Conceptual Rift: Mechanisms & Theory Change

• Conceptual Rift is a residual component in conceptual systems where current models fail to capture all aspects of experience, triggering potential growth in understanding.
• It spans both geometric frameworks using MDL-based basis extension and topological models in HDCS, illustrating how structured novelty emerges from unresolved errors.
• Practical insights include its role in theory change and innovation, linking persistent anomalies to low-complexity repairs that enable coherent conceptual evolution.

Conceptual rift denotes a structured failure of an existing conceptual organization. In the representational-geometric account of conceptual growth, it is the residual component that remains when experience is projected onto the current conceptual subspace; in Heraclitean Dialectical Concept Spaces, it is the open remainder produced when adjacent conceptual neighbourhoods partially overlap but do not fit together coherently. In both formulations, the rift is not mere absence. It is a positive locus of unresolved structure from which new concepts can emerge, either through low-rank basis extension governed by Minimum Description Length or through the stabilization of a new open region that inherits topology from parent concepts (Amornbunchornvej, 21 Dec 2025, Yildiz, 31 Dec 2025).

1. Residual and remainder formulations

In the MDL-constrained geometric framework, the current conceptual system is modeled as a finite-dimensional real vector space,

$\mathcal C = \mathrm{span}\{b_1,\dots,b_k\}.$

For an experience vector $u$, the residual relative to $\mathcal C$ is

$r_{\mathcal C}(u) := u - \Pi_{\mathcal C}u.$

This residual lies in $\mathcal C^\perp$ and represents what the current concepts cannot yet encode. The residual span over a dataset $D$ is

$$W := \mathrm{span}\{r_{\mathcal C}(u):u\in D\}\subseteq \mathcal C^\perp.$$

The paper states that conceptual rifts are precisely these residuals: the “space of ignorance” revealed when the current conceptual basis fails (Amornbunchornvej, 21 Dec 2025).

In HDCS, the formal object is topological rather than metric. A concept space is a triple $(C,\mathcal F,N)$, where $C$ is the set of concepts, $\mathcal F\subseteq\mathcal P(C)$ is a feasible family of coherent regions, and $u$0 assigns to each concept a nonempty family of neighbourhoods. The external topology $u$1 is defined by

$u$2

Within that topology, a conceptual rift is “a topological and structural rupture” in which existing conceptual regions no longer fit together coherently, leaving an open remainder that is not adequately covered by the parent concepts (Yildiz, 31 Dec 2025).

Framework	Formal object of the rift	Conceptual growth mechanism
Representational geometry	$u$3 and $u$4	$u$5
HDCS	$u$6	$u$7

These two formulations differ in mathematical apparatus, but both treat conceptual rupture as structured and generative rather than as an uninformative error term.

2. Basis extension under MDL

The geometric framework constrains conceptual change to admissible basis extension. A candidate extension is written

$u$8

Admissibility includes bounded novelty rank,

$u$9

and $\mathcal C$0-reducibility / closure, under which

$\mathcal C$1

is also admissible. The paper proves

$\mathcal C$2

Conceptual growth is therefore low-rank and structurally bounded (Amornbunchornvej, 21 Dec 2025).

Selection among extensions is governed by Minimum Description Length. For a subspace $\mathcal C$3, the description length is

$\mathcal C$4

where $\mathcal C$5 is nondecreasing and $\mathcal C$6 is the complexity penalty per new dimension. An admissible extension is MDL-accepted iff

$\mathcal C$7

For a one-dimensional extension, the paper gives the threshold rule

$\mathcal C$8

A central technical result is that novelty orthogonal to the residual span is rejected. If $\mathcal C$9, then

$r_{\mathcal C}(u) := u - \Pi_{\mathcal C}u.$0

for all $r_{\mathcal C}(u) := u - \Pi_{\mathcal C}u.$1, so the fit term does not change, while the complexity term increases by $r_{\mathcal C}(u) := u - \Pi_{\mathcal C}u.$2. Hence

$r_{\mathcal C}(u) := u - \Pi_{\mathcal C}u.$3

with strict inequality if $r_{\mathcal C}(u) := u - \Pi_{\mathcal C}u.$4. The paper formalizes this through the propositions “Residual-supported dominance” and “No MDL gain from purely $r_{\mathcal C}(u) := u - \Pi_{\mathcal C}u.$5 novelty,” culminating in the theorem “Counterfactual Basis Extension under MDL” (Amornbunchornvej, 21 Dec 2025).

The resulting picture is conservative: new concepts are warranted only when unresolved residual structure is coherent enough to pay for itself.

3. Topological rupture and emergence in HDCS

HDCS begins with locality rather than projection. The feasible family satisfies

$r_{\mathcal C}(u) := u - \Pi_{\mathcal C}u.$6

$r_{\mathcal C}(u) := u - \Pi_{\mathcal C}u.$7

and the neighbourhood assignment satisfies

$r_{\mathcal C}(u) := u - \Pi_{\mathcal C}u.$8

$r_{\mathcal C}(u) := u - \Pi_{\mathcal C}u.$9

$\mathcal C^\perp$0

Adjacency is defined by neighbourhood overlap,

$\mathcal C^\perp$1

Because adjacency is not transitive, concepts can be locally connected without collapsing into a single uniform structure. From adjacency the framework builds the overlap family

$\mathcal C^\perp$2

and then the channel ideal

$\mathcal C^\perp$3

A nontrivial channel ideal means there is some open conceptual corridor between the concepts (Yildiz, 31 Dec 2025).

The formal heart of conceptual rift is the remainder construction. For an interaction profile

$\mathcal C^\perp$4

with $\mathcal C^\perp$5, the remainder is

$\mathcal C^\perp$6

The paper emphasizes that

$\mathcal C^\perp$7

The rift is therefore a positive open region of unintegrated structure, not emptiness in a trivial sense. New concept formation is governed by the CCER mechanism, the Cumulative Core Emergence Rule. In its core form,

$\mathcal C^\perp$8

The emergent region is an open set and inherits topology from its parent regions. Its internal topology is the induced subspace topology,

$\mathcal C^\perp$9

Where the geometric account privileges residual alignment and complexity control, HDCS privileges local coherence, overlap, and remainder. Both nonetheless treat new concepts as constrained by the structure of prior failure.

4. Counterfactuals, imagination, and developmental continuity

The representational-geometric model distinguishes representational counterfactuals from causal or value-level counterfactuals. The question is not “What would happen if variable $D$0 were different?” but “What would become representable if we added a new conceptual dimension?” Internally generated counterfactual representations contribute to learning only insofar as they expose or amplify structured residual error. The paper identifies two mechanisms: directional enrichment, in which simulation enlarges the residual span, and threshold amplification, in which simulation strengthens residual evidence enough to satisfy the MDL threshold (Amornbunchornvej, 21 Dec 2025).

This yields a strong negative claim as well. Imagination cannot introduce arbitrary novelty. If a simulated scenario is already well explained by the current model, or if it produces only unstructured noise, it does not justify conceptual change. The paper characterizes imagination as constrained novelty rather than free invention (Amornbunchornvej, 21 Dec 2025).

HDCS supplies a different developmental machinery. Conceptual change is stage-based: each stage $D$1 has a concept space $D$2, and change is tracked by carry maps

$D$3

The evolution rule is

$D$4

where the events include emergences and edits. The global history is formed by the quotient

$D$5

with global inclusion

$D$6

and stage-to-global evolution map

$D$7

The paper’s claim is that even though a concept may split, transform, or give rise to a new open region, the carry maps and colimit topology keep all stages in a single connected historical structure (Yildiz, 31 Dec 2025).

Taken together, these accounts treat conceptual rift as a dynamic operator on conceptual systems: one via admissible basis extension, the other via staged topological inheritance.

5. Theory change, anomaly, and innovation pressure

The geometric framework explicitly generalizes conceptual rift to theory change. A theory is treated like a conceptual subspace that compresses observations, anomalies are residuals, and theory change occurs when anomalies accumulate in a structured way that makes a new representational dimension worth the complexity cost. The paper presents this as a formal version of the idea that scientific revolutions are driven by persistent anomalies, but only those that admit a low-complexity repair (Amornbunchornvej, 21 Dec 2025).

An empirical backdrop for this claim appears in the study of technological concept spaces built from TechNet, a technology semantic network containing 4,038,924 technology concepts derived from all 5,771,030 granted U.S. utility patents from 1976 to October 2017. That study reports that the cumulative number of technological concepts grows linearly rather than exponentially, that the mean semantic similarity of all concepts increased by 23% from 1981 to 2016, that the mean semantic similarity between new and prior concepts increased by 31%, and that the mean additional information content contributed by new concepts decreased by 21% (Sarica et al., 2023).

The same paper defines

$D$8

and for a new concept $D$9,

$$W := \mathrm{span}\{r_{\mathcal C}(u):u\in D\}\subseteq \mathcal C^\perp.$$0

Its interpretation is that innovation faces a paradox: accumulated prior art creates more opportunities for recombination but also raises the burden of knowledge and the difficulty of originality (Sarica et al., 2023).

This suggests an empirical condition under which conceptual rifts may become both more frequent and more difficult to resolve. Residual failure can accumulate in an ever-growing conceptual landscape, while the search for genuinely new, low-complexity repair becomes harder. The same study proposes Creative Artificial Intelligence as a possible way to alter those trends by absorbing prior knowledge, generating new combinations, and evaluating originality at scale (Sarica et al., 2023).

6. Related uses, extensions, and controversies

Not all uses of rift are internal to a single formal learning model. In the conceptuality interpretation of quantum mechanics, the apparent divide between quantum microphysics, cognition, and conceptual meaning is presented not as a gap to be bridged by a psychophysical collapse story, but as a sign that “quantumness” and “conceptuality” are “just two terms referring to a same reality, or nature, which manifests at different organizational levels.” The paper argues that quantum entities are conceptual-like, that concepts are “highly contextual entities,” and that quantum probabilities are epistemic rather than ontic in the relevant sense (Aerts et al., 2018).

A different extension appears in foundational physics, where the “Great Rift” is described as a real incompatibility between empirically verified quantum predictions and the locality principles Einstein built into relativity. The claim is not merely that there is a challenge in unifying two formalisms, but that Bell’s theorem and related arguments show that Einstein-style locality cannot be maintained if the quantum predictions are correct. On that reading, what has to give is the relativistic account of spacetime structure and dynamics understood as local causal structure (Maudlin, 25 Mar 2025).

These uses clarify an important misconception. Conceptual rift is not always treated as a deficit to be eliminated. In the representational and topological frameworks, the rift is productive: a residual or remainder that can support principled conceptual growth. In the broader theoretical literature, however, rift can denote a durable incompatibility between frameworks or ontologies. This suggests a family resemblance rather than a single universal definition.

The current literature therefore supports three tightly related meanings. First, conceptual rift can name representational failure made explicit as residual structure. Second, it can name a topological rupture that leaves an open remainder from which a new concept emerges. Third, it can designate a deeper discontinuity between established theoretical frameworks, where the issue is not local repair but revision of ontology, causation, or conceptual architecture itself (Amornbunchornvej, 21 Dec 2025, Yildiz, 31 Dec 2025, Maudlin, 25 Mar 2025).