Learning Curves: A Knowledge-Centric Perspective

Abstract

We address real-world applications of ....

1. Introduction

Few empirical regularities are as robust, or as theoretically undernourished, as the learning curve. Across settings as disparate as aircraft assembly, chemical plants, mining, and a single novelist's lifetime output, the cost or time required to complete one more unit of work falls in a strikingly regular way as cumulative output grows. It is the kind of cross-scale universality that ordinarily signals an underlying law rather than a coincidence of any one domain. Two research traditions have grown up around this regularity without fully meeting. In cognitive psychology, the power law of practice, consolidated by Newell and Rosenbloom and reinforced by single-subject data like Asimov's, describes how response time shrinks with practice as a power of the number of trials. In industrial and software engineering, the log-linear curve of Wright and its descendants describes how unit cost shrinks as a power of cumulative output What unites these literatures is that the curve is fitted, not derived. The question about which functional form the data prefer — power-law versus exponential — is settled by comparing residuals in log-log and semi-log plots. The catalogue of equations is a menu to be tried against data until one fits, not a family generated by a common principle and the parameters that carry the most interpretive weight are estimated rather than explained. The result is a mature descriptive science with a thin explanatory core: we know the shapes learning takes, but not why it takes them.

Learning-Curve Objects and Scope

The term learning curve is not used with one invariant meaning across the literature. It names a family of plots that share a broad intuition — performance changes with experience, data, repetition, or exposure — but they do not always share the same horizontal axis, vertical axis, statistical object, or causal interpretation. Before deriving any curve from the knowledge ledger, we therefore need to specify which learning-curve object is being explained.

The object derived in Sections 2–7 is the first-order practice/production learning curve: a curve in which comparable episodes are closed one after another, and the residual ignorance carried into the next episode changes because some of the knowledge discovered in the current episode is retained, lost, or made obsolete. Formally, the horizontal axis is not calendar time but cumulative surviving closures in a fixed task class:

The vertical axis (ordinate) is not raw time, raw cost, or raw output. It is the latent residual ignorance carried into the episode:

Observed cost, time, output, and quality are therefore treated as operational indicators of this residual ignorance only after a measurement model has been supplied. This is why the paper later distinguishes the latent knowledge curve from the observed effort curve and introduces KEDE as a normalized operational ordinate. The derivation does not claim that every published learning curve literally measures entropy. It claims that, for stable comparable task classes, the recurrent reduction of residual ignorance can generate the same families of shapes that practitioners often fit to cost, time, and productivity data.

Stable task classes versus changing objects

The practice/production object requires a fixed comparability frame. The episodes may differ in realized disturbance, duration, local context, and implementation details, but they must still ask the same regulatory question: given the available disturbance information, which response class must be selected to close the episode acceptably? If the disturbance variable, response-equivalence variable, response map, measurement resolution, or success criterion changes, then the observed series may no longer be one learning curve. It may be a splice of several curves, a curve with a reset, or a multivariate curve whose missing variables have been suppressed.

This is especially important for software and knowledge work. Calendar time, lines of code, function points, projects, commits, stories, or prompts may correlate with experience in one setting and fail badly in another. A project that changes domain, architecture, toolchain, team composition, or quality bar may appear to violate the learning curve when the real issue is simpler: the object being measured has changed. In such cases the analyst must either split the data into comparable task classes, condition on the changing variables, or treat the transition as a frame revision.

What is included in the main derivation

The main derivation covers the monotone, scalar, closure-indexed case first. That case is broad enough to include individual practice curves, industrial experience curves, software process curves, and professional-skill curves such as book-writing, provided that the episodes are comparable at the chosen level of abstraction. In this setting, learning means that the next comparable episode begins with less residual ignorance than the previous one:

This condition is not a universal definition of learning. It is the monotone-learning condition for the first model. Later extensions relax it by allowing knowledge loss, forgetting, interruptions, task-class resets, changing reservoirs, group knowledge transfer, and variance effects. Those cases are not exceptions to the ledger; they are reasons to use the full ledger rather than the pure monotone recurrence.

Consequence for the claim of the paper

The paper's claim is therefore deliberately staged. First, it derives the practice/production learning curve from the knowledge ledger under a stable comparable task class. Second, it shows how familiar production forms arise when prior coupling, irreducible execution work, incompressible components, or task-class resets are added. Third, it identifies which additional state variables are needed for broader cases: variance curves require second-moment accounting, group curves require knowledge-transfer terms, interrupted curves require forgetting and loss terms, and machine-learning generalization curves require an explicit predictive-risk model.

This scope discipline prevents an equivocation. The same phrase learning curve appears across psychology, production economics, software engineering, operations management, and machine learning, but the curve is not always the same mathematical object. The ledger is offered as a generative account of the closure-indexed residual ignorance first, and as an organizing language for the wider family second.

Solution

This paper offers an explanatory core. In prior work we developed a knowledge-discovery account of regulation in which a regulator closes episodes by discovering the information needed to select an acceptable response, and in which learning across comparable episodes is bookkept by a closure-indexed knowledge ledger. That ledger obeys an exact stock-flow identity: the residual ignorance a learner carries into the next comparable episode equals the ignorance carried into the current one, minus the knowledge retained, plus any knowledge lost. Here we show that this recurrence, read as a difference equation in the per-episode residual ignorance, generates the learning curve rather than presupposing it. The log-linear curve is not an empirical input to the theory; it is one of its solutions. Our central result is that the entire family of curves is governed by a single quantity: how the fraction of residual ignorance retained per episode scales with accumulated experience. When that fraction is constant, the recurrence yields exponential decay. When it dilutes with experience as roughly one over the trial number, the recurrence yields power-law asymptotics — the log-linear curve — with the Stanford-B, DeJong, and Kemerer forms recovered as parameter perturbations of the same equation: a nonzero starting coupling shifts the origin, an irreducible entropy floor lifts the asymptote, and a task-class reset reintroduces the lost-knowledge term that produces the adoption dip. The exponential-versus-power-law debate is thereby recast, from a contest between two fitted curves, into a statement about a single structural assumption on retention. We are deliberately modest about that assumption. We do not claim that the one-over-t dilution is the unique route to a power law, nor that a bounded-capacity learner must produce it. We claim only that it is one sufficient mechanism: an information-theoretically natural retention law, motivated by the fact that stored coupling is bounded above by the marginal variety of the task and so cannot keep absorbing a fixed fraction of ignorance indefinitely, which suffices to derive the log-linear curve from the ledger. Whether it is also necessary we leave open. The weaker claim is enough for the unification we seek, and it keeps the argument honest about what the accounting does and does not force. The paper proceeds as follows. Section 2 recapitulates the knowledge-ledger recurrence and the comparability conditions under which a learning curve is well-defined — conditions that, as a byproduct, supply the formal precondition for aggregating individual curves that the empirical literature has long known to be otherwise futile. Section 3 states the master recurrence and its regimes, and proves that the log-linear curve emerges precisely when retention dilutes as one over the trial number. Section 4 gives the information-theoretic motivation for that dilution law as one sufficient mechanism. Section 5 recovers the Stanford-B, DeJong, and S-curve forms as perturbations of the same recurrence. Section 6 connects the derived ignorance curve to the cost curves practitioners actually plot, via the operational one-bit effective-depth estimator and its error envelope, and argues that KEDE is the normalized, unit-free ordinate the learning-curve literature has lacked. Section 7 discusses what the derivation buys and where it breaks, and Section 8 concludes.

2. The Ledger Recurrence as a Difference Equation

Fix a comparable task class ${\tau }_j$ and index its episodes by $t=0,1,2,\dots$ in order of closure. The closure-indexed knowledge ledger attaches to each episode a starting Knowledge To Be Discovered, the residual response-selection uncertainty carried into the episode under the stored law of action:

The ledger obeys the exact stock-flow identity

where $G_t^K\ge 0$ is the retained knowledge gain and $L_t^K\ge 0$ the knowledge loss.

This is the object the rest of the paper studies. The classical learning curve plots cost or time against cumulative output; ours plots the same abscissa against residual ignorance, in bits. The generative claim is that the shapes catalogued in the empirical literature are the solutions of Definition 1 under particular retention laws $G_t^K$, which Sections 3–5 exhibit.

For pure learning we set $L_t^K=0$ and carry the loss term only where it is needed (Section 5). Under the constant-marginal-variety assumption $H_t\left(X\right)=H\left(X\right)$, the identity has the dual stored-coupling form

so that falling ignorance and rising coupling are two readings of one recurrence. The Learning Axiom, that a comparable episode begins with less residual ignorance than its predecessor, is just the requirement $H_{t+1}^{\text{start}}<H_t^{\text{start}}$, i.e. a monotone-decreasing $y_t$.

2.1 When is a learning curve well-defined?

The recurrence is meaningful only if the same quantity is being measured at every $t$. This is the comparability condition: the episodes indexed by $t$ must share a task-class frame that fixes the disturbance-information variable $Y$, the required response-equivalence variable $X$, the response-equivalence map $g:\mathcal{R}\to \mathcal{X}$, and the measurement resolution. Episodes may differ in realized disturbance, committed response, duration, and internal stage count; however they must ask the same regulatory question. When $Y$, $X$, or $g$ change meaning, the series is no longer a single learning curve but two curves spliced at a frame boundary — the situation Section 5 uses to derive the adoption dip.

This condition also answers a standing objection to individual learning curves. Baloff and Becker's remark on the futility of aggregating them is, in our terms, the observation that pooling episodes across incompatible task-class frames measures no well-defined $H\left(X\right|Y)$. Aggregation is valid exactly when the pooled episodes share a comparability frame; otherwise the averaged ordinate has no invariant referent. The frame condition is thus not a technicality but the precondition under which a learning curve exists at all.

3. The Master Recurrence and Its Regimes

Specialize Definition 1 to pure learning $L_t=0$ so that $y_{t+1}=y_t-G_t^K$ and write the retained gain as a fraction of current ignorance:

so the master recurrence is

The condition $0\le c_t<1$ keeps $y_t$ positive and non-increasing, satisfying the Learning Axiom. The entire curve family is fixed by the single sequence $c_t$: the fraction of residual ignorance a learner retains per episode.

3.1 Constant retention leads to exponential decay

If the retained fraction is constant, $c_t=\lambda$, the product collapses to a geometric law:

This is exponential decay in the trial number, linear in a semi-log plot of ${\log }_2{y}_t$ against $t$.

3.2 Experience-diluted retention leads to power law decay

If instead the retained fraction dilutes inversely with accumulated experience,

the product telescopes through the gamma function:

By the ratio asymptotics $\Gamma (t-\alpha )/\Gamma \left(t\right)\sim t^{-\alpha }$,

This is a power law in the trial number, asymptotically linear in a log-log plot with slope $-\alpha$ which is the log-linear learning curve $y=a{x}^n$ with $n=-\alpha$.

3.3 Cumulative-retention representation

Proof. Iterating the master recurrence gives

Taking logarithms yields

and therefore

The exponential and power-law results follow by substituting their respective asymptotic forms for $A_t$ into this exact identity.

For the sub-power case, $A_t\to \infty$ implies $y_t\to 0$. Moreover, for every $\epsilon >0$,

Since $A_t=o\left(\ln t\right)$, the exponent tends to $\infty$, which proves that the residual ignorance decays more slowly than every power law. The positive-floor and stretched-exponential results follow directly from the same representation. $\square$

The theorem identifies the cumulative retention hazard, rather than the pointwise retained fraction alone, as the quantity that governs learning-curve shape. Linear hazard growth gives exponential decay; logarithmic growth gives power-law decay; divergent but sub-logarithmic growth gives arbitrarily slow decay; and bounded growth leaves a positive residual ignorance. Intermediate hazard growth laws generate additional shapes, including stretched-exponential trajectories.

The power-law constant is written differently here and in the power-law regime (§3.2), but the two agree. Theorem 1 gives $C=y_0{e}^{-\beta }$ with $\beta :=\underset{t\to \infty }{\lim }(A_t-\alpha \ln t)$, whereas §3.2 gives the exact constant $C=\frac{y_1}{\Gamma (1-\alpha )}$ for the literal retention law $c_t=\alpha /t$. Equating the two representations fixes

The apparent discrepancy is only one of indexing. The pure law $c_t=\alpha /t$ is undefined at $t=0$ and is therefore based at $t=1$ with ordinate $y_1$, so it enters the $s=0$-indexed representation of Theorem 1 through the bounded-fraction tail form of §4.4 rather than as a literal $c_0$. The constant $\beta$ simply absorbs the logarithm of the gamma-ratio limit $\Gamma (t-\alpha )/\Gamma \left(t\right)\sim t^{-\alpha }$ established in §3.2.

4. A Sufficient Information-Theoretic Mechanism for Power-Law Learning

Section 3 established that a power-law residual ignorance follows if the retained fraction satisfies $c_t=\alpha /t+o(1/t)$. This section supplies one restricted information-theoretic mechanism that produces such a law. The mechanism is based on contraction of the set of task models still compatible with accumulated evidence.

The argument is deliberately conditional. It does not claim that bounded knowledge capacity, repeated exposure, or redundancy alone forces a power law. Instead, it identifies a class of learning problems for which an inverse-evidence learning curve has been derived independently, and then states the additional assumption needed to translate effective evidence into closure-indexed experience.

4.1 Bounded coupling does not by itself select a curve

The upper bound $I_t^{\text{start}}\le H\left(X\right)$ does not rule out constant fractional retention. If $G_t^K=\lambda y_t$ with $0<\lambda <1$, then $y_t=y_0{(1-\lambda )}^t$, and the total retained gain is finite:

The finite reservoir is therefore respected exactly. Constant fractional retention captures a constant fraction of a shrinking residual and produces a shrinking absolute gain. Boundedness alone consequently does not distinguish exponential from power-law learning. A further structural assumption is required.

4.2 Version-space contraction under entropic loss

Consider a restricted learning problem of the kind studied by Amari [8][9]. A fixed, noiseless task relation is represented by a family of dichotomy machines parameterized by $w\in {\mathrm{R}}^d$, where $d$ is the number of effective modifiable parameters. Let ${\mathcal{D}}_m$ denote the first $m$ independent training examples.

Let ${\mathcal{A}}_m$ be the set of parameter values that classify every example in ${\mathcal{D}}_m$ correctly. This is the learner's admissible model set, or version space. Under a nonsingular prior density $q\left(w\right)$, define its prior measure by

Each correctly labelled example removes incompatible models, so

A Gibbs learner samples a candidate model from the prior restricted to ${\mathcal{A}}_m$. Conditional on the next example, the probability that this model gives the correct response is the fraction of the current version space that survives that example:

The entropic prediction residual ignorance of the next example is therefore

This quantity is both the logarithmic predictive loss for the next response and the logarithmic contraction of the admissible model set caused by the next example. Let its expected value be

For regular noiseless dichotomy machines, independently sampled examples, effective parameters, and nonsingular input and prior distributions, Amari's universal theorem gives the asymptotic relation

when the logarithm is measured in natural units. In bits, the same expected residual ignorance is

Thus, within this restricted model, the expected information required to identify the correct response to the next disturbance decreases inversely with the amount of independent evidence already accumulated.

4.3 From independent examples to comparable closures

Amari's theorem is indexed by independent training examples. The ledger is indexed by comparable closures. These indices need not coincide. One closure may provide several independent pieces of evidence, only a fraction of one effective piece, or evidence that overlaps strongly with what is already stored.

Let $m_t$ be the effective-evidence clock: the effective number of independent, information-bearing examples accumulated by the beginning of closure $t$. The effective-evidence clock is a property of the episode sequence and its dependence structure, not merely a count of physical observations.

Closures is different than bits of retained knowledge. The former is a count of completed episodes, the latter is a measure of information content. Therefore, prior knowledge cannot be converted into a number of closures unless an explicit calibration function is assumed.

For the present mechanism, define the closure-indexed residual ignorance as Amari's expected entropic prediction residual ignorance evaluated on the effective-evidence clock:

Proof. Substitute the effective-evidence clock into Amari's inverse-evidence law:

Writing $C=d/\left(\kappa \ln 2\right)$ gives $y_t\sim C{(t+b)}^{-\alpha }$, the log-linear learning curve. $\square$

4.4 The retention law implied by the mechanism

The effective-evidence mechanism derives the residual ignorance curve first. The corresponding fractional retention law can then be read from the ledger recurrence. For the leading power-law term

the exact retained fraction is

This fraction is bounded between zero and one for all $\alpha >0$ and has the expansion

The retained knowledge gain is therefore

This distinction is important. The current per-episode residual ignorance, the absolute reduction in that residual ignorance, and the fractional reduction are three different quantities. In the basic Amari case, with one independent example per closure,

The inverse-evidence result applies to the current entropic residual ignorance $y_t$, not directly to the absolute retained gain $G_t^K$. The retained gain is the reduction in the residual ignorance carried into the next comparable episode.

4.5 Interpretation of the exponent

Under this mechanism, the exponent $\alpha$ is not the amount of information contained in one sample. It is the asymptotic elasticity of effective accumulated evidence with respect to recorded closures:

If every closure supplies one new independent example, then $m_t\propto t$ and $\alpha =1$. If successive closures supply strongly overlapping evidence, the effective-evidence clock may grow sublinearly and $0<\alpha <1$. The classical learning rate per doubling is then $2^{-\alpha }$.

The same construction also shows how exponential learning can arise. If effective evidence grows geometrically with closures,

then the inverse-evidence law gives

In this restricted account, the power-law-versus-exponential distinction can therefore be read as a distinction in how effective evidence accumulates with recorded experience: polynomial evidence growth produces a power law, while geometric evidence growth produces exponential decay.

4.6 Scope of the mechanism

The mechanism establishes a sufficient route to a power-law learning curve, not a universal explanation of every observed curve. Amari's theorem concerns expected entropic predictive loss for regular parametric, noiseless dichotomy machines using a particular learning procedure. It does not directly establish that industrial unit cost, human response time, software-project effort, or every instance of $H\left(X\right|Y)$ must decay as $1/t$.

The extension to practice and production requires two empirical identifications: first, that the ledger's per-episode residual ignorance is adequately represented by an entropic predictive residual ignorance; and second, that the relationship between recorded closures and effective evidence can be estimated by $m_t$. These are testable modeling assumptions, not consequences of the stock-flow identity.

Nor is this mechanism claimed to be necessary. Mixtures of learning components, stochastic selection, exhaustion of faster learning processes, and chunking mechanisms can also generate or approximate power-law practice curves [7]. Observing a power law therefore does not uniquely identify its cause. The result established here is narrower: version-space contraction gives an inverse-effective-evidence residual ignorance, and a polynomial effective-evidence clock converts that residual ignorance into the log-linear learning curve.

5. Perturbations and Compositions of One Recurrence

The learning curves' catalogue treats Stanford-B, DeJong, Plateau, the S-curve, and the dual-phase and group curves as separate equations to be tried against data. Under the ledger they fall into two families. The first are perturbations of a single recurrence: the power-law solution of Section 3 with one term switched on — a prior-evidence shift already carried by the mechanism, an entropy floor, or a reinstated loss term. The second are compositions of several ledgers: a superposition of sub-frames carried by one learner (Jaber–Glock), or an aggregation of individual ledgers coupled by knowledge transfer (the group curve). Nothing new is fitted in either family; each classical parameter becomes a bit-valued quantity in the same recurrence, or a structural relation among several copies of it.

5.1 Stanford-B: carried-over coupling is the prior-evidence parameter

Stanford-B needs no term switched on beyond the mechanism of Section 4. Proposition 1 already carries a prior-evidence parameter $b\ge 0$: the effective evidence a learner brings to the first recorded closure, expressed in prior-experience offset. Its power-law solution

is the Stanford-B curve $y=a{(x+b)}^n$ with $n=-\alpha$. The log-linear curve of Section 3 is the $b=0$ case; Stanford-B is the same solution read at $b>0$, not a separate perturbation of the recurrence.

What the classical constant $b$ measures is fixed by the mechanism rather than left free. A learner arriving with prior coupling $I_0^{\text{start}}>0$ — equivalently, with prior evidence $b>0$ in prior-experience offset, so that the effective-evidence clock starts at $m_0=\kappa b^{\alpha }$ — has already absorbed the equivalent of $b$ comparable closures. The Boeing finding that a team carries one to ten airframes of learning between airframe models, and Pierson's one-to-six months for miners, are estimates of exactly this quantity, not of a free fitting constant. The curve starts flat because the learner starts partway down it: a nonzero $b$ places the first recorded closure at effective evidence $m_0>0$ rather than at the singular origin.^†

The origin needs care. With $b=0$ the effective-evidence clock starts at $m_0=0$, so the inverse-evidence residual ignorance $y_0=d/\left(m_0\ln 2\right)$ diverges and the recurrence is indexed from $t=1$ i.e. the same convention under which the gamma-ratio derivation of §3.2 is written from $y_1$. The Stanford-B case $b>0$ removes the singularity, since $m_0=\kappa b^{\alpha }>0$ gives a finite starting residual ignorance. The asymptotics of Proposition 1 are unaffected either way; only the treatment of the first term differs.

5.2 DeJong and Plateau: one floor form across two observational layers

Section 3 assumed that the residual ignorance could be exhausted, $y_t\to 0$. This need not hold. Under a fixed task-class frame, observation channel, response-equivalence map, measurement resolution, and learner architecture, some residual response-selection uncertainty may remain after arbitrarily many comparable episodes. Define the frame-relative epistemic floor by

The floor is irreducible relative to the current frame: additional episodes sampled through the same information channel do not remove it. It need not be irreducible under every possible redesign. Adding a sensor, supplying a missing work instruction, changing the response-equivalence map, refining the task definition, or enlarging the learner's model class may lower the floor, but each of those interventions changes the frame rather than continuing along the same learning curve.

Only the reducible excess above the floor,

obeys the retention recurrence. Thus

When the retention law has the power-law tail of Section 3, the reducible component satisfies ${\tilde{y}}_t\sim A{t}^{-\alpha }$, and therefore the closure-level residual ignorance follows

5.2.1 The closure-level DeJong form

Define the normalized epistemic incompressibility factor

The floor solution can then be written as

This has the algebraic form of the DeJong curve $y=a[M+(1-M\left)x^n\right]$, with $a=y_1$, $M=M_K$, and $n=-\alpha$. The identification is an information-theoretic reinterpretation of DeJong's mathematical structure. In the original production interpretation, $M$ is the fraction of observed task time or cost associated with a component that does not improve through worker learning, often because it is machine-limited. In the closure-level ledger, $M_K$ is instead the fraction of the initial Knowledge To Be Discovered that remains unresolved under the fixed frame.

This does not identify machine cycle time itself with entropy. It states that the same normalized-floor equation can describe a closure-level epistemic asymptote when its ordinate is residual response-selection uncertainty rather than elapsed production time.

5.2.2 The observed Plateau form

The distinction matters because the classical literature usually observes total task time, cost, or effort, whereas the present ledger defines closure duration as the duration of response selection and commitment. Post-closure machine operation, physical unfolding, idle observation, and feedback latency are not part of the closure residual ignorance. Let the observed duration of episode $t$ be

where $F_t$ is post-closure feedback or physical duration, and $\psi \left(y_t\right)$ is the observable closure effort associated with the latent residual ignorance. Under the simple stationary mapping

the observed duration becomes

Define the observed asymptote

Then

which is the Baloff Plateau form $T_x=C+A{x}^{-b}$, with

The observed Plateau constant may therefore combine two different mechanisms:

The first term is outside the Knowledge To Be Discovered ledger. It records time that passes after the response has already been selected and committed. The second is the observable contribution of residual response-selection uncertainty. An elapsed-time curve may contain either term or both, whereas a closure-only KEDE curve contains only the epistemic component.

Consequently, the conventional observed DeJong factor is

while the closure-level epistemic factor is

The two are equal only when the post-closure floor has been removed from the measurement, $F_{\infty }=0$, or when the ordinate has otherwise been calibrated to closure effort alone.

5.2.3 Which response multiplicity creates an epistemic floor?

The existence of several concrete ways to produce an acceptable outcome does not by itself imply positive Knowledge To Be Discovered. Let $g:\mathcal{R}\to \mathcal{X}$ map concrete responses to response-equivalence classes. If two responses $r_1$ and $r_2$ are genuinely interchangeable under the adopted regulatory criterion, then

and uncertainty between them is implementation freedom inside one already determined response class. It does not contribute to $H\left(X\right|Y)$.

A genuine epistemic floor arises when distinctions between response classes matter, but the available information does not determine which class is required. For example, two procedures may both start the same machine, yet differ with respect to safety, auditability, maintenance policy, sequencing, or another goal-relevant constraint. If the applicable procedure is not specified by the available disturbance information or stored work instruction, then

If repeated experience through the same channel cannot reveal the missing criterion, this uncertainty contributes to the frame-relative floor $y_{\infty }$. A new instruction or additional observation may remove it, but that intervention revises the frame and begins a different curve.

A third case must be kept separate. If several response classes are intentionally acceptable and the task contains no requirement to choose one rather than another, then the regulatory target is set-valued rather than a single required class. The multiplicity should not be counted as ignorance merely because one concrete response must eventually be committed. A single-valued $H\left(X\right|Y)$ model applies only after an additional selection criterion has identified which response-equivalence class is required.

5.2.4 Reconciliation

DeJong and Baloff are therefore best understood as the same algebraic floor perturbation viewed through different parameterizations, but not necessarily as the same physical or epistemic mechanism. The Baloff constant writes the asymptote as an absolute observed minimum. The DeJong factor writes it as a fraction of initial observed performance. The closure-indexed ledger adds a third reading: a normalized frame-relative residual entropy that further comparable experience cannot remove.

The correspondence is

Thus, the catalogue's Plateau and DeJong entries belong to one offset-power family, while the ledger distinguishes what produces the offset: post-closure physical or feedback limits in the observed process, residual response-selection entropy at closure, or a combination of the two. Only the closure-level epistemic component is irreducible Knowledge To Be Discovered.

5.3 The S-curve: shift and floor together

Applying both perturbations gives the full S-curve $y=a[M+(1-M\left){(x+b)}^n\right]$:

Carried-over coupling and an irreducible floor are independent parameters of one recurrence, which is why the primer needs a combined equation for processes that both start flat and level off[2].

5.4 The dip: a task-class reset reinstates the loss term

The perturbations above keep $L_t^K=0$. Kemerer's adoption curve, which drops below the old-technology baseline before recovering, requires it back[3]. Introducing a new tool is a frame revision: the task class changes from $\tau$ to ${\tau }^{\prime }$, and coupling stored against $\tau$ is not comparable under ${\tau }^{\prime }$. At the reset episode $t_0$ the ledger books a one-time loss

raising the residual ignorance above where the old frame had driven it. The learner then climbs a fresh power-law curve under ${\tau }^{\prime }$ from that elevated start. Because performance is read against the old frame's attained level, the spliced curve dips below baseline over the interval where

and recovers once the new coupling overtakes the old. The depth of the dip is the reset loss $L_{t_0}^K$; its duration is set by the new exponent $\alpha$. What the adoption literature reports as an anomaly of the smooth-decline picture is, on the ledger, two power-law curves joined at a loss-bearing frame boundary — the same recurrence with $L_t^K$ momentarily nonzero.

5.5 Jaber–Glock: a composite episode expressed as a vector ledger

The perturbations so far modify a single scalar learning process. Jaber–Glock exposes a different case: the observed unit time may be the sum of two learning processes that are both present inside every repetition. In their model, a worker first refers to procedures, manuals, or process descriptions in order to determine how the task should be performed. They call this the build-up of knowledge. After that information has been processed, the worker performs the production steps, which they call the knowledge retrieval step. Learning can occur in both steps: the worker becomes faster at finding or recalling the required procedure, and also faster at physically executing the production process[14].

In the ledger language, a comparable episode is therefore not atomic. It decomposes into two sub-episodes: a cognitive sub-ledger and a motor sub-ledger. The cognitive sub-ledger tracks the residual ignorance of deciding, recalling, or reconstructing what must be done. The motor sub-ledger tracks the residual ignorance of executing the required action sequence. The two are not a frame splice, as in the technology-adoption reset; they are co-present components of each repetition.

Each component obeys its own recurrence:

When both sub-ledgers are in the power-law regime, their residual ignorances have distinct exponents:

To recover the Jaber–Glock form, express both ledgers in the observed unit-time scale and let $p\in [0,1]$ be the fraction of first-unit time assigned to the cognitive component. Then

This is not merely another scalar perturbation. It says that the apparent learning curve is an aggregate of two component curves: one for cognitive learning and one for motor learning. The single Wright exponent is therefore an effective exponent that hides the internal composition of the task. Jaber–Glock makes that composition visible by estimating the cognitive share, the cognitive learning exponent, and the motor learning exponent jointly.

The ledger interpretation is direct. The cognitive exponent $b_c$ measures how quickly the burden of reconstructing or retrieving task instructions declines. The motor exponent $b_m$ measures how quickly the execution ignorance declines. The weight $p$ measures how much of the first repetition is initially consumed by the cognitive component. If $b_c>b_m$, the cognitive burden fades faster and the long-run tail is governed by the slower motor component. If the motor component is small or learns slowly, it may still dominate the asymptote of the observed curve.

The crossover point occurs where the two components contribute equally:

Thus the Jaber–Glock model is not a new scalar curve; it is the vector-ledger case: a single observed task contains multiple retained-knowledge stocks, multiple retention laws, and multiple exponents.

Vector learning means that a task does not improve along one hidden dimension only. A single observed learning curve may be the projection of several underlying learning processes, each with its own residual ignorance, retention rate, and exponent. In the Jaber–Glock case, the worker is learning both how to retrieve or reconstruct the required procedure and how to execute the physical action sequence. The observed unit-time curve is therefore not one scalar learning process, but the weighted sum of cognitive and motor sub-curves. This turns Jaber–Glock into the first clear case where the scalar ledger may be expressed as a vector ledger.

5.6 The group curve: aggregated ledgers coupled by transfer

The literature distinguishes individual, team, and organizational experience. The group-learning model explicitly contains individual production plus knowledge-transfer terms[12]. The meta-analysis also finds that models developed for group learning often perform best on group data.

The curves so far describe one learner. A group of $n$ learners each carries a ledger, and the group curve is their aggregate — but two conditions from earlier sections govern whether that aggregate is well-defined and how it moves.

The first is comparability. By Section 2.1, pooling the individual residual ignorances $y_t^{\left(I\right)}$ measures a single group residual ignorance only when the learners share a task-class frame; otherwise the averaged ordinate has no invariant referent, which is precisely Baloff and Becker's futility of aggregation. When the frame is shared, the group residual ignorance is the weighted aggregate $y_t={\sum }_{i=1}^n{w}_i{y}_t^{\left(I\right)}$.

The second is transfer. A group is not $n$ isolated ledgers: coupling that learner $j$ has already stored against the shared frame can be adopted by learner $i$. Each individual recurrence then carries an extra reduction term sourced from the others:

where $T_{ij,t}^K\ge 0$ is the knowledge transferred from $j$ to $i$ at episode $t$, bounded by the coupling gap $T_{ij,t}^K\le {(I_t^{\text{start},j}-I_t^{\text{start},i})}^{+}$: a learner can receive only coupling another already holds and it lacks. This is the Stanford-B carryover of Section 5.1 generalized across learners — there, $b$ was coupling carried over from the same learner's earlier frame; here, $T_{ij}^K$ is coupling carried across from another learner's current stock in the shared frame. This subsection's transfer bound reuses $I_t^{\text{start}}$ notation from §2. That's deliberate (it's the stored-coupling reading of the same ledger)

Aggregating the transfer-coupled recurrences and adding the individual outputs recovers the group learning curve $Z\left(T\right)={\sum }_i{Y}_i\left(T\right)+{\sum }_{i,j}{X}_{i,j}\left(T\right)$: the first sum is the learners' own descent down their ledgers, the second is the output produced from transferred coupling $T_{ij}^K$. Because every transfer term is a further non-negative reduction in some learner's residual ignorance, the group descends faster than the same learners in isolation would — the acceleration the group curve is built to capture. When transfer is switched off, the group curve degenerates to the frame-conditioned average of Section 2.1, and Baloff and Becker's warning is the statement that even that average is meaningless without the shared frame.

5.8 Summary

The literature's canonical shapes resolve into two families of one recurrence. As single-ledger perturbations, they are one power-law solution under settings of two terms: log-linear ( $b=0$, $y_{\infty }=0$, $L_t^K=0$), Stanford-B ( $b>0$), DeJong and Plateau ( $y_{\infty }>0$, the one floor read as a fraction or as an absolute minimum), the S-curve ( $b>0$ and $y_{\infty }>0$), and the Kemerer dip ( $L_{t_0}^K>0$).

As multi-ledger compositions, two further shapes are structural relations among copies of the same recurrence: the Jaber–Glock dual-phase curve is a share-weighted superposition of two sub-frame ledgers with distinct exponents, and the group curve is a frame-conditioned aggregate of individual ledgers coupled by a cross-learner transfer term. The catalogue is not a list of rival equations but a coordinate chart on one ledger — and a partly redundant chart, since Plateau and DeJong name a single floor.

6. From Latent Bits to the Observed Cost Curve

Sections 3–5 derive a curve in the information-theoretical currency. The ordinate is $H_t^{\text{start}}$, latent bits of residual ignorance; the literature plots cost or effort per unit. This section connects the two through the operational one-bit effective-depth estimator, so the unification is anchored to a measurable quantity rather than to an unobservable entropy.

6.1 The operational estimator

Over a window $I$ of comparable episodes, the effective one-bit net estimator reads the per-closure residual ignorance off the execution ledger:

with $N\left(I\right)$ the counted channel capacity and $S_{\text{net}}\left(I\right)$ the surviving closures. Both are counts, not entropies. Theorem 2 of the prior work brackets the latent residual ignorance by this observable up to a calibration error and a sub-one-bit coding gap:

So the ledger recurrence in $H_t^{\text{start}}$ is observable, per window, as an effective action-units-per-closure ratio, within a bounded envelope.

6.2 Why the count ratio is the effort curve

The classical ordinate is effort per unit — action units expended per delivered unit. The estimator's numerator $Q_{\text{eff}}\left(I\right)$ is effective discriminating work and its denominator $S_{\text{net}}\left(I\right)$ is delivered units, so

The literature's $y=a{x}^n$ is a fitted proxy for exactly this ratio; the ledger supplies its generative law and the calibration theory relating the fitted ordinate to bits.

6.3 KEDE: the normalized, unit-free ordinate

The classical curve is unbounded and unit-dependent — dollars, hours, lines of code — which is why the literature warns that a line of code means different things in different phases and that measurement can mask the curve entirely. KEDE removes both problems by the bounded transform

The latent counterpart is ${\text{KEDE}}^{\ast }=1/(1+H(X\left|Y\right))$, and Corollary 2 pushes the Theorem 2 envelope through this transform, so the two differ by a controlled, mostly-observable amount. KEDE is therefore the learning-curve ordinate re-expressed on a common $(0,1]$ axis: a value near one means little remained to discover, near zero that most of the delivered work was discovery.

6.4 The derived curve in KEDE units

Substituting the power-law solution $H_t^{\text{start}}\sim C{t}^{-\alpha }$ of Section 3 into the latent transform gives the learning curve in its normalized form:

Efficiency rises toward one as residual ignorance decays as a power of experience — a bounded, unit-free curve derived from the ledger and readable off delivery counts. Where the classical literature fits an unbounded cost curve and struggles with what its ordinate means across contexts, the ledger produces a bounded efficiency curve whose ordinate is the same quantity — missing knowledge — in every context.

7. What the Observed Ordinate Can and Cannot Determine

Sections 3–5 derive trajectories in the latent ordinate $y_t$, and Section 6 connects that ordinate to a counted estimator. Between the two lie three conventions, each of which the practitioner must choose and none of which the learning-curve literature routinely reports: a measurement convention supplying the action unit and hence the coding gap, a windowing convention supplying the observational unit, and a counting convention supplying the rule by which gross closures become net closures.

This section asks what survives the composition of the three. The answer is not uniform. One parameter of the derived curve is identified and can even be estimated by a route the literature does not currently use; one is biased in a signed and computable way; and one is not identified at all below a resolution threshold, so that a published value of it may be entirely an artefact of the measuring instrument. Since the catalogue of Section 5 distinguishes its members precisely by these parameters, the result bears directly on the question of which learning-curve models can be told apart from data.

7.1 The composed observation model

Write the three layers explicitly. At the measurement layer, Theorem 2 of the prior work gives, for each surviving closure,

The calibration error ${ϵ}_t$ is signed and, in principle, estimable; the coding gap ${\delta }_t$ is one-signed and bounded by a single action unit per closure. At the windowing layer, Lemma W averages these over a block. At the counting layer, the analyst chooses whether the denominator is $S_{\text{net}}$ or $S_{\text{gross}}$. The reported ordinate of block $k$ is therefore

where $\Lambda$ is the identity under net counting and the yield distortion of §7.4 under gross counting. The structural object of interest is the four-parameter family of Section 5,

We ask, for each component of $\theta$, what the sequence $R_k$ determines.

7.2 What is identified: the exponent

The exponent is the robust parameter of the family, and it is robust for a structural reason: it is a property of the ratio of successive ordinates, whereas all three conventions contribute additive or multiplicative level terms that either vanish asymptotically or cancel in differences.

Three results already established combine to give this. Proposition W1(iii) shows that fixed-count blocking leaves the log-log slope intact, with block-level distortion $O\left({(L/m_k)}^2\right)$ vanishing in the block index. Proposition W2(i) shows the same for expanding windows in the sub-critical regime, where only the intercept moves. And the additive envelope ${ϵ}_t+{\delta }_t$ is bounded while $y_t$ is not, so on any range where $y_t\gg 1+\overline{ϵ}$ the envelope is a vanishing relative perturbation of the slope.

Two qualifications are essential, and they are the two negative results of §7.3 and §7.4 seen from the exponent's side. The envelope argument fails precisely where the curve approaches its floor, which is where the exponent and the floor trade off against each other in a fit; and the counting layer attenuates the slope whenever yield is improving. The exponent is therefore identified on the region of the curve where residual ignorance is large relative to one action unit per closure, under net counting, and only there.

Corollary W of §6.5.4 adds a second, independent route. Where a series is reported under both the episodic and the cumulative-average convention — the unit curve and the cumulative-average curve of the production literature — the exponent follows from the level ratio of the two ordinates at a single point, with no regression. Because that estimator uses levels rather than slopes, its bias profile is different from the regression estimator's, and disagreement between the two is diagnostic: under a single comparability frame Proposition W2 forces them to agree.

7.3 What is not identified: the floor

The floor is the parameter that distinguishes DeJong and Plateau from log-linear, and the combination of floor and offset that distinguishes the S-curve from both. It is also the parameter that the measurement layer is least able to determine, because a persistent coding gap is observationally indistinguishable from a persistent epistemic floor.

Proof. Taking limits in the composed observation model gives the first display. The map

is not injective on $[0,\infty )\times [-\overline{ϵ},\overline{ϵ}]\times [0,1)$; its fibre over ${\widehat{y}}_{\infty }$ projects onto the stated interval in the first coordinate, since ${\delta }_{\infty }$ may take any value in $[0,1)$ and ${ϵ}_{\infty }$ any value in its bound, subject to $y_{\infty }\ge 0$. $\square$

The consequence for Section 5.2 is immediate. The normalized incompressibility factor $M_K=y_{\infty }/y_1$ inherits the non-identification in its numerator and, by §7.5, additional width in its denominator. A reported $M_K$ is therefore an upper bound on the epistemic incompressibility of the task, never a point estimate, unless the observed asymptote exceeds one action unit per closure by more than the calibration bound. This does not affect the algebra of §5.2 — the DeJong form is still the floor perturbation of the recurrence — but it means that fitting DeJong rather than log-linear is not, by itself, evidence that an epistemic floor exists.

Non-identification here is a resolution limit rather than a permanent obstruction, and the ledger says how to lift it.

Corollary I1 converts a caveat into a test. A genuine frame-relative floor, in the sense of §5.2, is a property of the task frame and is invariant under refinement of the counting unit; it responds instead to frame interventions — adding a sensor, supplying the missing work instruction, refining the response-equivalence map. A coding gap is the opposite: it is invariant under frame intervention and shrinks under unit refinement. The two therefore have orthogonal comparative statics, and either intervention discriminates them:

Neither intervention is available retrospectively for a published curve, which is the honest conclusion: the epistemic floors reported in the historical literature are not recoverable from the published ordinates. They are recoverable prospectively, from a process instrumented to permit unit refinement.

7.4 What is biased: the counting layer

The third convention concerns which closures enter the denominator. Let

be the survival ratio of episode $t$, and let ${\widehat{y}}_t^{\text{gross}}$ denote the estimator computed with the gross denominator.

Proof. The identity is immediate from $1+\widehat{y}=N/S$ and $S_{\text{net}}={\sigma }_t{S}_{\text{gross}}$. Claim (i) follows since ${\sigma }_t\le 1$, and the stated threshold is where the right-hand side falls below one. Claim (ii) is the logarithmic derivative of the identity. Claim (iii) is direct differentiation. $\square$

The direction deserves emphasis, because the intuitive expectation runs the other way. One might suppose that a gross ordinate credits declining scrap to learning and so overstates the exponent. The opposite holds. A gross denominator counts attempts, and improving yield does not reduce effort per attempt; it is the net denominator that registers yield improvement as improvement. Gross counting therefore hides the yield component of learning rather than inflating it.

A numerical illustration makes this concrete. Suppose the window capacity is $N=100$ action units throughout, and that the process makes fifty attempts per window in both an early and a late window, of which twenty-five and fifty respectively survive. Then

a threefold reduction in residual ignorance, whereas

so the gross ordinate registers no learning whatsoever. All of the improvement in this example is yield improvement, and gross counting is blind to it.

A second consequence matters for diagnosis. Proposition W1 inflates the early blocks of a blocked series, while Proposition I2 deflates the early points of a gross series. On blocked gross data the two act in opposite directions on the same region of the curve, and the fitted exponent is a mixture whose components cannot be separated without either the block-level raw counts or an independently reported yield series. A gross blocked curve that appears well behaved may therefore be two substantial biases in partial cancellation, and the apparent absence of distortion is not evidence of its absence.

7.5 What is weakly identified: intercept and prior-experience offset

The remaining parameters, $y_1$ and the Stanford-B offset $b$, are determined mainly by the early part of the curve, which is exactly where every distortion catalogued above is largest: the block bias of Proposition W1 is $O\left({(L/m)}^2\right)$ and diverges as the first block is approached; the yield attenuation of Proposition I2 is concentrated there; and the frame is least likely to have stabilized.

Moreover $y_1$ and $b$ trade off against each other directly: over any finite range,

for a one-parameter family of pairs, the approximation improving as the observed range moves away from the origin. This is the ledger-side account of a well-known practical difficulty: first-unit cost estimates in the production literature are notoriously unstable, and the Boeing and Pierson estimates of carried-over experience are reported as ranges — one to ten airframes, one to six months — rather than as point values. Under §5.1 those ranges are estimates of a prior-experience offset, and Proposition W1 explains why the range is wide: the block that would determine it is the block least able to.

The practical recommendation of §2.2.7 follows directly. Exclude the first block of an equal-count blocking from intercept and offset estimation, and report $b$ as an interval.

7.6 Summary

7.7 A reporting protocol

The results above are negative only in the absence of metadata that costs nothing to record. We therefore state the minimal reporting protocol under which a learning curve becomes interpretable on the ledger.

1. The action unit $\mathrm{\Delta }$ and how it was fixed — externally, or by calibration to a best observed window.
2. The windowing mode (a)–(d) of §2.2.1 and, for blocked or rolling modes, the block width $L$ and the plotting abscissa (midpoint or endpoint).
3. The counting convention, gross or net, and the settling lag $\ell$ of Convention S.
4. The survival ratio series ${\sigma }_t$, or at minimum its endpoints, so that Proposition I2 can be inverted.
5. Whether the frame was revised during the observation window, and where.

Items 1–4 are sufficient to convert a published curve into a ledger ordinate with a stated identified set for each parameter. Their absence, not any deficiency of the underlying data, is what makes the historical corpus only partially recoverable.

7.8 Consequence for the catalogue

Section 5 demoted the learning curves catalogue from a menu of rival equations to a coordinate chart on one recurrence. The present section constrains how much of that chart is legible from data.

The distinction between log-linear and DeJong or Plateau turns entirely on $y_{\infty }>0$, which Proposition I1 shows is not identified below one action unit per closure. The distinction between log-linear and Stanford-B turns on $b>0$, which §7.5 shows is weakly identified and confounded with the intercept. The S-curve requires both. Only the exponent, which all four share, is securely determined. It follows that a substantial part of the model-selection literature — the comparison of residuals that decides which of the fifteen equations a dataset prefers — is adjudicating between parameters that the reported ordinate does not determine.

This suggests a reading of the meta-analytic evidence more specific than the one offered in §8.1. Grosse and colleagues report that different curve families win for different datasets. Part of that variation may be substantive, reflecting which ledger terms are active in which task class. But part of it is predicted here to be conventional: datasets reported on a cumulative-average ordinate, on gross output, or with coarse action units should systematically favour different members of the catalogue than the same processes measured otherwise, irrespective of the underlying learning. The prediction is testable wherever the source studies document their measurement protocol, and it is falsifiable in the useful direction — if observational convention turns out to carry no explanatory weight in the meta-analytic sample, the identification results below remain valid as bounds but lose their claim on the historical record.

We state it as a prediction rather than a finding. Recovering the measurement protocol of several hundred published curves is a distinct undertaking, and nothing in Sections 3–6 depends on its outcome.

8. Discussion

8.1 What the derivation buys

Three things follow from reading the learning curve as ledger dynamics rather than fitting it.

First, the curve's shape is explained, not assumed. The log-linear form is the solution of the knowledge-ledger recurrence under one retention law, and the exponential is the solution under another. The power-law-versus-exponential question — which Ohlsson settles for Asimov by comparing log-log and semi-log residuals — becomes a question about a single structural quantity, the retained fraction $c_t$, rather than a contest between two curves. Section 4 gives one sufficient reason $c_t=\alpha /t$ holds under stationary redundant sampling.

Second, the literature's catalogue of fifteen equations is demoted from a menu to a coordinate chart. Stanford-B, DeJong, and the S-curve are one solution under three parameter settings, and their classical constants acquire bit-valued meanings: carried-over experience is stored coupling, the incompressibility factor is irreducible entropy, the adoption dip is a loss-bearing frame reset. The meta-analytic finding that different curves win for different data — Grosse and colleagues report the S-curve best for a large share of time-reduction datasets and exponential models best for others — is on this reading not a competition between theories but a report of which parameters are active in which task class.

Third, the ledger uses the axis the empirical literature wants. Both Kemerer and the literature argue that cumulative output, not calendar time, is the correct abscissa, and that measuring against time is a source of the contradictory CASE evidence. The ledger is natively closure-indexed, and the Knowledge Discovery Rate layer reattaches time separately through the unit convention. The predictive superiority Everett and Farghal report for the simpler log-linear and Stanford-B forms — better forecasts even when richer equations fit the past better — is consonant with a generative account: the log-linear form is the bare recurrence, and the extra parameters of the richer curves are the perturbations of Section 5, which help in-sample but overfit out-of-sample when their governing terms are inactive.

8.2 Where it breaks

The $\alpha /t$ mechanism is sufficient, not necessary, and it is bounded by the comparability frame. Three limits matter.

The frame must hold. If $Y$, $X$, or $g$ drift, Assumption E fails, each episode samples a partly new relation, and the ledger measures no single $H\left(X\right|Y)$. The primer's own warnings — that a line of code means different things across phases, that seasonal and growth biases mask the curve — are exactly frame violations, and no derivation survives them. The theory locates the failure but does not repair it.

Retention need not dilute as $1/t$. Non-stationary evidence, a growing reservoir $H\left(X\right)$, or interference between episodes give other $c_t$ laws and other shapes. The hyperbolic, Gompertz, and plateau forms that win for certain task types in the meta-analysis presumably correspond to such laws; deriving them is future work, not a claim of this paper.

Forgetting is only sketched. Section 5 reinstates $L_t^K$ as a one-time reset to produce the Kemerer dip, but sustained forgetting — a positive $L_t^K$ at every episode — would compete with retained gain and could halt or reverse the curve. The ledger accommodates this term but this paper does not solve the resulting dynamics.

8.3 What is claimed

The claim is narrow and, we think, defensible: under a stable task class sampled redundantly, the knowledge-ledger recurrence reproduces the log-linear learning curve, with the exponential as the special case in which cross-episode redundancy is ignored, and with Stanford-B, DeJong, and the Kemerer dip as perturbations of the same equation. The empirical curve is thereby exhibited as a consequence of knowledge-discovery accounting rather than as a free-standing regularity. What remains open — whether $\alpha /t$ is also necessary, and which laws generate the non-power-law shapes — is the natural continuation.

9. Conclusion

The learning curve has been, for nearly a century, a regularity in search of a mechanism: measured everywhere, fitted routinely, derived nowhere. This paper supplies one mechanism. The closure-indexed knowledge ledger obeys an exact stock-flow identity, and read as a difference equation in the per-episode residual ignorance $H_t^{\text{start}}$, that identity generates the learning curve rather than presupposing it.

The result is a single quantity governing the whole family. When the fraction of residual ignorance retained per episode is constant, the recurrence decays exponentially; when it dilutes as $\alpha /t$, it follows the power law $H_t^{\text{start}}\sim C{t}^{-\alpha }$ — the log-linear curve $y=a{x}^n$. The power-law-versus-exponential debate, long adjudicated by comparing residuals, is thereby recast as a statement about how retention scales with experience, and Section 4 gives one sufficient reason — stationary redundant sampling of a fixed task relation — for the diluting case to hold.

Around that core, the primer's catalogue resolves into one solution under four settings of two terms: log-linear, Stanford-B with carried-over coupling shifting the origin, DeJong with irreducible entropy lifting the asymptote, and the Kemerer adoption dip with a loss-bearing task-class reset. Classical parameters become bit-valued quantities, and the KEDE transform re-expresses the whole curve on a bounded, unit-free $(0,1]$ axis whose ordinate — missing knowledge — means the same thing in every context, with the operational estimator tied to the latent quantity through the Theorem 2 envelope.

The claim is deliberately bounded. We show that $\alpha /t$ dilution is sufficient to derive the log-linear curve, not that it is necessary, and the account holds only where the comparability frame holds. What lies beyond — whether the diluting law is also forced, and which retention laws generate the hyperbolic, Gompertz, and plateau shapes that other task classes prefer — is the natural continuation of this line. But the central point stands: the learning curve is not a free-standing empirical law to be catalogued. It is what knowledge-discovery accounting looks like when a stable task is practiced, and its most familiar form is one equation of one ledger.

Applications

The Knowledge-Centric Perspective builds on Ashby's Law of Requisite Variety by emphasizing that successful outcomes depend not only on a system's range of possible responses, but also on its ability to select the right response for each disturbance. This requires internal “system knowledge” that maps disturbances to appropriate actions. As Francis Heylighen proposed in his “Law of Requisite Knowledge,” effective regulation demands more than variety—it demands informed selection[29]. This knowledge-centric lens provides a foundation for analyzing how systems—biological, technical, or organizational—achieve control not just through options, but through understanding. The model we present operationalizes this perspective by estimating the informational requirements a system must satisfy to achieve its observed level of regulatory performance.

In what follows, we apply this Knowledge-Centric Perspective to a range of domains, including motor tasks and manual assembly, industrial assembly lines, software development processes, speed of light in a medium, intelligence testing and sports performance. In each case, the model enables us to estimate, in bits of information, the amount of knowledge a system must lack to produce its observed level of performance. By quantifying the knowledge to be discovered H(X|Y), we assess how much uncertainty was there in the system's ability to select appropriate responses. This allows us to compare systems not by tangible outcomes, but by the hidden knowledge structures required to achieve them, offering a unified lens for analyzing adaptation, skill, and control across diverse contexts.