Learn the Directions, Normalize the Gains: Post-Training Normalization for LoRA

• Trained adapters exhibit adaptation imbalance: directional output energy is concentrated in a small subset of components, despite LoRA’s rank constraint.The method investigates deployment consequences by changing gains while keeping learned directions fixed.Source: §1 · p. 1
• LoRA-Norm applies fixed nonlinear transformations to singular values, restores each module’s original nuclear norm, and requires no calibration data, additional training, or inference operations.It keeps learned directions fixed and reconstructs the edited update using standard low-rank factors with the original adapter configuration and scaling.Source: §1 · p. 2
• Across 18 adapters, LoRA-Norm improves average target-task performance and capability retention, exceeding evaluated spectral-pruning and gradient-guided editing configurations on both measures.The evaluation spans two backbones and three tasks: coding, mathematical reasoning, and instruction following.Source: §1 · p. 2
• Matched controls recover much of LoRA-Norm’s benefit, while stronger functional equalization brings no consistent additional gains; balancing gains and equalizing responses are distinct objectives.A direction’s functional contribution depends on both its gain and the inputs that activate it.Source: §1 · p. 2

Spectral and direction-aware LoRA training

• Unlike training-time spectral and direction-aware methods, LoRA-Norm rebalances gains in a completed adapter’s learned basis without further optimization.Related approaches include PiSSA and MiLoRA’s singular-component initializations, DoRA’s magnitude-direction separation, SCLoRA’s spectral clipping, SVFT’s sparse singular-basis coefficients, and LoRA-XS’s small matrix between frozen SVD-derived factors (Meng et al., 2024; Wang et al., 2025; Liu et al., 2024; Lingam et al., 2024; Bałazy et al., 2024).Source: §2 · pp. 2–3

Post-training update editing

• LoRA-Norm replaces gradient-guided post-training reweighting with a fixed data-free map and uses scalar and exact-flattening controls to test update strength and spectral shape.Prior editing includes WiSE-FT scaling, spectral thresholding or filtering methods, safety-oriented pruning or attenuation, and Spectral Surgery’s calibration-gradient reweighting with nuclear-mass restoration (Wortsman et al., 2022; Abro & Tahir, 2026; Yang et al., 2026; Kumaravelu et al., 2026; Ao & Rumchurn, 2025; Marchenko Breneur et al., 2026; Tian et al., 2026). LoRA-Norm uses a finite nonlinear transformation with per-module nuclear-norm restoration and studies capability retention and matched-strength controls.Source: §2 · p. 3

Activation-aware adaptation and functional analysis

• LoRA-Norm uses activation statistics retrospectively to characterize completed adapters and construct gain controls, while its functional participation ratio is not an optimization objective.EVA and CorDA use activation information during adaptation, whereas LoRA-Norm’s activation-weighted effective-dimensionality measure applies the standard participation-ratio form and is not a new mathematical definition; the main edit remains calibration-free (Paischer et al., 2025; Yang et al., 2024).Source: §2 · p. 3

3.1 FIXED-DIRECTION ADAPTER EDITING

• LoRA-Norm changes only nonnegative gains, keeping them paired with the source singular directions while leaving base weights, adapter scaling, and rank budget unchanged.The transformation may change the gains’ relative ordering.Source: §3.1 · p. 3, Figure 2 · p. 4

3.2 HYBRID NEWTON–SCHULZ REBALANCING

• The hybrid rebalancing normalizes the source spectrum, applies fast and stable quintic stages, then restores the original nuclear-norm budget.The fast stage amplifies small positive inputs; the stable polynomial fixes 1 with zero derivative there to refine gains near unity. In exact arithmetic, the edited update preserves the original nuclear norm, but finite-step HNS need not yield a flat spectrum; reduced QR factorizations and an r × r core SVD avoid materializing the dense update.Source: §3.2 · p. 4

3.3 DIAGNOSING FUNCTIONAL CONCENTRATION

• Functional participation ratio measures how directional energy is spread, but it is a diagnostic rather than an optimization target and is invariant to uniform gain scaling.Its range is 1 to r for nonzero total energy, and module-level values are aggregated by the median within each adapter; energy is reported alongside FPR because FPR cannot capture overall update strength.Source: §3.3 · p. 5

4.1 EXPERIMENTAL SETUP

• The primary evaluation paired edits across 18 Qwen3-8B and Llama-3.1-8B-Instruct checkpoints adapted to Magicoder, MetaMath, or Tulu, scoring target benchmarks and Commonsense-8 retention.Target is the score on the target benchmark; Off averages three non-target categories, and FG measures category-level deficits from the base model. Fixed activation diagnostics used 256 source-task examples, but LoRA-Norm itself is data-free.Source: §4.1 · pp. 5–6

4.2 TARGET-TASK PERFORMANCE AND CAPABILITY RETENTION

• Across 18 checkpoints, LoRA-Norm raised Target from 68.00 to 72.08 (+4.09 percentage points) and Off from 69.55 to 72.79 (+3.24), while reducing FG from 2.71 to 0.34.Target improved on 16 checkpoints, tied on one, and decreased on one; Off improved on 15 and decreased on three; FG improved on 14 and tied on four.Source: Table 1 · p. 6, §4.2 · p. 6
• Qwen–Magicoder gained 10.37 Target points versus 0.20 for Llama–Magicoder; Off improved in five of six groups but fell for Qwen–Tulu from 85.63 to 84.93, where FG remained zero.The Qwen–Tulu result shows that staying above the base model does not imply preserving all positive transfer.Source: §4.2 · p. 6

4.3 COMPARISON WITH POST-HOC EDITING METHODS

• Among edited adapters, LoRA-Norm had the highest mean Target and Off and lowest FG; it exceeded PARA at ϵ = .90 by 3.31 Target and 2.78 Off points, and Spectral Surgery by 3.32 and 4.51 points.PARA retained 38.16–80.16% of parameters across settings. The Spectral Surgery comparison covers its tested configuration, which edits only o proj/down proj with calibration, rather than isolating gradient guidance.Source: §4.3 · pp. 6–7, Table 2 · p. 7

4.4 SENSITIVITY TO HNS ITERATION COUNTS

• Across six checkpoints, all nine nonzero HNS iteration configurations gained +3.42–+3.62 target-task points over LoRA; the main 4+1 setting gained +3.57 points, within 0.05 of the largest mean.Increasing fast or stable steps did not monotonically improve performance, and individual checkpoint responses were task-dependent.Source: §4.4 · p. 7

5.1 RQ1: WHAT CHANGES UNDER GAIN NORMALIZATION?

• LoRA-Norm raises mean module-median FPR from 2.02 to 7.35 and full-moment PR from 1.84 to 5.39, while mean functional-energy ratio falls to 0.0972.These measurements characterize local updates under fixed frozen-base inputs, not the edited network’s end-to-end activations.Source: §5.1 · p. 7, Figure 3 · p. 8

5.2 RQ2: CAN MATCHED-STRENGTH CONTROLS REPRODUCE THE GAINS?

• Energy-matched Scalar-E has differences of −0.62 Target and +0.32 Off points versus LoRA-Norm, with neither interval excluding zero; LoRA-Norm’s FG is 0.18 points lower.Against Flat-E, LoRA-Norm’s Off difference is +0.23 points and Target difference is −0.19; Scalar-E gains 4.70 Target and 2.92 Off points over LoRA while leaving module FPR unchanged. These metric-specific results do not establish uniform superiority or equivalence, and show FPR increases are not necessary for improvement under these interventions.Source: Figure 4 · p. 8, §5.2 · pp. 8–9
• Frobenius-matched Scalar-F trails LoRA-Norm by 1.44 Target points and 1.23 Off points, while its FG is 0.77 points higher.Scalar-F’s module-level functional energy relative to LoRA-Norm has median 2.88, showing that equal Frobenius norms do not imply equal functional strength.Source: §5.2 · p. 8
• Flat-Nuclear improves Target on 16 of 18 checkpoints and trails LoRA-Norm by 0.15 average points, limiting claims that the finite-step HNS map is essential.It preserves module nuclear mass while equalizing gains across learned singular directions; its paired wins/ties/losses against LoRA-Norm are 8/4/6. The controls do not establish a general shape advantage beyond energy matching, and matching uses fixed input statistics rather than a causal decomposition of end-to-end performance.Source: §5.2 · p. 9

5.3 RQ3: DOES STRONGER FUNCTIONAL EQUALIZATION HELP?

• Stronger functional equalization does not consistently help: partial weighting raises mean FPR to 12.73 with only a 0.025-point Target gain and a 0.21-point Off decrease versus LoRA-Norm.Full weighting and protected flattening raise FPR to approximately 16 but lower both mean Target and Off; mean FG decreases slightly under all three interventions. Higher FPR coincides with worse Target in 22 of 54 comparisons and worse Off in 37; full weighting helps Llama–MetaMath but harms Qwen–Magicoder. These activation-aware interventions change energy as well, so they do not isolate a causal effect.Source: Figure 5 · p. 9, §5.3 · p. 9
• Adding FPR to log functional-energy ratio changes held-out R2 by 0.0155 on Target and 0.0092 on Off, with both intervals crossing zero.Within the energy-matched cohort, the changes are −0.0767 on Target and −0.0319 on Off; full-moment PR also lacks stable improvement. This is a conditional within-checkpoint analysis of relative performance, not label-free prediction of absolute performance, and does not support maximizing FPR as a performance objective.Source: §5.3 · pp. 9–10

6 CONCLUSION

• Across 18 adapters spanning two backbones and three tasks, LoRA-Norm improves specialization and capability retention on average.Energy-matched controls recover much of the benefit, while stronger functional equalization yields no consistent further gains; the authors call for broader model and task coverage and further controlled replications.Source: §6 · p. 10

AI USE STATEMENT

• The authors conceived the method and experiments, reviewed and executed AI-assisted code, verified outputs, independently assessed conclusions, and take responsibility for the final paper, code, results, and claims.OpenAI Codex assisted with code implementation, analysis of experimental outputs, figure preparation, and manuscript editing.Source: AI USE STATEMENT · p. 10

REPRODUCIBILITY STATEMENT

• The paper documents implementation details, evaluation protocols, training configurations, checkpoint provenance, results, controls, iteration sensitivity, and statistical procedures across its appendices.Recorded protocol differences and gaps in archived training information are documented alongside the relevant results.Source: REPRODUCIBILITY STATEMENT · p. 10

A.1 COMPACT DECOMPOSITION AND RECONSTRUCTION / A.2 NORM BUDGETS AND EXACT PARAMETER FLATTENING

• Compact QR/SVD reconstruction retains source directions and adapter configuration with low-rank storage, while equal gains attain norm bounds at fixed nuclear mass without implying optimal task performance.The stated complexity bounds are arithmetic estimates, not measured wall-clock speedups; nuclear-budget and reconstruction checks follow saving and reloading edited factors.Source: §A.1 · p. 13, §A.2 · p. 13

A.3 FUNCTIONAL STATISTICS AND THEIR INVARIANCES

• 1.000, 0.288, 0.219, 0.198, and 0.197 are the mean RMS ratios for LoRA, HNS, Functional-HNS (0.5), Functional-HNS (1), and protected flat, respectively.Full-moment FPR is invariant to orthogonal coordinate changes, whereas raw FPR depends on the source basis; Flat-Fro and Flat-Nuclear have identical raw FPR despite different update magnitudes.Source: §A.3 · p. 14

A.4 PROTECTED FUNCTIONAL EDITS

• The main method applies H4+1 to gains, while Functional-HNS adds a functional weighting exponent and restores the original parameter-space nuclear sum.The protected flattening ablation has a closed form; invalid inputs or a nonpositive median trigger an error rather than an activation-free fallback.Source: §A.4 · p. 14

B.1 BENCHMARKS AND DECODING / B.2 ACTIVATION COLLECTION

• Evaluation used the recorded 541-example IFEval artifact and deterministic generation settings, while activation statistics came from sampled training-SFT tokens under frozen-base inputs.Edited-adapter functional statistics were not recollected on the edited network; the experiments establish neither independent retention-distribution calibration nor content-level calibration/evaluation overlap.Source: §B.1 · p. 15, §B.2 · p. 15

B.3 SEPARATE EVALUATION BLOCKS AND PARAMETER-SPACE CONTROLS

• The joint parameter-control block has its own LoRA and HNS outputs, so its scores are not interchangeable with the primary block’s; the different HNS aggregates reflect evaluation provenance.The primary block includes LoRA, HNS, two Functional-HNS variants, protected functional flat, and base references; the joint control block evaluates LoRA, Flat-Fro, Flat-Nuclear, DG-Hard, and HNS.Source: §B.3 · pp. 15–16
• In the separate joint block, LoRA-Norm exceeds Flat-Nuclear by 0.151 percentage points in mean Target; Flat-Nuclear improves 16 of 18 checkpoints over LoRA and wins/ties/loses against LoRA-Norm on 8/4/6 checkpoints.Flat-Fro and Flat-Nuclear preserve the Frobenius and nuclear norms, respectively, using the source singular basis and balanced reconstruction.Source: Table 4 · p. 16, §B.3 · p. 16
• DG-Hard is an identity intervention in this regime: the full-spectrum median is zero, so its threshold retains all 16 active singular values in every source module.The reported baseline uses the full spectrum, including zeros omitted by low-rank factorization; thresholding only the active values would define a different intervention.Source: §B.3 · p. 16

B.4 MATCHED-STRENGTH EVALUATION EXTENSION

• The eight-method extension comprises 584 unique method–category/reference cells, including Scalar-E and Flat-E for 18 checkpoints and Scalar-F for those same sources.The original five-method primary block accounts for 368 cells; these counts exclude the parameter-control block, archived step-grid evaluation, and later PARA/Spectral Surgery extension.Source: §B.4 · p. 16
• For 296 reused cached cells, input content and ordering were hashed against base-model references; these checks verify evaluation inputs, not equality of model predictions.The extension used recorded final prompt, template, parser, and scoring implementations, with a 4,096-token context and adapter blocks of five.Source: §B.4 · p. 16

B.5 TRAINING RUNS AND NUMERICAL CHECKS

• The numerical audit found no discrepancy in recomputed checkpoint medians across 368 primary cells and 21,420 module-level functional records; maximum combined post-save numerical error was 1.80 × 10−6.The α = 0 compatibility check had maximum spectral error 4.02 × 10−8.Source: §B.5 · p. 17

B.6 RECORDED TRAINING CONFIGURATIONS

• Runs 43 and 44 use rank 16, LoRA alpha 32, dropout 0.05, all-linear adaptation, bf16, and zero warmup; complete optimizer and schedule settings are unverified for archived Llama run-42 sources.The archived Llama/Math record verifies MetaMath 50K and rank 16, while Llama/Instruction verifies length 1024, global batch 128, rank 16, and seed 42.Source: §B.6 · pp. 17–18, Table 5 · p. 17

B.5 TRAINING RUNS AND NUMERICAL CHECKS

• Each backbone–task pair has three available runs, 42, 43, and 44, summarized by their mean and sample standard deviation; between-run variation cannot be attributed solely to random seeds.Runs 43 and 44 share settings within each backbone–task pair except for training seeds and output locations, while archived run-42 records are incomplete.Source: §B.5 · p. 17, §B.6 · p. 18

C COMPLETE PRIMARY RESULTS

• The evaluation retains all three source runs, including run 42, and estimates uncertainty using means and sample SDs.Performance comparisons and uncertainty estimation retain checkpoint pairing; scores are percentages and FG is in percentage points.Source: C COMPLETE PRIMARY RESULTS · p. 18
• The primary protocol reports five-method results averaged over 18 source checkpoints, with FPR and RE as descriptive diagnostics.Higher Target/Off and lower FG are favorable; the tables report Target, Off, and FG results, paired changes against HNS, and functional diagnostics.Source: Table 7 · p. 18, C COMPLETE PRIMARY RESULTS · p. 18, Table 8 · p. 18

D.1 DEFINITIONS AND NUMERICAL VERIFICATION

• Scalar-E and Flat-E match energy using the unfloored second moment from frozen-base inputs, while protected functional-flat instead equalizes protected directional energies.The controls retain original scaling, which cancels in the matching factors; Scalar-F matches actual HNS rather than an ideal exact-flat update.Source: §D.1 · p. 21
• Scalar-F preserves source spectral ratios while matching HNS’s Frobenius norm, whereas Scalar-E matches functional energy and Flat-E equalizes parameter gains under that constraint.All controls retain U, V, and original scaling; they do not restore nuclear norm afterward because that would alter the matching constraint.Source: §D.1 · p. 21
• Scalar-F is data-free, while Scalar-E and Flat-E use activation statistics; their matches are audited per module under fixed base-input conditions.Energy matching does not guarantee identical full-network changes under edited-model activations, and saved-factor errors include reconstruction and storage effects.Source: §D.1 · p. 21, Table 16 · p. 21
• The per-module Scalar-F/HNS functional-energy ratio ranges from 0.294708 to 14.773126, with a median of 2.883298, so Frobenius matching does not imply functional-energy matching.Historical ScalarShrink implementations matched different quantities, and their scores were not substituted for the current-protocol Scalar-F control.Source: §D.1 · p. 21

D.2 COMPLETE RESULTS AND PAIRED COMPARISONS

• Matched-strength comparisons cover Target, Off, and FG across 18 source checkpoints, with higher Target/Off and lower FG favorable.The four edited methods use the same final references; the current functional-energy results are not pooled with the earlier Flat-Fro/Flat-Nuclear/DG-Hard block.Source: §D.2 · p. 21, Table 17 · p. 22, Table 18 · p. 22
• Table 22 reports all six paired comparisons among four edits with mean differences and 95% percentile intervals in percentage points.Negative FG differences favor the left method.Source: Table 22 · p. 23
• Table 23 gives HNS-versus-control comparisons within backbone–task groups, using unadjusted descriptive intervals not intended to select favorable configurations.Each group contains three sources.Source: Table 23 · p. 24

E.1 PAIRED UNCERTAINTY ESTIMATION

• Overall performance intervals resample source checkpoints in paired 2,000-draw bootstrap replicates with seed 20260914, without additional stratification.Intervals are empirical 2.5th and 97.5th percentiles; group intervals are unadjusted and descriptive, and interpretation is limited by fixed test examples and archived training-recipe differences.Source: §E.1 · p. 25
• Figure 5 adds descriptive intervals from existing checkpoint records, sharing checkpoint-index resamples across variants and outcomes without new evaluations.Target and Off use variant minus HNS; FG reduction uses HNS minus variant, opposite to FG-change signs in Table 11. Figure 4 reproduces Table 22’s archived matched-strength intervals with HNS minus control for Target/Off and control minus HNS for FG reduction.Source: §E.1 · p. 25
• Correctness-transition uncertainty uses a separate conditional test-sample bootstrap fixing one checkpoint and resampling paired benchmark-example states.The 32 joint correctness states cover Base, LoRA, HNS, Scalar-E, and Flat-E; opportunity-set denominators are recomputed, undefined differences omitted, and valid-resample coverage recorded. The 2,000 draws use seed 20260914; conditional sample and source-checkpoint intervals represent different uncertainties, with no pooled interval treating checkpoint–sample events as independent.Source: §E.1 · p. 25

E.2 INCREMENTAL PREDICTION AND LEAVE-ONE-CHECKPOINT-OUT EVALUATION

• LOCO R2 is conditional within-checkpoint prediction, not label-free prediction of absolute performance on unseen checkpoints.The training-fold-only standardization audit reproduced 25,920 predictions with maximum difference 4.44 × 10−15; LOCO bootstrap intervals condition on saved predictions and do not capture retraining variation.Source: §E.2 · p. 26
• In cohort B, participation-ratio intervals include zero; in cohort C, both Target intervals lie below zero and both Off intervals include zero, showing no stable positive increment beyond energy.This applies to raw FPR and full-moment PR for Target and Off gains.Source: §E.2 · p. 26, Figure 6 · p. 27

E.3 CORRECTNESS TRANSITIONS AND AGGREGATION

• Correctness rates are hierarchically aggregated across benchmarks, families, and checkpoints; transition prevalence uses the full benchmark sample and is not opportunity-conditional.The transition analysis covers three edits, while Scalar-F appears in the performance comparison but not that fixed analysis.Source: §E.3 · pp. 29–30
• 62.351% Recovery and 62.288% Retention characterize LoRA-Norm’s overall transitions; neither LoRA-Norm nor Scalar-E has an established advantage on these measures.Their differences are −0.524 Recovery points (95% paired interval [−2.587, 1.770]) and −0.932 Retention points ([−2.657, 0.969]).Source: Table 28 · p. 30, §E.3 · p. 30

E.4 REPRODUCIBILITY RECORDS

• The appendix records method construction, evaluation provenance, and statistical outputs, but hashes establish artifact identity and internal consistency—not independent replication or a causal mechanism.The appendix tables required no retraining or new model inference; large weights, activations, and per-example predictions are not embedded.Source: §E.4 · p. 31

F.1 CONSTRUCTION AND EVALUATION SCOPE

• The extension evaluates three PARA configurations and one Spectral Surgery configuration on the 18 original checkpoints without retraining source adapters.PARA parameter retention is 38.16%, 54.28%, and 80.16% across its configurations; Spectral Surgery uses 128 training examples and a fixed gradient-guided configuration, so the comparison does not isolate gradient guidance alone.Source: §F.1 · pp. 32–33

F.2 EXECUTION AND NUMERICAL AUDIT

• An FP32 reconstruction precheck exceeded its initial 10−5 tolerance, which was changed to 5 × 10−5 after attributing the discrepancy to rounding.The audit explicitly retains the measured error; the tolerance change alone does not establish end-to-end prediction equivalence.Source: §F.2 · p. 33

F.3 INTERPRETATION AND REPRODUCIBILITY RECORDS

• PARA’s parameter savings must accompany its performance comparison, and Spectral Surgery’s Target gain with an Off decrease describes only the evaluated configuration.Neither comparison isolates functional energy from spectral shape; the experiment records are in the stated result directory.Source: §F.3 · p. 33

F.4 PAIRED COMPARISONS AND COMPLETE CHECKPOINT RESULTS

• Table 34 covers complete new-baseline results for all 18 source checkpoints.The caption defines Target and Off as percentages and FG as percentage points; Code, Math, and IF denote Magicoder, MetaMath, and Tulu training.Source: Table 34 · p. 33
• The supplied baseline-result rows expose Base, Task, Run, Method, Target, Off, and FG headers but no checkpoint values.The passages therefore do not support reporting specific baseline comparisons or numerical outcomes.Source: §F.4 · pp. 34–35
• Table 33 reports LoRA-Norm-minus-baseline cohort mean differences with 95% paired source-checkpoint bootstrap intervals; the intervals are descriptive and unadjusted.Positive differences favor LoRA-Norm for Target/Off, while negative differences favor it for FG.Source: Table 33 · p. 36

G.1 PROTOCOL AND COMPLETE GRID

• The archived grid covers six adapters from Qwen3-8B and Llama-3.1-8B-Instruct across Magicoder, MetaMath, and Tulu, testing nine nonzero settings.Each setting edits all LoRA modules at strength 1, retains rank 16 and original scaling, and restores each module’s nuclear norm; 4 + 1 and 8 + 2 were rebuilt and evaluated. The two Commonsense170K sources in the eight-checkpoint archive are excluded.Source: §G.1 · p. 37
• The 0 + 0 reconstruction control gains +0.47 points on average across six checkpoints, with five wins, one tie, and no losses.It performs compact SVD and balanced reconstruction without a nontrivial spectral edit; although an update-level identity in exact arithmetic, its stored factors and numerical execution differ.Source: §G.1 · p. 37
• The main 4 + 1 setting improves all six checkpoints, and every nonzero setting improves five or six without decreases.Both GSM8K checkpoints improve under every nonzero setting; task-specific descriptive maxima vary, and are observations on evaluated test sets rather than independently validated configuration choices.Source: §G.1 · p. 37

G.2 FIXED-CONFIGURATION AGGREGATES AND UNCERTAINTY

• The fixed 4 + 1 setting yields +3.57 points over original LoRA, within a +3.42–+3.62-point observed mean-improvement range across nonzero settings.Each setting is averaged equally over the same six checkpoints; the small observed differences do not establish significant differences, and increasing iteration counts does not yield monotonic task performance.Source: Table 36 · p. 38, §G.2 · p. 38
• Paired inference uses 20,000 example-level bootstrap draws and Holm-adjusted exact McNemar tests, estimating test-example uncertainty for fixed adapters rather than training-seed uncertainty.The ten comparisons against original LoRA within each checkpoint include 0 + 0; the retained intervals and adjusted p-values were reused after restricting the cohort.Source: §G.2 · p. 38
• Both GSM8K checkpoints have positive paired intervals and Holm-adjusted p < 0.05 across all nine nonzero settings, while other checkpoint findings are less consistent.Qwen HumanEval intervals are positive but none survive correction; all Llama HumanEval and IFEval intervals include zero, and Qwen IFEval has ordinary intervals excluding zero at 2+1, 4+0, and 8+1, with none surviving correction. Tests compare settings with original LoRA, not with one another.Source: §G.2 · p. 38

H EXPLORATORY SCALAR CALIBRATION AND NUMERICAL RECHECK

• The archived reconstruction changed only the source singular-value spectrum across 252 modules, retained original LoRA scaling, and was not pooled with the primary 18-source cohort.It used an all-module HNS reference specified as 8 + 2 rather than the main 4 + 1; existing-representation HNS was diagnostic only.Source: H EXPLORATORY SCALAR CALIBRATION AND… · p. 39
• The selected γ = 0.40 was the fixed grid’s lower boundary, so calibration did not identify the scalar optimum; matching total Frobenius norm instead gives γ ≈0.609.A global multiplier was selected by strict accuracy on 256 calibration questions, with ties favoring the value closest to one, then evaluated without reselection on a separate 512-question subset; per-module scaling matches archived HNS Frobenius norm, not functional energy.Source: H EXPLORATORY SCALAR CALIBRATION AND… · p. 39
• On the 512-question recheck, common-basis HNS differed from global scaling by +0.39 strict points and −0.20 numeric points, establishing neither superiority nor statistical equivalence.Against per-module scaling, the strict difference was +1.37 points (95% interval [−0.78, 3.52]).Source: Table 39 · p. 39, H EXPLORATORY SCALAR CALIBRATION AND… · p. 39
• Identical adapters reproduced sequences across three fresh processes, but alternative factorizations changed some sequences and zero rebuild shifted strict accuracy by −0.59 points.Thus repeatability does not establish representation invariance, and the zero-rebuild shift is not a universal noise floor.Source: H EXPLORATORY SCALAR CALIBRATION AND… · p. 39
• This exploratory target-task recheck uses one Qwen3-8B–MetaMath source and does not establish retention benefits, independent downstream confirmation, or superiority over independently validated scaling.Earlier evaluations already used all 1,319 GSM8K test questions, so this is not a clean held-out selection study.Source: H EXPLORATORY SCALAR CALIBRATION AND… · p. 39