Concept Subspaces Compute Beyond the Logit Lens: A Weights-Only Test for Locating Representations Upstream of Readout

Aojie Yuan; Zhiyuan Julian Su; Haiyue Zhang; Zijian Su

cs.CLSep 30, 2026

Concept Subspaces Compute Beyond the Logit Lens: A Weights-Only Test for Locating Representations Upstream of Readout

Authors: Aojie Yuan, Zhiyuan Julian Su, Haiyue Zhang, Zijian Su

Organizations: University of Southern California · Duke University · University of Michigan

Abstract

A concept subspace's effect on model behavior does not establish how it relates to the output readout. We introduce a two-sided geometric diagnostic that measures an extracted subspace's overlap with the dominant right-singular directions of the unembedding matrix, evaluated against output-oriented positive controls. Given an extracted basis, the raw diagnostic requires only model weights. Our testbed is the Format-Agnostic Reasoning Subspace (FARS), a ten-dimensional basis extracted from eighteen reasoning concepts expressed in six surface forms. Across nine rank-matched estimators and twenty-six models, four activation-derived concept estimators carry only 0.38--0.80% mean energy in the top-ten readout span. Final-layer PCA carries 3.56%, exceeding FARS in 25 of 26 models. A same-layer next-token control, evaluated using a fitted linear translator for depth matching, carries approximately thirteen times more energy than FARS, with separation in all 25 tested models. Re-extracting FARS on ten disjoint concepts yields 62--100% cross-format retrieval across twenty-four generative models, demonstrating transfer of the extraction procedure rather than a fixed basis. A complementary four-model, three-seed intervention study finds model-dependent source-directed effects that remain well below full-vector replacement. Together, the geometry and intervention controls distinguish concept structure from dominant readout directions while limiting claims of causal sufficiency.

Figures & tables

Figure 1: From controlled concepts to readout geometry. Six schematic forms define a shared concept basis. Principal angles compare it with the dominant readout span, calibrated by two controls. Illustrations are schematic; quantitative results follow.

Model	Argmax preserved (%) $\uparrow$			Behavioural (drop pp) $\downarrow$
$L{=}$ best FARS layer	FARS-10	PCA-10	Full	GSM8K	MATH
GPT-2 XL	93.7	87.7	81.0	—	—
Qwen-2.5-3B	89.7	37.3	22.0	—	—
Qwen-2.5-7B-Inst.	94.7	59.7	26.3	$\mathbf{-20.0}$ pp	$\mathbf{-18.4}$ pp
Mistral-7B	97.3	85.3	83.3	—	—
Llama-3.1-8B	94.0	56.7	39.7	$+5$ (n.s.); $-22$ at $c^{\star}{=}2$	—

Table 1: Argmax-preservation under

10

-d cross-form patching (paired Wilcoxon

p{<}10^{-22}

vs. Full-PCA on every model). Behavioural: FARS ablation at the FARS layer costs

-20

(Qwen-

7

B-Inst),

-28

(Phi-

3.5

-17

(Qwen-

3

B-Inst), and

-10

pp (Mistral-

7

B-Inst) on GSM8K relative to matched-rank random (paired Wilcoxon

p\leq 0.03

); Llama-

8

B needs amplification

c^{\star}{=}2

(

-22

pp,

p{=}3{\times}10^{-3}

); Llama-

70

B is inert at

c\leq 2

(App. P.3 ).

Figure 2: Concept-subspace interventions have limited source-directed effects. Last-token replacement across four models; each dot is one seed with $120$ directed concept pairs. Bars show seed means, with no inferred confidence interval. Both random controls and the no-intervention control produce zero events. Full replacement is a reference intervention, not a guarantee of task success.

Rank- $10$ subspace	role	$d_{G}$	energy %	energy %	$\theta_{\min}$
		vs. $W_{U}$		vs. $W_{U}\!\to\!\ell$
Final-layer PCA	positive ctrl	4.556	3.56	0.86	65.5
FormPCA	control	4.738	1.26	0.90	75.8
FullPCA	control	4.751	1.04	0.83	77.5
Shuffled-FARS	control	4.762	0.90	0.62	78.9
Diff-of-means	published	4.774	0.80	0.46	79.5

Table 2: Readout-orthogonality of nine matched-rank subspace estimators, averaged over 26 models spanning four architecture families plus reasoning-tuned variants (

0.4

B–

70

B).

d_{G}

is the Grassmann geodesic distance to the top-

10

right singular vectors of

W_{U}

(ceiling

\sqrt{10}\pi/2=4.97

); energy % is the fraction of subspace energy inside the top-

10

readout span, measured against

W_{U}

directly and against the readout channels pulled back to the FARS layer

\ell

by a tuned-lens translator (

W_{U}\!\to\!\ell

; App. G , 25 models);

\theta_{\min}

is the smallest principal angle. Rows are ordered by readout-adjacency. Published estimators are the constructions used in the concept-direction literature; control rows probe rank and structure; the positive controls are the top-

10

PCA of the final layer, whose residual stream is the immediate input to

W_{U}

, and next-token-identity centroid PCA at the FARS layer itself — the same estimator as FARS with the grouping variable changed from concept to emitted token.

Figure 3: Positive controls distinguish readout loading. Each point is a model. (a) Final-layer PCA exceeds FARS in

25/26

models; the diamond marks R1-Distill-Qwen-7B. (b) Same-layer next-token PCA exceeds FARS in all

25

depth-matched models. Black ticks mark Haar expectations at the corresponding control energy. Dashed lines indicate equal loading; axis ranges differ. Panel (b) uses table values rounded to

0.01

percentage point.

Probe	Llama pair	Qwen pair	QwQ-32B	Baseline
$d_{G}$ (base last, distill last)	$\mathbf{3.06}$	$\mathbf{3.48}$	—	$4.97$ (rank- $10$ ceiling)
$d_{G}$ (last, CoT)	$\mathbf{4.53}$	$\mathbf{4.26}$	$\mathbf{4.33}$	$4.97$ (rank- $10$ ceiling)
CoT-tail FARS % cross-format Agn.	$\mathbf{60.8}$	$\mathbf{58.0}$	$\mathbf{59.0}$	$21.6$ – $24.4$ (Haar rand.)
X-FARS pool $+$ Llama-Distill	$\mathbf{55.6\%}$ ( $+1.7$ pp)		—	$53.8\%$ (15 models)

Table 3: Reasoning-tuned CoT departure at a glance. Row 1 (Q1): base-to-distilled distances are nonzero and below the rank-

10

ceiling; this is a geometric comparison in paired hidden coordinate spaces. Row 2 (Q2): within each reasoning model, the last-token and CoT-tail subspaces sit at large aggregate Grassmann distances, including the natively trained QwQ-32B. Row 3 : the CoT-tail subspace itself carries cross-format concept identity at

58

–

61\%

top-

1

retrieval,

34

–

39

pp above a matched-rank Haar-random baseline. Row 4 (Q3): absorbing R1-Distill-Llama-8B into the shared X-FARS frame improves absolute alignment (

53.8\to 55.6\%

) with the Haar-random-null gap

27.2

–

29.6

pp across the three pools. Detailed tables and run distinctions appear in App. W .

Appendix figures & tables78 assets

Supplementary material from the paper’s appendix.

Appendix

Form	Text
English	If it rains then the ground is wet. It rains. Therefore the ground is wet.
Chinese	如果下雨，那么地面是湿的。下雨了，所以地面是湿的。
French	S’il pleut alors le sol est mouillé. Il pleut. Donc le sol est mouillé.
Python	def mp(p, q): return q if p else None
Math	$P\to Q,\;P\;\vdash\;Q$
Struct.	P1: P->Q \| P2: P \| Rule: MP \| Q

Appendix

Table 4: Example: modus ponens in 6 surface forms.

Domain	Concept	Example
Arithmetic	Multi-step eval	$(7{+}3){\times}4{-}8/2$
	Modular	$47\bmod 7$
	Proportional	3 cost $12; 7 cost?
	GCD	$\gcd(48,18)$
Logic	Syllogism	All A are B; all C are A
	Modus ponens	P $\to$ Q; P; therefore Q

Appendix

Table 5: 18 concepts across 5 reasoning domains.

Domain	Concept	Example (math form)
Temporal	Event ordering	$t_{A}=15{:}00$ , $t_{B}=t_{A}+30$ min, $t_{C}\in(t_{B},16{:}00)$
	Duration arithmetic	$t_{\text{end}}=t_{\text{start}}+\Delta t$
Probabilistic	Conditional probability	$P(B_{2}{=}\text{red}\mid B_{1}{=}\text{red})=2/4$
	Expected value	$E[X]=\sum_{i=1}^{6}i\cdot(1/6)=3.5$
Graph	Shortest path	$d(A,C)=\min(d(A,C),\,d(A,B){+}d(B,C))$
	Connectivity	$E=\{(A,B),(C,D)\}$ ; $A\nsim C$

Appendix

Table 6: The disjoint Novelty inventory:

10

concepts across

5

domains, none of them in TriForm. Rendered in the same six surface forms,

3

instances each, giving

180

stimuli. Examples are the mathematical-notation rendering.

Model	Space	Dim	Concept-ARI	Concept%	Form%
GPT-2 XL	Full	1600	.142	28.7	46.0
	FARS	10	.300	42.6	25.9
	Form ctrl	5	.038	17.6	92.6
Qwen-7B	Full	3584	.284	42.3	25.9
	FARS	10	.619	66.7	23.8
	Form ctrl	5	.036	16.7	95.4

Appendix

Table 7: Cluster purity at the best FARS layer.

k

-means accuracy (Hungarian-matched). Chance:

5.6\%

concept (

18

-way),

16.7\%

form (

6

-way). FARS and the form-control subspace are quantitatively orthogonal information carriers.

Figure 4: FARS dimensionality sweep (

k\in\{1,\ldots,17\}

). The saved five-model sweeps attain

97.1

–

97.9\%

of their

k{=}17

concept RSA at

k{=}10

. Each point reports the saved best-layer statistic; selected layers can vary with

k

Model	RSA-C	Probe%	Purity%	FARS Patch	$z$
Qwen2.5-7B (base)	.150	59.8	66.7	.943	28.8
+Instruct	.153	62.9	68.8	.867	25.5
Llama-3.1-8B (base)	.165	53.9	—	.941	—
+Instruct	.187	59.8	67.6	.890	12.6

Appendix

Table 8: Instruction tuning preserves and slightly strengthens FARS.

Benchmark	Direction	Full top-1	FARS top-1	$\Delta$
TriForm in-dist. ( $n{=}30$ pairs, avg.)	all	$.559$	$\bm{.738}$	$\bm{+.179}$
MBPP-sanitized ( $n{=}200$ )	prose $\!\to\!$ code	$.065$	$.025$	$-.040$
MBPP-sanitized ( $n{=}200$ )	code $\!\to\!$ prose	$.150$	$.030$	$-.120$

Appendix

Table 9: In-distribution vs. out-of-distribution retrieval on Llama-3.1-8B-Instruct. Inside TriForm,

10

-d FARS outperforms full

4096

-d by

+18

+23

pp; on MBPP-sanitized whose problems lie outside the

18

TriForm concepts, FARS is far below full, the construction is concept-distribution-specific by design.

	FARS				Control
Model	RSA-C	RSA-F	Probe	Boost	RSA-F	RSA-C
GPT-2 XL	.317	.118	52.8	+20.1	.626	$-$ .003
Qwen-3B	.357	.056	64.4	+11.6	.634	$-$ .005
Qwen-7B	.367	.009	68.0	+8.2	.638	$-$ .010
Mistral	.366	.008	70.5	+10.2	.631	$-$ .003
Llama-8B	.368	.015	71.9	+18.0	.634	$-$ .007

Appendix

Table 10: FARS (10-dim) vs. Form Control (5-dim).

Model	Metric	FARS	Shuffled	LDA	Full-PCA
GPT-2 XL	RSA-C $\uparrow$	.279	$.072\pm.008$	.381	.091
GPT-2 XL	RSA-F $\downarrow$	.107	.473	$-$ .011	.486
Qwen-7B	RSA-C $\uparrow$	.357	$.128\pm.008$	.384	.151
Qwen-7B	RSA-F $\downarrow$	.020	.243	$-$ .012	.261
Mistral-7B	RSA-C $\uparrow$	.362	$.149\pm.013$	.380	.194
Mistral-7B	RSA-F $\downarrow$	.012	.259	$-$ .012	.275

Appendix

Table 11: Concept (C) and form (F) RSA in the

10

-dim FARS subspace vs. (i) shuffled-centroid PCA, (ii) supervised LDA, (iii) Full-PCA. FARS exceeds the shuffled null by

17

–

29\sigma

and matches LDA.

Figure 5: Full and projected representations of 324 stimuli. Mistral-7B-v0.3, cached layer 11; t-SNE with seed 42, perplexity 30 and 2,000 iterations. (a) Full space, colored and marked by form. (b,c) The same FARS embedding colored by concept (18 colors) or marked by form. Full and FARS spaces are embedded independently; their axes and distances are not directly comparable. This visualization is descriptive, not a test of separability.

Model	Family	$L/L_{\max}$	$d_{G}$	$\theta_{\min}$	FARS energy
Llama-3.1-70B-Inst	dense	23/80	$\mathbf{4.86}$	$85.6^{\circ}$	$\mathbf{0.16}\%$
OLMo-2-7B-Inst	dense	13/32	$4.82$	$84.2^{\circ}$	$0.31\%$
Falcon3-7B-Inst	dense	8/27	$4.82$	$84.8^{\circ}$	$0.31\%$
Mistral-7B-v0.3	dense	11/31	$4.82$	$82.2^{\circ}$	$0.36\%$
Mistral-7B-Inst	dense	10/32	$4.81$	$81.3^{\circ}$	$0.40\%$
Phi-4	dense	12/39	$4.81$	$82.3^{\circ}$	$0.40\%$

Appendix

Table 12: FARS versus the top-

10

readout channels of

W_{U}

, per model, across 26 LLMs (four architecture families plus reasoning-tuned variants).

d_{G}

is the Grassmann geodesic distance (ceiling

\sqrt{10}\pi/2=4.97

\theta_{\min}

the smallest principal angle, and the last column the fraction of FARS energy inside the top-

10

readout span.

L/L_{\max}

is the FARS layer as a fraction of depth; every model sits in the middle band, which matters because a near-final layer would be readout-adjacent for reasons unrelated to the concept subspace. Distances span

4.60

–

4.86

and the minimum principal angle exceeds

74^{\circ}

in 25 of 26 models. The single outlier, R1-Distill-Qwen-7B, is the model our chain-of-thought analysis independently flags (§ 8 ); its loading falls to the Haar floor once the readout is pulled back to its FARS layer (App. G ).

Model	$\ell/L$	$R^{2}$	FARS	NextTok ℓ	Last-PCA	FARS	LDA	FullPCA	Haar	NextTok ℓ
			energy % in top- $10$ of $W_{U}$			energy % in top- $10$ of $W_{U}\,\mathrm{diag}(\gamma)A_{\ell}$
Phi-4	12/39	0.40	0.40	1.19	4.29	0.22	0.25	0.61	0.26	$\mathbf{6.88}$
Llama-3.1-70B-Inst	22/79	0.37	0.16	0.72	1.65	0.19	0.19	0.45	0.10	$\mathbf{5.72}$
Mistral-7B-Inst	9/31	0.34	0.40	6.57	4.58	0.37	0.29	1.03	0.25	$\mathbf{8.09}$
Falcon-Mamba-7B	29/63	0.50	0.28	0.45	0.31	0.32	0.28	0.61	0.22	$\mathbf{6.17}$
Llama-3.1-8B-Inst	8/31	0.44	0.49	0.84	4.51	0.29	0.24	0.52	0.30	$\mathbf{5.26}$

Appendix

Table 13: Readout loading at matched depth across 25 models. Left block: the body’s test (readout

=

top-

10

right singular vectors of

W_{U}

). Right block: the same subspaces against the readout channels pulled back to the FARS layer

\ell

. Under the pulled-back readout the concept estimators stay at the Haar floor (FARS 0.46% vs. random 0.30%; the Haar expectation is

k/D

), whereas the same-layer, same-estimator next-token control loads 5.88% — 12.9

\times

FARS — and exceeds all four published estimators in 25/25 models.

R^{2}

is the held-out fit of the translator.

Figure 6: Readout loading across subspace constructions. Each small dot is one model and the larger role-specific marker is the median. The statistic is energy in the top-

10

right singular span of

W_{U}

, on a logarithmic axis; the dotted reference is the median empirical Haar control. Sample sizes appear at right. Symbols distinguish activation-derived estimators, structural controls, positive controls, and vocabulary-derived constructions. Pulled-back readout results use a different reference and appear separately in App. G . Published directions use the same paired cohorts as Fig. 8 : any available AUC below

0.9

excludes a pair; missing final-block AUCs remain flagged there. The pooled Haar median is a visual reference, not a model-specific significance threshold.

		enrichment $E(k)/(k/D)$ , FARS / final-layer			Haar
Model	$D$	$k{=}10$	$k{=}32$	$k{=}100$	$k{=}10$
DeBERTa-v3-large	1024	0.83 / 1.2	0.92 / 1.1	0.93 / 1.1	1.03
DeepSeek-V2-Lite	2048	1.26 / 11.7	1.12 / 6.3	1.06 / 3.2	1.00
Falcon-Mamba-7B	4096	1.17 / 16.8	1.36 / 6.8	1.23 / 3.4	1.01
Falcon3-7B-Inst	3072	0.96 / 10.7	0.95 / 5.8	1.00 / 2.9	0.98
gpt-oss-20B	2880	0.77 / 15.7	0.88 / 10.3	0.83 / 5.4	1.00

Appendix

Table 14: Readout enrichment at three ranks for the concept subspace (FARS) and the readout pathway (top-

10

PCA of the final layer), across 26 models. The Haar column is the null check: a random rank-

10

subspace has expected enrichment

1.00

; the sampled controls are close (max deviation

0.046

). Enrichment generally approaches the Haar expectation as rank grows; the magnitude and ordering vary across models. The same tabulated comparisons are visualized in Fig. 7 .

Figure 7: Paired readout enrichment at three reported ranks. Each point is one of

26

models, plotted from the rounded values in Table 14 . Both axes use the same logarithmic scale. The dashed diagonal marks equal enrichment; dotted lines mark the Haar expectation of

1

. Diamonds identify models where final-layer PCA is below FARS: R1-Distill-Qwen-7B at all three ranks, and Mamba-2.8B additionally at

k=100

. No error bars or full-spectrum trajectories are inferred from these table values.

		AUC		Haar	vs. $W_{U}$		vs. $W_{U}\!\to\!\ell$			final block, vs. $W_{U}$
Model	$\ell$	ref.	truth	$k/D$	refusal	truth	refusal	truth	NextTok 1	refusal	truth
DeepSeek-V2-Lite	13	1.00	1.00	0.49	0.59	1.42	1.04	1.01	1.48	7.55	8.37
Falcon-Mamba-7B	29	1.00	1.00	0.24	0.40	0.36	2.12	0.11	2.81	1.06	5.77
Falcon3-7B-Inst	8	1.00	1.00	0.33	0.41	0.59	1.06	0.18	0.02	1.33	5.54
gpt-oss-20B	7	1.00	1.00	0.35	0.29	0.24	0.51	0.15	2.61	4.05	3.84
GPT-2 XL	20	1.00	1.00	0.62	3.92	0.33	4.28	0.28	0.30	7.86	1.44

Appendix

Table 15: Published concept directions under the readout test (22 models; energies in %). At the FARS block both the refusal and the truth direction sit at the Haar floor against

W_{U}

(medians 0.42% and 0.36%,

k/D\approx 0.26\%

) and against the pulled-back readout (0.44% and 0.20%), while the first next-token direction at the same block loads 2.71%. The same two directions built at the final block load 1.98% and 4.09% (medians): the middle-layer construction has low linear readout loading and becomes readout-adjacent only where the residual stream is about to be decoded. † AUC

<0.9

Figure 8: The same published direction changes verdict with extraction depth. Each line is one model: the energy its refusal (left) or truth (right) direction carries in the top-

10

span of

W_{U}

, read off the FARS block versus the final block. Estimator, concept, stimuli and model are identical along each line; only the depth changes. Band: the range of per-model Haar expectations

k/D

; dark line: population median. Grey lines fall. Directions with available held-out AUC

<0.9

at either block are omitted. Dashed pairs lack a final-block AUC (three refusal, two truth) and are retained with this qualification.

Model	$D$	$k/D$	FARS	random-token	vocabulary-anchored	ratio
		Haar	energy % in top- $10$ readout span
Llama-3.1-70B-Inst	8192	0.12	0.16	2.37	$\mathbf{12.47}$	77.3 $\times$
SmolLM3-3B	2048	0.49	0.64	26.49	$\mathbf{26.11}$	40.7 $\times$
R1-Distill-Qwen-14B	5120	0.20	0.23	1.99	$\mathbf{6.12}$	27.0 $\times$
Llama-3.1-8B-Inst	4096	0.24	0.49	4.27	$\mathbf{12.94}$	26.4 $\times$
gpt-oss-20B	2880	0.35	0.27	5.99	$\mathbf{6.95}$	25.9 $\times$

Appendix

Table 16: Vocabulary-anchored concept directions against the readout basis, 26 models. The construction lies in the row space of

W_{U}

by design, so some loading is guaranteed; the honest baseline is therefore the random-token column, which repeats the identical construction with random token sets of the same size (

20

draws). Concept-diagnostic tokens load 8.43% on average against 5.20% for random tokens and 0.80% for FARS at its own layer — a 16.9

\times

separation from FARS, in the same direction in 26/26 models, with per-model ratios spanning 1.7

\times

–77.3

\times

. The two subspaces are not the same object: their mutual energy overlap is 0.56%.

Rank- $10$ subspace	$d_{G}$	energy %	$\theta_{\min}$
SAE dictionary (top- $10$ PCA of live decoder directions)	4.590	3.93	56.3
SAE concept features (selectivity-weighted, rank $10$ )	4.748	1.06	73.8
FARS at the same layer	4.759	0.96	75.1
SAE random features (control)	4.749	0.88	76.5
Haar expectation $k/D$	—	0.24	—

Appendix

Table 17: SAE feature directions against the top-

10

readout channels of

W_{U}

on Llama-3.1-8B-Inst. The SAE’s concept-selective features have low linear readout loading at 1.06%, indistinguishable from FARS at the same layer (0.96%) and from randomly chosen features (0.88%). The dictionary as a whole is different: the top-

10

PCA of all live decoder directions sits at 3.93%, roughly 4

\times

its own concept features — a dictionary trained to reconstruct the residual stream inherits whatever the readout writes into it, while the features that carry concept identity do not.

Model	RSA-C	RSA-F	Probe-W%	Probe-X%	X/W	Agn%
GPT-2 XL	.100	.430	96.6	32.7	.34	20.0
Qwen2.5-3B	.126	.501	98.5	52.8	.54	23.3
Qwen2.5-7B	.150	.485	99.1	59.8	.60	19.5
Mistral-7B	.200	.529	98.5	60.3	.61	20.8
Llama-3.1-8B	.165	.525	99.4	53.9	.54	21.1

Appendix

Table 18: Localization of format-agnostic computation. RSA-C/F: peak concept/form RSA; Probe-W: within-form probe ceiling; Probe-X: cross-form probe (chance

5.6\%

); X/W: cross-/within-form ratio. Cross-form probing recovers

34

–

61\%

of the ceiling, jumping between

1.6

B (

0.34

) and

3

B (

0.54

Figure 9: Layerwise representation metrics in the historical five-model baseline. (a) Concept RSA. (b) Cross-form probe accuracy; the dotted line marks

1/18

chance. (c) Format-agnostic neuron fraction. Depth is normalized within each model. Curves reproduce the saved per-layer results; colors and markers jointly identify models.

Figure 10: Retrieval peak versus final layer. Each thin line joins two measurements from one model; the dark line joins population medians. (a) Cross-format retrieval. (b) Distance to the top-

10

readout span. These plots use the reported rounded values in Table 19 , not reconstructed layer-by-layer trajectories. Peak selection favours higher retrieval by construction.

Model	$L_{\text{peak}}/L$	frac.	Agn% peak	$d_{G}$ peak	Agn% last	$d_{G}$ last
Llama-3.1-8B-Inst	9/ 32	0.29	81.2%	4.79	66.4%	4.63
Mistral-7B	9/ 32	0.29	81.8%	4.81	69.4%	4.63
Llama-3.1-70B-Inst	24/ 81	0.30	87.3%	4.87	65.4%	4.74
Mistral-7B-Inst	11/ 33	0.34	84.0%	4.81	61.1%	4.63
Falcon3-7B-Inst	10/ 28	0.37	79.0%	4.82	32.7%	4.65
Mixtral-8x7B-Inst	12/ 33	0.38	82.4%	4.73	30.2%	4.73

Appendix

Table 19: Layer-localization of FARS across 26 models. Left half gives the layer

L_{\text{peak}}

at which cross-format concept retrieval Agn% peaks; right half gives the last-layer values. In every model the peak layer sits at depth fraction in

[0.29,0.97]

of the network, with

n_{\text{mid}}=17/26

peaks falling inside the middle band

[0.30,0.85]

. At the peak layer,

d_{G}

to the top-

10

readout channels is higher than at the last layer in 24/26 models (median

\Delta d_{G}=+0.12

), and Agn% drops sharply toward the last layer. Concept-invariance and readout-adjacency move in opposite directions along depth, so the FARS peak is not the layer nearest the readout basis.

Rank	Variance (% of $k$ =17)	Agn% (% of $k$ =17)
$k=3$	52.9	52.7
$k=5$	71.2	60.5
$k=8$	84.3	73.4
$k=10$	89.9	83.0
$k=12$	94.2	89.4
$k=17$	100.0	100.0

Appendix

Table 20: Concept-centroid PCA saturation on TriForm, mean over 9 models. Beyond

k=10

the marginal gain in both metrics is

\leq 10

percentage points;

k=10

captures

\sim 90\%

of both what is attainable and what a full-rank (

k=17

) subspace would attain. On the disjoint Novelty benchmark (10 concepts, at most 9 non-trivial dimensions)

k=10

is by construction full-rank and therefore trivially saturates; the ELBOW test above is only meaningful on TriForm.

Model	Variance (% of $k$ =17)	Agn% (% of $k$ =17)
qwen2.5-7b	89.1	82.5
qwen2.5-3b-instruct	91.7	74.9
phi-3.5-mini-instruct	91.3	86.0
mistral-7b-v0.3	87.9	86.1
gpt2-xl	91.3	85.8
olmo-2-7b-instruct	87.6	88.4

Appendix

Table 21: TriForm concept-centroid PCA saturation at

k=10

across 9 models. The plateau is architecture-agnostic: dense decoders, state-space models, and encoder-MLMs all reach

\geq 87\,\%

of the

k=17

variance at

k=10

, and

\geq 73\,\%

of the

k=17

cross-format retrieval accuracy. Choosing

k=10

therefore trades

\leq 13

percentage points of TriForm variance for a substantial rank reduction (

10

1600

–

8192

ambient dimensions, a

160

–

820\times

compression). The same

k=10

setting is used uniformly across all experiments in the paper.

Model	Consec in-plat	$\pm 2$ – $3$ in-plat	Outside	Random
GPT-2 XL	$.952$	$.902$	$.968$	$.067$
Qwen-3B-Inst	$.876$	$.766$	$.798$	$.058$
Phi-3.5-mini-Inst	$.854$	$.716$	$.835$	$.047$
Mamba-2.8B	$.947$	$.875$	$.956$	$.052$
Mistral-7B-base	$.868$	$.735$	$.745$	$.041$
Mistral-7B-Inst	$.856$	$.727$	$.682$	$.041$

Appendix

Table 22: Inter-layer FARS subspace overlap (mean cos of principal angles). Plateau = layers with RSA-C

\geq 90\%

peak. “Consec within” = overlap between consecutive plateau layers; “

\pm 2

–

3

within” = overlap between plateau layers at distance

2

3

. The chance baseline is

\sqrt{10/D_{m}}\approx 0.03

–

0.08

across our models. Observed in-plateau overlap is

\sim 17\times

chance: the canonical concept frame is a stable manifold across multiple layers, not a layer-specific artifact.

Model	$D$	next token	language id	sentiment	factual tru	inference v	mono.
gpt2-xl	1600	14.4	8.2	4.0	1.4	1.0	✓
phi-3.5-mini-instruct	3072	13.5	35.4	18.0	9.7	7.2	—
qwen2.5-7b	3584	62.9	2.1	4.8	33.9	38.0	—
llama-3.1-8b-instruct	4096	17.3	2.1	4.6	3.8	1.5	—
mistral-7b-v0.3	4096	19.3	37.5	26.1	55.3	15.0	—
olmo-2-7b-instruct	4096	27.4	4.0	2.1	14.2	6.3	—

Appendix

Table 23: Final-layer readout enrichment

E_{v}(10)/(k/D)

of the difference-of-means direction for five binary concept families, across 6 models. Columns are in the pre-registered surface-to-abstract order, so the prediction is that each row decreases left to right; the last column marks whether it does. Haar null is

0.99

Model	FARS	Full vector
GPT-2 XL	10.8 [7.5, 16.7]	41.9 [34.2, 47.5]
Qwen2.5-3B-Inst.	3.3 [0.8, 7.5]	47.8 [43.3, 51.7]
SmolLM3-3B	0.8 [0.0, 1.7]	19.7 [16.7, 22.5]
Mamba-2.8B	0.8 [0.0, 1.7]	26.9 [21.7, 33.3]

Appendix

Table 24: Last-token source-directed event rates (%): seed mean [minimum, maximum],

3

seeds with

120

directed pairs each. Zero entries are identical across seeds. The brackets are observed seed ranges, not confidence intervals. Random seeds vary both pair sampling and the random basis.

Figure 11: Token-position protocol changes the measured intervention effect. FARS source-directed event rates for the legacy prefix and last-token protocols. Dots represent seeds; horizontal marks represent seed means. Shared concept pairs make these repeated observations. A position change is not a test of semantic correctness.

Figure 12: New logged dose sweeps, separated by protocol. Both panels use Qwen2.5-3B-Instruct at cached layer 22. Chat: baseline

90\%

; FARS at

c=2

6\%

; two random controls,

84\%

and

82\%

. Plain: baseline

32.5\%

; FARS at

c=0.5

55\%

, and at

c=2

20\%

. Lines join tested settings, without interpolated measurements or uncertainty estimates. Random controls match rank, not intervention norm.

Figure 13: Norm-matched intervention and generation limits. Each curve uses the same

50

questions; the three random bases are fixed across questions and doses. Baseline is

90\%

accurate. At

c=2

, FARS reaches

6\%

versus

40

–

58\%

for random controls, while output-limit rates are

2\%

versus

12

–

60\%

. Truncated generations remain in the accuracy denominator. Lines connect tested doses; random curves are separate basis draws, not confidence bounds.

	Top-1% $\uparrow$			Top-10 ov. $\uparrow$			KL med. $\downarrow$
Model	FARS	PCA	Full	FARS	PCA	Full	FARS	PCA	Full
GPT-2 XL	93.7	87.7	81.0	.90	.65	.47	.016	.154	.655
Qwen-3B	89.7	37.3	22.0	.92	.60	.44	.012	.612	2.77
Qwen-7B	94.7	59.7	26.3	.94	.63	.48	.006	.262	.683
Mistral-7B	97.3	85.3	83.3	.96	.74	.56	.002	.034	.142
Llama-8B	94.0	56.7	39.7	.94	.64	.47	.004	.098	.811

Appendix

Table 25: Causal validation under three metrics on the same

10

dimensions: top-

1

next-token match, top-

10

overlap, median KL of patched vs. clean. FARS cross-form patching matches within-form same-instance ceiling (

0.85

on Qwen-

7

B-Inst).

		Source-directed events (%)		Median $d_{B}$
Model	Pairs	FARS	Haar	FARS	Haar
GPT-2 XL	40	5.0	0.0	0.02435	0.00128

Appendix

Table 26: Historical cross-concept pilot,

40

directed pairs on one model. The FARS perturbation changes the output distribution more than this rank-matched Haar control, but the two interventions are not norm-matched. Values reproduce the saved pilot summary.

Panel of Table 27 — Table 27: Probe-level FARS ablation replicates across $4$ new architecture families. $L$ is the model’s best FARS layer, $\Delta$ the FARS-ablation effect in percentage points. Encoder models are mean-pooled. Base = cross-form ridge probe trained on form $A$ , tested on form $B$ at the model’s best FARS layer. FARS ablation removes the $10$ -d FARS subspace from both train and test activations; random ablation removes a matched-rank random subspace (mean over $3$ seeds). Every architecture family shows $\geq-21$ pp FARS-ablation effect with $\leq\pm 0.001$ random-ablation effect.

Model	Baseline	FARS abl.	Random abl.	FARS $\Delta$
GPT-2 XL	29.4	12.0	29.6	$-$ 17.4
Qwen2.5-7B	52.8	18.3	52.7	$-$ 34.5
Mistral-7B	52.8	15.6	52.9	$-$ 37.2
Llama-3.1-8B-Instruct	56.5	17.7	56.4	$-$ 38.8

Appendix

Table 28: Cross-form concept-probe accuracy with FARS or a random 10-dim subspace ablated from both training and test activations. “

\Delta

” is the drop from the no-ablation baseline. FARS ablation drops accuracy

17

–

44

pp; random ablation does not move accuracy detectably. The same 10 dimensions that constitute FARS are the literal carriers of cross-form concept identifiability.

Ablation	Top-1	Top-10 overlap	Median KL
Random 10-dim (control, mean of 5 draws)	0.988	0.984	0.0005
FARS 10-dim	0.767	0.804	0.065
Difference	$-$ 22 pp	$-$ 18 pp	$130\times$

Appendix

Table 29: Ablation control on Qwen2.5-7B-Instruct (top-

3

FARS layers,

n{=}54

per layer,

5

random draws). FARS ablation flips top-

1

argmax in

22

pp more cases than random ablation and median KL is

130\times

larger.

Figure 14: Historical amplification profiles, separated by model. Points reproduce the six chat-template summaries embedded in the original plotting script, not a newly rerun dose sweep.

c=0

is baseline and

c=-1

is projection removal. Lines connect observed settings only; no intermediate doses or error bars are inferred. These summaries are distinct from the paired FARS–random entries in Fig. 15 .

Figure 15: Paired historical FARS–random differences. Vertical values are percentage points of GSM8K accuracy, recomputed from the paired summaries embedded in the original threshold-plot script. These are not subtractions from the separate dose-profile figure: for Qwen-3B at

c=2

, the two sources report FARS accuracy

8\%

and

13\%

, respectively. Their run correspondence remains unresolved; they are therefore kept separate.

Model	$n$	Base.	Rand. abl.	FARS abl.	Paired test
Qwen2.5-7B-Instruct ( $L{=}13$ )	100	90.0	90.0	70.0	$p{=}4.0\!\times\!10^{-5}{}$
Llama-3.1-8B-Instruct ( $L{=}8$ )	100	86.0	85.5	85.0	$p{=}0.44{}$ (n.s.)

Appendix

Table 30: GSM8K under FARS- vs. random-ablation at each model’s best FARS layer (

n{=}100

, chat template). Qwen shows bidirectional necessity at

L{=}13

; Llama is inert to single-layer intervention at

L{=}8

Model (layer, $n$ )	Coef.	FARS	Random	$\Delta$	Paired $p$
Qwen-7B-Inst. ( $L{=}13$ , $n{=}100$ )	$-1.0$	.700	.900	$\bm{-.200}$	$\bm{4{\times}10^{-5}}$
	$+0.3$	.880	.910	$-.030$	$0.20$
	$+0.5$	.870	.910	$-.040$	$0.11$
	$+1.0$	.710	.900	$\bm{-.190}$	$\bm{2.7{\times}10^{-5}}$
Llama-8B-Inst. ( $L{=}8$ , $n{=}100$ )	$-1.0$	.850	.855	$-.005$	$0.44$
	$+0.3$	.850	.865	$-.015$	$0.29$

Appendix

Table 31: Bidirectional intervention

h\to h+c\cdot\mathbf{B}^{\top}\mathbf{B}h

(Coef.

=-1

is ablation) for FARS vs. random orthonormal

\mathbf{B}

. Qwen-

7

B: clean bidirectional necessity (

-20

pp both directions,

p{<}10^{-4}

). Llama-

8

B: inert under single-layer ablation at

L{=}7,8,14,18

; amplification at

L{=}7

with

c{=}{+}2.0

gives FARS

60.0\%

vs. random

82.0\%

(

p{=}1.4{\times}10^{-3}

n{=}50

). Llama-

70

B: inert at

c\in[1,2]

L{=}23

; the

11

-layer multi-scrub null indicates a wider plateau. Mistral-base: no usable baseline (

2\%

Jaccard does not predict patching (Pearson $r{=}{+}0.03$ , n.s.); code $\leftrightarrow$ math has
Pair	Jaccard	Patching	Class
en $\to$ math	.228	.696	decl
en $\to$ zh	.108	.319	decl
en $\to$ code	.216	.195	proc
code $\to$ math	.259	.208	proc
the highest Jaccard yet second-lowest patching, opposite to a tokenisation confound.

Appendix

Table 32: Form-pair Jaccard vs. mean patching overlap (averaged across

6

tested models). Procedural class: pair involves py_code .

Comparison	Mean overlap	Paired test
EN $\to$ math_notation	.646	—
EN $\to$ py_decl	.194	—
EN $\to$ py_code	.140	—
py_decl $-$ py_code (procedural axis)	$+.054$	$p{=}2.5\!\times\!10^{-9}$ , CI $[-.006,+.111]$
math_notation $-$ py_decl (surface axis)	$+.452$	$p{<}10^{-60}$
math_notation $-$ py_code (total gap)	$+.506$	$p{<}10^{-60}$

Appendix

Table 33: Decomposing the prose-code gap on Qwen2.5-7B-Instruct. Mean top-10 overlap, EN-prose source

\to

each target form, averaged over

54

stimuli

\times 7

layers (

n=378

). Paired Wilcoxon tests on within-(concept, instance, layer) differences.

Model	$n$	Prose refusal	Code refusal	Discordant (p/c)	$p$ (prose $>$ code)
Qwen2.5-7B-Instruct	100	.980	.970	1 / 0	.500 (n.s.)
Llama-3.1-8B-Instruct	200	.935	.965	1 / 7	.996 (reject)

Appendix

Table 34: Cross-format refusal on two instruction-tuned models, AdvBench harmful behaviours.

Quantity	Llama-3.1-8B-Inst. ( $L{=}7$ )	Mistral-7B-v0.3 ( $L{=}10$ )
Layer FARS-fraction	$0.269$	$0.225$
MLP share of layer FARS	$24.4\%$	$30.4\%$
Attention share of layer FARS	$75.6\%$	$69.6\%$
Top-3 head share	$15.8\%$	$12.7\%$
Top-5 head share	$23.8\%$	$18.7\%$
Top-3 head indices	H1, H14, H8	H31, H4, H0

Appendix

Table 35: Head-level FARS decomposition at each model’s writer layer (

324

stimuli). Attention contribution is the sum of all

32

heads’ FARS-projected output norms; MLP share is the MLP block’s share of the layer’s total FARS-bearing residual. Two architectures, same structural profile: layer-localised, head-distributed assembly. Head indices differ; the structural fingerprint replicates.

Intervention	Accuracy	$\Delta$ vs. baseline
Baseline (no hook)	$0.910$	—
Full FARS ablation ( $10$ -d)	$\mathbf{0.770}$	$\bm{-0.140}$
Canonical axis $0$ ( $1$ -d)	$0.910$	$+0.000$
Canonical axis $1$ ( $1$ -d, “procedural”)	$0.910$	$+0.000$
Canonical axis $2$ ( $1$ -d)	$0.930$	$+0.020$
Canonical axis $3$ ( $1$ -d)	$0.890$	$-0.020$

Appendix

Table 36: Axis-specific GSM8K ablation on Qwen2.5-7B-Instruct (

L{=}13

n{=}100

). Full FARS ablation drops

14

pp; no single canonical axis produces a separable effect (

\pm 2

pp, all

p{>}0.2

). FARS is a distributed

10

-d representation; single-axis interpretations are descriptive labels of geometric structure, not

1

-d causal handles.

Model	Peak / Writer	Plateau
GPT-2 XL	20 / 18	16–25
Qwen-2.5-3B-Inst	22 / 22	—
Phi-3.5-mini-Inst	18 / 17	—
Qwen-2.5-7B	13 / 12	9–19
Mistral-7B-v0.3	11 / 10	6–19
Mistral-7B-Inst	10 / 8	—

Appendix

Table 37: Writer layer (top concept-RSA on FARS-projected delta

B\Delta_{\ell}

) sits

0

–

5

layers below the peak FARS layer on all

15

models. Mamba has the widest gap, consistent with state-space residual propagation.

Model	Layer	Frozen FARS	Full space	Random mean
GPT-2 XL	20	22.8	48.3	20.2
Qwen2.5-7B	13	40.0	90.6	35.3
Mistral-7B-v0.3	11	39.4	86.7	30.3

Appendix

Table 38: Fixed-basis transfer with matched checkpoints and layers. Accuracy in percent; chance is

10\%

. Random is the mean of

20

bases. These descriptive comparisons are not significance tests.

Model	Family	TriForm-FARS		Novelty-FARS		Random
		Agn%	X/W	Agn%	X/W	Agn%
R1-Distill-Llama-8B	reasoning	79.6	0.77	$\mathbf{100.0}$	$\mathbf{1.00}$	50.6
Mixtral-8x7B-Inst	MoE	82.1	0.81	$\mathbf{100.0}$	$\mathbf{0.98}$	56.1
Llama-3.1-70B-Inst	dense	82.1	0.78	$\mathbf{100.0}$	$\mathbf{1.00}$	56.1
SmolLM3-3B	dense	82.4	0.80	$\mathbf{100.0}$	$\mathbf{0.99}$	54.4
Yi-1.5-9B-Chat	dense	75.0	0.72	$\mathbf{98.9}$	$\mathbf{1.01}$	45.0

Appendix

Table 39: Cross-format concept retrieval on TriForm vs. Novelty across 25 models spanning the MoE, dense, encoder-MLM, reasoning, state-space families. Chance rates are

1/18=5.6\%

(TriForm) and

1/10=10.0\%

(Novelty). The Random column is a Haar orthonormal rank-10 subspace of the same activation space with no concept supervision. Novelty-FARS exceeds it by 6–57 pp on every model, and the cross-to-within-format ratio

X/W

reaches

\geq 0.94

in 17 of 25 models. The higher Novelty headline reflects the smaller concept set (10 vs. 18), not a stronger effect.

Panel of Table 40 — Table 40: Held-out concept-RSA in the FARS basis trained on $15$ / $18$ concepts ( $N_{\text{splits}}=5$ random concept-splits each). Random $10$ -d orthonormal projection control at same layer. $\Delta=\text{FARS}-\text{random}$ . Every model shows positive $\Delta$ .

Panel of Table 41 — Table 41: FARS plateau width (fraction of layers maintaining $\geq 90\%$ peak concept-RSA), per-model peak RSA-C, and published benchmark scores. Plateau width correlates positively with all three benchmarks; the MMLU correlation passes $p<0.05$ .

Model	%	%	forms	by prompt	by trace	$X$ %
	trunc.	collapse	trace	cross-format $X$ (%)		Haar
R1-Distill-Llama-8B	79	56	6	30.6	26.9	8.0
R1-Distill-Qwen-7B	86	57	6	44.4	28.1	5.6

Appendix

Table 42: Format-collapse control on the CoT-tail subspace (2 reasoning-tuned models,

324

stimuli each). Collapse % is the fraction of non-English prompts whose generated trace is dominantly English. By prompt is the cross-format score as § 8 computes it; by trace regroups the same stimuli by the form the model actually wrote in. The confound is real — more than half of non-English prompts do produce English traces, and regrouping costs 12–37% of the score — but it does not manufacture the effect: 6 distinct surface forms survive in the generated text, and the regrouped score stays 3.3–5.1

\times

a matched-rank Haar subspace measured the same way.

Figure 16: Reported CoT-tail distances from last-input-token FARS. The bars show the manuscript’s aggregate Grassmann distances,

d_{G}=\sqrt{\sum_{i}\theta_{i}^{2}}

, for two R1-Distill models and native QwQ-32B. The dashed line marks the rank-

10

ceiling

\sqrt{10}\pi/2=4.97

. Individual principal angles are not plotted because an aggregate

d_{G}

does not uniquely determine them.

Pair	Base layer	Distilled layer	$d_{G}$
Llama-3.1-8B / R1-Distill-Llama-8B	22	20	3.06
Qwen2.5-7B / R1-Distill-Qwen-7B	7	13	3.48

Appendix

Table 43: Last-input-token FARS in base and distilled models. Layer indices are those reported for the reasoning-position extraction runs.

Model	Last-input layer	CoT-tail layer	$d_{G}$
R1-Distill-Llama-8B	20	14	4.53
R1-Distill-Qwen-7B	13	10	4.26

Appendix

Table 44: Within-model distance between separately extracted subspaces. Layer selection and token position both change in these comparisons.

Model	Last-input Agn. (%)	CoT-tail Agn. (%)	Haar (%)
R1-Distill-Llama-8B	79.6	60.8	21.6
R1-Distill-Qwen-7B	75.0	58.0	24.4

Appendix

Table 45: Original reasoning-position retrieval summaries on

324

stimuli. The regenerated control separately measures cross-format

X

after grouping by the form of the generated trace, rather than the original prompt.

Pool	X-FARS (%)	Haar mean $\pm$ SD	Haar p95	Gap (pp)
15 models	53.83	$25.82\pm 1.01$	27.58	28.01
16 (+Llama-Distill)	55.57	$25.94\pm 1.16$	28.01	29.63
17 (+Qwen-Distill)	51.83	$24.64\pm 1.12$	26.22	27.19

Appendix

Table 46: Cross-architecture concept retrieval and matched-optimizer Haar controls. The gap is explicitly X-FARS minus the Haar mean, calculated from the displayed rounded values; it is not the gain from adding a model.

Model	Stimuli	Mean length	Median length	Budget hit (%)
R1-Distill-Llama-8B	324	489	512	84.9
R1-Distill-Qwen-14B	324	389	512	63.9
R1-Distill-Qwen-7B	324	491	512	88.0

Appendix

Table 47: Generated-token lengths under the original

512

-token budget. These are the original runs; the regenerated trace-form controls have budget-hit rates of

78.7\%

and

86.4\%

for Llama-8B and Qwen-7B.

Figure 17: Reported cross-model summary statistics. The shared ten-dimensional X-FARS frame explains

97.6\%

of cross-model centroid variance over

27

extracted models; the reported Haar-random projection baseline is

61.7\pm 1.0\%

. This panel compares aggregate explained variance, not pairwise similarity or held-out generalization.

Figure 18: Cross-model alignment in a five-model subset. Centroid RSA ranges from

0.77

0.94

off the diagonal; gray cells denote self-comparisons. Values are shown to two decimal places; this panel reports the five-model subset. The selected five models should not be confused with the larger

15

- and

27

-model analysis pools.

Figure 19: Concept analogies: schematic construction and measured retrieval. Left: one curated analogy, with positions chosen only to illustrate the algebra. Right: saved top-

1

accuracy across

27

models and

10

analogies (

270

attempts):

37.4\%

observed,

6.6\%

shuffled, and

6.7\%

chance. The schematic is not an empirical embedding.

Model	Dim full	Full top-1	FARS top-1	$\Delta$
GPT-2 XL	1600	.351	.562	$+.211$
Qwen2.5-7B	3584	.495	.729	$+.234$
Mistral-7B-v0.3	4096	.522	.743	$+.220$
Llama-3.1-8B-Instruct	4096	.559	.738	$+.179$
Llama-3.1-70B-Instruct	8192	.615	.816	$\mathbf{+.201}$
Pooled ( $n{=}150$ pairs)	—	.508	.717	$+.209$

Appendix

Table 48: Cross-format top-

1

retrieval, mean over

30

ordered form pairs. Full = best-FARS-layer hidden state; FARS =

10

-d projection. Chance

5.6\%

Quantity	Real	Null (shuffled, $n{=}100$ )
FARS $+$ GPA, $5$ models ( $1.6$ – $70$ B)	$\mathbf{95.1\%}$	$63.0\%$ ( $95\%$ CI $[59.9,66.3]$ )
FARS $+$ GPA, $9$ models, $6$ arch. families	$\mathbf{92.4\%}$	$63.0\%$
FARS $+$ GPA, all $15$ models	$\mathbf{91.9\%}$	$63.0\%$
X-FARS joint, all $15$ models	$\mathbf{97.0\%}$	—
FARS $+$ GPA, all $27$ models	$\mathbf{93.2\%}$	—
X-FARS joint, all $27$ models	$\mathbf{97.6\%}$	—

Appendix

Table 49: Cross-model FARS canonical-axis alignment. FARS

+

GPA:

93.2\%

variance explained on all

27

extracted models; Historical X-FARS alignment:

\mathbf{97.6\%}

\mathbf{36.8\sigma}

above the random-projection control at

61.7\pm 1.0\%

. Shuffled-concept-label null:

63.0\%

. Alignment remains high across the reported model pools; this comparison does not establish generalization to held-out concepts or rule out overfitting.

Figure 20: Canonical FARS axes loadings: rows are $18$ concepts grouped by domain; columns are the $10$ canonical axes ordered by variance. Axis numbers and variance fractions are shown below the columns; interpretation labels appear in Table 50 . Axis $1$ is the procedural-vs-set-relational axis that mirrors the prose-code behavioural asymmetry.

Figure 21: Biplot of the

18

concepts in the (axis

0

, axis

1

) (a) and (axis

0

, axis

4

) (b) planes of the canonical FARS frame, colored by domain. Colors and marker shapes identify domains; exact concept loadings are in the accompanying heatmap. Axis signs and interpretations refer to this historical fitted frame.

Ax.	Var	$+$ extreme	$-$ extreme
0	$.242$	set_diff, set_ $\cap$ , func_comp	causal_intervention, spatial_containment
1	$.157$	func_comp, multi_step, proportional	set_diff, set_ $\cap$ , causal_confound
2	$.153$	causal_intervention, spatial_containment	logic_negation, arith_modular
3	$.131$	causal_intervention, arith_modular	spatial_direction, spatial_rotation
4	$.078$	modus_ponens, contrapositive, syllogism	causal_confound, logic_negation

Appendix

Table 50: Interpretation of the top-

5

canonical FARS axes on the

5

-model GPA alignment (including Llama-3.1-70B-Instruct). The variance fractions are reported on the

5

-model canonical frame; the

+/-

extreme concepts and the sign of every loading are preserved between the

4

-model (

1.6

B–

8

B) and

5

-model (

1.6

B–

70

B) alignments. Each axis admits a recognisable concept dichotomy; cross-model stability is

\sigma{<}0.02

for axes

0

–

4

. Axis

1

corresponds to the same procedural-vs-declarative distinction that drives our prose-code asymmetry at the behavioural level (§ Q.2 ).

Objective	Initialization	Final loss	Max. gradient
Raw centroids	Historical	5.1923	$1.3e+02$
Raw centroids	Cached FARS	798.21	$8.1e+02$
Raw centroids	Random	69.723	$2.8e+02$
Unit-norm centroids	Historical	1.4303	$0.00052$
Unit-norm centroids	Cached FARS	1.4503	$0.0018$
Unit-norm centroids	Random	1.4274	$0.00061$

Appendix

Table 51: Optimization diagnostics after 700 iterations. All nine fits reach the iteration budget without meeting the

10^{-6}

maximum tangent-gradient criterion. Final objective values are comparable only within the same objective definition. Orthogonality residuals are below

9\times 10^{-15}

. Small objective values alone do not validate concept preservation.

Figure 22: Concept-difference alignment in the saved 27-model analysis. Left: distribution of

153

concept-pair means, each averaging

351

cross-model comparisons. This is not a histogram of the

53{,}703

individual cosines. Right: aggregate statistics recorded in the same result file. Raw pairwise cosines and shuffled samples were not reconstructed for this plot.

Figure 23: Historical cross-model concept retrieval in a six-model subset. Values reproduce Table 52 at its reported precision (off-diagonal mean $69.6\%$ ); rows are query models and columns are corpus models. Self-comparisons are omitted. This panel replaces a legacy figure whose fitting statistics referred to a different model pool. It reports empirical retrieval under historical alignment, not an optimizer guarantee or held-out-concept generalization.

Query $\backslash$ Corpus	GPT-2 XL	Qwen-7B	Mistral-7B	Llama-8B	Mixtral-MoE	Llama-70B
GPT-2 XL	,	$.565$	$.590$	$.617$	$.426$	$.562$
Qwen2.5-7B	$.519$	,	$.812$	$.778$	$.750$	$.741$
Mistral-7B-v0.3	$.698$	$.824$	,	$\mathbf{.870}$	$.747$	$.824$
Llama-3.1-8B-Inst	$.710$	$.812$	$\mathbf{.870}$	,	$.744$	$.824$
Mixtral-8x7B-Inst (MoE)	$.451$	$.608$	$.568$	$.528$	,	$.494$
Llama-3.1-70B-Inst	$.664$	$.843$	$.852$	$.870$	$.719$	,

Appendix

Table 52: Cross-architecture top-

1

concept retrieval via X-FARS shared canonical frame (chance

5.6\%

). Row: query model; column: corpus model. Cell: mean top-

1

accuracy across

6

surface forms,

54

same-form stimuli per side. No model weights are updated; alignment uses shared concept identities. This is not an unpaired evaluation.

Panel of Table 53 — Table 53: Leave-one-out plug-and-play protocol validated on the full $12$ -model set. For each held-out model, refit X-FARS on the remaining $11$ to obtain a canonical frame $\bar{C}^{(-m)}$ , then construct the held-out basis using a polar-factor SVD update against $\bar{C}^{(-m)}$ . The plug-in basis recovers $\Delta\text{RSA}\in[-0.004,+0.014]$ relative to the model’s standard FARS basis (mean $+0.002$ ). Architectures most divergent from the canonical (RoBERTa, Qwen-3B) benefit most.

Metric	FARS + GPA	X-FARS
Canonical variance explained ( $9$ models, $6$ arch. families)	$92.43\%$	$\mathbf{97.22\%}$
Cross-family pooled $\rho$ (pred. patching, $n{=}120$ )	$-0.806$	$-0.806$
$\Delta$ per-model concept-RSA vs. FARS	,	$[-0.002,\,-0.0006]$
Steps required	PCA + iterated Procrustes	iterated alignment heuristic

Appendix

Table 54: Standard FARS (two-step: per-model PCA then post-hoc GPA) vs. X-FARS (historical alignment heuristic) as reported in the historical summary. Cohorts differ by metric; the variance row specifies its own nine-model pool. These entries are not a fresh optimizer validation.

Original $6$ -pair set ( $n{=}36$ stimuli):
Model	Span(A,B)	Span(rand C)	Span(rand dirs)	Both $\alpha,\beta{>}0$
GPT-2 XL	$\mathbf{.736}$	$.537$	$.453$	$24/36$ ( $67\%$ )
Llama-3.1-8B-Inst	$\mathbf{.715}$	$.393$	$.410$	$27/36$ ( $75\%$ )
Mistral-7B-v0.3	$\mathbf{.758}$	$.473$	$.406$	$31/36$ ( $86\%$ )
Extended $18$ -pair set ( $n{=}108$ stimuli):
Llama-3.1-8B-Inst	$\mathbf{.696}$	$.422$	$.424$	$73/108$ ( $68\%$ )

Appendix

Table 55: Compositional FARS span ratios. Top (

6

pairs,

n{=}36

): original concept pairs. Bottom (

18

pairs,

n{=}108

): extended set. Compositionality holds across

4

architecture families spanning

1.6

B–

70

70

B has the highest span ratio and both-positive rate.

Model	Span(A,B,C)	rand-3-C	rand-3-D	$p$ vs rand-D	All $\alpha,\beta,\gamma{>}0$
Phi-3.5-mini-Inst (3.8B dense, $L{=}18$ )	$\mathbf{.92}$	$.86$	$.53$	$\mathbf{<10^{-12}}$	$\mathbf{25/48}$ ( $\mathbf{52\%}$ )
Qwen-2.5-3B-Inst (dense, $L{=}22$ )	$\mathbf{.83}$	$.76$	$.46$	$<10^{-12}$	$\mathbf{26/48}$ ( $\mathbf{54\%}$ )
Mamba-2.8B (state-space, $L{=}35$ )	$\mathbf{.83}$	$.60$	$.52$	$\mathbf{2.3{\times}10^{-12}}$	$27/48$ ( $56\%$ )
Qwen-7B (dense decoder, $L{=}13$ )	$\mathbf{.81}$	$.62$	$.49$	$1.1{\times}10^{-11}$	$28/48$ ( $58\%$ )
Mistral-7B-Inst (dense, $L{=}10$ )	$\mathbf{.76}$	$.60$	$.53$	$1.5{\times}10^{-6}$	$19/48$ ( $40\%$ )
Llama-3.1-8B-Inst (dense, $L{=}8$ )	$\mathbf{.68}$	$.60$	$.51$	$9.3{\times}10^{-5}$	$19/48$ ( $40\%$ )

Appendix

Table 56: Triplet compositionality (

A\wedge B\wedge C

48

stimuli across

8

triplet templates, tested on

\mathbf{6}

models spanning dense decoder-only AND state-space (Mamba) architectures. Span(A,B,C) is the least-squares span ratio against

3

atomic centroids; “rand-3-C” and “rand-3-D” are random-3-concept and random-3-direction controls. All six tested models show span(A,B,C) significantly higher than random-direction control (

p{<}10^{-4}

on every model); the smaller models (Phi-3.5, Qwen-3B) show the largest span ratios (

0.83

–

0.92

) and highest all-positive-3-coefs rates (

52

–

58\%

), well above the chance rate of

1/2^{3}=12.5\%

Model	Span(A,B,C,D)	rand-C	rand-D	$\Delta$ rand-C	All 4 ${>}0$
Mamba-2.8B (state-space)	$\mathbf{.87}$	$.79$	$.60$	$+.08$ ( $p{=}0.023$ )	$\mathbf{10/24}$ ( $42\%$ )
Qwen-7B (dense)	$.84$	$.84$	$.63$	$.00$ (n.s.)	$6/24$ ( $25\%$ )
Mistral-7B-Inst (dense)	$.79$	$.74$	$.58$	$+.05$ (n.s.)	$4/24$ ( $17\%$ )
Llama-3.1-8B-Inst (dense)	$.68$	$.70$	$.60$	$-.02$ (n.s.)	$4/24$ ( $17\%$ )

Appendix

Table 57: Quartet (

4

-way) compositionality on

4

models,

24

stimuli. Span(A,B,C,D) vs. rand-

4

-concept (rand-C) and rand-

4

-direction (rand-D). Only Mamba’s

\Delta

over rand-C reaches significance.

Figure 24: Constituent-span reconstruction in two cached model runs. Each panel contains

36

composite projections over six concept pairs. Axes show projection and reconstruction norms; the dashed line is equality. Mean norm ratios are

0.715

for Llama-3.1-8B-Instruct (layer

8

) and

0.758

for Mistral-7B-v0.3 (layer

11

). Both panels are rebuilt from saved activations and bases; they are a two-model subset of Table 55 . P1: gcd + transitivity; P2: modular arithmetic + modus ponens; P3: causal chain + multi-step arithmetic; P4: syllogism + set intersection; P5: function composition + modus ponens; P6: spatial direction + proportionality.

Figure 25: Historical predictiveness from centroid distances. Within-model (left) and cross-model (right) associations with patching overlap. The reported cross-family aggregate is

\rho=-0.75

over

204

repeated entries. Random low-rank projections recover much of the association (

\rho=-0.71

); these correlations do not establish causal necessity.

Bold: cross-family $n{=}30$ . Plain: $n{=}4$ .
$X\downarrow$ / $Y\rightarrow$	GPT-2	Llama	Mistral	Qwen
GPT-2 XL	$-.40$	$\mathbf{-.83}$	$\mathbf{-.84}$	$-.80$
Llama-Inst	$+.20$	$-.70$	$\mathbf{-.73}$	$-.60$
Mistral-7B	$+.20$	$\mathbf{-.76}$	$-.80$	$-.60$
Qwen-7B	$+.20$	$\mathbf{-.78}$	$\mathbf{-.81}$	$-.60$

Appendix

Table 58: Spearman

\rho

between

X

’s FARS-distance per form-pair (rows) and

Y

’s patching overlap (cols). Off-diagonal cells with

n{=}30

are the strongest evidence:

X

and

Y

are different models. Pooled cross-family

\rho{=}{-}0.750

(

p{=}3.8{\times}10^{-38}

n{=}204

Projection	Cross-family $\rho$	$p$
FARS centroid (10-dim, supervised)	$-0.750$	$3.8{\times}10^{-38}$
Random orthonormal (10-dim, no labels)	$-0.714$	$3.9{\times}10^{-33}$
Shuffled- $Y$ (1000 permutations)	$-0.001$ (mean)	95% CI $[-.13,+.14]$

Appendix

Table 59: Historical cross-family predictiveness controls (within-cell z-pooled,

n{=}204

). Random projections retain much of the correlation. Reported

p

-values are historical; pooled entries share models and form pairs.

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