Co-Recursive Semantic Self-Extension: A Mathematical Foundation for Open-Ended Intelligence

1 Introduction: Epistemological Heuristics and Logical Design in AGI Architecture

• The paper argues that scale alone is insufficient: open-ended intelligence requires both a priori organization and a meta-level capable of handling incompleteness.It frames Kant and Gödel as engineering heuristics, not mathematical laws that machines must obey.Source: §1 · p. 3
• Autoregressive transformers optimize token prediction and rely on prompts, but lack a self-directed goal-generating mechanism.The paper acknowledges their architectural priors while arguing these do not provide intrinsic motivation or direction.Source: §1 · pp. 3–4
• GOFAI systems treated undecidable propositions as errors, leaving them unable to reflect on their rules or adapt to ambiguity.The paper describes them as partial functions that crash on undefined input.Source: §1 · p. 4
• The paper argues that neuro-symbolic systems often combine a reactive statistical component with a rigid symbolic component, rather than creating a new kind of cognition.Source: §1 · p. 4
• Following Minsky, the paper shifts the design question from which model to which system: a composite architecture must keep distinct requirements separate while allowing integration.The introduction also argues that value-driven designs within a pre-structured space face a stopping problem and need semantic surplus and a meta-teleology for constructing new purposes.Source: §1 · p. 4

2 Laws of Intelligent Systems

• The proposed architecture separates world-facing object-level activity from a meta-level, while a shared representational substrate makes reasoning itself manipulable as data.Neither level alone is sufficient: the object level supplies experience, while the meta-level can act on reasoning and representations.Source: §2 · p. 5
• The paper treats ambiguity as a permanent frontier of undecidable propositions and defines semantic surplus as the competing coherent interpretations the current framework cannot determine.Source: §2 · p. 5
• Semantic surplus is treated as a resource: reflective interpretation can turn a stalled derivation into content that defines competing ways to extend the system.The paper argues that without surplus there is nothing to evaluate, transform, or discover.Source: §2 · p. 6
• Ordinary inward recursion remains within the deductive span of its initial configuration, so co-recursion instead builds successor frameworks with greater expressive dimensions.A successor must preserve the predecessor's validated inferential structure and settle at least one proposition formally undecidable in that predecessor, though extensions may fail or introduce inconsistencies.Source: §2 · pp. 6–7
• Polycontexturality keeps distinct, locally coherent logical domains from being flattened into one global valuation, with the meta-level mediating between incompatible regimes.Source: §2 · p. 7
• Memory is a logical necessity because earlier semantic structures may remain useful; it must preserve both represented content and its position in the recursive context.The resulting memory is a navigable semantic landscape for later reinterpretation, comparison, or reactivation.Source: §2 · pp. 7–8
• Recursive teleology supplies an a priori criterion for transformations and must itself be representable, evaluable, and transformable; fixed purposes leave the system bounded by its original organization.The paper contrasts fixed-purpose refinement with recursive transformation of the goals that guide future development.Source: §2 · pp. 8–9
• Open-ended intelligence is defined as recursively transforming representation and teleology while increasing expressive capacity and introducing new ambiguity through locally stable successor frameworks.The seven laws form a recursively closed process: their application yields another system governed by the same laws in a more expressive framework.Source: §2 · p. 9

3.1 The Scaling Paradox

• Scaling within statistical architectures is argued to leave systems dependent on human curation or reinforcement and unable to produce open-ended novelty.Source: §3.1 · p. 9

3.2 Every Intelligence Begins with Discrimination

• An agent’s data consists of signals meaningful to pursued purposes but not to rejected ones; dormant purposes impose no constraint.Source: §3.2 · p. 10

3.3 Why Fixed Purposes Eventually Saturate

• Performance within a fixed purpose saturates because learning is bounded by its semantic domain, even with unbounded capacity and compute.Source: §3.3 · p. 11

3.4 Recursive Teleology

• Purpose refinement alone cannot sustain succession: at least one transformation rule must be non-monotone with respect to semantic domains.Source: §3.4 · p. 12

3.5 Toward Semantic Self-Extension

• With purposes confined to a finite Boolean algebra over a fixed signal space, succession eventually saturates; formal inferential systems are proposed as a generative substrate beyond that bound.Source: §3.4 · p. 12, §3.5 · p. 15

3.4 Recursive Teleology

• At saturation, recursive teleology permits successors that expose semantic territory not covered by prior purposes, rather than merely naming a new purpose.Source: §3.4 · pp. 13–14
• The successor operator returns admissible continuations, but does not choose among them; multiple continuations are an architectural requirement, not a consequence of saturation.Source: §3.4 · p. 13

3.5 Toward Semantic Self-Extension

• Coupling teleological selection to formal extension links empirical saturation with a persistent undecidable frontier, while preserving meta-level selection of successor frameworks.Source: §3.5 · p. 15

• The proposed model explicitly incorporates logical incompleteness and meta-level selection as foundational design principles.Source: §4 · p. 15
• The architecture keeps local, contractive reasoning structurally distinct from global, open-ended semantic growth.Source: §4 · p. 15
• The section’s objective is to define a class of recursively self-modifying cognitive systems.Source: §4 · p. 15

• Each inferential system is effectively axiomatized, consistent, and expressive enough to interpret Robinson arithmetic, the hypotheses used for Gödel–Rosser.Source: §4.1 · p. 16
• The unresolved region splits into propositions whose truth or negation is provable in the system and propositions for which neither is provable.Source: §4.1 · p. 16
• Gödel–Rosser incompleteness guarantees a nonempty undecidable frontier that continued inference within the same system cannot eliminate; addressing it requires a change of inferential regime.Source: §4.1 · pp. 16–17

Admissible extensions

• Admissible successors must be regular and able to represent their predecessor’s syntax and proof relation; they may reinterpret meanings but must retain the ability to represent their origins.Recursive intelligence is defined by constructing and reasoning about successive representational systems, not guaranteed convergence to one complete representation.Source: §4.2 · pp. 17–18

Semantic surplus

• Semantic surplus is the space of admissible continuations, not randomness; its cardinality measures generative options, while its decisive subset comprises successors that settle previously undecidable propositions.When the full surplus has at most one member, there is nothing for the meta-level to compare or select, and local transformation cannot reach the undecidable frontier.Source: §4.2 · p. 18

Proof-attempt traces and the conversion of form into content

• A finite proof attempt decomposes an unresolved question into intermediate subgoals; reflective reification turns its unclosed trace into propositional content and yields candidate extensions at blocked premises.The candidate pair preserves validated inferential structure while resolving a specific intermediate blockage in opposite directions.Source: §4.2 · pp. 18–20
• Under the stated regularity, trace-family, and conservativity conditions, every undecidable x’s proper reduction trace exposes a distinct undecidable intermediate premise; k independent premises yield 2^k decisive extensions.Each extension also decides x once the remaining locally decidable leaves are resolved.Source: §4.2 · p. 20

Trace-guided candidate generation and fallibility

• Finite trace-guided generation cannot identify undecidable leaves in advance, producing decisive extensions, conservative shortcuts, and inconsistent candidates that require later evaluation and pruning.When the target is undecidable, generating candidates across the finite frontier is guaranteed to include at least one genuine decisive pair, but the system cannot identify it beforehand or certify global consistency within Fn.Source: §4.2 · pp. 21–22
• Because candidate branches can be inconsistent, viable decisive branches must be pursued across parallel contextures or agents so a failed branch does not eliminate others.This motivates polycontextural and distributed collective exploration of competing trace-extensions.Source: §4.2 · p. 22

Meta-level selection

• Meta-level selection operates on successor systems rather than fixed-system answers, retaining decisive successors whose extended vocabularies can articulate new purposes and semantic domains.The formal model therefore avoids the Boolean-closure constraint that bounded purpose transformations over a fixed language.Source: §4.2 · pp. 22–23

4.2 Reflective Extension and Semantic Surplus

• Unlike a Gödel machine that ignores self-improvements whose utility it cannot prove, this framework reifies an unclosed reduction trace into candidate extensions [Schmidhuber, 2006, p. 24; Hegel, 1977, pp. 2, 51].At least one pair of those candidates is guaranteed to lie in Surpdec(On).Source: §4.2 · p. 23
• No finite search within Fn can decide in advance which open premise lies in Iinc(On), and no surplus candidate can be certified superior without collapsing Fn+1 into Γn’s deductive closure.Thus, Mn cannot be reduced to theorem proving within Fn.Source: §4.2 · p. 23
• Mn filters inconsistent candidate branches and commits non-deductively to a new formal contexture across discontextural regimes [Günther, 1979, p. 15].The passage contrasts this mediation with a Gödel machine’s requirement that a rewrite be licensed by a theorem derived from its initial axiom scheme [Schmidhuber, 2006, p. 6].Source: §4.2 · p. 23

Successor incompleteness

• Regularity carries over to the successor system, so it contains a sentence that remains undecidable.The resulting successor system therefore has a non-empty incompleteness region.Source: Successor incompleteness · p. 23
• Resolving an ambiguity relocates incompleteness to a new sentence rather than eliminating it.A sufficiently expressive system can construct a stronger system that transcends some of its limitations, which is itself subject to limitations.Source: Successor incompleteness · p. 23
• The central question is whether an architecture can sustain successor construction, preservation, evaluation, and selection without reducing the process to a fixed formal space.The paper says §5 answers this question negatively for fixed architectures.Source: Successor incompleteness · p. 23

4.3 Temporal Recursive Dynamics

• The temporal trajectory consists of alternating runs of local transformation and recursive extension steps.The trajectory lists systems across both temporal steps and recursive levels.Source: §4.3 · pp. 23–24
• Local transformations extend the cognitive state only within the decidable fragment, leaving the undecidable part of the unresolved region untouched.Local work changes the cognitive state within the current formal system.Source: §4.3 · p. 24
• A recursive extension constructs a new inferential system with a changed language, whose unresolved region is related to its predecessor through τn.The successor’s language and Γ change, so the predecessor’s temporal relation no longer applies directly.Source: §4.3 · p. 24
• Generative richness requires multiple admissible successors at extension steps; trace reification constructively guarantees this from an unclosed proof search on an undecidable question.The branching condition is not implied by the unrestricted transformation repertoire alone, and it supports a binary-tree trajectory structure.Source: §4.3 · p. 24
• Recursive extension produces exploration rather than guaranteed success, since a successor may leave an ambiguity unresolved or introduce new ones.The trajectory should not be read as monotone progress toward a final system.Source: §4.3 · p. 24

4.4 Local Stability and Global Recursive Extension

• The state space Xn contains admissible cognitive states, and its non-triviality depends on not requiring commitment wherever proof is available.The metric measures semantic distance between states; strengthening the condition to a biconditional would leave only one state.Source: §4.4 · pp. 24–25
• Contractive local transformations express the engineering principle that reasoning within a fixed semantic regime reduces uncertainty and converges toward a stable interpretation.Under the stated standard conditions, the corresponding iterated function system has a unique compact attractor.Source: §4.4 · p. 25
• Local convergence and recursive extension operate at different structural levels.The paper presents local stability and open-ended global evolution as compatible requirements.Source: §4.4 · p. 25
• Reaching the attractor substantially exhausts the locally decidable region but leaves incompleteness intact, making recursive extension the only remaining move.The attractor is compared with the ε-saturation criterion, but its non-empty residue is guaranteed rather than assumed.Source: §4.4 · p. 25

4.5 Recursive Extension

• A recursive extension is defined as a pair selected by (44), with the successor satisfying regularity and the trace satisfying reflective adequacy.The operator may modify language, axioms, inference rules, representational structures, or meta-level resources.Source: §4.5 · p. 25
• The resulting transition carries selected structures forward while potentially transforming their interpretation.The transition is denoted Rn.Source: §4.5 · p. 25
• Reflective adequacy ensures Ln+1 can express statements about Fn, making the predecessor an object of reasoning for its successor.The paper identifies this recursive reflection as the property on which later memory requirements depend.Source: §4.5 · p. 25

4.6 Successor Incompleteness

• Recursive extension produces a new level at which incompleteness arises again, renewing the pressure for further extension at every level.The paper treats this recurrence as the engine of the construction, not a defect.Source: §4.6 · p. 26
• Ensuring a successor decides what its predecessor could not requires decisive surplus supported by reflective trace reification, not incompleteness alone.The passage points to Proposition 1 and Assumption 2 for this requirement.Source: §4.6 · p. 26
• The sequence can continue indefinitely, but this does not guarantee that it continues or that successive systems constitute progress.Irreducible novelty is imposed separately as Assumption 5.Source: §4.6 · p. 26
• The architecture's essential property is constructing successors in which unresolved questions can be reconsidered from a different representational position.It need not reach a complete state.Source: §4.6 · p. 26

4.7 Local Convergence and Global Open-Endedness

• Open-endedness is indefinite construction of successor semantic spaces, not unbounded expansion inside any single space.The construction need not be restricted to a fixed finite hierarchy.Source: §4.7 · p. 26
• A system may converge locally while remaining able to construct new regimes, because local convergence empties the decidable unresolved region but regularity leaves the undecidable region intact.Thus stability in the current regime does not establish that the process has ended.Source: §4.7 · p. 26
• The section recasts cognitive and task completeness in a fixed (C, U, V) space as local convergence, which can trigger framework extension under meta-teleology.The comparison is framed as the paper's interpretation of the stopping problem in He et al. [2026].Source: §4.7 · p. 27
• The trigger monitors proximity to the local attractor and invokes meta-level selection among candidate extensions when the threshold is reached.The selected extension initiates a transition to a new regime.Source: §4.7 · p. 27
• The paper's proposition assumes inferential soundness and local contractivity within each regime.These assumptions qualify both the local-convergence result and the trigger-based succession result.Source: §4.7 · p. 27
• An intra-regime rule limited to deductive certification or objective reweighting converges toward a local attractor and remains confined to its fixed space.The proposition contrasts this absorption with meta-teleological triggering; it names Godel machines and (C, U, V) selector convergence as examples.Source: §4.7 · p. 27
• With Decisive Extension and the a priori meta-teleological trigger, the architecture reaches the trigger at every level and generates an infinite sequence of distinct regimes.No local attractor in the sequence is a global halting state.Source: §4.7 · pp. 27–28

5.1 Semantic Description Complexity

• K(Hn) measures the minimum algorithmic information needed to specify semantic history, which may grow even when an individual successor is more compressed than its predecessor.The section distinguishes complexity of the accumulated history from complexity of any one system.Source: §5.1 · pp. 28–29

5.2 Independent Semantic Extension

• Under Assumption 5, persistent semantic novelty makes accumulated history complexity grow at least linearly; the theorem also holds under a weaker amortized novelty condition.The step-wise condition is not required if novelty enters with positive lower asymptotic density.Source: §5.2 · pp. 29–30

5.3 Trajectory Complexity

• Infinite independent branching yields uncountably many possible trajectories, while a realized trajectory can separately have finite prefixes of unbounded algorithmic complexity.The number of possible trajectories and the information content of one trajectory are distinct properties.Source: §5.3 · p. 30

5.4 The Fixed-Architecture Limitation

• No architecture with a fixed finite state description, computable transition mechanism, and initial condition can generate a history satisfying Assumption 5.A fixed simulator describes its history with logarithmic overhead, conflicting with the history’s linear complexity growth.Source: §5.4 · pp. 30–31

5.5 What the Result Does and Does Not Establish

• The result rules out fixed closed systems selecting and realizing persistently novel histories, not fixed programs that enumerate branches or generate apparent complexity.Recursive self-extension distributes representational burden across successor systems; coupled agents can also supply information not derivable from an individual’s history.Source: §5.5 · pp. 31–32

5.6 Compression and Semantic Growth

• Compression within a semantic regime is compatible with extension: individual systems may shrink while accumulated history gains irreducible information.Semantic extension enlarges the space of representable distinctions rather than merely minimizing the current representation.Source: §5.6 · p. 32

5.7 Relation to Open-Ended Intelligence

• Open-ended intelligence requires constructing a successor regime with distinctions not derivable from its predecessor, rather than optimizing within a fixed representational space.The physical-system implication is that novelty must come from outside an individual architecture or the assumption fails in real systems.Source: §5.7 · pp. 32–33

6 Collective Intelligence and Distributed Semantic Extension

• Theorem 1 applies to every bounded individual, and a finite collective without persistent external information remains a fixed architecture subject to the same bound.The section identifies external information as the escape route: a collective can receive representations not derivable from an agent’s own history.Source: §6 · p. 33

6.1 Collective Cognitive System / 6.2 Distributed Semantic Surplus

• Partial translations let agents communicate across different languages while retaining nonshared representations; one agent’s surplus can become another’s unresolved region and prompt further extensions.Content resolved by one agent can exert recursive pressure on another and return in transformed form.Source: §6.1 · p. 33, §6.2 · p. 34

6.3 Collective Co-Recursion

• Collective co-recursion allows transformations to cross agent boundaries, but interaction among finitely many bounded agents cycles unless persistent external memory accumulates their semantic history.With persistent memory that agents can read and write, the collective history is not bounded by any individual state space or by their finite joint state space.Source: §6.3 · p. 34
• Heterogeneous agents can supply bounded agents with distinctions unavailable from their own experience, while teleological homogenization removes the irreducible information needed for continued novelty.When cross-agent and memory-mediated distinctions have positive asymptotic algorithmic density, a bounded agent can realize a trajectory satisfying the amortized novelty condition.Source: §6.3 · p. 35

6.4 Collective Memory as Extended Cognitive Space

• Collective memory must preserve both generated content and its semantic coordinates, including its origin, inferential context, and transformation history.Without those coordinates, retrieved representations cannot be correctly reinterpreted because meaning may change across recursive extension.Source: §6.4 · p. 36

6.5 Collective Teleology / 6.9 Proxy Teleology and Autonomous Intelligence

• Collective purposes need not coincide, but agents must evaluate their relations and make teleology recursively accessible rather than externally fixed.The architecture allows a system to select successor purposes without specifying which admissible purpose it adopts; autonomy is underdetermination, not arbitrariness.Source: §6.5 · p. 36, §6.9 · pp. 37–38

6.6 Open-Endedness Beyond the Individual / 6.7 A Hegelian Interpretation / 6.8 From Understanding to Speculative Intelligence

• Open-endedness means continuing to construct semantically relevant successor states without a fixed lifetime bound, despite finite physical substrates at each moment.The section frames recursive preservation and transformation as a formal analogue of Hegelian Aufhebung, and describes speculative intelligence as treating distinctions, ambiguities, and purposes as objects of further determination.Source: §6.6 · p. 36, §6.7 · p. 37, §6.8 · p. 37

6.10 Diversity and Common Ground / 6.11 Civilizational Co-Recursion

• Productive collective interaction requires both shared representational ground for translation and difference that can generate transformations neither agent reaches independently.Civilizational language, artifacts, institutions, and persistent records let transformations pass between bounded people and persist beyond any participant’s lifetime.Source: §6.10 · p. 38, §6.11 · p. 38

6.11 Civilizational Co-Recursion

• Different recursive extension selections yield different directed systems, so X∞ is one realized trajectory rather than an architectural invariant, with continuum-many trajectories possible.The translation τ_n need not preserve interpretation, and the successor adopted is underdetermined by M_n.Source: §6.11 · p. 39
• Subject to these qualifications, civilization lets finite cognitive systems participate in a semantic process whose representational history is not bounded by any one component’s state.Source: §6.11 · p. 39

• A finite cognitive system can participate in open-ended intelligence when its semantic representations, memory, transformations, and teleological structures can be recursively coupled to those of other bounded systems.Source: §7 · p. 39
• Individual recursion, memory, polycontexturality, teleology, and collective coupling respectively support self-extension, semantic history, coexisting interpretations, transformation selection, and cross-agent processes.Source: §7 · p. 39
• Intelligence can remain physically finite while its semantic process remains open-ended.Source: §7 · p. 39

8 Biological Inspirations and Architectural Motifs

• Biological analogies do not validate the formal theorems; they suggest plausible counterparts for the framework’s structural requirements.The paper explicitly distinguishes analogy from empirical support for its mathematical claims.Source: §8 · p. 39, §8.1 · p. 40, §8.8 · p. 45

8.1 Neural Analogues of Semantic Surplus

• Retaining unintegrated episodic information resembles Iloc, but the findings do not imply a biological counterpart to formal undecidability.Discarding unintegrated information as noise would lose material for later reorganization.Source: §8.1 · p. 40

8.2 Statistical Processing with Structured Scaffolds

• Biological cognition combines statistical inference with hierarchical, distributed, and structurally constrained representations; these processes can operate within structures that higher-order mechanisms transform.The paper distinguishes statistical computation from recursive transformation of the representations in which it occurs.Source: §8.2 · p. 41, §8.5 · p. 43

8.3 Reference Frames as Semantic Coordinates

• Recursive extension can change what stored content means, so memory must preserve relational coordinates as well as content; whether reference frames also support selection remains unsettled.The biological analogy is not evidence for a literal implementation of semantic organization.Source: §8.3 · pp. 41–42

8.4 Geometric and Multi-Scale Organization

• The biological examples span periodic coordinate systems, morphological branching, and scale-free dynamics, but do not establish fractal semantic computation.The paper treats these as distinct phenomena and limits their relevance to analogies for geometric or multi-scale organization.Source: §8.4 · pp. 42–43

8.5 Hierarchical Constraints in Language

• Biological language processing offers an analogy in which statistical prediction operates within hierarchical, recursively organized structures rather than replacing them.Higher-order mechanisms can work on structures that organize statistical computations.Source: §8.5 · p. 43

8.6 Biological Safeguards for Recursive Learning

• Recursive learning requires both plasticity and protection: systems must modify representations without overwriting semantic history needed for future recursion.A robust initial structure can be reorganized over time rather than built anew from scratch.Source: §8.6 · pp. 43–44

8.7 Relationship to the Free Energy Principle

• The Free Energy Principle describes inference and action within a generative model, while this framework addresses recursive modification of representational structure.The paper proposes that local prediction-error reduction can coexist with using persistent errors or unexplained observations to extend representation.Source: §8.7 · p. 44

9 Discussion: The Problem of Epistemic Convergence

• Open-ended intelligence depends on preserving the capacity to generate new distinctions, not merely accumulating or compressing knowledge.Accumulation retains empirical information; compression reduces representational and search costs within a regime; generation constructs new semantic structures when ambiguity persists.Source: §9 · p. 45

9.1 Collective Generative Capacity

• Collective generation requires enough overlap for transfer and enough distinctiveness for new continuations; neither maximum overlap nor maximum divergence is productive.The section treats generative capacity as a trade-off, not a scalar to maximize.Source: §9.1 · p. 46

9.2 AI and the Convergence of Semantic Space

• Shared models, data, or evaluation criteria can correlate pursued continuations, including when agents generate diverse options but select similarly.The authors do not claim contemporary systems inevitably converge; they identify convergence as a structural risk under rewards that consistently favor it.Source: §9.2 · pp. 46–47

9.3 Epistemic Convergence and Open-Endedness

• If agents converge on the same regime and transformations, interaction adds no successor options; collective open-endedness instead depends on unresolved structures and variability among agents.The paper distinguishes solving more problems within Xn from developing Xn+1.Source: §9.3 · p. 47

9.4 The Epistemic Drive

• An open-ended system must couple local exploitation with co-recursive exploration when decidable structure saturates, even though the transition cannot be certified within the current formalism.The paper distinguishes this structural exploration from expected-utility exploration within a fixed formalism.Source: §9.4 · pp. 47–48

9.5 Implications for AGI

• AGI evaluation should assess trajectories across recursive regime transitions, including whether failed derivations are transformed into new representational primitives and purposes.The paper argues that a static benchmark measures performance within one frozen regime rather than this movement.Source: §9.5 · pp. 48–49