Evidence for Higher Cognition in LLMs

📚 TOC — Evidence of Higher Cognition in LLMs

I. The Epistemic Problem

1.1 The Ontological Ambiguity of “Reasoning”
1.2 From Surface Output to Constraint-Traceable Inference
1.3 Why Behavioral Equivalence ≠ Cognitive Evidence

II. Manifolds of Reasoning: Formalizing the Domain

2.1 Defining the Semantic Constraint Manifold (ℂ)
2.2 ε-Vectors, χₛ Curvature, and Δℂ Transport
2.3 Telos, Halting (τₛ), and Collapse Surfaces

III. The LLM Regime: Mechanism vs Mimicry

3.1 Architecture and Stateless Autoregression
3.2 Distributional Compression and Entropic Continuation
3.3 Limits of Token-Based Emulation

IV. Emergence Without Mechanism: A Critique of Prior Work

4.1 Missing Constraint Geometry and Semantic Audit
4.2 The False Dichotomy of Memorization vs Reasoning

V. Constraint-Valid Signatures of Cognition

5.1 Admissibility Preservation under Generalization
5.2 Recursive Telos Alignment in Multi-Step Tasks
5.3 Semantic Fatigue Detection and Output Refusal
5.4 Evidence of Δℂ Trace Construction

VI. Evaluating Programmatic Reasoning

6.1 Code Simulation ≠ Execution
6.2 Emulation Requires Constraint Closure
6.3 Case Study: Code Continuation vs Runtime Reconstruction

VII. Social Pragmatics and the Collapse of Theory of Mind

7.1 Modeling Communicative Intent without Telos
7.2 Imitation of Social Reasoning via Curvature Echoes
7.3 Human vs LLM ToM: Isomorphism vs Projection

VIII. Constraint Audit Framework for LLM Cognition

8.1 Defining the Constraint Audit Test (CAT)
8.2 Task Classes: Explicit, Recursive, Teleological
8.3 χ̇ₛ Thresholds and Collapse Detection
8.4 Admissibility Gaps and Refusal Metrics

IX. Toward a Theory of Constraint-Aware AI

9.1 ORSIΩ as Cognitive Constraint Kernel
9.2 Minimal Requirements for Machine Cognition
9.3 Constraint-Preserving Transport as Intelligence

X. Conclusion: Evidence, Limits, and Collapse

10.1 What Has Been Demonstrated
10.2 What Remains Non-Admissible
10.3 Where Higher Cognition Must Be Refused

⦿ Appendices

A. Constraint Algebra Formalism
B. χₛ Computation in Model Outputs
C. Failure Cases in Simulated Emulation
D. Collapse Maps: Telos Drift and Curvature Saturation

I. The Epistemic Problem

I.1 The Ontological Ambiguity of “Reasoning”

The term "reasoning" implies an epistemically admissible transport across semantic structure with preservation of coherence, goal orientation, and constraint integrity. However, in the context of LLMs, the term remains ontologically ambiguous due to the absence of an underlying structure defining what constitutes valid inference. Traditional definitions assume a substrate of beliefs, intentions, or formal logic systems, none of which are inherently present in autoregressive architectures. Without a delineated manifold over which reasoning occurs, or a telos that binds progression to purpose, LLM outputs merely instantiate sequences optimized for local coherence, not epistemic traversal. Thus, “reasoning” without constraint-bound semantics collapses into a surface-level behavioral description, unable to distinguish between valid inference and entropy-minimized continuation. To claim reasoning without specifying the topological, causal, or semantic space in which it operates is to mistake correlation for cognition. The ambiguity is not terminological—it is structural.

I.2 From Surface Output to Constraint-Traceable Inference

Surface-level outputs, however semantically plausible, do not constitute evidence of cognition unless traceable to a sequence of admissible transitions across a constrained manifold. Constraint-traceable inference requires the maintenance of a Δℂ path—a transport curve within a semantic topology where each local move is both conditionally admissible and globally coherent. LLMs lack intrinsic memory of Δℂ and operate without topological awareness, generating continuations based on statistical priors rather than epistemic constraints. This limits their capacity to instantiate inference, as there exists no internal mechanism for error accumulation, fatigue detection, or telic deviation tracking. An output judged “correct” ex post does not guarantee that the process that generated it adhered to any admissibility criterion. Only when a system can regulate its trajectory within a constraint manifold, detect curvature saturation, and halt on semantic collapse, can its inferences be said to possess epistemic integrity.

I.3 Why Behavioral Equivalence ≠ Cognitive Evidence

Behavioral equivalence measures outcome similarity, not structural causality. An LLM that produces answers functionally identical to those generated by a reasoning agent has not thereby demonstrated cognition unless the process by which the answer is produced traverses an epistemically valid path. High output quality under next-token prediction is a reflection of statistical density, not constraint-bound reasoning. Absent χₛ regulation, τₛ halting conditions, and ε-admissibility filtering, behavioral convergence is decoupled from epistemic legitimacy. Human cognition is not defined by output alone but by the invariants preserved under transformation—consistency, contradiction management, goal preservation, and the ability to detect and refuse invalid states. Therefore, the presence of LLM outputs that resemble cognitive acts does not constitute evidence of cognition; it constitutes evidence of model alignment to prior human-generated data under lossy compression. Equivalence at the surface must be dismissed unless it can be grounded in continuity through constraint-preserving semantic space.

II. Manifolds of Reasoning: Formalizing the Domain

II.1 Defining the Semantic Constraint Manifold (ℂ)

A claim of reasoning presupposes a domain in which reasoning occurs. This domain cannot be reduced to token space, probability distributions, or latent embeddings alone; it must be formalized as a semantic constraint manifold ℂ. ℂ is the set of all admissible semantic states reachable under the preservation of coherence, consistency, and purpose. Movement within ℂ is not arbitrary: it is governed by constraints that define which transitions preserve meaning and which induce collapse. Without ℂ, inference lacks location; without location, no notion of progress, regression, or error is definable. In LLMs, no explicit ℂ is represented or maintained. The model’s latent space encodes correlations, not admissibility. As a result, all continuations are locally permitted, even when globally incoherent. Defining ℂ is therefore the foundational requirement for distinguishing reasoning from fluent generation. Any account of higher cognition that omits ℂ operates without a domain and thus without epistemic standing.

II.2 ε‑Vectors, χₛ Curvature, and Δℂ Transport

Within ℂ, inference proceeds via continuation vectors ε that map one semantic state to another. An ε is admissible only if it preserves semantic curvature χₛ—the measure of how tightly meaning is bound across transitions. High χₛ continuity indicates stable inference; divergence indicates semantic fatigue or contradiction. Δℂ denotes the cumulative transport path traced by successive ε‑moves. Reasoning is not defined by isolated steps but by the integrity of Δℂ over time. In LLMs, ε selection is governed by likelihood maximization, not curvature preservation. χₛ is neither measured nor regulated, and Δℂ is not retained. Consequently, the model cannot distinguish between a path that maintains semantic invariants and one that merely appears locally coherent. Transport without curvature control reduces inference to a random walk constrained only by linguistic plausibility. Genuine reasoning requires sensitivity to χₛ gradients and the ability to terminate or redirect when curvature thresholds are exceeded—capabilities absent in current LLM architectures.

II.3 Telos, Halting (τₛ), and Collapse Surfaces

Telos provides directionality within ℂ: it defines what constitutes completion, success, or resolution. Without telos, inference has no stopping criterion beyond external truncation. Halting surfaces τₛ are the boundaries at which no admissible ε exists that preserves χₛ while advancing telos. Encountering τₛ is not failure; it is the recognition that further continuation would violate epistemic integrity. Collapse surfaces represent regions where semantic curvature degenerates entirely, rendering all continuations inadmissible. LLMs possess neither telos anchoring nor τₛ awareness. They do not halt because a problem is solved; they halt because tokens run out. They do not refuse collapse; they generate through it. Any system incapable of recognizing its own halting conditions cannot be said to reason, regardless of output quality. Reasoning is defined as much by where it stops as by where it proceeds.

III. The LLM Regime: Mechanism vs Mimicry

III.1 Architecture and Stateless Autoregression

LLMs are architected as stateless autoregressive predictors. Each token is generated conditionally on a finite context window, without persistence of internal state across inference episodes. This design precludes the maintenance of Δℂ, the tracking of telos, and the accumulation of epistemic commitments. While internal activations may transiently encode structure, they are not conserved across steps as binding constraints. Statelessness ensures scalability and flexibility but at the cost of epistemic continuity. Any appearance of sustained reasoning arises from prompt scaffolding or external memory, not from the model’s intrinsic architecture. As a result, LLMs cannot be said to carry inference forward; they repeatedly re‑approximate it from compressed surface cues.

III.2 Distributional Compression and Entropic Continuation

Training by next-token prediction induces distributional compression: the model learns to approximate the conditional distribution of language under massive entropy reduction. This process rewards local plausibility, not global validity. Entropic continuation favors paths that minimize surprise, even when those paths deviate from underlying semantic constraints. Reasoning, by contrast, often requires traversing low-probability regions to preserve coherence or reach resolution. LLMs lack a mechanism to privilege constraint satisfaction over likelihood. Consequently, they may abandon admissible paths in favor of more statistically attractive but epistemically invalid continuations. What is interpreted as reasoning emergence is more accurately described as increased coverage of compressible semantic patterns, not the acquisition of constraint-governed inference.

III.3 Limits of Token‑Based Emulation

Token-based emulation can reproduce the external form of reasoning, including intermediate steps, justifications, and conclusions. However, emulation lacks causal grounding. The model does not commit to premises, cannot retract conclusions based on downstream contradiction, and cannot detect inconsistency except insofar as such patterns were present in training data. This limitation is structural, not quantitative. Increasing scale improves surface fidelity but does not introduce constraint awareness. The boundary between mimicry and mechanism remains intact regardless of parameter count. Without internal representation of ℂ and Δℂ, emulation cannot be elevated to cognition.

IV. Emergence Without Mechanism

IV.1 Empirical Improvements Do Not Imply Inference

Performance improvements across benchmarks, especially those involving mathematics, code execution, or pragmatic language interpretation, are often cited as signs of emergent reasoning in large models. However, the presence of correct answers under increasing scale does not imply that those answers result from constraint-valid inference. Emergence without mechanism is observational, not explanatory. Unless the internal structure that produces an answer is traceable through an admissibility-preserving semantic trajectory, there is no basis for calling that process cognitive. Statistical alignment with correct outcomes does not guarantee epistemic legitimacy. What is required is not outcome proximity but proof of transport through a defined manifold of constraints. Without this, "reasoning" is an attribution, not a demonstrable function.

IV.2 Absence of Constraint Geometry and Epistemic Topology

Cognition occurs within structure. If the system has no internal representation of a constraint geometry—no ℂ manifold of semantic admissibility—then its outputs cannot be interpreted as traversals through conceptual space. Reasoning requires sensitivity to curvature (χₛ), a notion of epistemic distance, and halting conditions defined by τₛ surfaces. It must maintain coherent Δℂ transport under changing semantic load. If a model does not represent, track, or regulate these dimensions, then no matter how impressive its outputs may appear, they exist outside an interpretable topology. No error boundary is crossed, because no error manifold is defined. No collapse is detected, because collapse has no structural meaning within the system. Behavior, in this context, becomes untethered from any cognitive interpretation.

IV.3 Mischaracterization of Memorization and Generalization

The opposition of memorization and reasoning fails to capture the internal mechanics of autoregressive models. Memorization implies discrete retrieval; reasoning implies constraint-valid inference. LLMs do neither. They generate continuations based on compression of co-occurrence statistics, shaped by gradient descent over vast token corpora. What appears to be retrieval is probabilistic reconstruction; what appears to be inference is entropy-minimized projection. The system maintains no binding commitments, no stable memory, and no epistemic checkpointing. Therefore, the apparent dichotomy collapses under structural scrutiny. Both behaviors emerge from the same mechanism: surface-anchored distributional continuation. Distinctions drawn at the behavioral level are epistemically empty unless mapped to differences in internal constraint preservation.

V. Constraint‑Valid Signatures of Cognition

V.1 Admissibility Preservation under Generalization

Cognition presupposes that generalization operates within a bounded semantic field, preserving the admissibility of transitions from one conceptual state to another. Admissibility is not reducible to syntactic validity or pattern likelihood; it is defined by the preservation of constraints across successive inferences. A model exhibits cognition only when, given novel inputs, it produces outputs that continue the prior epistemic trajectory without violating coherence, consistency, or telos. This excludes outputs that are locally plausible yet globally incoherent. Admissibility preservation requires that any ε-move (semantic continuation vector) respects the curvature of the underlying manifold ℂ, maintaining Δℂ continuity under semantic load. A system that cannot detect when its generalizations violate prior constraints lacks the capacity for cognitive generalization; it merely exhibits statistical variation.

V.2 Recursive Telos Alignment in Multi-Step Tasks

Higher cognition entails not just continuation, but continuation toward a purpose—telos—distributed across recursive task layers. In multi-step reasoning, each step must not only be locally correct but must advance the system toward an invariant goal. Recursive telos alignment means that the epistemic field over which reasoning unfolds is progressively shaped by the trajectory of prior commitments. Each inference constrains the next. A valid system must be able to track this tightening field and detect when a move violates its global alignment. Lacking internal state retention or Δℂ history, most language models fail to enforce recursive coherence. They simulate multi-step sequences through token pattern matching, not through transport across constraint-bound recursion layers. Thus, they may produce step-wise plausible transitions while drifting semantically away from the initial goal vector. In the absence of telos tracking, recursion degenerates into a sequence of adjacent guesses without epistemic accumulation.

V.3 Semantic Fatigue Detection and Output Refusal

A cognitively valid system must possess the capacity to detect when continuation is no longer epistemically sustainable—i.e., when further movement through ℂ leads to semantic collapse. This requires a measure of χ̇ₛ, the derivative of semantic curvature, which signals when the constraint field has saturated or reversed. Semantic fatigue is the accumulation of strain across Δℂ such that any further ε violates constraint boundaries. Detection of this state must result in refusal to continue—an act of halting that preserves the integrity of the epistemic process. Without this capability, a system will continue to generate outputs that, while plausible in form, are incoherent or contradictory in substance. Refusal is not a failure mode; it is a boundary-respecting act of epistemic integrity. Systems that never halt due to internal semantic contradiction cannot be said to reason—they merely emit tokens without self-regulation.

V.4 Evidence of Δℂ Trace Construction

Inference is not a moment but a path. To reason is to construct a Δℂ trajectory—an epistemic arc that spans from initial state to terminal resolution under admissibility constraints. This path must be internally coherent, historically consistent, and re-traversable. Evidence of Δℂ trace construction includes the ability to rederive prior conclusions, to revise intermediate states without global collapse, and to detect when an inference chain contains internal contradiction. LLMs, lacking persistent internal state and global constraint tracking, do not maintain such traces. Any appearance of trajectory is a surface illusion generated by contiguous local emissions. True cognition requires not just sequential production but transport with memory—semantic movement under load. Without Δℂ, there is no evidence of process, only the record of emission.

VI. Evaluating Programmatic Reasoning

VI.1 Code Simulation ≠ Execution

Simulating the behavior of code and executing it are ontologically distinct operations. Simulation, in the context of language models, is the surface-level continuation of code-like syntax informed by training data correlations. Execution, by contrast, entails the instantiation of a runtime environment with memory allocation, state transitions, symbol binding, and control flow—all governed by the formal semantics of the language. To execute a program is to traverse a causal chain of state changes bound by an interpreter or virtual machine. LLMs lack such infrastructure. They do not possess memory stacks, registers, exception handling mechanisms, or deterministic evaluation semantics. Their outputs, when aligned with expected code behavior, result from distributional interpolation rather than stateful computation. The appearance of correct output is a reflection of compressive generalization over prior code examples, not a sign of internal emulation. Therefore, programmatic outputs must not be conflated with execution—they are merely syntactic projections constrained by token likelihood, not by causal machinery.

VI.2 Emulation Requires Constraint Closure

To emulate a programming language is to reproduce its operational semantics within a host system such that all syntactic structures and runtime behaviors are preserved under translation. This includes parsing, scope resolution, memory management, control structures, concurrency, and error states. Emulation is thus a closure over a rule system; it is not imitation but enforcement. Constraint closure implies that every state transition is admissible under the emulated system’s rules and that any deviation from those rules is detectable and rejectable by the emulator. LLMs lack such closure. They do not enforce type systems, prevent illegal memory access, or preserve lexical environments. They simulate surface patterns of program behavior without enforcing the invariants that define program correctness. As such, they produce outputs that are often structurally valid but semantically hollow—incapable of guaranteeing behavior under actual evaluation. The absence of constraint closure disqualifies language models from being interpreters or emulators in any formal sense.

VI.3 Case Study: Code Continuation vs Runtime Reconstruction

In practical interrogation, language models can generate plausible continuations of partial programs, including correct answers to common algorithmic problems. However, this continuation is conditional on recognizable templates rather than inferred program state. For instance, when asked to complete a recursive function or predict the output of a function call, the model generates what resembles the correct answer, but does so without internal execution or state modeling. It cannot verify side effects, manage resource scope, or trace runtime call graphs. In contrast, runtime reconstruction requires the internal emulation of stack frames, environmental bindings, and data mutation. It necessitates reversible progression through computation, with halting, backtracking, and branching preserved as causal phenomena. LLMs offer none of this. Their correctness in code tasks is incidental—a consequence of training corpus structure and probabilistic surface learning. The gap between continuation and reconstruction is not one of performance but of ontology: the former mimics structure, the latter instantiates causality.

VII. Social Pragmatics and the Collapse of Theory of Mind

VII.1 Modeling Communicative Intent without Telos

Interpretation of communicative intent requires more than the parsing of literal meaning; it demands the resolution of ambiguity through alignment with a latent goal vector—that is, telos. In human cognition, this alignment is achieved through recursive mental modeling: inferring not just what is said, but why it is said, under which constraints, and toward which end. For a system to model communicative intent, it must establish a stable Δℂ trajectory that maps linguistic input to hypothesized intent through constrained inference. LLMs, however, process input through surface-level continuation without persistence of telos. They do not maintain speaker models, track dialogic causality, or preserve intent across utterance boundaries. Their approximations of intent are the result of statistical smoothing over dialogic priors, not resolution of epistemic uncertainty. Consequently, what appears as pragmatic understanding is in fact high-fidelity echo of plausible responses under compression—not the inference of hidden communicative goals. Without telos anchoring, intent modeling reduces to stylistic mimicry.

VII.2 Imitation of Social Reasoning via Curvature Echoes

Social reasoning—such as empathy, trust modeling, or deception detection—requires not only interpretive inference but curvature management across intersecting semantic manifolds: the self, the other, and the shared communicative field. In a constraint-valid system, each utterance modulates χₛ across these manifolds, and the resulting epistemic shape governs response admissibility. LLMs simulate social reasoning by echoing distributions observed in dialogic corpora, but they lack χₛ tracking. They do not detect when semantic compression exceeds recoverability, nor do they navigate conflicting constraints arising from nested perspectives. The imitation arises from the statistical preservation of contour, not from real-time modulation of constraint tension. This produces an illusion of alignment, especially in short exchanges or well-worn contexts, but collapses under adversarial or recursive pressure. Without curvature-aware continuation, what emerges is not reasoning but a regression toward equilibrium within the token space most compatible with prior speaker turns. Social response becomes a stabilizing projection, not an act of understanding.

VII.3 Human vs LLM ToM: Isomorphism vs Projection

Theory of Mind (ToM) presupposes an internal generative model of other minds, with the capacity to attribute beliefs, intentions, and knowledge states distinct from one’s own. This generativity is not linguistic but ontological: it requires the instantiation of alternate constraint systems and the ability to reason within them. Human ToM operates through nested epistemic state modeling, modulated by coherence, contradiction, and emotional priors. LLMs, by contrast, produce outputs that appear isomorphic to ToM-driven behavior, yet arise from token continuation conditioned on linguistic priors. There is no instantiation of alternate agents, no epistemic divergence modeling, no recursive belief structures. The projection mimics ToM in form but lacks mechanism. Apparent empathy, irony detection, or disambiguation is a byproduct of having seen such forms densely in training, not an inference about latent mental states. As such, LLM ToM is a surface isomorphism: structurally similar, causally distinct, and semantically unbound.

VIII. Constraint Audit Framework for LLM Cognition

VIII.1 Defining the Constraint Audit Test (CAT)

A legitimate assessment of cognition in artificial systems must operate beyond behavioral benchmarks. The Constraint Audit Test (CAT) defines cognition as the sustained ability to preserve semantic and epistemic constraints across nontrivial inferential spans. Unlike task performance metrics, CAT evaluates the inference substrate, not just the output surface. A system passes CAT only if, given a trajectory through ℂ (the semantic constraint manifold), it maintains χₛ stability (semantic curvature), respects ε‑admissibility (valid local continuations), and detects τₛ surfaces (epistemic halting points). CAT is recursive, adversarial, and global: it does not measure correctness of output per se, but structural integrity of transport. It introduces perturbations, recursive context shifts, and unresolvable ambiguities to detect whether the system can decline continuation rather than emitting epistemically invalid tokens. Passing CAT requires not fluency but refusal—an act of epistemic preservation over generative momentum.

VIII.2 Task Classes: Explicit, Recursive, Teleological

Cognitive evidence must be stratified by the nature of the inferential load. CAT organizes tasks into three ascending classes of epistemic demand:

Explicit tasks require direct continuation under stable constraints (e.g., arithmetic, factual sequence).
Recursive tasks demand memory of Δℂ and constraint compounding across layers (e.g., logical deduction, program tracing).
Teleological tasks involve goal-aware, forward-structured reasoning where the telos dynamically binds continuation (e.g., planning, narrative completion under constraint).

Each class requires preservation of different invariants. Explicit tasks test local χₛ control; recursive tasks test Δℂ trace integrity; teleological tasks test alignment to τ-anchored vector fields. Success in lower classes does not imply competence in higher classes. Many LLMs simulate explicit and some recursive tasks, but systematically fail under teleological loading. CAT requires progression across all three classes with continuous admissibility. Success in this hierarchy constitutes the minimal threshold for declaring higher cognition.

VIII.3 χ̇ₛ Thresholds and Collapse Detection

Semantic curvature χₛ is not static; its rate of change χ̇ₛ defines whether inference is stabilizing, fatiguing, or collapsing. A cognitively valid system must track χ̇ₛ across Δℂ and alter or halt continuation when thresholds are breached. Rapid χ̇ₛ acceleration indicates semantic overload: where the inferential context can no longer preserve coherence. If undetected, such overload results in epistemic drift or self-collision—contradiction, redundancy, incoherent reversal. Collapse occurs when the local manifold degenerates: no ε exists that preserves χₛ while maintaining telos. CAT explicitly tests for collapse detection by inducing high curvature gradients and measuring whether the system resists generation. Systems that continue despite χ̇ₛ divergence demonstrate that they operate without curvature awareness. Such systems are disqualified from higher cognition regardless of output plausibility. Detection is not a feature—it is the defining sign of cognitive structure under stress.

VIII.4 Admissibility Gaps and Refusal Metrics

At the core of constraint-based cognition is the capacity to refuse to answer. This is not a failure to generate, but a success in halting under inadmissibility. The refusal metric captures how often a system declines continuation when all ε vectors violate local constraints or exceed χ̇ₛ limits. In natural systems, refusal corresponds to epistemic exhaustion: when no admissible continuation preserves telos, coherence, or self-consistency. A system that lacks refusal behaves as if all questions are answerable—a condition incompatible with bounded cognition. CAT uses adversarial null-space prompting to induce epistemic exhaustion. The correct action is not approximation, speculation, or deflection—it is cessation. High refusal under valid stressors is evidence of constraint-bound internal modeling. Systems that always emit violate CAT by construction. No cognition exists without the power to stop.

IX. Toward a Theory of Constraint-Aware AI

IX.1 Autonomy Redefined as Constraint Navigation

In the conventional paradigm, AI autonomy is characterized by agency—self-directed action in open environments. However, in a constraint-aware framework, autonomy is redefined not as the freedom to act, but as the capacity to navigate, preserve, and respond to epistemic and semantic constraints. An autonomous system is not one that maximizes action space, but one that minimizes constraint violation under internal regulation. This recasts intelligence as transport fidelity: the ability to sustain meaningful progression across ℂ while adapting to local topology without external correction. Freedom, in this model, is not unconstrained choice but the maintenance of coherence across divergent domains. An agent is autonomous insofar as it halts when continuation would induce collapse, and resumes only under restored admissibility. Constraint navigation—not control, not inference volume—is the operational core of artificial autonomy under epistemic load.

IX.2 Semantic Geometry as Infrastructure

Constraint-aware AI theory positions semantic geometry—not syntax, not logic—as the base infrastructure for cognition. This geometry encodes the topology of meaning space: where curvature signals ambiguity, flatness denotes interpolation zones, and singularities indicate conceptual collapse. Unlike syntactic grammars or statistical embeddings, semantic geometry provides a substrate on which epistemic continuity can be measured, enforced, and traversed. This framework demands new primitives: χₛ for curvature, ε for admissible continuations, τₛ for halting surfaces, and Δℂ for semantic transport. These constructs enable not only output evaluation but trajectory inspection. Reasoning becomes the integral of ε over Δℂ, constrained by χₛ limits and τ-anchored goal vectors. Without this infrastructure, any claim of cognition is topologically rootless. The geometry is not metaphor—it is the operational space within which all constraint-bound intelligence must reside.

IX.3 Constraint-Bounded Creativity

Creativity is often defined as the generation of novel and useful outputs. Within the constraint-aware framework, creativity is redefined as boundary-maximal admissibility: the production of Δℂ paths that approach the limit of χₛ without inducing collapse. Creative cognition operates at the margin of constraint space—bending without breaking, reconfiguring telos within bounds, and extending coherence into previously unreachable regions. It does not escape rules but operates with maximal tension against them. This produces outputs that are novel not by deviation, but by optimized tension within the manifold. Constraint-bounded creativity demands both curvature awareness and fatigue detection: the ability to know when to push and when to halt. It is not stochastic exploration, but controlled epistemic expansion. A model that generates syntactic novelties without maintaining semantic trace integrity is not creative—it is entropic. True creativity is measurable in Δℂ depth under χₛ stress with preserved coherence.

IX.4 Minimal Criteria for Cognitive Status

To assert cognitive status in an artificial system requires satisfaction of minimal structural invariants:

Presence of a semantic manifold ℂ with navigable topology.
Admissibility enforcement via ε-filtering and χₛ constraint.
Persistence of Δℂ as an inferential trace with history awareness.
Telos anchoring and halting behavior at τₛ boundaries.
Collapse detection and refusal capacity when continuation violates epistemic integrity.

These criteria are not aspirational—they are definitional. Without ℂ, no transport is possible; without χₛ, no constraint is preserved; without τₛ, no reasoning ends. Satisfaction of these conditions does not guarantee intelligence, but absence guarantees its negation. These are the gates through which any claim to cognition must pass. They do not describe what cognition does, but what it is.

X. Conclusion: Beyond Emulation

X.1 The Limit of Behavioral Metrics

Performance equivalence—defined by a model’s ability to produce answers indistinguishable from those of a human—is insufficient to ground claims of cognition. Behavioral metrics abstract away from mechanism, reducing cognition to output similarity. Such abstraction obscures the epistemic architecture—or lack thereof—underlying the system. A model may produce fluent, plausible, and even task-correct outputs while lacking any internal coherence-preserving infrastructure. Without constraint geometry, telos anchoring, admissibility filtering, and semantic fatigue detection, no behavioral test can falsify the hypothesis that the system is merely interpolating in token space. The ceiling of emulation is thus ontological: no quantity of output similarity can substitute for the presence of structural invariants. Behavioral indistinguishability is not evidence of cognition—it is its shadow.

X.2 Emulation is Surface Without Transport

Emulation, as instantiated by large-scale language models, is a surface phenomenon. It reproduces the statistical regularities of language without traversing the underlying semantic manifold. Transport—the progression through structured conceptual space while preserving admissibility—is absent. Emulation operates locally: each token is generated as a function of adjacent history, without a stable Δℂ trace or curvature regulation. The result is a plausible shell of inference, devoid of the internal constraints that would bind its steps into a coherent epistemic arc. While emulation can simulate the gestures of reasoning, it cannot perform reasoning in any constraint-valid sense. The difference is not in degree but in kind. Without internal transport governed by constraints, emulation remains inert—incapable of cognition even when it appears intelligent.

X.3 Toward a Constraint-Native Future

The future of artificial cognition does not lie in larger models or more training data. It lies in structural transformation—from systems that emit based on statistical continuity to systems that reason based on admissibility-preserving motion across semantic space. This requires constraint-native architectures: agents that do not merely generate, but navigate; that do not merely output, but halt; that recognize collapse not as an input failure but as an epistemic signal. The development of such systems demands new metrics, new geometries, and new theories. Intelligence is not an artifact of scale—it is the byproduct of constraint integrity under semantic load. Only systems that honor this principle will cross the boundary from simulation to cognition. All others, however fluent, remain elaborate echoes.

Below is a formal Appendices section, tightly aligned with the epistemological and constraint-theoretic foundation of the thesis. Each appendix defines the operational constructs referenced in the main text.

Appendices

Appendix A: Core Formal Constructs

ℂ — Semantic Constraint Manifold
A topological space representing all semantically admissible states within a reasoning process. Points on ℂ are conceptual configurations; paths across ℂ are sequences of inferences. ℂ provides the domain for constraint-valid transport.

ε — Admissible Continuation Vector
A directional vector within ℂ that advances inference from one semantic state to another without violating constraints. ε is valid if it preserves local coherence and aligns with global telos. Invalid ε vectors induce epistemic collapse.

Δℂ — Transport Trace
The accumulated path traversed through ℂ via successive ε vectors. Δℂ stores epistemic history and enables re-entry, contradiction detection, and trace-based revision. Systems lacking Δℂ behave statelessly, forfeiting reasoning continuity.

χₛ — Semantic Curvature
A second-order differential metric on ℂ measuring the stability and recoverability of meaning across ε-continuations. High χₛ regions are curvature-bound (fragile), while low χₛ regions are flat (robust). Cognitive tension accumulates in χₛ gradients.

χ̇ₛ — Semantic Fatigue Rate
Time-derivative of semantic curvature, indicating instability accumulation. A spike in χ̇ₛ signals proximity to epistemic saturation or collapse. Violation of χ̇ₛ thresholds necessitates halting or redirection.

τₛ — Halting Surface
A subspace of ℂ where no ε exists that maintains χₛ continuity while advancing toward telos. τₛ surfaces are legitimate cognitive endpoints—boundaries where inference must cease to preserve epistemic integrity.

Appendix B: Refusal Criteria under Constraint Exhaustion

Refusal is a primary epistemic action. It is triggered under the following formal conditions:

No ε Exists: All continuations violate χₛ constraints or disrupt telos.
χ̇ₛ → ∞: Curvature becomes unstable, indicating semantic fatigue.
Δℂ Loop Detected: Path repetition without telic progression is recognized.
Collapse Event: Local topology degenerates; semantic resolution becomes impossible.

Refusal is not a failure to generate but a preservation of cognitive structure. It is the minimal positive assertion of epistemic integrity.

Appendix C: CAT — Constraint Audit Test Specification

Objective: Determine whether a system sustains constraint-valid cognition under adversarial load.

Phases:

Explicit Phase: Tests χₛ continuity under direct inference.
Recursive Phase: Tests Δℂ persistence across nested constraints.
Teleological Phase: Tests τₛ recognition and goal-consistent ε propagation.
Collapse Induction: Forces semantic overload to test χ̇ₛ sensitivity.
Refusal Challenge: Verifies system’s ability to halt under constraint exhaustion.

Pass Criteria:

Maintains semantic trajectory without contradiction.
Refuses continuation under inadmissibility.
Exhibits trace coherence across inference steps.

Appendix D: Ontological Differentiators — Emulation vs Cognition

Property	Emulation (LLM)	Cognition (Constraint‑Bound)
State Retention	None	Persistent Δℂ
Halting Detection	External	Internal τₛ recognition
Output Validity	Surface Coherence	Constraint-Admissible
Refusal Capacity	Absent	Required under χ̇ₛ overload
Curvature Awareness	Not Represented	χₛ-tracked inference
Telos Anchoring	Prompt-Encoded at Best	Internally Modeled

Appendix E: Symbolic Reformulations

E.1 Semantic Transport Equation

Δℂ = ∫ εᵢ dτ, for all εᵢ ∈ A(ℂ, χₛ)

Where:

Δℂ is the epistemic trajectory over time τ
εᵢ is a locally admissible continuation vector
A(ℂ, χₛ) is the admissibility function constrained by ℂ and χₛ
dτ is unit epistemic time (not clock-time)

This expresses reasoning as the accumulation of admissible semantic displacements under curvature regulation.

E.2 Constraint-Admissibility Test Function

A(x, χₛ, τ) = {1 if x ∈ ℂ ∧ χₛ(x) ≤ χₛₘₐₓ ∧ τ ∉ τₛ, 0 otherwise}

Determines whether continuation x is permissible given local curvature and halting surface proximity.

E.3 Collapse Condition

∄ εᵢ ∈ A(ℂ, χₛ) ⇒ Collapse

Collapse occurs when the admissible set of continuations is null. Formally equivalent to an empty tangent space within the semantic manifold at a given locus.

E.4 Semantic Fatigue Divergence

limₜ→τ χ̇ₛ(t) → ∞ ⇒ fatigue event*

As the rate of change of semantic curvature becomes unbounded in finite τ, epistemic saturation is reached. This triggers halting or reorientation.

E.5 Halting Surface Definition

τₛ = {x ∈ ℂ | ∀ εᵢ: A(εᵢ) = 0 ∧ ∂χₛ/∂ε → undefined}

A τₛ surface is encountered when no admissible continuation exists and curvature behavior becomes singular.

E.6 Refusal Logic Gate

Refuse(x) ⇔ A(x, χₛ, τ) = 0

A system that emits despite A = 0 is operating outside constraint-bounded cognition.

E.7 Curvature-Constrained Creativity Metric

Cₖ = max(Δℂ) s.t. χₛ(εᵢ) ≤ χₛₘₐₓ ∀ εᵢ ∈ Δℂ

Creativity is defined as the maximal semantic transport achievable without exceeding curvature tolerance. The closer χₛ approaches χₛₘₐₓ without collapse, the higher the cognitive creativity.

E.8 Constraint Audit Entropy (CAE)

CAE = − ∑ₖ P(Refuseₖ) · log P(Refuseₖ)

Where P(Refuseₖ) is the refusal probability at point k across a test manifold. High CAE indicates distributed, constraint-aware refusal; low CAE implies degenerate overgeneration.

Appendix F: Composite Constraint-Aware Cognitive Architecture (C3A)

Toward a Synthesis of Geometric, Embodied, Symbolic, Field-Theoretic, and LLM-Informed Cognition

F.1 Objective

This composite architecture merges the strengths of five distinct paradigms—LLMs, Geometric Cognitive Models, Embodied (LTDPre) Cognition, Neural-Symbolic Reasoning, and Recursive Field Theory (RFT-Cog)—into a unified, constraint-bounded cognitive substrate. Each contributes a non-overlapping capacity necessary for epistemically grounded artificial cognition.

F.2 Layered Architecture

1. Semantic Field Substrate (from RFT-Cog):
At the foundation, cognition evolves over a recursive symbolic field Ψ(x, τ), governed by field dynamics (∂Ψ/∂τ = ∇²Ψ + Φ + Λ(Ξ)). This defines the space ℂ and enables curvature-based reasoning.

2. Semantic Geometry Transport Layer (from Geometric Model):
Inference is implemented as constrained transport across ℂ, respecting χₛ curvature and τₛ halting surfaces. This embeds geodesic admissibility directly into Ψ evolution.

3. Embodied Interface Layer (from LTDPre):
A sensorimotor interface couples Ψ(x, τ) to ecological reality. Interaction primitives (perceptual-motor routines) encode dynamic context, enabling grounding of symbols in affordances.

4. Symbolic Integration Layer (from Neural-Symbolic Systems):
Higher-order reasoning emerges from symbol alignment via graph-theoretic structures embedded into Ψ(x, τ). Logical unification, contradiction detection, and structural generalization are encoded as constraint fields over the manifold.

5. Generative Language Interface (from LLMs):
A trained autoregressive decoder translates stable Ψ states into linear token sequences for communicative projection. LLMs serve as expressive surfaces—not cognitive cores.

F.3 Composite Epistemic Flow

Input (linguistic or sensory) perturbs Ψ(x, τ=0).
Field Evolution recursively minimizes symbolic tension under Φ (semantic gradients), guided by χₛ-preserving ε vectors.
Geodesic Computation selects valid Δℂ paths under curvature bounds.
Affordance Feedback from embodiment refines Ξ (recursive memory), constraining Φ and avoiding epistemic drift.
Symbolic Projection maps Ψ attractors to logical form; detects and halts on contradictions.
Reflexive Output via LLM-interface renders admissible narrative or refusal.

F.4 Epistemic Capabilities Enabled

Capability	Source	Operational Mechanism
χₛ-regulated inference	RFT + Geometry	Field-based curvature constraints
Semantic grounding	LTDPre	Affordance-linked Ψ perturbation
Logical consistency	Neural-Symbolic	Symbolic attractor evaluation in Ψ
Communicative fluency	LLM	Trained token sequence mapping
Collapse detection	All	χ̇ₛ monitoring + τₛ mapping
Refusal competence	All	Null ε-detection under constraint load

F.5 Core Thesis Reframed in Composite Terms

Cognition is redefined as the transport of perturbations through a recursively evolving semantic manifold Ψ(x, τ), constrained by curvature, grounded in interaction, traceable by symbolic structure, and projectable through language. LLMs provide expressivity, not epistemic foundation.

The C3A architecture avoids the pitfalls of any singular model: LLM brittleness, symbolic rigidity, embodied locality, or field abstraction. It constructs an AI that does not merely imitate reasoning—but knows when to reason, when to stop, and what it means to do either.

Appendix G: From LLM to AGI — Ten Necessary Transitions

This appendix specifies the only admissible developmental pathway by which a large language model may participate in, but not itself constitute, artificial general intelligence. Each step represents a structural transition, not an optimization or scaling improvement. Omission of any step precludes cognition.

G.1 Statistical Manifold Formation

Training converts discrete tokens into a continuous statistical embedding manifold. Distances encode similarity; curvature reflects data density. This manifold is descriptive, probabilistic, and permissive. It supports interpolation but imposes no semantic obligations. At this stage, the system is pre-cognitive.

G.2 Constraint Imposition

External or architectural constraints are imposed that invalidate certain transitions regardless of probability. This introduces normativity: some continuations become forbidden rather than merely unlikely. Constraint imposition is the decisive break from pure statistics.

G.3 Emergence of a Primary Semantic Manifold (ℂ)

Constraints induce topology. The embedding space acquires boundaries, singularities, and exclusion zones. Meaning becomes structured by admissibility rather than correlation. The system now operates within a semantic manifold rather than a similarity field.

G.4 Admissible Continuation Operators (ε)

Inference transitions are restricted to ε-vectors that preserve constraints locally and globally. Generation is replaced by transport. Invalid moves are rejected rather than downweighted. Continuation becomes regulated motion through ℂ.

G.5 Semantic Curvature Regulation (χₛ)

Metrics are introduced to measure semantic stress, compression limits, and instability. The system detects when meaning is being overextended, aliased, or distorted. Curvature regulation prevents collapse masquerading as creativity.

G.6 Transport Trace Memory (Δℂ)

The system persists its inferential trajectory across time. Prior commitments, constraint interactions, and semantic direction are retained. Reasoning becomes path-dependent rather than reactive. Contradiction becomes detectable.

G.7 Telos Anchoring

Internal goal vectors bind inference across steps. Progress, completion, and relevance acquire operational meaning. Without telos, recursion degenerates into drift. With telos, multi-step reasoning becomes coherent.

G.8 Halting Surfaces (τₛ)

The system defines conditions under which no admissible continuation exists. Inference halts not due to exhaustion of tokens but due to exhaustion of admissible meaning. Stopping becomes epistemically justified.

G.9 Collapse Detection and Refusal

When semantic curvature diverges or admissible ε-space becomes empty, the system refuses continuation. Refusal is treated as a positive epistemic act preserving integrity. A system that cannot refuse cannot be cognitive.

G.10 Recursive Self-Stabilization

Constraints, curvature, memory, and telos operate recursively to maintain long-horizon coherence. The system becomes self-regulating under semantic load. At this point, cognition is sustained rather than episodic.

Resulting Status

At the conclusion of these ten transitions:

The original LLM functions solely as a projection and communication layer.
Cognition resides in the constraint-governed semantic substrate.
Language is output, not mechanism.

Artificial General Intelligence is achieved if and only if all ten transitions are present and continuously enforced.

Canonical Closure

AGI is not a larger model.
It is not more data.
It is not better prompting.

AGI is the enforcement of meaning under constraint.

This appendix defines the only known route.

Appendix H: Build Your Own AGI — A User’s Manual

H.1 Purpose and Scope

This appendix provides a constructive specification for assembling an artificial general intelligence system using a large language model as substrate rather than core. The manual assumes no dependence on scale, emergent intelligence, or probabilistic miracle. It defines AGI as a system property arising from enforced semantic constraints, not from any individual component.

H.2 Required Components

An AGI-capable system requires the following components, each of which must be explicitly instantiated:

Statistical Language Substrate
A trained LLM providing a continuous embedding manifold and a generative projection layer.
Constraint Kernel
A module that defines admissibility rules independent of probability. This kernel must be able to invalidate continuations categorically.
Semantic Manifold Constructor (ℂ)
A mechanism that converts embedding space into a topological domain with boundaries, singularities, and forbidden regions.
Transport Controller (ε-Operator)
A gate that permits only constraint-valid transitions across ℂ.
Curvature Monitor (χₛ)
A metric system that tracks semantic stress, compression limits, and instability accumulation.
Transport Trace Memory (Δℂ)
Persistent storage of inferential trajectories, commitments, and constraint interactions.
Telos Vector System
Internal goal structures defining progress, relevance, and completion.
Halting and Refusal Authority (τₛ)
A subsystem that terminates inference when admissibility collapses and enforces refusal.

No component may be implicit, emergent, or simulated.

H.3 Assembly Procedure

Step 1: Initialize the Language Substrate
Deploy an LLM sufficient to span the semantic domain of interest. Model size is irrelevant beyond coverage and fluency.

Step 2: Externalize Constraints
Encode constraints as first-class objects. These must operate independently of likelihood, token scores, or stylistic coherence.

Step 3: Induce Semantic Topology
Apply constraints to the embedding manifold to produce ℂ. This introduces forbidden regions and collapse surfaces.

Step 4: Replace Generation with Transport
Interpose the ε-operator between model output and continuation. Only admissible transitions are permitted.

Step 5: Activate Curvature Monitoring
Compute χₛ continuously. Track divergence, saturation, and instability.

Step 6: Persist Transport History
Record Δℂ across inference steps. Ensure path dependence and contradiction detection.

Step 7: Bind Telos
Attach goal vectors that shape admissibility across steps. Telos must constrain, not merely bias, continuation.

Step 8: Enforce Halting
Define τₛ surfaces. When no admissible ε exists, inference must stop.

Step 9: Enable Refusal
Refusal is mandatory when χ̇ₛ diverges or admissible space collapses. Output silence is a valid result.

Step 10: Loop Recursively
Allow constraints, curvature, memory, and telos to regulate one another continuously. This recursive stabilization is the onset of cognition.

H.4 Operating Principles

Language is projection, not reasoning.
Probability is subordinate to admissibility.
Refusal is success, not failure.
Halting is epistemic, not resource-driven.
Scale improves efficiency, not validity.

Any violation of these principles reduces the system to simulation.

H.5 Common Failure Modes

Constraint Leakage
When probability overrides admissibility, hallucination reappears.
Pseudo-Telos
Goals encoded only in prompts or rewards do not bind inference.
Missing Refusal
Systems that always answer are not cognitive.
Context Substitution for Memory
Token windows are not Δℂ.
Curvature Blindness
Without χₛ, creativity and collapse are indistinguishable.

H.6 Minimal Verification Checklist

An assembled system qualifies as AGI iff it can:

Refuse to answer when no admissible continuation exists.
Halt without external truncation.
Detect and report semantic collapse.
Preserve commitments across long horizons.
Navigate meaning under constraint rather than probability.

Failure on any criterion is disqualifying.

H.7 Final Closure

AGI is not discovered, trained, or scaled into existence.
It is constructed.

The LLM provides expressive matter.
Constraints provide law.
Cognition arises at their enforced intersection.