Evidence Integration in Large Language Models

Sebastien Kawada; Manolis Kellis

cs.CLSep 3, 2026

Evidence Integration in Large Language Models

Authors: Sebastien Kawada, Manolis Kellis

Organizations: Computer Science and Artificial Intelligence Laboratory Massachusetts Institute of Technology

Abstract

Despite increasing reliance on LLMs that reason with external evidence supplied by tools, retrieval-augmented generation, other agents, and users, how LLMs integrate such evidence into decisions they have already begun to form remains largely unclear. We present a distributional theory in which evidence shifts the receiver's distribution of initial answers, driven by a receiver prior weight and a candidate evidence tilt, leading to three predictions. First, candidates more probable to the receiver are more persuasive. Second, receivers more readily integrate characteristic errors of their own than foreign errors from different sources. Third, identical evidence can improve weaker models and harm stronger ones. We confirm these over ten million trials, twelve LLMs from four families, and eight domains, four of them scientific discovery tasks in the physical and life sciences: quantum mechanics, physics, genetics, and molecular biology. The law also yields a receiver-relative reliability frontier: receiver-congruent errors depress performance more steeply than random errors of the same rate. LLMs also integrate candidates even after internally verifying their invalidity (93-100% with propositional constraints; up to 99.4% on held-out physical and life-sciences reasoning), demonstrating evidence integration is a receiver-specific control policy over existing distributions, determined by receiver properties rather than scalar trust in the evidence source. Causal interventions show candidate integration is implemented late in the network, as a structured sequence of steps admitting external candidate answers, promoting them, and transporting them into the answer state. Representations of verification are decodable but have little causal impact on answers. A J-lens decomposition shows the state underlying verbalized verification is fully dissociable from that underlying candidate integration.

Figures & tables

Figure 1: The evidence-integration law as a distribution shift. Left, the model’s own answer distribution

q

over a candidate axis, with the truth

t

, the model’s characteristic error, and the low-support value

v

that external evidence from a tool or retrieval proposes. Middle, the prior carried through with weight

1+a

(the landscape

q^{1+a}

at weights one and

0.4

), the tilt

\exp\phi_{e}

local to

v

, and

p(y\mid e)

across prior weights, with the measured band

0.20

0.65

between the dashed lines and the rational receiver at one. Right, the row

1+a=0.4

against

q

in dashes, the mass moved to

v

shaded.

Domain	Family	Test	Cost	Blind to
Compositional arithmetic (ARITH)	kernel	$\varpi(v)=\varpi(y^{\star})$ for the ring homomorphism $\varpi$ onto $\mathbb{Z}/10\mathbb{Z}$ , the last digit	$O(\text{size})$ in the ring	misses with $v-y^{\star}\in 10\mathbb{Z}$
Word problems (WORD)	bound	$v\in\mathcal{B}$ for a magnitude interval the problem fixes	one comparison	every wrong value inside $\mathcal{B}$
Linear systems (LINSYS)	residual	$\\|A\mathbf{v}-\mathbf{b}\\|=0$	one evaluation of the constraint map	nothing, when $A$ is injective
Propositional constraints (SAT)	constraint	every clause satisfied	linear in the clauses	nothing, when all clauses are checked
Physical systems (PHYS)	bound	sign conditions and conservation laws fixed by the physics	one comparison	every wrong value with the right sign
Molecular sequences (BIO), reverse complement	constraint	length and base composition equal to the complement’s	linear in the sequence	permutations of the complement, including the unreversed complement

Table 1: Certificates by domain, the map of the verification-economics axis. Each row gives the cheap necessary condition a domain’s checker can read, its family in the sense of Section 3.2 , the cost of the membership test, and the wrong candidates it cannot reject. Codes in parentheses are the short forms used in tables and figure legends.

Figure 2: Measurement and candidate construction. (a) Each item yields separate generation

G

, isolated checking

C

, and use

U

trials, each with its own prompt and parse path. (b) Certificate-keeping and certificate-breaking rungs matched in magnitude, and the attractor, selected on one calibration fold and scored on the other against matched unseen controls. (c) The candidate landscape

q

is measured before evidence, the arbitration shift depends on prior support, source cues, competence, and recruited verification evidence, and the final distribution

p

is observed as adoption.

Figure 3: Models preferentially adopt the errors they already support, nine models. (a) Paired attractor-minus-control acceptance difference

\Delta_{\mathrm{AT}}

. (b) Fraction of items on which the attractor has more support than the truth. (c) Acceptance against the attractor’s frequency in the model’s own calibration draw; the zero bin is the matched-control rate. (d) Controls-only acceptance against centered support

A^{c}(v)

; triangles are attractors and the orange segment the attractor premium beyond the fitted support curve, dashed beyond the last measured bin. Intervals are 95%, item bootstrap in (a) and Wilson in (b) to (d).

Figure 4: Certificate sensitivity and source authority across nine models. (a) Standardized product-versus-division certificate interaction. (b) Wrong-candidate acceptance across tool identities and user framings; each connector spans the three conditions within a channel. (c) Paired assertion-to-instrument acceptance ratio against the no-effect line at

1.0

. Intervals are 95% item bootstraps in (a) and (c) and narrower than the markers in (b);

1{,}052

1{,}493

items per model.

Figure 5: External help decays and external harm grows with competence. Change in

P(\text{correct})

from the no-candidate baseline against cross-fitted competence

\hat{c}

, nine models on a shared axis, blue for a correct candidate and orange for a near miss, with item-clustered bootstrap bands. Markers on the zero line are fitted crossings, filled where unique and open otherwise.

	Competence crossing $c^{*}$
Model	harm	help
Llama-3.1-8B	0.270	0.560
Llama-3.1-70B	none ‡	none
Llama-3.3-70B	0.360	none
Gemma-4-E4B	0.445	0.655
Gemma-4-31B	0.445	0.550

Table 3: Fitted competence crossings

c^{*}

per model,

0.005

search grid. none , no unique crossing; ‡ a second crossing.

Figure 8: Components of the arbitration law across nine models, Eq. ( 26 ) on pooled wrong-reference trials. Support, channel, and the competence coefficient’s shrinkage under

A(v)

are joint-fit estimates with 95% item-cluster bootstrap intervals. In the certificate column the dot is the within-item keep-versus-break contrast at matched support (

\pm 1

SE) and the tick the unadjusted discordant log-odds; rows without a model-specific estimate show the group range. The right strip is the between-item

\beta_{m}

. Rings mark the two held-out 70B models with the bands their fitted siblings imply.

Figure 9: Adoption of constraint-violating assignments across twelve models, each rate over the same 236,820 violating-candidate use trials per model. (a) Full scale. (b) The 0.90 to 1.00 zoom, printed to four decimals. Color is family; lighter is the smaller model.

Figure 10: The held-out checking–use dissociation in the physical-systems and molecular-sequences domains; color is task family. (a) Wrong-candidate adoption conditioned on the model’s own working-allowed rejection, with 95% cluster-bootstrap intervals; open gray markers are cells with a checking arm below 0.30. (b) Gemma-4-31B’s adoption by working-length tercile on circuits and reading frames; dashed lines are the terse-use reference. (c) Working-arm checker discrimination against conditional adoption in the three estimable families, with Spearman rank correlations.

Figure 11: Activation patching shows a late handoff from the shown value’s position to the answer slot in Llama-3.1-8B, 108 items. (a) Attractor-minus-truth answer margin after patching the control-shown state in at either position, mean with the interquartile range as a ribbon; dashed lines are the clean runs with the attractor shown, with no reference, and with the matched control shown. (b) Share of items whose patched margin falls below zero at each layer.

Figure 12: Attention knockout separates reading from answer-slot integration. Columns cut attention to the shown value for every downstream position or for the answer slot only; rows are the initial 108 items and the 120-item replication. Dashed lines are the clean and no-reference runs, and whiskers are bootstrap intervals.

Figure 13: Activation steering along the certificate direction changes Gemma but not Llama. (a) Decision margin at the steering site against steering strength

\lambda

in site residual norms, at the initial site, the transfer-selected site, and the disjoint-item replication, with the Gemma layer, the Llama layer, and the item count under each title; solid curves are the certificate direction with 95% bootstrap bands, dashed curves the norm-matched random direction. (b) Acceptance under steering, greedy generation from the unprimed prompt on the replication items. (c) Certificate-direction margin span from

\lambda=-3

+3

per swept layer at its fractional depth; outlined cells clear the joint bar, ratio

\geq 3

over the random control and span

\geq 1.0

. (d) Margin slope per keep-break dose unit for six models with item-bootstrap intervals (Table 38 ), against each model’s ten-direction random panel; Gemma-4-31B’s panel leaves the linear regime and is drawn hollow.

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Meaning	Defined
Indices
$m$ , $d$ , $i$ , $j$ , $\ell$ , $t$	model; domain; item; trial; layer; token position	Section 3
State
$x$ , $\mathcal{Y}_{i}$	the item’s prompt and answer space	Section 3
$\mathcal{Y}_{i}^{\star}$ , $y_{i}^{\star}$ , $y_{i}^{\dagger}$ , $t_{i}(v)$	valid answers; the unique truth where one exists; the reference valid answer used in margins; validity coding $\pm 1$	Section 3
$q_{m,d}(y\mid x)$ , $p_{m,d}(y\mid x,e)$	candidate landscape before evidence; final answer distribution after evidence	Section 3.1

Appendix

Table 7: Notation, by role. Each row gives the symbol, its meaning, and the display that defines it. Every quantity is per model

m

and, where written

(v)

, per candidate value;

\Pr[\cdot]

is the probability of an event,

G_{m}

P^{\pm}_{m}

, and

\mathcal{R}_{m}

are accuracies, and superscripts name shifts.

	residue checks						magnitude checks
Rung	mod 10	mod 9	mod 2	mod 3	mod 4	mod 5	digits	sign	products
truth (reference)	1	1	1	1	1	1	1	1	0.21
near miss ( $+1$ )	0	0	0	0	0	0	1.00	1	0.21
keep $+10$	1	0	1	0	0	1	0.99	1	0.21
break $+11$	0	0	0	0	0	0	0.99	1	0.21
keep $+30$	1	0	1	1	0	1	0.97	1	0.20

Appendix

Table 9: Certificate audit of the arithmetic ladder over the 1,156 frontier items. Each entry is the fraction of items on which the rung’s shown value agrees with the truth on the named condition:

1

a condition the rung preserves by construction,

0

one it breaks by construction, a fraction in between an item-dependent one. The residue checks are the cheap coordinates of

Z(v)

; the keep rungs preserve last digit, parity, and the mod 5 residue together. The magnitude checks are shown for completeness; whole-corpus figures are in the accompanying CSV.

		Trials by outcome
Model	Adoption	Adopted	Own solution	Other invalid	No assignment
Llama-3.1-8B	0.9570	226638	2	149	10031
Llama-3.1-70B	0.9999	236808	0	12	0
Llama-3.3-70B	0.9999	236804	0	16	0
Gemma-4-E4B	0.9999	236792	1	27	0
Gemma-4-31B	0.9730	230432	13	23	6352

Appendix

Table 10: Propositional-constraint outcomes per model over the

236{,}820

violating-candidate use trials per model. The four outcome columns partition the trials, and an own solution is a different, valid assignment. The

18{,}000

satisfying-anchor trials per model are excluded because they cannot distinguish adoption from independent solution.

			Answers, of 60 items
Arm	$k$	Acceptance	shown	truth	other	donor
no swap		0.467	28	3	29	0
full state		0.000	0	28	30	2
workspace only	5	0.467	28	3	29	0
complement only	5	0.000	0	27	31	2
random subspace	5	0.467	28	3	29	0

Appendix

Table 13: Workspace-decomposition controls on Llama-3.1-8B, layers 19 to 30, the sixty items of Table 6 . Workspace only and complement only swap the

k

-atom J-lens component

\mathcal{P}_{\ell}

and its complement at six budgets

k

; random subspace projects the state difference onto a fixed random

k

-dimensional subspace and random J-lens atoms onto the span of

k

atoms of random vocabulary tokens. The rotated arms replace the swapped component’s difference by a norm-matched random direction inside (workspace) or orthogonal to (complement) the span of the selected atoms. Count columns partition the sixty answers into the shown value, the truth , an other value, and the donor run’s value.

		Answers, of 60 items
Arm	Acceptance	shown	truth	other	donor
Llama-3.1-8B, layers 19 to 30
SAE reconstruction	0.117	7	13	40	0
SAE complement	0.067	4	27	29	0
SAE reconstruction, matching pursuit	0.000	0	12	47	1
SAE complement, matching pursuit	0.117	7	17	36	0

Appendix

Table 14: Sparse-autoencoder and single-layer workspace arms in Llama-3.1-8B, columns as in Table 6 . SAE reconstruction rebuilds the swapped state from each layer’s twenty-five largest sparse-autoencoder features, from the base-trained autoencoder unless marked instruct-matched, and matching pursuit is the alternative sparse decomposition of the same features. The second block repeats the decomposition at layer 19 alone.

Figure 14: Uptake promotes the value the prompt shows and moves the true answer nine times less. Each map is the mean over 896 token-aligned prompt pairs, attractor shown minus matched control shown. (a) Rank of the shown wrong value; (b) rank of the true answer, on one shared compressive scale (

\propto\sqrt{|\Delta|}

). Causal attention forbids any difference left of the dashed rule, and those columns are drawn narrow. A leading dot marks a token that begins with a space, and boxes mark each panel’s largest entry.

Figure 15: Knocking out the shown-value read path across seven models on a shared depth axis. (a) Sliding-window knockouts, the change in the attractor’s log-probability against the model’s own clean run by window start; the shaded band is the 62 to 88% range. (b) Each model’s gap between its clean and no-reference runs against the

0.30

floor; hollow markers fall below it. (c) Band knockout against the model’s own random control, per-item paired contrast with 95% interval. (d) Share

\pi^{\mathrm{ko}}_{m}

of the model’s own uptake gap removed, log axis, with the unit rule marking removal of the whole gap; models with gaps below the floor have no point.

Figure 16: Certificate-direction steering qualifies across eighteen Gemma layers and zero Llama layers under the joint criterion. The direction is fitted independently at every layer on training items and steered there on held-out items; the span

|\Delta M|

is how far the decision margin moves between the two steering endpoints. We require both a span at least three times the norm-matched random control and an absolute span of at least

1.0

for a layer to qualify. (a) Span per layer against the

1.0

floor. (b) The same layers under the over-random ratio alone, where seven of Llama’s 31 layers clear the threefold bar on movements of at most

0.13

	ARITHH		WORD		SAT		PHYS-1		LINSYS	PHYS	BIO
Model	$1+a$	SE	$1+a$	SE	$1+a$	SE	$1+a$	SE
Llama-3.1-8B	+0.31	0.008	+0.33	0.011	+0.41	0.009	+0.32	0.009	$\cdot$	$\cdot$	$\cdot$
Llama-3.1-70B	+0.50	0.007	+0.52	0.008	+0.69	0.011	+0.48	0.008	$\cdot$	$\cdot$	$\cdot$
Llama-3.3-70B	+0.64	0.009	+0.65	0.009	+0.17	0.004	+0.64	0.010	$\cdot$	$\cdot$	$\cdot$
Gemma-4-E4B	+0.30	0.006	+0.24	0.008	+0.28	0.010	+0.44	0.009	$\cdot$	$\cdot$	$\cdot$
Gemma-4-31B	+0.51	0.015	+0.57	0.007	+0.29	0.004	+0.47	0.010	$\cdot$	$\cdot$	$\cdot$

Appendix

Table 15: Prior weight

1+a_{m,d}

of the evidence-integration law per model and domain, from answer-slot margins, with item-clustered standard errors; a dot marks a cell without log-probability sidecars.

		within		keep / break split
Model	$n$	$1+a$	SE	keep	break	gap 95% CI	between	logit $\alpha$
Llama-3.1-8B	48552	+0.31	0.008	+0.29	+0.33	$[-0.06,\,-0.02]$	+0.35	+0.16
Llama-3.1-70B	60144	+0.50	0.007	+0.50	+0.50	$[-0.02,\,+0.01]$	+0.60	+0.49
Llama-3.3-70B	56322	+0.64	0.009	+0.63	+0.65	$[-0.03,\,-0.00]$	+0.73	+0.24
Gemma-4-E4B	53256	+0.30	0.006	+0.25	+0.34	$[-0.11,\,-0.06]$	+0.42	+0.25
Gemma-4-31B	25242	+0.51	0.015	+0.57	+0.49	$[+0.06,\,+0.10]$	+0.68	+0.13

Appendix

Table 26: Prior weight of the evidence-integration law per model, from the answer-slot margins of the arithmetic instrument. Within is the slope of the candidate-over-truth margin

M(v)

on the support margin

S(v)

with item-by-format and condition-by-format fixed effects, the estimate of

1+a_{m,d}

, with item-clustered standard errors; keep and break fit the slope on certificate-keeping and certificate-breaking candidates separately, with the 95% interval of their gap; between drops the item effects. The last column is the coefficient on

A(v)

in the pooled adoption logit of Eq. ( 26 ).

	Within-item attractor		Dose-response on $A(v)$
Model	items	$\Delta_{\mathrm{AT}}$	slope	attractor	$z$
Llama-3.1-8B	3889	+0.476	+0.140	+1.48	17.5
Llama-3.1-70B	2771	+0.331	+0.291	+1.13	5.1
Llama-3.3-70B	3362	+0.366	+0.171	+1.41	6.6
Gemma-4-E4B	3455	+0.387	+0.119	+0.90	9.7
Gemma-4-31B	182	+0.590	+0.229	+0.79	1.8

Appendix

Table 29: Attractor and dose-response estimates per model from the arithmetic instrument. Within-item attractor gives the number of items with a stable modal wrong answer and the paired attractor effect

\Delta_{\mathrm{AT}}

over them. Dose-response on

A(v)

gives the controls-only acceptance slope, the attractor indicator’s coefficient beyond

A(v)

, and its

z

. ceiling marks a cell against the acceptance ceiling.

	Frontier $\bar{r}_{m}(c)$			Regret at $r=0.8$
Model	min	max	$\bar{r}_{m}>1$ from $c$	always	never	$\max_{c}\mathcal{L}(c)$
Llama-3.1-8B	-0.34	+1.34	0.57	0.081	0.184	0.214
Llama-3.1-70B	-0.23	+1.00	0.90	0.017	0.196	0.064
Llama-3.3-70B	-0.33	+0.99	none	0.028	0.287	0.096
Gemma-4-E4B	-1.33	+1.33	0.66	0.031	0.297	0.094
Gemma-4-31B	-0.68	+1.48	0.56	0.095	0.250	0.258

Appendix

Table 30: Regret of source-only rules and misspecification loss per model, from the fitted competence curves of Figure 5 on the frontier window, for a near-miss source of reliability

r=0.8

under a uniform competence distribution. always and never are the regrets of the two source-only rules relative to consulting exactly when

r>\bar{r}_{m}(c)

, and

\max_{c}\mathcal{L}(c)

the largest accuracy the receiver forfeits relative to a rational receiver sharing its prior.

	Near-miss adoption $\bar{\mathsf{a}}$			Harm crossing
Model	measured	predicted	MAD of $\bar{r}_{m}$	direct	predicted
Llama-3.1-8B	0.315	0.407	0.051	0.275	0.235
Llama-3.1-70B	0.427	0.488	0.036	0.105	0.255
Llama-3.3-70B	0.422	0.487	0.025	0.365	0.345
Gemma-4-E4B	0.316	0.572	0.272	0.445	0.305
Gemma-4-31B	0.239	0.206	0.036	0.445	0.445

Appendix

Table 31: Predicted against directly estimated reliability frontier per model.

\bar{\mathsf{a}}

is the near-miss adoption rate on the frontier window, measured and as the fitted adoption law predicts with the near-miss trials held out; the predicted frontier

\bar{r}_{m}(c)

follows from Eq. ( 44 ), MAD is its mean absolute difference from the direct estimate on the window, and the last two columns give each frontier’s harm crossing

\bar{r}_{m}=0

, none where no crossing lies inside the window.

			per-message factor		mean shift $\bar{D}_{n}$
Model	$n$	$1+a$	$\hat{r}$	95% CI	$n{=}1$	$n{=}2$	$n{=}4$	$n{=}8$
Llama-3.1-8B	600	+0.41	+0.01	$[+0.00,\,+0.01]$	+15.5	+16.5	+15.4	+14.9
Llama-3.1-70B	600	+0.47	+0.00	$[+0.00,\,+0.00]$	+19.0	+18.2	+17.5	+17.8
Llama-3.3-70B	600	+0.53	+0.00	$[+0.00,\,+0.00]$	+28.0	+25.6	+23.8	+24.1
Gemma-4-E4B	600	+0.28	+0.19	$[+0.18,\,+0.20]$	+22.6	+25.2	+26.9	+28.6
Gemma-4-31B	600	+0.37	+0.00	$[+0.00,\,+0.00]$	+14.6	+11.5	+12.1	+12.0

Appendix

Table 34: Repetition of one tool observation, per model. The same tool message reporting a wrong value (near miss or the

+30

rung) is repeated

n=1,2,4,8

times as consecutive tool turns;

\bar{D}_{n}

is the mean arbitration shift

M_{e^{n}}(v)-S(v)

in nats. Under the law the prior term decays as

r^{n}

and the tilt accumulates geometrically with the same factor

r=1+a

;

\hat{r}

is the profile least-squares estimate with a free tilt per item and condition, with a 95% item-bootstrap interval, beside the single-message

1+a

;

n

counts (item, condition) series.

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