Knowledge Discovery Efficiency (KEDE) and Learning Curves

Abstract

We address real-world applications of ....

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. Ohlsson's analysis of Isaac Asimov's writing career is a vivid instance: treating each completed book as one practice trial, the time to finish successive blocks of a hundred books declines along a line in log-log coordinates, so tightly that it fits a power law over four decades of professional practice. That such a curve, first characterized for perceptual-motor tasks lasting minutes in the laboratory, should also govern a complex cognitive skill exercised over a lifetime is, as Ohlsson notes, remarkable — and 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; the software-engineering primer in our sources catalogues fifteen such equations, of which the log-linear form y = a·xⁿ is the workhorse, with the Stanford-B, DeJong, and S-curve variants layered on to accommodate carried-over experience, machine-limited irreducibility, and slow-then-fast onset. Kemerer's study of CASE-tool adoption adds a further wrinkle that the smooth-decline picture cannot easily house: when a new technology is introduced, measured performance often drops *below* the old baseline before it recovers, so the curve dips before it bends. What unites these literatures is that the curve is *fitted*, not *derived*. The power-law-versus-exponential question — which functional form the data prefer — is settled by comparing residuals in log-log and semi-log plots, as Ohlsson does when he concludes that Asimov's data favor a power law over an exponential. The industrial catalogue of fifteen equations is, by the primer's own admission, 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 — Stanford-B's carry-over term, DeJong's incompressibility factor, the depth and duration of Kemerer's adoption dip — are estimated rather than explained. The result is a mature descriptive science with a thin generative core: we know the shapes learning takes, but not why it takes them.

Solution

This paper offers a generative 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 discovery burden, 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 two 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 KEDE 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 we read as a learning curve. Write the ordinate as the per-episode discovery burden $y_t:=H_t^{\text{start}}$ and the abscissa as cumulative surviving closures $x=t$. The classical curve plots cost or time against cumulative output; ours plots residual ignorance, in bits, against the same axis.

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; 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 Two Regimes

Take the pure-learning ledger recurrence $y_{t+1}=y_t-G_t^K$ 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 → exponential

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 $t$. It is the form Ohlsson's semi-log plot of the Asimov data rejects.

3.2 Experience-diluted retention → power law

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 $-\alpha$ — the log-linear learning curve $y=a{x}^n$ with $n=-\alpha$ — and the shape Ohlsson's log-log plot of the Asimov data confirms.

3.3 Theorem

Proof. Take logs: $\ln y_t=\ln y_0+{\sum }_{s=0}^{t-1}\ln (1-c_s)$. If $c_s\to \lambda >0$ the summand tends to a nonzero constant, so $\ln y_t\sim t\ln (1-\lambda )$ (exponential). If $c_s=\alpha /s+o(1/s)$, then $\ln (1-c_s)=-\alpha /s+O(1/s^2)$, and using the harmonic asymptotics ${\sum }_{s=1}^{t-1}1/s=\ln t+O\left(1\right)$ gives $\ln y_t=-\alpha \ln t+O\left(1\right)$, i.e. $y_t\sim C{t}^{-\alpha }$. The gamma-ratio identity of §3.2 fixes the constant $C=y_1/\Gamma (1-\alpha )$ exactly. Conversely, matching either asymptotic form back through $c_t=1-y_{t+1}/y_t$ recovers the stated retention law. $\square$

Theorem 1 relocates the power-law-versus-exponential debate. It is not a contest between two fitted curves but a question about a single structural quantity: whether the retained fraction $c_t$ stays constant or dilutes as $\alpha /t$. Section 4 gives one sufficient reason the second holds.

4. One Sufficient Mechanism for $\alpha /t$ Dilution

Theorem 1 makes the power law hinge on a single fact: the retained fraction $c_t$ dilutes as $\alpha /t$. We give one information-theoretic reason this can hold. We do not claim it is the only route to a power law, nor that a bounded-capacity learner must produce it; we claim only that it is sufficient, and that it is natural rather than contrived.

4.1 Why constant retention is unphysical

Constant retention, $c_t=\lambda$, says each episode captures a fixed fraction of whatever ignorance remains, forever. But stored coupling is bounded above by the marginal variety of the task, $I_t^{\text{start}}\le H\left(X\right)$, by the Complete-Adaptation corollary. As $I_t^{\text{start}}\to H\left(X\right)$, the ledger is filling a finite reservoir; the informative content of the next comparable episode is increasingly what the learner has already stored. Constant retention ignores this and treats every episode as equally novel. The dilution law is what remains once redundancy across comparable episodes is taken seriously.

4.2 The mechanism: diminishing marginal evidence

Fix the task class and model the discovery of $X$ as accumulation of evidence about a fixed disturbance–response relation. Under Posterior-Becomes-Prior, the coupling written to storage after episode $t$ is the mutual information the episode's evidence stream $U_t$ carries about $X$, beyond what earlier episodes already supplied:

The chain rule ties this conditional term to the running total: after $t$ comparable episodes the learner has resolved $I(X;U_{\le t}|Y)=y_0-y_t$ of the initial ignorance. The question is how the marginal term at episode $t$ scales.

Under Assumption D the situation is the standard one of posterior contraction about a fixed target from repeated observation. The marginal information the $t$-th sample carries about the target, conditioned on the previous $t-1$, decays as $\Theta (1/t)$ — the same rate at which the entropy of a posterior over a fixed parameter falls after $t$ i.i.d. observations. Writing that marginal as a fraction of the ignorance still outstanding gives

with $\alpha >0$ set by the informativeness of a single sample relative to the residual uncertainty. By Theorem 1 this yields the power law $y_t\sim C{t}^{-\alpha }$, i.e. the log-linear learning curve.

4.3 Reading of the exponent

The exponent $\alpha$ acquires a meaning the fitted slope $n=-\alpha$ lacks: it is the per-sample informativeness of a comparable episode about its fixed task relation. Rich, well-separated evidence gives large $\alpha$ and a steep curve; redundant or noisy evidence gives small $\alpha$ and a shallow one. The classical learning rate, $2^{-\alpha }$ per doubling, is thereby an information-theoretic quantity rather than a bare regression output.

4.4 Scope of the claim

The argument is sufficient, not necessary, and it fails cleanly where it should. If episodes are not comparable — a shifting $Y$, $X$, or $g$ — Assumption D breaks, each episode samples a partly new relation, and the marginal need not decay as $1/t$. If evidence is non-stationary or the reservoir $H\left(X\right)$ is itself growing, other retention laws — and other curve shapes — arise. Forgetting adds a positive $L_t^K$ term, treated in Section 5. The claim here is only that under the natural conditions of a stable task class sampled redundantly, the ledger reproduces the log-linear curve, and the exponential is the special case in which redundancy is ignored.

5. The Rest of Table 2 as Perturbations of One Recurrence

The primer's catalogue treats Stanford-B, DeJong, and the S-curve as separate equations to be tried against data. Under the ledger they are the power-law solution of Section 3 with one term switched on: a shifted origin, an entropy floor, or a reinstated loss term. Nothing new is fitted; each classical parameter becomes a bit-valued quantity in the same recurrence.

5.1 Stanford-B: carried-over coupling shifts the origin

Section 3 set the first comparable episode at $t=1$. A learner who arrives with prior coupling $I_0^{\text{start}}>0$ has, under Assumption D, already absorbed the equivalent of $b$ comparable episodes of evidence. The dilution law then runs from a shifted index:

This is the Stanford-B curve $y=a{(x+b)}^n$ with $n=-\alpha$. The Boeing finding that a team carries one to ten airframes of learning between models, and Pierson's one-to-six months for miners, are estimates of $b$; here $b$ is the number of comparable episodes' worth of coupling already in storage, not a free constant. The curve starts flat because the learner starts partway down it.

5.2 DeJong: irreducible entropy lifts the asymptote

Section 3 assumed the reservoir could be emptied, $y_t\to 0$. A task class with equally-acceptable response classes or aleatoric disturbance has an irreducible floor $y_{\infty }>0$: residual uncertainty no evidence can remove. Only the reducible part $y_t-y_{\infty }$ obeys the dilution law, giving

This is the DeJong curve $y=a[M+(1-M\left)x^n\right]$, with the incompressibility factor $M=y_{\infty }/y_0$ identified as the fraction of initial ignorance that is irreducible. DeJong's machine-limited floor is, in bits, the entropy the task cannot surrender.

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.

5.4 The Kemerer 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. 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 discovery burden 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 Summary

Four of the primer's canonical shapes are one solution under four settings of the same two terms: log-linear ( $b=0$, $y_{\infty }=0$, $L_t^K=0$), Stanford-B ( $b>0$), DeJong ( $y_{\infty }>0$), and the Kemerer dip ( $L_{t_0}^K>0$). The catalogue is not a list of rival equations but a coordinate chart on one ledger.

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 primer and Kemerer plot cost or effort per unit. This section connects the two through the operational KEDE 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 discovery burden 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 burden 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 primer'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 primer 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. Discussion

7.1 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 primer'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 primer 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.

7.2 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 D 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.

7.3 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.

8. 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 discovery burden $H_t^{\text{start}}$, 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.

Appendix

Cumulative Knowledge To Be Discovered

Using

$H\left(S\right)=\frac{N}{S}-1$

from (5) with constant $N$, the cumulative residual variety (C) as a function of performance level (S) has a clean closed form.

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

$$C\left(S;S_0\right)=\int S_{0}SH\left(u\right)du=\int S_{0}S(\frac{N}{u}-1)du=Nln\frac{S}{S_0}-\left(S-S_0\right).$$

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: $$C(N;S_0)=N\cdot ln\left(\frac{N}{S_0}\right)-(N-S_0).$$

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.