Low-Rank Dynamics-Effective Latent Carriers for Counterfactual Rollout in Learned World Models

Yang Liu; Yuming Chen

Source-linked AI summary

Low-Rank Dynamics-Effective Latent Carriers for Counterfactual Rollout in Learned World Models

Yang Liu, Yuming Chen

arXiv:2608.15156v1Aug 15, 2026cs.ROcs.AI

TL;DR

The paper asks whether a small, directly addressable hidden-state change can place a learned world model on a counterfactual trajectory and sustain it autonomously. Using low-rank hidden carriers in a controlled recurrent world model, it finds that a single rank-4 patch can launch a 12-step autonomous velocity-edit rollout, with replication and specificity controls supporting the effect.

Problem
The paper asks whether a small, directly addressable hidden-state change can implement a local physical counterfactual and guide the model’s future predictions.
Method
The study evaluates low-rank hidden-state interventions in a recurrent world model using a controlled two-object environment and one-shot autonomous rollout criteria.
Results
Rank 4 is the smallest tested rank passing the registered criteria, and one patch launches a 12-step autonomous counterfactual rollout with replication and specificity controls supporting the effect.
Takeaways & Limitations
The findings support a compact, addressable intervention interface for the tested local velocity-edit family, while showing that rollout improvement alone is insufficient evidence.
Takeaways & Limitations
The evidence is limited to one recurrent model family in a deterministic two-object environment with numeric observations and bounded local velocity edits.

Abstract

from arXiv · show

World models may predict the future without making clear which parts of their hidden state actually drive those predictions. We ask whether a small, directly addressable hidden-state change can place a learned world model on the intended counterfactual trajectory and then let the model continue that future on its own. We study a recurrent world model with a 192-dimensional hidden state in a controlled two-object, two-dimensional collision environment. For a bounded family of local velocity edits, we first verify that the model can natively represent and roll out the edited future. We then construct candidate low-rank carriers from training-only factual-to-counterfactual hidden differences and learn a map from the factual state and requested edit to carrier coefficients. On the registered rank grid, rank 4 is the smallest tested rank that satisfies the full development-panel criteria. A single rank-4 patch at the anchor is sufficient to redirect a 12-step autonomous rollout, with no future observations, teacher forcing, or repeated correction. The frozen procedure satisfies the preregistered replication rule across independently trained checkpoints and remains usable across nearby intervention times. Random equal-norm, wrong-object, and wrong-time controls do not explain the effect. A position-edit stress test provides a negative contrast: the intended position patch can pass the raw rollout criteria, but no-patch and random controls can pass the same criteria, and wrong-object specificity is not established. Thus, successful editing alone is not enough. We use dynamics-effective to describe an intervention that changes the model's future computation in a sustained and target-specific way under autonomous rollout. The rank-4 result identifies a compact intervention interface for the tested velocity-edit family, not a closed four-dimensional state or an intrinsic state dimension.

1 Introduction

This section asks whether hidden-state interventions can place a learned world model on a counterfactual trajectory that it can sustain autonomously, rather than merely changing its next output. In a controlled two-object collision world model, a rank-four carrier redirects a twelve-step counterfactual rollout from a single patch.

Motivation: The central question is whether changing a hidden state produces a self-sustaining counterfactual future, not merely a changed next output or predicted distribution.The intervention must place the model in a state that supports its later predictions.
Experimental setting: The study uses a 192-dimensional recurrent hidden state in a deterministic two-dimensional collision environment with two moving objects.The model is trained on factual trajectories and locally edited counterfactual trajectories, with native counterfactual rollout checked first.
Main result: Rank four is the smallest tested rank that passes the full preregistered test set for the velocity-edit carrier.The carrier reliably redirects the model’s future dynamics on the preregistered rank grid.
Main result: A single hidden-state patch launches a twelve-step autonomous counterfactual rollout without future observations or repeated correction.The effect replicates across independently trained checkpoints and remains effective across several intervention times.

2 Related Work

The related work spans latent-state sufficiency, causal representation intervention, and physical interpretation in predictive world models. This paper complements these directions by testing whether an existing recurrent state supports a compact, addressable edit for autonomous counterfactual rollout.

Latent-state sufficiency: Latent-state research seeks compact representations that preserve information needed for prediction, control, planning, memory, grounding, or counterfactual reasoning.DeepMDP preserves reward and transition structure relevant to control, while later work emphasizes judging latent states by the functions they support.
Latent-state sufficiency: Action-relevant structure may depend more on temporal prediction than reconstruction quality, while forward prediction does not automatically make physical state identifiable or usable.These findings motivate evaluating what latent states enable rather than assuming that predictive training alone yields accessible physical variables.
Representation intervention: Probing and editing methods range from linear concept readouts and erasure to constructive activation steering and representation swapping, but probe accuracy alone can mislead.Control tasks and intervention-based evaluations distinguish recoverable information from information that meaningfully affects model behavior.
Representation intervention: Causal editing requires completeness and selectivity, motivating downstream tests that assess whether interventions change targeted behavior without broad, unrelated damage.The paper applies this principle more strictly to world models by requiring effects beyond an immediate readout or one-step output.
Physical world-model interpretation: Physical-prediction benchmarks and structured world models assess behavioral competence, but generally do not identify the internal state variables used by the model.Recent work finds both high-dimensional physical codes and functionally used probe-derived directions, while this paper tests a single compact edit followed by autonomous rollout with specificity controls.

3 Problem Formulation

This section formulates whether a small, directly addressable hidden-state change can implement a local physical counterfactual and autonomously sustain its future. It defines dynamics-effective low-rank carriers within a bounded state, intervention, and time domain.

3 Problem Formulation: The goal is not to recover a physics state, but to determine whether a small hidden-state change can implement and continue a local counterfactual.The intervention is evaluated by whether the model continues the new future on its own.
3.1 World-model setting: The controlled environment contains two circular objects moving in a bounded two-dimensional arena with deterministic dynamics and noisy numeric observations.The restricted setting permits exact definition of physical interventions and counterfactual futures.
3.1 World-model setting: The learned world model maintains a 192-dimensional recurrent hidden state, and factual history through the anchor produces the state from which autonomous prediction begins.Derived physical quantities are used for evaluation rather than treated as additional independent state variables.
3.2 Local physical counterfactual: A counterfactual edits exactly one velocity component of exactly one object at the anchor while leaving positions and other primitive velocity components unchanged then.The native counterfactual state is obtained by passing the edited anchor observation through the model’s normal observation/update path.
3.2 Local physical counterfactual: The edit is local only at the intervention instant; later positions, contact time, and collision-coupled variables may change through the resulting dynamics.Only variables truly unaffected by the intervention are required to remain equal to factual values.
3.3 Dynamics-effective latent intervention: A hidden-state intervention is dynamics-effective only when one patched anchor state autonomously produces a simulator-grounded counterfactual trajectory that passes registered agreement and validity checks.The checks include target, derived-state, downstream, preservation, and physical-validity criteria.
3.4 Low-rank and addressable latent carrier: A rank-r carrier represents the hidden perturbation as U_r c, where U_r spans an r-dimensional subspace of the 192-dimensional hidden state.The basis is constructed using SVD, and its columns are not assigned fixed physical meanings.
3.4 Low-rank and addressable latent carrier: The addressable operator may use native counterfactual hidden states during training, but those states are unavailable at evaluation time.The operational basis and coefficient map are learned from training data and applied at evaluation without native counterfactual access.

4 Method

The method verifies native counterfactual capability, learns checkpoint-specific low-rank carriers and an addressable edit-to-coefficient map, then applies one hidden patch for autonomous evaluation. Rank selection and replication use preregistered criteria, reference routes, and specificity controls.

Procedure: The six-stage procedure verifies counterfactual representability, learns training-only carriers and an addressable map, applies one patch, evaluates controls, and tests persistence.The stages proceed from native capability checks through carrier construction, mapping, autonomous rollout evaluation, and cross-condition persistence testing.
Carrier construction: At t = 7, the carrier-fit matrix contains 1024 192-dimensional factual-to-counterfactual hidden differences spanning the registered local velocity-edit family.The matrix samples different factual states, edited objects, velocity axes, signs, and amplitudes rather than alternative solutions to one fixed edit.
Carrier construction: The candidate basis uses the first r right-singular-vector columns from an uncentered SVD of the raw difference matrix, with checkpoint-specific latent directions.No mean hidden difference is subtracted, and the basis columns are not named physical coordinates.
Addressable mapping: The addressable intervention predicts carrier coefficients from four sparse velocity-edit slots and eight factual-state features using standardized inputs and ridge regression with λ = 10^-4.Exactly one edit slot is nonzero; standardization uses carrier-fit statistics, and the affine design has 13 columns including the intercept.
Autonomous rollout: At β = 1, the predicted 192-dimensional perturbation is added once at the factual anchor, followed by 12 future transitions without future observations, teacher forcing, or per-step clamps.The rollout contains 13 decoded states including the anchor, and acceptance requires registered scientific, engineering, and autonomy criteria.

5 Results

The model natively represented and autonomously propagated the tested counterfactuals, enabling a low-rank intervention study. Rank 4 was the smallest tested dynamics-effective carrier on development data, replicated across fresh checkpoints and nearby intervention times, while specificity controls constrained alternative explanations.

Model adequacy: All three development checkpoints passed all six required G0, G1, and G2 route-by-stratum cells, establishing model adequacy and hidden-state patching capability.The model could continue factual and edited counterfactual futures in both non-contact and future-contact strata, while complete hidden differences reproduced the native counterfactual state and rollout.
Rank sweep: Rank 1 failed both strata for all three development checkpoints, rank 2 passed both for one of three, and rank 4 passed both with mandatory controls for all three.The preregistered rule therefore selected rank 4, which was also selected by both oracle-capacity and addressable-operator analyses.
Rank sweep: Rank 4 supported a single anchor patch for the full 12-step autonomous future and registered controls, making it the smallest tested addressable, dynamics-effective scale.This result concerns the tested velocity-edit family and is not an estimate of intrinsic state dimension.
Fresh-checkpoint replication: Fresh models 1, 2, and 3 achieved joint coverage of 0.867/0.914, 1.000/0.922, and 0.711/0.973 in S1/S2, respectively.The confirmatory rule was satisfied because at least two of three fresh models passed both strata; fresh model 3 missed only S1.
Temporal generalization: Every tested anchor from t = 5 through 9 met the 2-of-3 fresh-model criterion, and unchanged t = 7 procedures met it when transported to t = 5, 6, 8, and 9.B1 allowed anchor-specific refits, whereas B2 forbade carrier, mapper, standardization, rank, and intervention-strength changes, supporting temporal stability within the bounded window.

6 Discussion

The discussion supports a compact, dynamics-effective intervention interface for bounded velocity edits, while emphasizing that it is neither a reduced four-dimensional state nor a general result. Its evidence depends on sustained autonomous rollout and target-specific controls, with important limits from the controlled setting, joint edits, and untested dynamical closure.

Interpretation: The velocity intervention is addressable, remains effective for twelve autonomous transitions, and passes registered specificity and integrity controls.The authors use “dynamics-effective” operationally for a compact, addressable edit that changes the intended future in a sustained, target-specific way.
Limits of the rank-4 result: Rank 4 is a tested intervention-interface rank, not a physical or intrinsic state dimension; the model still evolves its full 192-dimensional recurrent state.Rank four is the smallest passing value on the tested grid, while rank three was not tested and the carrier basis is not unique.
Temporal and checkpoint replication: The t = 7 rank-4 carrier and addressable operator remain useful at nearby anchors, and independently trained checkpoints recover the structure functionally but with variable strength.This reduces concern that the result depends on one accidental time point or exact internal coordinates shared across models.
Specificity and controls: The position stress test shows that raw improvement is insufficient because intended, no-patch, random, and wrong-object alternatives are not separated under registered controls.Controls are treated as part of the evidence standard for target, timing, autonomous continuation, and resistance to unrelated perturbations.
Scope limitations: The confirmatory claim is limited to bounded local velocity edits, a fixed rank grid, and a short-tested window in one deterministic two-object recurrent environment with zero evaluated actions.It does not establish the same structure for raw-video models, action-conditioned agents, larger scenes, real-world systems, or other architectures and widths.
Open boundaries: Simultaneous velocity components remained representable, but tested maps lacked full specificity, so the study does not claim general compositional addressability or full dynamical closure.A closure test would require the compact representation to update itself and reproduce the relevant future without returning to the full recurrent state.

7 Conclusion

For bounded local velocity edits, a 192-dimensional recurrent world model exposes a compact rank-4 intervention interface that can launch a 12-step autonomous counterfactual rollout. Replication, temporal reuse, and specificity controls support the result, while the position-edit stress test shows that rollout improvement alone is insufficient and the finding does not establish a closed four-dimensional state.

Core result: Rank 4 is the smallest tested rank passing the registered development criteria, and one patch launches a 12-step autonomous counterfactual rollout.The interface is demonstrated for a bounded family of local velocity edits in a 192-dimensional recurrent world model.
Validation: Fresh-checkpoint replication, temporal reuse, and specificity controls support the intervention result.These checks support the result across independently trained checkpoints, nearby intervention times, and control interventions.
Stress test: The position-edit stress test shows that apparent rollout improvement is insufficient when controls can reproduce it.This stress test does not establish wrong-object specificity for the position edit.
Interpretation and next steps: The result supports a low-dimensional dynamics-effective intervention interface, not a closed four-dimensional state.Testing closure, history/belief-state compression, and transfer across architectures and hidden widths are identified as next steps.

AI Assistance Disclosure · Appendix A. Formal Definitions and Notation

The appendix discloses AI-assisted research workflows while defining the notation, intervention construction, evaluation roles, and bounded scope used throughout the paper. Its formalism emphasizes low-rank perturbation subspaces, autonomous rollout, and claims limited to the tested operational domain.

AI Assistance Disclosure: Generative AI tools supported literature synthesis, protocol refinement, coding, analysis organization, and manuscript preparation, with authors reviewing scientific decisions, formal runs, interpretations, and final claims.The disclosure states that AI-generated code, references, analyses, and text were reviewed by the authors.
Appendix A. Formal Definitions and Notation: Appendix A defines the factual state, local counterfactual, native counterfactual hidden state, one-shot intervention, low-rank carrier, and bounded operational domain.Environment and model details, counterfactual construction, carrier derivation, addressable ridge operator, and rollout evaluation are deferred to Appendices B–F.
A.1 Core notation: Table A1 summarizes the core notation and dimensions used by the formal definitions.It serves as the appendix’s reference table for symbols and dimensionalities.
A.2 Factual state and autonomous rollout: The recurrent hidden state is obtained from observation history, and the rollout operator generates the autonomous future from the factual anchor state.Derived quantities are deterministic functions of the joint two-object state rather than additional independent coordinates.
A.3 Same-history local counterfactual: The factual and counterfactual branches share pre-anchor history, while the anchor differs only in one registered velocity component before the normal model update produces the native counterfactual hidden reference.This defines the same-history local counterfactual construction.
A.4 Low-rank and addressable intervention: A rank-r intervention restricts the hidden perturbation to a carrier subspace, with coefficients predicted from the factual physical state and edit specification.The relevant object is the supported subspace and intervention, not a uniquely oriented set of carrier axes; rotations preserve the same 192-dimensional perturbation.
A.5 Dynamics-effective criterion and variable roles: Evaluation distinguishes direct targets, contemporaneously derived variables, dynamically downstream variables, and unaffected variables, using a multi-part rule rather than a single Euclidean-distance test.Exact metrics, physical-validity checks, control comparisons, and pass thresholds are defined in Appendix F.
A.6 Bounded operational domain: Intervention claims apply only within the evaluated bounded domain and do not automatically extend to larger edits, unseen regimes, longer time scales, architectures, or environments.Those settings remain extrapolations until separately tested, even if the fitted operator can computationally reach them.

Appendix B. Environment, Observation Interface, Model Architecture, and Training … B.3 Recurrent world model

The appendix specifies the implemented simulator, observation interface, and recurrent world-model intervention point. The model exposes a 192-dimensional carrier patched once after anchor assimilation and before autonomous rollout.

Appendix B. Environment, Observation Interface, Model Architecture, and Training: The simulator description reports only implemented behavior and assigns no explicit mass or restitution parameters absent from the code.This appendix avoids adding unrecorded physical assumptions.
B.1 Simulator and physical state: Collision detection finds the analytic earliest circle impact, applies the implemented normal impulse update, and advances through the remaining step.Arena violations reflect the out-of-range coordinate and matching velocity component until the center returns inside the allowed range.
B.2 Observation interface: Factual and counterfactual paired interventions share the same realized observation-noise array, preventing independent noise redraws from driving hidden differences.Velocity and position are directly available whenever their corresponding frames are visible.
B.2 Observation interface: The observation interface exposes position and velocity directly when the corresponding frame is visible, so the study does not require velocity inference from images or finite differences.This clarifies the information available to the recurrent model during observation assimilation.
B.3 Recurrent world model: The Deterministic Belief-like Model has 496200 trainable parameters and exposes a 192-dimensional carrier composed of object-specific and global recurrent components.The carrier concatenates these recurrent components at the intervention interface.
B.3 Recurrent world model: At time index t, the model performs recurrent updating, computes prior or posterior quantities, selects visible/update information, packs the full carrier, and stores it.This sequence defines how the intervention carrier is formed at each step.
B.3 Recurrent world model: The one-shot patch is added to the complete 192-dimensional carrier after recurrent updating and observation assimilation, before the autonomous future rollout.Thus the interface is post-update and post-assimilation through t, but pre-transition for the future rollout.

B.4 Training data and optimization … C.3 Simulator truth and model-native counterfactual

The study trains a 192-dimensional recurrent world model with specified auxiliary losses, then evaluates single-anchor velocity counterfactuals using matched histories and model-native reference routes. The construction distinguishes simulator truth, native counterfactual adequacy, and hidden-state interface control.

B.4 Training data and optimization: Training combines rollout mean-squared error with auxiliary anchor-state, filtering-state, relative-state, and delta-velocity losses using frozen weights.The weights are 1.0, 0.2, 0.2, 0.25, and 0.25; the delta-velocity loss uses q90 scale 1.2762992187.
B.4 Training data and optimization: Training excludes contact/event labels, later low-rank carrier supervision, rank supervision, and hyperparameter search over the intervention method.
B.5 Checkpoint roles: All six checkpoints use the same architecture and 192-dimensional carrier interface, while carrier bases and addressable operators remain checkpoint-specific.Numerical basis coordinates are not assumed to align across independently trained checkpoints.
Appendix C. Counterfactual Pair Construction and Intervention Semantics: The primary counterfactual intervention changes one velocity component of one object at a single anchor time, while specifying what remains fixed and how reference routes are constructed.
C.1 Same-history pairing: Each factual/counterfactual pair shares the same episode history, identical pre-anchor physical trajectory, and the same realized observation-noise sequence.The counterfactual branch differs only because the registered edit is applied at the anchor.
C.2 Velocity-edit family: The edit uses four velocity slots with exactly one nonzero entry, identifying the edited object and axis.The slots represent horizontal and vertical velocity edits for objects A and B, using three absolute increments with positive and negative signs.
C.2 Velocity-edit family: At the anchor, positions and the other three primitive velocity components stay fixed, while derived quantities are recomputed and later outcomes follow simulator evolution.The construction also enforces an edited-anchor speed suppo…
C.3 Simulator truth and model-native counterfactual: The G1 model-native route supplies the edited anchor observation through the ordinary observation/update path and then runs without future counterfactual observations.G2 injects the complete native hidden difference into the factual hidden state; Eq. (C4) verifies exact reproduction at the post-observation/pre-transition interface, making G2 an interface control and G1 against simulator truth the model-adequacy test.

C.4 Evaluation strata … E.6 Oracle capacity versus addressability

The evaluation separates non-contact and contact-sensitive regimes, while the low-rank carrier is constructed from training-only hidden differences and addressed from factual physical states plus requested edits. Oracle projection measures carrier capacity, whereas the frozen affine operator performs a single autonomous test-time patch without counterfactual hidden-state access.

C.4 Evaluation strata; C.5 Anchor and rollout support: S1 and S2 are evaluated separately, so success in the easier non-contact regime cannot compensate for failure in the contact-sensitive regime.The primary experiment uses anchor t = 7 and 12 autonomous future transitions; temporal extension tests anchors t = 5, 6, 7, 8, and 9.
C.6 Position-edit diagnostic boundary: The position intervention is development-only, changing one coordinate by 0.05, 0.10, or 0.20 while holding velocities and other primitive anchor fields fixed.Simulator truth is regenerated from the edited state to test whether evaluation distinguishes a physically meaningful position intervention.
C.7 Development-only simultaneous velocity-vector extension: The simultaneous velocity-vector extension changes both components of one object’s velocity within the primary family’s largest supported Euclidean magnitude and remains separate from confirmatory rank selection.It uses ordinary S1 and S2 semantics and cannot redefine the selected rank or primary claim.
Appendix D. Low-Rank Carrier Construction; D.1 Hidden-difference matrix: The candidate carrier uses raw hidden differences that the trained model produces for registered local counterfactuals, with 1,024 fit-split vectors per checkpoint and no test units.The construction asks whether these differences share low-dimensional linear geometry without using development or confirmatory results to choose the basis.
D.2 Uncentered singular-value decomposition: An uncentered SVD places the candidate rank-r basis in the original 192-dimensional hidden coordinate system with orthonormal columns and deterministic sign conventions.The sign convention has no scientific meaning because reversing a basis vector and its coefficient leaves the perturbation unchanged.
D.3 Projection and oracle capacity; D.4 Descriptive second-moment capture; E.6 Oracle capacity versus addressability: Oracle reconstruction is the closest rank-r Euclidean projection of each native hidden difference, making it a capacity diagnostic rather than an addressable test-time operator.Second-moment capture is descriptive only and is not interpreted as a physical dimension or used for rank selection.
D.5 Registered rank sets: The primary rank grid is fixed before development outcomes: rank 0 is the no-patch reference, rank 64 is a geometry diagnostic, and rank 192 is the full-hidden reference.The method reports the smallest passing tested rank and does not exclude solutions from untested ranks, nonlinear coordinates, or other carrier families.

E.7 Joint-edit composition and operator extensions · Appendix F. Rollout Metrics, Eligibility Gates, and Registered Thresholds

The joint-edit extension verifies algebraic composition under the frozen affine operator, while the rollout appendix defines shared scientific metrics and separates eligibility engineering from performance. Operator-only development tests retain the fixed world model and rank-4 carrier while comparing additive and minimal non-additive coefficient models.

E.7 Joint-edit composition and operator extensions: Joint requests can contain two nonzero velocity entries belonging to the same object under the original four-slot edit vector.
E.7 Joint-edit composition and operator extensions: Direct coefficients matched component-wise composed coefficients to numerical precision under the original frozen affine operator.
E.7 Joint-edit composition and operator extensions: Direct and composed rollouts were numerically identical, verifying affine algebraic composition but not target-specific dynamics-effective composition.
E.7 Joint-edit composition and operator extensions: The operator-only development test held the world model and checkpoint-specific rank-4 carrier fixed.
E.7 Joint-edit composition and operator extensions: One candidate refit the same affine feature family using equal numbers of original single-component and new joint-edit carrier-fit units.
E.7 Joint-edit composition and operator extensions: A second candidate added two same-object joint-product features to permit a minimal non-additive correction.
Appendix F. Rollout Metrics, Eligibility Gates, and Registered Thresholds: Five scientific metrics jointly score the decoded anchor, released twelve-step future, and post-intervention physical consistency.
Appendix F. Rollout Metrics, Eligibility Gates, and Registered Thresholds: Numerical eligibility, autonomy metadata, stratum definitions, and aggregation rules are reported separately from scientific performance.

F.1 Evaluation notation and frozen normalization scales

Evaluation uses a 12-transition autonomous horizon with decoded 2 × 4 object states indexed from the intervention anchor. Frozen primitive and derived-representation scales are applied without additional normalization or metric-specific weighting.

Evaluation indexing: H = 12 autonomous future transitions are evaluated from anchor t_i using decoded 2 × 4 states for objects A and B.Primitive columns are ordered x, y, v_x, v_y.
Evaluation indexing: Relative index j = 0 denotes the decoded anchor, while j = 1, … ,12 denote the twelve released future states.The metric kernel uses the corresponding reference state for each route.
Primitive edits: The edited object is o_i ∈ {0,1}, and velocity-axis index a_i ∈ {0,1} selects primitive column c_i = 2 + a_i.The frozen primitive scale matrix s_P is applied coordinatewise.
Derived representation and normalization: M2 evaluates a 13-dimensional derived vector containing relative, center, summed-velocity, energy-like, distance, and contact features under frozen scales.Continuous scales use max(IQR/1.349, 1.4826 MAD, 0.01), the contact-indicator scale is fixed to 1, and no evaluator adds epsilon, clipping, amplitude normalization, or metric-specific weighting.

F.2 Anchor fidelity and future rollout metrics (M1-M4) … Appendix G. Control Routes and Temporal Generalization Protocols

The evaluation combines anchor fidelity, autonomous-rollout, law-consistency, numerical-eligibility, and structural checks under registered strata and panel-level decision rules. Appendix G adds controls and temporal protocols to distinguish targeted dynamics-facing edits from generic perturbations, interface errors, timing errors, or leakage.

F.2 Anchor fidelity and future rollout metrics (M1-M4): M1 measures edited-velocity fidelity at the decoded anchor, while M2 measures fidelity of the complete 13-dimensional derived representation.M1 is a single normalized absolute error; M2 is an unweighted RMS across 13 normalized derived-feature errors.
F.2 Anchor fidelity and future rollout metrics (M1-M4): M3 evaluates autonomous future fidelity across 12 future states, and M4 takes the maximum normalized anchor error over unaffected primitive coordinates.M3 uses 96 normalized scalar errors per unit; M4 has no future-preservation term.
F.3 Post-edit law and geometry consistency (M5): M5 is the maximum of one-step replay, overlap, arena-boundary, per-step displacement, and intervention-declaration consistency terms.The terms are neither summed nor weighted; replay uses the exact simulator, including collision handling and reflecting walls.
F.4 M6E numerical eligibility; F.5 M7 autonomous-rollout structure: M6E is an engineering eligibility gate requiring finite final hidden state and decoded trajectory, while M7 records autonomous-rollout structure including zero future-oracle reads.M6E failures remain in the denominator and block all-eligible status; M7 includes one anchor decode plus one decode after each of 12 transitions.
F.6 Full-hidden structural equivalence (G2): G2 copies the native counterfactual hidden state, making hidden discrepancy exactly zero by construction and gating only rollout discrepancy at tolerance 10^-6.The rollout discrepancy is the maximum absolute difference across all decoded states, objects, primitive fields, and units in the stratum.
F.7 Programmatic S1/S2 strata: S1 requires no detected contact in either branch, whereas S2 requires factual contact and a sufficiently large post-anchor factual-counterfactual trajectory difference.The detector flags center distance ≤0.500001 or velocity-array jumps >10^-5, and S2 does not require counterfactual contact or changed contact timing.
F.8 Registered thresholds and unit/cell decision rule: Registered thresholds use inclusive comparisons and separate S1/S2 decisions, with M6E and M7 included in joint unit passes and failed units retained in fixed denominators.Cell passing additionally requires expected counts, eligibility, Coverage ≥0.8, finite threshold-compliant M1-M5 medians, and mandatory controls.
F.9 Checkpoint and panel aggregation; F.10 Joint-edit scoring and paired-specificity calibration; Appendix G. Control Routes and Temporal Generalization Protocols: Checkpoint panels require two passing checkpoints out of three, while temporal aggregation requires every registered anchor or transport-target panel to pass; Appendix G controls test specificity against generic, interface, timing, and leakage explanations.Joint-edit controls compare intended and matched-control median direct-target and future errors, requiring at least 0.80 paired units worse for controls on both summaries.

G.1 Positive and structural reference routes … G.6 Joint-edit controls and development boundary

The appendix defines reference, control, and temporal assays for evaluating low-rank interventions, then bounds interpretation of transported carriers and joint-edit results. Across these analyses, successful rollout is distinguished from target-specific control and from claims of globally invariant latent coordinates.

G.1 Positive and structural reference routes: The assay combines factual, external-physical, native-counterfactual, hidden-injection, zero-edit, no-patch, random, wrong-object, and wrong-time routes to separate valid rollout from control effects.It also verifies that intervention construction requires no counterfactual future payload and checks object-slot identity and decoder ordering.
G.2 B1: per-anchor carrier existence: B1 tests whether an addressable rank-4 carrier can be established independently at anchors t = 5, 6, 7, 8, and 9 using anchor-specific fitting and frozen confirmatory evaluation.The assay asks an existence question at each time and does not require one anchor’s numerical basis to be reused elsewhere.
G.3 B2: exact transport of the t = 7 coordinate system: B2 transports the exact t = 7 rank-four artifact to anchors t = 5, 6, 8, and 9 without refitting, restandardization, rank changes, or strength changes.B1 at each target anchor is the refitted reference, while shared M1–M5, M6E, M7, control, coverage, and 2-of-3 checkpoint rules govern both assays.
G.4 B3: event-relative phase hypothesis: B3 deconfounds event-relative phase from absolute time by balancing reference contact times 8 and 9 and applying registered velocity edits across pre-contact and post-contact phases.This phase assay is developmental and does not alter the frozen B1/B2 temporal result or primary t = 7 carrier definition.
G.5 Temporal interpretation boundary: Repeated anchor success and exact t = 7 transport provide bounded temporal evidence, limited to the registered model family, physical support, edit family, and tested time range.Neither B1 nor B2 establishes globally time-invariant latent coordinates, a unique physical gauge, a closed low-dimensional Markov state, belief-state sufficiency, or transfer across architectures and environments.
G.6 Joint-edit controls and development boundary: The joint velocity analysis uses matched no-patch, random equal-norm, wrong-object, and matched-wrong-vector controls to test whether joint target-specific effects exceed generic perturbation or baseline behavior.These controls respectively probe factual proximity, unrelated perturbations, object specificity, and edit-vector direction.
G.6 Joint-edit controls and development boundary: 3/3 development checkpoints met the intended joint-rollout criterion for the original frozen affine operator, joint-aware affine refit, and minimal joint-correction ridge map, but none passed the complete paired-specificity panel.Consequently, joint edits are not treated as a second positive intervention family or nominated for confirmatory or fresh-checkpoint testing.