Knowledge Discovery Efficiency (KEDE) and Ashby's Law of Requisite Variety

In section 2, we provide a detailed overview of Ashby's Law of Requisite Variety, including its mathematical formulation and implications for system regulation. We also discuss the present day understanding of residual variety and its significance in the context of Ashby's Law. In section 3, we discuss the challenges of applying Ashby's Law to real-world systems, including combinatorial explosion, dual control dilemma, and resource constraints. We propose a solution to these challenges by treating the system as a black box, observing probability of successful outcomes to disturbances, and estimate the gap in its internal variety based on that. In section 4, we start establishing the solution by introducing the Knowledge Discovery Process, which is narrowing down and selecting the appropriate response from its set of alternative responses. We then label H(X|Y) as the knowledge to be discovered, which is the gap in internal variety that had to be compensated by selection. In section 5, we show how to quantify the knowledge to be discovered H(X|Y) based on the observable outcomes E. In section 6, we generalize the Knowledge-Discovery Efficiency (KEDE) - scalar metric that quantifies how efficiently a system closes the gap between the variety demanded by its environment and the variety embodied in its prior knowledge. Finally, in section 7, we explore applications of KEDE in various domains, demonstrating its utility as a diagnostic tool for evaluating system performance and adaptability.

Mathematical Formulation

In Ashby's terms, we can think of the system as a transducer:

Where:

• D be the set of disturbances, representing the possible states of disturbances that a system may experience.
• R be the set of responses, representing the possible regulatory actions that counteract disturbances.
• O be the set of realized outcomes, representing the possible outcomes that can result from the disturbance-response pairs (D, R).
• E be the subset of acceptable outcomes, representing the values / essential-variable states that the system aims to achieve, induced by a valuation mapping v : O → E. E may be as simple as the 2-element set {good, bad}, and is commonly an ordered set. Some subset of E can be defined as the “goal”[8].

In the deterministic pay-off-matrix case, the environment is the fixed function T : D x R → O, and evaluation is v : O → E. If D and R are treated as random variables over episodes, then the induced conditional distribution is P(E | D, R) determined by E = v(T(D,R)). In a noisy environment, replace T by a channel P(O | D, R).

This can be represented as a Table of Outcomes (T) where:

• T be the pay-off matrix, i.e. a fixed mapping T : D x R → O or the fixed transition rule of the environment.
• Rows represent disturbances D
• Columns represent regulatory responses R
• Entries represent realized outcomes O = {o₁₁, o₁₂, o₁₃, ..., o₂₁, o₂₂, o₂₃, ..., o₃₁, o₃₂, o₃₃, ..., ...}
in the cells of the matrix, O = {o} = T(d,r) ∈ O, where d is a disturbance and r is a regulatory response.

The Table of Outcomes (T) is a fixed, structured space of possibilities, or the fixed transition rule of the environment, which directly determines the essential variables E. In simplified analyses, O and E are sometimes conflated, but they are conceptually distinct. In the deterministic pay-off-matrix case, the environment is the fixed function T : D x R → O, and evaluation is v : O → E. If D and R are treated as random variables over episodes, then the induced conditional distribution is P(E | D, R) determined by E = v(T(D,R)).

The regulator's accumulated structure M is its learned law of action. In the deterministic case, M : D → R. In general (learning, uncertainty), treat it as a policy P(R | D), i.e. how probability mass is allocated across responses for each disturbance.

Across time, the learning object M is not fixed. If the table O is the space of possibilities, then M is the mechanism that induces a measure over possible paths through that space.

Learning and rework do not alter the structure of the Table of Outcomes T. Instead, they modify the regulator's accumulated structure M. Changes in M alter which disturbance-action pairs become more or less probable and therefore how the system's realized trajectories are distributed over the table O.

Rework corresponds to a revision accumulated structure M: probability mass is shifted away from previously selected or ineffective responses and reassigned to alternative responses for the same disturbance. Thus, the Table of Outcomes (T) remains a fixed space of possibilities, while learning and rework change only the regulator's induced distribution of trajectories through that space, not the space itself.

Each input from D is transformed via selection into a response $r\in R$, which is then processed to produce an outcome $o\in O$. which is then evaluated as $e\in E$, which is the essential variable that the system aims to achieve, induced by a valuation mapping v : O → E.

If R does nothing, i.e. keeps to one value, then the variety in D threatens to go through O to E, contrary to what is wanted. It may happen that O, without change by R, will block some of the variety and occasionally this blocking may give sufficient constancy at E. More commonly, a further suppression at E is necessary; it can be achieved only by further variety at R.

Under idealized assumptions, Ashby showed that a lower bound on the achievable variety in the outcomes is given by the smallest variety that can be achieved in the set of actual outcomes cannot be less than the quotient of the number of rows divided by the number of columns. $$\mathrm{V}\left(\mathrm{E}\right)\ge \frac{\mathrm{V}\left(\mathrm{D}\right)}{\mathrm{V}\left(\mathrm{R}\right)}$$

Where:

• V(D) = variety in the disturbances
• V(R) = variety in the regulatory responses
• V(E) = variety in the outcome (essential variables)

This inequality shows that the variety in the outcomes cannot be reduced below the division between the variety in the disturbances and the variety in the regulatory responses.

Perfect regulation here means deterministic essential outcome, not merely acceptable performance. For perfect regulation, we need V(E) to be stable or close to 1 (only one possible outcome):

The law reflects a fundamental insight: control is about the ratio of disturbances to regulatory responses, not just their arithmetic difference.

Information-Theoretic Formulation

All acts of regulation can be related to the concepts of communication theory by noticing that the “disturbances” correspond to noise, and the “goal” is a message of zero entropy, because the target value E is constant. Thus, the law of Requisite Variety says that R 's capacity as a regulator cannot exceed R 's capacity as a channel of communication.

The variety is measured by the logarithm of its value. If the logarithm is taken to base 2, the unit is the bit of information. In practice, we use the Shannon information entropy, denoted by H. For a quantifiable variable, entropy is just another measure of variance.

Applying this to Ashby's Law we get:

If we assume equiprobable disturbances/responses and treat variety as cardinality, then H reduces to ${\log }_{2}V$, yielding:

Ashby's law can thus be reformulated clearly:

The information-processing capacity (entropy) of a control system must be at least as large as the information (entropy) in the system it regulates.

Ashby's Law can be interpreted as a cybernetic analogue of Shannon's "Noisy channel coding theorem" which states that communication through a channel that is corrupted by noise may be restored by adding a correction channel with a capacity equal to or larger than the noise corrupting that channel. The disturbance D, which threatens to get through to the outcome E, clearly corresponds to the noise; and the correction channel is the system R, which is supposed to restore the outcome E[8].

It has been shown shown that the law of requisite variety can be extended to include knowledge or ignorance by simply adding this conditional uncertainty term[31] When buffering is present, part of the environmental variety is absorbed passively before reaching the regulator. This reduces the effective disturbance entropy by an amount K, which is the buffering capacity:

Where:

• H(E) is the residual variety i.e. the realized essential-variable distribution, not the size of the set E
• H(R) is the entropy of the regulator, representing its information-processing capacity.
• H(D) is the entropy of the disturbances, representing the complexity of the environment.
• H(R|D) is the conditional entropy of the regulator given disturbances, representing the lack of requisite knowledge i.e. the ignorance of the regulator about how to react correctly to each appearance of a disturbance D. Only a regulator that knows how to use the available regulatory variety H(R) to react correctly to each disturbance D will reach the optimal result of regulation,
• K is the buffering capacity measured in bits of disturbance variety absorbed before reaching the regulator. Buffering is the passive absorption or damping of disturbances i.e. the amount of noise that a system can absorb without requiring an active regulatory response.

A necessary but not sufficient condition for effective control is: $H\left(R\right)\ge H\left(D\right)$ . Sufficiency additionally requires $H\left(R|D\right)\approx 0$

Successful (essential) outcomes E do not depend solely on the variety of responses H(R) available to a regulator R; the system must also know which response to select for a given disturbance. Effective compensation of disturbances requires that the system possess the ability to map each disturbance to an appropriate response from its repertoire. The absence or incompleteness of such knowledge can be quantified using the conditional entropy H(R|D)[31]. In other words, H(R|D) measures how much the regulator R lacks the requisite knowledge to match responses to disturbances. In the absence of such requisite knowledge, the system would have to select responses, until eliminating all disturbances. Thus, merely increasing the response variety H(R) is not sufficient; it must be complemented by a corresponding increase in selectivity, that is, reduction in H(R|D) i.e. increasing knowledge. H(R|D) = 0 represents the case of no uncertainty or complete knowledge, where the action is completely determined by the disturbance. This requirement may be called the law of requisite knowledge[29].

H(R|D) reminds us that response alone is not sufficient: if the regulator does not know which response is appropriate for the given disturbance, it can only try out regulatory actions at random, in the hope that one of them will be effective and that none of them would make the situation worse. The larger the H(R|D), the larger the probability that the regulator would choose a wrong regulatory response, and thus fail to reduce the variety in the outcomes H(E). Therefore, this term H(R|D) has a “+” sign in the inequality: more uncertainty (less knowledge) produces more variation in the essential variables E[54].

To achieve control, the regulator R must possess sufficient information-processing capacity (entropy) such that the following is achieved: $$H\left(R\right)\ge H\left(D\right)$$ $$H\left(R|D\right)=0$$

In other words, the complexity of the environment D can not exceed that of the system R, which means the system R fully matches the environment D[7].

Since $H\left(R\right)-H\left(R|D\right)=I(R:D)$ the law simplifies to: $$H\left(E\right)\ge H\left(D\right)-I\left(R:D\right)-K$$

The mutual information I(R:D) represents the requisite knowledge of the regulator R about how to react correctly to each disturbance D, i.e. the amount of regulatory variety that is effectively correlated with and therefore absorbs the variety in the disturbances. Such knowledge may be realized structurally as the regulator's learned law of action M represented by a mapping M: D → R , by which disturbances are mapped to regulatory responses[54]. The mutual information I(R:D) quantifies how much of the regulator's learned law of action M: D → R effectively couples disturbances to responses, while the remaining uncertainty H(R|D) quantifies the lack of requisite knowledge[29].

Solution

These challenges significantly hinder the practical application of Ashby's Law. Whether considering a human, an AI model, or an organization, we are typically limited to observing external behavior rather than internal mechanisms—unless we are able to "open the box."

Ashby himself emphasized that all real systems can be considered black boxes. He argued that while black boxes mimic the behavior of real objects, in practice, real objects are black boxes: we have always interacted with systems whose internal workings are, to some extent, unknown.

This leads to what Ashby termed the black box identification approach [2], which involves:

1. Perturbing the system by applying external disturbances,
2. Measuring the system's responses to these perturbations, and
3. Inferring the internal variety or capacity from the observed input-outcome relationships.

In most practical scenarios, we are only able to observe the outcomes of a system. These observable outcomes can be used to infer bounds on the system's internal variety—specifically, the extent of variety it must possess or lack in order to exhibit the observed behavior.

We propose such an approach: to treat the system as a black box, observe the probability of successful outcomes to disturbances, and estimate the gap in its internal variety based on that. Let E denote the event that the system gives a correct response to disturbance D, and let R be the regulator's action. In information-theoretic terms, perfect regulation requires H(R|D) = 0[31]. Using our novel information-theoretic estimator, empirical estimates of P(E=1) are used to quantify H(R|D) in bits of information. This quantifies how much disorder the regulator still permits and, conversely, how close the system comes to meeting Ashby's requisite-variety condition.

Multi-stage selection

We model each stage outcome as S₁, S₂, …, Sₖ. When mapping one stage of selection with one selection signal Z_i we have:

Staged selection is additive only in incremental terms. If stages share information or impose overlapping constraints, summing their marginal 'reductions' overcounts. The correct decomposition credits each stage only for the reduction of residual uncertainty left by previous stages Then he per-stage contributions must be conditional (incremental) to avoid double counting, because the stages may share information or constrain overlapping parts of X. The correct staged accounting is $$I\left(X;S_1,\dots ,S_k\right)=\sum _{i=1}^{k}I\left(X;S_i|S_{<i}\right)$$ i.e., each stage is credited only for reducing the residual uncertainty left by earlier stages.

As selections accumulate, the remaining uncertainty H(X | Y, Z₁…Zᵢ) shrinks. Successful adaptation occurs when this uncertainty collapses to zero, meaning the system now knows exactly which response works for the given disturbance. The total amount of selection Ashby describes is mathematically identical to the total mutual information accumulated across stages.

Shannon's chain rule for mutual information is:

After k selections we have:

Given the success criterion of determining the correct response X for a given disturbance Y, Ashby's Law of Requisite Variety for a staged selection process is:

Or equivalently, the sum of the bits removed by selection by every stage must at least equal the bits of uncertainty injected by the original range of possibilities or by disturbances.

If there can be multiple acceptable responses for a given disturbance, then the “right object” is not “entropy of X”, but entropy of success i.e. of essential variables E. Thus, we would replace X with an equivalence class of actions or replace the criterion with the entropy of success H(E|Y,Z)=0.

Regulation

Regulation uses staged information $Z_{1:k}$ to eliminate uncertainty within a stage, where k is the number of selection signals within a stage.

During the stage, staged selection supplies $Z_{1:k}$ until the choice is effectively determined:

Learning (across stages)

Regulation can use within-stage evidence Z to determine the correct response X for a given disturbance Y during the stage. Learning makes that success persistent by updating the stored structural coupling (mapping) M so future stages start with less uncertainty.

We treat M_t as the system's stored mapping at stage t (a parameter, not a random variable). To avoid conditioning on a function, we write entropies and mutual informations as induced by M_t:

So: regulation uses realized evidence Z to drive uncertainty down within a stage (e.g., it is H(X|Y, Z=z) that collapses after observing a particular z), while learning stores the effect of that evidence by updating M so future stages start with lower H_t(X|Y).

Rework as negative contribution to accumulated coupling (negative learning)

Sometimes what was stored in the mapping later proves incorrect (e.g., failing test, production defect, requirement change), forcing a revision of the mapping: M_t+1 = Revise(M_t, E_t). We do not model memory mechanics (what is overwritten, versioned, or kept). We only track the net effect of the mapping change on the system's induced uncertainty/coupling measures.

Coupling and uncertainty measures (induced by the mapping). Let:

Here, K_t is “stored requisite coupling” and U_t is “lack of requisite knowledge” (remaining uncertainty about which response works given the disturbance), both evaluated under the mapping M_t.

Exact knowledge ledger

Define the net change in stored coupling as the actual difference:

Then define gross learned bits and retracted (rework) bits as the positive and negative parts of that change:

Equivalently (as difference measures on the mapping update):

With these definitions, the additive ledger is exact (a tautology, not an assumption):

Therefore:

• If G_t > 0, the mapping update yields net learning (stored coupling increases).
• If L_t > 0, the mapping update yields net rework (stored coupling decreases).
• Mixed stages are allowed: the net is captured by Δ_tK.

Uncertainty ledger (parallel view)

We can track lack of requisite knowledge” directly as U_t = H_t(X|Y). Define Δ_tU = U_t+1 - U_t, and split it into its positive/negative parts:

This view is always valid. If we want to link it to the coupling ledger via U_t = H_t(X) - K_t, we must be explicit about whether and how the marginal H_t(X) changes as the mapping changes. Absent constraints on the marginal H_t(X), the coupling and uncertainty ledgers are complementary but not interchangeable.

Rework criterion. Rework at stage t is simply: L_t > 0 (equivalently K_t+1 < K_t ).

Rework is measured directly as the drop in the coupling measure induced by the mapping change M_t → M_t+1.

Link to linear information flow, structural coupling (mapping) impact

The evidence stream is linear (stages), but the object edited is the entire structural coupling (mapping) M. The per-stage chain rule quantifies how many bits the stage could contribute via staged selection, while the retraction term quantifies how many previously stored bits must be removed due to replacement. Rework is not a separate phenomenon; it is the negative part of the same additive knowledge accounting.

Residual Variety

In the context of Ashby's Law of Requisite Variety, residual variety refers to the variety that the regulator fails to absorb. In other words, the remaining uncertainty or uncontrolled states in a system after a regulator has applied its available counter-actions. Information-theoretically, this can be understood in two ways:

• The residual variety is the uncertainty H(X|Y) about the regulator's response given the disturbance i.e., the regulator's remaining uncertainty about what action to take given a known disturbance D. It focuses on the input side - how much the regulator doesn't know about the disturbance that is hitting the system.
• The residual variety is the uncertainty H(E) in essential variables, or operationally H(B) when outcomes are binarized into success/failure. It focuses on the outcome side - how much uncertainty persists in what we care about the essential variable.

Both interpretations quantify how much uncertainty is left once the regulator has made its move, because H(X|Y) upper-bounds the achievable reduction in H(E) given the fixed table T. If the regulator perfectly counters every disturbance (full requisite variety), the residual variety in both forms would be zero.

Observability of Residual Variety

Now we focus on one important consideration, which is that E (essential variable) values are observable. These are the outcomes we can measure and care about - system performance, outcome quality, stability measures, etc. Being observable, we can empirically estimate H(E) by collecting data on how E varies given different regulator states R This makes H(E) (residual variety) a measurable quantity in practice.

H(X|Y) presents observability challenges, as disturbances D may not be directly observable i.e. they could be internal system dynamics, environmental factors we can't measure, or complex interactions we can't decompose. Even if some disturbances are observable, the full set D might include hidden or latent factors that we cannot directly measure or quantify. This makes H(X|Y) potentially unobservable or only partially estimable and often theoretical or abstract.

This creates an important asymmetry in cybernetic analysis:

• H(E) is observable and measurable via H(B), allowing us to empirically assess how well the regulator is performing in controlling outcomes.
• H(X|Y) may not be fully observable, making it difficult to quantify the regulator's knowledge of disturbances it's regulating.

This is why H(E) is often the more practically useful measure - it tells us what we can observe about system performance. That is reflected in the literature as well, where work focuses on observable outcomes rather than unobservable disturbances[29,30,35,36,37,38,39,40,41,42,43,44,45].

Defining Knowledge to be Discovered

Ashby's Law of Requisite Variety concerns capacity: the regulator must have a sufficiently large repertoire of possible responses[1]. Heylighen's Law of Requisite Knowledge adds the missing condition for effective regulation: it is not enough to have many possible actions; the regulator must know which action to select for the given disturbance. Otherwise, increased action variety increases the chance of choosing the wrong action, forcing trial-and-error selection[29].

In information-theoretic terms, this “knowing which action to select” is exactly the reduction of uncertainty about the response X once the disturbance Y is observed. That uncertainty is the conditional entropy H(X|Y).

Knowledge To Be Discovered. We define Knowledge To Be Discovered as the conditional entropy H(X|Y): the expected uncertainty about which response(s) will achieve success, given that the disturbance is known. Operationally, it is the amount of selection (in bits) that must still be supplied in order to determine a successful response.

Thus, Knowledge To Be Discovered is the precise difference between the action variety available and the coupling/selectivity actually achieved:

Here:

• H(X) is the regulator's available response variety (the size/diversity of the action repertoire), or the prior knowledge.
• I(X;Y) is the regulator's requisite knowledge: how much the disturbance actually informs the choice of action, or the required knowledge.
• H(X|Y) is the remaining uncertainty after the disturbance is known, i.e. the lack of requisite knowledge, or the knowledge to be discovered.

Requisite knowledge is not an independent thing but the total selectivity actually available at decision time makes H(X|Y) small which can be stored structure M or online evidence Z:

• Offline (stored) coupling: learning updates the mapping M so that future stages start with higher I_t(X;Y) and therefore lower H_t(X|Y).
• Online (within-stage) selection: evidence Z (tests, observations, constraints) supplies additional conditional information I(X;Z|Y), driving H(X|Y,Z) toward 0 during regulation.

In the ideal case of complete adaptation for the task class, the regulator's selectivity is perfect: H(X|Y) = 0 (the response is effectively determined by the disturbance). If multiple responses are acceptable, we will replace X with the equivalence class of acceptable actions or measure success directly via the goal variable.

This yields the intended conceptual pairing:

• Residual variety H(E): what remains uncontrolled in outcomes.
• Knowledge To Be Discovered H(X|Y): what remains undetermined about which action succeeds, given the disturbance.

This framing suggests that reducing "knowledge to be discovered" is a pathway to reducing "residual variety" - which is exactly the cybernetic insight that better information about disturbances enables better control. It emphasizes the constructive, learning-oriented aspect of cybernetic systems rather than just their current limitations.

Rework, waste, and throughput reduction under fixed capacity

Under fixed execution capacity, rework reduces effective throughput because some capacity is spent revising prior commitments rather than closing new stages. In an information-theoretic view, this is the case where an attempted “learning update” produces negative or zero net retained information over the chosen ledger horizon:

Key operational distinction. The execution channel observes attempted stage-closes (commitment emissions), but the model’s “realized outcomes” are net stage closures that survive the ledger horizon. Let:

• N (or N_max) be anchored action capacity in the window (selections + outcome-emissions).
• S be the number of outcome-emissions (stage-closing commitment attempts) observed in the window.
• W be the number of wasted outcome-emissions in that same window: commitments whose retained contribution over the ledger horizon is ≤ 0 (e.g., reverted/rolled back/fully undone).
• Then S_net = S - W is the number of realized (net) outcomes attributable to that window under the ledger rule.

Because wasted outcomes still consumed capacity, waste must be carved out of observed outcomes, not out of “idle time.” This forces the bookkeeping identities (under the single-channel / no-idle assumption):

• Net throughput decreases: $S_{\text{net}}=S-W$
• Net non-outcome debits increase (selection-equivalent depth): $Q_{\text{net}}=N_{\text{max}}-S_{\text{net}}=N_{\text{max}}-(S-W)=(N_{\text{max}}-S)+W=Q+W$

Operational missing information with rework. Rework is not a hidden variable in this model: it appears immediately as reduced $S$ (net stage closures) under fixed capacity. The operational missing information is the average non-outcome debits per net stage closure:

Then, we can state the Knowledge To Be Discovered theorem:

Consider a regulator operating under the following conditions:

• There is a single shared execution channel with no idle time, so that each discrete time unit is used either to:
  1. perform one binary selection (one yes/no question that halves the remaining candidate set), or
  2. emit one externally visible outcome.
• For each disturbance realization $Y=y$, there exists a finite candidate set ${\Omega }_y$ of admissible responses $X$, with prior distribution $P\left(X\right|Y=y)$.
• The regulator applies an optimal binary questioning strategy for identifying the correct response $X$ given $Y=y$.

Then the expected number of binary selections required to identify $X$ satisfies the Shannon bound:

$$H\left(X|Y=y\right)\le E\left(k|Y=y\right)<H\left(X|Y=y\right)+1$$

where $k$ is the number of binary selections preceding an outcome. Averaging over disturbances yields:

$$H\left(X|Y\right)\le E\left(k\right)<H\left(X|Y\right)+1$$

Consequently, over any fully-utilized observation window of length $N$ time units on the shared execution channel, let:

• $S$ be the number of outcome-emissions (stage-closing commitments) in the window.
• $Q$ be the number of binary selections in the window.
• $W$ be the number of wasted outcome-emissions (commitments that do not count as realized outcomes), with $0\le W\le S$. The number of realized outcomes is therefore $S-W$.

Since each time unit is used either for one selection or one outcome-emission (no idle time), the channel identity is:

$$N=(Q+W)+(S-W)$$

Define the empirical mean number of selections per realized outcome:

$$E^{^}\left(k\right)=\frac{Q+W}{S-W}=\frac{N}{S-W}-1$$

This ratio is directly observable from execution counts and provides an operational estimator of the expected search depth.

Under the stated assumptions, this estimator satisfies:

$$H\left(X|Y\right)\le \frac{N}{S-W}-1<H\left(X|Y\right)+1$$

(4)

and therefore bounds the conditional entropy — the knowledge to be discovered — up to an additive constant.

The result is operational: it links an abstract information-theoretic quantity $H\left(X|Y\right)$ to directly observable execution counts $(N,S)$, without assuming access to the true distribution. It provides an operational upper bound on conditional entropy under idealized assumptions (optimal binary selection, single shared channel, no idle capacity). It does not claim to recover the true entropy exactly in arbitrary systems.

If there is no knowledge to be discovered, i.e. there is no need to make selections at all, then S equals N and H(X|Y) is zero. This is not redefining entropy — it is measuring it operationally.

We can rearange the bounds to be:

Conservative Estimator (Operational Upper Bound)

In the idealized case where the posterior over Ω_y is uniform and |Ω_y| is a power of two, the optimal binary decision tree is perfectly balanced and the bound is tight: E[k] = H(X|Y).

Operationally, over a time window with maximum action rate N and observed outcome rate S, we define the estimator:

$$\widehat{H}\left(X|Y\right)=\frac{N}{S-W}-1$$

(5)

We call Ĥ(X|Y) a conservative estimator of the Knowledge To Be Discovered if: (i) it is computable solely from observables (N, S), and (ii) it upper-bounds the latent conditional entropy H(X|Y).

Formally, Ĥ(X|Y) is an operational upper bound on the true conditional entropy, with a guaranteed approximation error of less than one bit:

This bound follows from the optimality of binary decision trees (or prefix codes): the expected decision depth cannot be smaller than the conditional entropy H(X|Y), and integer decision depths imply an excess of at most one bit.

Interpretation. Ĥ(X|Y) measures the empirically observed expected decision depth per outcome. It is therefore a conservative (never-underestimating) operational proxy for the true Knowledge To Be Discovered H(X|Y). In the idealized uniform, power-of-two case it is exact; otherwise it overestimates by less than one bit. Whatever the true H(X|Y) is, it cannot exceed what is implied by the observed execution capacity.

Anchoring KEDE to Natural Constraints

In our model, N is always the theoretical maximum action rate (selections + outcomes) in an unconstrained environment, and S is the observed outcome rate under specific conditions over a given interval.

A key question is how to assign a natural constraint to N. That is, what constitutes an appropriate reference value for the maximum action rate (selections + outcomes)?

We may turn to physics for an instructive analogy. A quantum (plural: quanta) represents the smallest discrete unit of a physical phenomenon. For instance, a quantum of light is a photon, and a quantum of electricity is an electron. In this context, the speed of light in a vacuum serves as a fundamental upper bound for N. However, identifying an analogous natural constraint for human activity—particularly knowledge work—presents greater challenges.

Consider the example of typing. Here, the quantum can reasonably be defined as a symbol, since it is the smallest discrete unit of text. A symbol may be a letter, number, punctuation mark, or whitespace character. To determine the appropriate bin width Δt, we refer to empirical data on the minimum time required to produce a single symbol. Typing speed has been subject to considerable research. One of the metrics used for analyzing typing speed is inter-key interval (IKI), which is the difference in timestamps between two keypress events. We see that IKI is defined equal to the symbol duration time t. Hence we can use the research of IKI to find the symbol duration time t. Studies have reported an average IKI of 0.238 seconds [26], yielding a maximum human typing rate of approximately N=1/t=1/0,238=4.2 symbols per second

A similar approach can be applied to tasks such as furniture assembly. In this case, a plausible quantum is a single screw tightened, since it represents a minimal, repeatable unit of outcome. We then identify Δt as the average time required to tighten one screw. Empirical studies report that this task typically takes between 5 and 10 seconds[34]. Using the upper bound, we estimate the maximum screw-tightening rate as N=1/t=1/10=0.1 screws per second.

This methodology offers a principled way to estimate N using domain-specific quanta and empirically grounded time durations, enabling the application of our model to a broad range of human tasks.

The next question concerns the appropriate definition of outcome for measuring S and N.

Both N and S can always be discretized—or “binned”—in a way that preserves the total information rate, regardless of whether the outcome arises from natural processes, human behavior, or machines. By choosing a bin width Δt small enough (e.g., milliseconds), the range of possible tangible outcomes within each bin shrinks dramatically. This reduced range leads to less uncertainty in each bin, which compensates for the smaller time interval. Yet the ratio

remains an accurate measure of information rate.

As Δt becomes smaller, the measurements of S and N become more precise, as they reflect outcome over finer time intervals. But how small should Δt be? This dilemma is resolved by considering the granularity of outcomes associated with the outcome. The set E of outcomes can be thought of as the effects of the regulation process — the resulting states after the regulator responds to disturbances. In our model E is a sequence of {0,1}, where 0 = wrong outcome(failure to regulate) and 1 = correct outcome(successful selection). So the presence of a concrete outcome leads to a natural binning of the outcomes, It also enables a clear distinction between signal (the entropy associated with producing the outcome) and noise (the residual variability unrelated to success or failure).

For example, two distinct symbols typed (e.g., ‘a' vs. ‘b') are clearly different outcomes. However, if one symbol is typed in 91 milliseconds and another in 92 milliseconds, this minute variation is inconsequential to the outcome. Such timing fluctuations are typically unintentional, irrelevant to task performance, and should not be considered part of the outcome. In practical terms, if the theoretical upper bound N is known—for instance, 4.2 symbols per second as derived from human typing speed, and the observed rate is S=1 symbol per second, then time should be partitioned into one-second bins. Each bin then yields a single outcome: either 1 (a symbol was successfully typed) or 0 (no symbol typed or incorrect input).

This binning principle generalizes beyond typing. Whether analyzing foot strikes in trail running (where negligible spatial change occurs over milliseconds) or the discrete moves in solving a Rubik's cube (where each turn resolves multiple potential states into a single action), binning ensures that no intermediate state need be modeled explicitly.

Tightening screws

We can apply our model to motor tasks such as furniture assembly. In this context, a natural unit of outcome — or “quantum” — is the tightening of a single screw.

Skilled workers engaged in manual assembly tasks can typically insert and tighten standard screws at a rate of 6'-12 screws per minute under optimal, repetitive conditions — such as those found in furniture construction or industrial assembly lines. In contrast, automated screw-tightening machines can achieve significantly higher rates, often between 30 and 60 screws per minute [34] More complex manual tasks, such as high-torque applications involving ratchets or Allen keys, typically reduce the rate to 2'-4 screws per minute due to the increased effort and precision required. In surgical or medical contexts, such as orthopedic screw insertion, accuracy and the avoidance of overtightening are paramount; here, rates often fall to 1'-2 screws per minute, or approximately one screw every 30'-60 seconds [46].

The key observation is that rates decrease as torque, task complexity, or required precision increases. If we take the machine rate as the maximum possible outcome N and the observed human rate as S, we can estimate the average number of bits of information H(X|Y) that the human operator must process per action.

This implies that the human must absorb approximately 4 bits of information, on average, to tighten a single screw under typical conditions.

The rate at which a person tightens screws depends on various factors, including:

• Screw type and size
• Material being fastened
• Required torque
• Tool used (screwdriver, ratchet, etc.)
• Operator skill and fatigue

This interpretation aligns with existing research, which suggests that task difficulty directly influences the amount of information a task imparts [47, 48]. When difficulty is appropriately matched to the individual's skill level, the task yields maximal informational value [49], and the time required reflects the interaction between task complexity and the individual's regulatory capacity [50].

Using our model, we transform a sequence of real-world actions in furniture assembly into a granular, time-based measure of regulatory capacity. This enables us to quantify — in bits — how much variety the individual must absorb in order to successfully complete the task.

Typing the longest English word

Let's use an example scenario to see Ashby's law applied to human cognition and knowledge work.

For that we'll have myself executing the task of typing on a keyboard the word “Honorificabilitudinitatibus”. It means “the state of being able to achieve honours” and is mentioned by Costard in Act V, Scene I of William Shakespeare's “Love's Labour's Lost”. With its 27 letters “Honorificabilitudinitatibus” is the longest word in the English language featuring only alternating consonants and vowels.

The way I will execute this task is to go to the "play text" or "script" of “Love's Labour's Lost”, look up the word and type it down. The manual part of the task is to type 27 letters. The knowledge part of the task is to know which are those 27 letters.

In order to track the knowledge discovery process I will put "1" for each time interval when I have a letter typed and "0" for each time interval when I don't know what letter to type.

I start by taking a good look at the word “Honorificabilitudinitatibus” in the script of “Love's Labours' Lost”. That takes me two time intervals. Then I type the first letters “H”, “o”, and “n”.I continue typing letter after letter: “o”, “r”. At this point I cannot recall the next letter. What should I do? I am missing information so I go and open up the script of “Love's Labours Lost” and I look up the word again. Now I know what the next letter to type is but acquiring that information took me one time interval. This time I have remembered more letters so I am able to type “i”,”f”,”i”,”c”,”a”,”b”,”i”. Then again I cannot continue because I have forgotten what were the next letters of the word, so I have to look it up again.in the script. That takes two more time intervals. Now I can continue my typing of “l”,”i”,”t”. At this point I stop again because I am not sure what were the next letters to type, so I have to think about it. That takes one time interval. I continue my typing with “u”,”d”,”i”. Then I stop again because I have again forgotten what were the next letters to type, so I have to look it up again in the script of “Love's Labours Lost”. That takes two more time intervals. Now I know what the next letter to type is so I can continue typing “n”,”i”.At this point I cannot recall the next letter. so I have to look it up again in the script. That takes two more time intervals. After I know what the next letter to type is I can continue typing “t”,”a”,”t”,”i”,”b”,”u”,”s”. Eventually I am done!

At the end of the exercise I have the word “Honorificabilitudinitatibus” typed and along with it a sequence of zeros and ones.

In the table we have separated the manual work of typing from the knowledge work of thinking about what to type.

We made visible both the manual work and the knowledge discovery parts of a Knowledge Discovery process.

The first row of the table shows the knowledge I manually transformed into tangible outcome - in this case the longest English word. The second row of the table shows the way I discovered that knowledge. There is a "0" for each time interval when I was missing information about what to type next. There is "1" for each time interval when I had prior knowledge about what to type next. Each "0" represents a selection I needed to ask in order to acquire the missing information about what letter to type next. Each "1" represents prior knowledge.

We know that there is knowledge applied when we see the tangible outcome of the process. We know there was knowledge discovered when we see there was at least one selection made.

In the exercise above we witnessed the discovery and transformation of invisible knowledge into visible tangible outcome.

KEDE calculation

We can calculate the KEDE for this sequence of outcomes.

We can also calculate the knowledge discovered H(X|Y) in bits of information.

We've turned a real-world sequence of action and hesitation into a fine-grained, time-based measurement of regulatory capacity — effectively measuring how much variety I needed to absorb with external help i.e. my knowledge discovered.

Measuring software development

In order to use the KEDE formula (6) in practice we need to know both S and N. We can count the actual number of symbols of source code contributed straight from the source code files. For N we want to use some naturally constrained value.

N is the maximum number of symbols that could be contributed for a time interval by a single human being.

To achieve this, the following estimation is performed. We pick T = 8 hours of work because that is the standard length of a work day for a software developer.

To calculate the value of r we need to pick the symbol duration t.

The value of the symbol duration time t is determined by two natural constraints:

1. the maximum typing speed of human beings
2. the capacity of the cognitive control of the human brain

Typing speed has been subject to considerable research. One of the metrics used for analyzing typing speed is inter-key interval (IKI), which is the difference in timestamps between two keypress events. We see that IKI is defined equal to the symbol duration time t. Hence we can use the research of IKI to find the symbol duration time t. It was found that the average IKI is 0.238s [26]. There are many factors that affect IKI [6]. It was also found that proficient typing is dependent on the ability to view characters in advance of the one currently being typed. The median IKI was 0.101s for typing with unlimited preview and for typing with 8 characters visible to the right of the to-be-typed character but was 0.446s with only 1 character visible prior to each keystroke [7]. Another well-documented finding is that familiar, meaningful material is typed faster than unfamiliar, nonsense material[8]. Another finding that may account for some of the IKI variability is what may be called the “word initiation effect”. If words are stored in memory as integral units, one may expect the latency of the first keystroke in the word to reflect the time required to retrieve the word from memory[55].

Cognitive control, also known as executive function, is a higher-level cognitive process that involves the ability to control and manage other cognitive processes that permit selection and prioritization of information processing in different cognitive domains to reach the capacity-limited conscious mind. Cognitive control coordinates thoughts and actions under uncertainty. It's like the "conductor" of the cognitive processes, orchestrating and managing how they work together. Information theory has been applied to cognitive control by studying the capacity of cognitive control in terms of the amount of information that can be processed or manipulated at any given time. Researchers found that the capacity of cognitive control is approximately 3 to 4 bits per second[32][33], That means cognitive control as a higher-level function has a remarkably low capacity.

Based on the above research we get:

1. Maximum typing speed of human beings to be r=1/t=1/0,238=4.2 symbols per second
2. Capacity of the cognitive control of the human brain to be approximately 3 to 4 bits per second. Since we assume one question equals one bit of information we get 3 to 4 questions per second.
3. Asking questions is an effortful task and humans cannot type at the same time. If there was a symbol NOT typed then there was a question asked. That means the question rate equals the symbol rate, as explained here.

In order to get a round value of maximum symbol rate N of 100 000 symbols per 8 hours of work we pick symbol duration time t to be 0.288 seconds. That is a bit larger than what the IKI research found but makes sense when we think of 8 hours of typing. Having t of 0.288 seconds makes a symbol rate r of 3.47 symbols per second. That is between 3 and 4 and matches the capacity of the cognitive control of the human brain.

We define CPH as the maximum rate of characters that could be contributed per hour. Since r is 3.47 symbols per second we get CPH of 12 500 symbols per hour. We substitute T = h and r=CPH and the formula for N becomes:

where h is the number of working hours in a day and CPH is the maximum number of characters that could be contributed per hour. We define h to be eight hours and get N to be 100 000 symbols per eight hours of work.

Total working time consist of four components:

• Time spent typing (coding)
• Time spent figuring out WHAT to develop
• Time spent figuring out HOW to code the WHAT
• Time doing something else (NW)

Let us assume an ideal system where the time spent doing something else T_NW is zero. Using the new formula for N the formula for H becomes

Note, that since N is calculated per hour so S also needs to be counted in an hour.

We see that the more symbols of source code contributed during a time interval the less missing information was there to be acquired. We want to compare the performance of different software development processes in terms of the efficiency of their knowledge discovery processes. Hence we rearrange the formula to emphasize that.

The right hand part is the KEDE we defined earlier. Thus, we define an instance of the metric KEDE - the general metric that we introduced earlier. This version of KEDE is for the case of knowledge workers that produce tangible outcome in the form of textual content:

KEDE from (8) contains only quantities we can measure in practice. KEDE also satisfies all properties we defined earlier. it has a maximum value of 1 and minimum value of 0; it equals 0 when H is infinite; it equals 1 when H is zero; it is anchored on a natural constraint—the maximum typing speed of a human being.

If we convert the KEDE formula into percentages then it becomes:

We can use KEDE to compare the knowledge discovery efficiency of software development organizations.

Testing Intelligence

Today all measure intelligence by the power of appropriate selection (of the right answers from the wrong). The tests thus use the same operation as is used in the theorem on requisite variety, and must therefore be subject to the same limitation. (D, of course, is here the set of possible questions, and R is the set of all possible answers). Thus what we understand as a man's “intelligence” is subject to the fundamental limitation: it cannot exceed his capacity as a transducer. (To be exact, “capacity” must here be defined on a per-second or a per-question basis, according to the type of test.)[3]

We can also use our model to the testing of human and AI intelligence. We infer this capacity from performance under variety — i.e., how many different problems a system or a person can solve correctly.

The dominant mathematical models for testing intelligence by the number of answered problems are benchmark datasets like MMLU, GSM8K, MATH, and FrontierMath. These models measure intelligence by the raw count or percentage of correctly solved problems, with more advanced benchmarks designed to minimize guessing and require deep reasoning.

From the knowledge-centric perspective:

• The disturbances are the questions $Q=\{q_1,q_2,q_3,...q_n\}$
• The person gives responses $R=\{r_1,r_2,r_3,...r_n\}$
• The outcomes are $E=\{e_1,e_2,e_3,...e_n\} \in \{0,1\}$

Several mathematical models and benchmark datasets are used to evaluate intelligence—especially artificial intelligence (AI)—by measuring the number and complexity of math problems answered correctly. These models serve as standardized tests for both AI and, by analogy, human intelligence[52].

Massive Multitask Language Understanding (MMLU):

• MMLU is a widely used benchmark that tests AI models on a broad range of subjects, including mathematics at various levels (high school, college, abstract algebra, formal logic).
• The test is typically formatted as multiple-choice questions, and performance is measured by the percentage of correct answers out of the total number of questions
• For example, advanced AI models have achieved up to 98% accuracy on math sections of MMLU, indicating high proficiency in standard math tasks but not necessarily deep reasoning

Grade School Math 8K (GSM8K)

• GSM8K is a dataset of 8,500 high-quality, grade school-level word problems designed to test logical reasoning and basic arithmetic skills.
• Evaluation is based on exact match accuracy: the number of problems answered exactly correctly divided by the total number attempted
• This benchmark is used to assess step-by-step reasoning and the ability to handle linguistic diversity in problem statements.

MATH (Mathematics Competitions Dataset)

• MATH consists of problems from high-level math competitions (e.g., AMC 10, AMC 12, AIME), focusing on advanced reasoning rather than rote computation.
• Performance is measured by the percentage of correct answers, with human experts (e.g., IMO medalists) providing a reference for top-level performance
• The dataset is challenging for both humans and AI, with LLMs typically scoring much lower than expert humans.

FrontierMath[53]

• FrontierMath is a new benchmark featuring hundreds of original, expert-level math problems spanning major branches of modern mathematics.
• Problems are designed to be "guessproof" and require genuine mathematical understanding, with automatic verification of answers
• The benchmark is used to assess how well AI models can understand and solve complex mathematical problems, similar to human performance.

In human intelligence testing, Psychometric models such as IQ tests or psychometric approaches also use the number of correctly answered problems as a key metric. These tests are standardized, and the raw score (number of correct answers) is often converted into a scaled score or percentile.

As an example we will use the Exact Match metric as the evaluation method[52]. Given that each question in our benchmark dataset has a single correct answer and the model produces a response per query, Exact Match ensures a rigorous evaluation by comparing the extracted answer to the ground truth.

Let ${ŷ}_i$ represent the extracted answer from the model's outcome for the $i^{\mathrm{th}}$ question, and let $y_i$ be the corresponding ground truth answer. The Exact Match accuracy is computed as:

where:

• $N$ is the total number of evaluated questions.
• $𝟙(\cdot )$ is the indicator function, returning 1 if the extracted model response matches the ground truth after preprocessing, and 0 otherwise.
• $\text{normalize}(\cdot )$ is a function that standardizes formatting, trims spaces, and normalizes numerical values.

The knowledge discovery efficiency of an LLM can be calculated as:

Let's pick the case of the performance of GPT-4o on the MATH benchmark, which achieved a significantly lower accuracy of 64.88%, lagging behind its peer models[52]. Now, we can calculate the average knowledge discovered H(X|Y).

Basketball Game

We can also use this model to assess the performance of a basketball player.

• Timeframe is a basketball game.
• We observe N total shot attempts.
• S of them are successful (shot made).
• We record a binary outcome sequence
  $E\in \{0,1{\}}^N$
• The empirical success rate:
  $\theta =\frac{S}{N}$
  is our observed probability of success.

Interpretation using Ashby's Law

The basketball shot is a regulation problem: the player must control their body and respond to the game environment to produce the desired outcome. The player is faced with a series of disturbances (D) in the form of different shots to make under different conditions. The player responds with a selection, drawn from their internal skills (regulatory variety R) in the form of different shooting techniques. Each shot is uncertain whether it will be successful. The outcome E is whether the shot is made (1) or missed (0).

Over N shots, the success rate

In this case, θ becomes a practical proxy for how often the regulator (player) has sufficient internal variety to absorb the disturbance presented by the game. However, it is important to note that this is a simplified model and does not account for all the complexities of basketball performance. For example, the player may have different success rates depending on the type of shot, the position on the court, or the level of defense. These factors can all affect the player's ability to regulate their performance and should be considered when interpreting the results. Thus, as explained here θ is a useful heuristics for P(E=1), but the full picture includes the quality of mapping, not just quantity.

Applying the Model

NBA keeps track of field goal attempts and makes for each player. The most field goal attempts by a player in a single NBA game is 63, achieved by Wilt Chamberlain during his legendary 100-point game against the New York Knicks on March 2, 1962 We take this as the natural constraint so N=63. We can also take the number of successful shots S=36, which is the most field goals made in a single game by a player[13].

We can calculate the KEDE for this sequence of outcomes.

We can also calculate the knowledge discovered H(X|Y) in bits of information.

That means that the player needed to absorb 0.75 bits of information on average to make the shot.

We've turned a real-world sequence of basketball shots into a fine-grained, time-based measurement of a regulatory capacity — effectively measuring how much variety the player needed to absorb.

We can also use this model to assess the performance of a basketball team. In this case the success rate coincides with the field goal percentage (FG%) of the team which is the percentage proportion of made shots over total shots that a player or a team takes in games. There is a statistical distribution for NBA field goal percentage (FG%) [10]. Analysts and researchers often study the distribution of FG% across players or teams to understand scoring efficiency and trends[11]. The NBA record for the highest FG% in a single game by a team is 69.3%, set by the Los Angeles Clippers on March 13, 1998, when they made 61 of 88 shots[12].

For example, in the 2023-24 season, team FG% ranged from about 43.5% (lowest) to 50.6% (highest), with the league average typically falling in the mid-to-high 40% range[11]. if we take the average FG% of 45% , we can calculate the average knowledge discovered H(X|Y).

That means that a team needed to absorb 1.22 bits of information on average to make a shot.

Assembly Line

We can also use this model to assess the knowledge discovery efficiency of an assembly line.

The assembly line is a system that transforms raw materials into finished products. The assembly line has a set of disturbances (D) in the form of different raw materials, machines, and processes. The assembly line responds with a selection, drawn from its internal structure (R) in the form of different machines, processes, and workers.

From a knowledge-sentric perspective most of the knpwledge discovery happens in the design phase of the assembly line. This is the planning for design, fabrication and assembly. This activity has also been called design for manufacturing and assembly (DFM/A) or sometimes predictive engineering. It is essentially the selection of design features and options that promote cost-competitive manufacturing, assembly, and test practices[51]. Thus most of the disturbances D are already absorbed by the design of the assembly line. That means when the workers have most of the knowledge built into the assembly line and the operational procedures.

Assembly line efficiency (AE) is the ratio of the outcome to the maximum possible outcome, often expressed as a percentage.

The efficiency of the assembly line can be calculated as:

We can assume that an assembly line is designed to produce a certain number of successful products (S) with a maximum rate of N products per hour. So for example, a shoe manufacturer has an actual outcome of 100 shoes per day, and a maximum potential outcome of 120 shoes per day. Their production line efficiency would be 83%. Now, we can calculate the average knowledge discovered H(X|Y).

To optimize the AE, companies can apply DFA guidelines, such as minimizing the number and variety of parts, standardizing the fasteners and connectors, and simplifying the assembly sequence and orientation[51].

Interpreting the results involves a comprehensive analysis of the data to understand where and why inefficiencies occur. In general, the higher the AE, the better the design. On the other hand, AE close to 100% might indicate under-utilised capacity. It's essential to compare high efficiency with industry capacity standards to determine if an increase in production is feasible and beneficial.

If AE is consistently below industry benchmarks, this could highlight several potential issues:

• Machinery: It may indicate that machines are outdated, malfunctioning, or not suitable for the required tasks.
• Labour Skills: Low efficiency might be due to workforce training gaps.
• Process Design: Sometimes, the workflow or layout of the production line itself causes inefficiencies.

Speed of Light in Medium

We can also use this model to support an interpretation of Ashby's Law of Requisite Variety to assess the speed of light in a medium where the medium acts as a disturbance to photon flow. Here's how this perspective aligns with the physics of light-matter interactions:

• Disturbance: The medium's atomic/molecular structure introduces spatial and electromagnetic inhomogeneities (e.g., refractive index variations, turbulence).
• Control Mechanism: Photons' ability to "counteract" disturbances through wavelength compression and phase synchronization.
• Requisite Variety: Photons require sufficient adaptability (e.g., frequency range, polarization states) to navigate the medium's complexity without scattering or losing coherence.

The speed of light in a vacuum is 299,792,458 m/s. In a medium, the speed of light is reduced by a factor n, called the refractive index defined as:

The refractive index is a measure of how much the speed of light is reduced in the medium. The higher the refractive index, the more the speed of light is reduced.

For example, the refractive index of water is 1.33, which means that the speed of light in water is:

The knowledge discovery efficiency of the speed of light in a medium can be calculated as:

Now, we can calculate the average knowledge discovered H(X|Y) by a photon in water:

What learning could also do (but we are explicitly excluding)

Not every form of learning improves regulation H(E|Y) in Ashby's sense. Other possibilities include:

1. Expanding action variety without selectivity

   Learning might increase $H\left(X\right)$ (more possible actions, tools, behaviors) without reducing $H\left(X\right|Y)$.

   • The system becomes more capable in principle
   • But still does not know which action to take
   • Regulation does not improve

   This violates Ashby's requirement that variety must be constrained, not merely expanded.

2. Improving buffering instead of knowledge

   Learning might increase buffering capacity $K$ (delay, slack, tolerance), so disturbances are absorbed without better action selection.

   • Outcomes may improve
   • But $I(X:Y)$ does not increase
   • Regulation improves without learning the mapping

   This is explicitly separated from knowledge in Ashby's extended formulation.

3. Changing goals or success criteria

   Learning could redefine what counts as success $E$.

   • Apparent performance improves
   • But the structural coupling (mapping) is unchanged
   • Information-theoretically, nothing about $H\left(X\right|Y)$ need change

   This is semantic drift, not cybernetic learning.

4. One-off adaptation without structural retention

   The system may succeed through exploration $Z$ without storing the result.

   • Regulation succeeds this time
   • Next encounter repeats the same uncertainty
   • No accumulation of $I(X:Y)$

   This is regulation, not learning.

Cumulative Knowledge To Be Discovered

Using

Cumulative w.r.t. S
Choose a baseline $S_0$ > 0.
Define:

Key properties:

• $\frac{dC}{dS}=H\left(S\right)=\frac{N}{S}-1$
• $\frac{d^{2}C}{d^{2}S}=-\frac{N}{S^2}<0$ → $C$ is concave in $S$.

Domain: $S$ ∈ (0, N]. Since $H\left(S\right)$ > 0 for $S<N$, $C(S;S_0)$ increases with $S$ (for $S\ge S_0$) and is finite as long as $S_0>0$.

Useful normalizations:
Dimensionless form with $\widehat{S}=\frac{S}{N}$: $$\frac{C\left(\widehat{S};{\widehat{S}}_0\right)}{N}=ln\frac{\widehat{S}}{{\widehat{S}}_0}-\left(\widehat{S}-{\widehat{S}}_0\right).$$

Total cumulative up to completion S = N: $$C(N;S_0)=Nln\frac{N}{S_0}-\left(N-S_0\right).$$

This can be thought of as the total knowledge-effort curve or “cumulative residual variety as a function of performance level” i.e. how much “knowledge work” has been consumed to reach performance level S.