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

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

cs.AIOct 27, 2026

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

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

Organizations: Universitat Autònoma de Barcelona · Apple

Abstract

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.

Figures & tables

Figure 1: Overview of the experimental framework. Crossing input modality ( $\mathsf{T},\mathsf{I}$ ) with operation ( \(\mathbin{\vbox{\hbox{\hbox to4.71pt{\vbox to4.71pt{\pgfpicture\makeatletter\hbox{\hskip 2.35277pt\lower-2.35277pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.98 0 C 2.98 1.65 1.65 2.98 0 2.98 C -1.65 2.98 -2.98 1.65 -2.98 0 C -2.98 -1.65 -1.65 -2.98 0 -2.98 C 1.65 -2.98 2.98 -1.65 2.98 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.79 0 L 1.79 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 -1.79 L 0 1.79}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}},\mathbin{\vbox{\hbox{\hbox to4.71pt{\vbox to4.71pt{\pgfpicture\makeatletter\hbox{\hskip 2.35277pt\lower-2.35277pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.98 0 C 2.98 1.65 1.65 2.98 0 2.98 C -1.65 2.98 -2.98 1.65 -2.98 0 C -2.98 -1.65 -1.65 -2.98 0 -2.98 C 1.65 -2.98 2.98 -1.65 2.98 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.79 0 L 1.79 0}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}}\) ) yields four conditions reaching the same number. In each condition, we learn a $22$ -dimensional causal subspace ${\bm{R}}_{i}$ that governs generation ( Section 3 ). While direct reuse fails across modalities, scaled rotations $\alpha{\bm{Q}}$ restore causal control to $91.4\%$ ( Section 4 ). Decomposing ${\bm{R}}_{i}$ into a $9$ -dimensional readout subspace ${\bm{D}}$ and a $13$ -dimensional latent subspace ${\bm{L}}$ reveals that numerical geometry survives outside the digit readout and synchronizes into a single shared latent frame ( Section 5 ). Finally, latent numerical identity in ${\bm{L}}$ is scheduled across time, redirecting the token emitted at the subsequent autoregressive step ( Section 6 ). — Figure 1: Overview of the experimental framework. Crossing input modality ( $\mathsf{T},\mathsf{I}$ ) with operation ( $\mathbin{\vbox{\hbox{\hbox to4.71pt{\vbox to4.71pt{\pgfpicture\makeatletter\hbox{\hskip 2.35277pt\lower-2.35277pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.98 0 C 2.98 1.65 1.65 2.98 0 2.98 C -1.65 2.98 -2.98 1.65 -2.98 0 C -2.98 -1.65 -1.65 -2.98 0 -2.98 C 1.65 -2.98 2.98 -1.65 2.98 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.79 0 L 1.79 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 -1.79 L 0 1.79}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}},\mathbin{\vbox{\hbox{\hbox to4.71pt{\vbox to4.71pt{\pgfpicture\makeatletter\hbox{\hskip 2.35277pt\lower-2.35277pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.98 0 C 2.98 1.65 1.65 2.98 0 2.98 C -1.65 2.98 -2.98 1.65 -2.98 0 C -2.98 -1.65 -1.65 -2.98 0 -2.98 C 1.65 -2.98 2.98 -1.65 2.98 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.79 0 L 1.79 0}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}}$ ) yields four conditions reaching the same number. In each condition, we learn a $22$ -dimensional causal subspace ${\bm{R}}_{i}$ that governs generation ( Section 3 ). While direct reuse fails across modalities, scaled rotations $\alpha{\bm{Q}}$ restore causal control to $91.4\%$ ( Section 4 ). Decomposing ${\bm{R}}_{i}$ into a $9$ -dimensional readout subspace ${\bm{D}}$ and a $13$ -dimensional latent subspace ${\bm{L}}$ reveals that numerical geometry survives outside the digit readout and synchronizes into a single shared latent frame ( Section 5 ). Finally, latent numerical identity in ${\bm{L}}$ is scheduled across time, redirecting the token emitted at the subsequent autoregressive step ( Section 6 ).

Figure 2: Causal transfer across conditions. (a) Direct versus aligned reuse (fitting

\alpha{\bm{Q}}

) for changes in operation, modality, or both. (b) Cross-modal transformation controls. A map of the form

\alpha{\bm{Q}}

aligns subspaces nearly as well as an unconstrained map. Rotation alone improves recovery, and the fitted scale

\alpha

provides a further gain. (c) Pairwise and composed transports. Bars average four direction means. Scores are destination-normalized autoregressive recovery on held-out expressions.

Figure 3: Numerical geometry beyond readout supports a common frame. (a) No-fit RSA in

{\bm{L}}

(

\rho=0.765

) versus random

13

-dimensional readout-orthogonal subspaces (

\rho=0.545

). (b) Unseen-value retrieval after alignment (

75.5\%

), with random-space (

38.3\%

) and shuffled-label (

5.2\%

) controls. (c) Pairwise maps (

73.4\%

) versus full synchronization (

74.8\%

), withheld-pair reconstruction (

71.1\%

), and modality-only synchronization (

18.1\%

). Points average repeats within each relation and value split; half-violins show distributions, and diamonds and numbers show means. The dashed line marks the

5\%

chance baseline among

20

candidate values.

Figure 4: The latent subspace

{\bm{L}}

sets the digit written next. Each trial edits the activation of a base prompt ( e.g.,

17{+}14=31

) at layer

43

of the last prompt position

t_{0}

, using donors with answers

y

and

\tilde{y}

that share their first digit and differ in the second ( e.g.,

52

and

57

). (a) Current digit. The first digit of

y

differs from that of the base. Editing

{\bm{D}}

alone with the donor

y

makes the model write the first digit of

y

with probability

94.7\%

, editing

{\bm{L}}

alone only

14.0\%

. (b) Full answer. Matched takes

{\bm{D}}

and

{\bm{L}}

from

y

and yields

y

79.9\%

of cases, the ceiling of a

22

-dimensional edit. Swap

{\bm{L}}

takes

{\bm{D}}

from

y

and

{\bm{L}}

from

\tilde{y}

( eq. 6 ), and yields

\tilde{y}

69.2\%

. Random

{\bm{L}}

takes

{\bm{L}}

from an unrelated value and yields

\tilde{y}

in only

8.9\%

. (c) Delayed readout. At the next step

t_{1}

, the cosine between the swap’s effect and the natural change in readout directions is near zero at layer

44

and jumps at layer

45

, once the edited KV-cache is read.

Appendix figures & tables27 assets

Supplementary material from the paper’s appendix.

Appendix

Position	Semantic token	Position	Semantic token
0	<bos>	9	-
1	<\|turn>	10	6
2	user	11	=
3	Output	12	<turn\|>
4	ONLY	13	<\|turn>
5	a	14	model

Appendix

Table 1: Example semantic-token alignment for the textual prompt Output ONLY a number. 10 - 6 = . When a semantic element spans multiple model tokens, the final subtoken is used.

Figure 5: Epoch-matched DAS dimension selection. Autoregressive IIA for

k\in\{16,22,32\}

across text and image addition and subtraction. Using

k=32

as the reference, we compare retention across the tested dimensions. A dimension of

k=16

falls below the

90\%

retention threshold in two conditions, whereas

k=22

satisfies it across all four. We therefore retain

k=22

for the main analyses, retraining the final subspaces with the longer

15

-epoch budget.

Figure 6: A base-10 result geometry becomes progressively organized across layers. Residual stream activations for addition and subtraction projected onto the orthonormalized period- $10$ Fourier plane at layers $34$ , $35$ and $38$ , coloured by result modulo $10$ .

Figure 7: The numerical representation stabilizes into partially shared, operation-dependent late states. (Left) Cross-layer stability of the Fourier representation, averaged first over periods $T\in\{5,10,20,50,100\}$ and then over comparison layers. Layers $42$ to $44$ achieve the highest mean overlap. (Middle) Symmetric overlap between independently learned $k=22$ DAS subspaces. Addition and subtraction share a substantial component ( $0.45$ ) while remaining below within-variable consistency across seeds. The multiplication variables form a second family. (Right) Cross-variable Fourier plane overlap between the multiplication intermediates and the final product. All panels use activations at semantic position $17$ .

Panel of Figure 8 — Figure 8: Multiplication develops from coarse magnitude to exact classification. Left: Top- $k$ accuracy for the result-balanced exact-product linear probe. The increase with $k$ shows that the correct product is often restricted to a smaller candidate set before becoming the top prediction. Right: Held-out mean $R^{2}$ for a long-period Fourier probe spanning the multiplication result range. The earlier Fourier signal is consistent with approximate magnitude information preceding exact product classification.

Figure 9: DAS layer and dimensionality sweeps for multiplication variables. Rows show the final result,

\hat{c}_{1}

, controlled

\hat{c}_{1}

, and

\hat{c}_{0}

; columns report autoregressive, teacher-forced, and variable-level

\mathrm{IIA}

for

k=16

and

k=32

. In the controlled

\hat{c}_{1}

condition, base and donor examples share

r\bmod 100

but differ in

\hat{c}_{1}

. The intervention achieves near-perfect variable-level control in the late layers, but substantially weaker answer-level control.

Figure 10: The causal role of Fourier geometry differs across models. (Left) In Llama3.1-8B , directions most aligned with the Fourier subspace recover most of the full DAS effect. (Right) Gemma4-12B-it is more distributed: Fourier-aligned directions recover a minority of the effect and substantial recovery needs directions outside the visible Fourier geometry. The

x

-axis reports

m/k

, where

m

is the number of retained DAS directions ordered by Fourier alignment and

k

is the full DAS dimensionality.

Figure 11: Arithmetic result subspaces show partial direct causal transfer across operations and modalities. Raw $\mathrm{IIA}_{\mathrm{AR}}$ when the subspace learned on the source condition is used to intervene on base and donor activations drawn from the destination condition. Entries report mean $\pm$ standard deviation across the $3\times 3$ combinations of source and destination DAS seeds.

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2.09 C -1.15 2.09 -2.09 1.15 -2.09 0 C -2.09 -1.15 -1.15 -2.09 0 -2.09 C 1.15 -2.09 2.09 -1.15 2.09 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.25 0 L 1.25 0}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}}}$
Source seed	0	1	2	0	1	2	0	1	2	0	1	2
0	.306	.323	.312	.535	.517	.505	.582	.610	.615	.691	.691	.685
1	.245	.258	.250	.512	.494	.484	.527	.552	.558	.700	.700	.694
2	.337	.355	.344	.605	.584	.571	.627	.657	.663	.691	.691	.685

Appendix

Table 2: Per-seed destination-normalized direct transfer. Rows denote source DAS seeds and columns destination DAS seeds. Each

3\times 3

block reports one same-operation cross-modal transfer direction.

Figure 12: Cross-modal alignment of result Fourier planes. Min-rank containment between textual and visual addition-result Fourier planes across layers and periods. Cross-modal alignment remains weak, including between matched periods, despite the periodic numerical structure visible separately within each modality.

Method	$\mathsf{T}_{\mathbin{\vbox{\hbox{\hbox to3.41pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 1.70694pt\lower-1.70694pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.09 0 C 2.09 1.15 1.15 2.09 0 2.09 C -1.15 2.09 -2.09 1.15 -2.09 0 C -2.09 -1.15 -1.15 -2.09 0 -2.09 C 1.15 -2.09 2.09 -1.15 2.09 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.25 0 L 1.25 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} 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Direct transfer	0.303	0.534	0.599	0.692
Absolute-state ${\bm{Q}}$	0.546	0.772	0.708	0.804
Centroid-displacement ${\bm{Q}}$	0.531	0.795	0.705	0.816
Displacement ${\bm{Q}}$	0.531	0.811	0.704	0.828
Scaled displacement $\alpha{\bm{Q}}$	0.791	0.983	0.950	0.932

Appendix

Table 3: Comparison of cross-modal alignment targets. Destination-normalized transfer, averaged over the

3\times 3

source and destination DAS seed combinations.

Method	$\mathsf{T}_{\mathbin{\vbox{\hbox{\hbox to3.41pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 1.70694pt\lower-1.70694pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.09 0 C 2.09 1.15 1.15 2.09 0 2.09 C -1.15 2.09 -2.09 1.15 -2.09 0 C -2.09 -1.15 -1.15 -2.09 0 -2.09 C 1.15 -2.09 2.09 -1.15 2.09 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.25 0 L 1.25 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 -1.25 L 0 1.25}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}}}\!\to\!\mathsf{I}_{\mathbin{\vbox{\hbox{\hbox to3.41pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 1.70694pt\lower-1.70694pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.09 0 C 2.09 1.15 1.15 2.09 0 2.09 C -1.15 2.09 -2.09 1.15 -2.09 0 C -2.09 -1.15 -1.15 -2.09 0 -2.09 C 1.15 -2.09 2.09 -1.15 2.09 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.25 0 L 1.25 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 -1.25 L 0 1.25}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}}}$	$\mathsf{T}_{\mathbin{\vbox{\hbox{\hbox to3.41pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 1.70694pt\lower-1.70694pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.09 0 C 2.09 1.15 1.15 2.09 0 2.09 C -1.15 2.09 -2.09 1.15 -2.09 0 C -2.09 -1.15 -1.15 -2.09 0 -2.09 C 1.15 -2.09 2.09 -1.15 2.09 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.25 0 L 1.25 0}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}}}\!\to\!\mathsf{I}_{\mathbin{\vbox{\hbox{\hbox to3.41pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 1.70694pt\lower-1.70694pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.09 0 C 2.09 1.15 1.15 2.09 0 2.09 C -1.15 2.09 -2.09 1.15 -2.09 0 C -2.09 -1.15 -1.15 -2.09 0 -2.09 C 1.15 -2.09 2.09 -1.15 2.09 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.25 0 L 1.25 0}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}}}$	$\mathsf{I}_{\mathbin{\vbox{\hbox{\hbox to3.41pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 1.70694pt\lower-1.70694pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.09 0 C 2.09 1.15 1.15 2.09 0 2.09 C -1.15 2.09 -2.09 1.15 -2.09 0 C -2.09 -1.15 -1.15 -2.09 0 -2.09 C 1.15 -2.09 2.09 -1.15 2.09 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.25 0 L 1.25 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 -1.25 L 0 1.25}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}}}\!\to\!\mathsf{T}_{\mathbin{\vbox{\hbox{\hbox to3.41pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 1.70694pt\lower-1.70694pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.09 0 C 2.09 1.15 1.15 2.09 0 2.09 C -1.15 2.09 -2.09 1.15 -2.09 0 C -2.09 -1.15 -1.15 -2.09 0 -2.09 C 1.15 -2.09 2.09 -1.15 2.09 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.25 0 L 1.25 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 -1.25 L 0 1.25}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}}}$	$\mathsf{I}_{\mathbin{\vbox{\hbox{\hbox to3.41pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 1.70694pt\lower-1.70694pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.09 0 C 2.09 1.15 1.15 2.09 0 2.09 C -1.15 2.09 -2.09 1.15 -2.09 0 C -2.09 -1.15 -1.15 -2.09 0 -2.09 C 1.15 -2.09 2.09 -1.15 2.09 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.25 0 L 1.25 0}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}}}\!\to\!\mathsf{T}_{\mathbin{\vbox{\hbox{\hbox to3.41pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 1.70694pt\lower-1.70694pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.09 0 C 2.09 1.15 1.15 2.09 0 2.09 C -1.15 2.09 -2.09 1.15 -2.09 0 C -2.09 -1.15 -1.15 -2.09 0 -2.09 C 1.15 -2.09 2.09 -1.15 2.09 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.25 0 L 1.25 0}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}}}$
Direct	0.303	0.534	0.599	0.692
Displacement ${\bm{Q}}$	0.531	0.811	0.704	0.828
Scale only $\alpha{\bm{I}}$	0.004	0.004	0.003	0.012
Random rotation ${\bm{Q}}_{\mathrm{rand}}$	0.001	0.004	0.000	0.001
Scaled random $\alpha{\bm{Q}}_{\mathrm{rand}}$	0.001	0.011	0.002	0.004
Full $\alpha{\bm{Q}}$	0.791	0.983	0.950	0.932

Appendix

Table 4: Full scaled-displacement ablation. Destination-normalized transfer, averaged over the

3\times 3

seed combinations.

Figure 13: Full diagnostics for cumulative rank-

m

transport. From top to bottom: cumulative singular-value mass, cosine alignment on test expressions, and destination-normalized causal transfer. Geometric quantities saturate earlier than causal performance, showing that the leading matched modes are not sufficient to reproduce the full intervention effect.

Transport	Normalized transfer
$\mathsf{I}_{\mathbin{\vbox{\hbox{\hbox to3.41pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 1.70694pt\lower-1.70694pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.09 0 C 2.09 1.15 1.15 2.09 0 2.09 C -1.15 2.09 -2.09 1.15 -2.09 0 C -2.09 -1.15 -1.15 -2.09 0 -2.09 C 1.15 -2.09 2.09 -1.15 2.09 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.25 0 L 1.25 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 -1.25 L 0 1.25}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}}}\rightarrow\mathsf{T}_{\mathbin{\vbox{\hbox{\hbox to3.41pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 1.70694pt\lower-1.70694pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.09 0 C 2.09 1.15 1.15 2.09 0 2.09 C -1.15 2.09 -2.09 1.15 -2.09 0 C -2.09 -1.15 -1.15 -2.09 0 -2.09 C 1.15 -2.09 2.09 -1.15 2.09 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.25 0 L 1.25 0}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}}}$	$0.847\pm 0.057$
$\mathsf{I}_{\mathbin{\vbox{\hbox{\hbox to3.41pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 1.70694pt\lower-1.70694pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.09 0 C 2.09 1.15 1.15 2.09 0 2.09 C -1.15 2.09 -2.09 1.15 -2.09 0 C -2.09 -1.15 -1.15 -2.09 0 -2.09 C 1.15 -2.09 2.09 -1.15 2.09 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.25 0 L 1.25 0}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}}}\rightarrow\mathsf{T}_{\mathbin{\vbox{\hbox{\hbox to3.41pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 1.70694pt\lower-1.70694pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.09 0 C 2.09 1.15 1.15 2.09 0 2.09 C -1.15 2.09 -2.09 1.15 -2.09 0 C -2.09 -1.15 -1.15 -2.09 0 -2.09 C 1.15 -2.09 2.09 -1.15 2.09 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.25 0 L 1.25 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 -1.25 L 0 1.25}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}}}$	$0.842\pm 0.082$
$\mathsf{T}_{\mathbin{\vbox{\hbox{\hbox to3.41pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 1.70694pt\lower-1.70694pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.09 0 C 2.09 1.15 1.15 2.09 0 2.09 C -1.15 2.09 -2.09 1.15 -2.09 0 C -2.09 -1.15 -1.15 -2.09 0 -2.09 C 1.15 -2.09 2.09 -1.15 2.09 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.25 0 L 1.25 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 -1.25 L 0 1.25}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}}}\rightarrow\mathsf{I}_{\mathbin{\vbox{\hbox{\hbox to3.41pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 1.70694pt\lower-1.70694pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.09 0 C 2.09 1.15 1.15 2.09 0 2.09 C -1.15 2.09 -2.09 1.15 -2.09 0 C -2.09 -1.15 -1.15 -2.09 0 -2.09 C 1.15 -2.09 2.09 -1.15 2.09 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.25 0 L 1.25 0}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}}}$	$0.839\pm 0.047$
$\mathsf{T}_{\mathbin{\vbox{\hbox{\hbox to3.41pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 1.70694pt\lower-1.70694pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.09 0 C 2.09 1.15 1.15 2.09 0 2.09 C -1.15 2.09 -2.09 1.15 -2.09 0 C -2.09 -1.15 -1.15 -2.09 0 -2.09 C 1.15 -2.09 2.09 -1.15 2.09 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.25 0 L 1.25 0}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}}}\rightarrow\mathsf{I}_{\mathbin{\vbox{\hbox{\hbox to3.41pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 1.70694pt\lower-1.70694pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.09 0 C 2.09 1.15 1.15 2.09 0 2.09 C -1.15 2.09 -2.09 1.15 -2.09 0 C -2.09 -1.15 -1.15 -2.09 0 -2.09 C 1.15 -2.09 2.09 -1.15 2.09 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.25 0 L 1.25 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 -1.25 L 0 1.25}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}}}$	$0.699\pm 0.069$
Mean	$0.807$

Appendix

Table 5: Direct transports that change both modality and operation. Each map is fitted directly between the indicated source and destination condition using matched ordered numerical transitions. Entries report mean

\pm

sample standard deviation of destination-normalized autoregressive transfer across the

3\times 3

source–destination DAS seed combinations.

Transport	Direct	Modality first	Operation first
$\mathsf{I}_{\mathbin{\vbox{\hbox{\hbox to3.41pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 1.70694pt\lower-1.70694pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.09 0 C 2.09 1.15 1.15 2.09 0 2.09 C -1.15 2.09 -2.09 1.15 -2.09 0 C -2.09 -1.15 -1.15 -2.09 0 -2.09 C 1.15 -2.09 2.09 -1.15 2.09 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.25 0 L 1.25 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 -1.25 L 0 1.25}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}}}\rightarrow\mathsf{T}_{\mathbin{\vbox{\hbox{\hbox to3.41pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 1.70694pt\lower-1.70694pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.09 0 C 2.09 1.15 1.15 2.09 0 2.09 C -1.15 2.09 -2.09 1.15 -2.09 0 C -2.09 -1.15 -1.15 -2.09 0 -2.09 C 1.15 -2.09 2.09 -1.15 2.09 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.25 0 L 1.25 0}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}}}$	$0.847\pm 0.057$	$0.845\pm 0.043$	$0.846\pm 0.049$
$\mathsf{I}_{\mathbin{\vbox{\hbox{\hbox to3.41pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 1.70694pt\lower-1.70694pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.09 0 C 2.09 1.15 1.15 2.09 0 2.09 C -1.15 2.09 -2.09 1.15 -2.09 0 C -2.09 -1.15 -1.15 -2.09 0 -2.09 C 1.15 -2.09 2.09 -1.15 2.09 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.25 0 L 1.25 0}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}}}\rightarrow\mathsf{T}_{\mathbin{\vbox{\hbox{\hbox to3.41pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 1.70694pt\lower-1.70694pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.09 0 C 2.09 1.15 1.15 2.09 0 2.09 C -1.15 2.09 -2.09 1.15 -2.09 0 C -2.09 -1.15 -1.15 -2.09 0 -2.09 C 1.15 -2.09 2.09 -1.15 2.09 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.25 0 L 1.25 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 -1.25 L 0 1.25}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}}}$	$0.842\pm 0.082$	$0.841\pm 0.071$	$0.835\pm 0.074$
$\mathsf{T}_{\mathbin{\vbox{\hbox{\hbox to3.41pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 1.70694pt\lower-1.70694pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.09 0 C 2.09 1.15 1.15 2.09 0 2.09 C -1.15 2.09 -2.09 1.15 -2.09 0 C -2.09 -1.15 -1.15 -2.09 0 -2.09 C 1.15 -2.09 2.09 -1.15 2.09 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.25 0 L 1.25 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 -1.25 L 0 1.25}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}}}\rightarrow\mathsf{I}_{\mathbin{\vbox{\hbox{\hbox to3.41pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 1.70694pt\lower-1.70694pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.09 0 C 2.09 1.15 1.15 2.09 0 2.09 C -1.15 2.09 -2.09 1.15 -2.09 0 C -2.09 -1.15 -1.15 -2.09 0 -2.09 C 1.15 -2.09 2.09 -1.15 2.09 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.25 0 L 1.25 0}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}}}$	$0.839\pm 0.047$	$0.817\pm 0.037$	$0.833\pm 0.043$
$\mathsf{T}_{\mathbin{\vbox{\hbox{\hbox to3.41pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 1.70694pt\lower-1.70694pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.09 0 C 2.09 1.15 1.15 2.09 0 2.09 C -1.15 2.09 -2.09 1.15 -2.09 0 C -2.09 -1.15 -1.15 -2.09 0 -2.09 C 1.15 -2.09 2.09 -1.15 2.09 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.25 0 L 1.25 0}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}}}\rightarrow\mathsf{I}_{\mathbin{\vbox{\hbox{\hbox to3.41pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 1.70694pt\lower-1.70694pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{{}}{}{{{}}{\lx@inpgf@ignorespaces}{}{\lx@inpgf@ignorespaces}{}{}{}{}{}}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 0 M 2.09 0 C 2.09 1.15 1.15 2.09 0 2.09 C -1.15 2.09 -2.09 1.15 -2.09 0 C -2.09 -1.15 -1.15 -2.09 0 -2.09 C 1.15 -2.09 2.09 -1.15 2.09 0 Z M 0 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -1.25 0 L 1.25 0}{fill:none} \lx@inpgf@ignorespaces {{\lx@inpgf@ignorespaces}{}}{{}}{} {{\lx@inpgf@ignorespaces}{}}{}{}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 -1.25 L 0 1.25}{fill:none} \lx@inpgf@ignorespaces \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}}}}$	$0.699\pm 0.069$	$0.695\pm 0.074$	$0.682\pm 0.083$
Mean	$0.807$	$0.799$	$0.799$

Appendix

Table 6: Causal performance of direct and composed transport paths. Entries report mean

\pm

sample standard deviation of destination-normalized autoregressive transfer across the

3\times 3

source–destination DAS seed combinations. The composed paths use only the two simpler pairwise maps along the corresponding modality-first or operation-first route.

Figure 14: Complete composed-path comparison. For each cross-operation direction, the figure compares the directly fitted transport with the modality-first and operation-first compositions. Panels report destination-normalized causal transfer, cosine to the true destination displacement, cosine between the composed and direct-map predictions, and normalized Frobenius distance to the direct operator.

Figure 15: Final-position causal control depends specifically on digit-readout coupling, but is carried by neither component alone. (a) Cumulative autoregressive recovery with the

22

DAS directions ordered from most to least digit-readout aligned, and in reverse. The dashed line marks the rank-

9

boundary between

{\bm{D}}

and

{\bm{L}}

defined in Section 5 . Neither component is sufficient by itself. (b) Removing

{\bm{U}}_{\mathrm{dig}}

from the complete DAS state almost eliminates autoregressive control in all four conditions, whereas ablating a dimension-matched random span preserves a substantial fraction of the original effect.

Representation	Transition cosine	Top-1	RSA $\rho$
Projected DAS (22d)	$0.908$	$98.1\%$	$0.808$
Exact ${\bm{L}}$ (13d)	$0.807$	$75.5\%$	$0.765$
Random readout-orthogonal (13d)	$0.558$	$38.3\%$	$0.545$
Shuffled value correspondence	$-0.003$	$5.2\%$	–

Appendix

Table 7: Cross-condition numerical geometry after separating the current digit readout. Transition cosine and retrieval are evaluated only on numerical values excluded from the alignment fit. Top-1 retrieval is among

20

held-out values, so chance is

5\%

. A dash indicates that RSA was not evaluated.

Map	Transition cosine	Top-1	Top-5
Direct pairwise	0.801	73.4%	98.4%
Full synchronization	0.791	74.8%	98.9%
LORO synchronization	0.762	71.1%	98.3%
Modality-only frames	0.249	18.1%	55.3%

Appendix

Table 8: Global consistency of the exact

{\bm{L}}

spaces. All metrics are evaluated on numerical values excluded from the pairwise alignment fits.

Figure 16: A local successor relation transfers across the shared numerical frame. After registering the four full

22

-dimensional DAS spaces into common coordinates, an orthogonal successor operator

G_{+1}

is fitted using text addition only and frozen. On held-out numerical values, the same operator moves representations toward the corresponding

y+1

centroid in text/image addition and subtraction, while identity and random-orthogonal controls fail. Transfer is strongest for one local step and degrades under repeated application.

Condition	$y$	Hybrid	Base
Matched	74.5	1.7	7.8
Swap ${\bm{L}}$	4.7	51.6	11.8
${\bm{D}}$ only	10.4	7.5	17.4
${\bm{L}}$ only	4.4	1.2	75.9
Random ${\bm{L}}$	8.7	49.1	10.9

Appendix

Table 9: Mismatched edits on pairs with different first digits. Percentage of generations equal to

y

, to the hybrid (tens digit of

y

and units digit of the value supplying

{\bm{L}}

, which is

\tilde{y}

except for Random

{\bm{L}}

), and to the base answer. For Matched,

{\bm{D}}

only and

{\bm{L}}

only, the hybrid column counts how often the output coincides with the hybrid defined from the trial’s

y

and

\tilde{y}

Figure 17: The readout-coupled and readout-orthogonal components have distinct temporal causal roles. (a) The immediate digit follows

{\bm{D}}

. When the values supplying

{\bm{D}}

and

{\bm{L}}

require different first digits, the current prediction follows the value inserted through

{\bm{D}}

; an

{\bm{L}}

-only intervention does not comparably redirect that digit. (b) Numerical identity in

{\bm{L}}

redirects the continuation. For values

y

and

\tilde{y}

sharing their first digit but differing in their second, the matched intervention recovers

y

, whereas replacing only

\Delta h_{{\bm{L}}}^{b\rightarrow y}

with

\Delta h_{{\bm{L}}}^{b\rightarrow\tilde{y}}

redirects autoregressive behaviour toward

\tilde{y}

. A random latent displacement does not reproduce this value-specific effect.

Figure 18: The readout-orthogonal numerical geometry extends beyond arithmetic. (a) The repeat task supplies the numerical value directly. No repeat-specific DAS space is trained and no intervention is performed; the layer-

43

final-position activation is projected into the arithmetic

{\bm{L}}

basis and compared with arithmetic states representing the same values. (b) Estimator-matched comparison over

200

samples of one arithmetic expression per numerical value. Violin distributions show held-out transition cosine and no-fit RSA for arithmetic-to-arithmetic and repeat-to-arithmetic comparisons, with markers indicating the mean.

Condition	TF-IIA, seeds 0/1/2	TF mean $\pm$ std	AR-IIA, seeds 0/1/2	AR mean $\pm$ std
$T_{+}$	.326/.326/.324	$.325\pm.001$	.312/.297/.289	$.299\pm.012$
$T_{-}$	.500/.504/.486	$.497\pm.009$	.375/.414/.414	$.401\pm.023$
$I_{+}$	.318/.299/.318	$.312\pm.011$	.312/.297/.305	$.305\pm.008$
$I_{-}$	.473/.459/.469	$.467\pm.007$	.336/.320/.336	$.331\pm.009$

Appendix

Table 10: Final

k=32

DAS results at layer 37 and the final pre-generation position, across three independently initialized seeds. Standard deviations are sample standard deviations across seeds.

Representation/control	Transition cosine	Top-1	Top-5
Exact $L$ (23d)	$.8963\pm.0463$	$93.35\%$	$99.85\%$
Random readout-null (23d), mean	$.6950$	$72.30\%$	$94.17\%$
Label permutation, mean	$\approx 0.0000$	$5.00\%$	$25.01\%$

Appendix

Table 11: Held-out numerical geometry in the Ministral readout-free space. Top-1 and top-5 retrieval are among 20 unseen candidate values, giving chance levels of

5\%

and

25\%

. The random row is the mean over 200 independent dimension-matched readout-null projections; the permutation row is the mean over 500 numerical-label permutations of the real

L

spaces.

Map	Transition cosine	Top-1	Top-5
Direct pairwise	$.89642\pm.04611$	$93.33\%$	$99.82\%$
Full synchronization	$.89482\pm.04515$	$93.84\%$	$99.86\%$
LORO synchronization	$.89020\pm.04357$	$93.89\%$	$99.91\%$

Appendix

Table 12: Global consistency of the Ministral

L

spaces. LORO removes both directions of one complete condition relation before synchronization and reconstructs the missing relation only through the remaining task graph.

Task	Matched $P(y_{2})$	Mismatch $P(\widetilde{y}_{2})$	Matched $\rightarrow y$	Mismatch $\rightarrow\widetilde{y}$	$\Delta P(y_{1})$
$I_{+}$	.257	.263	.174	.206	$-.010$
$I_{-}$	.371	.375	.286	.339	$-.017$
$T_{+}$	.262	.249	.268	.279	$-.006$
$T_{-}$	.440	.405	.414	.430	$+.003$

Appendix

Table 13: Per-condition consistency of the same-first-digit latent mismatch, averaged over the three DAS seeds.

\Delta P(y_{1})

is the mismatch-minus-matched change in probability of the shared first digit. The qualitative continuation effect appears in every modality and operation.

Multimodal Model Interpretability

Multimodal model interpretability studies how AI systems combine and use images, text, audio, video, and other signals, and how to explain their decisions.

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Multiplication in Multimodal LLMs: Computation with Text, Image, and Audio Inputs
Apr 20, 2026Samuel G. Balter, Ethan Jerzak, Connor T. JerzakMultimodal Large Language Models Multimodal Reasoning
University of Texas at Austin · National University of Singapore (NUS)
What MLLMs Learn about When they Learn about Multimodal Reasoning
Oct 2, 2025Jiwan Chung, Neel Joshi, Pratyusha Sharma +2Multimodal Reasoning Multimodal Model Evaluation
Microsoft Research AI Frontiers · Yonsei University · Seoul National University
Reading, Not Thinking: Understanding and Bridging the Modality Gap When Text Becomes Pixels in Multimodal LLMs
Mar 10, 2026Kaiser Sun, Xiaochuang Yuan, Hongjun Liu +4Multimodal Large Language Models Modality Gap in VLMs
Johns Hopkins University · Amazon · New York University +1

Apr 20, 2026J/K move · Enter open · S save

Multiplication in Multimodal LLMs: Computation with Text, Image, and Audio Inputs

Samuel G. Balter, Ethan Jerzak, Connor T. Jerzak

University of Texas at Austin · National University of Singapore (NUS)

Multimodal Large Language Models Multimodal Reasoning Multimodal Foundation Models Multimodal Model Evaluation Multimodal Understanding

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Multimodal LLMs can accurately perceive numerical content across modalities yet fail to perform exact multi-digit multiplication when the identical underlying arithmetic problem is presented as numerals, number words, images, or in audio form. Because existing benchmarks often lack systematically paired instances across modalities, it remains difficult to compare genuine arithmetic limits within and across model families. We therefore introduce a controlled multimodal multiplication benchmark that factorially varies digit length, digit sparsity, representation (e.g., numerals vs. number words), and modality (text, rendered images, audio), with paired instances from a reproducible generator. We also define arithmetic load, C, as the product of the total and non-zero digit count as a compact, mechanistically motivated proxy for operation count. Across evaluations, accuracy falls sharply as C grows, often nearing zero by C > 100. Indeed, C remains predictive of performance across modalities and models, with R-squared often > 0.5, nearing the value from more complex measures of arithmetic load that count the number of intermediate arithmetic steps. A separate perception-versus-computation decomposition shows that multimodal degradation is primarily computational rather than perceptual: on matched-perception checks, models are near-perfect (> 99%) across modalities, even when multiplication accuracy drops. Beyond measuring when models fail, we ask which procedures they are predisposed to follow. We introduce a forced-completion loss probe that scores heuristic-specific reasoning prefixes--including columnar multiplication, distributive decomposition, and rounding/compensation. Here, decomposition is favored in both text and vision modalities; heuristic-specific LoRA adapters produce near-orthogonal updates yet degrade accuracy, indicating the base model maintains a well-tuned internal router.

Abstract

Multimodal Model Interpretability

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