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DODR: Deterministic Operator-Driven Reasoning in Latent Space

Weicai Huang

arXiv:2609.04782v1cs.AI

TL;DR

Autoregressive reasoning relies on probabilistic token sampling, motivating concerns about hallucination, error accumulation, and linear-chain constraints. DODR instead computes over latent-space snapshots with deterministic matrix operators and graph structure. Across dedicated and end-to-end experiments, it reports strong deduction and induction results, abduction above a random baseline, and limitations concentrated in encoder resolution and manual graph scheduling.

  • Problem

    Autoregressive models use probabilistic token generation, while the paper identifies hallucination, error accumulation, and linear-chain topology as limitations for complex reasoning.

  • Method

    DODR represents reasoning as graph computation over latent-space snapshots using deterministic matrix operators for deduction, induction, and pseudo-inverse abduction.

  • Results

    Across 218 end-to-end samples, DODR reports deduction at 100% (60/60), induction hard vetoes at 12/12, and abduction judgment accuracy at 81.7% (49/60).

  • Takeaways & Limitations

    The results support the feasibility of combining the three inference modes, while frozen operators also attain 100% (60/60) on brand-new cross-domain deduction problems.

  • Takeaways & Limitations

    Abduction failures are attributed to an encoder bottleneck, and graph topologies are manually specified because the scheduling controller has not yet been implemented.

Abstract

from arXiv · show

Autoregressive (AR) large language models formulate reasoning as token-level probabilistic sampling, which induces three fundamental defects in complex logical reasoning: error accumulation, probability substituting necessity, and the linear-chain information bottleneck. This paper proposes the Deterministic Operator-Driven Reasoning in Latent Space architecture (DODR), which reconstructs reasoning as reasoning-graph computation in a high-dimensional linear-algebraic space. Reasoning states are represented as snapshot vectors whose primitives are semantic units (phrases or sentences) rather than tokens, and each inference step is a deterministic matrix operation with no token sampling. Peirce's three inference types are formalized as three trainable matrix operators: a rank-deficient deduction operator (information collapse), a full-rank induction operator (information expansion), and an abduction operator defined as the Moore-Penrose pseudo-inverse of deduction (information hypothesizing). We prove that the operator set is minimal and complete given Peirce's trichotomy, that no single "super-operator" can realize all three types (a rank obstruction), and that reasoning graphs are Turing-complete with contractive backflow converging by Banach's fixed-point theorem. Experiments on 503 sample records (420 deduplicated samples) across dedicated and end-to-end settings show: deduction loss converges to 1.40e-05; induction achieves 0.9996 generalization coverage with 20/20 hard vetoes on counterexamples; abduction solutions exceed the random baseline by 28x with judgment accuracies of 72.5% (58/80, Wilson 95% CI [61.9%, 81.1%]) and 81.7% (49/60, CI [70.1%, 89.4%]); frozen operators attain 100% (60/60) on unseen cross-domain deduction. The architecture provides a structural zero-hallucination guarantee and a three-layer continual-learning mechanism. All data and code are released.

1 Introduction

DODR replaces probabilistic token-generation reasoning with deterministic latent-space operators and graph-structured computation. It formalizes deduction, induction, and abduction as distinct trainable matrix operations, including pseudo-inverse abduction and hard-veto induction.

  • Proposed architecture: DODR represents reasoning states as deterministic latent-space snapshots and performs inference with trainable matrix operators rather than token sampling.The three operators correspond to deduction, induction, and abduction, with switching implemented by changing matrices.
  • Deduction: Rank-deficient deduction formalizes irreversible information loss through null-space components that no inverse process can recover.The construction links information-flow direction, matrix rank, and non-invertibility.
  • Induction: A hard-veto induction mechanism uses counterexamples and Gram–Schmidt projection to overturn conclusions and reconstruct finer ones.A counterexample below the cosine-similarity threshold can veto the original conclusion by a single vote.
  • Abduction: Abduction is defined as the Moore–Penrose pseudo-inverse of deduction, producing a minimum-norm best hypothesis rather than the unrecoverable true explanation.The residual quantifies hypotheticality because rank deficiency hides explanation components.
  • Operator theory: The three operators are presented as an irreducible, complete set, while rank incompatibility rules out a single super-operator.Deduction requires rank reduction, whereas induction requires full rank, so one matrix cannot satisfy both requirements.
  • Reliability: DODR claims a structural 0.0% hallucination rate because its reasoning process uses frozen encoders, deterministic matrix operations, and no token sampling.The architecture removes the sampling mechanism rather than merely reducing hallucination probability.

4 Core Theoretical Foundations: The Five Postulates

DODR’s five postulates recast reasoning as deterministic geometric and graph computation: high-dimensional snapshots support linear operators, while graph structure supports branching, composition, and convergent backflow. Experiments report that the postulates were not falsified, alongside explicit resource, rank, granularity, and scheduling trade-offs.

  • 4.1 Postulate One: Dimensionality Expansion in Place of Multi-Layer Stacking (Cover’s Theorem): A single ultra-wide linear transformation can replace a multi-layer nonlinear network after dimensionality expansion makes low-dimensional data nearly linearly separable.The design trades predictable memory, bandwidth, and parameter overhead for analyzable algebraic structure.
  • 4.1 Postulate One: Dimensionality Expansion in Place of Multi-Layer Stacking (Cover’s Theorem): 1.40e-05 final Wstep loss, down from 2.09e−01, supports linear expression of logical state transitions in the 1536-dimensional snapshot space.The reported decrease is 14932-fold and approached the numerical convergence floor under the training configuration.
  • 4.2 Postulate Two: Snapshot State Representation: DODR concatenates layer activations into static snapshot vectors, replacing temporal unfolding with geometric projections and representing transitions at semantic-primitive granularity.Snapshots correspond to words, phrases, or entire sentences rather than tokens; end-to-end deductive accuracy was 100%.
  • 4.3 Postulate Three: Direction of Information Flow (Replacing the Irreversibility Assumption): The three reasoning types are characterized by information-flow direction: deduction decreases information, induction increases it, and abduction hypothesizes it.Reversibility follows from matrix rank rather than serving as an independent core assumption.
  • 4.3 Postulate Three: Direction of Information Flow (Replacing the Irreversibility Assumption): The deduction operator was rank-deficient at rank 384 < 3072, while the induction operator was full-rank at 1536/1536, matching the design intent.These algebraic results were reported as not falsifying Postulate Three.
  • 4.4 Postulate Four: Local Linearization: Multi-step reasoning approximates nonlinear computation through small local linear steps, with bounded error required to avoid divergence.Two-step transitive reasoning produced cosine similarities of 0.9898 and 0.9861, and the error did not diverge.
  • 4.5 Postulate Five: Graph-Structured Reasoning: Reasoning graphs support branching, merging, and cross-referencing, with path-independent computation and successful compositional execution.The graph experiment completed an induction → deduction → abduction → re-induction cycle, while scheduling complexity and storage remain costs.

5 Formal Definitions of the Three Basic Operators

DODR assigns deduction, induction, and abduction distinct algebraic roles: rank-deficient collapse, full-rank expansion, and pseudo-inverse hypothesization. The deduction operator deliberately discards information, while its low-rank structure makes that collapse and irreversibility explicit.

  • 5 Formal Definitions of the Three Basic Operators: DODR formalizes deduction, induction, and abduction as matrix operators whose algebraic properties encode collapse, expansion, and hypothesization.The correspondence is deduction = rank-deficient collapse, induction = full-rank expansion, and abduction = pseudo-inverse hypothesization.
  • 5.1 The Deduction Operator Wstep (Information Collapse): Deduction maps the concatenated state and input from R2N to RN, compressing joint information into a conclusion state.In abduction experiments, the input is instead a single state, so Wstep is RN×N while retaining rank deficiency.
  • 5.1 The Deduction Operator Wstep (Information Collapse): Low-rank parameterization guarantees rank(Wstep) ≤ r < N regardless of optimization outcomes.Writing Wstep = A·B with r < N structurally prevents accidental full rank and reduces parameter count relative to a full-rank parameterization.
  • 5.1 The Deduction Operator Wstep (Information Collapse): 87.5% of the 3072-dimensional input information is discarded in the reported deduction design with N = 1536 and r = 384.The operator is characterized as a lossy compressor whose collapse rate is controlled by the design rank.
  • 5.1 The Deduction Operator Wstep (Information Collapse): Rank deficiency makes deduction irreversible: no left inverse can uniquely recover the premises and new evidence from the conclusion.This follows from the rank inequality and is treated as a consequence of defining deduction as information collapse.
  • 5.1 The Deduction Operator Wstep (Information Collapse): 83.3% is the experimentally measured mean pseudo-inverse reconstruction error, compared with a theoretical isotropic expectation of approximately 0.9354.The expectation is not a pointwise lower bound; individual errors can be lower or higher depending on row-space energy.
  • 5.3 The Abduction Operator Wabduce (Information Hypothesization): Pseudo-inverse reconstruction projects inputs onto Wstep’s row space, leaving null-space components as irrecoverable error.For abduction, the minimum-norm solution provides the shortest explanation consistent with the evidence, while the residual represents information that cannot be recovered.
  • 5.3 The Abduction Operator Wabduce (Information Hypothesization): Abduction therefore produces a best hypothesis rather than the true explanation when deduction has discarded null-space information.The paper reports that the irrecoverable residual accounts for 85.9% in the abduction setting.

6 Completeness Argument for the Minimal Set of Basic Operators

The paper argues that DODR’s three operators are sufficient and necessary under Peirce’s tripartition, while no single linear super-operator can replace them. Generalization is instead assigned to composition of fixed operators with task-specific snapshots, subject to encoder and training-data limits.

  • 6 Completeness Argument: Under Peirce’s accepted tripartition, the three operators form a complete set covering deduction, induction, and abduction.The paper distinguishes this philosophical completeness premise from the section’s mathematical proofs of operator sufficiency and necessity.
  • 6.2 Reducibility of Candidate Operators: Wmerge and Wcross are reducible to compositions of Wstep because they introduce no new rank structure, inverse problem, or information-flow direction.Reducibility prevents variants with identical algebraic signatures from being counted as independent basic operators.
  • 6.3 Irreducibility of the Three Operators: Each operator is mutually non-expressible because its rank and information-flow signature cannot be replicated by compositions of the other two.The rank-product inequality preserves the relevant distinction between rank-deficient and full-rank maps.
  • 6.4 Minimal Complete Set: Removing Wstep, Winduce, or Wabduce respectively eliminates deduction, induction, or inverse-problem solving.Thus the set is minimal as well as complete: three operators, no more and no fewer.
  • 6.5 The Impossibility of a Super Operator: No single linear Wsuper can realize information collapse, preservation, and inverse-problem solving because rank and domain-codomain requirements conflict.Deduction, induction, and abduction require incompatible rank conditions and dimension signatures.
  • 6.5 The Impossibility of a Super Operator: The minimal implementation therefore uses two trained matrices plus one SVD when abduction is derived from deduction via the pseudo-inverse.Plurality is presented as an algebraic constraint rather than an engineering choice.
  • 6.5 The Impossibility of a Super Operator: DODR’s system-level unity resides in a reasoning graph and scheduling controller that compose fixed operators rather than merge them into one matrix.The operators function as an alphabet, the graph as a language, and the controller as grammar.
  • 6.6 Universality of Operator Instances: Frozen operators separate reasoning mechanism from task content, allowing domain changes through re-encoding rather than operator retraining.The paper connects this design to a universal Turing machine, with operators as a fixed transition function and snapshots as task-specific tape content.

7 System Architecture Design

DODR separates deterministic reasoning from heuristic graph scheduling: semantic snapshots flow through fixed operators, while encoders, projections, controllers, and decoders serve distinct roles. The design supports multimodal inputs, operator-generated intermediate states, and reproducible decoding without token sampling.

  • Snapshot Encoder: Semantic primitives, rather than tokens, are the basic reasoning units; each operator produces the next complete semantic expression as a snapshot state.Tokens are used only inside encoders, while graph nodes represent words, phrases, sentences, or sentence groups.
  • Multimodal Snapshot Encoder: Multimodal encoders align modality-specific snapshots through projection layers, leaving the three-operator reasoning kernel in a unified space.The operators therefore retain the same algebraic conclusions across modalities, while cross-modal training is confined to semantic alignment.
  • System Modules: The deterministic kernel comprises the encoder and operators, whereas heuristic decisions reside in the graph controller and translation-only decoder.This separation is intended to optimize algebraic reasoning and graph scheduling independently.
  • Graph Scheduling Controller: The controller uses layered scheduling with type-signature dispatch and learned policy mechanisms to select reasoning-graph paths.Its reward combines correctness, efficiency, and redundancy: R = α · correctness + β · efficiency −γ · redundancy.
  • Decoder: End-to-end decoding uses nearest-neighbor matching against candidate conclusion snapshots, with no generative decoder or token sampling.This prioritizes fidelity over the greater expressiveness of autoregressive decoding.
  • Encoder Choice and Limitations: The current system’s main practical limitation is encoder snapshot resolution, and the Transformer choice is replaceable without retraining frozen operators.The paper identifies specialized semantic-discrimination training as a possible encoder improvement.
  • Operator-Generated Snapshots: Operator-generated snapshots can support subsequent reasoning: similarity remains 0.9861 in a multi-step chain, and unseen cross-domain deduction reaches 60/60 correct.The experiments also generate complete compositional chains from observations through revised rules.

8 Experimental Validation of the Deduction Operator Wstep

The deduction experiment tests whether a rank-deficient linear operator can learn premise-to-conclusion mappings while exhibiting structural information loss and limited reversibility. Results support accurate forward inference, rank-controlled collapse, and local robustness across a short inference chain.

  • 1.40e-05 final loss demonstrates that Wstep learns the forward mapping from premise states to conclusion states.The loss decreased from 2.09e-01 by a factor of 14932×.
  • Numerical rank 384 exactly matches the design rank, with a five-order singular-value gap after the 384th component.The largest singular value is 4.2366, while the 385th is 1.06e-06.
  • 87.5% of input dimensions lie in the null space, so deduction permanently discards information outside the 384-dimensional row space.The input dimension is 3072, with null-space dimension 2688.
  • Pseudoinverse reconstruction has 0.8330 relative error versus an expected 0.9354, supporting irreversible information loss without treating 0.9354 as a universal lower bound.Individual inputs can have zero or complete reconstruction error depending on their row-space or null-space alignment.
  • Cosine similarities above 0.98 across two transitive steps show that Wstep supports short multi-step inference with modest accumulated degradation.Similarity decreases from 0.9898 to 0.9861 on the second step.
  • Structural rank and null-space conclusions are guaranteed by the low-rank parameterization, whereas numerical validations depend on training convergence and experimental conditions.This separates algebraic guarantees from empirical measurements.

9 Dedicated Operator Experiment for the Induction Operator

The induction experiment evaluates a full-rank operator that broadly covers positive examples, then uses hard vetoes and reconstruction to exclude counterexamples while preserving most of the original concept. Abduction results additionally show strong separation from random baselines but shallow discrimination margins and encoder-resolution limitations.

  • Full-rank Winduce achieves numerical rank 1536/1536 with pseudoinverse reconstruction error 3.37e-05, contrasting with irreversible deduction.The comparison treats induction as reversible and deduction as information-collapsing.
  • Unseen-positive coverage reaches 0.9993, while training-positive coverage reaches 0.9996, indicating minimal coverage decay without counterexamples.Counterexample similarity is also 0.9974 before vetoing, showing broad over-coverage.
  • 20/20 counterexamples trigger the hard veto at threshold τ = 0.5, producing a 100% veto rate.The mean is 2.0 veto-triggering counterexamples per category across 10 categories.
  • After rebuild, counterexample similarity changes from 0.9974 to −0.0099 while refinement degree reaches 0.9496.The rebuild excludes counterexample directions while preserving most of the original concept.
  • The reported abduction bottleneck is insufficient encoder snapshot resolution, associated with highly regular sentence templates and compressed semantic differences.The paper identifies stronger encoders or contrastive objectives as direct remedies.
  • Abductive solutions reach 0.5117 similarity to true explanations versus a 0.0181 random baseline, a 28.3× improvement, but the discrimination margin is only +0.0051.Dedicated abduction judgment accuracy is 72.5% (58/80), with failed cases having margins near zero.

11 End-to-End Real Inference Validation

The end-to-end evaluation validates DODR’s three inference modes and their composition across new, multi-domain tasks, while exposing encoder resolution as the main abduction constraint.

  • Deduction: 100% (60/60) deductive accuracy held on brand-new cross-domain problems with frozen operators.The problems covered 8 domains and had zero overlap with the dedicated deduction experiment.
  • Induction: 0.9994 mean training coverage and 12/12 hard vetoes were achieved across six end-to-end induction categories.The 12/12 veto result had a Wilson 95% CI lower bound of about 75.7%, reflecting the small sample size.
  • Abduction: 81.7% (49/60) end-to-end abduction accuracy exceeded random discrimination, but margins remained small.The mean discrimination margin was +0.0082, with 11 failed problems and a maximum absolute negative margin of 0.0056.
  • Conclusion: The three operators form a complete closed loop whose compositions are Turing-complete and experimentally support emergent reasoning capabilities.The architecture’s composition is dynamically scheduled, and scientific discovery, analogical reasoning, and creative thinking emerge through operator interleaving.
  • Compositional reasoning: 100% (20/20) of newly constructed compositional tasks succeeded across scientific discovery, diagnostic, analogical, and creative reasoning.The tasks covered four categories and collectively mapped to all 8 human thinking-activity categories.
  • Compositional reasoning: The compositional chain retained similarities above 0.97 for induction, deduction, and contradiction, while refinement degrees exceeded 0.81.Abduction similarity remained 0.38–0.49 and was constrained by encoder resolution without changing overall task-success judgments.

14 Reasoning-Trajectory Topology: From Parallel to Graph-Interleaved

DODR replaces independent parallel trajectories with graph-interleaved reasoning, combining deterministic state transitions, cross-links, merges, and contractive backflow.

  • From parallel to graph-interleaved: Cross edges, convergence points, and backflow edges show why independent parallel trajectories cannot express rigorous reasoning interactions.These structures allow one trajectory’s intermediate conclusion to become another trajectory’s premise.
  • Graph operations: Serial, fork, merge, cross-reference, backflow, and mesh interleaving form a complete vocabulary for compositional thinking.Every operator sequence listed for the paper’s thinking-activity categories can be expressed using these six graph operations.
  • Deterministic graphs: DAG states are uniquely determined by initial nodes and operator parameters, with no probabilistic sampling or hallucination channel.The determinism follows jointly from DAG topology and deterministic matrix operations.
  • Backflow convergence: Contractive backflow converges to a unique fixed point, and the induction–deduction–abduction–re-induction loop converges when revisions are contractive.The convergence condition is a contraction mapping with 0 < q < 1.
  • Expressive power: Reasoning graphs are Turing-complete, although computational universality does not by itself require all three operator classes.The paper notes that reversible computations can be represented using full-rank permutation matrices alone.
  • Closed-loop reasoning: The closed loop enables self-updating knowledge by induction, deduction, abduction, and re-induction.Induction forms regularities, deduction applies them, abduction discovers exceptions, and re-induction revises them.
  • Continual learning: Continual learning separates snapshot accumulation from operator changes, so new knowledge can be written without perturbing existing operator parameters.The encoder is frozen and new knowledge is added to an explicit snapshot library through forward encoding.

16 Complexity Analysis and Performance Bounds

DODR’s matrix-vector execution offers lower nominal and measured costs than autoregressive decoding, while practical speedups remain bounded by serial graph stages and representation trade-offs.

  • Complexity: About 4× lower single-step complexity and about 40× total speedup for 10 parallel subtasks are reported under the experimental configuration.DODR uses O(384×1536) versus the AR model’s O(1536^2) single-step complexity.
  • Constant factors: DODR’s effective constant is approximately 1, whereas autoregressive decoding adds layered attention, feed-forward, normalization, softmax, cache, and sampling costs.Autoregressive generation also proceeds serially token by token.
  • Measured runtime: All dedicated experiments took about 5 minutes, and the end-to-end experiment took about 2 minutes in the reported CPU environment.The end-to-end timing includes snapshot extraction, 200-epoch operator training, inference, and evaluation.
  • Storage and parameters: DODR uses fewer operator parameters than a same-dimension L-layer Transformer and typically fewer graph nodes than tokens for equivalent reasoning content.A semantic snapshot is described as carrying information equivalent to tens to hundreds of tokens.
  • Performance bounds: Actual parallel speedup is limited by Amdahl’s law and by synchronization, serial backflow iterations, and graph-decision overhead.The serial fraction includes inter-layer dependencies and fixed-point correction rounds.
  • Hallucination: The architecture claims structural zero hallucination because reasoning contains no token-sampling step.The paper contrasts this with probabilistic AR generation and prompting methods that reduce rather than eliminate hallucination probability.
  • Comparison: DODR combines determinism, graph topology, three reasoning types, interpretability, and autonomous learning, while related paradigms match only subsets of these dimensions.The paper attributes these properties to snapshots plus three rank-structured operators.
  • Novelty boundary: The paper positions its novelty in the formalization and combination of components, not in inventing linear maps, random lifting, latent reasoning, or geometric representations individually.The comparison explicitly identifies precursors for those individual components.

A Complete Summary of Experimental Data

The experiments evaluate DODR’s rank-designed operators, pseudo-inverse reconstruction, and end-to-end reasoning across dedicated and integrated settings. Results align with the stated rank structure and show strong deduction and induction behavior alongside measurable abduction limitations.

  • Rank structure: The deduction operator has null-space dimension 2688, the abduction operator 1152, and the full-rank induction operator 0, exactly matching rank–nullity predictions.The dimensions correspond to 87.5%, 75.0%, and 0% of the relevant input spaces, respectively.
  • Abduction: 28.3-fold improvement over the random baseline of 0.0181 was obtained by abduction, with mean similarity 0.5117.This result supports recovery of semantically relevant explanatory directions rather than random guesses.
  • End-to-end results: 100% deduction accuracy (60/60) was achieved on brand-new cross-domain end-to-end problems, with mean similarity 0.9980.The 60 problems had zero overlap with the dedicated deduction experiment.
  • End-to-end results: 20/20 compositional reasoning tasks succeeded in the end-to-end experiment.Each task traversed the integrated reasoning pipeline across 4 categories.
  • Reconstruction: Under isotropic inputs, expected pseudo-inverse reconstruction error equals the null-space fraction, not an individual-input lower bound.For isotropic distributions, the expected relative error is dim(Null(W))/n.

D Glossary

The glossary defines DODR’s operator vocabulary and connects its three operators to Peirce’s deduction, induction, and abduction. It also records the reasoning graph, hard-veto, reconstruction-error, and fixed-point concepts used throughout the paper.

  • Inference types: Deduction derives a result from a rule and case, whereas induction derives a rule from cases and results.Deduction is characterized as necessary and truth-preserving; induction is probabilistic and ampliative.
  • Operator concepts: DODR represents deduction, induction, and abduction as collapse, expansion, and hypothesis operations, respectively.The corresponding matrices are rank-deficient deduction, full-rank induction, and pseudo-inverse abduction.
  • Induction control: The hard-veto threshold is τ = 0.5, and a counterexample exceeding that similarity threshold vetoes the inductive conclusion.Gram–Schmidt orthogonalization removes the counterexample direction to reconstruct a finer conclusion.
  • Inference types: Abduction derives a hypothesis from a rule and result and corresponds in DODR to pseudo-inverse inversion.Its output is an optimal hypothesis rather than a necessarily justified fact because null-space components may be unobserved.
  • Reasoning cycle: DODR’s three operators correspond to a closed inquiry cycle of hypothesis formation, prediction, testing, and revision.The end-to-end compositional experiment executed this cycle successfully on 20/20 tasks.
  • Architecture: The architecture uses a frozen Transformer encoder plus linear-algebraic operators in one latent space, avoiding a separate neural-symbolic interface.Its granularity remains limited by the discriminative capacity of snapshot representations.

H.3 Abduction Experiment Data Set (80 Abduction Problems)

The abduction dataset contains 80 deduplicated problems spanning medical, causal, and physical explanations. Its construction uses explicit phenomenon–explanation relationships, distractors, and multiple domains to test abductive discrimination.

  • Dataset composition: 80 deduplicated abduction problems cover medical diagnosis, causal explanation, and physical phenomena.The dataset contains 30 medical, 30 causal, and 20 physical problems.
  • Examples: The dataset includes medical examples such as cough and weight loss mapped to tuberculosis, alongside physical examples such as melting ice cream mapped to sun exposure.These examples illustrate the cross-domain phenomenon-to-explanation format.
  • Construction principles: Each problem pairs one phenomenon with one manually verified correct explanation.The construction explicitly tests abductive explanation rather than simple classification.
  • Construction principles: Multiple possible causes and constructed distractors test the hypothetical nature of abduction and measure explanatory margins.The dataset includes examples where a phenomenon can have more than one plausible explanation.
  • Overlap: Five of the 60 end-to-end abduction problems overlap with this dataset, while 55 are newly constructed.The 80 problems in this table are otherwise described as deduplicated independent samples.

H.4 Physics Experiment Data Set (55 Samples)

The physics dataset contains 55 deduplicated samples covering conservation, dissipation, reversibility, and multimodal alignment. It contributes to the broader experimental accounting and supports the paper’s physical-process interpretation of operator types.

  • Dataset composition: 55 physics samples comprise 15 conservation-law, 15 dissipative, 15 reversible-evolution, and 10 multimodal-alignment cases.The four categories total 55 independent samples.
  • Examples: Examples include a ball bouncing after impact, coffee cooling, a pendulum returning to its start, and a text–image pair describing a ball rolling downhill.The examples instantiate conservation, dissipation, reversibility, and multimodal alignment.
  • Construction principles: The dataset covers conserved quantities, entropy-increasing processes, idealized periodic processes, and text–image descriptions of the same event.These construction principles define the four physical sample categories.
  • Overlap and accounting: The 55 physics samples have no overlap with other experiment groups and are all deduplicated independent samples.They form a separate dataset component in the experimental accounting.
  • End-to-end context: The end-to-end experiment contains 218 sample-instances across deduction, induction, abduction, and compositional tasks.Its pipeline follows induction, deduction, contradiction detection, abduction, and re-induction.

I.5 End-to-End Induction Reasoning Samples (6 Classes)

The six-class induction set combines existing and newly added classes, with partial overlap in its sample pool and no item-by-item inventory of that overlap.

  • Six classes include existing vehicles and colors plus newly added geometric shapes, academic disciplines, weather, and emotions.
  • The six-class samples partially share the existing sample pool and are not counted as newly added deduplicated independent samples.
  • Sample accounting is deferred to Tables 1-2 in Section 1.4.

I.6 End-to-End Abduction Reasoning Samples (60)

End-to-end abduction was evaluated on 60 samples using similarity against distractors, with 49/60 judgments correct and performance above the dedicated-experiment result.

  • 81.7% of end-to-end abduction judgments were correct, with 49/60 samples exceeding distractor similarity.
  • The end-to-end result improved over the dedicated abduction experiment’s 72.5% accuracy on 80 problems.
  • The minimum end-to-end margin was −0.0056, and 11 samples had negative margins.
  • The dedicated abduction experiment reported a 28.3× improvement over the random baseline, with 58/80 judgments correct.

I.7 End-to-End Compositional Reasoning Tasks (20)

The 20 compositional reasoning tasks were newly constructed, with category-wise means reported elsewhere in the paper.

  • All 20 tasks in this table were newly constructed.
  • The tasks are organized into four categories whose means are reported in Tables 11-4 and 12-2.
  • Category-wise means are given in Section 11.6 and Section 12.3.

I.8 Summary of All Test Samples

The paper reports 503 experimental sample-instances and 420 deduplicated independent samples, while end-to-end abduction reaches 81.7% and unseen cross-domain deduction reaches 100%.

  • 503 experimental sample-instances comprise 285 dedicated samples and 218 end-to-end samples, yielding 420 deduplicated independent samples.
  • Dedicated abduction reached 72.5% judgment accuracy on 80 problems, compared with 81.7% for end-to-end abduction.
  • Both abduction evaluations had small margins near ±0.01, and the paper identifies encoder representational resolution as the judgment bottleneck.
  • 81.7% end-to-end abduction judgment accuracy was obtained on 60 samples, with 49/60 correct and mean margin +0.0082.
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