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Beyond the Readout: A Latent Numerical Frame Shared Across Modality and Operation

Oriol Juan Sabater, Alex Ferrando de las Morenas, Pau Rodríguez, Xavier Suau

arXiv:2610.37422v1cs.AI

TL;DR

The paper asks whether text and image arithmetic share an internal concept of number, a question passive similarity metrics cannot resolve. It tests this with causal subspaces and interventions, finding shared numerical geometry in distinct coordinates and a latent signal that redirects later digit generation.

  • Problem

    Similarity metrics cannot distinguish shared coordinates from rotated representations or rule out similarity caused by a shared output-token interface.

    Source: §1 · p. 2

  • Method

    The authors use causal interventions on answer-controlling subspaces across text and image arithmetic, then separate digit-readout directions from latent directions.

    Source: §1 · p. 2, §4 · p. 4

  • Results

    91.4% recovery with scaled rotations versus 53.2% with direct reuse shows shared geometry in different coordinates, while latent swaps redirect the subsequent digit in 77.6% of cases.

    Source: §4 · p. 5, §6 · p. 9, §7 · p. 10

  • Takeaways & Limitations

    The findings support an upstream numerical identity outside the digit readout that is unfolded sequentially across autoregressive steps.

    Source: §7 · p. 10

    Experiments cover addition and subtraction of integers in [0, 99]; multiplication uses distributed, depth-dependent representations, and the specific attention heads for delayed readout remain unidentified.

    Source: §7 · p. 10

Abstract

from arXiv · show

When a multimodal model computes 17+35 from text or 91-39 from a rendered image, it produces the same answer, 52. Does it use one internal concept of number for both, or separate machinery for each path? Similarity metrics such as RSA or CKA cannot settle this question. They give the same score to identical and to rotated directions, and two representations can look alike only because both prepare the same output digits. We therefore use causal interventions. For four conditions (text or image, addition or subtraction), we learn a small subspace of Gemma4-12B-it's residual stream in which swapping coordinates swaps the model's answer. The subspaces answer three questions. First, do the conditions write numbers along the same directions? They do not. Reusing a subspace from text in an image on the same operation recovers only 53.2% of its causal effect. A single rotation and rescaling maps one subspace onto the other and raises recovery to 91.4%, so the conditions share one geometry expressed in different coordinates. Second, does this shared geometry only reflect the digit the model is about to print? To test this, we split each subspace into 9 directions that the output layer reads to choose the next digit and 13 directions that it cannot see. The 13 hidden directions still carry the number, yet changing them barely alters the digit being written. Third, what are these hidden directions for? A model must emit a multi-digit answer one digit at a time, so it can compute the full result early and keep it in reserve. When two numbers share their first digit, swapping the number in the hidden directions leaves the digit being written intact and redirects the next digit in 77.6% of cases. The model thus forms an abstract numerical concept upstream and unfolds it across time: multimodal representations share meaning through relational geometry across distinct coordinate frames.

1 INTRODUCTION

The introduction asks whether different routes to the same number share an internal representation, and argues that passive similarity cannot distinguish shared coordinates, rotated geometry, or an output-token artifact. It introduces causal interventions to test these possibilities across text and image arithmetic.

1 INTRODUCTION

  • Similarity metrics cannot distinguish identical coordinates from rotated ones or rule out apparent similarity caused by a shared output token; causal interventions test whether representations share causal function.The paper also notes that representational agreement need not imply shared causal function.Source: §1 · p. 2
  • The study crosses text and image inputs with addition and subtraction, learning a 22-dimensional causal subspace in each condition to control answer generation.The four conditions can reach the same numerical result by different input and operation routes.Source: §1 · p. 2
  • Direct cross-modal reuse recovers 53.2% of the intervention effect, while scaled rotations raise same-operation recovery to 91.4%, supporting shared geometry in different coordinates.The introduction presents the coordinate-identity hypothesis as false and relational isomorphism as supported.Source: §1 · p. 2, Figure 1 · p. 2
  • The 13-dimensional latent subspace preserves numerical geometry beyond the 9-dimensional digit readout, and swapping its number redirects the subsequent digit in 77.6% of cases.The abstract number is formed upstream and read out sequentially across generation steps.Source: Figure 1 · p. 2, §1 · p. 3

2 RELATED WORK

Prior work studies arithmetic representations, causal methods for identifying computation, and theories of cross-modal representational convergence. The paper positions its approach as testing causal function rather than relying on representational similarity alone.

2 RELATED WORK

3 EXPERIMENTAL SETUP AND CAUSAL SUBSPACES

The experiments compare text and image arithmetic in a shared residual stream and use causal interventions to identify answer-controlling subspaces. The analysis focuses on correctly answered expressions and chooses a compact subspace dimension that retains most of the baseline causal effect.

3 EXPERIMENTAL SETUP AND CAUSAL SUBSPACES

  • The four conditions pair text or image inputs with addition or subtraction, using the same expressions across modalities and restricting comparisons to expressions answered correctly in both.Operands are integers from [0, 99], with valid addition and subtraction domains, and 5,050 expressions per operation.Source: §3 · p. 3
  • At layer 43 immediately before generation, Distributed Alignment Search learns a 22-dimensional subspace whose coordinate swaps replace the base answer with a donor answer.The model weights remain frozen while the subspace is optimized.Source: §3 · pp. 3–4
  • The 22-dimensional subspace is the smallest tested dimension retaining at least 90% of baseline autoregressive intervention accuracy across all four conditions.The baseline used for dimension selection is the 32-dimensional subspace with the highest preliminary intervention accuracy.Source: §3 · p. 4
  • Fourier-aligned directions alone recover only 23% of the full autoregressive intervention effect in Gemma4-12B-it, compared with 84% in Llama3.1-8B.The result motivates analyzing the complete learned causal subspaces rather than only Fourier features.Source: §3 · p. 4
  • The full 22-dimensional causal subspaces achieve 69.3%–86.2% autoregressive accuracy, identifying directions that mediate arithmetic answers.Source: §3 · p. 4

4 NUMERICAL CHANGE TRANSPORTS ACROSS MODALITY AND OPERATION

Direct reuse of causal subspaces transfers incompletely across modalities, but scaled rotations restore much of the causal control. The fitted maps also compose across modality and operation, while transfer scales indicate smaller visual displacements without establishing their architectural cause.

4 NUMERICAL CHANGE TRANSPORTS ACROSS MODALITY AND OPERATION

  • 77.3% mean direct reuse across operations within one modality contrasts with 45.5% mean direct reuse across all cross-modal condition pairs.Cross-modal reuse is directional: image-trained subspaces recover 57.7% in text, while text-trained subspaces recover 33.4% in images.Source: §4 · pp. 4–5
  • 91.4% recovery with scaled rotations exceeds 53.2% direct reuse for same-operation cross-modal transfer, showing that causal coordinates differ but numerical changes align geometrically.An orthogonal rotation alone gives 0.718 normalized recovery; adding a uniform scale raises mean recovery by 0.196 to 0.914.Source: Figure 2 · p. 5, §4 · p. 5
  • Text-to-image displacement scales of 0.639 ± 0.006 and 0.658 ± 0.011 contrast with image-to-text scales of 1.283 ± 0.019 and 1.271 ± 0.021.The fitted scale magnitude follows transfer direction rather than arithmetic operation; the scales alone do not establish an architectural cause.Source: §4 · p. 5
  • 0.799 recovery from either two-step transport chain is close to the 0.807 mean recovery of directly fitted maps when both modality and operation change.The chained displacement predictions have mean cosine similarities of 0.968 and 0.969 to directly fitted maps.Source: Figure 2 · p. 5, §4 · p. 6
  • The results support scaled-rotation isomorphism for numerical changes, while leaving open whether the shared geometry represents abstract numbers or the next digit being prepared.Source: §4 · p. 6

5 BEYOND READOUT: A SHARED LATENT FRAME

This section tests whether numerical geometry survives outside the digit readout and whether latent representations across conditions form a coherent shared frame.

5 BEYOND READOUT: A SHARED LATENT FRAME

  • Removing the digit-readout span collapses IIAAR from 0.826, 0.865, 0.747, and 0.695 to 0.008, 0.000, 0.005, and 0.003.Ablating a random rank-9 subspace preserves substantial causal efficacy: 0.634, 0.791, 0.339, and 0.560.Source: §5 · p. 6
  • The 22-dimensional causal subspace splits exactly into a 9-dimensional readout subspace D and a 13-dimensional latent subspace L orthogonal to digit unembeddings.L encodes values from 0 to 99 without coupling to immediate token logits; D is close to, but not identical to, the full-stream digit-readout span.Source: §5 · pp. 6–7
  • 75.5% top-1 retrieval of unseen values in latent subspaces exceeds random-space retrieval (38.3%) and shuffled-label retrieval (5.2%).The latent geometry also reaches 98.8% top-5 retrieval; the reported chance baseline is 5%.Source: Figure 3 · p. 7, §5 · p. 7
  • 74.8% top-1 retrieval after full synchronization is comparable to pairwise maps (73.4%), while modality-only synchronization falls to 18.1%.With one condition pair withheld, reconstruction reaches 71.1% top-1 retrieval; the results support condition-dependent frames that cohere globally.Source: Figure 3 · p. 7, §5 · p. 8
  • The synchronized latent frames retain unseen-value retrieval while eliminating 711 degrees of freedom, and reconstruct a withheld condition pair from the other ten relations.Predicting unseen-condition coordinates from modality and operation shifts achieves 0.629 cosine and 58.9% top-1 retrieval; a successor operator transfers without refitting, identifying y + 1 centroids in 65.0% to 70.0% of held-out cases.Source: §5 · p. 8

6 LATENT NUMERICAL IDENTITY STEERS SUBSEQUENT GENERATION

This section tests whether latent numerical identity can be separated from the digit emitted now and influence later generation. Swapping latent identity redirects the next digit even when the current digit remains unchanged.

6 LATENT NUMERICAL IDENTITY STEERS SUBSEQUENT GENERATION

  • The hybrid intervention pairs readout displacement from y with latent displacement from ỹ, choosing answers that share their first digit but differ in the second.This makes both components agree on the current tens digit while disagreeing on the next units digit.Source: §6 · p. 8
  • 69.2% of mismatched states produce the latent donor answer ỹ, versus 8.9% for random-L controls; matched edits produce y in 79.9% of cases.The first-digit probability is nearly unchanged between matched and mismatched states (0.963 versus 0.965); when both digits differ, swaps produce the hybrid in 51.6% of cases.Source: Figure 4 · p. 9, §6 · p. 9
  • 77.6% of mismatched-state cases greedily predict the latent donor’s second digit after the shared first digit is supplied.The next-step probability of that digit is 0.737, versus 0.040 for the readout donor’s second digit; readout-only edits leave the two probabilities unresolved.Source: §6 · p. 9
  • The swap’s effect on readout coordinates is near zero at layer 44 (γ44 ≈ 0.113), then rises at layer 45 (γ45 ≈ 0.722) when attention reads the edited KV-cache.The median cosine reaches 0.852 in the final layer, consistent with latent identity converting into readout directions across intervening layers.Source: §6 · p. 9

7 DISCUSSION AND CONCLUSION

The discussion argues that addition and subtraction across text and images share causal numerical meaning without using identical residual-stream coordinates. It also locates numerical computation upstream of sequential digit readout and states the study’s scope and implementation context.

7 DISCUSSION AND CONCLUSION

  • A scaled rotation maps condition-specific subspaces onto shared geometry, while direct reuse recovers only 45.5% of the causal effect on average.All four conditions synchronize into one relational frame; the shared 13-dimensional latent space is orthogonal to digit unembeddings.Source: §7 · p. 10
  • The 13-dimensional latent subspace holds numerical identity and steers a subsequent token through the attention cache, while the readout subspace drives the immediate token.The authors describe the model as solving the arithmetic problem before reading out the answer across autoregressive steps.Source: §7 · p. 10
  • The experiments cover addition and subtraction on integers in [0, 99]; multiplication has distributed, depth-dependent representations across intermediate layers.Identifying the specific attention heads that perform the cache-mediated readout remains open.Source: §7 · p. 10

AI USE STATEMENT; ETHICS STATEMENT

  • The authors report using language models for writing and coding assistance, while taking responsibility for the content and reporting no human subjects, personal data, or crowdsourcing.The studied models are described as the objects of the experiments.Source: AI USE STATEMENT · p. 10, ETHICS STATEMENT · p. 10

A.1 PROMPTS, DATASETS, AND ACTIVATION EXTRACTION

  • Prompts use addition and subtraction records, and analyses retain model-correct examples, with matched cross-modal comparisons using examples answered correctly in every compared format.The records include operands, results, and modular labels.Source: §A.1 · p. 13
  • Text activations are aligned by semantic element, using the final subtoken when an element spans multiple tokens; the main analyses use the final pre-generation position.The final prompt position is the common reference position across text and image conditions.Source: §A.1 · p. 13, §A.2 · p. 14

A.2 REFERENCE LAYER AND POSITION

  • The final pre-generation position is the strongest tested causal-control site and has a common meaning across modalities after the complete input is processed.Earlier text positions correspond to elements without visual counterparts.Source: §A.2 · p. 14

A.3 DAS TRAINING AND CAUSAL DIMENSION

The appendix selects a common causal-subspace dimension for the four conditions and characterizes how base-10 result geometry emerges and stabilizes in text arithmetic. Result information becomes organized around layer 35, while its coordinate system consolidates later.

A.3 DAS TRAINING AND CAUSAL DIMENSION

  • k = 22 meets the 90% retention threshold in all four conditions, whereas k = 16 falls below it in two; the main analyses use k = 22.The comparison uses k = 32 as its reference, and final subspaces are retrained with a 15-epoch budget.Source: Figure 5 · p. 15, §A.3 · p. 15

B.1 EMERGENCE AND CONSOLIDATION OF A BASE-10 CODE

  • Result-modulo-10 structure is weak at layer 34, emerges as a circular ordering at layer 35, and is clear by layer 38.Probe accuracy and periodic structure rise sharply around layer 35.Source: §B.1 · p. 16, Figure 6 · p. 16
  • Layers 42 to 44 have the greatest average overlap with surrounding late-layer Fourier planes, marking consolidation after the code first appears.The main analyses use this stable late-layer range.Source: §B.1 · p. 16
  • Result probes transfer strongly between addition and subtraction, although their Fourier planes at layers 42 to 44 align only weakly.The result is compatible information in non-identical coordinates.Source: §B.1 · p. 16

B.2 MULTIPLICATION

Multiplication develops through coarse numerical structure before exact product classification, with causal roles that vary across depth. Across modalities, arithmetic states emerge late, while direct cross-modal reuse is limited and learned alignment improves transfer.

B.2 MULTIPLICATION

  • Long-period Fourier signal and rising top-k accuracy are consistent with coarse product magnitude information preceding exact product classification.Exact top-1 classification remains weak at depths where the Fourier probe is already informative.Source: Figure 8 · p. 17, §B.2 · p. 17
  • Multiplication intermediates have distinct, depth-dependent causal roles: ˆc0 peaks near layer 36, while ˆc1 has weaker, more sustained influence.When lower two answer digits are fixed, ˆc1 control is nearly perfect at the variable level but substantially weaker for the complete answer.Source: §B.2 · pp. 17–18

B.3 FOURIER GEOMETRY IS A SIGNATURE, NOT A CAUSAL BASIS

  • In Gemma4-12B-it, the 11 most Fourier-aligned DAS directions recover 23% of the full effect; substantial recovery requires combining them with other directions.In Llama3.1-8B, the corresponding 11 directions recover about 84%.Source: §B.3 · p. 18, Figure 10 · p. 19

C.1 VISUAL ARITHMETIC DEVELOPS A COMPARABLE LATE CAUSAL STATE

  • Image arithmetic develops strong late-layer causal states, but within-condition IIAAR is lower for image than text addition and subtraction.Image addition and subtraction reach 0.747 and 0.693, versus 0.831 and 0.862 for text.Source: §C.1 · p. 20

C.2 DIRECT REUSE: RAW CAUSAL EFFECTS AND CONTROLS

  • Image and text Fourier planes align weakly, including matched layer-44, period-20 planes, despite periodic structure in both modalities.Mean text–image min-rank containment for image addition is 0.0076.Source: §C.2 · pp. 20–21, Figure 12 · p. 21

C.3 CHOICE OF ALIGNMENT TARGET

  • 0.914 mean normalized recovery is reached by adding a global scale to a learned rotation for cross-modal alignment.A rotation alone reaches 0.718, while direct reuse reaches 0.532.Source: Table 3 · p. 21, §C.3 · p. 21

C.4 TRANSFORMATION ABLATIONS

Cross-modal causal transfer depends on learning a coordinate transformation, not merely matching magnitude or retaining a few dominant modes. Numerical-transition geometry also transfers across operations, and compositions of simpler maps closely approximate direct transport.

C.4 TRANSFORMATION ABLATIONS

  • 0.914 mean normalized recovery follows from a learned rotation plus global scale, versus 0.718 for rotation alone and essentially no transfer from scale alone.Random orthogonal maps fail, while an unrestricted linear map reaches 0.963 mean recovery.Source: Table 4 · p. 22, §C.4 · p. 22

C.5 RANK-RESTRICTED TRANSPORT

  • 16–19 matched modes are needed to reach 95% of full-map causal recovery across the four directions, despite earlier geometric saturation.The leading modes alone do not reproduce the full intervention effect.Source: §C.5 · p. 22, Figure 13 · p. 23

C.6 CROSS-OPERATION AND COMPOSED TRANSPORTS

  • When only operation changes, fitted transport reaches 0.828 mean normalized recovery, compared with 0.773 for direct reuse; these means use different sampling protocols.Relations changing both modality and operation reach 0.807 mean normalized recovery.Source: §C.6 · p. 23, Table 5 · p. 23
  • 0.799 mean destination-normalized transfer is reached by either composition, close to 0.807 for direct transport across relations changing both factors.On held-out transitions, composed-path predictions have mean cosine 0.968 and 0.969 with direct-map predictions; this does not establish equality of the full transformations.Source: Table 6 · p. 24, §C.6 · p. 24, Figure 14 · p. 24

D.1 DIGIT-READOUT OVERLAP AND ABLATION CONTROLS

The causal subspaces overlap the digit readout, but that relatively small component carries nearly all immediate autoregressive control. Arithmetic-trained directions also transfer to a non-arithmetic digit-output task.

D.1 DIGIT-READOUT OVERLAP AND ABLATION CONTROLS

  • 41.65 times the analytic random residual-subspace expectation is the digit-readout containment observed for text-addition DAS at layer 43.Untrained PCA and random-in-PCA controls reach 22.78 and 23.90 times that expectation.Source: §D.1 · p. 25
  • Removing the digit-readout component drops mean self-IIAAR from 0.826, 0.865, 0.747, and 0.695 to 0.008, 0.000, 0.005, and 0.003 across the four conditions.Dimension-matched random rank-9 ablations preserve 0.634, 0.791, 0.339, and 0.560, respectively.Source: §D.1 · p. 25, Figure 15 · p. 25
  • Arithmetic-trained directions reach 0.489 teacher-forced IIA on a one-digit repeat task that requires digit output but no arithmetic computation.Transfer under the stricter complete-answer autoregressive criterion is 1.5%.Source: §D.1 · p. 25

D.2 READOUT-FREE GEOMETRY

After separating digit-readout directions, the remaining latent spaces retain numerical relations and align across conditions. This shared geometry supports approximate held-out-condition closure, transfer of a local successor relation, and globally consistent full-space transports, with important scope limits.

D.2 READOUT-FREE GEOMETRY

  • The exact 13-dimensional latent component retains transition cosine 0.807 and 75.5% top-1 retrieval among unseen values, exceeding dimension-matched random readout-orthogonal controls.Shuffling numerical identities reduces retrieval to 5.2%, approximately chance.Source: §D.2 · p. 26, Table 7 · p. 27

D.3 SYNCHRONIZATION PROTOCOL

  • A stricter two-frame synchronization control reaches only 0.249 transition cosine and 18.1% top-1 retrieval; leave-one-relation-out retrieval remains 71.1%.The four-frame result therefore cannot be reduced to one text–image transformation.Source: §D.3 · p. 27, Table 8 · p. 27

D.5 A LOCAL SUCCESSOR RELATION TRANSFERS ACROSS CONDITIONS

  • A successor operator learned from text addition identifies the correct y + 1 centroid in 65.0%–70.0% of cases across all four conditions.Complete-answer autoregressive accuracy is 9.5%, and repeated application becomes less accurate for larger shifts.Source: Figure 16 · p. 28, §D.5 · p. 29

D.4 HELD-OUT-CONDITION ADDITIVE CLOSURE

  • Held-out-condition predictions reach transition cosine 0.629 and 58.9% top-1 retrieval among 20 unseen values, but represent approximate—not exact—additive closure.No autoregressive intervention is performed on the predicted held-out latent geometry.Source: §D.4 · p. 28

D.6 FULL-DAS SYNCHRONIZATION AS A CAUSAL CONTROL

  • Synchronized full-DAS transports reach mean IIAAR 0.6359, close to 0.6385 for independently fitted pairwise transports; predicted displacement cosine is 0.989.This supports global organization but does not imply improved performance for every individual relation.Source: §D.6 · p. 29

E.1 MATCHED AND MISMATCHED D/L INTERVENTIONS

The experiments separate the immediate digit readout from numerical identity used for later generation, compare arithmetic and repeat-number geometry, and test the central mechanism in Ministral.

E.1 MATCHED AND MISMATCHED D/L INTERVENTIONS

  • 0.692 IIAAR toward ỹ after replacing only L, versus 0.799 toward y when matched; the shared first-digit probability remains 0.965 versus 0.963.A random latent displacement reaches only 0.089 toward the selected alternative value.Source: §E.1 · p. 30
  • 51.6% of different-first-digit mismatches produce the hybrid answer, while matched edits reproduce y in 74.5% of cases.In swap trials, the units digit follows L in 57.7% of cases, while the tens digit follows D in 83.9%.Source: §E.1 · p. 30, Table 9 · p. 31

E.2 CACHE PROPAGATION PROTOCOL

  • With the same first digit supplied in both conditions, an earlier L edit propagates through the retained cache and redirects the next digit without an intervention at that position.The unrelated-latent control raises the alternative units-digit probability only to 0.020, versus 0.737 for the value-specific mismatch.Source: §E.2 · pp. 31–32

E.3 REPEAT-NUMBER COMPARISON

  • 0.690 repeat-to-arithmetic held-out transition cosine exceeds 0.655 arithmetic-to-arithmetic, but no-fit RSA favors arithmetic at 0.633 versus 0.518.Repeat-to-arithmetic top-1 retrieval is 65.1%, versus 57.9% for arithmetic-to-arithmetic; repeat generation is correct for all values 0–99 under the stated criterion.Source: §E.3 · p. 32, Figure 18 · p. 33

F SECOND-MODEL REPLICATION IN MINISTRAL

  • The Ministral replication tests whether a late causal numerical state, its readout-free geometry, global alignment, and later-generation role recur in a second multimodal architecture.It evaluates text/image inputs crossed with addition/subtraction, with thinking disabled and digit-only output.Source: F SECOND-MODEL REPLICATION IN MINISTRAL · p. 34

F.1 LOCALIZATION AND A CAUSAL NUMERICAL SUBSPACE

  • At layer 37, the 32-dimensional subspace retains 90.9% of the T+ and 93.0% of the I+ k = 64 effects, while its full-answer causal control remains moderate.Mean IIAAR ranges from 0.299 to 0.401 across the four conditions, so the learned space is not a complete bottleneck.Source: §F.1 · pp. 34–35

F.2 EXACT SEPARATION OF THE CURRENT DIGIT READOUT

  • The learned space separates exactly into a rank-9 digit-readout-coupled component and a 23-dimensional component in the readout null space.The decomposition discards no part of the learned causal subspace at the reported numerical precision.Source: §F.2 · p. 35

F.3 SHARED NUMERICAL GEOMETRY IN THE READOUT-FREE COMPONENT

The readout-free components preserve numerical relations across conditions and synchronize into a global frame; changing their numerical identity can redirect later digits while leaving the current digit largely intact.

F.3 SHARED NUMERICAL GEOMETRY IN THE READOUT-FREE COMPONENT

  • 0.8963 ± 0.0463 held-out transition cosine accompanies 93.35% top-1 and 99.85% top-5 retrieval across condition relations.Fitting-free cross-condition RSA is ρ = 0.8541 ± 0.0898.Source: §F.3 · p. 36, Table 11 · p. 36
  • None of 200 random readout-null projections or 500 label permutations reaches the observed L scores, supporting broad residual geometry enriched in the causal-space remainder.The authors qualify this as enrichment, not uniqueness, because random readout-free projections also preserve numerical structure.Source: §F.3 · p. 36

F.4 THE PAIRWISE RELATIONS SYNCHRONIZE INTO ONE LATENT FRAME

  • 0.89020 LORO transition cosine reconstructs a withheld relation after synchronization, only 0.00623 below direct fitting, with retrieval unchanged.The result supports global consistency up to fitted task-specific rotations, scales, and offsets, not identical residual-space subspaces.Source: §F.4 · p. 36, Table 12 · p. 36

F.5 THE READOUT-FREE IDENTITY CAUSALLY REDIRECTS THE CONTINUATION

  • Replacing only L reverses the preferred second digit and redirects free generation, while the shared first-digit probability changes by only −0.00746.An unrelated latent donor compatible with the same first digit does not produce the selected-value continuation effect.Source: §F.5 · p. 37

F.6 SUMMARY

  • The replication distinguishes moderate control of the full autoregressive answer from strong organization and functional use of the underlying numerical state.The readout-free identity affects later continuation even though it does not directly couple to the current digit readout.Source: §F.6 · pp. 37–38
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