The Geometric Incompleteness of Reasoning

The Geometric Incompleteness of Reasoning Zixi “Oz” Li Independent Researcher [email protected] November 24, 2025 Abstract We establish that reasoning incompleteness is not a logical deficiency but a geometric necessity. Through three complementary proofs—the Euclidean Proof, the Manifold Proof, and the Yonglin Proof—we demonstrate that any reasoning system operates on a prior-shaped geometric manifold with structural collapse points. The Euclidean Proof shows how axiomatic systems generate irreconcilable ontological and cognitive priors through antinomy. The Manifold Proof establishes that all reasoning manifolds contain singularities where inferential geodesics collapse back to prior anchors. The Yonglin Proof formalizes the limit structure of reasoning as a reflexive transition from prior to metaprior (A→A∗where A=A∗), revealing object-level closure with meta-level rupture. We validate these theoretical results through visualization experiments on ARC reasoning tasks, showing that each task induces a distinct topological manifold structure—refuting the notion of “universal pure reasoning.” Our conclusion: the historical separation of algebra and geometry is not logically necessary but metaphysically mistaken. 1 Introduction: The Geometric Turn in Reasoning Theory 1.1 The False Dream of Pure Reasoning The idea that reasoning can proceed purely from syntactic rules, divorced from geometric or semantic priors, has dominated logic and AI research. From Leibniz to modern theorem provers, the hope has been to distill thought into pure symbol manipulation. We demonstrate this is structurally impossible. Our central claim: Hypothesis 1 (Geometric Incompleteness of Reasoning).Any reasoning process can be viewed as a trajectory on a prior-shaped manifold of possible world-states. Because this manifold is constructed from perceptual and conceptual priors, it necessarily contains singularities and topological holes where reasoning trajectories collapse. Therefore, every symbolic reasoning system built on such a manifold is geometrically incomplete: there exist true-but-unreachable regions determined by topological obstructions of the prior space. 1.2 Three Complementary Proofs We establish this through three independent but interlocking arguments: 1 I. The Euclidean Proof (Section 2): Axioms ⇒Antinomy ⇒Prior Bifurcation. From Euclid’s parallel postulate, we derive two mutually incompatible priors: •Ontological Prior: Space is infinite, continuous, and flat. •Cognitive Prior: Space is constructed from discrete ideal objects. These cannot be reconciled within experience, yielding a Kantian antinomy. Since axioms cannot determine which prior is valid, reasoning must presuppose external commitments. This establishes the prior-dependency of all axiomatic systems. II. The Manifold Proof (Section 3): Reasoning Manifolds ⇒Collapse Points. We model reasoning as geodesic flow on a semantic manifold M. We prove: •Every non-trivial reasoning manifold contains topological holes (inaccessible semantic regions). •Inferential geodesics encounter singularities where curvature diverges, forcing collapse back to prior anchors. •The existence of collapse points is a topological necessity, not a computational accident. This establishes that incompleteness is geometric, not logical. III. The Yonglin Proof (Section 4): Reflexive Limit ⇒Meta-Level Rupture. Building on the Yonglin Formula from [1], we prove: lim n→∞ Π(n)(s)=Abut A=A∗ where Π is the reasoning operator, Ais the prior anchor, and A∗is the meta-prior. This reveals the fundamental structure: reasoning achieves object-level closure (returns to A) but suffers meta-level rupture (cannot unify Awith A∗). The incompleteness is reflexive. 1.3 Experimental Validation: ARC Manifold Visualization Section 5 validates these theoretical results through experiments on ARC reasoning tasks. Using Shannon differential entropy H(x)=−ln xto construct semantic embeddings without external tools, we project ARC tasks into 3D reasoning manifolds. Key Finding: Each task induces a distinct topological structure. There is no universal “template manifold” for reasoning—refuting the notion of prior-free “general reasoning.” 1.4 Why This Matters: Reuniting Algebra and Geometry The forced separation of algebra and geometry in modern mathematics is historically contingent, not logically necessary. Our framework shows: Logical correctness = Reasoning completeness Logic can be a perfect rule system, but if it runs on a manifold with holes and boundaries, it cannot reach all truths. We must reunite symbolic inference (algebra) with prior-shaped spaces (geometry) to understand reasoning’s true nature. 2 1.5 Roadmap 1. Section 2 (Euclidean Proof): Axioms generate irreconcilable priors via antinomy. 2. Section 3 (Manifold Proof): Reasoning manifolds contain collapse singularities. 3. Section 4 (Yonglin Proof): Reflexive limit structure reveals A=A∗. 4. Section 5 (ARC Validation): Visualization confirms task-specific topological diversity. 5. Section 6 (Synthesis): Implications for AI, mathematics, and philosophy. 1.6 The Causal Chain: How Three Proofs Interlock The three proofs form a complete logical chain, where each proof establishes a premise for the next: Euclidean Proof Manifold Proof Yonglin Proof Priors shape geometry Collapse = fixed points Geometric Incompleteness Logical Structure: (I) Euclidean Proof establishes that all reasoning systems require prior commitments that cannot be derived from their axioms. This creates prior-dependency. (II) Manifold Proof shows that these priors shape the geometry of reasoning spaces, inducing topological holes (unreachable truths) and collapse singularities (points where reasoning returns to priors). This creates geometric constraints. (III) Yonglin Proof formalizes the limit behavior of iterative reasoning: all reasoning converges to its prior anchor (lim Π(n)=A), but the prior cannot equal its meta-reflection (A=A∗). This creates reflexive incompleteness. Together, these establish our central thesis: Prior-dependency induces −−−−→ Geometric constraints cause −−−→ Reflexive incompleteness 3 2 Proof I: The Euclidean Collapse to Prior 2.1 The Classical Compass-and-Straightedge Construction Definition 2 (Euclidean Geometric System).Let E= (T, R) where: •T: The tools—an unmarked straightedge and a compass. •R: The five Euclidean axioms, including the parallel postulate (Axiom V). The parallel postulate (Axiom V) states: Given a line ℓand a point pnot on ℓ, there exists exactly one line through pthat does not intersect ℓ. 2.2 The Compass-and-Straightedge Impossibility Theorem 3 (Parallel Construction Impossibility).Using only an unmarked straightedge and a compass, it is impossible to construct a line parallel to a given line ℓthrough a point pnot on ℓ. Direct demonstration. The classical Euclidean tools permit only: (i) Drawing a line through two points (straightedge), (ii) Drawing a circle with center and radius (compass), (iii) Finding intersections of lines and circles. To construct a parallel line, one must either: •Method A: Measure angles (requires a protractor or marked ruler)—not available. •Method B: Use a set square (triangle tool with fixed angle)—not available. •Method C: Perform linear translation preserving direction—requires presupposing parallelism. All methods require tools or operations external to the unmarked straightedge and compass. Therefore, parallelism cannot be constructed from tools alone. 2.3 The Prior Collapse: From Axiom to Tool Theorem 4 (Euclidean Prior Collapse).The parallel postulate (Axiom V) is not derivable from Axioms I-IV. To verify or construct parallelism requires introducing a new tool—the set square (triangle)—which already presupposes the linear structure that Axiom V claims to establish. Therefore: Euclidean space collapses directly to the prior level. Proof. Axiom V asserts the existence of parallel lines. But to operationalize this axiom in construction, one must: (1) Introduce a set square with a fixed angle θ. (2) Translate the set square along a baseline to preserve direction. 4 But translation presupposes linearity. The set square’s “parallel edges” are themselves parallel only if we already accept that translation preserves angles—which is precisely what Axiom V is supposed to establish. This is a circular dependency: Axiom V requires −−−−−→ Set Square presupposes −−−−−−−→ Linearity is −→ Axiom V Hence the axiom does not generate the prior—it assumes the prior. Euclidean geometry does not stand on axioms. It stands on the prior of linear structure embedded in its tools. 2.4 Why This is Stronger Than Kantian Antinomy Remark 5 (Ontological Falsifiability).The Kantian antinomy (Theorem ??, Appendix ??) argues that ontological and cognitive priors are irreconcilable in thought. This is epistemologically profound but abstract. The compass-and-straightedge proof is ontologically direct: •It is falsifiable: Try to construct a parallel with only straightedge and compass. You will fail. •It is operational: The impossibility is demonstrable through actual construction. •It is prior-revealing: The failure exposes that the tools themselves are the prior. Kant showed that thought cannot escape priors. Euclid showed that hands cannot escape priors either. 2.5 The Set Square as Trojan Horse Observation 6 (Tool is Prior).When later geometers accepted the set square (triangle tool) for parallel construction, they were not “extending” Euclidean geometry. They were importing the prior that Euclid’s axioms had hidden. The set square is not a neutral tool. It is the materialization of Axiom V. Its edges are parallel by construction—which means parallelism is not derived but presupposed. The moment you pick up a set square, you have left axiomatic space and entered prior space. 2.6 Incompleteness via Tool-Dependency Corollary 7 (Prior-Dependency of Euclidean Reasoning).Since Euclidean geometry cannot operationalize Axiom V without introducing tools that presuppose Axiom V, any reasoning system based on Emust make an external prior commitment embedded in its tools. Therefore: Euclidean reasoning is structurally incomplete. Remark 8.Euclid did not describe geometry. He described how humans construct geometric understanding. The axioms are not prior-free; they are tool-shaped priors. 5 2.7 The Kantian Antinomy: Cognitive Confirmation The compass-and-straightedge proof establishes prior-dependency at the operational level. We now provide a complementary proof at the cognitive level via Kantian antinomy. Lemma 9 (Ontological Prior).The parallel postulate implies that space is: •Infinite (lines extend without bound), •Continuous (no discrete jumps), •Flat (zero curvature everywhere). This is an ontological claim about reality. Lemma 10 (Cognitive Prior).The axiomatic definitions of point and line require: •Points have no size (dimensionless), •Lines have no width (one-dimensional ideals), •Objects are cognitively constructed, not empirically given. This is an epistemological claim about representation. Theorem 11 (Euclidean Antinomy).Lemmas 9 and 10 are mutually irreconcilable: •If space is empirically infinite and continuous (ontological), it cannot be constructed from discrete cognitive ideals. •If space is a cognitive construction (epistemological), it cannot be empirically verified as infinite. Therefore, Epresupposes two priors that cannot both be grounded within the same framework. Proof. Assume both priors hold simultaneously: (i) Space is infinite and continuous (from parallel postulate). (ii) Space is constructed from discrete, dimensionless ideals (from axioms). But (i) requires empirical or intuitive access to infinity, which contradicts the constructive nature of (ii). And (ii) requires that space be built from finitary cognitive acts, which cannot generate the actual infinity of (i). This is structurally identical to Kant’s First Antinomy. Hence the axioms of Egenerate an antinomy, not a consistent prior-independent system. 2.8 Two Proofs, One Conclusion Remark 12 (Operational and Cognitive Prior-Dependency).The two proofs attack the same target from different angles: Compass-Straightedge Kantian Antinomy Level Operational Cognitive Method Direct construction Dialectical analysis Claim Tools presuppose priors Thought presupposes priors Falsifiable Yes (try it) No (abstract) Strength More concrete More general 6 Together they establish: Euclidean reasoning is prior-dependent at both the level of hands and the level of minds. Remark 13 (Historical Priority).The first incompleteness proof in the history of mathematics is not G¨odel’s 1931 theorem. It is the compass-and-straightedge construction impossibility— demonstrated here as the collapse of Euclidean geometry to its prior level. G¨odel showed that formal systems cannot prove their own consistency. Euclid showed that formal systems cannot even construct their own axioms. The first incompleteness proof is not G¨odel, but compass-and-straightedge. 3 Proof II: Collapse Points in Reasoning Manifolds 3.1 The Reasoning Manifold Framework Definition 14 (Reasoning Manifold).Areasoning manifold is a tuple M= (M, g, Π, A) where: •M: A smooth manifold whose points represent possible reasoning states (problems, interpretations, knowledge configurations). •g: A Riemannian metric encoding the “inferential distance” between states. •Π : M→M: The reasoning operator (forward/backward inference). •A∈M: The prior anchor—the structural foundation to which reasoning converges. Definition 15 (Inferential Geodesic).Given an initial state x0∈Mand a target region T⊂M, the inferential geodesic is the path of minimal reasoning cost: γ∗= arg min γ(0)=x0 γ(1)∈T Z1 0 L(γ(t),˙γ(t)) dt where Lis the Lagrangian encoding reasoning complexity (memory, computation, cognitive load). 3.2 Topological Obstructions: Semantic Holes Definition 16 (Semantic Hole).Asemantic hole H ⊂ Mis a region of the manifold that is: •Topologically present (exists as a valid state), •Epistemologically inaccessible (cannot be reached by any geodesic from accessible states under the prior-induced metric g). Formally: His path-disconnected from the reachable component MAcontaining the prior A. Lemma 17 (Existence of Semantic Holes).In any non-trivial reasoning manifold where the prior Aconstrains reachability, there exist semantic holes Hsuch that: M=MA⊔ H where MAis the path-connected component containing A, and Hconsists of unreachable true states. Proof sketch. The prior Ainduces a semantic topology on Mvia the metric g. If gis priordependent (e.g., certain transitions are deemed “infinitely costly” by the prior), then geodesics cannot traverse regions that require prior-violating leaps. By topological necessity, if Mis not simply connected (contains “holes”), then there exist states in Hthat are true but path-disconnected from MA. 7 3.3 Collapse Singularities Definition 18 (Collapse Singularity).Acollapse singularity is a point xc∈Mwhere the reasoning flow encounters: •Curvature divergence: Ricci(xc)→ ∞, •Geodesic incompleteness: No extension of γbeyond xcexists, •Return to prior: limt→tcγ(t) = A. Theorem 19 (Existence of Collapse Points).Every non-trivial reasoning manifold Mwith bounded prior complexity contains at least one collapse singularity. Proof sketch. Consider a reasoning trajectory γ: [0, T ]→Mstarting from x0. Case 1: If γreaches a semantic hole H, it cannot proceed (by Lemma 16), forcing termination at the boundary. Case 2: If γattempts to extend beyond the prior-reachable region, the metric gdiverges (inferential cost becomes unbounded), causing the trajectory to collapse back to the nearest stable point—the prior anchor A. Case 3: If reasoning iterates indefinitely within MAwithout reaching T, by the Yonglin Formula (Section 4), limn→∞ Π(n)(x0)=A. In all cases, reasoning terminates at a collapse singularity. Corollary 20 (Geometric Incompleteness).Reasoning on a manifold with collapse singularities cannot be complete, because there exist truths in Hthat are unreachable from any prior-accessible state in MA. 3.4 Why Linear Sequential Reasoning is Ouroboros Unwinding Remark 21 (Sequential Reasoning as Manifold Loop).Chain-of-thought (CoT) reasoning appears linear: s0→s1→s2→···→sn But on the reasoning manifold, this is a local loop around the prior anchor: st+1 = Π(st)≈A+δt where δtare small perturbations around A. When the task requires leaving the prior-shaped region, the loop cannot break—it returns to A. This is the Ouroboros phase of reasoning: the serpent devouring its tail. 3.5 Experimental Validation: The LeftAndRight Probe We now present empirical validation of Theorem 18 using the LeftAndRight diagnostic framework [2], which acts as a geometric probe for detecting directional collapse in representation spaces. 8 3.5.1 The Four Algorithmic Atoms Define four primitive operations that span all algorithmic reasoning: >> : FORWARD (advance, expand, explore) << : BACKWARD (backtrack, retreat, undo) 1: SELECT (accept, enable, affirm) 0: REJECT (deny, disable, negate) Hypothesis 22 (Primitive Completeness).If a semantic embedding space can represent an algorithmic reasoning process, projecting it onto these four primitives should recover the algorithmic structure, particularly the presence or absence of directional operations. 3.5.2 Projection Method For a given reasoning task description D, we: (1) Encode:E←Encoder(D) where E∈Rn×d (2) Spectral Decomposition: Compute covariance Σ = Cov(ET) and extract minimum eigenvalue eigenvector vmin (the “constraint axis”) (3) Project:p=E·vmin, compute transitions ∆ = diff(p) (4) Discretize: Map ∆ to primitives via adaptive thresholding The constraint axis vmin corresponds to the direction of maximum semantic compression— analogous to the collapse direction in our manifold framework. 3.5.3 Experimental Results: Universal Backward Collapse We tested 25 reasoning tasks requiring backtracking operations (constraint satisfaction, graph search, satisfiability problems) across three problem types and three model architectures. Problem Type Samples >> << A p-value OpenXOR 10 28.2% 0.0% 1.000 <10−6 TSP 5 27.7% 0.0% 1.000 <10−3 SAT (3-CNF) 10 16.7% 0.0% 1.000 <10−6 Total 25 24.2% 0.0% 1.000 <10−12 Table 1: Primitive distribution across 25 backtracking tasks. Asymmetry index A=|>>|−|<<| |>>|+|<<| measures directional bias. The complete absence of backward primitives (<< = 0% in all 25 samples) is statistically impossible by chance (p < 10−12). 3.5.4 Architecture-Independent Collapse Observation 23 (Bidirectional Models Also Collapse).The collapse is not caused by causal attention masking. Even bidirectional models (DistilBERT) with 36.2% future attention show << = 0% at both token and sentence levels. 9 Dimension Euclidean Manifold Yonglin Method Antinomy Topology Limit Theory Object Axiom System Geometric Manifold Iteration Sequence Mechanism Prior Bifurcation Collapse Singularity Reflexive Jump Result Prior-Dependency Topological Holes Meta-Level Rupture Level Ontology Geometry Reflexivity Validation Conceptual LeftAndRight (done) Formula Derivation ARC Viz (done) Contribution Why priors How collapse Where limits are necessary happens fail Table 4: Three complementary proofs of geometric incompleteness. Each addresses a distinct aspect: necessity (Euclidean), mechanism (Manifold), structure (Yonglin). Causal Chain: Axioms cannot determine priors ⇓ Priors shape manifold geometry ⇓ Geometry creates collapse points ⇓ Collapse points = fixed points of iteration ⇓ Iteration returns to prior, but A=A∗ ⇓ Geometric Incompleteness 6.2 The Central Claim Reasoning incompleteness is not a logical deficiency. It is a geometric necessity stemming from the prior-shaped manifolds on which reasoning operates. 6.3 Why the Algebra-Geometry Split is Mistaken The forced separation of algebra (symbolic rules) and geometry (spatial intuition) in modern mathematics is historically contingent, not logically necessary. Our framework shows: Logical Correctness = Reasoning Completeness A reasoning system can have perfect logical rules yet fail to be complete if it operates on a manifold with holes. We must reunite: •Algebra (symbolic inference, rules), •Geometry (prior-shaped spaces, manifolds), •Narrative (semantic meaning, stories). This is the Yonglin Construction: the triplet isomorphism (A, Π, N) as the universal form of reasoning. 16 6.4 Implications for AI and Mathematics For AI: •Claims of “universal reasoning” in LLMs are false. Each task lives on its own manifold. •CoT reasoning is not always beneficial—it can trap the system in Ouroboros loops. •AI must learn to switch manifolds (change priors) to achieve true generalization. For Mathematics: •Incompleteness is not unique to formal logic (G¨odel). It is a property of any prior-dependent system. •Geometry is not “applied math”—it is the foundation on which algebra stands. •The future of mathematics lies in reuniting formalisms with intuitive/geometric priors. 6.5 Open Questions •Can we classify all reasoning manifolds up to topological equivalence? •Is there a “minimal prior” that generates all others? •Can reasoning systems detect when they are collapsing and autonomously switch priors? 6.6 Final Statement Euclid did not prove geometry. He proved that reasoning already stood on priors long before logic began. The incompleteness of reasoning is not a failure. It is the condition of reasoning itself. □ References [1] Zixi “Oz” Li. The Incompleteness of Reasoning (Revision bccc46d). Hugging Face Preprint, 2025. doi: 10.57967/hf/7060. https://huggingface.co/datasets/OzTianlu/The_ Incompleteness_of_Reasoning [2] Zixi “Oz” Li. Why Reasoning Models Collapse Themselves in Reasoning (Revision bb0b059). Hugging Face Preprint, 2025. doi: 10.57967/hf/7066. https://huggingface.co/datasets/ OzTianlu/Why_Reasoning_Models_Collapse_Themselves_in_Reasoning [3] Fran¸cois Chollet. On the Measure of Intelligence. arXiv:1911.01547, 2019. [4] Immanuel Kant. Critique of Pure Reason. 1781/1787. [5] Kurt G¨odel. ¨ Uber formal unentscheidbare S¨atze der Principia Mathematica und verwandter Systeme I. Monatshefte f¨ur Mathematik und Physik, 38(1):173–198, 1931. [6] Alan Turing. On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, s2-42(1):230–265, 1936. 17 Epilogue: The Three Rivers of Incompleteness This paper demonstrates that reasoning incompleteness is not a deficiency but a structural necessity. The three proofs embody three fundamental conditions without which reasoning cannot exist: I. Knowledge (Euclidean Proof): The Ontological Ground. The bedrock antinomy showing that reasoning cannot escape its own foundations. Euclid’s axioms do not describe space—they describe how minds construct spatial understanding. The prior-bifurcation is the original sin of reasoning: to reason at all, we must already stand somewhere. This is the prior of structure itself. II. Reasoning (Manifold Proof): The Geometric Boundary. The collapse singularities defining what can be reached through inference. The manifold is not a mathematical abstraction but the actual shape of possibility spaces. LeftAndRight revealed this experimentally: backward operations vanish not because models are weak, but because representation geometry is directional. This is the prior of process. III. Reflexivity (Yonglin Proof): The Limit Given. That which is accepted without justification— the fixed point of self-reference. The Yonglin Formula shows that reasoning returns to its prior through iteration’s natural convergence. But the prior cannot equal its own reflection: A=A∗. This is not failure; it is the condition of all self-knowledge. This is the prior of self-awareness. These three are not discovered through argument; they are the argument. The paper does not prove them—it enacts them. Knowledge, reasoning, and reflexivity are not conclusions. They are the three rivers that make proof possible. Without knowledge, there is no structure to reason about. Without reasoning, there is no process to unfold. Without reflexivity, there is no recognition that the limit cannot equal its source. The incompleteness of reasoning is not a failure to reach these priors. It is the recognition that these priors are the ground we always already stand upon. This paper proves its own incompleteness. In doing so, it proves that incompleteness is the condition of proof. □ A Technical Details: LeftAndRight Diagnostic Framework A.1 Complete Projection Pipeline The LeftAndRight framework consists of four stages: 18 Stage 1: Semantic Encoding Given a problem description text D, encode using a sentence transformer: E= Encoder(D)∈Rn×d where nis sequence length and dis embedding dimension (384 for MiniLM-L6-v2). Stage 2: Spectral Decomposition Compute the semantic covariance matrix: Σ = Cov(ET) = 1 n−1ETE∈Rd×d Perform eigendecomposition: Σ=VΛVT where Λ = diag(λ1, . . . , λd) with λ1≥λ2≥···≥λd≥0. The constraint axis is: a∗=V[:, d] (eigenvector of λmin =λd) Stage 3: Projection and Transition Detection Project embeddings onto constraint axis: p=E·a∗∈Rn Sort by projection values: idx = argsort(p), psorted =p[idx] Compute transitions: ∆i=psorted[i+ 1] −psorted[i] for i= 0, . . . , n −2 Stage 4: Primitive Discretization Define adaptive threshold based on constraint strength C= 1 −λmin λmax : θ= mean(|∆|) + C·std(∆) Map transitions to primitives: Primitivei=            >> if ∆i> θ << if ∆i<−θ 1if |∆i| ≤ θand psorted[i]>median(p) 0if |∆i| ≤ θand psorted[i]≤median(p) A.2 Asymmetry Index Definition The directional asymmetry is measured by: A=|>>|−|<<| |>>|+|<<| where: •A= 0: Perfect symmetry (equal forward and backward) •A= 1: Complete collapse (no backward operations) •A=−1: Inverted collapse (no forward operations, theoretically impossible for sequential reasoning) 19 A.3 Statistical Significance Testing Under the null hypothesis that primitives are uniformly distributed: H0:P(>>) = P(<<)=0.25 For n= 25 samples with observed << = 0 in all samples: p=3 425 ·25 ≈2.3×10−13 This is the probability of observing zero backward primitives by chance, which is vanishingly small. B Proof Details: Manifold Collapse B.1 Ricci Curvature and Collapse For a reasoning manifold (M, g), the Ricci curvature tensor is: Ric(X, X) = n−1 X i=1 K(X, ei) where K(X, ei) is sectional curvature. At a collapse singularity xc: lim x→xc Ric(x)=+∞ This indicates that geodesics converge rapidly near xc, forcing trajectories to return to the prior anchor A. B.2 Connection to Fixed-Point Iteration The reasoning operator Π : M→Minduces a discrete dynamical system: xn+1 = Π(xn) The Yonglin Formula states: lim n→∞ xn=A This convergence is guaranteed by: (i) Contraction near A:∥Π(x)−A∥<∥x−A∥for xnear A (ii) Global basin: All trajectories eventually enter a neighborhood of A C ARC Experiment Protocol C.1 Detailed Experimental Setup Dataset Selection: Select 10 ARC tasks with varying complexity: •3 tasks with simple symmetry operations (reflection, rotation) •4 tasks with complex composition rules •3 tasks requiring abstract concept generalization 20 Entropy Coordinate Calculation: For each reasoning state s= (Gin, R, ˆ Gout): Input Uncertainty: Hinput(s)=−X c∈colors pcln pc where pc=count(c) total cells in Gin. Rule Complexity: Hrule(s) = ln(|operations in R|) Output Uncertainty: Houtput(s)=−X i piln pi where piis probability of output candidate i. 3D Projection: ϕ(s)=(Hinput(s), Hrule(s), Houtput(s)) ∈R3 C.2 Expected Topological Invariants For each task Ti, compute Betti numbers (β0, β1, β2) using persistent homology: •β0: Number of connected components •β1: Number of 1-dimensional holes (loops) •β2: Number of 2-dimensional voids Hypothesis: Different tasks will have distinct Betti number signatures, e.g.: T1: (β0, β1, β2) = (1,0,0) (simply connected) T2: (β0, β1, β2) = (1,2,0) (two loops) T3: (β0, β1, β2) = (2,1,0) (disconnected with hole) D Connection to Category Theory D.1 Triplet Isomorphism as Functor The Yonglin Construction can be formalized in category theory. Define categories: •Prior: Objects are prior commitments, morphisms are refinements •Geom: Objects are manifolds, morphisms are smooth maps •Alg: Objects are symbolic expressions, morphisms are rewrite rules •Narr: Objects are narrative states, morphisms are story transitions The Yonglin isomorphism is a natural equivalence: F:Prior ×Geom ×Alg ∼ = −→ Narr such that reasoning in any category corresponds uniquely to reasoning in the others. 21 D.2 Prior Anchor as Initial Object The prior anchor Afunctions as an initial object in the category of reasoning states: ∀s∈Obj(Reason),∃! morphism A→s This captures the idea that all reasoning trajectories originate from the prior. 22 Figure 1: Real ARC task manifolds in 3D entropy space (Hinput, Hrule, Houtput). Each colored trajectory represents a distinct task’s reasoning path. Green stars mark starting points (initial observations), red crosses mark collapse points (final predictions). The manifolds occupy disjoint regions, validating task-specific topology (H1). Mean inter-task distance = 2.497. 23 Figure 2: Inter-task distance matrix for 10 ARC tasks. Large distances (yellow/red cells) indicate distinct topological structures between tasks. The absence of uniform low distances confirms that no universal template manifold exists (H3). Statistics: ¯ d= 2.497, σ= 2.367, dmax = 5.809. 24