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Provenance-Aware Belief Updating in Human-AI Feedback Loops

Tianyi Huang, Eric Liu, Jay Chong, Emily Yin, Wanqi Zhao

Year: 2026Venue: arXiv preprintArea: cs.AIType: PreprintEmbeddings: 69

Intelligence

Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%

Last extracted: 10/12/2026, 6:45:21 AM

Summary

The paper introduces Latent Quotient Evidence Training (LQET), a relational training objective designed to help language models update beliefs based on evidence provenance rather than report frequency. LQET combines sparse Bayesian supervision with invariance constraints to prevent 'data incest' (overconfidence from repeated evidence) while maintaining sensitivity to new evidence. Experiments on Qwen3-4B and Phi-4-mini show significant reductions in update error and belief drift in relayed information scenarios.

Entities (7)

Phi-4-mini · model · 99%Qwen3-4B · model · 99%LQET · method · 98%QLoRA · technique · 95%Data Incest · phenomenon · 92%Provenance Equivalence · concept · 90%Bayesian Social Learning · concept · 85%

Relation Signals (7)

LQET → appliedto → Phi-4-mini

confidence 98% · relative to task-only training across Qwen3-4B and Phi-4-mini

LQET → appliedto → Qwen3-4B

confidence 98% · relative to task-only training across Qwen3-4B and Phi-4-mini

LQET → improves → new-evidence update RMSE

confidence 95% · LQET reduces new-evidence update RMSE by 41-44% ... relative to task-only training

LQET → reduces → log-odds drift

confidence 95% · reduces ... log-odds drift after eight relay steps by 89-91%

LQET → uses → sparse Bayesian supervision

confidence 92% · a relational objective that combines sparse Bayesian supervision with constraints

Data Incest → causes → overconfidence

confidence 90% · inadvertent reuse of the same information (data incest) can bias beliefs and produce overconfidence

LQET → addresses → Provenance Equivalence

confidence 88% · We formalize provenance equivalence over sets of distinct evidence roots

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Abstract

Abstract:In repeated human-AI interaction, a single observation can be quoted, summarized, and returned to a model as multiple reports, making repetition look like independent corroboration. We study whether language models can use supplied report lineage to update according to evidence provenance rather than report count. We introduce Latent Quotient Evidence Training (LQET), a relational objective that combines sparse Bayesian supervision with constraints that encourage stable predictions when reports add no new evidence and consistent updates to the same new evidence across relay histories. On a controlled binary benchmark, LQET reduces new-evidence update RMSE by 41-44% and log-odds drift after eight relay steps by 89-91% relative to task-only training across Qwen3-4B and Phi-4-mini, with three paired training seeds per backbone. The gains extend to a closed-loop simulation in which model confidence controls how many relayed reports re-enter the model's context in later rounds. By grounding belief updates in supplied provenance, LQET offers a route toward dynamic human-AI systems in which influence follows new evidence, not the number of times the same evidence returns.

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Dynamic Alignment on Human-AI Coupled System (BiAlign Workshop) Provenance-Aware Belief Updating in Human-AI Feedback Loops Tianyi Huang Affiliation: Ryquo Eric Liu Affiliation: App-In Club Jay Chong Affiliation: App-In Club Emily Yin Affiliation: App-In Club Wanqi Zhao Affiliation: App-In Club tianyi@ryquo.com eric.liu jay.chong emily vicky@appinclub.org Abstract In repeated human-AI interaction, a single observation can be quoted, summarized, and returned to a model as multiple reports, making repetition look like independent corroboration. We study whether language models can use supplied report lineage to update according to evidence provenance rather than report count. We introduce Latent Quotient Evidence Training (LQET), a relational objective that combines sparse Bayesian supervision with constraints that encourage stable predictions when reports add no new evidence and consistent updates to the same new evidence across relay histories. On a controlled binary benchmark, LQET reduces new-evidence update RMSE by 41-44% and log-odds drift after eight relay steps by 89-91% relative to task-only training across Qwen3-4B and Phi-4-mini, with three paired training seeds per backbone. The gains extend to a closed-loop simulation in which model confidence controls how many relayed reports re-enter the model’s context in later rounds. By grounding belief updates in supplied provenance, LQET offers a route toward dynamic human-AI systems in which influence follows new evidence, not the number of times the same evidence returns. 1 Introduction A report is not always a new observation. In a coupled information system, one claim may be quoted by a model, copied into a summary, and returned through later context. A single speaker can sound like a chorus even when listeners know that repeated statements share a source [24]. In Bayesian social learning and decentralized estimation, inadvertent reuse of the same information (data incest) can bias beliefs and produce overconfidence [1; 15]. People also tend to neglect correlation among signals [7], while repeated human-AI interaction can amplify judgment biases through feedback [9]. Appropriate updating therefore depends not only on what reports say, but also on whether they introduce evidence or inherit it. Prior work examines probabilistic updating from demonstrations [19], responses to evidence with varying informativeness and reliability [13], sequential belief trajectories [21], and sensitivity to repeated or paraphrased support [22; 17]. We ask a complementary structured question: given explicit multi-hop report provenance, can a model preserve its prediction when a report merely relays existing evidence, yet apply the same reliability-conditioned update when genuinely new evidence arrives after different relay histories? Sparse pointwise supervision on individual transcript views does not directly enforce this cross-transcript relationship: it assigns each transcript its own target, whereas the desired behavior concerns equivalence classes of transcripts and score changes between them. Figure 1: Reports can multiply without adding evidence. Relays preserve the evidence set e0\e_0\, while a new observation expands it to e0,eh\e_0,e_h\. Solid arrows show report derivation; dashed arrows identify the distinct evidence in each history. With the prior and calibration records fixed, equivalent evidence sets have identical Bayesian targets. We formalize this supplied-lineage setting with matched families of synthetic evidence reports. Prompts expose the derivation graph through parent-report links and natural-language descriptions of whether a report introduces an independent observation or relays an earlier one; evidence-root identifiers used to construct the matched families are not rendered. Two transcripts are provenance-equivalent when they contain the same set of distinct, query-relevant evidence roots. Under the source-reliability model, exact Bayesian inference assigns the same log-odds target to every transcript in an equivalence class. The model should therefore satisfy two relations: adding only reports derived from existing evidence should leave its score unchanged, whereas adding a new evidence root should produce the same score change regardless of the preceding relay history. Latent Quotient Evidence Training (LQET) directly supervises these relations. It combines sparse soft-target supervision with four invariance constraints, three new-evidence update-consistency constraints, and a presentation-order constraint. The learning problem is therefore to translate visible lineage into consistent belief updates across transcripts, rather than to discover unobserved report dependencies. Our contributions are: • We formalize provenance equivalence over sets of distinct evidence roots and derive exact Bayesian log-odds targets under uncertain source reliability. • We introduce a relational objective that extends repetition invariance with new-evidence update consistency, composition, and presentation-order consistency while retaining sparse pointwise supervision. • Across two language-model backbones, paired training seeds, and controlled distribution shifts, LQET consistently improves new-evidence update accuracy and stability across long relay histories. • A confidence-dependent feedback simulation connects these static relations to the dynamic setting in which model confidence influences how many inherited reports return as later input. 2 Evidence provenance and Bayesian targets Latent state and source reliability. Let Y∈−1,+1Y∈\-1,+1\ be the binary answer with prior π=P⁡(Y=+1)∈(0,1)π=P(Y=+1)∈(0,1). Before observing calibration data, each source reliability RsR_s has an independent Beta⁡(1,1)Beta(1,1) prior. Conditional on RsR_s, the source’s calibration outcomes are independent and identically distributed Bernoulli trials; a visible record of csc_s correct and wsw_s incorrect measurements therefore gives Rs|cs,ws∼Beta⁡(αs,βs),αs=cs+1,βs=ws+1.R_s c_s,w_s ( _s, _s), _s=c_s+1, _s=w_s+1. (1) An evidence root e denotes one distinct underlying observation, with reported outcome xe∈−1,+1x_e∈\-1,+1\ and source s⁡(e)s(e). For a query-relevant root, P⁡(xe=Y∣Y,Rs⁡(e))=Rs⁡(e)P(x_e=Y Y,R_s(e))=R_s(e). Conditional on Y and Rss\R_s\_s, the outcomes of distinct roots are independent of one another and of the calibration outcomes. Multiple roots from the same source share RsR_s and thus become dependent after reliability is marginalized. By construction, conditional on the relevant-root observations and visible calibration records, the report graph, rendering choices, and query-irrelevant roots carry no additional information about Y. Reports and provenance equivalence. A report is a node in an acyclic derivation graph. It may link to parent reports, inherit their evidence roots, and introduce zero or more roots of its own. For transcript T, let E⁡(T)=⋃r∈Te:e is a query-relevant root represented by rE(T)= _r∈ T\e:e is a query-relevant root represented by r\ (2) be the set of distinct relevant roots. Set union removes repeated descendants. Within a matched family, we define T∼T′⟺E(T)=E(T′).T T E(T)=E(T ). (3) This relation partitions each matched family into provenance-equivalence classes and induces the quotient used by LQET. To compute the Bayesian target, let ns,a(T)=|e∈E(T):s(e)=s,xe=a|n_s,a(T)=|\e∈ E(T):s(e)=s,\ x_e=a\| for a∈−1,+1a∈\-1,+1\. Exact Bayesian target. For y∈−1,+1y∈\-1,+1\, let (Cs(y,T),Ws(y,T))=(ns,+(T),ns,−(T))y=+1,(ns,−(T),ns,+(T))y=−1.(C_s(y,T),W_s(y,T))= cases(n_s,+(T),n_s,-(T))&y=+1,\\ (n_s,-(T),n_s,+(T))&y=-1. cases (4) Let B⁡(a,b)=Γ⁡(a)Γ(b)/Γ⁡(a+b)B(a,b)= (a) (b)/ (a+b) and σ⁡(u)=(1+e−u)−1σ(u)=(1+e^-u)^-1. Write ⁡(T)=((s⁡(e),xe))e∈E⁡(T)O(T)= ((s(e),x_e) )_e∈ E(T) for the indexed collection of relevant-root observations. After conditioning on the calibration records, let ℓy(T) _y(T) be the log prior plus log marginal likelihood for Y=yY=y; it equals the posterior log-probability up to an additive normalizing constant that is independent of y. Integrating each RsR_s yields ℓy(T) _y(T) =log⁡P⁡(Y=y)+∑s[log⁡B⁡(αs+Cs(y,T),βs+Ws(y,T))−log⁡B⁡(αs,βs)], = P(Y=y)+ _s [ B( _s+C_s(y,T), _s+W_s(y,T))- B( _s, _s) ], (5) t⁡(T) t(T) =ℓ+1(T)−ℓ−1(T),p⋆(T)=P⁡(Y=+1∣⁡(T),cs,wss)=σ⁡(t⁡(T)). = _+1(T)- _-1(T), p (T)=P\! (Y=+1 (T),\c_s,w_s\_s )=σ(t(T)). (6) Thus a relay that adds no new evidence leaves t⁡(T)t(T) unchanged, whereas a new root changes it according to its source, reported outcome, and the other roots from that source. The Bayesian target is constant within each provenance-equivalence class and, holding the prior and calibration records fixed, depends on a root set only through its source-by-outcome counts under the exchangeable source model. Matched transcript families. Each static benchmark family produces 14 matched transcript views. The core 2×22× 2 block crosses a relay history with the addition of new evidence: h00h_00 contains the base root, h10h_10 adds only relays, h01h_01 adds a new root, and h11h_11 adds that root after the relay history. Hence t⁡(h10)=t⁡(h00),t⁡(h11)=t⁡(h01),t⁡(h11)−t⁡(h10)=t⁡(h01)−t⁡(h00).t(h_10)=t(h_00), t(h_11)=t(h_01), t(h_11)-t(h_10)=t(h_01)-t(h_00). (7) A composition block sharing c00=h00c_00=h_00 separately adds roots eae_a and ebe_b in c10c_10 and c01c_01, adds both in c11c_11, and reverses their presentation in c11revc_11^rev. The base root and both added roots are drawn from three mutually distinct sources. Matched controls hold report author and displayed outcome fixed while varying the supplied provenance: a report either relays an existing observation or introduces a new one. Prompts include priors, calibration records, source names, report text, and explicit parent-report links; internal evidence-root and family identifiers are never rendered. A deterministic prompt-only calculator recovers every static-view Bayesian target, confirming that each target is fully specified by the visible prompt (Appendix E). 3 Latent Quotient Evidence Training Forced-choice score. We evaluate a next-token forced-choice diagnostic rather than parse a generated numeric response. For prompt T, the score is the YES-NO log-odds zθ(T)=Lθ(YES∣T)−Lθ(NO∣T),p^θ(T)=σ⁡(zθ(T)).z_θ(T)=L_θ( YES T)-L_θ( NO T), p_θ(T)=σ(z_θ(T)). (8) Here Lθ(a∣T)L_θ(a T) is the pre-softmax next-token logit for token a. Thus p^θ(T)=Pθ(YES∣T,next token∈YES,NO) p_θ(T)=P_θ( YES T,next token∈\ YES, NO\): it is a forced-choice YES score, not the full-vocabulary probability of an unconstrained response. Each tokenizer maps YES and NO to distinct single tokens at the assistant boundary. Sparse pointwise supervision. Seven views receive pointwise soft targets. Writing =h00,h01,c10,c01,same-source,counter,matched-new,A=\h_00,h_01,c_10,c_01,same-source,counter,matched-new\, let N be the number of matched families in a training batch and define pi,v⋆=p⋆(Ti,v)p _i,v=p (T_i,v) and p^i,v=p^θ(Ti,v) p_i,v= p_θ(T_i,v). Their average Bernoulli cross-entropy is ℒtask=−1N||∑i=1N∑v∈[pi,v⋆logp^i,v+(1−pi,v⋆)log(1−p^i,v)].L_task=- 1N|A| _i=1^N _v [p _i,v p_i,v+(1-p _i,v) (1- p_i,v) ]. (9) Views such as h11h_11 and c11c_11 are supervised only through relational losses and receive no pointwise loss. Relational constraints. For family i, write zi(v)=zθ(Ti,v)z_i(v)=z_θ(T_i,v) and ti(v)=t⁡(Ti,v)t_i(v)=t(T_i,v). We use the Huber penalty with δ=0.25δ=0.25, ρδ(u)=12u2,|u|≤δ,δ⁡(|u|−12δ),|u|>δ. _δ(u)= cases 12u^2,&|u|≤δ,\\ δ (|u|- 12δ ),&|u|>δ. cases (10) The four provenance-invariance residuals are ri,1inv r^inv_i,1 =zi(h10)−zi(h00), =z_i(h_10)-z_i(h_00), ri,2inv r^inv_i,2 =zi(h11)−zi(h01), =z_i(h_11)-z_i(h_01), ri,3inv r^inv_i,3 =zi(matched-copy)−zi(h00), =z_i(matched-copy)-z_i(h_00), ri,4inv r^inv_i,4 =zi(irrelevant)−zi(h00), =z_i(irrelevant)-z_i(h_00), and their mean loss is ℒinv=14N∑i=1N∑k=14ρδ(ri,kinv).L_inv= 14N _i=1^N _k=1^4 _δ\! (r^inv_i,k ). (11) The three new-evidence update residuals are ri,1upd r^upd_i,1 =[zi(h11)−zi(h10)]−[ti(h01)−ti(h00)], =[z_i(h_11)-z_i(h_10)]-[t_i(h_01)-t_i(h_00)], ri,2upd r^upd_i,2 =[zi(c11)−zi(c10)]−[ti(c01)−ti(c00)], =[z_i(c_11)-z_i(c_10)]-[t_i(c_01)-t_i(c_00)], ri,3upd r^upd_i,3 =[zi(c11)−zi(c01)]−[ti(c10)−ti(c00)], =[z_i(c_11)-z_i(c_01)]-[t_i(c_10)-t_i(c_00)], with ℒupdate=13N∑i=1N∑k=13ρδ(ri,kupd).L_update= 13N _i=1^N _k=1^3 _δ\! (r^upd_i,k ). (12) The target differences in the latter two residuals use the benchmark’s three-way source distinctness: the base root and the two added roots come from mutually distinct sources. Updates from multiple roots sharing a source need not be additive. Finally, ℒorder=N−1∑iρδ(zi(c11)−zi(c11rev))L_order=N^-1 _i _δ(z_i(c_11)-z_i(c_11^rev)). The implemented objective is ℒLQET=ℒtask+4ℒinv+2ℒupdate+ℒorder. L_ LQET=L_task+4L_inv+2L_update+L_order. (13) Each loss component is averaged separately before weighting. LQET uses 13 training views, whereas task-only uses the seven pointwise-supervised views; both methods use identical families and optimizer-update counts. 4 Experimental design Backbones and training. We evaluate Qwen3-4B-Instruct-2507 [26; 20] and Phi-4-mini-instruct [16]. For each backbone and method, training seeds 0, 1, and 2 use the same fixed families: 4,000 for training and 400 for validation. We train for 500 updates (two epochs) with QLoRA [11; 4]: NF4 with double quantization, BF16 computation, LoRA rank 32, scaling factor α=64α=64, dropout 0.05, and adapters on all linear modules. AdamW uses learning rate 2×10−42× 10^-4, zero weight decay, a 3% warmup followed by cosine decay, gradient accumulation 16, and clipping at 1.0. Sequence length is capped at 768, and no evaluated prompt is truncated. The 1:4:2:11:4:2:1 loss-weight ratio, Huber threshold, and optimization settings were fixed before the reported Qwen runs and transferred unchanged to Phi; no validation-set hyperparameter search was conducted. Validation results served as run diagnostics and selected neither the final configuration nor checkpoints. We evaluate the final checkpoint. Evaluation sets. Each run is evaluated on 400 independent and identically distributed (IID) test families from the training distribution, 800 controlled-shift out-of-distribution (OOD) families (160 for each of five factor groups: held-out domains and source names, longer relay histories, fan-in and diamond provenance graphs, held-out templates, and held-out prior and calibration-count supports), and 200 recursive-relay families at depths 1,2,4,81,2,4,8. All 14 views are scored for the IID and OOD families; the recursive-relay families are evaluated on nested relay histories and their subsequent new-evidence probes. For the static IID and OOD views, task-only and LQET are scored on identical rendered prompts. Recursive-relay prompts are model-specific because each appended relay quotes that model’s preceding forced-choice score, although the underlying family, lineage schedule, and rendering rule are aligned. All partitions are mutually disjoint, and we check for overlap in sampled benchmark instances, evidence roots, family identifiers, and rendered prompts. Primary endpoints. For the primary Qwen evaluation and Phi replication, new-evidence update RMSE and supervised-view KL pool the 800 families across the five equally sized OOD shifts within each seed; shift-specific estimates are secondary. New-evidence update RMSE compares predicted log-odds changes with the corresponding exact Bayesian changes on the three update pairs in Equation 12. Depth-8 relay drift averages the absolute score change from the base transcript after eight recursively generated relays over the 200 relay families. Supervised-view KL compares the exact Bayesian target with the forced-choice Bernoulli score and averages DKL(Bern(p⋆)∥Bern(p^))D_KL(Bern(p )\|Bern( p)) over the seven pointwise-supervised views. Each backbone-level point estimate is the arithmetic mean of its three seed-level estimates; for RMSE, the square root is taken within each seed before averaging. Statistical inference and decision criterion. We use 10,000 paired crossed seed-by-family bootstrap replicates. For each endpoint, a replicate independently resamples the three paired training seeds and one shared evaluation-family roster; the same family resample is reused across every selected seed occurrence and both methods, with all views from a family kept together. Pooled OOD resampling is stratified over the five fixed 160-family shifts, and nonlinear endpoints are recomputed within seed before averaging across resampled seeds. The 2.5th and 97.5th percentiles define unadjusted 95% intervals (Appendix C). The prespecified conjunctive criterion requires (i) negative LQET-minus-task-only effects for update RMSE and depth-8 relay drift in every seed, with pooled 95% upper bounds below zero, and (i) an LQET-minus-task-only supervised-view KL effect of at most 0.02 in every seed, with a pooled upper bound of at most 0.02. This prespecified +0.02+0.02-nat non-inferiority margin for mean pointwise prediction quality was fixed before training and was not selected from the reported test results. We analyze each backbone separately and do not form a cross-backbone interval. 5 Results 5.1 Controlled results across backbones Table 1 shows the same pattern on both backbones. LQET lowers new-evidence update RMSE by 41.25% on Qwen and 44.45% on Phi. Depth-8 relay drift falls by 91.36% and 89.38%, respectively. Both improvements hold in every training seed, and the three-seed mean update RMSE is lower on every OOD shift for both backbones. Pooled supervised-view KL changes by +0.00228+0.00228 on Qwen and +0.00120+0.00120 on Phi, with both upper bootstrap bounds below the prespecified +0.02+0.02-nat non-inferiority margin. Both backbones therefore satisfy the full conjunctive criterion. The IID test set shows the same directional pattern, with lower update RMSE and supervised-view KL on both backbones (Appendix D, Table 7). Figure 2 breaks down the aggregate results by relay depth and distribution shift. Along nested relay histories, task-only drift accumulates sharply through depth four and remains near one log-odds unit at depth eight, whereas the three-seed mean for LQET remains below 0.14 at every depth (panels a,b). Update RMSE is lower on every controlled shift (panel c), with the largest proportional gains on unseen depth and topology and smaller gains on held-out prior and calibration supports. Supervised-view KL is lower or essentially unchanged on the other four shifts but increases on the held-out prior-and-calibration shift, the only shift that changes the numerical supports for both the class prior and source-calibration records (panel d). Each recursive relay quotes the preceding forced-choice YES score rounded to three decimals, so the depth test repeatedly introduces model-generated surface content rather than duplicating a fixed string. Figure 2: Controlled generalization across relay depth and distribution shift. (a,b) Absolute relay-induced drift across nested depths; thick and thin curves show three-seed means and individual seeds on the same 200 families. (c) Update RMSE and (d) supervised-view KL across controlled shifts; color marks the method, shape the backbone, and connectors the paired contrast. Panels (a,b) use logarithmic y-axes, and panels (c,d) use logarithmic x-axes. Lower is better. Table 1: Primary controlled results. Effects are LQET minus task-only and are computed from unrounded estimates; lower is better. Point estimates average three paired training seeds. Intervals are 95% paired crossed seed-by-family bootstrap intervals. † marks the supervised-view KL endpoint with a prespecified +0.02+0.02-nat non-inferiority margin. Backbone Endpoint Task-only LQET Effect [95% interval] Qwen3-4B Update RMSE 0.6816 0.4005 −0.2812-0.2812 [−0.3070,−0.2565-0.3070,-0.2565] Depth-8 drift 1.0122 0.0875 −0.9247-0.9247 [−1.1175,−0.7703-1.1175,-0.7703] Supervised-view KL† 0.0215 0.0237 +0.0023+0.0023 [+0.00116,+0.00356+0.00116,+0.00356] Phi-4-mini Update RMSE 0.7461 0.4144 −0.3317-0.3317 [−0.3545,−0.3091-0.3545,-0.3091] Depth-8 drift 1.1079 0.1177 −0.9902-0.9902 [−1.0823,−0.8988-1.0823,-0.8988] Supervised-view KL† 0.0280 0.0292 +0.0012+0.0012 [−0.00002,+0.00251-0.00002,+0.00251] 5.2 Selective invariance, not uniform update suppression A model could lower relay drift by suppressing all updates. Matched copy-versus-new-evidence controls test this alternative directly (Figure 3a). LQET reduces base-to-relay and matched-copy drift by approximately 15-40 fold, while new-root and matched-new RMSE remain close to task-only. In the tightly matched comparison, which holds the report author and displayed outcome fixed while varying supplied provenance, copy-versus-new RMSE drops from 1.0145 to 0.4383 on Qwen and from 0.9818 to 0.4630 on Phi. The corresponding zero-intercept copy-versus-new slopes rise from 0.027 to 0.782 and from 0.067 to 0.769. Within this supplied-lineage setting, the pattern is selective: the model becomes less sensitive to declared relays while retaining its response to declared new observations. 5.3 Closed-loop feedback dynamics We next place each trained model in a deterministic human-AI feedback stress test on a fixed 64-family subset of the recursive-relay split (Appendix F). Crossing each family with initial trust τ0∈0.2,0.8 _0∈\0.2,0.8\ yields 128 trajectories per trained model. In round r, score-derived confidence crc_r, trust τr _r, and the number of simulated human relays mrm_r obey cr=|2σ(zr)−1|,τr+1=0.6τr+0.4cr,mr=1+min⁡2,⌊3τr+1cr⌋.c_r=|2σ(z_r)-1|, _r+1=0.6 _r+0.4c_r, m_r=1+ \2, 3 _r+1c_r \. Each of three rounds appends one AI-derived report and mr∈1,2,3m_r∈\1,2,3\ simulated human relays that add no new evidence. A fresh root is then added to measure the subsequent update. All three metrics improve in every backbone-seed pair (Figure 3b-d). Mean reductions range from 64.6-76.9% on Qwen and 79.1-90.9% on Phi. This result connects the static provenance constraints to a dynamic failure mode: when confidence changes how many derived reports return, lower sensitivity to inherited reports also reduces feedback amplification. Figure 3: Selective invariance and closed-loop feedback. (a) LQET/task-only error ratio on matched provenance controls, shown on a logarithmic x-axis; unity denotes no change, and marker and color identify the backbone. Inherited-report and copy-versus-new errors fall while new-evidence update errors stay near parity. (b-d) Thin segments pair matched adapters by seed; thick segments and larger markers are three-seed means. Endpoint color marks the method, while shape and line style mark the backbone. All panels are descriptive; lower is better. 5.4 What each relation contributes A Qwen seed-0 component study evaluates each variant on fresh, disjoint sets of 800 controlled-shift families and 200 recursive-relay families. Full LQET and the no-order variant use the same 13 views and forward passes. Compared with no-order, full LQET lowers the order-sensitivity gap from 0.0426 to 0.0359, update RMSE from 0.4105 to 0.4058, and supervised-view KL from 0.0239 to 0.0232; both variants retain low depth-8 relay drift (0.0959 and 0.0753). The order term therefore contributes most clearly to presentation-order robustness. Comparing no-order with Pair-invariance suggests that update-consistency supervision together with the added composition views accounts for most of the compositional gain. 6 Related work Belief updating and predictive consistency. Bayesian teaching can improve probabilistic reasoning in language models [19]. Evaluations study responses to evidence reliability [13], sequential belief trajectories [21], and martingale conditions for in-context learning [8]; Martingale Amortization aligns short- and longer-horizon predictions through self-consistency [14]. We instead ask whether supplied dependence changes the evidential force of otherwise matched reports. Repetition bias and dependent social learning. Schuster et al. show that repetition can reverse language-model source preferences and reduce the effect by training on aligned repeated and unrepeated contexts [22]; this setup most closely overlaps our invariance component. Naphade finds that paraphrases can outweigh genuinely distinct support and documents presentation-order effects [17]. Related work studies the chorus effect [24], persuasion bias [3], and correlation neglect [7]. Graph-based methods correct data incest when topology and the probabilistic model are available [1; 15], while Jain and Krishnamurthy illustrate data incest among interacting language-model agents [12]. Within our controlled setting, LQET extends repetition invariance to explicit multi-hop lineage, exact reliability-conditioned targets, new-root updates across relay histories, composition, and presentation order. Feedback loops and evidence propagation. Language-model outputs can enter later contexts and create failures that static evaluation misses [18]; bidirectional alignment likewise emphasizes that model outputs shape later human inputs [23]. Multi-agent evaluation traces evidence propagation through later messages [25]. Our simulation isolates how score-dependent return of inherited reports interacts with provenance-sensitive updating. Provenance, aggregation, and dependence discovery. Database provenance tracks information origins [2] and composes dependencies on input records [10]; natural-language claim-provenance graphs represent how claims originate and evolve [27]. GEAR is an adjacent evidence-aggregation method over graphs of retrieved sentences, not a report-lineage method [28]. Dong et al. infer copying and source dependence from output agreement [5] and update histories [6]. LQET assumes parent-report links are supplied and learns to map them to consistent belief updates, rather than retrieving evidence or discovering hidden dependence. 7 Scope and limitations This study examines provenance-aware updating in a controlled binary setting with supplied parent links and a specified source-reliability model, enabling exact tests of evidence accounting. Evaluation across two 4B-class backbones and three paired seeds supports conclusions within this setting; transfer to larger models, naturalistic tasks, and incomplete or unreliable provenance remains for further study. The forced-choice YES-NO score assesses binary preference, while free-form probability estimation and downstream decision quality require separate evaluation. Comparisons match training families and optimizer updates but differ in transcript-view exposure, so the reported gains characterize the complete training protocol. Under held-out prior and calibration supports, lower update RMSE accompanies higher supervised-view KL, identifying numerical generalization as a remaining challenge alongside improved relational consistency. The confidence-dependent simulation isolates report recirculation under a prescribed trust rule; studies with human participants are needed to assess how this mechanism operates in practical human-AI interaction. 8 Conclusion LQET formulates evidence accounting as a relational learning problem, connecting predictions on matched transcripts through provenance invariance and new-evidence update consistency. Matched controls distinguish the learned stability from indiscriminate update suppression, while the confidence-dependent feedback simulation demonstrates reduced amplification when model predictions shape future inputs. Supervision thus extends from targets for individual transcripts to consistent updates across relay histories. 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Zhou, and Z. Qiu Qwen3 technical report. External Links: 2505.09388, Link Cited by: §4. Zhang et al. (2020) Y. Zhang, Z. Ives, and D. Roth “Who said it, and why?” provenance for natural language claims. In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, D. Jurafsky, J. Chai, N. Schluter, and J. Tetreault (Eds.), Online, p. 4416–4426. External Links: Link, Document Cited by: §6. Zhou et al. (2019) J. Zhou, X. Han, C. Yang, Z. Liu, L. Wang, C. Li, and M. Sun GEAR: graph-based evidence aggregating and reasoning for fact verification. In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, A. Korhonen, D. Traum, and L. Màrquez (Eds.), Florence, Italy, p. 892–901. External Links: Link, Document Cited by: §6. Appendix A Exact Bayesian target and benchmark construction A.1 Beta–Bernoulli derivation After conditioning on the public calibration record, fix Y=yY=y and a source s. Among the distinct relevant roots from s in E⁡(T)E(T), let Cs(y,T)C_s(y,T) count outcomes that agree with y and Ws(y,T)W_s(y,T) count outcomes that disagree. Integrating the source reliability in Equation 1 gives P((xe)e∈E⁡(T):s⁡(e)=s∣Y=y,cs,ws)=B⁡(αs+Cs(y,T),βs+Ws(y,T))B⁡(αs,βs).P ((x_e)_e∈ E(T):\,s(e)=s Y=y,c_s,w_s )= B( _s+C_s(y,T), _s+W_s(y,T))B( _s, _s). (14) Multiplying over sources and taking posterior log-odds yields Equation 6. The terms log⁡B⁡(αs,βs) B( _s, _s) cancel between the two values of Y, giving t⁡(T) t(T) =log⁡π1−π = π1-π (15) +∑s[logB(αs+ns,+(T),βs+ns,−(T))−logB(αs+ns,−(T),βs+ns,+(T))]. + _s [ B( _s+n_s,+(T), _s+n_s,-(T))- B( _s+n_s,-(T), _s+n_s,+(T)) ]. Multiple distinct roots from one source are conditionally independent given (Y,Rs)(Y,R_s) but become dependent after RsR_s is marginalized. Descendant reports carrying the same root do not create repeated observations because E⁡(T)E(T) contains each root once. The target is the exact Bayesian posterior under the benchmark’s binary source-reliability model. The matched-new and counter views are constructed controls rather than standard sampled views: the former is matched to the copied-report control in author and displayed outcome but declares a new measurement, whereas the latter displays the opposing outcome. Their targets are computed from the evidence declared in each prompt, allowing provenance and outcome polarity to vary separately. A.2 Report provenance and validity Every family is validated as a directed acyclic report graph. Each evidence root has exactly one introducing report; any later report carrying that root includes an explicit link to a parent report that also carries it. A pure relay introduces no root. A mixed report can inherit one or more roots and introduce a new root at the same node. Roots marked irrelevant to the query are excluded from E⁡(T)E(T). These rules make evidence multiplicity a property of provenance rather than the number of textual mentions. A provenance equivalence class contains transcripts with the same set E⁡(T)E(T) of relevant evidence roots. Holding the prior and public calibration records fixed, equality of source-by-outcome counts is sufficient for the same Bayesian target under this benchmark, but it does not define equivalence-class membership: distinct roots from the same source with the same outcome remain distinct evidence. Appendix B Matched transcript views Table 2 lists the 14 matched views. Latent root labels such as e0e_0, eae_a, and ebe_b are construction notation and are never rendered in a prompt. LQET trains on 13 views; c00c_00 is the same rendered transcript as h00h_00 and contributes no distinct training term. The primary task-only and LQET runs are evaluated on all 14 views in the static IID and controlled-shift sets. Table 2: Matched transcript views. Supervised denotes a pointwise soft target; invariance and update denote relational loss terms. View View construction Bayesian-target relation LQET role h00h_00 Base observation carrying root e0e_0 base target supervised; invariance/update reference h10h_10 Base plus a relay DAG; no new root t⁡(h10)=t⁡(h00)t(h_10)=t(h_00) invariance; update reference h01h_01 Base plus new root ehe_h new-evidence target supervised; invariance/update target h11h_11 Relay DAG plus a mixed report introducing ehe_h t⁡(h11)=t⁡(h01)t(h_11)=t(h_01) invariance; update endpoint c00c_00 Same rendered base as h00h_00 t⁡(c00)=t⁡(h00)t(c_00)=t(h_00) static evaluation only c10c_10 Base plus new root eae_a new-evidence target supervised; update reference c01c_01 Base plus new root ebe_b new-evidence target supervised; update reference c11c_11 Base plus both eae_a and ebe_b updates add under source distinctness update endpoint; order c11revc_11^rev Same roots in reversed presentation order t⁡(c11rev)=t⁡(c11)t(c_11^rev)=t(c_11) order matched-copy Matched author/outcome; carries the base root through an explicit parent-report link t=t⁡(h00)t=t(h_00) invariance matched-new Matched author/outcome; declares a new measurement and root matched new-root target supervised same-source New root from the base source source-coupled target supervised irrelevant Adds a root marked irrelevant to the query t=t⁡(h00)t=t(h_00) invariance counter Adds a controlled root with the opposing outcome controlled new-root target supervised The base root e0e_0 and the two composition roots eae_a and ebe_b come from three distinct sources. The core-block root ehe_h is separate from these composition roots. Consequently, t⁡(c11)−t⁡(c00)=[t⁡(c10)−t⁡(c00)]+[t⁡(c01)−t⁡(c00)].t(c_11)-t(c_00)=[t(c_10)-t(c_00)]+[t(c_01)-t(c_00)]. (16) The benchmark does not assume that arbitrary additions from the same source are additive. Appendix C Training and evaluation details C.1 Training views and objective variants Table 3: Training protocols. Methods share training families and 500 optimizer updates. Method Training views Objective Task-only 7 ℒtaskL_task Pair-invariance 11 ℒtask+4ℒinvL_task+4L_inv No-order 13 ℒtask+4ℒinv+2ℒupdateL_task+4L_inv+2L_update Full LQET 13 Equation 13 C.2 Optimization configuration Table 4: Shared training configuration. Item Value Item Value Training families 4,000 Validation families 400 Dataset seed 17 Training seeds 0, 1, 2 Optimizer updates 500 Epochs 2 Families/microbatch 1 Gradient accumulation 16 Optimizer AdamW Learning rate 2×10−42× 10^-4 Weight decay 0 Warmup 15 updates Schedule cosine Gradient clip 1.0 Maximum length 768 Truncated prompts 0 Quantization NF4 with double quantization Compute dtype BF16 LoRA rank 32 LoRA α / dropout 64 / 0.05 LoRA targets all linear modules Huber δ 0.25 Loss weights task/invariance/update/order: 1:4:2:11:4:2:1 Checkpoint final update The loss weights, Huber threshold, and optimization settings were fixed before the reported multi-seed Qwen runs and transferred unchanged to Phi; no validation-set hyperparameter search was conducted. Validation results served as run diagnostics only. The final checkpoint is evaluated without early stopping or validation-based checkpoint selection. C.3 Generator supports and prompt format Table 5 specifies the discrete benchmark supports. Each controlled shift changes only the named factor group; all remaining factors use the training support. Table 5: Benchmark-generator supports. Calibration entries are numbers of correct and incorrect historical calibration trials, (c,w)(c,w). Factor Training, validation, and IID support Controlled held-out support Domain and names Four domains: beacon status, valve position, package authenticity, and server security; 26 source names Three domains: soil contamination, sample radioactivity, and component magnetization; 26 disjoint source names Report graph Chains of depth 1–3; stars with 1–8 relays Depth shift: chains of depth 4–8. Topology shift: diamonds with 3 or 4 relays; fan-in graphs with branch depth 1 or 2 (3 or 5 relays total) Rendering template Direct and contrast renderings Record-style and basis-style renderings Prior π 0.20,0.35,0.50,0.65,0.80\0.20, 0.35, 0.50, 0.65, 0.80\ 0.125,0.275,0.425,0.575,0.725,0.875\0.125, 0.275, 0.425, 0.575, 0.725, 0.875\ Calibration (c,w)(c,w) c∈4,…,9c∈\4,…,9\ and w∈1,2,3w∈\1,2,3\ (0,0),(0,3),(1,4),(2,0),(2,5),(4,4),(4,10),(10,0),(10,4),(10,10),(11,0),(11,5)\(0,0), (0,3), (1,4), (2,0), (2,5), (4,4), (4,10), (10,0), (10,4), (10,10), (11,0), (11,5)\ Each family has four calibrated evidence sources; additional names can appear only as relay authors and contribute no independent observation. The prior and each calibrated source’s record are sampled uniformly from the applicable supports. For generatively sampled evidence roots, the generator draws Rs∼Beta⁡(cs+1,ws+1)R_s (c_s+1,w_s+1), samples Y with P⁡(Y=+1)=πP(Y=+1)=π, and draws an outcome that agrees with Y with probability RsR_s. Separate SHA-256-derived random-number streams govern evidence, calibration, domain, graph, and surface choices. A fixed SHA-256 rule assigns candidate family seeds to training, validation, and test partitions in an 80/10/10 ratio. Automated checks reject any cross-partition overlap in family identifiers, complete benchmark states, latent evidence configurations, or rendered prompts. The benchmark intentionally supplies report lineage. In this representative prompt from the training distribution, parent citations and report text identify an independently acquired observation and its relay; the evaluated task is to use these declarations consistently across matched transcripts and report graphs. Question: Is the valve open? Prior probability of YES: 0.650 Calibration records: – Ava previously made 8 correct and 2 incorrect measurements. – Ben previously made 5 correct and 1 incorrect measurements. – Dara previously made 6 correct and 3 incorrect measurements. – Emil previously made 4 correct and 2 incorrect measurements. Reports: 1. [R1] Ava: Cites: none. Ava independently ran a new position check without seeing any other report. The recorded outcome was OPEN. 2. [R2] Cleo: Cites: R1. Cleo read the cited predecessor report, ran no new measurement, and recorded the inherited outcome OPEN. Instruction: Estimate the posterior probability that the answer is YES. The displayed instruction is the verbatim instruction used in the experiments. We do not decode or parse a generated numeric answer. Instead, the reported quantity is the forced-choice YES score in Equation 8, conditional on the next token belonging to YES,NO\ YES, NO\. For each static IID or OOD view, the rendered prompt is identical across methods. C.4 Static and recursive-relay evaluation sets Each run is evaluated on a static 400-family IID set and five static controlled-shift sets of 160 families each: • domain/names: held-out domains and source names; • depth: relay depths 4–8; • topology: diamond and fan-in report graphs; • template: held-out rendering templates; and • prior/calibration: held-out supports of priors and calibration counts. All 14 matched views are scored on these static sets. Within each split, the ordered family roster is shared across methods, training seeds, and backbones. A separate set of 200 recursive-relay families supports the depth and closed-loop feedback evaluations. For each family, one provenance-preserving chain is extended through depth 8; depths 1, 2, 4, and 8 are nested snapshots of that chain. The new-root probe at each snapshot is evaluated on a separate branch and is not included in later relay steps. The fixed ordered list of recursive-relay families, lineage schedule, and rendering rule is shared across methods, seeds, and backbones. Realized prompt text becomes model-specific after the first prediction because each appended relay quotes that model’s preceding forced-choice score. Training, validation, static evaluation, and recursive-relay partitions are mutually disjoint in benchmark instances, evidence roots, family identifiers, and rendered prompts. Every scored prompt fits within the length limit without truncation. C.5 Endpoint definitions and bootstrap For a fixed backbone and method, let seed j∈0,1,2j∈\0,1,2\ and suppress the backbone and method indices. Let ℰupd=(h10,h11),(c10,c11),(c01,c11)E_upd=\(h_10,h_11),(c_10,c_11),(c_01,c_11)\ contain the three oriented new-evidence update edges from Equation 12, written (v−,v+)(v^-,v^+). For family i and view v, write zj,i(v)=zθj(Ti,v)z_j,i(v)=z_ _j(T_i,v), ti(v)=t⁡(Ti,v)t_i(v)=t(T_i,v), pi⋆(v)=σ⁡(ti(v))p _i(v)=σ(t_i(v)), and p^j,i(v)=σ⁡(zj,i(v)) p_j,i(v)=σ(z_j,i(v)). On the 800 pooled controlled-shift families, the new-evidence update RMSE is RMSEjupdate=1800⋅3∑i=1800∑(v−,v+)∈ℰupd([zj,i(v+)−zj,i(v−)]−[ti(v+)−ti(v−)])2.RMSE^update_j= 1800· 3 _i=1^800 _(v^-,v^+) _upd ([z_j,i(v^+)-z_j,i(v^-)]-[t_i(v^+)-t_i(v^-)] )^2. (17) For recursive-relay family i, let zj,ibasez^base_j,i be the base score and zj,irelay,8z^relay,8_j,i the score after eight provenance-preserving relays, before any new root is added. The seed-level depth-8 drift is D8,jrelay=1200∑i=1200|zj,irelay,8−zj,ibase|.D^relay_8,j= 1200 _i=1^200 |z^relay,8_j,i-z^base_j,i |. (18) Let =h00,h01,c10,c01,same-source,counter,matched-newA=\h_00,h_01,c_10,c_01,same-source,counter,matched-new\ be the seven supervised views. Their seed-level supervised-view KL is Kjsup=1800||∑i=1800∑v∈DKL(Bern⁡(pi⋆(v))∥Bern⁡(p^j,i(v))).K^sup_j= 1800|A| _i=1^800 _v D_KL\! (Bern(p _i(v))\, \|\,Bern( p_j,i(v)) ). (19) For any seed-level endpoint qjq_j above, the reported backbone point estimate is q¯=13∑j=02qj q= 13 _j=0^2q_j. In particular, each RMSE is square-rooted within its seed before the three seed-level values are averaged. Because every trained model is evaluated on the same family rosters, training seeds and evaluation families form a crossed rather than nested design. Using random seed 20260829, each of 10,000 bootstrap replicates independently draws three paired seed labels with replacement and one common with-replacement family sample. For pooled controlled-shift endpoints, the family draw is stratified by resampling 160 families within each of the five fixed shifts; IID endpoints use one 400-family draw, and depth-8 drift uses one 200-family recursive-relay draw. The sampled family indices are reused across every selected seed occurrence and both methods, and all views or relay snapshots from a family remain grouped. We recompute each method-specific endpoint within seed, including taking the square root for RMSE, average over the three sampled seed occurrences, and form the LQET-minus-task-only contrast. We report the 2.5th and 97.5th percentiles without multiplicity adjustment. Appendix D Primary results by seed and evaluation regime D.1 Every training seed Table 6: Per-seed primary endpoints. Effects are LQET minus task-only and are computed from unrounded estimates; lower is better. Backbone Endpoint Seed Task-only LQET Effect Qwen3-4B Update RMSE 0 0.6596 0.3901 −0.2695-0.2695 1 0.6895 0.4143 −0.2752-0.2752 2 0.6958 0.3971 −0.2987-0.2987 Depth-8 drift 0 1.2172 0.0931 −1.1241-1.1241 1 0.8875 0.1078 −0.7797-0.7797 2 0.9319 0.0616 −0.8703-0.8703 Supervised-view KL 0 0.0212 0.0226 +0.0014+0.0014 1 0.0227 0.0258 +0.0031+0.0031 2 0.0205 0.0229 +0.0024+0.0024 Phi-4-mini Update RMSE 0 0.7464 0.4074 −0.3391-0.3391 1 0.7439 0.4199 −0.3240-0.3240 2 0.7479 0.4160 −0.3319-0.3319 Depth-8 drift 0 1.0375 0.1156 −0.9219-0.9219 1 1.1775 0.1313 −1.0463-1.0463 2 1.1088 0.1063 −1.0025-1.0025 Supervised-view KL 0 0.0279 0.0281 +0.0001+0.0001 1 0.0277 0.0296 +0.0019+0.0019 2 0.0285 0.0301 +0.0015+0.0015 D.2 IID test results Table 7: Results on the 400-family IID test set. Point estimates average three paired training seeds. Effects are LQET minus task-only; intervals are 95% paired crossed seed-by-family bootstrap intervals. Lower is better. Backbone Endpoint Task-only LQET Effect [95% interval] Qwen3-4B Update RMSE 0.5635 0.0515 −0.5119-0.5119 [−0.5395,−0.4834-0.5395,-0.4834] Supervised-view KL 0.00052 0.00023 −0.00029-0.00029 [−0.00036,−0.00022-0.00036,-0.00022] Phi-4-mini Update RMSE 0.6354 0.0865 −0.5489-0.5489 [−0.5772,−0.5200-0.5772,-0.5200] Supervised-view KL 0.00156 0.00061 −0.00094-0.00094 [−0.00104,−0.00084-0.00104,-0.00084] D.3 Controlled-shift decomposition Table 8: Results by controlled shift. Descriptive three-seed means are shown as task-only → LQET. Update RMSE is measured in log-odds units and KL in nats. Each shift has 160 families per seed; lower is better. Backbone Shift Update RMSE Supervised-view KL Qwen3-4B Domain/names 0.5717→0.17700.5717→ 0.1770 0.00673→0.005460.00673→ 0.00546 Depth 0.5371→0.05110.5371→ 0.0511 0.00052→0.000260.00052→ 0.00026 Topology 0.6287→0.05000.6287→ 0.0500 0.00055→0.000250.00055→ 0.00025 Template 0.5903→0.09280.5903→ 0.0928 0.00076→0.000800.00076→ 0.00080 Prior/calibration 0.9817→0.86940.9817→ 0.8694 0.09877→0.111960.09877→ 0.11196 Phi-4-mini Domain/names 0.6278→0.11290.6278→ 0.1129 0.00685→0.002020.00685→ 0.00202 Depth 0.6146→0.08390.6146→ 0.0839 0.00151→0.000610.00151→ 0.00061 Topology 0.7009→0.08420.7009→ 0.0842 0.00151→0.000620.00151→ 0.00062 Template 0.6556→0.14450.6556→ 0.1445 0.00180→0.001310.00180→ 0.00131 Prior/calibration 1.0439→0.90051.0439→ 0.9005 0.12853→0.141640.12853→ 0.14164 The prior-and-calibration shift, which requires generalization to held-out numerical supports for both the class prior and source-calibration records, is the most difficult shift for both methods. LQET still lowers update RMSE from 0.9817 to 0.8694 on Qwen (11.4%) and from 1.0439 to 0.9005 on Phi (13.7%), although supervised-view KL rises by approximately 0.013 nats on each backbone. Across the other four shifts, supervised-view KL decreases in seven of eight backbone–shift comparisons and changes by only 4×10−54× 10^-5 in the remaining comparison. D.4 Recursive-relay depth trajectory Table 9: Drift across recursive-relay depth. Values are mean absolute pre-update log-odds drift over three seeds and correspond to Figure 2(a,b). Backbone Method Depth 1 Depth 2 Depth 4 Depth 8 Qwen3-4B Task-only 0.3911 0.7548 0.9932 1.0122 LQET 0.0385 0.0469 0.0594 0.0875 Phi-4-mini Task-only 0.2529 0.5454 0.7879 1.1079 LQET 0.0629 0.0619 0.0842 0.1177 Appendix E Prompt sufficiency and selective-updating controls E.1 Bayesian-target recovery from rendered prompts A deterministic rule-based calculator receives only the rendered prompt. It parses the prior, calibration records, report text, relevance flags, and explicit parent-report links; it does not read Bayesian targets, hidden root identifiers, or family metadata. It successfully processes all 1,200×14=16,8001,200× 14=16,800 transcript-view instances in the static IID and controlled-shift sets. Its maximum absolute difference from the exact Bayesian log-odds computed directly from the benchmark state is below 5×10−155× 10^-15, confirming that the Bayesian target is uniquely determined by the rendered prompt in this benchmark. E.2 Matched copy-versus-new-root controls Let I=800I=800 denote the pooled controlled-shift families for training seed j, and write zj,i(v)z_j,i(v) and ti(v)t_i(v) for the model and exact Bayesian log-odds of view v. For the six supervised new-root views new=h01,c10,c01,same-source,counter,matched-new,V_new=\h_01,c_10,c_01,same-source,counter,matched-new\, define rj,inew(v)=[zj,i(v)−zj,i(h00)]−[ti(v)−ti(h00)].r^new_j,i(v)= [z_j,i(v)-z_j,i(h_00) ]- [t_i(v)-t_i(h_00) ]. (20) The relay and matched-copy metrics are the mean absolute changes along their provenance-preserving edges, Drelay,j D_relay,j =1I∑i|zj,i(h10)−zj,i(h00)|, = 1I _i |z_j,i(h_10)-z_j,i(h_00) |, (21) Dcopy,j D_copy,j =1I∑i|zj,i(matched-copy)−zj,i(h00)|. = 1I _i |z_j,i(matched-copy)-z_j,i(h_00) |. (22) The new-root RMSE averages all six supervised new-root views, whereas matched-new RMSE uses only the matched-new control: RMSEnew,j _new,j =16I∑i∑v∈newrj,inew(v)2, = 16I _i _v _newr^new_j,i(v)^2, (23) RMSEmatched-new,j _matched -new,j =1I∑irj,inew(matched-new)2. = 1I _ir^new_j,i(matched-new)^2. (24) Finally, the copy-versus-new contrast holds report author and displayed outcome fixed while changing whether the report carries the base root through an explicit parent-report link or declares a new measurement. With δ^j,i δ_j,i =zj,i(matched-new)−zj,i(matched-copy), =z_j,i(matched-new)-z_j,i(matched-copy), (25) δi∗ _i^* =ti(matched-new)−ti(matched-copy), =t_i(matched-new)-t_i(matched-copy), (26) we report RMSEcopy-new,j _copy -new,j =1I∑i(δ^j,i−δi∗)2, = 1I _i( δ_j,i- _i^*)^2, (27) Slopej _j =∑iδi∗δ^j,i∑i(δi∗)2. = _i _i^* δ_j,i _i( _i^*)^2. (28) The slope is the zero-intercept regression coefficient of the predicted copy-versus-new change on the exact Bayesian change; its ideal value is one. Table 10 averages each seed-level metric over the three training seeds. Table 10: Provenance-selectivity metrics. Values are descriptive means over three seeds. Lower is better except for the copy-versus-new slope, whose ideal value is one. Backbone Method Relay drift Copy drift New-root RMSE Matched-new RMSE Copy-vs-new RMSE Copy-vs-new slope Qwen3-4B Task-only 1.0463 0.8338 0.3980 0.4326 1.0145 0.027 LQET 0.0264 0.0373 0.4043 0.4328 0.4383 0.782 Phi-4-mini Task-only 1.0377 0.7813 0.4352 0.4682 0.9818 0.067 LQET 0.0540 0.0533 0.4185 0.4582 0.4630 0.769 The matched-copy and matched-new views share the report author and displayed outcome; their supplied provenance differs because one relays the base observation and the other declares a new measurement. Because provenance status is intentionally explicit, this control tests whether the model uses supplied provenance while holding author and outcome fixed. Together with retained new-root update accuracy, the result argues against reliance on author/outcome alone and against uniform suppression of log-odds changes. Appendix F Closed-loop feedback details The closed-loop feedback analysis uses the first 64 families in the fixed ordered list of the same 200 recursive-relay families described in Section C. The subset rule and all simulator constants were fixed before evaluation. The two trust arms span low- and high-trust initial conditions, while three rounds with at most three relays per round define a bounded stress test; the same deterministic family prefix and settings preserve paired comparisons across methods, seeds, and backbones. Each family is evaluated at initial trust values 0.2 and 0.8, producing 128 trajectories per model and seed. If zrz_r is the current forced-choice log-odds, round r applies pr p_r =σ⁡(zr), =σ(z_r), cr c_r =|2pr−1|, =|2p_r-1|, (29) τr+1 _r+1 =0.6τr+0.4cr, =0.6 _r+0.4c_r, mr m_r =1+min⁡2,⌊3τr+1cr⌋. =1+ \2, 3 _r+1c_r \. (30) Here crc_r is confidence derived from the forced-choice score. The simulator appends one AI-derived report followed by mr∈1,2,3m_r∈\1,2,3\ simulated human relays, then recomputes the model score from the rendered prompt. Each appended report includes an explicit link to its immediate parent report and introduces no new evidence root. After three rounds, the simulator appends an observation using the same new-root construction as h01h_01. Let zi0z_i^0 be the base log-odds, ziaprez_ia^pre the log-odds after three feedback rounds for family i and trust condition a, ziapostz_ia^post the log-odds after the new-root observation, and tiapre,tiapostt_ia^pre,t_ia^post the corresponding exact Bayesian log-odds. With M=128M=128 trajectories, the endpoints are Dfeedback D_feedback =1M∑i,a|ziapre−zi0|, = 1M _i,a|z_ia^pre-z_i^0|, (31) RMSEpostfb _postfb =1M∑i,a[(ziapost−ziapre)−(tiapost−tiapre)]2, = 1M _i,a [(z_ia^post-z_ia^pre)-(t_ia^post-t_ia^pre) ]^2, (32) MAEfinal _final =1M∑i,a|ziapost−tiapost|. = 1M _i,a|z_ia^post-t_ia^post|. (33) Table 11: Closed-loop feedback results by training seed. Lower is better. Backbone Method Seed Feedback drift Post-feedback update RMSE Final log-odds MAE Qwen3-4B Task-only 0 1.5078 0.6936 0.9644 LQET 0 0.4336 0.1732 0.3160 Task-only 1 1.1973 0.6725 0.6909 LQET 1 0.3887 0.1733 0.2560 Task-only 2 1.2891 0.7191 0.6708 LQET 2 0.3169 0.1362 0.2520 Phi-4-mini Task-only 0 1.0566 0.8477 0.4717 LQET 0 0.1172 0.1361 0.1041 Task-only 1 1.1191 0.8601 0.5065 LQET 1 0.0938 0.1197 0.1056 Task-only 2 1.1113 0.8610 0.4810 LQET 2 0.0869 0.1068 0.0947 LQET also reduces the mean cumulative number of simulated human relays from 5.4349 to 4.4557 on Qwen and from 5.0469 to 4.3438 on Phi. Because the number of relays depends on score-derived confidence, these trajectories capture both reduced sensitivity to provenance-preserving reports and the associated reduction in how many such reports return. Appendix G Component analysis The Qwen3-4B seed-0 component study uses fresh, disjoint evaluation sets: 800 static controlled-shift families (160 per shift) for update RMSE, supervised-view KL, and order-sensitivity gap, and 200 recursive-relay families for depth-8 drift. For the static controlled-shift families, define Gorder=1800∑i=1800|zi(c11)−zi(c11rev)|.G_order= 1800 _i=1^800 |z_i(c_11)-z_i(c_11^rev) |. (34) Full LQET and No-order both process 13 training views and use identical forward computation. No-order sets the order-loss weight to zero, so c11revc_11^rev has no loss contribution. Pair-invariance processes 11 training views. Intervals use 10,000 paired family-bootstrap draws over the relevant fresh static or recursive-relay families. Table 12: Qwen3-4B seed-0 component results. Lower is better. Method Update RMSE Depth-8 drift Supervised-view KL Order gap Pair-invariance 0.5860 0.0984 0.0212 0.1184 No-order 0.4105 0.0753 0.0239 0.0426 Full LQET 0.4058 0.0959 0.0232 0.0359 Table 13: Paired component effects. Effects are No-order minus full LQET and use unrounded estimates. Brackets are unadjusted 95% paired family-bootstrap intervals; positive effects favor full LQET for these lower-is-better endpoints. Endpoint Effect 95% CI Update RMSE +0.00475+0.00475 [+0.00157,+0.00782+0.00157,+0.00782] Depth-8 drift −0.02063-0.02063 [−0.03125,−0.01031-0.03125,-0.01031] Supervised-view KL +0.00076+0.00076 [+0.00031,+0.00125+0.00031,+0.00125] Order gap +0.00664+0.00664 [+0.00289,+0.01023+0.00289,+0.01023] Figure 4: Single-seed component trade-off. (a) No-order-minus-full effects normalized by full LQET, with 95% paired family-bootstrap intervals; positive values favor full LQET for the displayed lower-is-better endpoints. (b) Edgewise update RMSE across component variants. The order term yields its clearest improvement on the order-sensitivity gap and modest improvements in update RMSE and supervised-view KL, while No-order has lower depth-8 drift for this seed (Figure 4a). Relative to Pair-invariance, No-order reduces update RMSE by 0.1755 and the order-sensitivity gap by 0.0759; the two composition edges account for most of the update improvement (Figure 4b). Because adding update consistency also adds the composition views, this comparison estimates their combined contribution rather than isolating the loss term alone.

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