The R/K/G Framework

Definitions

Assumptions

A.1 — Lineage: fluid and crystallized intelligence

The decomposition of intelligence into independent factors has a long empirical history. In 1941, Raymond Cattell and Donald Hebb independently proposed that intelligence comprises two separable components: a capacity for novel reasoning (which Cattell named fluid intelligence, G_f) and an accumulated store of knowledge and learned procedures (crystallized intelligence, G_c). This distinction was refined through decades of factor-analytic research, culminating in the Cattell–Horn–Carroll (CHC) framework — the most empirically validated psychometric model of human intelligence.

Three findings from the CHC tradition are directly relevant to cognitive engineering:

Diminishing returns on fluid intelligence. G_f peaks in early adulthood and declines thereafter. Its contribution to task performance is strongest on novel problems and weakest on tasks where domain knowledge dominates. Beyond a moderate threshold, additional G_f produces marginal improvements.

Compounding returns on crystallized intelligence. G_c accumulates throughout the lifespan with no observed ceiling. Each piece of knowledge connects to existing knowledge, creating combinatorial growth in the problem space an individual can address. G_c is superlinear in its contribution to practical task performance.

The investment hypothesis. Cattell proposed that G_f ‘invests’ in G_c — higher fluid intelligence accelerates the rate of knowledge acquisition. This means G_f has both a direct effect on reasoning quality and an indirect effect through its influence on the growth rate of G_c. However, once knowledge is acquired, it has independent value regardless of the G_f that produced it.

These findings have been replicated across thousands of studies and multiple cultures. They are not controversial in psychometrics. What they describe, however, is a two-variable model of intelligence — the interaction of reasoning capacity and knowledge. We propose a three-variable extension.

A.2 — What if we add another axis: reasoning structure (G)

The CHC framework treats the structure of the reasoning process as a byproduct of fluid intelligence. In the psychometric model, how you reason is a consequence of how much G_f (D1) you have. A person with high G_f naturally decomposes problems, checks intermediate results, and considers alternatives. A person with low G_f relies on pattern matching and cached heuristics. We assume the reasoning structure is entangled with the reasoning capacity (A2).

If that assumption holds, the entanglement is a property of biological cognition, not of intelligence itself.

In engineered reasoning systems, we can treat the structure of the reasoning process as an independent, designable variable (A3). An IRG graph specifies how the system reasons — the topology, the node types, the iteration depth, the convergence criteria — separately from the model’s base reasoning capability (R, D3) or knowledge (K, D4). This separation is what would make cognitive engineering possible, and it is the contribution we propose relative to the CHC tradition.

We propose a three-factor decomposition of cognitive system output quality:

The key claim — and it is a claim, resting on A2 and A3 rather than a proof: in innate, unaided cognition, G is entangled with R. You cannot simply decide to hold more items in working memory, or to run a cleaner search over hypotheses; how well you reason and how you reason come as a package.

The obvious objection is that humans plainly do control the shape of their reasoning — we follow recipes, work through algorithms, do long division, apply IRAC, run the scientific method. That objection is the point. Every one of those is an external procedure, a reasoning structure invented and written down precisely because it cannot be varied natively. Long division lets a person of ordinary capacity execute a computation reliably; a checklist lets a surgeon of ordinary memory not skip a step. These cultural technologies are humans supplying G from outside the individual — manually, one procedure at a time. So separability is not a novel discovery; people have been hand-engineering reasoning structure for millennia. What is new is that in an engineered system, G stops being a fixed recipe you follow and becomes a programmable variable you can specify, measure, and vary while holding R and K constant.

If that separability survives testing, it is not merely a technical convenience. It is the structural precondition for effective reasoning graphs.

A.3 — The System 1 / System 2 correspondence

Daniel Kahneman’s dual-process theory provides an intuitive bridge between the CHC tradition and the cognitive engineering framework. System 1 (fast, automatic, pattern-matching) corresponds to a shallow graph: stimulus → cached response, with no decomposition, no verification, no iteration. System 2 (slow, deliberate, effortful) corresponds to a deeper graph with evaluation nodes, revision loops, and convergence checks.

When cognitive scientists advise people to ‘think more carefully,’ they are prescribing deeper graph execution. When educators teach ‘show your work,’ they are requiring trace production. When critical thinking curricula instruct ‘consider the counterargument,’ they are specifying an adversarial evaluation node. These are all intuitive descriptions of graph design. What they lack is formalization.

This correspondence suggests that what is colloquially called ‘intelligence’ in humans may conflate three separable factors: reasoning capability (R), knowledge (K), and the quality of internal reasoning structures (G). People we call ‘smart’ may disproportionately possess well-developed internal graphs — through education, practice, or disposition — rather than categorically superior raw reasoning capacity. In humans, we cannot test this hypothesis because G is not independently observable. In engineered systems, we can.

A.4 — A proposed equation

Before writing anything down, it is worth asking what shape the relationship should take — and three everyday observations point the way. Picture a brilliant reasoner with almost no relevant knowledge: the 200-IQ physicist asked to rule on a Reg E dispute. The raw reasoning is dazzling and the answer is often wrong, because there is little correct knowledge for it to operate on. Now picture the opposite: a high-schooler of ordinary cleverness with a well-indexed library. On most real tasks the high-schooler wins, because the knowledge is there to be retrieved and combined. And picture either of them made to show their work — decompose the problem, check each step, consider the counterargument — and watch the answer improve without anyone getting smarter or learning anything new.

Those three observations suggest three different functional shapes. Reasoning capability (R) shows diminishing returns: past a threshold, more raw horsepower buys little — the physicist was already smart enough. That is the signature of something log-like. Knowledge (K) shows compounding returns: each fact is worth more in the presence of the others it connects to, so the library beats the genius — the signature of a power that accelerates. And structure (G) is not a fourth ingredient competing with the other two; it amplifies both, and it does the most amplifying when the base is weakest — showing your work helps the struggling student more than the expert.

Turning those three shapes into one expression gives a candidate family of models for cognitive system output quality (Q, D6). We write one representative of it below. A companion note — Why the Curve Has This Shape — derives it from the three assumptions just stated, and is explicit about the catch: each assumption pins down a shape (rising-and-flattening, accelerating, gap-proportional), not an exact function, and other curves satisfy the same shapes. Read the equation as the simplest member of that family, meant to be fit to data rather than asserted (A4):

Where Q(t) is output quality for task t; R is the base reasoning capability of the system; K is knowledge sufficiency for task t — the accurate, relevant, retrievable knowledge available to it, not raw corpus size; G is graph design quality (0 = no graph / unstructured inference); α is the reasoning sensitivity coefficient (controls steepness of diminishing returns); and β is the knowledge compounding exponent (β > 1 captures superlinear returns on knowledge). All variables are normalized 0–1; the log term is divided by log(1 + α) so that it, too, runs 0–1 and Q stays genuinely normalized. The companion note works the shape of each term out from its assumption.

The amplification terms G·(1−R) and G·(1−K) encode the property the model is built around: if the model holds, graph design helps weak systems more than strong ones. When R is already high, (1−R) is small, so G adds little. When R is low, (1−R) is large, so G provides substantial amplification. This is consistent with our observation that structured reasoning (IRG scaffolding) disproportionately benefits smaller language models — though consistency is not yet calibration (A4).

Behavioral properties

The equation produces the following qualitative behaviors, each of which is independently testable:

A.5 — The three regimes of investment

The marginal returns on investing in each factor reveal three distinct regimes of optimization priority:

Marginal return on R: ∂Q/∂R is proportional to α / (1 + αR). This is a monotonically decreasing function — each additional unit of reasoning capability is worth less than the last. Investment in R has the highest ROI when R is low, and diminishing ROI thereafter.

Marginal return on K: ∂Q/∂K is proportional to β · K^β−1. For β > 1, this is a monotonically increasing function — each additional unit of knowledge is worth more than the last. Investment in K has increasing ROI with scale, which would explain why knowledge-rich systems compound their advantage.

Marginal return on G: ∂Q/∂G is proportional to both (1−R) and (1−K). Graph design investment has highest ROI when reasoning and knowledge are both moderate — because G amplifies both simultaneously. When either R or K is already near ceiling, G adds less.

Critical nuance: these regimes are not mutually exclusive. All three factors can be optimized simultaneously, and in practice the AI field has been pulling all three levers at once. The regimes describe which lever yields the highest marginal return at a given point in the optimization landscape, not which lever is exclusively valuable. The scaling laws accurately describe R and K investment returns when G is held constant. They are a special case of a three-variable optimization, not the complete picture.

A.6 — Four proposed tenets of cognitive engineering

From the framework above we propose four tenets. We state them as propositions, not laws: each is an intuition we find compelling and a hypothesis we intend to test, and we note after each where existing work is consistent with it — and, for the one where it is contested, where it is not.

TENET 1 — Diminishing returns on reasoning. Beyond a moderate threshold of reasoning capability, the highest-return investment in cognitive system quality may be the structure of the reasoning process itself — not the capability of the reasoner or the volume of available knowledge, because graph design (G) can raise effective reasoning and effective knowledge at once. Consistent with the finding that adding reasoning structure to a fixed model improves its output: chain-of-thought prompting (Wei et al., 2022), self-consistency (Wang et al., 2022), and tree-of-thoughts search (Yao et al., 2023) all raise quality without changing model weights.

TENET 2 — Knowledge compounding. Knowledge sufficiency may compound superlinearly with output quality: each unit of relevant knowledge creates combinatorial connections with what is already present, producing accelerating returns, so that across most practical task distributions a well-informed moderate reasoner outperforms a brilliant uninformed one. Consistent with the effectiveness of retrieval-augmented generation (Lewis et al., 2020), where supplying relevant knowledge to a fixed model reliably improves knowledge-bound tasks.

TENET 3 — Graph amplification. A well-designed reasoning graph may raise the effective capability floor of both reasoning and knowledge, with disproportionate benefit to less capable base systems — making optimal graph depth inversely related to base model capability. This is the tenet whose direction is genuinely contested, and we flag it as such. The emergent-abilities literature finds that unprompted chain-of-thought helps larger models more, with little or negative benefit below a scale threshold (Wei et al., 2022). Against that, explicit task decomposition and verification appear to help smaller models close specific gaps. The two findings may not conflict — emergent CoT and engineered decomposition are different interventions — but which effect dominates for IRG-style graphs is exactly what this tenet leaves open and testable.

TENET 4 — Separability. What is observed as ‘intelligence’ in natural systems may be the entanglement of reasoning capability (R), knowledge (K), and reasoning structure (G): in innate cognition these are hard to vary independently, whereas in engineered systems they can be. On this reading, cognitive engineering is the discipline that becomes possible once G is separable. Grounded in the CHC tradition’s separation of G_f from G_c (A1), and in the everyday existence of externalized reasoning procedures — algorithms, proofs, checklists — through which humans already supply reasoning structure from outside the individual (A.2).

Tenet 4 is the load-bearing one. Without it, the other three would be observations about a single entangled phenomenon called intelligence, and there would be no discipline — only a capability to improve. With it, G could be treated as an independent design variable, the other three tenets would describe the return curves on three separable investments, and cognitive engineering would have a theoretical basis. Whether it holds is the question the empirical program is meant to answer.

What would validate each: Tenet 1 requires demonstrating that graph design improvements outperform equivalent investment in model scaling on matched tasks. Tenet 2 requires showing superlinear K contribution across multiple domains. Tenet 3 requires resolving the contested direction above — showing whether, and when, structured reasoning benefits smaller models disproportionately. Tenet 4 requires demonstrating that G can be varied independently of R and K with predictable, measurable effects on output quality. The framework’s empirical program is designed to test all four.

A.7 — Relationship to scaling laws

The neural scaling laws (Kaplan et al., 2020; Hoffmann et al., 2022) describe the relationship between model size, training data, and loss. They have been the dominant framework for understanding AI capability improvement. Our framework does not contradict scaling laws. It proposes to contextualize them.

Scaling laws describe the returns on R and K investment when G is held constant (G ≈ 0, unstructured inference). In the scaling era (Regime 1), this was the correct optimization frame because G was approximately zero for everyone — all models ran unstructured. The returns on R (model scale) and K (training data) were high because both factors were below their diminishing-returns thresholds.

The ‘scaling is hitting a wall’ narrative reflects the approach to Regimes 2 and 3 — the point where marginal returns on R and K diminish and the G variable becomes the dominant lever. From this perspective, scaling laws are not wrong. They are a two-dimensional projection of a three-dimensional optimization surface. They accurately describe the R–K plane. They say nothing about the G axis because it was zero during the period they were measured.

The cognitive engineering hypothesis, in scaling-law terms: the loss surface may have a third axis. On this reading, the models are not hitting a wall so much as approaching a ridge in the R–K plane, with the next improvement direction orthogonal to it — along G. We offer this as a testable prediction, not a conclusion.

A.8 — Scope and limitations

This theoretical framework has deliberate boundaries:

It is a model of cognitive system output quality, not a general theory of intelligence. It does not address consciousness, qualia, intentionality, or other philosophical dimensions of mind. It addresses the engineering question: how do measurable design choices affect measurable output quality?

The functional forms are proposed, not proven. The logarithmic contribution of R, the power-law contribution of K, and the linear amplification of G are initial hypotheses based on qualitative observation. The actual functional forms may be different. The empirical program will calibrate them.

The model assumes task-specificity of K. Knowledge sufficiency K is defined relative to a specific task, not as a global property. A system may have high K for medical diagnosis and low K for legal analysis. This is deliberate: it captures the reality that knowledge relevance is task-dependent.

The model does not address learning. The Cattell investment hypothesis (G_f invests in G_c) describes how reasoning capability accelerates knowledge acquisition over time. Our static model captures a snapshot: given current R, K, and G, what is the output quality? A dynamic extension incorporating learning trajectories is a natural next step but is beyond the current scope.

Human cognition is not the target. While the System 1/System 2 correspondence and the educational instruction mapping are suggestive, this framework is designed for engineered systems where G is independently observable and manipulable. Claims about human intelligence are illustrative, not prescriptive.