mathbb R^{d_m}. |
| \] |
|
|
| Define the corpus-averaged input-output Jacobian: |
|
|
| \[ |
| J^\ell_m |
| = |
| \mathbb E_{x\sim\mathcal C} |
| \left[ |
| \frac{\partial h^L_m(x)}{\partial h^\ell_m(x)} |
| \right], |
| \] |
|
|
| where \(\mathcal C\) is an independent calibration corpus. |
|
|
| The transported representation is: |
|
|
| \[ |
| \boxed{z^\ell_{m,f,q}=J^\ell_m h^\ell_{m,f,q}} |
| \] |
| |
| Do **not** compute: |
| |
| \[ |
| W_UJ^\ell_m h^\ell_{m,f,q}. |
| \] |
|
|
| The official ISS therefore operates in the final-layer residual basis rather than vocabulary-logit space. |
|
|
| --- |
|
|
| # 2. Formal Metric Versus Practical Estimator |
|
|
| The formal metric uses the exact population average \(J^\ell_m\). The practical implementation uses a finite-corpus, finite-rank estimator: |
| |
| \[ |
| \widehat J^{\ell,(n,k)}_m. |
| \] |
|
|
| The practical transported state is: |
|
|
| \[ |
| \widehat z^\ell_{m,f,q} |
| = |
| \widehat J^{\ell,(n,k)}_m h^\ell_{m,f,q}. |
| \] |
| |
| The paper should distinguish: |
| |
| - \(\operatorname{ISS}\): the formal metric; |
| - \(\widehat{\operatorname{ISS}}_{n,k}\): the empirical estimator. |
|
|
| The approximation is acceptable only if it satisfies the validation criteria below. |
|
|
| --- |
|
|
| # 3. Why Full Jacobians Are Not the General Solution |
|
|
| For hidden dimension \(d\): |
|
|
| \[ |
| J^\ell_m\in\mathbb R^{d\times d}. |
| \] |
| |
| Materializing this matrix is feasible only for small models. For large models with \(d=4096\), \(5120\), or more, exact recovery across many layers and calibration prompts is prohibitively expensive. |
| |
| Therefore: |
| |
| 1. full Jacobians are used only for small-model calibration; |
| 2. large models use a randomized factorized estimator; |
| 3. approximation quality must be validated for every model rather than inferred from a small model. |
| |
| --- |
| |
| # 4. Overall Engineering Pipeline |
| |
| ```text |
| Stage A: Full-matrix calibration on a small model |
| | |
| v |
| Stage B: Per-model approximation validation |
| | |
| v |
| Stage C: Freeze estimator settings |
| | |
| v |
| Stage D: Official ISS evaluation |
| ``` |
| |
| The final test set must not be used to choose estimator hyperparameters. |
| |
| --- |
| |
| # 5. Stage A: Small-Model Full-Matrix Calibration |
| |
| Use Qwen2.5-0.5B as the primary exact-reference model. |
| |
| ## 5.1 Exact finite-corpus reference |
| |
| For a fixed calibration corpus of \(n\) prompts, compute: |
| |
| \[ |
| J^{\ell,(n)}_{\mathrm{full}} |
| = |
| \frac{1}{n} |
| \sum_{i=1}^{n}J^\ell_{x_i}. |
| \] |
| |
| This is an exact empirical Jacobian for that finite corpus. |
| |
| Do not call it the exact population Jacobian or universal ground truth. Use one of the following terms: |
| |
| - full empirical Jacobian reference; |
| - exact finite-corpus Jacobian. |
| |
| ## 5.2 Layer coverage |
| |
| At minimum, compute full references for: |
| |
| - one early layer; |
| - one middle layer; |
| - one late layer. |
| |
| If feasible on Qwen2.5-0.5B, compute every layer. |
| |
| ## 5.3 Candidate ranks |
| |
| Evaluate: |
| |
| \[ |
| k\in\{32,64,128,256,512,768\}, |
| \] |
| |
| subject to \(k<d\). |
| |
| Use nested random probes so that larger ranks contain the directions used at smaller ranks. |
| |
| ## 5.4 Required validation statistics |
| |
| ### Matrix approximation error |
| |
| \[ |
| E_F(k) |
| = |
| \frac{\|J_{\mathrm{full}}-\widehat J_k\|_F} |
| {\|J_{\mathrm{full}}\|_F}. |
| \] |
| |
| ### Transported-vector relative error |
| |
| For held-out activation vectors \(h_i\): |
| |
| \[ |
| E_{\mathrm{vec}}(k) |
| = |
| \frac{1}{M} |
| \sum_{i=1}^{M} |
| \frac{\|J_{\mathrm{full}}h_i-\widehat J_kh_i\|_2} |
| {\|J_{\mathrm{full}}h_i\|_2+\epsilon}. |
| \] |
| |
| ### Transported-vector cosine agreement |
| |
| \[ |
| C_{\mathrm{vec}}(k) |
| = |
| \frac{1}{M} |
| \sum_{i=1}^{M} |
| \cos(J_{\mathrm{full}}h_i,\widehat J_kh_i). |
| \] |
| |
| ### ISS absolute error |
| |
| \[ |
| E_{\mathrm{ISS}}(k) |
| = |
| \left| |
| \operatorname{ISS}_{\mathrm{full}} |
| - |
| \widehat{\operatorname{ISS}}_k |
| \right|. |
| \] |
| |
| ### Per-fact ISS rank agreement |
| |
| \[ |
| \rho_{\mathrm{fact}}(k) |
| = |
| \operatorname{Spearman} |
| \left( |
| \{\operatorname{ISS}^{\mathrm{full}}_f\}, |
| \{\widehat{\operatorname{ISS}}^{(k)}_f\} |
| \right). |
| \] |
| |
| A low Frobenius error is not sufficient. The transported vectors and the final ISS must also be stable. |
| |
| --- |
| |
| # 6. Randomized Factorized Jacobian Estimator |
| |
| The estimator must define a linear operator that can be applied to arbitrary hidden states. Computing only \(J\Omega\) is not enough. |
| |
| Use randomized range finding with a two-sided operator estimate. |
| |
| ## 6.1 Random probe matrix |
| |
| For target rank \(k\), draw: |
| |
| \[ |
| \Omega\in\mathbb R^{d\times(k+p)}, |
| \] |
| |
| where: |
| |
| - \(p\) is an oversampling dimension; |
| - recommended \(p\in\{32,64\}\); |
| - \(\Omega\) uses Gaussian or Rademacher entries; |
| - all random seeds are fixed and recorded. |
| |
| ## 6.2 Estimate the Jacobian output range |
| |
| Compute: |
| |
| \[ |
| Y |
| = |
| J^\ell\Omega |
| = |
| \mathbb E_x[J^\ell_x\Omega]. |
| \] |
| |
| Use batched Jacobian-vector products. |
| |
| Then compute: |
| |
| \[ |
| Q=\operatorname{qr}(Y), |
| \] |
| |
| with: |
| |
| \[ |
| Q\in\mathbb R^{d\times(k+p)}. |
| \] |
| |
| ## 6.3 Estimate the reduced operator |
| |
| Compute: |
| |
| \[ |
| B=Q^\top J^\ell. |
| \] |
| |
| Equivalently: |
| |
| \[ |
| B^\top=J^{\ell\top}Q. |
| \] |
| |
| Use batched vector-Jacobian products. |
| |
| The final estimator is: |
| |
| \[ |
| \boxed{\widehat J^\ell=QB} |
| \] |
| |
| and: |
| |
| \[ |
| \widehat J^\ell h=Q(Bh). |
| \] |
| |
| ## 6.4 Storage |
| |
| Store only: |
| |
| \[ |
| Q\in\mathbb R^{d\times r}, |
| \qquad |
| B\in\mathbb R^{r\times d}, |
| \] |
| |
| where \(r=k+p\). |
| |
| Do not reconstruct and store the complete \(d\times d\) matrix. |
| |
| Since \(Q\) has orthonormal columns: |
| |
| \[ |
| \cos(Qy_1,Qy_2)=\cos(y_1,y_2). |
| \] |
| |
| Therefore downstream cosine computations may use: |
| |
| \[ |
| y^\ell=Bh |
| \] |
| |
| directly. Residualization and whitening must then be performed consistently in that reduced coordinate space. |
| |
| --- |
| |
| # 7. Per-Model Validation Is Mandatory |
| |
| A rank that works for Qwen2.5-0.5B must not be assumed to work for larger models. |
| |
| Different models can have different: |
| |
| - hidden dimensions; |
| - Jacobian singular-value decay; |
| - effective rank; |
| - depth; |
| - normalization; |
| - architecture; |
| - post-training behavior. |
| |
| The small-model experiment validates the estimator implementation. It does not determine one universal rank. |
| |
| ## 7.1 Direct action validation |
| |
| For every evaluated model, select representative: |
| |
| - early layers; |
| - middle layers; |
| - late layers. |
| |
| For held-out activation vectors \(h_i\), directly compute: |
| |
| \[ |
| u_i=J^\ell h_i |
| = |
| \frac1n\sum_{x\in\mathcal C}J^\ell_xh_i. |
| \] |
| |
| This does not require materializing the full Jacobian. |
| |
| Compare with: |
| |
| \[ |
| \widehat u_i=\widehat J^\ell_kh_i. |
| \] |
| |
| ## 7.2 Validation activation set |
| |
| Use a calibration activation set independent of the final ISS test set. |
| |
| Recommended: |
| |
| - 64–128 activation vectors per representative layer; |
| - multiple prompt types if benchmark-like prompts are used; |
| - or a separate set of held-out natural questions. |
| |
| ## 7.3 Rank candidates |
| |
| Use a nested grid such as: |
| |
| \[ |
| k\in\{64,128,256,512,1024\}, |
| \] |
| |
| bounded by model dimension and compute resources. |
| |
| Choose the smallest \(k_{m,\ell}\) that satisfies the fixed validation criteria. |
| |
| Recommended initial criteria: |
| |
| \[ |
| \operatorname{median}_i\cos(u_i,\widehat u_i)\ge0.99, |
| \] |
| |
| \[ |
| \operatorname{median}_i |
| \frac{\|u_i-\widehat u_i\|_2} |
| {\|u_i\|_2+\epsilon} |
| \le0.05, |
| \] |
| |
| and: |
| |
| \[ |
| \left| |
| \widehat{\operatorname{ISS}}_{2k} |
| - |
| \widehat{\operatorname{ISS}}_k |
| \right| |
| \le0.01. |
| \] |
| |
| The exact numeric thresholds may be adjusted after Stage A, but they must be frozen before final evaluation. |
| |
| ## 7.4 Model-specific rank is allowed |
| |
| Different models and layers may use different ranks: |
| |
| \[ |
| k_{m,\ell}. |
| \] |
| |
| This is methodologically acceptable because all models use the same approximation-quality standard. |
| |
| Always report: |
| |
| - \(k_{m,\ell}\); |
| - \(k_{m,\ell}/d_m\); |
| - validation cosine; |
| - validation relative error; |
| - rank-stability error. |
| |
| --- |
| |
| # 8. Calibration Corpus Size |
| |
| Rank error and corpus-sampling error are separate. |
| |
| The empirical Jacobian is: |
| |
| \[ |
| J^{\ell,(n)} |
| = |
| \frac1n\sum_{i=1}^{n}J^\ell_{x_i}. |
| \] |
| |
| A large \(k\) does not compensate for insufficient corpus size \(n\). |
| |
| ## 8.1 Corpus requirements |
| |
| The Jacobian corpus must: |
| |
| - be independent of the factual benchmark; |
| - contain general pretraining-style or natural text; |
| - use the same construction rule for all models; |
| - use a fixed sequence-length and position convention; |
| - record all sample IDs and random seeds. |
| |
| ## 8.2 Corpus-size convergence |
| |
| On representative models, compare: |
| |
| \[ |
| n\in\{32,64,128,256,512\}. |
| \] |
| |
| Measure: |
| |
| \[ |
| \cos(\widehat J^{(n)}h,\widehat J^{(2n)}h), |
| \] |
| |
| and: |
| |
| \[ |
| \left| |
| \widehat{\operatorname{ISS}}_n |
| - |
| \widehat{\operatorname{ISS}}_{2n} |
| \right|. |
| \] |
| |
| Select \(n\) using a fixed convergence rule. Do not select \(n\) based on preferred final benchmark conclusions. |
| |
| --- |
| |
| # 9. Random-Seed Stability |
| |
| For representative model-layer-rank settings, use at least three sketch seeds. |
| |
| Report mean and standard deviation for: |
| |
| - transported-vector cosine; |
| - transported-vector relative error; |
| - ISS; |
| - KTS if KTS uses the same transported states. |
| |
| A rank is not considered stable if results vary materially across seeds. |
| |
| Preferred final reporting: |
| |
| - average the result across three validated seeds; or |
| - use one preregistered seed after proving seed variance is negligible. |
| |
| Never choose the seed that gives the preferred scientific result. |
| |
| --- |
| |
| # 10. Official ISS Computation |
| |
| After freezing the estimator, compute: |
| |
| \[ |
| \widehat z^\ell_{m,f,q} |
| = |
| \widehat J^\ell_m h^\ell_{m,f,q}. |
| \] |
| |
| Then run the standard ISS pipeline. |
| |
| ## 10.1 Relation and condition residualization |
| |
| \[ |
| \bar z^\ell_{m,f,q} |
| = |
| z^\ell_{m,f,q} |
| - |
| \mu^\ell_{m,r_f} |
| - |
| \mu^\ell_{m,t(q)} |
| + |
| \mu^\ell_m. |
| \] |
| |
| ## 10.2 Regularized whitening |
| |
| \[ |
| \tilde z^\ell_{m,f,q} |
| = |
| (\Sigma^\ell_m+\lambda I)^{-1/2} |
| \bar z^\ell_{m,f,q}. |
| \] |
| |
| If the reduced coordinate \(y=Bh\) is used directly, estimate \(\Sigma\) in that coordinate system. |
| |
| Do not fit a separate whitening transform for each condition family. |
| |
| ## 10.3 Family centroid |
| |
| \[ |
| v^\ell_{m,f,t} |
| = |
| \operatorname{Normalize} |
| \left( |
| \frac1{|Q_{f,t}|} |
| \sum_{q\in Q_{f,t}} |
| \tilde z^\ell_{m,f,q} |
| \right). |
| \] |
| |
| ## 10.4 Same-fact cross-condition similarity |
| |
| \[ |
| S^+_{m,f,\ell} |
| = |
| \frac1{|\mathcal P_f|} |
| \sum_{(t,t')\in\mathcal P_f} |
| \cos(v^\ell_{m,f,t},v^\ell_{m,f,t'}). |
| \] |
| |
| ## 10.5 Relation-matched background |
| |
| \[ |
| S^-_{m,f,\ell} |
| = |
| \mathbb E_{\substack{g\neq f,\ r_g=r_f\\t\neq t'}} |
| \cos(v^\ell_{m,f,t},v^\ell_{m,g,t'}). |
| \] |
| |
| ## 10.6 Official ISS estimator |
| |
| \[ |
| \boxed{ |
| \widehat{\operatorname{ISS}}_{m,f,\ell} |
| = |
| \frac{S^+_{m,f,\ell}-S^-_{m,f,\ell}} |
| {1-S^-_{m,f,\ell}+\epsilon} |
| } |
| \] |
| |
| Average over the preregistered layer window: |
| |
| \[ |
| \widehat{\operatorname{ISS}}_{m,f} |
| = |
| \frac1{|\mathcal W_m|} |
| \sum_{\ell\in\mathcal W_m} |
| \widehat{\operatorname{ISS}}_{m,f,\ell}. |
| \] |
| |
| Then average over the fixed fact set: |
| |
| \[ |
| \widehat{\operatorname{ISS}}_m |
| = |
| \frac1{|\mathcal D|} |
| \sum_{f\in\mathcal D} |
| \widehat{\operatorname{ISS}}_{m,f}. |
| \] |
| |
| --- |
| |
| # 11. Raw-ISS |
| |
| Raw-ISS uses: |
| |
| \[ |
| z^\ell_{m,f,q}=h^\ell_{m,f,q}. |
| \] |
| |
| It should be computed because it is inexpensive and useful for: |
| |
| - debugging hidden-state extraction; |
| - validating family aggregation; |
| - validating negative sampling; |
| - checking whitening and residualization; |
| - ablation; |
| - testing whether Jacobian transport changes the conclusion. |
| |
| However: |
| |
| \[ |
| \boxed{\text{Raw-ISS is not the official ISS result.}} |
| \] |
| |
| Recommended table terminology: |
| |
| | Metric | Representation | Role | |
| |---|---|---| |
| | Raw-ISS | \(h^\ell\) | Identity-transport ablation | |
| | ISS | \(\widehat J^\ell h^\ell\) | Official metric | |
| | Token readout | \(W_U\widehat J^\ell h^\ell\) | Not used | |
| |
| --- |
| |
| # 12. Failure Policy |
| |
| Do not silently report an unvalidated approximation. |
| |
| A model-layer estimator fails validation if: |
| |
| - rank convergence is not achieved; |
| - action-vector cosine remains below threshold; |
| - relative action error remains above threshold; |
| - ISS changes materially across ranks; |
| - ISS changes materially across random seeds. |
| |
| ## 12.1 Allowed responses |
| |
| In order: |
| |
| 1. increase \(k\); |
| 2. increase oversampling \(p\); |
| 3. increase corpus size \(n\); |
| 4. add power iterations; |
| 5. restrict official ISS to validated layers; |
| 6. mark the model-layer estimator as unresolved. |
| |
| Raw-ISS may still be reported as an ablation, but it must not be relabeled as official ISS. |
| |
| ## 12.2 Optional fallback estimator |
| |
| If low-rank reconstruction consistently fails, consider a geometry-preserving output sketch: |
| |
| \[ |
| RJ^\ell h, |
| \] |
| |
| where: |
| |
| \[ |
| R\in\mathbb R^{k\times d}. |
| \] |
| |
| This avoids assuming that \(J^\ell\) itself is low rank. However, it is a different estimator and must be separately validated against the full small-model reference. |
| |
| Do not switch to this estimator silently. |
| |
| --- |
| |
| # 13. Data Separation |
| |
| Use three disjoint resources. |
| |
| ## 13.1 Jacobian corpus |
| |
| Used to estimate \(J^\ell_m\). It must contain independent general text. |
| |
| ## 13.2 Estimator calibration set |
| |
| Used to select: |
| |
| - corpus size \(n\); |
| - rank \(k\); |
| - sketch seed; |
| - oversampling; |
| - power iterations; |
| - approximation thresholds. |
| |
| It must not overlap with the final benchmark evaluation facts. |
| |
| ## 13.3 Final ISS benchmark |
| |
| Used only after all estimator settings are frozen. |
| |
| Do not tune \(n\), \(k\), seeds, layer windows, or whitening choices on the final benchmark. |
| |
| --- |
| |
| # 14. Required Output Files |
| |
| ```text |
| jacobian_iss/ |
| ├── configs/ |
| │ ├── corpus_config.yaml |
| │ ├── estimator_config.yaml |
| │ ├── validation_thresholds.yaml |
| │ └── layer_windows.yaml |
| ├── full_reference/ |
| │ └── qwen2.5_0.5b/ |
| ├── low_rank_factors/ |
| │ └── {model}/{layer}/ |
| │ ├── Q.pt |
| │ ├── B.pt |
| │ └── metadata.json |
| ├── validation/ |
| │ ├── small_model_full_comparison.json |
| │ ├── per_model_action_validation.jsonl |
| │ ├── rank_convergence.jsonl |
| │ ├── corpus_convergence.jsonl |
| │ └── seed_stability.jsonl |
| ├── transported_states/ |
| ├── iss_per_fact_layer.jsonl |
| ├── iss_per_fact.jsonl |
| ├── iss_model_summary.json |
| └── raw_iss_ablation.json |
| ``` |
| |
| --- |
| |
| # 15. Required Metadata |
| |
| For every estimator, store: |
| |
| ```json |
| { |
| "model": "model_name", |
| "model_revision": "revision", |
| "tokenizer_revision": "revision", |
| "layer": 16, |
| "hidden_dimension": 4096, |
| "rank": 512, |
| "oversampling": 64, |
| "power_iterations": 0, |
| "calibration_corpus_size": 256, |
| "calibration_sequence_length": 128, |
| "random_seed": 42, |
| "validation_cosine_median": 0.993, |
| "validation_relative_error_median": 0.041, |
| "iss_rank_difference": 0.006, |
| "validated": true |
| } |
| ``` |
| |
| --- |
| |
| # 16. Minimum Required Experiments |
| |
| ## Experiment A: Full-reference calibration |
| |
| Model: |
| |
| ```text |
| Qwen2.5-0.5B |
| ``` |
| |
| Compare: |
| |
| ```text |
| full empirical Jacobian |
| vs. |
| rank 32/64/128/256/512/768 approximations |
| ``` |
| |
| Report: |
| |
| - Frobenius error; |
| - transported-vector error; |
| - transported-vector cosine; |
| - ISS absolute error; |
| - per-fact ISS Spearman correlation. |
| |
| ## Experiment B: Medium-scale transfer check |
| |
| Use at least one medium-sized model. |
| |
| At representative layers: |
| |
| - compute high-accuracy direct \(Jh\) actions; |
| - validate rank behavior; |
| - confirm that the small-model rank does not automatically transfer. |
| |
| ## Experiment C: Per-model validation |
| |
| For every evaluated model: |
| |
| - representative early, middle, and late layers; |
| - 64–128 held-out activation vectors; |
| - nested candidate ranks; |
| - fixed validation thresholds; |
| - at least three random seeds on representative settings. |
| |
| ## Experiment D: Official evaluation |
| |
| Only after freezing all estimator choices: |
| |
| - compute transported states; |
| - compute official ISS; |
| - compute Raw-ISS; |
| - compare trends; |
| - report approximation uncertainty. |
| |
| --- |
| |
| # 17. Recommended Decision Rule |
| |
| Do not use: |
| |
| ```text |
| k = 512 for every model |
| ``` |
| |
| Use: |
| |
| ```text |
| Choose the smallest k for each model-layer that satisfies |
| one fixed approximation-quality standard. |
| ``` |
| |
| Recommended initial rule: |
| |
| ```text |
| median action cosine >= 0.99 |
| median relative action error <= 0.05 |
| |ISS(2k) - ISS(k)| <= 0.01 |
| seed standard deviation of ISS <= 0.005 |
| ``` |
| |
| The exact thresholds may be refined after the small-model calibration, but they must be frozen before final evaluation. |
| |
| --- |
| |
| # 18. Uncertainty Reporting |
| |
| Separate the following sources of uncertainty: |
| |
| 1. finite Jacobian-corpus error; |
| 2. low-rank approximation error; |
| 3. random-sketch error; |
| 4. benchmark sampling uncertainty; |
| 5. relation-level heterogeneity. |
| |
| At minimum, report: |
| |
| - rank sensitivity; |
| - corpus-size sensitivity; |
| - random-seed sensitivity; |
| - per-model action validation; |
| - relation-clustered bootstrap confidence intervals for final ISS. |
| |
| Do not present one confidence interval that silently mixes all uncertainty sources. |
| |
| --- |
| |
| # 19. Paper-Ready Method Description |
| |
| > We define Internal State Stability on Jacobian-transported residual states \(z^\ell=J^\ell h^\ell\), where \(J^\ell\) is the corpus-averaged input-output Jacobian from layer \(\ell\) to the final residual stream. We do not apply the vocabulary unembedding matrix, thereby retaining a continuous hidden representation compatible with multi-token and multilingual answers. Since explicitly materializing \(J^\ell\in\mathbb R^{d\times d}\) is prohibitive for large models, we estimate it using randomized range finding, yielding a factorized operator \(\widehat J^\ell=Q^\ell B^\ell\). We validate this approximation against full empirical Jacobians on Qwen2.5-0.5B and against directly computed Jacobian-vector products on held-out activations for every evaluated model. The sketch rank is selected independently for each model and layer using a fixed preregistered approximation-error tolerance rather than a universal rank or final benchmark performance. |
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| Continuation: |
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| > We separately evaluate calibration-corpus convergence, rank convergence, and random-seed stability. Raw hidden-state ISS is reported only as an identity-transport ablation. |
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| --- |
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| # 20. Final Engineering Decisions |
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| ```text |
| Official ISS: |
| Jacobian-transported hidden-state ISS |
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| Transport: |
| z = J h |
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| Unembedding: |
| Do not multiply by W_U |
|
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| Small-model full Jacobian: |
| Finite-corpus exact reference only |
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| Large-model implementation: |
| Randomized factorized operator estimate |
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| Rank selection: |
| Per model and per layer |
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| Rank criterion: |
| Shared fixed approximation-quality threshold |
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| Validation: |
| Full matrix comparison on Qwen2.5-0.5B |
| Direct J h comparison on every evaluated model |
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| Raw-ISS: |
| Debugging and ablation only |
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| Final test set: |
| Never used to tune n, k, seeds, layers, or estimator settings |
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| Failure: |
| Do not report unvalidated Jacobian ISS |
| ``` |
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| --- |
|
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| # 21. Immediate Implementation Order for Claude |
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| 1. Finish Raw-ISS to validate data loading and metric aggregation. |
| 2. Implement exact finite-corpus Jacobian recovery on Qwen2.5-0.5B. |
| 3. Implement the randomized factorized estimator \(\widehat J=QB\). |
| 4. Compare candidate ranks against the full small-model reference. |
| 5. Measure matrix, vector-action, ISS-value, and fact-ranking errors. |
| 6. Freeze approximation-quality thresholds. |
| 7. Implement direct \(Jh\) validation for large models. |
| 8. Select \(k_{m,\ell}\) using the fixed thresholds. |
| 9. Run corpus-size and seed-stability checks. |
| 10. Freeze all estimator configurations. |
| 11. Compute official ISS on the final benchmark. |
| 12. Report Raw-ISS only as an ablation. |
| 13. Save all factors, configurations, validation logs, and checksums. |
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| The objective is not to recover every Jacobian entry exactly. The objective is to guarantee that the transported representations and the resulting ISS remain within a preregistered numerical tolerance. |
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