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What to Preserve in Recursive Computation: A Local Predictive Sufficiency Principle

Peilin Wang, Feng Shiyang, Hongfu Gao, Cencheng Zhao, Di Yuan, Hui Chen, Guiguang Ding

arXiv:2610.04303v1cs.LGcs.AI

TL;DR

Recursive computation repeatedly compresses intermediate states, while existing approaches do not provide a tractable criterion for what each interface should preserve for subsequent prediction. The paper develops local predictive sufficiency and a training procedure that constrains task updates using estimated predictive loss across compression. Across four recursive settings, it reports improved host performance under matched compression budgets, with larger gains under heavier recursive demands.

  • Problem

    Recursive compression creates a need for a tractable criterion specifying what intermediate states should preserve for subsequent prediction.

    Source: §1 · p. 1

  • Method

    The method estimates empirical predictive deficiency at recursive interfaces and uses its sensitivities to constrain task-driven parameter updates.

    Source: §1 · p. 2

  • Results

    Across temporal graphs, language memory, vision-language-action control, and recursive self-improvement, the method improves host models under matched compression budgets, with larger gains under longer recursive demands.

    Source: §1 · p. 2, §4 · p. 8, §5 · p. 9

  • Takeaways & Limitations

    The results support predictive preservation at recursive interfaces as a principle for compact states across domains.

    Source: §5 · p. 9

Abstract

from arXiv · show

Recursive computation repeatedly compresses or reuses intermediate states, creating a simple tension: information that must remain useful across longer recursive paths is also exposed to more opportunities for loss before reaching the final prediction. Existing reconstruction or local-prediction objectives provide tractable supervision, but do not ensure that the retained information remains sufficient for subsequent recursive computation. We identify local predictive sufficiency with recursive predictive closure: controlling local predictive deficiencies at individual interfaces controls the resulting discrepancy at the root. We then turn this principle into a tractable training procedure. Starting from a variational characterization, we derive finite predictive tests and an empirical predictive deficiency that measures predictive value retained across compression. Its predictive sensitivities define margin-relaxed half-space constraints on parameter updates, and we project the host optimizer's proposed update onto their intersection only when predictive preservation would otherwise be violated. Across temporal graphs, language memory, vision-language-action control, and recursive self-improvement, the method matches or improves the corresponding host models under matched compression budgets, with larger gains under heavier recursive or memory demands, while better preserving predictive information across successive transformations. Crucially, the same task-agnostic predictive-preservation principle is instantiated across all four settings through host-compatible interventions while keeping the endpoint task, backbone, and evaluation protocol fixed. These results establish predictive preservation at recursive interfaces as a general training principle for recursive compression.

1 Introduction

Recursive systems repeatedly compress intermediate states, so information needed across longer paths faces more opportunities for loss. The paper develops local predictive sufficiency and a training procedure for preserving predictive information at recursive interfaces, then evaluates it across four settings.

1 Introduction

  • Longer recursive paths expose information needed for the final prediction to more opportunities for loss during repeated aggregation and compression.The paper frames this as a shared structure across graph and tree propagation, recurrent context compression, embodied memory, and recursive self-improvement.Source: §1 · p. 1
  • Reconstruction, predictive-state, and information-theoretic approaches do not provide a tractable criterion for what each interface should preserve for subsequent prediction.The paper notes that information-theoretic criteria are difficult to estimate and optimize for high-dimensional intermediate states.Source: §1 · p. 1
  • Local predictive sufficiency asks whether compression retains the predictive information available before compression for the adjacent predictive pair at that interface.The paper connects these local conditions to protection of the root target through recursive predictive closure and derives empirical predictive deficiency.Source: §1 · p. 2
  • Across temporal graphs, language memory, vision-language-action control, and recursive self-improvement, the method improves hosts with larger gains under longer recursive demands.On LIBERO-Mem T3, full-task success rises from 1.7 ± 2.9% to 11.7 ± 2.9% across three independently trained seeds; the paper also reports slower predictive-statistic attenuation across successive interfaces.Source: §1 · p. 2

2 Preliminaries and Problem Formulation

The paper models recursive computation as compression across interfaces in a rooted tree, with predictive events associated with recursive stages and a root target. Local predictive sufficiency for adjacent events can compose toward root-target sufficiency under stated structural conditions.

2.1 Recursive Compression Setup

  • Recursive computation is unrolled as a finite rooted tree, with each non-root node an interface and the root the target site.At each interface, U_v is the pre-compression state, C_v is retained directly across the update, and Γ produces a width-r representation Z_Γ.Source: §2.1 · p. 2
  • The compression map is intended to preserve predictive information required for the root target across repeated compression.Each recursive interface has an associated predictive event Y_v, with Y_ρ denoting the root target.Source: §2.1 · p. 2

2.2 Predictive Preservation under Recursive Structure

  • As depth increases, the root target becomes a more indirect guide to what an intermediate state should retain because its relation is mediated by more transformations.The passage attributes this to the data-processing principle: mutual information cannot increase along the chain and may contract under repeated transformations.Source: §2.2 · p. 3
  • Adjacent predictive events along a rootward path form pairs S_v=(Y_v,Y_pa(v)) that motivate local predictive sufficiency at each non-root interface.The tree structure and stated local-separation condition support using nearby predictive events; consecutive pairs share an intermediate event.Source: §2.2 · p. 3
  • Under conditions (M1)–(M3), local predictive sufficiency at recursive interfaces implies sufficiency for the root target.The proposition's stated conditions include the predictive-tree condition, recursive host construction, and an interface screening condition.Source: §2.2 · p. 3
  • The local-to-root argument relies on shared predictive events that let sufficiency compose toward Y_ρ.The paper motivates local deficiency I(S_v;U_v|Z_Γ,C) as the quantity to control; driving these local deficiencies to zero recovers the root-level guarantee.Source: §2.2 · p. 3

3 Learning for Predictive Preservation

The learning procedure estimates predictive information lost across compression, turns its sensitivities into margin-relaxed constraints, and projects task-driven updates when necessary. The section also gives a local-to-global guarantee relating interface deficiencies to root predictive discrepancy.

3.1 Predictive Deficiency Estimation

  • Predictive deficiency measures the predictive information about adjacent events lost when the pre-compression state is replaced by its compressed representation.The method moves from an exact criterion to finite predictive tests, profiled ridge readouts, covariance-balanced predictive value, and empirical covariance estimates over an optimization window.Source: §3.1 · pp. 3–4, Figure 1 · p. 3
  • The empirical deficiency compares balanced predictive value before and after compression, treating the pre-compression value as a fixed reference for optimization sensitivities.The main method uses full-covariance balancing; the diagonal form is evaluated only as a covariance approximation in an ablation.Source: §3.1 · p. 4

3.2 Margin-Relaxed Predictive Constraints

  • Predictive sensitivities are pooled into grouped directions that define half-space constraints on the task proposal, with tolerance κ controlling the allowed first-order increase in deficiency.The intersection of these constraints is the predictive-feasible region; Figure 2 depicts the task proposal and constrained update in that region.Source: §3.2 · p. 4, Figure 2 · p. 4
  • The projection correction lies in the span of predictive constraint directions, and its dual optimization uses the constraint space rather than the full parameter space.The dual problem is m-dimensional instead of p-dimensional, and the full-covariance choice sets b_t=κ∥t_t∥_2 1_m.Source: §3.2 · p. 4

3.3 Predictive Feasibility Optimization

  • Within each optimization window, predictive statistics form the grouped constraints, the host optimizer proposes an update, and projection returns a constrained update for the compressor.The host forward computation and task objective remain unchanged; the optimizer state advances once per window, and the update is written back to compressor parameters.Source: §3.3 · p. 5, Figure 3 · p. 5

3.4 Properties and Predictive Guarantees

  • Under (M1)–(M3), root predictive-law change is controlled by the local predictive deficiencies at individual recursive interfaces.When every local deficiency vanishes, the compressed computation preserves the root target exactly; the guarantee concerns population-level deficiencies when they are controlled.Source: §3.4 · pp. 5–6
  • Root-target sufficiency alone can hold while local predictive deficiency remains positive, so the paper constrains grouped changes in a predictive surrogate.At zero margin, each grouped constraint has a non-positive linearized contribution, while a feasible task proposal is unchanged.Source: §3.4 · p. 6

4 Experiments

Experiments test predictive preservation across four recursive settings using matched host models and fixed evaluation protocols. Results favor the method across these systems, with stronger gains under longer histories or deeper recursive dependence.

Experimental Setup

  • Experiments cover temporal graphs, language memory, vision-language-action control, and recursive self-improvement with matched hosts and fixed evaluation protocols.Checkpoint selection uses development data, followed by frozen evaluation on separate held-out data.Source: §4 · p. 6

Main Findings

  • The method consistently improves corresponding hosts across four recursive systems; language-memory gains are largest at longer histories, while the UCI gap to DyGFormer closes at deeper propagation.At the deepest UCI setting, the method reaches comparable AP and higher AUC than DyGFormer.Source: Table 1 · p. 7, §4 · p. 8

Mechanism Analysis

  • Mechanism diagnostics associate stronger task performance with retained predictive value across interfaces and greater usefulness of retained information in later recursive updates.The diagnostics do not directly estimate population predictive deficiency.Source: §4 · p. 8, Figure 4 · p. 9

Optimization and Ablations

  • Constraint violations persist in unconstrained task proposals, and controlled ablations show substantial degradation when predictive criteria or interface-wise constrained updates are removed.The ablations use the UCI depth-3 setting.Source: §4 · p. 8, Table 2 · p. 9

Robustness and Efficiency

  • Feasibility tolerance has an interior optimum, while increasing eligible-interface coverage improves test AP before approaching saturation.The study selects the tolerance by a small validation sweep and retains broad interface coverage.Source: §4 · p. 8

5 Conclusion

The conclusion presents local predictive preservation as a principle for learning compact recursive states and reports gains across four settings. It also identifies estimator stability, tolerance calibration, and the scope of VLA evaluation as limitations; the AI-use statement describes supporting uses reviewed by the authors.

5 Conclusion

  • The framework combines local predictive deficiency, recursive predictive closure, and feasibility projection, and improves host models across four recursive settings under matched compression budgets.The conclusion presents predictive preservation at recursive interfaces as a principle for compact states across domains.Source: §5 · p. 9
  • Estimating local predictive constraints may be less stable in data-sparse regimes, and feasibility tolerance κ requires validation-based calibration that may vary across tasks and domains.VLA evidence is limited to held-out LIBERO-Mem T3 evaluation initializations, not broad out-of-distribution generalization.Source: §5 · p. 9

A AI Use Statement

  • Generative AI assisted manuscript polishing, literature and repository discovery, descriptive-text drafting, intermediate mathematical work, and code implementation, with authors reviewing or validating the final material.The authors state that they take full responsibility for the final content.Source: A AI Use Statement · p. 15

B Related Work

Prior approaches offer reconstruction, prediction, information-theoretic compression, predictive-state approximation, and constrained optimization, but do not specify what must remain predictive across recursive interfaces. The paper formulates that condition and supplies a reproduction path across the experimental domains.

B Related Work

  • Reconstruction and fixed future-prediction targets do not by themselves ensure that compressed states preserve distinctions needed after subsequent recursive aggregation and compression.Information-bottleneck and predictive-state approaches offer relevance or future predictability criteria, while this work adds a compositional requirement across interfaces.Source: B Related Work · p. 15
  • Empirical predictive deficiency approximates a recursively specified preservation requirement using finite tests, while predictive sensitivities define data-dependent constraints on host-optimizer proposals.The contribution is the predictive constraint and its interface-wise application, not projection or quadratic programming itself.Source: B Related Work · p. 16

Reproduction Guide

  • The reproduction guide traces each domain from dataset and cohort construction through matched-host training, validation-only selection, and frozen evaluation.Appendix references and supplementary configurations connect protocols, results, and implementations.Source: Table 3 · p. 17, Reproduction Guide · p. 17

C.1 Temporal Graphs

This section describes matched evaluation protocols for temporal-graph prediction, recursive language-memory compression, and vision–language–action memory, varying temporal depth or accumulated history while holding comparison conditions fixed. It also defines the MemoryVLA–Ours evaluation and completion metrics, treating published comparisons as contextual rather than protocol-matched evidence.

C.1 Temporal Graphs

  • Wikipedia node classification and UCI temporal link prediction use TGN as the host and uncompressed baseline across matched, architecture-specific propagation depths.Depth counts successive representation stages; TGN neighborhoods use only interactions before the query time, and data splits are chronological.Source: §C.1 · pp. 17–18

C.2 Language Memory

  • DailyDialog and MSC assess language-memory compression with CCM on LLaMA-7B and Gemma-4, using controlled history-depth comparisons and cross-session prediction with session openings as context.DailyDialog varies L ∈ {4, 6, 8, 11, 13}, holding cohort, context, target, and scored positions fixed; MSC uses each session’s opening user utterance as context and its response as target, retaining the same conversation cohort.Source: §C.2 · pp. 19–21

C.3 Vision–Language–Action

  • On LIBERO-Mem’s repeated-bowl-lifting T3 task, Ours trains only a new compressor Γ over a frozen MemoryVLA host and is compared on matched held-out initializations.The host uses AVG aggregation, replaced by Γ for Ours; evaluation uses the 20 official validation initializations, separate from the internal checkpoint-selection split.Source: §C.3 · pp. 22–23
  • Matched MemoryVLA–Ours comparisons reuse fixed held-out initializations, with uncertainty computed across independently trained runs.Reusing evaluation cases does not count as additional training runs.Source: §C.3 · p. 23
  • SR measures rollouts completing all three repetitions, while SGC measures the share of credited repetitions across 20 held-out rollouts.Each rollout uses an official LIBERO-Mem validation initialization, and the original evaluator determines repetition credit.Source: §C.3 · p. 23
  • The original LIBERO-Mem evaluator is unchanged, with no method-specific criteria for full or partial success.Repetition counting, stability checks, and overshoot handling follow the original evaluator.Source: §C.3 · p. 23
  • Published π0, SlotVLA, and E-SlotSSM results are contextual references, while method-effect claims rely on matched MemoryVLA–Ours comparisons.E-SlotSSM uses oracle subgoal information and is marked as an oracle-assisted reference.Source: §C.3 · p. 24

C.4 Self-Recursive Improvement

The SRI comparison holds the realized starting state, retained-history capacity, recursive-search opportunity, backbone, and evaluation protocol fixed, changing which available execution traces are retained. The extended results include recursive-conflict diagnostics and domain-specific analyses.

C.4 Self-Recursive Improvement

  • Ours changes which already available execution traces are retained, while sharing Metan’s history pool, retention capacity, solver stack, and context construction.Predictive preservation receives no held-out information and creates no extra candidate proposals, evaluator queries, retrieval calls, or external LLM requests.Source: §C.4 · p. 25, Table 7 · p. 26
  • Both arms receive equal candidate-generation and evaluation opportunities; failed generations count toward the budget without replacement proposals.Internal predictive-preservation computation is accounted for separately from the external recursive-search budget.Source: §C.4 · p. 25
  • Held-out evaluation follows frozen development-side selection, and the reported Dev Gain and Held-out Score measure distinct stages of recursive refinement.Held-out scores are not used for candidate selection or Dev Gain; frozen solutions are evaluated on the same official held-out instances.Source: §C.4 · pp. 25–26
  • Violation density on UCI 3-hop (R0, κ = 0.05) falls from 0.020 in the first training third to 0.012 in the final third.The appendix reports that conflicts remain throughout training, with a temporary mid-training rise before the decline.Source: Figure 5 · p. 28

D Extended Results and Diagnostic Analyses

  • The extended appendix provides domain-specific diagnostics, including five-seed temporal-graph results, language-memory depth comparisons, VLA rollout analyses, and complete SRI results.It also reports predictive-constraint and training-time predictive diagnostics.Source: D Extended Results and Diagnostic… · p. 28

D.1 Temporal Graph Results

Temporal-graph diagnostics show fewer late-training constraint conflicts and close agreement between the constraint surrogate and measured window-statistic changes over the tested perturbations.

D.1 Temporal Graph Results

  • Constraint conflicts persist through training, rise temporarily in the middle, then decline; the final-third violation density is lower than the first-third density.The lower late-training density is consistent with task proposals becoming more aligned with the predictive-feasible region, although violations remain.Source: §D.1 · p. 28
  • The audit defines q as the local predictive sensitivity represented by the constraint surrogate, δ as a parameter perturbation, and ΔbJwin using PairKFWindow’s tree-weighted aggregation.Complete temporal-graph results are averaged over five random seeds and reported as mean ± standard deviation.Source: §D.1 · pp. 28–29
  • At base scale, measured change is 1.111× the first-order prediction; with 16× larger perturbation, the ratio is 1.097 and measured change grows 15.8× versus 16.0× predicted.Table 9 compares the constraint surrogate’s prediction with the measured finite change of the PairKFWindow statistic on UCI.Source: §D.1 · p. 29, Table 9 · p. 29
  • The optimizer constrains a local first-order surrogate rather than the finite post-update statistic; the roughly 10% scale difference remains stable across perturbation sizes.This real-window audit supports the practical bridge from the predictive statistic to the linearized half-space constraints.Source: §D.1 · p. 29

D.2 Language Memory Results

The language-memory evaluation compares Ours with a fixed CCM-merge host across two backbones and datasets. Training-time predictive-preservation signals stay positive as validation improvement remains above the host baseline, with broadly aligned trajectories.

D.2 Language Memory Results

  • The comparison covers DailyDialog and Multi-Session Chat across LLaMA-7B and Gemma-4-E4B-It, using matched examples, targets, memory depths, and perplexity aggregation.Ours starts from the fixed validation-selected CCM-merge checkpoint and is trained under three independent seeds.Source: §D.2 · p. 29
  • Training diagnostics measure correctly paired versus shuffled future observations, including a within-depth shuffle control, and smooth noisy measurements with a window of 9.Figure 6 places these predictive-preservation signals alongside validation gains on the same training trajectory; higher values indicate better behavior in both panels.Source: §D.2 · p. 29, Figure 6 · p. 31
  • Predictive-preservation signals stay positive throughout training while validation improvement remains above the CCM-merge baseline, with both trajectories strengthening and then weakening together.Both reach their strongest region around the later-middle training windows; the validation gain increases toward s40 and weakens toward s50.Source: §D.2 · p. 30

D.3 Vision–Language–Action Results

Two paired held-out LIBERO-Mem T3 rollouts show more ordered repetitions completed by Ours than MemoryVLA. The snapshots are sampled regularly from training run 1 under the same task setting.

D.3 Vision–Language–Action Results

  • Each comparison is one closed-loop T3 rollout from a held-out evaluation initialization, with up to three ordered repetitions completed sequentially.The broader evaluation protocol uses 20 official held-out initializations and selects checkpoints on the internal development split.Source: §D.3 · p. 31
  • Ours completes 2/3 ordered repetitions versus MemoryVLA’s 1/3 on held-out initialization 15, and 3/3 versus 2/3 on initialization 16.These are two paired rollouts from training run 1, sampled at regular intervals.Source: §D.3 · p. 31

D.4 Recursive Self-Improvement Results

In recursive self-improvement, paired held-out rollouts and solver traces illustrate differences in task completion and strategy evolution. The implementation sections describe how predictive preservation is inserted into temporal graphs, language memory, VLA control, and discrete recursive-history retention.

D.4 Recursive Self-Improvement Results

  • For DeepSeek-V4.1-Flash, Ours and the baseline begin from the same depth-1 state but retain different candidate counts and reach different development scores.The reported candidate-evolution table summarizes retained candidates by recursive depth.Source: §D.4 · p. 31, Table 15 · p. 35
  • The gemini-3.1-pro aircraft solver progressively separates discrete landing-sequence search from landing-time optimization, ultimately retaining the Gen. 5 simulated-annealing-plus-exact-LP strategy.Later responses explore wider candidate funnels, but the selected deployed solver retains the Gen. 5 strategy.Source: §D.4 · pp. 31–32, Table 16 · p. 36
  • On held-out T3 initialization 15, Ours completes 2/3 repetitions versus MemoryVLA’s 1/3; on initialization 16, it completes 3/3 versus 2/3.These are qualitative paired rollouts from training run 1.Source: Figure 7 · p. 34, Figure 8 · p. 34

E Implementation, Ablations, and Reproducibility

  • The implementation appendix describes a shared predictive-preservation framework instantiated with host-compatible interventions and reports separate ablations and hyperparameter sensitivity.Dataset construction, split rules, and evaluation protocols are in Appendix C; complete results and diagnostics are in Appendix D.Source: E Implementation, Ablations, and… · p. 37

E.1 Temporal-Graph Implementation

  • For temporal graphs, predictive preservation acts on intermediate TGN representations reused by parent recursion, with predictive sensitivities constraining updates to compression parameters Γ.The host temporal memory, neighborhood construction, aggregation, and benchmark task pipelines are retained.Source: §E.1 · p. 37

E.2 Language-Memory Implementation

  • For language memory, CCM’s compressed dialogue state is reused across turns, and predictive observations pair each memory state with downstream utterances and the eventual response.The intervention trains the CCM compression parameters while freezing the base language model; the task and evaluation protocol remain unchanged.Source: §E.2 · pp. 37–38

E.3 Vision–Language–Action Implementation

  • For VLA control, MemoryVLA’s compressed working memory is paired with recorded future actions, while the original encoders, action pathway, task objective, and execution procedure remain intact.Future actions provide training-time supervision only; they are not supplied to memory, compression, or policy inputs.Source: §E.3 · pp. 38–39

E.4 Self-Recursive Improvement Implementation

  • For Metan, a learned reducer selects host-produced history items under the existing context budget, and predictive training uses recovered archive lineages offline.The language model and item encoder remain fixed, no additional Ω calls are made during training, and child and grandchild observations are unavailable at evaluation.Source: §E.4 · pp. 39–40

E.5 Training and Compute Configuration

The appendix reports formal training configurations and measures computational overhead on TGN under matched structural settings. The added computation is concentrated in training, while inference uses the learned compressor without feasibility projection.

E.5 Training and Compute Configuration

  • Table 17 summarizes the run-level configurations for the formal experiments, with method-specific statistical and feasibility hyperparameters reported separately.The table records the actual experimental configuration rather than host repository defaults.Source: §E.5 · p. 41, Table 17 · p. 41
  • VLA’s 25k-step budget is an upper bound; checkpoints are selected on an internal development split and evaluated on separate held-out T3 initializations.The reported success and completion rollouts are used for neither training nor checkpoint selection.Source: §E.5 · p. 41

E.6 Efficiency and Model Overhead

  • Training epoch time is 1.9–2.2× the vanilla host, while test-time evaluation stays within about ±3% of vanilla.The predictive-feasibility machinery is training-only; inference uses no projection or replay.Source: §E.6 · p. 41, Table 18 · p. 41, Table 19 · p. 41

E.7 Ablation Study

In a fixed seed-0 UCI depth-3 ablation, the full method achieves the highest AP and AUC. The ablations test the preservation objective, supervision structure, statistical realization, and constrained optimization.

E.7 Ablation Study

  • Each ablation configuration was run once, so the study diagnoses mechanisms rather than estimating cross-seed variability.The main comparison’s multi-seed results are reported separately.Source: §E.7 · p. 42
  • Direct Future Prediction reaches 92.76 AP and 92.24 AUC, below the full method’s 94.52 AP and 94.25 AUC.History reconstruction also falls below Task-Only Training, supporting the distinction between reconstructing the past and preserving information useful for subsequent prediction.Source: §E.7 · p. 42
  • Single-event, cross-level mispaired, and context-removed supervision each underperform the full method, supporting aligned adjacent-pair supervision with retained context.The single-event criterion weakens local supervision; mispairing predictive events and removing context also leave a gap to the full method.Source: §E.7 · p. 42
  • Per-tree estimation, removing balancing, and diagonal-covariance approximation each reduce performance, supporting the shared balanced statistic with cross-dimensional dependence.The authors interpret the per-tree result as evidence for pooling aligned observations across recursive instances.Source: §E.7 · p. 42
  • The additive predictive-gradient update and global constraint both underperform the full method, supporting interface-wise constrained optimization over either alternative.The global constraint has the largest drop; the authors link this to local predictive loss arising at individual recursive interfaces.Source: §E.7 · p. 42
  • The full method achieves the highest AP and AUC across the four groups in the fixed seed-0 run.The table covers UCI at depth 3, with each variant changing one component while keeping the remaining protocol fixed.Source: §E.7 · p. 42, Table 20 · p. 43

E.8 Method Hyperparameters and Sensitivity

The appendix reports frozen method-specific hyperparameters and examines UCI sensitivity to feasibility tolerance and eligible-interface support. Performance is stable near the formal operating region, while more extreme settings or reduced support lower it.

E.8 Method Hyperparameters and Sensitivity

  • Feasibility tolerance κ controls admissible predictive degradation and is the principal geometric hyperparameter; eligible-interface fraction varies statistical support.Hyperparameters were selected on development or validation data and frozen before formal evaluation; formal TGN and VLA runs do not use predictive coefficient λ.Source: §E.8 · p. 43
  • UCI performance remains stable near the formal operating region, while substantially tighter or looser feasibility and reduced statistical support lower performance.Figure 9 varies feasibility tolerance κ and the eligible-interface fraction while holding remaining settings fixed.Source: §E.8 · p. 43, Figure 9 · p. 44

F Theoretical Results and Proofs

F.1 Proof of Recursive Closure

This section proves how local predictive deficiencies support recursive closure and root-law or root-stability preservation under stated assumptions, while characterizing the relevant sufficiency conditions and why root-target sufficiency alone is insufficient. It also examines deficiency estimation and projected updates, including their calibration, approximation limits, and finite-horizon conditions.

F.1 Proof of Recursive Closure

  • The proof assumes M1’s predictive-boundary Markov tree, M2’s aggregation through compressed child states, and M3’s complete-interface screening.M2 does not require independence between the host pre-compression state and incoming compressed child states.Source: §F.1 · pp. 45–46
  • Zero local predictive deficiency and complete-interface screening imply local sufficiency still holds when the incoming compressed child tuple is exposed.This augmented closure follows from the original local defect and screening condition; no additional interaction or node-coupling defect is assumed.Source: §F.1 · p. 47
  • Subtree reconstruction depends on the compressed subtree and retained context, not the external boundary, while allowing the parent reconstruction to depend on incoming child states.The local reconstruction kernel is independent of the adjacent predictive pair once the compressed state, incoming child tuple, and context are fixed.Source: §F.1 · pp. 47–48
  • Subtree closure composes over the root’s conditionally independent child subtrees, yielding global conditional independence at the root.The induction establishes closure for every non-root subtree before the root composition.Source: §F.1 · p. 49
  • The proof combines two conditional-independence equalities to show that preserving the root prediction law follows from the recursive closure conditions.The proof then invokes the conditional Kullback–Leibler characterization of mutual information to conclude the theorem.Source: §F.1 · p. 50

F.2 Local Variational Characterization

  • The local variational characterization makes zero predictive deficiency equivalent to equality of the pre- and post-compression conditional laws of S_v given context.When the optimum is finite, minimizers are unique up to conditioning-law null sets; the appendix presents this as an exact reformulation of the same deficiency used in recursive closure.Source: §F.2 · p. 51

F.3 From Local Predictive Deficiency to Constrained Updates

  • The computable statistic approximates, rather than exactly estimates, the ideal local deficiency through finite tests, a restricted readout class, and empirical full-covariance moments.The analysis tracks separate finite-test, readout-class, and finite-sample residuals; the root-level guarantee applies to ideal deficiencies, while the pooled statistic is handled separately.Source: §F.3 · p. 52

F.3.1 Exact Local Criterion and Context Localization

  • Context localization identifies the main-text and interface-visible conditioning choices; without it, the construction is only a local-context surrogate.With full observation, the variational identity remains exact after localization; censoring requires an additional selection assumption to identify the eligible-interface defect with the unconditional one.Source: §F.3.1 · pp. 52–53

F.3.2 Finite Predictive Tests

  • Finite predictive tests give an exact value gap for the selected tests, but their use in place of the full conditional law introduces a coverage residual.Under the stated coverage condition, that residual tends to zero pointwise for a nested complete test family; it is a test-representation approximation, not sampling error.Source: §F.3.2 · pp. 53–54

F.3.3 Profiled Predictive Value

  • Profiling is exact within the fixed ridge readout class, but restricting predictors to that class is an approximation.The width of a transmitted state alone does not imply a rank-matched orthogonal projection in balanced covariance geometry, so the bridge does not assume a Ky–Fan equality.Source: §F.3.3 · p. 55

F.3.4 Empirical Predictive Deficiency

  • Diagonal covariance is an optional variant, with perturbation bounds for its discrepancy from the full-covariance geometry under the stated conditions.The variant adds a diagonal-approximation residual; relative bounds require the corresponding covariance-departure conditions.Source: §F.3.4 · p. 56
  • The empirical deficiency bridges the ideal local criterion to the implemented statistic through identifiable residuals, without asserting equality or monotonic decrease of the Shannon defect.Residuals shrink with richer tests, a better readout class, and more statistical support; the optional diagonal variant adds a covariance-geometry residual.Source: §F.3.4 · p. 58

F.3.5 Interface Constraints and Projection

  • The projection returns the feasible update closest to the task proposal and leaves a feasible proposal unchanged, but group feasibility is only a first-order guarantee.It does not guarantee the same bound for every individual contribution within a group or monotonic decrease of the ideal mutual-information defect.Source: §F.3.5 · p. 59

F.4 Finite-Horizon Control along Predictive Feasibility Optimization

  • Projected updates yield a finite-horizon drift bound for the pooled predictive statistic, not monotonicity; the bound accounts for margins, linearization remainder, and window-refresh change.The bridge to population root stability also requires controlled cumulative drift, local calibration residuals, and pooled-to-local discrepancy.Source: §F.4 · pp. 60–61

F.5 Proof of Root Predictive Stability

  • Conditioning on the full compressed-state tuple permits each host state to depend on incoming child states, without assuming host states are independent given the predictive tree and context.The proof uses the child-before-parent construction and its conditional kernels under assumptions (M1)–(M3).Source: §F.5 · pp. 61–62

2. Compare with a reference that retains the incoming child states. For fixed (z, c), define

  • Retaining incoming child states in the comparison yields a conditional-information term no larger than the local predictive deficiency; uncompressed nodes contribute zero.The bound follows from the chain rule and the fact that each compressed state is a function of its host state.Source: §2 · p. 62
  • For each fixed Γ under (M1)–(M3), conditional Pinsker and Jensen yield Proposition 3’s bound.The empirical calibration additionally requires test coverage, readout and covariance conditions, and a suitable independent evaluation sample; Proposition 2 alone does not bound the statistic at a later iterate.Source: §2 · p. 62

F.6 Root-Target Supervision Does Not Imply Local Predictive Sufficiency

  • Under Equation 55, Δ = 0 exactly when the local posterior R is measurable with respect to σ(Q, C).The root-target posterior Q is a function of the local posterior R and context C; when the downstream task posterior collapses distinct local posteriors, Δ can be positive.Source: §F.6 · pp. 63–64
  • Root-target sufficiency can fail to ensure local predictive sufficiency, and there is no distribution-free reverse control of local deficiency from root-target deficiency.With independent fair bits A and B, U = S = (A, B), T = A, and Z = A, the root-target deficiency is 0 while the local deficiency is log 2 nats.Source: §F.6 · pp. 63–64
  • If U → S → T conditional on C, then δS(Z) = δT(Z) + I(S; U | T, Z, C) ≥ δT(Z) for every admissible representation Z.Local predictive sufficiency therefore implies root-target sufficiency, but the converse need not hold.Source: §F.6 · p. 64
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