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The information geometry of large language models is shared, learned, and controllable

Dario Picozzi

arXiv:2609.11063v1cs.LGcs.CL

TL;DR

It remains unclear what behavioral structure language models share or how to change one behavior without disturbing others. This paper characterizes next-token output geometry and uses it to derive minimum-disturbance interventions; across architectures, same-language models recover nearly identical geometry, while changing the language law changes it.

  • Problem

    It remains unclear what behavioral structure language models share and how to change a target behavior while minimizing disturbance elsewhere.

    Source: Discussion · p. 15

  • Method

    The paper characterizes shared next-token output geometry and uses a regularized Fisher pullback to derive minimum-local-disturbance activation interventions for desired behavior changes.

    Source: Discussion · p. 15, §1.2 · p. 44

  • Results

    Across architectures, models trained on the same language recover nearly identical output geometry, while changing the language law changes it; shared geometry also aligns with semantic categories and human predictions.

    Source: Discussion · p. 14

  • Takeaways & Limitations

    The geometry offers reusable minimum-disturbance local control across prompts and multiple operations.

    Source: Discussion · pp. 15–16

    The simple margin equation does not guarantee identical arbitrary structures under RMS population blur without specifying the observable, Lipschitz constant, decision margin, and error type.

    Source: §4.2 · p. 54

Abstract

from arXiv · show

Large language models learn similar behaviours, yet it remains unclear what structure they share or how to change one behaviour without disturbing others. The Fisher-Rao geometry of next-token probabilities connects these questions: behaviour determines this geometry up to output-preserving symmetries, whereas activation geometry depends on coordinates. Across transformer, state-space and recurrent models, output geometries agree more strongly than activation geometries, and shared geometry supports semantic-category transfer. Agreement with human word choices increases with predictive accuracy, scale and training, and improves further after model-only calibration. Token probabilities and read-out geometry jointly predict the spectrum and its effective dimension. Controlled language assignments show that geometry follows the language law across architectures. Pretraining corpus statistics predict held-out fact acquisition without recalibration, while randomised experiments show that deeper evidence substantially delays acquisition across every tested architecture and evidence construction. Finally, the geometry prescribes minimum-disturbance local interventions, predicts their relative cost, and supports reusable control: updates learned on donor prompts transfer to unseen prompts while better preserving behaviour on reference prompts than Euclidean control. The same geometric correction improves steering, editing, attribution, dictionary learning and fine-tuning.

Results

The Results examine how output Fisher geometry is shared and learned from predictive laws, predicts language- and corpus-dependent structure, and supports lower-disturbance interventions that transfer across prompts. They also assess geometric corrections across model operations and how intervention gains depend on anisotropy while local predictions attenuate as intervention magnitude grows.

Predictive behaviour fixes a canonical, identifiable output geometry

  • Across 99 model–depth–objective cells, local Fisher geometry predicted realized output divergence at median ratio 1.000; behavior also identifies the resolved read-out subspace.The assigned language law recovered more than 99% of imposed squared geometric separation, while architecture changes within a law left geometry nearly unchanged; output geometry is invariant to behavior-preserving reparameterization, unlike activation geometry.Source: Figure 1 · p. 3, Predictive behaviour fixes a… · pp. 3–4

Independently trained models share the geometry

  • Model scale and predictive fit tracked human completion structure, and model-only calibration improved held-out human log score in six of seven families.On independent narratives, semantic distance fell with log parameter count (slope −0.00872); calibration gains were 0.0196–0.0507 nats, with Qwen2 at −0.0030 nats.Source: Independently trained models share the… · pp. 5–6
  • Across ten models, output-geometric rank agreement was 0.88, versus 0.62 mid-layer and 0.61 last-layer activation agreement.The shared consensus geometry supported semantic-probe transfer at 0.70 eight-way accuracy, near the within-model 0.72 and above the 0.125 chance level; cross-tokenizer agreement in a common byte outcome space was 0.91.Source: Independently trained models share the… · p. 5, Figure 2 · p. 7

The geometry inherits the statistics of language

  • Without fitted parameters, token-profile predictions reproduced held-out spectra and effective-dimension curves across five external model families.Effective-dimension RMSEs were 0.0270–0.0521, and spectral errors were 4.89–9.22-fold lower than a same-trace flat-spectrum control; probability-only profiles predicted effective dimension better, while weighted profiles predicted spectral exponents better in all seven families.Source: The geometry inherits the statistics… · pp. 6, 8, Figure 3 · p. 9

Corpus n-gram statistics predict acquisition, and evidence depth shifts its timing

  • Ex-ante corpus n-gram margins predicted held-out fact-acquisition trajectories, while randomized deeper evidence delayed persistent acquisition across tested architectures and constructions.Trajectory R2 was 0.775–0.792; in one controlled language, deep-evidence acquisition-time quantiles were about 4.3 times shallow-evidence quantiles, and a crossed study found delays of 0.81–1.51 log2 training-step units in every cell.Source: Corpus n-gram statistics predict… · p. 8, Figure 4 · p. 10

The geometry determines minimum-disturbance interventions and their relative cost

  • Fisher updates learned from donor prompts transferred positive objective changes to unseen prompts in all 24 settings while reducing reference-sequence KL.At matched donor changes, the reference metric reduced KL about 3–6 times versus Euclidean control and at least 2 times versus a donor-only Fisher metric; instruction-tuned natural-gradient frontiers also reduced off-target divergence across objectives.Source: The geometry determines… · p. 11, Figure 5 · p. 12
  • Across 3,515 matched-response measurements, the geometry’s preregistered finite-damping cost ratio predicted Euclidean-versus-Fisher disturbance with median measured-to-predicted ratio 0.967.Along finite intervention paths, Fisher updates accumulated about 11–127 times less local KL than Euclidean updates across 216 prompts, with positive paired intervals in every cell and target.Source: The geometry determines… · p. 11, Figure 5 · p. 12

One geometric correction spans model operations

  • One pullback-metric correction improved matched endpoints, including edits with 10.57-fold lower other-token divergence and Fisher feature-cost rankings correlated 0.997 with exact divergence.For sparse-autoencoder features, activation magnitude correlated 0.896; objective-effect-to-Fisher-cost predicted selected-token probability-change fraction at 0.935 versus 0.506 for attribution patching [41].Source: One geometric correction spans model… · p. 13
  • Across 11 models, every measured-to-predicted path decreases over three intervention fractions, with model-balanced geometric means 0.918, 0.860 and 0.755, quantifying attenuation as intervention magnitude grows.Supplementary Note 10 analyzes the finite-magnitude correction to cubic order.Source: One geometric correction spans model… · p. 13
  • The correction's advantage is largest at early-to-mid injection layers with the most anisotropic pulled-back metrics, shrinks toward near-linear read-out, and vanishes in the isotropic limit.Source: One geometric correction spans model… · p. 13
  • Output sensitivity, residual cross-model sharing and answer-associated probability motion concentrate in the same leading spectral modes; cyclic concepts trace nearly closed rotations, whereas analogy relations share a probability-displacement direction.Source: One geometric correction spans model… · p. 13
  • At 410M, code-Gram preconditioning raises Hungarian-matched mean absolute decoder–ground-truth cosine from 0.167 to 0.480 (+0.313), while natural-gradient LoRA yields 60–250-fold less off-target change than Adam.At 70M, it improves the preservation-KL frontier 1.57-fold relative to tuned AdamW with explicit KL regularisation; the code-Gram result used a planted-signal/background-RMS ratio β=0.5.Source: One geometric correction spans model… · p. 13, Table 1 · p. 14

Discussion · Methods

The Discussion considers how predictive behaviour, token statistics, and corpus evidence relate to shared output geometry, fact acquisition, and reusable calibrated control across the tested models and operations. The Methods describe protocols for forecasting and testing fact acquisition, computing and calibrating Fisher-based interventions, evaluating geometric methods across editing and learning tasks, and reporting statistical and computational procedures.

Discussion

  • Predictive behaviour identifies a canonical output geometry shared across architectures, and models with lower prediction error align more closely with human word choices.Agreement also increases with scale and training, and a calibration learned only from model laws improves predictions across the coarse distribution of human word choices.Source: Discussion · pp. 14, 16
  • Token statistics predict the output-Fisher spectrum and effective dimension; corpus n-gram evidence predicts fact acquisition, while randomized evidence depth delays it.The weighted read-out profile improves spectral-shape prediction across seven families, and pretraining corpus evidence predicts individual-fact acquisition with median absolute errors of 0.77–0.96 on the log2 training-step scale at all three sizes. The damped natural-gradient solve has a proved width-independent worst-case iteration bound; at c=10^-2 and Euclidean relative residual 10^-6, the exact-arithmetic bound is 85 iterations.Source: Discussion · p. 15
  • Geometry-based control reduces reference-sequence change about three- to sixfold versus Euclidean control at matched donor effects, while transferring updates to unseen prompts.Updates learned from four donor prompts transferred to eight unseen prompts per objective in four base models; the same geometry also improved the aggregate objective-change/off-target-change trade-off and persisted along relinearised finite paths.Source: Discussion · p. 15
  • Fisher-weighted feature sensitivity predicts intervention cost and selectivity, while output geometry also suggests uses for model auditing, factual editing, and behaviour-preserving post-training.The metric measures a feature’s functional consequence but does not explain what it represents; the held-out LoRA comparison is an initial local-preservation result, not evidence of safety-specific retention or broader capability preservation.Source: Discussion · pp. 15–16

Models, datasets and geometric definitions

  • The analyses use public checkpoints and controlled synthetic systems across transformer, recurrent, and state-space architectures, with interventions generally at a middle residual-stream layer.Experiments include Pythia, Qwen, GPT-2, GPT-Neo, Mistral, Mamba, RWKV, StarCoder2, BLOOM, and OLMo; feature-importance analyses use a Pythia-410M sparse autoencoder at layer 12.Source: Models, datasets and geometric… · p. 16

Behavioural identification and causal language assignment

  • Behavioural identification profiles held-out KL loss along candidate subspaces, while causal language-assignment experiments compare geometries learned under different laws and architectures.The controlled assignment uses three architectures at two capacities across eight test language pairs, with training schedules selected using separate pilot pairs.Source: Behavioural identification and causal… · p. 17

Cross-model geometry and human completions

  • Cross-model comparisons use rank agreement between Fisher–Rao distance matrices, semantic-anchor transfer probes, and human-completion comparisons across independent datasets.Human-alignment prediction fits a relation on Pythia alone and applies it to OLMo and external models without refitting; model-only calibration uses no human responses in fitting or model selection.Source: Cross-model geometry and human… · pp. 17–19

Spectral measurements and effective-dimension prediction

  • Spectrum prediction evaluates a prespecified weighted read-out profile on text-disjoint model-context data and scores spectral, exponent, and effective-dimension errors against flat-spectrum and probability-only comparisons.The mechanism comparison tests whether probability concentration explains effective dimension and whether weighted read-out information improves spectral-shape prediction across seven model families.Source: Spectral measurements and… · pp. 20–21

Acquisition forecasting and causal evidence depth

  • A model-size-specific n-gram predictor is fitted on a discovery battery and evaluated without refitting on two fresh fact batteries, alongside constant-time and label-permutation baselines.The held-out batteries contain 180 facts at Pythia 70M and 160M, and 300 facts at Pythia 410M; acquisition time is scored in log2 training-step units. Cumulative acquisition curves and observational composition use additional fixed batteries, seeds, checkpoints and held-out Wikitext loss, while causal effects are reserved for randomised experiments.Source: Acquisition forecasting and causal… · pp. 21–22
  • Randomised evidence-depth experiments test acquisition delays across depth rungs and across decoder architectures and controlled evidence constructions.The first uses paired arms with byte-identical initial weights and batch streams; the separate crossed study tests GPT-2-, GPT-NeoX- and Llama-style decoders on copy-back-reference and parity languages while matching key training conditions.Source: Acquisition forecasting and causal… · p. 21

Intervention computation, calibration and reusable control

  • Intervention computation applies Fisher operators matrix-free, with damped solves, resolved-eigenspace handling and matched-response calibration for method comparisons.Exact full-vocabulary products use Jacobian- and vector–Jacobian products with conjugate gradients; a top-token low-rank approximation uses a Woodbury solve. Damping scales with the largest generalised eigenvalue, and methods are compared at matched objective changes.Source: Intervention computation, calibration… · p. 22
  • Intervention evaluations quantify off-target disturbance and predicted relative cost, test finite paths, and assess donor-derived updates on unseen prompts without target gradients.Reusable control compares Euclidean, donor-Fisher and reference-Fisher directions, matching donor log-odds changes before applying updates to eight unseen prompts per objective. Finite-path comparisons recompute directions within a prompt-KL trust region; held-out cost prediction compares damped natural-gradient and Euclidean interventions at matched effects.Source: Intervention computation, calibration… · pp. 22–23

Editing, attribution, dictionary learning and fine-tuning

  • Knowledge-editing comparisons reuse Fisher and Euclidean displacements calibrated to the same target shift, while feature-ablation studies assess Fisher-predicted cost and held-out causal outcomes.The editing protocol covers 30 held-out CounterFact edits on GPT-2; the separate GPT-2-XL benchmark uses distinct access and amortisation conditions. Feature studies use discovery contexts for feature and objective selection and 18 held-out evaluation-context clusters for outcomes.Source: Editing, attribution, dictionary… · p. 24
  • Dictionary learning preconditions decoder updates in code geometry, while LoRA comparisons test exact natural-gradient updates against approximate-curvature, coarse-grained and tuned AdamW methods.The LoRA evaluation uses rank-4 adapters, matched target gains, eight outer seeds and 16 held-out preservation prompts; its primary endpoint is area under the preservation-KL frontier, with broader transfer claims restricted by the reported endpoints.Source: Editing, attribution, dictionary… · pp. 24–25
  • Cyclic-concept and relation analyses measure metric-isometry defects, closure, and probability-displacement alignment against a permuted-geometry control.For cyclic registries, the fitted advance-one-step map is a Fourier-1 phase rotation compared with the ideal 2π/n; relation alignment is evaluated across instances.Source: Editing, attribution, dictionary… · p. 25

Statistical analysis and reproducibility

  • Statistical intervals preserve experiment-specific pairing and replication units, while the paper records computational settings, model revisions, prompts and objective definitions.Most GPU-resident model-forward runs use an RTX 5000 Ada, with additional A100 and CPU analyses; full GPU pullback solves extend through 2.8B parameters, while 6.9B analyses use specified alternatives.Source: Statistical analysis and reproducibility · p. 25

Data availability

Public pretrained checkpoints and benchmark datasets used in the analyses are identified, while derived data are available from the corresponding author on reasonable request.

Data availability

  • Public pretrained checkpoints include Pythia, GPT-2, GPT-Neo, Qwen2, Qwen1.5, Mistral-7B, Mamba, RWKV, StarCoder2, BLOOM, OLMo, Sentence-BERT and DINOv2.The passage lists these checkpoints with bibliography markers 47, 50, 52, 48, 49, 53, 54, 55, 56, 51, 64, 31 and 32, respectively.Source: Data availability · p. 25
  • The reported analyses use WikiText, LAMBADA, SST-2, TruthfulQA, CounterFact, CIFAR-100 and human prediction datasets.The human prediction datasets are by Peelle et al.33, de Varda et al.34 and DERCo35.Source: Data availability · pp. 25–26
  • Derived data supporting the findings are available from the corresponding author on reasonable request.Source: Data availability · p. 26

Extended Data … Supplementary Note 2

The supplementary material establishes the Fisher–Rao metric’s statistical and coordinate properties, links it to cross-entropy training, and tests how evidence depth and behavior type relate to geometry. Extended Data also examines shared output structure, human-choice alignment, and acquisition across models.

Extended Data

  • Output-Fisher agreement stayed constant under hidden-coordinate reparameterisation while activation CKA, nearest-neighbour overlap and cosine RSA degraded; token-to-byte push-forward raised aggregate agreement from 0.899 to 0.910.Transporting the reference metric reproduced the native damped direction to at most 7.4 × 10−13 relative error; resetting it to identity produced a median error of 1.23.Source: Figure 1 · p. 32
  • On five previously unused stories, semantic distance declined with model scale (slope −0.00872) and training (changes −0.12080 and −0.10962); calibration helped six of seven families.For bounded effective dimension, held-out median error was 0.00493 versus 0.0814 for an equal-parameter unbroken power law, with every one of 80 curves below equality.Source: Figure 2 · p. 33, Figure 3 · p. 34
  • Deep evidence delayed persistent acquisition by 2.10 log2 training-step units (4.3×) in the transformer experiment; across architectures and evidence constructions, all 48 seed means were positive.The cross-architecture delays ranged from 0.810–1.514 log2 training-step units; fixed-treatment and label-randomisation control intervals contained zero.Source: Figure 4 · p. 35

Extended Data note: cyclic concepts and two geometries of behaviour

  • Across cyclic concepts, natural-gradient steps disturbed unrelated tokens less than Euclidean steps; analogy relations moved mass along a shared probability-displacement direction (0.49 versus 0.33 for the permuted control), unlike cyclic concepts.Cyclic concepts were near-isometries of the metric, with isometry defect 0.29 versus 1.02 for relations; ordinal concepts fell between the two classes.Source: Extended Data note: cyclic concepts… · pp. 40–41, Figure 10 · p. 40

Supplementary Note 1

  • Supplementary Note 1 supports the paper’s definition of the metric and its pullback instrument.It states the metric’s canonicity and the exact chart-covariance of the pullback solve.Source: The canonical output metric and its… · p. 43

1.1 Canonicity of the Fisher–Rao metric (Chentsov’s theorem)

  • Up to a positive multiplicative constant, Fisher–Rao is the unique metric on probability simplices invariant under congruent Markov embeddings.Such embeddings split outcomes into sub-outcomes with fixed conditional probabilities while preserving information and admitting a stochastic left inverse.Source: §1.1 · p. 43

1.2 The pullback metric and reparametrisation invariance

  • The pullback metric transforms tensorially, so transporting the reference under a chart change preserves the damped solve; replacing it with a fresh identity changes the problem.The intrinsic natural gradient is a unique class modulo the read-out kernel, whereas finite damping depends on the declared reference metric.Source: §1.2 · p. 44

Supplementary Note 2

  • At each affine-softmax checkpoint, the output Fisher is the conditional curvature of the cross-entropy-induced Gibbs law, paired with class-conditional hidden covariance as the opposite curvature.Excess predictive risks bound battery-averaged profile, response and fixed-read-out metric discrepancies at rate O(√ε_m + √ε_m′); the identity requires no particular optimiser.Source: Cross-entropy induces and determines… · pp. 44–46

Identification of the read-out geometry · Supplementary Note 3

Supplementary Note 3 establishes local and global identification of read-out geometry while showing that activation geometry is not determined by behaviour. In controlled synthetic systems, learned geometry follows the assigned language law.

Supplementary Note 3

  • The profiled Hessian's kernel is exactly the tangent space of M_d, with strictly positive curvature in every normal direction.Profiling context coordinates yields a Fisher Schur complement that supplies the normal curvature.Source: Identification of the read-out geometry · p. 47
  • The profiled risk is zero exactly when T⊂S: rank-matched read-outs identify T uniquely, while overcomplete read-outs identify it up to the containment fibre M_d.The global theorem also gives a quantitative lower bound on risk away from that zero set.Source: Identification of the read-out geometry · p. 48
  • The recovery corollary gives a finite-constant quantitative guarantee translating predictive error into read-out subspace error.Its constant is the near/far global margin from Theorem 5, not an additional loss from the final scalar step.Source: Identification of the read-out geometry · pp. 48–49
  • If the model core is within chordal distance B_m of T, mean capture is at least q_G(T) − B_m√r, and minimum capture at least q_min,G(T) − min{1, 2B_m}.These bounds transfer partition capture from the language core to a model core.Source: Identification of the read-out geometry · p. 49
  • The measured profiled-curvature margin is positive in every evaluated configuration.The supplementary note distinguishes these identification measurements from fractional-frame measurements of spectral resolution.Source: Identification of the read-out geometry · p. 49
  • Behaviour does not identify a unique Euclidean activation geometry: invertible hidden-layer reparameterisations preserve conditional laws but can change activation relational geometry.Output geometry, by contrast, is a functional of conditional laws.Source: Identification of the read-out geometry · p. 49

3.1 Causal language assignment

  • 0.997836 (95% language-pair bootstrap interval 0.997739–0.997933) is the mean assigned-law advantage; geometry change exceeds same-law cross-architecture change by 0.995678 (0.995485–0.995871).Both contrasts were positive in all eight independent pairs (exact one-sided sign P=1/256); test runs met the shared accuracy requirement ε<Δ2/32.Source: §3.1 · p. 50

Part II: Shared · Supplementary Note 4

This supplementary part examines the stability of output-based relational geometries under model disagreement and excess risk, while distinguishing them from hidden geometries. It also tests an exchangeable formula for predicting held-out behavior during training and checks an attenuation identity and its bounds, finding prediction accurate only after a training threshold and measured-distance variance floors vacuous.

4.1 Stability of the shared output geometry

  • The shared affinity geometry is determined by conditional output laws, and its entrywise change is bounded by the sum of the two contexts’ root-probability deviations.Those deviations are controlled by KL divergence and can also be bounded through a common reference law.Source: §4.1 · p. 51
  • If reference ties vanish, vanishing excess risk forces population rank agreement to one, with convergence rate determined by the reference margin distribution.The bound is 1 − τ ≤ 2F_q(t) + 32(R_m + R_m′)/t^2; when F_q(t) ≤ Lt^κ near zero, the rate is O((R_m + R_m′)^(κ/(κ+2))).Source: §4.1 · p. 52
  • Coarse-graining model outputs to a common outcome space extends the stability results across tokenizers, while agreement in native vocabularies remains empirical.For deterministic byte decoding, mapping tokens to the first newly decoded byte provides one such common outcome.Source: §4.1 · p. 53

4.2 The resolution law

  • The resolution law distinguishes exact threshold-count agreement under a margin band from smooth-response error bounds; aggregate RMS blur alone does not guarantee uniform contextwise identity.A valid stability claim must specify the observable, its Lipschitz constant, decision margin, and whether error is pointwise or aggregate.Source: §4.2 · p. 54
  • In trained-model pairs, realised affinity perturbations were only 2–3% of worst case, consistent with low coherence between model disagreement and relational directions.Well-trained pairs had median coherence 0.024–0.034; the weakest 70M-versus-6.9B pair had coherence 0.085 and lower agreement.Source: §4.2 · p. 54
  • As the weaker Pythia model improved from 70M to 1.4B, its median KL to 6.9B fell from 1.07 to 0.23 while Spearman agreement rose from 0.78 to 0.93.After first-byte coarse-graining, different-tokenizer agreement was 0.910 (80%-subsample stability interval 0.896–0.922).Source: §4.2 · p. 55

4.3 The attenuation identity and floors

  • A six-scalar identity expresses output-geometry agreement through shared signal, each model’s error variance and covariance with the signal, and cross-model error covariance; excess-risk bounds yield conditional attenuation floors.For conditionally independent, identically distributed runs measured against an external reference, run-averaged cross covariance is nonnegative, though an individual pair may be negative.Source: §4.3 · p. 56
  • For hidden-layer geometries, no positive agreement floor follows from identical output behaviour: an explicit integer construction at identical outputs attains Pearson agreement −0.2053387… and Spearman agreement exactly −4/17.Source: §4.3 · p. 57
  • The identity reconstructs all measured agreements, theorem identities and inequalities hold in 40 evaluations, and all 60 cross-covariances are nonnegative, but measured-distance variance floors are vacuous.Realised fluctuation variances are near 0.005 against bounds near 1.3.Source: §4.3 · p. 57
  • From step 256, exchangeable-estimator median absolute prediction error is 0.0036–0.014 (pooled defined median 0.00938; substream 95th percentile 0.01437), but earlier predictions are inaccurate or undefined.The estimator is undefined in 24 of 120 full-sample cells; median error is 0.25–0.31 at steps 1–16 and 0.056 at step 64.Source: §4.3 · pp. 57–58
  • Independent model deviations show Proposition 2’s signature: +0.57 and −0.51 on the cross-architecture trio, and +0.487 and −0.487 across PolyPythias seeds.Source: §4.3 · p. 58

Shared geometry carries meaning and transfers · Supplementary Note 5

The shared Fisher–Rao geometry transfers across models and aligns with external semantic structure, while model-specific residuals transfer poorly. It also extends to visual concept geometry and concentrates in the spectral bands carrying steering weight.

Supplementary Note 5

  • Residual alignment with Sentence-BERT was −0.001, +0.003, and +0.051 on natural, templated, and factual-relation batteries, versus shared-component alignment of 0.044, 0.102, and 0.440.Source: Shared geometry carries meaning and… · p. 58
  • +0.231 (95% interval +0.159 to +0.316) of shared-component alignment remained on the factual-relation battery after partialling out surface form.The shared component also contains structure beyond its alignment with the external meaning reference.Source: Shared geometry carries meaning and… · p. 58
  • Eight-way accuracy was 0.724 within model, 0.659 across models, and 0.700 using consensus alone, against chance 0.125.Transfer was tested by training a multinomial probe on Fisher–Rao distances to 40 fixed anchors in one model and evaluating it on another.Source: Shared geometry carries meaning and… · p. 58
  • The residual supported 0.407 within its own model but only 0.091 across models, whereas the shared component transferred across models.Source: Shared geometry carries meaning and… · p. 58
  • Across 100 concrete concepts, last-token, last-hidden-state activation geometry from Pythia-1.4B and GPT-2 Large agreed with vision-only DINOv2 geometry at 0.385 (permutation P< 0.001).Source: Shared geometry carries meaning and… · p. 58
  • 84% of the activation-to-vision association remained after controlling textual co-occurrence, while the first five spectral modes carried about half of steering weight and the first 40 about 78%.Sharing and answer-relevant motion decayed over the same spectral bands on templated and natural batteries, linking transferable geometry to minimal-disturbance control.Source: Shared geometry carries meaning and… · p. 58

Part III: Learned · Supplementary Note 6

This part examines how token probabilities and read-out weighting shape output Fisher spectra and effective dimension, including conditional profile-based predictions, finite-width constraints, and singleton-head/cluster-bulk structure. It tests these relationships across external model families and contexts, establishing an exact interchangeability result while finding no universal fixed partition.

6.1 Spectral constraints and effective dimension

  • Conditional spectral bounds relate eigenvalues to the weighted token profile through measurable read-out constants, while finite-width majorization constrains profile mass without locating individual crossover ranks.The bounds require no randomness or independence assumption; the majorization result applies when the vocabulary has at least as many outcomes as the read-out width.Source: §6.1 · pp. 58, 61
  • For a power-law spectrum with exponent s > 1, effective dimension scales with response exponent 1/s; at s = 1 it includes a finite-width logarithmic factor.At fixed width, the exact response tends to rank(H) as damping α approaches zero.Source: §6.1 · p. 62
  • Across 56 evaluated curves, the spectral-to-profile ratio had per-curve median 0.87 and median extremes [0.69, 1.13], with measured exponent inheritance of 0.04 median.Theorem 12 had zero violations across included curves, grid values, and ranks; the stated fractional-frame certificate also held in all 56 curves in both read-out representations.Source: §6.1 · p. 64
  • The profile-only crossover statistic predicts measured crossover rank, with median ratio 0.93 and Spearman correlation 0.48 across 56 curves.It is a mean-field predictor, not a consequence of the separate conditional theorems; the reported width examples show imperfect agreement at some dimensions.Source: §6.1 · pp. 62, 64
  • On the held-out battery, profile predictions reproduced spectral profiles and effective-dimension curves and outperformed three matched alternatives in every external family.Across 1,664 cells, the stated consistency checks had zero violations; exact certificates were available in 98.4–100% of cells in each family where an admissible truncation existed.Source: §6.1 · pp. 64–65, Table 2 · p. 64
  • The measured spectrum determined effective dimension with median held-out error 0.00245, versus 0.00473 for the fitted bounded law; the fitted crossover matched the half-ceiling resolution (Spearman 1.000, median ratio 1.06).The analysis included 72 curves from nine models; eight RWKV curves were excluded. The dense core exponent was 1.155, confirming the previously measured 1.14, and the two-parameter Hill curve accurately interpolated the spectrum.Source: §6.1 · p. 65
  • Tail steepening survived a random read-out with the real profile (+0.47), disappeared with a pure power-law profile and random read-out (−0.07), and the real read-out added a secondary component (+0.27).The crossover-rank log–log slope was 1.1–1.3 across every cell: width set the band-edge position, while high-rank profile depletion shaped the tail.Source: §6.1 · pp. 65–66

6.2 Separating probability concentration from read-out structure

  • Mean log-error ratios were 0.51068 for effective dimension, favoring the probability-only profile, and −0.37863 for spectral shape, favoring the weighted profile (P=1/128 for each contrast).The comparison used a common 27,450-outcome representation across 14 models and fresh contexts; it complements the native-vocabulary external-family prediction in main Figure 3a,b.Source: §6.2 · p. 66

6.3 Singleton tokens, interchangeable clusters and multiscale inheritance

  • Maximizing partition capture is weighted K-means, and when qmin(L)=1, resolved observables are invariant to within-cell redistribution, although the full output distribution may still distinguish tokens.The interchangeability statement concerns resolved geometry, not the full output distribution.Source: §6.3 · pp. 66–67
  • Weighted K-means captured mean 0.718 and worst-direction 0.252; 26 of 128 cells had fewer than eight tokens, and singleton cells held probability mass 0.3315.Every cell with fewer than eight tokens contained a token in the top 158 by reference probability; specificity, reference-swap, and outer-half replications recovered the same structure. An exact decomposition separates captured information from unresolved residuals, with the exceptional singleton contribution bounded by the singleton budget.Source: §6.3 · p. 67
  • The proposed within-cell attenuation failed in all six model-halves, while cross-model bulk agreement was 1.0 versus a matched-null median of 0.979 and remained descriptive evidence.Actual-Fisher inheritance recovered all 176 theorem certificates and the primary inheritance effect, but stronger half-sample directionality estimates did not support a directional effect in any arm; the results do not establish a universal fixed partition across contexts.Source: §6.3 · p. 67

Acquisition time on a logarithmic scale · Supplementary Note 7

The section distinguishes observational predictions from randomized tests: corpus n-gram statistics predict acquisition before training, while deeper evidence causally delays persistent commitment across architectures. Held-out loss also aligns acquisition trajectories across model sizes more accurately than training step.

Supplementary Note 7

  • Logarithmic acquisition time tracks the first checkpoint after which a fact’s signed answer margin remains above a fixed threshold set before model evaluation.The section distinguishes statistical schedules measured on pretrained-model checkpoint ladders from causal depth interventions.Source: Acquisition time on a logarithmic scale · p. 67

7.1 Corpus 𝑛-gram statistics predict acquisition before training

  • Without refitting on fresh facts with zero prefix overlap, corpus n-gram margins predicted acquisition trajectories across Pythia-70M, Pythia-160M and Pythia-410M.The predictor explained 0.775–0.792 of margin-trajectory variance, achieved acquisition-status agreement of 0.792–0.854, and predicted acquisition time with error 0.77–0.96 in τ. Baseline errors were 1.04, 2.20 and 0.98 for constant acquisition time, and 2.02, 2.56 and 2.34 for label permutation; at final checkpoints, trigram-to-unigram weight ratios ranged from 2.7 to 4.9.Source: §7.1 · p. 68

7.2 Evidence depth causes a multiplicative delay

  • Deepest evidence shifted lower-tail acquisition by Δ = 2.103 in logarithmic time (about 4.3× in steps); the R4 − R2 difference was 1.365 [1.14, 1.59] in τ.Random reassignment of the same facts across depth arms, with shared initial weights and batch streams, identifies evidence depth as the cause. The shift varies by only 0.167 in τ across analysed quantiles.Source: §7.2 · p. 68

7.3 Generalisation across architectures and evidence constructions

  • Across GPT-2, GPT-NeoX and Llama, all six depth-delay intervals and 48 seed means were positive; parity exceeded copy-back-reference by 0.414 [0.291, 0.536] in τ on average.For copy-back-reference and four-cue parity, paired differences were 1.017 and 1.205 in GPT-2, 0.996 and 1.345 in GPT-NeoX, and 0.810 and 1.514 in Llama. The twelve fixed-treatment and randomised-label control intervals contained zero.Source: §7.3 · p. 68

7.4 Held-out loss aligns cumulative acquisition curves across scale

  • Held-out loss aligned 70M-to-160M acquisition trajectories about three times more accurately than training step, with gaps of 0.130 versus 0.413.This is a descriptive comparison of real-checkpoint results; per-fact additive fits reached R2 of 0.72–0.83 with a measured n-gram-margin-by-depth interaction.Source: §7.4 · p. 69

Supplementary Note 8

Supplementary Note 8 defines exact accounting for a fixed weighted margin projection and derives conditional relations between its trajectory and acquisition time. It distinguishes what follows from the accounting identities from claims requiring additional persistence, rate, or scaling assumptions.

Supplementary Note 8

  • The estimand fixes the fact population, weights, and response-independent evidence coordinates, measuring the trajectory of the declared weighted margin projection.This is a model-relative accounting framework for a fixed weighted margin projection.Source: Projected-margin dynamics and… · p. 69

8.1 Exact continuous and finite-step accounting

  • Under exact endpoint accounting, the weighted projection’s finite-step change telescopes without smoothness; tangent-based accounting is exact only when its remainder is retained.For continuous trajectories, the rate identity holds almost everywhere under the stated absolute-continuity and differentiability assumptions. Checkpoint-dependent evidence coordinates add a product-rule term and can create apparent movement even when margins do not move.Source: §8.1 · pp. 69–70

8.2 Signed rate in a positive-semidefinite geometry

  • The signed projected-margin rate factorises in a declared positive-semidefinite geometry, with its sign indicating whether the update increases or decreases the projection.For singular metrics, the solve certificates require the projection and update covectors to lie in the metric’s range; if either squared norm is zero, the rate is zero and normalised alignment is undefined.Source: §8.2 · p. 71

8.3 Conditional first-passage and logarithmic-time relations

  • Cumulative-change dominance orders ordinary first-passage times, but does not order persistent acquisition times without an additional persistence argument.Sparse local rate measurements alone do not establish cumulative-change dominance.Source: §8.3 · p. 71
  • For positive quantiles, full population-level scaling gives a constant logarithmic-time translation, but a shift over one quantile band does not establish full distributional scaling.The result concerns population-level scaling and need not be a same-fact multiplier.Source: §8.3 · pp. 71–72
  • Nominal evidence depth alone cannot order acquisition time across the positive-rate cumulative-change model without assumptions on learner-relative effective rates or an equivalent accessibility object.The theorem allows opposite acquisition orders for the same shallow/deep evidence pair under different positive rate profiles.Source: §8.3 · p. 72

Supplementary Note 9

Supplementary Note 9 gives bounds linking probability changes to discrete decisions, learning rates, and read-out rotation, and tests geometric descriptions of cyclic concepts and relations. Its experiments show that margin, objective–update alignment, language residuals, and probability-displacement alignment capture distinct aspects of these behaviors.

Discrete decision

  • On held-out LAMBADA, flip frequency falls from 0.519 to zero as normalized margin z=m/(2ρ) crosses one, with zero flips forced by the theorem.The continuous bound had no violations in 5,704 implementation cases; the discrete margin theorem had zero violations across 45,315 observed cross-seed flips, or 90,630 oriented checks.Source: Geometric motion, margins and local… · p. 72

Objective rate

  • Cauchy–Schwarz gives objective rate^2/sensitivity ≤ update norm2, while held-out Pythia-70M examples show mean alignment squared 0.6811 (95% interval 0.6503–0.7117).Across 80 examples, the correlation between log sensitivity and log squared objective rate was 0.541; equal sensitivity and squared update norm can still yield objective rates of one or zero.Source: Geometric motion, margins and local… · pp. 72–73

Read-out subspace kinematics

  • Only the read-out velocity’s normal component rotates its subspace; updates confined to the current column span, including scalar decoupled weight decay, produce no rotation.The proposition identifies the normal component as B=(I−P)Q̇.Source: Geometric motion, margins and local… · p. 73

Language residual and read-out rotation

  • The same approximation residual governing static recovery bounds how unresolved language rotates the read-out subspace, and scalar weight decay contributes exactly zero.A controlled smooth-path experiment supported the projector identity, scalar-decay null, and bound in all 24 held-out seeds.Source: Geometric motion, margins and local… · pp. 73–74

Cyclic concepts and relations

  • For cyclic concepts, the fitted weekday rotation angle is 0.902 versus the ideal 2π/7 = 0.898, with metric-isometry defect 0.29.The rotation is fitted in the Fourier-1 phase representation; closure defect is defined as the Fisher–Rao distance from the start point to the composition of n steps.Source: Geometric motion, margins and local… · p. 74
  • Relation probability-displacement alignment is 0.49 versus near zero for cyclic concepts, compared with 0.33 under a permuted-geometry control preserving marginals.The reported metric-isometry defect is 1.02 for relations and 0.29 for cyclic concepts.Source: Geometric motion, margins and local… · p. 74

Part IV: Controllable · Supplementary Note 10

The supplement develops minimum-disturbance, coordinate-aware control and tests its cost predictions, finite interventions, and reuse on unseen prompts. It also examines preference optimisation, interpretable Fisher coarse-graining, and predictors of intervention erosion.

Supplementary Note 10

  • The minimum-cost feasible intervention is the natural step under A, and the local off-target decomposition measures excess disturbance from that step; increasing damping removes anisotropy and the natural-gradient advantage.For A = G + αR, the decomposition applies on the resolved range of positive-semidefinite G when q lies in its range. With Aα = B + αI, the Euclidean-to-natural cost ratio and its worst-case ceiling are nonincreasing in α and converge to one.Source: Minimum-disturbance control and… · pp. 74–75
  • Across 3,515 usable held-out responses, measured-to-predicted advantage has median 0.967 and log–log slope 0.982, while prediction attenuates as intervention magnitude grows.Coverage is 98.6%; at intervention fractions ΔC/C = 0.01, 0.03, and 0.10, model-balanced geometric mean ratios are 0.918, 0.860, and 0.755, respectively. All 11 within-model paths decrease at both adjacent steps. The predictor compares Euclidean and damped natural-gradient G-costs at fixed A = G + αI; the ratios generally differ at non-zero damping.Source: Minimum-disturbance control and… · pp. 75–76
  • Under invertible chart changes, objective change, cost, and the natural step are invariant, but coordinate-Euclidean control generally depends on the chosen chart.Covariance of a damped metric requires transforming the reference metric too; resetting it to a new-chart identity corresponds to a native reference of S⊤S.Source: Minimum-disturbance control and… · p. 76

10.1 Relation to direct preference optimisation

  • The inference-time activation-space natural-gradient update for preference objectives uses the activation pullback metric and preference covector, unlike DPO’s parameter update.The update is δh = (G + αI)^−1∇h[log p(yw | x) − log p(yl | x)].Source: §10.1 · p. 76

10.2 Mass-preserving Fisher coarse-graining and interpretable partitions

  • Fisher coarse-graining preserves tilt information when token values are constant within cells, but reproducing the full steering update also requires retaining read-out covariance.For sentiment, interpretable cells correspond approximately to sentiment level crossed with lexical role; at matched update fidelity, tested token groupings require several times more cells than fixed-tolerance Krylov iterations.Source: §10.2 · p. 76

10.3 The full cubic correction and the erosion predictor

  • In the full-curvature 410M analysis, adding the network-Hessian term reduces mean absolute error from 0.345 to 0.0176, while erosion prediction is regime-dependent.Across 410M–2.8B, the first cubic component’s rank correlation with gentle-to-aggressive erosion is +0.49 to +0.75 in the moderate-intervention regime, and the association reverses beyond it. The network-Hessian term vanishes at affine read-out; affine anchors return it exactly zero.Source: §10.3 · p. 77

10.4 Relinearised finite intervention paths

  • Across 216 prompts and 864 prompt–target pairs, Euclidean/Fisher path-cost ratios have cell–target means of 2.4403–4.8452 and geometric means of 11.48–127.12.All 72 paired 95% prompt-bootstrap intervals are positive in log space; the paths use relinearisation after each accepted step and matched objective checkpoints.Source: §10.4 · p. 77

10.5 Reusable interventions with a reference metric

  • Across 24 settings, the Euclidean/reference-Fisher cost ratio is 2.8986–6.2882, while reference-Fisher updates also transfer to unseen prompts and reduce reference-sequence KL versus Euclidean composition.All six model-bootstrap intervals for the Euclidean/reference-Fisher log ratio are positive; in a two-objective solve, both mean target effects are positive in all four models. The reusable update is constructed within a model and does not require aligned hidden coordinates between models.Source: §10.5 · p. 78

Matrix-free computation · Supplementary Note 11

Matrix-free operator methods make intervention solves practical without materialising large matrices. With relative damping, conjugate gradients have a worst-case iteration bound independent of model width.

Supplementary Note 11

  • Intervention solves avoid materialising H, J, and G=J^T HJ, applying operators to vectors; a general reference R uses whitening or generalised preconditioning.Source: Matrix-free computation · p. 78
  • At c=10^-2, the standard bound gives at most 85 iterations for Euclidean relative residual 10^-6; fixed relative damping and tolerance make the worst-case bound independent of width.The regularised condition number is at most (1 + c)/c; the Euclidean relative residual bound incurs at most a factor of √κ.Source: Matrix-free computation · p. 78
  • For the top-|S| output tokens by probability mass, truncated-support Fisher factorization enables |S| Jacobian-vector products and an |S| × |S| Woodbury solve.The probabilities are renormalised before forming the truncated-support approximation.Source: Matrix-free computation · p. 78
  • For the full vocabulary, each Gv product uses a JVP, a closed-form output-Fisher product, then a VJP; conjugate gradients solve the damped native-reference system.The implementation uses forward-mode JVP and reverse-mode VJP automatic-differentiation primitives.Source: Matrix-free computation · p. 78
  • Lanczos range restriction projects onto the numerically retained eigenspace, so the intervention step has no null-space, output-inert component.Source: Matrix-free computation · p. 79
  • Because G is severely ill-conditioned (κ∼10^5–10^7), solves use float32; bfloat16 and float32 recovered directions had absolute cosines of 0.056–0.542.Additive F-vector products are accumulated in example chunks, keeping only one example's autograd graph in memory at a time.Source: Matrix-free computation · p. 79
  • At evaluated widths d≤1024, dense operators are faster for spectral computation, while matrix-free additional operator storage scales linearly with activation dimension rather than quadratically.Matrix-free storage omits a vocabulary-by-activation Jacobian, although output evaluation still scales with vocabulary size.Source: Matrix-free computation · p. 79

Part V: Evidence, translation and reproducibility … Supplementary Note 19

Across the supplementary analyses, Fisher geometry predicts behavioral and intervention costs, and its output-space structure is more stable across models than coordinate-dependent activation geometry. Human-completion analyses further show that semantic alignment tracks predictive fit and training, while model-only calibration improves held-out human log scores in most tested families.

Knowledge-editing details

  • Under a shared gating rule and fixed-edit protocol, the Euclidean-to-Fisher ratio of other-token KL was 10.57 (95% interval 8.52–13.37) across 30 held-out edits.Both operators achieved the fixed +5 log-odds shift. The separate routed EasyEdit benchmark reports S=100.0 for the Fisher system, but its routing and access conditions differ from those of persisted editors.Source: Knowledge-editing details · pp. 80–81, Table 3 · p. 81

Supplementary Note 14; 12.2 Matched-effect comparisons

  • At equal aggregate behaviour change, natural-gradient steering reduced off-target KL by factors of 8–74 for sycophancy, 30–190 for truthfulness and 4–550 for style.Contrastive activation addition often did not reach the target; the same untuned preference covector was used across objectives, and results replicated on Qwen1.5-1.8B-Chat. Matched-response comparisons use equal realised objective change, while instruction-model frontiers match interpolated aggregate change.Source: §12.2 · p. 80, The instruct-model programme · pp. 81–82, Figure 2 · p. 82

Supplementary Note 15

  • Across 303 feature ablations, Fisher-predicted cost correlated 0.997 with measured KL, versus 0.896 for activation magnitude.The objective-effect-to-Fisher-cost ratio also correlated more strongly than attribution-patching magnitude with selected-token probability-change fraction (0.935 versus 0.506) and sign agreement (0.746 versus 0.354), but not absolute signed effect.Source: Sparse-autoencoder feature ablations · p. 82

Supplementary Note 16

  • In the evaluated 410M configuration, natural-gradient preconditioning raised mean matched cosine from 0.167 to 0.480 (+0.313), with reconstruction variance explained rising from 0.535 to 0.622.The fraction of features activated at least once rose from 21.8% to 62.3%; all three paired seeds improved in matched cosine.Source: Dictionary-learning details · pp. 82–83

Cross-model convergence: protocols and controls

  • On templated contexts, output-distribution geometry agreed more strongly across models than mid-layer activations across three measures: 0.778 versus 0.710, 0.975 versus 0.935, and 0.935 versus 0.755.Output agreement remained constant under independent invertible transformations of hidden states, while activation measures degraded. After calibration, Fisher–Rao agreement retained 99% of its score (0.879 to 0.869), whereas mid-layer centred-kernel alignment retained 74%.Source: Cross-model convergence: protocols and… · pp. 83–84

The displacement-magnitude-matched null and the anisotropy analysis; 12.1 Experimental design and statistical controls

  • After matching each word’s displacement magnitude, residual top-six cross-family eigenvector overlap was +0.063 (95% interval +0.056 to +0.070), about 9% of raw overlap.Objective-level Fisher-cost–sharing and objective-sensitivity–sharing associations survived adjusted tests, but direct item-level cost–meaning association did not. Experimental controls fixed evaluation batteries and resampling units in advance and used nulls that preserved relevant nuisance structure.Source: §12.1 · p. 79, The displacement-magnitude-matched… · p. 84

Supplementary Note 19; 19.1 Native-sample human semantic geometry; 19.2 Predictive fit and developmental human alignment

  • Semantic alignment of consensus geometry rose when next tokens directly encoded factual answers, while consensus-relative residuals had negligible alignment on natural and templated batteries.Native-sample model–human semantic distance decreased with scale (slope −0.011809), and a predictive-risk relation applied without refitting across other checkpoints and model families yielded correlation 0.758320 and RMSE 0.029465. OLMo alignment increased across checkpoint order; the Pythia trajectory slope was not resolved.Source: Meaning decomposition and transfer… · p. 85, §19.1 · p. 85, §19.2 · p. 86

19.3 Leading directions in an independent human probability geometry; 19.4 Independent human-completion replication and model-only calibration

  • Across seven families, directional lower bounds were positive for six (0.31937–0.41116) and −0.00120 for BLOOM; independent DERCo data showed Pythia’s equal-story log-size distance slope was −0.008725 (95% interval −0.012022 to −0.005556).Pythia distance nevertheless rose from 0.20084 at 1.4B to 0.20335 at 2.8B, so the negative slope does not imply a monotonic decline. Directional recovery establishes shared leading structure, not full-law recovery.Source: §19.3 · p. 86, §19.4 · p. 86

Supplementary Note 20 · Reproducibility map · Supplementary References

The reproducibility map organizes evidence for the paper’s central claim chain; the accompanying details specify model sources, execution conditions, scale limits, and reproducibility materials.

Reproducibility map

  • Supplementary Table 4 maps each strand of the central claim chain to its evidence and role in the manuscript.The supplied passage describes the table’s scope, not its individual entries.Source: Table 4 · p. 88
  • Pretrained experiments use public checkpoints spanning Pythia, GPT-2, Qwen2, Qwen1.5, GPT-Neo, Mistral, Mamba, RWKV, BLOOM and StarCoder2.The main-text randomized causal acquisition experiments instead use explicitly controlled synthetic systems.Source: Reproducibility map · p. 88
  • Model execution and intervention solves use float32, except analyses specifying float64 dense spectral arithmetic.Most GPU-resident model-forward runs use one 32-GB NVIDIA RTX 5000 Ada; additional runs use NVIDIA A100 GPUs, with CPU execution for dense statistical analyses.Source: Reproducibility map · p. 88
  • A full GPU-resident pullback solve fits through 2.8 billion parameters; at 6.9 billion, evidence is limited to spectra, CPU calibration and descriptive CPU-offload analysis.The 6.9-billion-parameter results comprise graph-free output-Fisher spectra, CPU-only mid-layer full-vocabulary pullback calibration, and descriptive late-layer CPU-offload iteration analysis.Source: Reproducibility map · p. 88
  • Given a random seed, conjugate-gradient and Woodbury solvers are deterministic.Seeds are logged for bootstrap resampling, dictionary learning and Monte Carlo rollout values.Source: Reproducibility map · p. 88
  • Supporting materials, including model revisions, prompt, pair and objective-set listings, analysis code and summary logs, are available from the corresponding author on reasonable request.The passage says code and data availability are stated in the main text.Source: Reproducibility map · p. 88

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