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DODR: Deterministic Operator-Driven Reasoning in Latent Space

Weicai Huang

arXiv:2609.04782v1cs.AI

TL;DR

Complex reasoning in autoregressive models is limited by hallucination and linear-chain information bottlenecks. DODR replaces token sampling with deterministic latent-space operator computation, and end-to-end experiments validate deduction, induction, abduction, and combined inference across eight domains and 218 samples.

  • Problem

    Autoregressive reasoning faces hallucination risks and linear-chain bottlenecks that limit parallel computation, cross-reference, and expressiveness.

    Source: §1.1 · p. 2, §2.3 · p. 11

  • Method

    DODR represents thought as high-dimensional snapshot vectors and applies three matrix operators corresponding to deduction, induction, and abduction.

    Source: §4.2 · p. 24, Conclusion · p. 37, §6.5 · p. 38

  • Results

    End-to-end experiments validate all three inference modes and combined reasoning across 8 domains and 218 samples, including 60/60 deduction and 49/60 abduction.

    Source: §11.7 · p. 71, §20 · p. 100

  • Takeaways & Limitations

    DODR’s three operators form a minimal complete reasoning basis, while its demonstrated validation supports deterministic latent-space reasoning within the tested scope.

    Source: Conclusion · p. 37, §11.7 · p. 71

    The graph-scheduling controller is not implemented, so current reasoning-graph topologies are manually specified and lack empirical controller evidence.

    Source: §19.1 · p. 98

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 · 2 Problem Definition: The Fundamental Defects of the Autoregressive Paradigm · 3 Mathematical Preliminaries

These sections identify token-level probabilistic sampling, probability substituting necessity, and linear-chain topology as fundamental defects of autoregressive reasoning, and present DODR’s deterministic matrix-based reasoning graphs as the alternative. They define the problem and formalize the linear-algebraic, pseudo-inverse, dimensionality, Peircean, projection, and contraction foundations underlying its operators and convergent graph computation.

1.1 Research Background and Motivation

  • DODR replaces probabilistic token sampling with deterministic snapshot vectors, trainable matrix operators, and graph-structured reasoning in latent space.The architecture is designed to eliminate sampling from the reasoning process rather than merely reduce hallucination probability.Source: §1.1 · p. 3

1.2 Core Contributions and Originality

  • DODR formalizes deduction, induction, and abduction as rank-deficient, full-rank, and pseudo-inverse trainable matrices, respectively.Its claimed contributions also include rank-based irreversibility, pseudo-inverse abduction with an 85.9% residual, hard-veto induction with 20/20 counterexample vetoes, a minimal complete operator set, structural zero hallucination, multimodal encoders, and Turing-complete reasoning graphs with convergent backflow.Source: §1.2 · pp. 3–5

1.3 Comparison of Key Metrics

  • DODR claims a structural 0.0% hallucination rate because its frozen-encoder reasoning chain uses deterministic matrix multiplication and no token sampling.The abductive solution-similarity metric is 0.5117 versus the random baseline 0.0181, while abductive judgment accuracy is independently reported as 72.5% (58/80) and 81.7% (49/60) end-to-end.Source: §1.2 · p. 6, §1.3 · pp. 7–8

1.4 Sample Statistics and Overlap Relationships

  • The experiments comprise 420 deduplicated independent samples and 503 experimental sample-instances, with overlaps documented across dedicated and end-to-end groups.End-to-end deduction uses 60 entirely new problems, while end-to-end abduction includes 55 new problems and five overlapping dedicated-abduction problems.Source: §1.4 · p. 8, Table 1 · p. 8

1.5 Organization of This Paper

  • The paper proceeds from problem definition and mathematical preliminaries through operator theory, architecture, dedicated and end-to-end experiments, applications, complexity, limitations, and conclusion.Appendices provide derivations, experimental data, code, glossary material, philosophical correspondence, operator analysis, datasets, and samples.Source: §1.5 · p. 9

2 Problem Definition: The Fundamental Defects of the Autoregressive Paradigm

  • The three autoregressive defects—error accumulation, probability substituting necessity, and the linear-chain information bottleneck—are presented as consequences of token-level probabilistic sequence generation.Section 2.4 maps these defects to DODR’s corresponding solutions.Source: §1.1 · p. 2, §2 · p. 9

2.1 Error Accumulation

  • For t reasoning steps, independent sampling noise yields accumulated standard deviation √t·σ, reaching approximately 4.47σ beyond 20 steps and worsening under correlated error contamination.A wrong token corrupts subsequent context, making covariance terms positive and turning the independent-noise estimate into a lower bound on true error growth.Source: §2.1 · p. 10

2.2 Probability Substituting Necessity

  • A t-step autoregressive chain has error-free probability at most (1 − p)^t, so reliability decays exponentially whenever each wrong-token probability p > 0.The logarithm gives ln P(error-free) ≈ −t · p, while mitigation methods only reduce p without eliminating sampling.Source: §2.2 · p. 11

2.3 The Linear-Chain Information Bottleneck

  • Reasoning graphs support branching, merging, cross-reference, and backflow, whereas autoregressive chains serialize k subtasks from O(d) to O(k · d).For k = 10 and d = 20, this is 20 graph steps versus 200 autoregressive steps; backflow convergence is guaranteed by Banach’s fixed-point theorem.Source: §2.3 · p. 11, §2.4 · p. 12

2.4 DODR’s Solutions

  • DODR replaces probabilistic sampling with deterministic matrix-vector updates, eliminating architectural sources of random error and hallucination.The reasoning update is S_t+1 = W_step · S_t, producing a unique vector rather than a probability distribution.Source: §2.4 · p. 12
  • DODR’s rank-deficient deduction operator mathematically enforces information collapse, with unrecoverable null-space components representing deductive one-wayness.The same rank-deficiency principle applies to concatenated-input and single-state configurations, although their input dimensions differ.Source: §2.4 · p. 12

3 Mathematical Preliminaries

  • The mathematical preliminaries define snapshot-space notation, linear mappings, rank, null spaces, SVD, pseudo-inverses, projections, contractions, and supporting matrix operations.These definitions and theorems provide the formal vocabulary for DODR’s reasoning operators and graph dynamics.Source: §3 · p. 13, §3.2 · p. 14, §3.3 · p. 15

3.3 SVD Decomposition

  • SVD identifies rank through nonzero singular values and the null space through corresponding zero-singular-value vectors, enabling DODR’s residual and pseudo-inverse analyses.For the measured deduction configuration, rank r = 384 and input dimension 2N = 3072 yield a null space of 2688 dimensions, or 87.5% of inputs.Source: §3.2 · p. 14, §3.3 · p. 15

3.4 Moore–Penrose Pseudo-Inverse

  • The Moore–Penrose pseudo-inverse gives the minimum-norm least-squares backtracking solution, explicitly zeroing null-space directions; full-rank matrices instead provide complete recoverability.This grounds abduction’s null-space-free solution and induction’s full-rank operator design.Source: §3.4 · p. 16

3.5 Cover’s Theorem

  • Cover’s theorem states that linear-separability probability rises with dimension and reaches P = 1 when D ≥ N − 1, motivating high-dimensional linear state transitions.DODR uses 1536-dimensional snapshots from the first two DistilBERT layers, where forward-reasoning loss drops from 2.09e−01 to 1.40e−05.Source: §3.5 · p. 17

3.7 Block Matrices

  • Block matrices decompose DODR deduction into separate transformations of the previous state and new evidence, while exposing rank-deficient information filtering and layer-wise attribution.For Wstep = [Wa, Wb], Wa carries the old state and Wb absorbs new evidence; rank(Wstep) = r < N constrains the joint column space.Source: §3.7 · p. 19

3.8 Orthogonal Projection Matrices

  • The 85.9% hypothesis residual is the projection-decomposition ratio measuring true-explanation components outside the pseudo-inverse solution’s row space.Orthogonal projection also underlies the inductive hard veto through Gram–Schmidt removal of a conclusion’s component along a counterexample direction.Source: §3.8 · pp. 20–21

3.9 Contraction Mappings and Fixed Points

  • Backflow converges to a unique fixed point when the loop operator satisfies σ1(Wloop) < 1, independently of the initial state, whereas rank-deficiency alone does not ensure contractivity.A separate spectral-norm constraint is required because zero smallest singular values do not imply that the largest singular value is below one.Source: §3.9 · pp. 21–22

4 Core Theoretical Foundations: The Five Postulates

The section grounds DODR in five falsifiable postulates that replace deep temporal computation with high-dimensional linear states, directional information flow, local transitions, and deterministic reasoning graphs. Each postulate specifies its theoretical basis, operational falsification test, empirical verification, and design trade-offs.

4.1 Postulate One: Dimensionality Expansion in Place of Multi-Layer Stacking (Cover’s Theorem)

  • 1.40e-05 forward-reasoning loss verifies that a 1536-dimensional linear matrix expresses logical state transitions, supporting dimensionality expansion over multilayer stacking.The loss fell from 2.09e-01 by 14932-fold, approaching the numerical convergence floor; the trade-off is higher memory, bandwidth, and parameter cost.Source: §4.1 · pp. 22–23

4.2 Postulate Two: Snapshot State Representation

  • 100% deductive accuracy verifies that concatenated layer activations preserve information needed for reasoning in a static snapshot representation.The snapshot S = Concat(h1, h2, . . . , hL) has dimension N = L × d, preserving hierarchical features while trading space for time and retaining redundancy for stability.Source: §4.2 · pp. 23–25

4.3 Postulate Three: Direction of Information Flow (Replacing the Irreversibility Assumption)

  • Rank = 384 < 3072 for deduction and rank = 1536/1536 for induction verify the postulate that reasoning is organized by information decrease, increase, and hypothesization.Irreversibility follows from rank deficiency rather than serving as an independent assumption; abduction uses the Moore–Penrose pseudo-inverse for minimum-norm hypothesization, while r = 384 and N = 1536 balance robustness and capacity.Source: §4.3 · pp. 25–26

4.4 Postulate Four: Local Linearization

  • 0.9898 and 0.9861 cosine similarities in two-step transitive reasoning show that local linear transitions do not produce divergent chain error.The postulate decomposes complex reasoning into small tangent-space projections, trading smaller per-step approximation error for more steps and greater computational overhead.Source: §4.4 · pp. 26–27

4.5 Postulate Five: Graph-Topology Computation

  • Multi-step scientific-discovery reasoning successfully executes an induction → deduction → abduction → re-induction graph, supporting deterministic graph-topology computation.The graph treats intermediate states as objective facts, supports branching, merging, and cross-referencing, and is theoretically grounded in path-independent commutative diagrams; scheduling and storage remain costs.Source: §4.5 · pp. 27–28

5 Formal Definitions of the Three Basic Operators

Section 5 defines deduction, induction, and abduction algebraically through rank, null-space, and pseudo-inverse properties. These operators respectively collapse information, expand coverage, and generate minimum-norm hypotheses under irreversible information loss.

5 Formal Definitions of the Three Basic Operators

  • The three operators encode distinct information flows: deduction is rank-deficient collapse, induction is full-rank expansion, and abduction is pseudo-inverse hypothesization.Their algebraic properties provide the architecture’s analyzable correspondence between cognitive semantics and linear operations.Source: §5 · p. 28

5.1 The Deduction Operator Wstep (Information Collapse)

  • Deduction maps concatenated state and evidence from R2N to RN through a rank-deficient operator, discarding null-space directions and making premises irrecoverable.The low-rank factorization guarantees rank(Wstep) ≤ r < N independently of training outcomes.Source: §5.1 · pp. 28–30
  • 87.5% information collapse ratio results when N = 1536 and r = 384, while pseudo-inverse reconstruction error is 83.3% under the reported experiments.The 87.5% ratio is structural; the 83.3% reconstruction error is a sample mean below the isotropic expectation because real snapshots are anisotropic.Source: §5.1 · p. 30

5.2 The Induction Operator Winduce (Information Expansion)

  • Induction uses a full-rank operator to map positive examples toward a general concept, preserving recoverability while expanding semantic coverage beyond the samples.In implementation, positive examples are aggregated by mean plus orthogonal projection, and counterexamples trigger hard-veto refinement.Source: §5.2 · pp. 32–33
  • A single counterexample can veto the initial positive-example mean, after which orthogonal exclusion reconstructs a more precise inductive conclusion.This deterministic hard-veto mechanism operationalizes falsifiability through overthrow, reconstruction, and refinement.Source: §5.2 · pp. 32–33

5.3 The Abduction Operator Wabduce (Information Hypothesization)

  • Abduction defines Wabduce as the Moore–Penrose pseudo-inverse of Wstep, selecting the minimum-norm explanation compatible with an observed phenomenon.It lifts observations along row-space directions and assigns zero to null-space directions, implementing an Occam-style hypothesis.Source: §5.3 · p. 33
  • Because deduction is rank-deficient, abduction cannot recover null-space components; multiple explanations are deductively equivalent, so its output remains a best hypothesis rather than the true explanation.The irrecoverable component is quantified in expectation as (2N −r)/(2N) in the concatenated configuration, linking hypotheticality to information loss and capacity control [22].Source: §5.3 · p. 34

5.4 Summary of the Algebraic Properties of the Three Operators

  • Table 5-1 and its synthesis place deduction and induction at opposite rank extremes, while abduction shares deduction’s rank deficiency but reverses its inference direction.Together they occupy three mutually irreplaceable positions defined by rank property and information-flow direction.Source: Table 5 · p. 34, §5.4 · p. 34

6 Completeness Argument for the Minimal Set of Basic Operators

DODR’s three operators are sufficient and necessary under Peirce’s deduction–induction–abduction premise, while reducible variants add no algebraic type. A single super-operator is impossible, whereas frozen operator instances have architectural and formal universality but remain empirically bounded.

6.2 Reducibility of Wmerge and Wcross

  • Wmerge and Wcross are not basic operators because constrained Wstep instantiations reproduce their collapse rank structure and introduce no new algebraic signature.Their apparent differences, including asymmetric input transformations, are already represented by Wstep’s block structure.Source: §6.2 · pp. 35–36

6.3 Irreducibility of the Three Operators

  • The three operators are mutually non-expressible: rank-deficient collapse, full-rank expansion, and pseudo-inverse reverse inference cannot be replicated by compositions of the other operators.Wabduce additionally differs by expanding dimensionality and solving a minimum-norm inverse problem rather than a forward map.Source: §6.3 · pp. 36–37

6.4 Conclusion: The Minimal Complete Set

  • {Wstep, Winduce, Wabduce} is DODR’s minimal complete operator set: three operators cover all Peircean reasoning types, and removing any one makes a reasoning direction inexpressible.Completeness relies on accepting Peirce’s Proposition 3.6 as a philosophical premise; the section supplies the algebraic sufficiency and necessity arguments.Source: §6.1 · p. 35, Conclusion · p. 37, §6.4 · p. 37

6.5 The Impossibility of a Super Operator: Plurality Is a Structural Necessity

  • No single linear Wsuper can realize deduction, induction, and abduction because their required ranks and dimension signatures are mutually incompatible.Deduction requires rank < N and maps R2N → RN, induction requires rank = N and maps RN → RN, while abduction maps RN → R2N.Source: §6.5 · p. 38
  • The correct unity locus is compositional: reasoning graphs and controllers combine plural operators, rather than collapsing their incompatible algebraic roles into one matrix.The section identifies operator plurality as mandatory while associating system-level unity and Turing completeness with composition.Source: §6.5 · pp. 38–39

6.6 Universality of Operator Instances: Train Once, Freeze Forever

  • Frozen operator instances are architecturally universal because operators encode reasoning mechanisms while task-specific content remains in snapshots, with deduction supported by finite rule schemata and abduction derived by pseudo-inverse.The formal argument uses first-order natural-deduction completeness to bound deduction patterns and requires only two trainable matrix types, with deduction represented by rule instances or one conditional universal instance.Source: §6.6 · pp. 39–40
  • Frozen applicability remains an open empirical proposition: snapshot discriminability and training coverage limit universality, despite 100% (60/60) unseen cross-domain deduction and 81.7% (49/60) abduction after broader training.The strongest evidence comes from operators frozen after 200 epochs, evaluated on 60 new deductive problems across 8 domains and 60 abduction problems, 55 of them new.Source: §6.6 · p. 40

7 System Architecture Design

This section presents DODR’s system architecture, covering its modular separation of latent-space reasoning, scheduling, and translation. It also details snapshot encoding, deterministic decoding, application scenarios, snapshot composition, experimental methodology, and encoder non-uniqueness.

7 System Architecture Design

  • DODR implements a deterministic reasoning kernel of snapshot encoder and operators, while the controller contains heuristic decisions and the decoder only translates terminal states.The four modules are the snapshot encoder, three-operator network, graph scheduling controller, and decoder.Source: §7 · p. 41

7.1 The Snapshot Encoder

  • Snapshots represent complete semantic primitives rather than tokens, enabling operator algebra to model reasoning over meaningful expressions.In the experiments, each text block is a complete sentence, and every reasoning-graph node corresponds to a semantic expression rather than a token prefix.Source: §7.1 · p. 41

7.1.1 Reusing Existing Pre-trained LLMs (Text Modality)

  • Decoder-only LLM snapshots are deterministic under teacher forcing because DODR performs fixed forward propagation without autoregressive sampling or feedback.The encoder is frozen to stabilize the snapshot space and permit operator reuse across multiple encoders.Source: §7.1.1 · p. 42, §7.1 · p. 42
  • 0.0909 snapshot discriminability for GPT-2 is reported as 8.2 times DistilBERT’s [14], supporting decoder-only LLMs as superior text snapshot sources.The selection criterion is hidden-state discriminability for reasoning-relevant semantics rather than generation capability.Source: §7.1.1 · pp. 42–43

7.1.2 The Multimodal Snapshot Encoder

  • Dedicated modality encoders project snapshots into a unified R^N space, allowing shared operators while confining modality differences to encoders and projection layers.The algebraic conclusions therefore hold across modalities, while projection layers alone require cross-modal semantic-alignment training.Source: §7.1.2 · pp. 43–44

7.2 The Three-Operator Network

  • The three operators are trained independently: low-rank W_step for deduction, full-rank W_induce for induction, and an SVD-computed pseudo-inverse W_abduce for abduction.A mandatory algebraic self-check rejects assemblies that violate the required rank structure.Source: §7.2 · pp. 44, 46

7.3 The Graph Scheduling Controller

  • The controller selects operators, target nodes, branching, merging, and termination, while three-layer scheduling combines type dispatch, geometric triggers, and learned PPO/PRM planning.Because operators remain deterministic, controller errors affect path efficiency rather than the correctness of individual executed steps.Source: §7.3 · pp. 46–47

7.4 The Decoder

  • All end-to-end experiments use deterministic nearest-neighbor decoding, matching the final state by cosine similarity without generative decoding or token sampling.The decoder is explicitly restricted to translation; seq2seq decoding is reserved as an extension for generation tasks because it introduces probabilistic behavior.Source: §7.4 · pp. 48–49
  • Nearest-neighbor decoding preserves determinism and zero hallucination, while controlled templates or argmax-only seq2seq translation extend expressiveness without semantic completion beyond the final snapshot.Template-and-slot filling uses restricted lexical retrieval with no decoding probability; deterministic seq2seq permits training randomness but remains deterministic at inference.Source: §7.4 · p. 49, §7.6 · p. 51

7.5 Application Scenarios and Input/Output Conventions

  • DODR supports question answering and scientific exploration through the same snapshot space and frozen operators, differing only in input sources and output readouts.Question answering returns an answer and discriminability score, whereas exploration runs an induction–deduction–abduction loop and outputs induced rules, revised knowledge, and reasoning trajectories.Source: §7.5 · p. 50

7.6 Overview of the Experimental Methodology (Shared by All Experiments; Reproduction Guide)

  • The shared methodology freezes DistilBERT snapshot extraction, trains operators with specified low-rank or full-rank parameterizations, evaluates cosine-based criteria, and decodes by deterministic nearest-neighbor matching.Experiments use fixed seeds, defined hard-veto and abduction rules, deposited JSON records, and Wilson intervals plus paired significance tests; cross-seed variance was not measured.Source: §7.6 · pp. 50–51

7.7.1 Non-Uniqueness of the Encoder: The Transformer Is an Engineering Choice, Not a Theoretical Necessity

  • The encoder need only be deterministic, semantically discriminative, and domain-covering; therefore Transformer self-attention is an engineering choice rather than a theoretical necessity.Alternative encoders include static vectors, sentence encoders, convolutional or recurrent networks, and classical features; operators remain usable after re-extraction and alignment.Source: §7.7.1 · pp. 51–52

7.7.2 Operator Generation of Snapshots: Very Few Raw Snapshots Are Truly Needed

  • Operator-generated snapshots supply most new knowledge: deduction, induction, abduction, and graph composition derive states without repeatedly encoding raw text.Generated states supported 0.9861 similarity after a second reasoning step and achieved 60/60 correct unseen cross-domain deductions, but approximately 0.01 single-step drift can accumulate and require re-grounding.Source: §7.7.2 · p. 52

7.7.3 Compositional Construction of High-Dimensional Snapshots: Five Schemes for Assembling High Dimensions from Low Dimensions

  • High-dimensional snapshots can be assembled from low-dimensional components through layer concatenation, random expansion, recursive semantic composition, or multi-encoder ensembles.These schemes provide training-free or low-cost dimensional expansion, hierarchical composition, and redundancy for robustness, while the paper adopts layer-wise concatenation.Source: §7.7.3 · p. 53
  • Only empirical facts and human-defined concepts require raw snapshots; all logically derivable knowledge can be generated by operators or composition within the snapshot system’s logical closure.This establishes the external interface and an engineering rule: spend encoding capacity on genuinely new observations and definitions.Source: §7.7.3 · p. 54

8 Experimental Validation of the Deduction Operator Wstep

Experiments with real DistilBERT snapshots and PyTorch backpropagation validate Wstep’s forward fitting, rank deficiency, information loss, irreversibility, and multi-step inference behavior. Results support the operator’s linear state-transition capacity and structural unidirectionality, while robustness claims remain bounded by the single-run design and tested conditions.

8 Experimental Validation of the Deduction Operator Wstep

  • Wstep validated five experimental properties: forward convergence, rank deficiency, null-space dimension, backward irreversibility, and end-to-end multi-step inference.The experiments used real DistilBERT snapshots and PyTorch backpropagation rather than simulated data or textual prediction.Source: §8 · p. 54

8.2 Validation 1: Forward-Inference Loss Convergence

  • 1.40e-05 final loss followed a 14932× reduction from 2.09e-01, showing that a rank-deficient linear operator fit the forward mapping.The result supports a single-layer high-dimensional linear transition within the training distribution, but does not by itself establish out-of-distribution generalization.Source: §8.2 · pp. 55–56

8.3 Validation 2: Matrix Rank Verification (Rank Deficiency)

  • 384 numerical rank exactly matched the design rank for Wstep, whose 385th singular value was 1.06e-06 versus 0.0998 at rank 384.The spectral gap confirms strict rank deficiency induced by the low-rank factorization.Source: §8.3 · p. 56

8.4 Validation 3: Null-Space Dimension

  • 2688 of 3072 input dimensions, or 87.5%, lay in Wstep’s null space, quantifying structural information deletion in deduction.Only the 384-dimensional row-space component is retained, while premise details in the null space cannot be recovered by subsequent operations.Source: §8.4 · p. 56

8.5 Validation 4: Irreversibility of Backward Inversion

  • 0.8330 measured pseudoinverse reconstruction error was comparable to the 0.9354 expected relative error under isotropic inputs, indicating substantial irreversible information loss.The 0.9354 value is an expectation, not a universal lower bound; the measured deviation is attributed to sample anisotropy and 20-sample variability.Source: §8.5 · pp. 57–58

8.6 Validation 5: End-to-End Inference Demonstration

  • 0.9898 and 0.9861 cosine similarities across two inference steps showed that Wstep executed a transitive chain on a shifted intermediate snapshot.The second-step input was not in training, supporting local continuity; similarity declined slightly from 0.9898 to 0.9861.Source: §8.6 · p. 58

8.7 Summary of the Deduction Experiments

  • Structural rank and null-space conclusions are parameterization-guaranteed, while numerical results retain large margins but were tested in one run without grid search or multi-seed statistics.The reported margins include 10^-1 to 10^-5 loss convergence, approximately 83% optimal-reconstruction loss, and 0.98 inference similarity against a 0.9 threshold.Source: §8.7 · p. 59

9 Dedicated Operator Experiment for the Induction Operator · 10 Dedicated Operator Experiment for the Abduction Operator

Dedicated operator experiments validate induction and abduction as distinct inference modes, examining their reversibility, hypothesis correction, solution quality, equivalence, and closed-loop consistency. They also identify insufficient semantic separation in the encoder snapshot space as the primary limitation in fine-grained judgment.

9.2 Validation 1: Full-Rankness and Invertibility of the Induction Operator

  • Winduce is full-rank and reversible, with rank 1536/1536 and pseudoinverse reconstruction error 3.37e-05, unlike rank-deficient irreversible deduction.The comparison supports the proposed symmetry between information-preserving induction and information-collapsing deduction.Source: §9.2 · p. 60

9.3 Validation 2: Broadest Induction in the Absence of Counterexamples

  • 0.9996 training-positive coverage and 0.9993 unseen-positive coverage coexist with 100% vetoes on 20/20 counterexamples, validating broad induction with corrective rebuilding.Counterexample similarity falls from 0.9974 to −0.0099 after rebuilding, while refinement remains 0.9496.Source: §9.3 · p. 60, §9.4 · p. 61

9.5 Hard-Veto Results by Category

  • Across all 10 induction categories, counterexample similarity drops from 0.9974 to −0.0099 and refinement degrees remain above 0.91, demonstrating robust minimal correction.Pre-veto similarities range from 0.9955–0.9987, while post-veto similarities range from −0.0227 to −0.0028.Source: §9.5 · pp. 61–62

9.6 Measured Evidence of the Encoder Bottleneck

  • Encoder snapshot resolution, not operator algebra, limits fine-grained judgments because highly regular templates compress semantic differences to only a few thousandths.The induction validations pass, while abduction failures include 22/80 negative-margin cases with maximum negative margin −0.0086; stronger encoders or contrastive learning are proposed remedies.Source: §9.6 · p. 62, §10.2 · p. 65

10.2 Validation 1: Optimality of Abductive Solutions

  • 28.3× improvement over the 0.0181 random baseline yields abductive similarity 0.5117 to the true explanation versus 0.5066 to the distractor.Rule-based judgment accuracy is 72.5% (58/80), with mean discrimination margin +0.0051 and Wilson 95% CI [61.9%, 81.1%].Source: §10.2 · pp. 64–65

10.3 Validation 2: Hypothetical Nature of Abductive Solutions (Null-Space Components)

  • 85.9% of the true-explanation norm lies in the 1152/1536 null space, so pseudoinverse abduction returns a minimum-norm hypothesis that cannot recover null-space information.The measured null-space residual norm is 15.34 versus a true-explanation norm of 17.85.Source: §10.3 · pp. 65–66

10.4 Validation 3: Multi-Explanation Equivalence

  • 1.01e-02 relative deduction-result difference after adding null-space components validates approximate multi-explanation equivalence rather than strict equivalence.Numerical rank truncation and the final Wstep loss of 1.89e-04 account for the residual difference.Source: §10.4 · p. 66

10.5 Validation 4: Abduction–Deduction Closed Loop

  • The abduction–deduction loop has measured reconstruction error 0.0200 versus expected relative error 0.8660 because real phenomenon snapshots lie near the operator’s column space.The two values use different input conventions: arbitrary-direction expectation versus in-distribution measured inputs.Source: §10.5 · pp. 66–67
  • 0.0200 closed-loop residual confirms approximate abduction–deduction consistency, with the nonzero error attributed to projection truncation and training convergence precision rather than structural inconsistency.The loop infers an explanation backward from a phenomenon and derives the phenomenon forward again; Wstep final loss is 1.89e-04.Source: §10.5 · p. 67

10.6 Comparison of the Three Basic Operators

  • At snapshot dimension N = 1536, deduction and abduction share rank 384 and are mutually opposite, irreversible directions of one rank-deficient mapping, whereas induction is full-rank and invertible.These algebraic properties distinguish Peirce’s trichotomy: rank deficiency corresponds to unidirectional collapse, while full rank corresponds to bidirectional information preservation.Source: §10.6 · p. 67

10.7 Summary of the Abduction Experiments

  • 0.5117 abductive-solution quality exceeds the random baseline 0.0181 by 28.3×, while rule-based judgment accuracy reaches 72.5% (58/80; Wilson 95% CI [61.9%, 81.1%]).The validations also report 85.9% residual contribution for hypothetical nature and a 1.01e-02 multi-explanation approximate-equivalence residual; paired testing gives t = 6.32, p < 10−4, and sign-test p ≈ 7 × 10−5.Source: §10.7 · pp. 67–68

11 End-to-End Real Inference Validation · 12 Compositional Thinking and Coverage Analysis · 13 Physical Transformations and Cognitive Unification

This section validates end-to-end operator-based reasoning across multiple domains and analyzes how ternary operator composition supports compositional thinking and Turing-complete reasoning. It further examines a proposed correspondence between physical transformations and cognitive operators, supported by small-scale experiments while noting unresolved questions about their deeper unity.

11.2 Real Training of the Three Operator Matrices

  • The trained matrices reached losses of 8.22e-04, 7.43e-05, and 9.53e-04 for deduction, induction, and abduction, while preserving the predicted rank structure.Wstep was rank-deficient at 384 < 1536, Winduce was full-rank at 1536, and Wabduce was obtained by SVD pseudoinverse rather than separately trained.Source: §11.2 · p. 69

11.5 Abductive Inference Validation

  • 81.7% (49/60) abduction accuracy exceeded distractors, but small discrimination margins traced failures to encoder representation bottlenecks rather than operator defects.The mean discrimination margin was +0.0082, with failed cases near ±0.006; this configuration must not be mixed with the separate 72.5% (58/80) experiment.Source: §11.5 · pp. 70–71

11.7 Summary of the End-to-End Validation

  • End-to-end validation covered 20 entirely new combined tasks, with 20/20 successful, across experiments spanning 8 domains and 218 samples.The pipeline started from scratch with real DistilBERT snapshots, trained three operator matrices with real gradient descent, and used no simulation or textual prediction.Source: §11 · p. 68, §11.6 · p. 71

12.1 Sources of Uncertainty in Composition

  • Composition is dynamically scheduled because operator order, graph topology, and operator choice depend on evolving state content, while each matrix operation remains deterministic.The semi-formal category-theoretic construction models snapshots as objects, operator applications as morphisms, and matrix multiplication as associative composition; forks and merges form a monoidal reasoning graph.Source: §12.1 · pp. 72–73, §12.4 · p. 76

12.2 Theoretical Analysis of Coverage of Human Thinking Activities by Composition

  • Turing-complete composition and ternary information-flow directions imply coverage of all 8 listed human-thinking categories and any algorithmically describable thinking process.Activities from samples to regularities require induction, regularities to conclusions require deduction, and phenomena to explanations require abduction.Source: §12.2 · pp. 73–74

12.3 Experimental Validation of Compositional Thinking

  • 100% (20/20) compositional tasks succeeded across scientific discovery, diagnosis, analogy, and creativity, with induction, deduction, and contradiction similarities above 0.97.Refinement degrees exceeded 0.81, whereas abduction similarities of 0.38–0.49 reflected approximation constrained by encoder resolution.Source: §12.3 · pp. 75–76

12.4 A Semi-Formal Discussion from the Perspective of Category Theory and Monoidal Categories

  • The correction iteration S(t+1) = f(S(t)) converges to a unique fixed point because f is a contraction mapping, as guaranteed by Banach’s fixed-point theorem [24].The paper identifies this fixed-point convergence with the trace axiom’s witness in its category-theoretic framing.Source: §12.4 · p. 77
  • Abduction is a relaxed adjoint of deduction: because Wstep is rank-deficient at r = 384 < 1536, its Moore–Penrose pseudo-inverse provides optimal least-squares, minimum-norm reconstruction rather than strict inversion.The measured reconstruction error is 0.0200, and the null-space dimension quantifies the degeneration from exact to approximate mutual invertibility.Source: §12.4 · p. 77

12.5 Conclusion on Compositional Coverage

  • The three-operator composition covers all eight human-thinking categories, succeeds on four representative compositional tasks, and is Turing-complete through dynamic scheduling and associative matrix composition.Emergent capabilities arise from state handoff and recombination across heterogeneous information-flow directions rather than from any single operator.Source: §12.5 · p. 77

13.1 Necessary Laws of State Transformation in the Physical World

  • Physical transformations are modeled by orthogonal or unitary matrices for conservation and reversible evolution, rank-deficient matrices for dissipation, and projection or pseudo-inverse operators for measurement.These forms correspond respectively to information preservation, information loss, and information collapse; the dissipation–deduction connection shares rank-deficient information erasure as its mathematical root.Source: §13.1 · p. 78

13.2 The Physics-Cognition Unification Proposition

  • The physics–cognition proposition maps conservation and reversible evolution to information-preserving induction, dissipation to information-losing deduction, and measurement to abductive information collapse.The correspondence is classified by information-flow direction, while induction’s knowledge increase is semantic and its matrix representation remains full-rank; the claim is a structural analogy, not a formal derivation.Source: §13.2 · pp. 79–80

13.3 Experimental Validation of Physical Transformation Operators

  • Conservation, dissipative, and reversible-evolution operators each achieved 100% inference accuracy, while multimodal alignment reached 100% with average alignment similarity of 0.9958.The dissipative operator exhibited rank deficiency 384/1536, matching Wstep after training without an artificial constraint.Source: §13.3 · p. 80

13 Physical Transformations and Cognitive Unification

  • The physical-transformation correspondence is a testable structural analogy supported by Noether’s theorem [12], Landauer’s principle, and the shared classification of state transformations by information flow.Its proposed deeper unity is not established, and the paper’s algebraic and experimental conclusions do not depend on it.Source: §13 · p. 77, §13.2 · p. 79, §13.4 · p. 80

14 Reasoning-Trajectory Topology: From Parallel to Graph-Interleaved · 15 The Three-Operator Closed Loop: Induction–Deduction–Abduction

DODR replaces independent parallel trajectories with deterministic reasoning graphs that support cross-reference, convergence, and iterative correction. Its induction–deduction–abduction loop enables falsification-driven self-update, approximate mathematical closure, and continual learning through separated storage layers.

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

  • Reasoning graphs replace “parallel trajectories” with cross edges, convergence points, and backflow, the irreducible structures required for rigorous reasoning.The six operations—serial, fork, merge, cross-reference, backflow, and mesh interleaving—form a complete vocabulary for compositional thinking.Source: §14 · p. 80, §14.1 · p. 81, §14.2 · p. 81

14.3 Formal Definition of the Reasoning Graph

  • A reasoning graph is a labeled directed graph whose snapshot nodes are connected by deterministic operator edges, comprising DAGs for one-way reasoning and backflow graphs for iterative correction.Backflow connects different states across time; its time-unfolded representation remains acyclic.Source: §14.3 · pp. 81–82

14.4 Graph-Theoretic Properties of Rigorous Reasoning

  • DAG execution uniquely determines every node without sampling, while contractive backflow converges to a unique fixed point and reasoning graphs are Turing complete.The convergence result follows from Banach’s fixed-point theorem; computational universality does not by itself establish completeness of reasoning types.Source: §14.4 · pp. 82–83

14.5 Comparison Between Reasoning Graphs and Parallel Trajectories

  • Compared with Petri nets and dataflow graphs, reasoning graphs carry continuous semantic snapshots and operator-labeled deterministic transformations rather than discrete token competition or ordinary data channels.The dataflow correspondence transfers scheduling methods, while reasoning graphs additionally provide latent-space backflow convergence theory.Source: §14.5 · p. 84, Table 14 · p. 84

14.6 Worked Example: The Reasoning Graph of the Scientific Discovery Loop

  • In the scientific-discovery example, induction forms g0, deduction predicts S1, contradiction detection yields S2, abduction explains it as S3, and re-induction produces revised regularity g1 via backflow correction.Similarities are 0.9925 for induction, 0.9864 for deduction, 0.9867 for contradiction detection, and 0.3849 for abduction.Source: §14.6 · pp. 84–85

15.1 Operating Mechanism of the Closed Loop

  • The three operators form a closed loop in which induction produces regularities, deduction applies them, abduction explains exceptions, and re-induction revises knowledge.Induction vetoed 20/20 counterexamples, deduction achieved 100% accuracy (60/60; mean cosine similarity 0.9980), and contractive revisions converge to a stable regularity.Source: §15 · p. 85, §15.1 · pp. 85–86

15.2 Mathematical Consistency of the Closed Loop

  • Abduction approximately inverts deduction with reconstruction error 0.0200, although rank deficiency permanently loses 75% of latent dimensions and makes closure dependent on the observation distribution.The pseudo-inverse recovers only the row-space component, so residual error rises when observations contain more null-space information.Source: §15.2 · p. 86

15.3 Continual Learning Capability: A Three-Layer Mechanism

  • DODR’s three-layer continual-learning mechanism stores new snapshots first, condenses them into regularities with veto correction second, and updates operators only as an occasional fallback.The design separates content, inductive conclusions, and reasoning forms, while operator updates must preserve the deductive, inductive, and abductive rank signature.Source: §15.3 · pp. 87–88

16 Complexity Analysis and Performance Bounds … 20 Conclusion

This section analyzes DODR’s complexity, performance bounds, deployment positioning, and engineering feasibility, including its deterministic operator guarantees and experimental support. It also outlines limitations involving scalability, validation, statistical robustness, decoding, and distribution shift.

16.1 Time Complexity

  • 5.9×10^5 versus 2.4×10^6 single-step complexity makes DODR about 4× faster; with k = 10 subtasks, total speedup reaches about 40×.DODR’s effective constant is approximately 1, while autoregressive generation incurs multilayer attention, feed-forward, normalization, softmax, KV-cache, and sampling costs.Source: §16.1 · pp. 88–89
  • 5–10× additional GPU speedup is expected, while dedicated experiments took about 5 minutes and end-to-end reasoning about 2 minutes on CPU.End-to-end cost was dominated by 200-epoch operator training; inference matrix–vector multiplication was negligible.Source: §16.1 · p. 89

16.2 Space Complexity

  • DODR uses O(r·N) low-rank deduction and abduction plus N^2 induction parameters, while semantic snapshots require fewer nodes than equivalent token sequences.A single snapshot can encode information equivalent to tens to hundreds of tokens, and the operator parameter count is far smaller than L·d^2 for a same-dimension Transformer.Source: §16.2 · p. 89

16.3 Parallel Speedup

  • 10 parallel subtasks of 20 steps take 20 DODR steps versus 200 autoregressive steps, yielding a 10× speedup under ideal parallel execution.For a reasoning DAG, nodes at the same depth can execute concurrently, with theoretical speedup equal to width W.Source: §16.3 · p. 89
  • With p = 0.9 and W = 10, Amdahl’s law limits speedup to approximately 5.3× rather than 10×, and backflow-intensive tasks are instead governed by contraction-based convergence.Serial constraints include inter-layer synchronization, fixed-point backflow iterations, scheduling, and global reductions; autoregressive models have p ≈ 0 and an upper bound near 1×.Source: §16.3 · p. 90

17.1 Dimension-by-Dimension Argument

  • DODR uniquely combines determinism, structural zero hallucination, graph topology, three reasoning types, white-box interpretability, and autonomous learning across the compared paradigms.Its distinctions include deterministic matrix transitions, rank-structured deduction–induction–abduction, explicit convergence, traceable null-space information loss, and fixed-point-supported revision.Source: §17.1 · pp. 91–92

17.2 Precise Delimitation from the Nearest-Neighbor Technical Routes

  • DODR’s claimed novelty lies in combining concrete deterministic, rank-structured reasoning formalizations rather than inventing individual components such as embeddings, latent spaces, or geometric representations.The comparison distinguishes DODR from knowledge-graph embeddings, differentiable logic, COCONUT, random features, conceptual spaces, and ProofWriter by operator semantics and reasoning guarantees.Source: §17.2 · pp. 92–94

18 Application Systems and Adaptation Schemes

  • 72.5% abductive-assertion judgment accuracy rises to 81.7% end-to-end, defining DODR as a consistency guardian limited by snapshot resolution and knowledge-base coverage.The deterministic core verifies, corrects, and plans through matrix operations; null-space sampling permits equivalent imaginative variants, while deployment exposes auditable verify/correct/plan primitives to LLMs or agents.Source: §18.1 · p. 95, §18.2 · p. 96

19.1 Current Limitations

  • The encoder bottleneck limits fine-grained judgment: abduction reaches 81.7% (49/60) end-to-end and 72.5% (58/80) dedicated, while coarse-grained deduction remains at 100%.Failed-sample discrimination margins are only a few thousandths, indicating representational resolution rather than an operator defect; stronger or domain-fine-tuned encoders are proposed.Source: §19.1 · p. 97
  • The evaluation remains limited by 420 deduplicated samples, a non-implemented graph-scheduling controller, single-seed runs, and untuned τ = 0.5, r = 384, and L = 2 hyperparameters.The induction setting has k = 10 examples per category versus a 1536 × 1536 Winduce with about 2.4×10^6 parameters; proposed remedies include larger proof-library datasets, reinforcement-learning scheduling, multiple seeds, and grid search.Source: §19.1 · pp. 97–98
  • Results lack recognized benchmarks and external baselines, use restricted nearest-neighbor candidate decoding, and test largely isomorphic training and evaluation templates rather than distribution-shifted reasoning.The system therefore does not yet demonstrate open-ended generation or robustness to multi-premise, negated, quantified, and non-monotonic reasoning; ProofWriter and LogiQA comparisons are proposed.Source: §19.1 · pp. 98–99

20 Conclusion

  • DODR formalizes Peirce’s three reasoning types as trainable operators and derives rank-based irreducibility, a 0.0% structural hallucination rate, Turing completeness, and Banach-based backflow convergence.Abduction has closed-loop reconstruction error 0.0200; physical operators achieved 100% inference accuracy on 55 samples with multimodal alignment 0.9958.Source: §20 · pp. 99–100
  • End-to-end compositional reasoning succeeded on 20/20 cases, while frozen operators achieved 100% cross-domain deduction (60/60) and abduction reached 81.7% (49/60).Across eight experiment groups, dedicated induction achieved 0.9996 coverage with 20/20 hard vetoes, and end-to-end induction achieved 12/12 hard vetoes with mean training coverage 0.9994.Source: §20 · p. 100

A Complete Summary of Experimental Data … C.4 Reproducibility Notes

The appendices consolidate dedicated and end-to-end experimental results, mathematical guarantees, implementation mechanisms, and reproducibility procedures for DODR. Together, they connect operator rank structure and pseudo-inverse theory to measured performance and provide runnable, deterministic replication materials.

B.1 The Rank–Nullity Theorem

  • The deduction operator’s rank r = 384 implies a 2688-dimensional null space for Wstep ∈ R^1536×3072, whereas full-rank induction has null space 0 and abduction has effective null space 1152.These dimensions match rank–nullity predictions and formalize deduction’s information collapse.Source: A Complete Summary of Experimental Data · p. 103, §B.1 · p. 104

A Complete Summary of Experimental Data; B.5 Derivation of the Expected Reconstruction Error of the Pseudo-Inverse (Supporting Theorem 5.3)

  • 83.3% measured deduction-direction reconstruction error approached the isotropic expectation of 93.5%, while abduction similarity reached 0.5117 versus a 0.0181 random baseline.The appendix attributes the deduction deviation to snapshot anisotropy and finite sample size; induction reconstruction error was 3.37e-05.Source: A Complete Summary of Experimental Data · p. 103

A Complete Summary of Experimental Data

  • 100% (60/60) deduction accuracy and 81.7% (49/60) abduction accuracy were achieved end-to-end, while all 20 compositional tasks succeeded.The end-to-end configuration used frozen DistilBERT snapshots with N = 1536, r = 384, 200 epochs, 218 samples, and 8 domains.Source: A Complete Summary of Experimental Data · pp. 103–104

B Mathematical Derivations and Theorem Proofs; B.1 The Rank–Nullity Theorem; B.2 Existence of the SVD; B.3 Cover’s Theorem; B.4 Gram–Schmidt Orthogonalization; B.5 Derivation of the Expected Reconstruction Error of the Pseudo-Inverse (Supporting Theorem 5.3)

  • The mathematical appendices establish rank–nullity, SVD existence, Cover separability, Gram–Schmidt correctness, and the expected pseudo-inverse reconstruction-error formula.These are presented as classical results or paper-specific derivations supporting DODR’s rank accounting, abduction construction, hard vetoes, and error analysis.Source: B Mathematical Derivations and Theorem… · p. 104, §B.5 · p. 106

B.2 Existence of the SVD; C.3 Core Snippet of Abduction Operator Pseudo-Inverse Computation Code

  • The abduction operator is built from the deduction operator’s truncated SVD, retaining rank 384 and inverting nonzero singular values to produce a minimum-norm least-squares explanation.Measured numerical ranks matched the designed ranks, with no additional rank degeneracy.Source: §B.2 · p. 104, §C.3 · p. 108

B.3 Cover’s Theorem

  • With snapshot dimension 1536 exceeding every experiment’s sample size, Cover’s theorem gives separability probability 1 for random labellings, but not generalization by itself.The appendix states that generalization still requires PAC/VC support and experiments.Source: §B.3 · p. 105

B.4 Gram–Schmidt Orthogonalization; C.2 Core Snippet of Induction Operator Hard-Veto Code

  • The induction hard-veto mechanism projects each veto-triggering concept away from counterexample directions, enforcing zero similarity and producing slight negative post-veto correlations after renormalization.The implementation uses a veto threshold τ = 0.5 and sequential Gram–Schmidt-style orthogonalization.Source: §B.4 · pp. 105–106, §C.2 · p. 108

C Experimental Code and Reproducibility; C.1 Core Snippet of Deduction Operator Training Code; C.4 Reproducibility Notes

  • The released code preserves operator construction, training, inference, fixed seeds, experiment records, and environment requirements for reproducibility.All numerical values should match digit for digit under identical dependencies and seeds, with only last-decimal floating-point differences expected across versions.Source: C Experimental Code and Reproducibility · p. 107, §C.4 · p. 109

D Glossary … H.2 Induction Experiment Data Set (10 Classes, 130 Samples)

DODR maps Peirce’s deduction, induction, and abduction onto deterministic latent-space operators whose algebraic, geometric, and Bayesian properties define reasoning as collapse, expansion, and hypothesis recovery. The surrounding analyses relate this design to representation learning, attention, neuro-symbolic systems, and process reward models, while the datasets operationalize its evaluation across syllogistic and inductive categories.

D Glossary; E.1 Peirce’s Three Types of Inference

  • DODR defines deduction as rank-deficient information collapse, induction as full-rank information expansion, and abduction as pseudo-inverse hypothesis recovery within Peirce’s three-inference framework [7].Deduction preserves necessary consequences while discarding null-space information; induction adds revisable knowledge; abduction returns a minimum-norm explanation.Source: §E.1 · pp. 109–111

E.2 Correspondence Between the Three DODR Operators and Peirce’s Three Inferences

  • The three correspondences are algebraically testable: deduction is deterministic on its range, induction preserves directions for revision, and abduction selects the minimum-norm pseudo-inverse solution with null-space components set to zero.This minimum-norm choice functions as an Occam-style default rather than a claim that the recovered prior is the true explanation.Source: §E.2 · p. 111

E.3 Peirce’s “Surprise–Hypothesis–Deduction” Cycle of Scientific Inquiry

  • DODR implements Peirce’s surprise–hypothesis–deduction–induction cycle through abduction, prediction unfolding, hard-veto contradiction detection, and hypothesis revision.In the end-to-end compositional experiment, all 20/20 tasks completed this pipeline successfully.Source: §E.3 · pp. 111–112

F.1 Relationship with Representation Learning Theory

  • DODR concatenates frozen Transformer multi-layer [CLS] activations into 1536-dimensional snapshots, relying on high-dimensional linear separability while inheriting representation quality from pre-training [5, 6].The snapshot dimension exceeds the experiment’s maximum sample size of 130, but the frozen encoder is identified as a system bottleneck.Source: §F.1 · p. 112

F.2 Relationship with the Attention Mechanism; F.3 Relationship with Neuro-Symbolic Systems

  • Unlike attention, DODR performs reasoning through deterministic operator multiplication, projection, and pseudo-inverse inversion; unlike neuro-symbolic systems, its encoder and operators share one latent space without a semantic interface.This improves reproducibility and auditability, but performance remains limited by snapshot discriminative capacity, including 72.5% judgment accuracy and 22 negative-margin samples.Source: §F.2 · p. 112, §F.3 · pp. 112–113

F.4 Relationship with Process Reward Models

  • Compared with process reward models, DODR scores deterministic DAG nodes rather than probabilistically sampled token sequences, so identical inputs necessarily produce identical reasoning graphs.The distinction is deterministic computation plus deterministic scoring versus probabilistic generation plus deterministic scoring.Source: §F.4 · p. 113

G.1 Information-Theoretic Analysis of the Deduction Operator; G.2 Generalization Error Analysis of the Induction Operator; G.3 Bayesian Interpretation of the Abduction Operator

  • The PAC/VC framework is numerically vacuous and does not explain strong observed generalization despite measured unseen-positive coverage of 0.9993; abduction error is inevitable information loss.Source: §G.2 · p. 113, §G.3 · p. 114

G.4 Geometric Interpretation of the Three Operators; H.1 Deduction Experiment Data Set (20 Syllogisms); H.2 Induction Experiment Data Set (10 Classes, 130 Samples)

  • Geometrically, deduction compresses, induction excludes and revises through counterexample constraints, and abduction inverse-maps observations into the recoverable row space, making the latter two spaces complementary.The datasets contain 20 cross-domain syllogisms with verifiable first-order-logic conclusions and 130 induction samples spanning 10 classes, each with 10 positives, 1 unseen positive, and 2 counterexamples; counterexamples test semantic rather than syntactic discrimination, with coverage 0.9986–0.9997.Source: §G.4 · pp. 114–115, §H.1 · p. 115, §H.2 · pp. 116–117

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

The abduction dataset contains 80 deduplicated problems spanning medical diagnosis, causal explanation, and physical phenomena. Manually verified explanations, alternative causes, domain variation, concise wording, and semantically related distractors support testing abductive hypothesis selection.

Dataset composition

  • 80 problems comprise 30 medical-diagnosis, 30 causal-explanation, and 20 physical-phenomenon cases, each pairing one phenomenon with one correct explanation.All 80 samples are deduplicated independent records.Source: §H.3 · pp. 117, 120

Data construction

  • Each phenomenon–explanation relationship is explicitly causal, and correct explanations were manually verified.The dataset defines one correct explanation for each problem while preserving abduction’s causal structure.Source: §H.3 · pp. 117, 120
  • Alternative causes make explanations non-unique, while same-semantic-field distractors with invalid causal directions measure the selection margin through distractor similarity.The construction example contrasts rain and sprinkling as alternative explanations for wet ground.Source: §H.3 · p. 120
  • The three domains expose domain dependence in the encoder bottleneck, and 10–15-word samples facilitate encoding.Snapshot-distribution differences across medicine, everyday causality, and physics are used for this diagnostic purpose.Source: §H.3 · p. 120

Overlap statement

  • Five of the 60 end-to-end abduction problems overlap with this dataset, leaving 55 newly constructed samples.The 80 problems in this table remain deduplicated independent samples.Source: §H.3 · p. 120

H.4 Physics Experiment Data Set (55 Samples) … I.4 End-to-End Deduction Reasoning Samples (60)

The appendices define physics, end-to-end, and dedicated experiment datasets, totaling 503 experimental sample-instances and 420 deduplicated independent samples. They also document abduction performance, sample overlap conditions, and the scope of measured-result tables.

H.4 Physics Experiment Data Set (55 Samples)

  • The physics dataset comprises 55 deduplicated, non-overlapping samples spanning conservation laws, dissipation, reversible evolution, and multimodal alignment.The type counts are 15, 15, 15, and 10, respectively; the temporal categories correspond to distinct physical-operator spectral properties, while multimodal pairs test snapshot modality invariance.Source: §H.4 · p. 121

H.5 End-to-End Experiment Data Set (218 Samples)

  • The end-to-end dataset contains 218 real samples across eight domains: 60 deduction, 78 induction, 60 abduction, and 20 compositional tasks.It includes 135 newly added deduplicated samples; deduction problems are entirely new, abduction includes 55 new and 5 overlapping problems, and compositional tasks are entirely new.Source: §H.5 · pp. 121–122

I All Test Samples and Measured Results

  • The complete experiment aggregation contains 503 sample-instances, which reduce to 420 deduplicated independent samples after accounting for overlap.The total combines 285 dedicated or physics instances with 218 end-to-end instances; the end-to-end addition includes 60 new deduction, 55 new abduction, and 20 new compositional samples.Source: I All Test Samples and Measured Results · p. 122

I.1 Dedicated Induction Experiment Samples (10 Classes, 130 Samples)

  • Table I-1 covers 130 samples in the dedicated induction experiment: 100 positive examples, 10 unseen positive examples, and 20 counterexamples across 10 classes.The supplied passage identifies the table’s scope but does not provide its individual measured values.Source: Table I-1 · p. 122

I.2 Dedicated Abduction Experiment Samples (80)

  • 58/80 = 72.5% of dedicated abduction samples satisfy the similarity rule, while mean similarity reaches 0.5117 versus a 0.0181 random baseline, a 28.3× improvement.The 22 failed samples have margins near zero, indicating encoder indistinguishability between true and distractor explanations rather than failure of pseudo-inverse recovery.Source: §I.2 · p. 123

I.3 Physical-World Multimodal Reasoning Samples (55)

  • The physical-world multimodal reasoning appendix preserves per-sample records in the experimental data package and reports aggregate summary metrics in its table.The supplied passage specifies that the dataset contains 55 physics samples but does not provide the summary metric values.Source: §I.3 · p. 123

I.4 End-to-End Deduction Reasoning Samples (60)

  • The 60 end-to-end deduction problems are newly constructed cross-domain samples with zero overlap with the dedicated deduction dataset.Deduction is judged by whether output-to-conclusion cosine similarity exceeds an unrelated-text baseline.Source: §H.5 · p. 121, §I.4 · p. 123

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

The six-class end-to-end induction samples combine two existing classes with four newly added classes, while their sample accounting and one refinement value require qualification.

Class composition

  • The six classes include existing vehicle and color classes plus newly added geometric-shape, academic-discipline, weather, and emotion classes.The existing classes share definitions with the dedicated 10-class induction dataset.Source: §I.5 · p. 123

Sample accounting

  • The six-class samples partially reuse the existing pool and are not counted as newly added deduplicated independent samples.Their item-by-item inventory is incomplete; accounting is provided in Table 1-2 of Section 1.4.Source: §I.5 · p. 123

Refinement

  • The academic-disciplines class has a refinement value of 1.0031, slightly above 1.Its monitoring implication is discussed in Section 15.3.Source: §I.5 · p. 123

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

This section examines 60 end-to-end abduction samples across everyday, mathematical, physical, and chemical domains, reporting judgment outcomes and highlighted solution confidence scores.

End-to-End Abduction Results

  • 81.7% of end-to-end abduction judgments were correct (49/60), compared with 72.5% (58/80) in the dedicated experiment.The two accountings are mutually independent and should not be conflated.Source: §I.6 · p. 123

Judgment Rule

  • The judgment rule counted a sample as correct when abduction similarity exceeded distractor similarity.This criterion produced 11 negative-margin samples among the 60 end-to-end cases.Source: §I.6 · p. 123

Training Distribution

  • The minimum end-to-end margin was −0.0056, while broadened training across 8 domains was associated with improved abduction judgment over the dedicated experiment.The reported improvement is attributed to the broadened training distribution.Source: §I.6 · p. 123

Cross-domain samples

  • End-to-end reasoning produced correct conclusions across business, civic, geometric, arithmetic, mathematical, physical, and chemical examples.Examples include a business owner filing returns, a rectangle having four sides, a force having magnitude, and vinegar having pH below 7.Source: §I.6 · pp. 127–128

Mathematical reasoning

  • 0.9985 confidence was reported for the conclusion that a quadratic has roots.The result follows the supplied premise that all polynomials have roots and that a quadratic is a polynomial.Source: §I.6 · p. 128

Physical reasoning

  • 0.9980 confidence was reported for the conclusion that a force has magnitude.The sample states that a force is a vector and that all vectors have magnitude.Source: §I.6 · p. 128

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

The section presents 20 newly constructed end-to-end compositional reasoning tasks, with category-wise means reported separately in Tables 11-4 and 12-2.

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

  • 20 tasks are newly constructed, while category-wise means for the four task categories appear in Tables 11-4 and 12-2.The means are reported in Section 11.6 and Section 12.3, respectively.Source: §I.7 · p. 129

I.8 Summary of All Test Samples

This section summarizes 503 experimental sample-instances across eight experiment groups, comprising 420 deduplicated independent samples. It distinguishes dedicated and end-to-end abduction evaluations and reports complete success on 20 compositional reasoning tasks.

Sample aggregation

  • 503 experimental sample-instances yielded 420 independent samples after deduplication across dedicated and end-to-end experiments.The accounting is 285 dedicated instances plus 218 end-to-end instances, with 135 newly added deduplicated samples.Source: §I.8 · p. 129

Abduction accounting

  • End-to-end abduction achieved 81.7% judgment accuracy (49/60), versus 72.5% (58/80) in the dedicated experiment.These accountings are independent and use different training configurations: 200 epochs of mixed end-to-end training versus 1000 epochs of standalone training.Source: §I.8 · p. 129, Table I-6 · p. 130

Compositional reasoning

  • 20/20 end-to-end compositional reasoning tasks succeeded across four categories and five reported metrics.The metrics were induction similarity, deduction similarity, contradiction similarity, abduction similarity, and refinement.Source: Table I-7 · p. 132

I.8 Summary of All Test Samples

  • 503 experimental sample-instances comprised 285 + 218 records, yielding 420 deduplicated independent samples across 8 experiment groups.Source: Table I-8 · p. 133

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