Appendix E
Revision history

E.1 History of versions

This is a theory under construction, and its structure and content have been remade several times. The principal stages are recorded here. The names given to the versions are matters of convenience.

The version number corresponds to the stage: stage $n$ is v0.$n$.0. A change of structure, the addition or withdrawal of a proposition, or new empirical work raises the stage; corrections of slips and adjustments of format are handled in the third digit (v0.8.1, for instance).

E.2 Claims withdrawn

Claims have been withdrawn several times in the course of this work. What was withdrawn and why is recorded here. Since the body retains only the corrected description, which claims were once thought to hold appears in this appendix alone.

So that a reader partway through is not misled, the body is kept consistent with the corrected content. What follows is a record of how this work erred, and is not needed to understand the body.

E.3 Corrections deriving from measurement

The first of these is the weightiest. Having observed only the deterioration of $\mathrm{CCC}$, the work came close to concluding that $g^{\star }$ had fallen; but Corollary 5.7 is a ratio, and no conclusion follows from a change in the denominator alone. The error would have been avoided by obtaining $m$, and in fact the work had halted with the note that “interpretation is reserved because the change in $m$ has not been measured”. Reserving judgement turned out to be right.

E.4 Corrections deriving from reasoning

E.5 A record of pre-registration at work

The following are places where, had the implication not been fixed in writing beforehand, the work would likely have drifted into retrospective interpretation.

(1)

The disappearance of capped-scheme fees. That 99.7% of the capped scheme has disappeared in family 5 could have been read as circumstantial support for the measurability hypothesis. In fact the price level of the capped scheme is simply an order of magnitude lower, and it has nothing to do with measurability.

(2)

The contemporaneous negative correlation of $\mathrm{CCC}$ with sales. A negative correlation was obtained in 31 of 45 industries ($p=0.008$). This could have been read as “$\mathrm{CCC}$ works as a business-cycle indicator”, but it is not distinguished from the denominator effect.

(3)

The credit position of large firms. That $\mathrm{DSO}-\mathrm{DPO}$ is large for large firms could have been interpreted as “large firms support their counterparties”. In fact it is the result of implication F being rejected, sign and all.

The retrospective narrativization described in Section 12.6 of others’ accounts arises equally in the author’s own interpretation.

E.6 Making assumptions explicit

Several propositions here were originally stated without making their independence assumptions explicit. In the course of verification it emerged that quantities in fact determined jointly were being treated as independent, and the assumptions were written into each proposition.

The endogeneity of $\mathcal{𝒢}$ (Remark 4.3) was added at the same time. None of the proofs was changed: only the range of application was narrowed.

A passage using the undefined symbol $\mathcal{𝒳}$ as the domain of the production function was rewritten without naming a domain. Since this work does not treat the internal structure of $f$, no space of inputs need be introduced. $\mathcal{𝒟}$, $\mathcal{F}$, $H$ and ${\varphi }_{\mathit{𝑟𝑖𝑠𝑘}}$ were added to the notation list.

E.7 Introducing the lag

The ${\tau }_i$ of Section 3.1.4 did not exist in the original version. It was introduced in the course of working on weakness (C) organized in Section 19.4.3, the failure to separate level from change.

The decision to introduce it was made after first confirming that it would be used in several places: the explanation of differences in level (Corollary 5.17), the impossibility of cross-sectional comparison (Corollary 5.18), and the decomposition of change (Corollary 5.20).

Distortion of $\mathrm{CCC}$ during growth and a reinterpretation of the $\beta$ estimate were originally also expected as uses, but the rejection of implications O and P showed that these cannot be detected in annual data (Section 16.9.2). That use was removed from the grounds for introduction.

At the same time, Example 4.9 was corrected where it used $\mathrm{CCC}$ and $W$ before their definitions, and rewritten in terms of $\kappa$ alone.

E.8 Re-evaluating the relation to existing theory

In the draft of this work, some of the terms of ${\kappa }_i$ were expected to be territory specific to it, without checking whether existing theory covered them. On checking the literature, theory existed for every one.

This check moved where the contribution lies. What was originally sought was territory without existing theory; in fact the contribution is introducing existing theory as the explanation of each term and treating them as a sum (Remark 3.3). This is why Appendix A.4.1 was recast in three layers.

The same failure has recurred. When implication B was set out in Section 15.3.5.0, the existence of the risk–incentive trade-off dispute had not been checked.

E.9 Filling in definitions

The following are not corrections of errors but additions where the definitions were thin.

The revision of $g^{\star }$ changes the measured values. In the stratum below ¥10 million, against a value of 20.9% from $m$ alone, the nine-year average including $I$ is 3.1%. The conclusion of Proposition 14.6 is maintained, but the burden of investment is added to the cause.

As to the split of costs, “$C$ does not depend on $\mathrm{\Phi }$” was originally proved as a proposition. But that $\mathrm{\Phi }$ does not appear once $C(\delta )$ has been defined is a consequence of the definition, and the proof was circular. It was withdrawn, rewritten as the definition of the split, and the eight routes by which $\mathrm{\Phi }$ generates costs enumerated.

Defining ${\kappa }_K$ revealed that subjectivity divides into three stages (Remark 3.23). The subjectivity of the valuation map $v$ is asymmetry of information; the subjectivity of the value $V_j$ of the residual claim is disagreement of probability measures. The former can be resolved by disclosure; the latter cannot.

E.10 Examining reversibility, and a correction

The framework was reviewed from the standpoint of irreversibility and checked against existing theory. One correction and two additions resulted.

Correction. In Remark 2.11, the mechanism making $\rho$ endogenous was described as “objects with low correlation are exhausted”. According to [30] the correct mechanism is different: it is not that the objects of pooling run out, but that the act of pooling itself creates correlation among those who pool. If two agents each hold half of two risks, each becomes individually safer while the two become perfectly correlated.

Additions. The switching cost of $\mathrm{\Phi }$ (Remark 2.18) and the social optimality of pooling (Remark 2.12) were recorded. The former corresponds to the theory of irreversible investment, the latter to research on systemic risk.

The framework of classifying reversibility by the size of a cost is itself already held by the theory of irreversible investment and is not a contribution of this work. The formulation that a decision is irreversible not because it cannot physically be undone but because undoing it is costly belongs to the line of [8].

E.11 Examination from the worker’s viewpoint

Whether to treat workers as an independent subject was examined, and the conclusion was not to.

The organization initially attempted was “a worker is a special case of the one-person business”: a one-person business whose $\mathrm{\Phi }$ is given and whose customers concentrate in a single firm. It emerged, however, that the constraints differ in both $\mathrm{\Phi }$ and $\varphi$, and this organization was withdrawn (Remark 7.3). The Labour Standards Act constrains five of the six degrees of freedom of Section 2.5, and ${\varphi }_{\mathrm{prod}}$ and ${\varphi }_{\mathrm{cog}}$ have no counterparts.

Four implications were registered in the course of this examination.

Implication T is untestable because $\kappa$ for sole proprietors is not published. In this process an obstacle was found that does not fit the four classes of Section 13.2: the form in which a survey is taken but not tabulated (Remark 15.18).

E.12 Extending the taxonomy of obstacles

The four classes of Section 13.2 were constructed from the initial experience of investigation. Later investigation produced two obstacles that do not fit them.

(vi) in particular carries large implications for the method of this work. Part III attributed the failure to verify to four causes: measurement, theory, identification and access. The case in which reality does not behave as the theory posits had not been anticipated.

At the stage of setting out implication B for the measurability hypothesis, it was written that $b$ could be obtained only as a binary. Later investigation revealed that the realized fee rate is published as a continuous quantity, making the test possible. The obstacle was not the granularity of the data.

E.13 Extending the taxonomy of biases

The selection biases of Part III originally consisted only of those concerning the selection of the sample. Two distortions that the selection of the sample cannot explain emerged in the course of the investigation.

Both were added as sections of Chapter 12 (Sections 12.7 and 12.8), and the opening of that chapter was changed from “six kinds” to “eight kinds”.

The former concerns the unit of observation rather than the selection of the sample, the latter variables rather than samples. Both lay outside the range Chapter 12 originally treated.

The grounds for the former were replaced at stage 17. They originally rested on the behaviour that, in retrieval specifying a licence number directly, a number that failed to resolve returned the immediately preceding result. This admits interpretation as a measure limiting automated retrieval, and erecting a general obstacle from a single observation was a leap of the same form as in Section 16.11. The grounds were changed to the fact that in Section 15.3.5.0 the presence or absence of consolidation reversed the sign of the correlation. The description of that behaviour and the accompanying $n=2$ table were removed.

At the same time, the point that a complete frame does not yield power if the population is small was added to Section 12.9. Section 15.3.5.0 had referred to that section with “as stated”, but the statement referred to did not exist there.

E.14 Reorganizing the list of open problems

The list, extended from 12 items to 21 at stage 13, was reorganized into 19 at stage 20.

(1)

A completed item remained. Geographical extension of the sample was completed when the national sample was retrieved for the test of Section 15.3.5.0. It nevertheless remained first among the items that could be started.

(2)

One task was split in two. Testing implication Z and collecting the refund rates and periods that are its regressor were listed as separate items. A test and the data missing for it are two faces of the same task; they were merged.

(3)

A classification was wrong. The curvature of the age effect was placed under “items with no obstacle”, but pursuing it requires a sample carrying firm age. The Financial Statements Statistics used here has three axes — industry, size and year — and does not include the year of founding. It was moved to (iv) access constraint.

Items with no obstacle thereby fell from six to three. A list drifts from the actual state not when items are added but when finished items are left undeleted.

E.15 The means of reproduction were not included

Appendix E.33 stated that the processing used in the calculations was “attached”. Not one file was in the repository. A claim of the body was not satisfied by what was published.

Data and processing were included at stage 19, and three defects were fixed at the same time.

(1)

The parse.py that Appendix E.33 named did not run. Its inputs were still absolute paths from an old environment, and running it failed with files not found. It was rewritten to read raw/hojin/.

(2)

Some intermediate data had no means of regeneration. No script existed to build the series for officers’ remuneration and depreciation, and they could not be restored from a fresh clone. parse.py was extended to all six series, and the regenerated values were confirmed to agree throughout with the previous ones.

(3)

A collision of output paths was resolved: the results of the test of implication E were being written under the same name as intermediate data.

After inclusion, it was confirmed from a state with derived/ deleted that parse.py and all 14 analyses run.

E.16 Removing a description resting on a single observation

Chapter 13 recorded the behaviour that, in retrieval specifying a licence number directly, a number that failed to resolve returned the immediately preceding result, and generalized this into the obstacle “retrieval succeeds but the object is mistaken”.

It admits interpretation as a measure limiting automated retrieval, and other explanations were not excluded. Erecting a general obstacle from a single observation is a leap of the same form as the one Remark 16.10 warns against. The description and the accompanying $n=2$ table were removed at stage 18.

The section of Chapter 12 itself was retained. Its grounds were changed to the fact that in Section 15.3.5.0 the presence or absence of consolidation reversed the sign of the correlation, and its subject changed from “mistaking” to the mismatch between what a row denotes and what its values denote. That appears at a magnitude sufficient to reverse the conclusion in a sample of 1,402 firm rows across four occupations (1,130 distinct firms), and is not a single observation.

E.17 Setting theorem-type body text in roman

Propositions and corollaries were being set in the plain style, which italicizes the body text. Japanese has no italic form, so it stayed upright, and only the Latin letters and digits became italic, mixed in among it. In places even the figures in tables placed inside propositions were italicized.

They were unified in the same roman as definitions, remarks and examples. The headings remain bold and numbered. Emphasis by \emph in the body that contained Latin letters and became italic was likewise changed to bold.

The heading of proofs was also changed. The default in amsthm is the English Proof, which does not suit a Japanese document; \proofname was set accordingly.

Italics thereby remain only in the punctuation attached to the proof heading and in journal names in the bibliography.

E.18 Testing implication Y, and a defect in the registration

At stage 17 a national sample was retrieved from the Jinzai Service General Site and implication Y tested on the part excluding Tokyo (Section 15.3.5.0). The verdict is partial support.

The registration was defective. Table 15.19 laid down that offices with amount-displayed fees would be used for implication Y, but the reference values of Table 15.18 against which the comparison was made had been computed on a sample excluding them (Remark 15.12). The composition of the reference values had not been checked at the time of registration. Both rules are reported, and the matter handled by showing that the verdict does not depend on the rule.

Three points emerged during retrieval.

First, rows of the search results are offices while the values are at firm level (Remark 15.11). Taking correlations without consolidating gives, among physicians, a single company 134 times the weight, and the sign of the correlation turns from positive to negative. Consolidation was not preprocessing but a procedure determining the conclusion.

Second, the displayed turnover rate cannot be computed from the placements and separations in the same row. For physicians the two agree in only 115 of 479 rows. The composition matches Section 15.3.5.0 and does not impede comparison, but it was recorded that size and turnover are not quantities over the same period (Remark 15.13).

Third, the initial tabulation dropped part of the sample. Extraction was restricted to the licence-number letters Yu and Fu, so 200 Mu (free placement office) and 7 Toku records were missed. This was wrongly described as loss during copying, and up to 11.2% was reported as lost. The number of data rows agrees with the search results; nothing was lost.

The error changed the verdict. With the dropped sample, the ordering excluding Tokyo did not agree with the reference values, and it had been recorded that the ordering is not preserved. After correction the ordering agrees perfectly under the registered rule. The verdict was set to partial support not because the ordering is not preserved but because the agreement of the ordering depends on the rule of tabulation, and that rule had not been aligned with the composition of the reference values at the time of registration.

The reference values of Table 15.18 agree, on the Tokyo portion of the corrected sample under the same composition, to within 0.02 in correlation; nurses agree to three decimal places.

E.19 A route for testing the deductions of Part II

The deductions of Part II had almost no pre-registered implications up to stage 15. Of the 16 implications then registered, only D and T clearly targeted that part; D was supported as to direction only, and T is untestable.

At stage 16, implications $\mathrm{\Phi }$1–$\mathrm{\Phi }$4 were registered for the “inside” partition of Chapter 10 (Section 18.1). In that partition the record of the procedure is itself the population, and halted procedures remain, so the survival conditioning of Chapter 12 does not arise. The reason for abandonment given in Chapter 17 — “no population frame” — disappears here.

Three points were distinguished before registering (Table 10.7).

(1)

Quantities the procedure enforces are not objects of verification. The collateral constraint of Proposition 10.9 is such a quantity: observing that collateral is posted is merely reading the rules. Only quantities appearing as the result of an operator’s choice are objects of verification.

(2)

The population differs (Remark 10.14). What can be verified is the correctness of the deduction, not whether the deduction applies to real one-person businesses.

(3)

Quantities at the level of agents cannot be recovered (Remark 10.5). What can be measured is only the distribution at the level of identifiers.

For $\mathrm{\Phi }$1 it was noted that, because the number of dependencies may correlate with usage, the alternative is hard to reject (Remark 18.1).

E.20 Registering implications Y and Z

The observation of Remark 15.8 — that the correlation of size with turnover differs greatly by occupation — remained recorded as a secondary observation with no use up to stage 14.

At stage 15 a route was set out for reading it as a proxy for the B1 side, the informativeness of effort that Table 15.13 records as lacking a proxy, and implications Y and Z were registered (Section 15.3.5.0).

Two points were fixed at registration.

First, the test must not use the same sample (Remark 15.10). The correlation was observed after the fact in the 307-firm sample, and testing on the same sample would be an error of the same form as retrospective narrativization. What is required is reproduction out of sample.

Second, the mechanism is not identified (Remark 15.9). Thinning of input and looser screening give the same sign, so the variable can be used as a proxy but no claim about the mechanism can be made.

There is a literature in labour economics on the relation between size and the quality of placements which this work has not surveyed; it was added to “theory that should be connected but is not” in Appendix A.4.1.

E.21 Checking the scope of inventories

Chapter 16 computed $\mathrm{DIO}$ from “finished goods or merchandise” alone, excluding work in process and raw materials. The limitation was recorded as “the effect on analyses of differences and correlations is small, but the absolute level cannot be interpreted”, with no grounds given.

At stage 14 the difference was measured against “inventories (end of period)” in the Financial Statements Statistics (Section 16.10). The result supports that statement: the level moves by more than 30% in some industries, but the correlation of $\mathrm{\Delta }\mathrm{CCC}$ is 0.976.

Incidentally, the elasticity $\beta$ rose from 0.900 to 0.960 and the coefficient of determination improved from 0.092 to 0.213. Working capital is more nearly proportional to sales when work in process and raw materials are included. The contribution of the denominator effect moves smaller, and the conclusion of Section 16.9 is strengthened.

During estimation it emerged that, because the observations with $W>0$ differ by definition, the sign comes out reversed unless the sample is aligned (Remark 16.9). Without alignment the values are 0.894 and 0.787, giving the opposite conclusion that the denominator effect increases.

E.22 Filling in operationalization and reorganizing the list of open problems

Classifying each item against the five stages of Chapter 11 revealed that items halted at stage 1 (operationalization) were not registered in the list of open problems. Although a defect at stage 1 requires no data retrieval and is filled by definition, the list was skewed towards problems of measurement and access.

The following were done at stage 13.

(1)

Availability was written as an equation (Definition 9.4). Previously it was stated only as “consuming the constraint” and had no form as an inequality. Whereas the capacity constraint bounds the total volume of delivery, availability bounds the length of an interval during which no response can be made. Both have the dimension of time.

(2)

The reason the degree of automation cannot be defined was identified (Remark 9.9). It had been recorded that “$\delta$ has no unit”, but $\delta$ carries monetary units through the valuation map. Correctly, what $v$ measures is value, not effort; and measuring on the response side leaves no measure defined on $\mathrm{\Omega }\setminus \widehat{\mathrm{\Omega }}$. The obstacle is the absence of a measure, not of a unit, and is not resolved by refining the definition.

(3)

Rules for assignment to types were laid down (Section 7.9). Using the catalogue empirically requires a rule for assigning an observed $\mathrm{\Phi }$ to a type, and none existed. Since the conditions identifying families are not exclusive, an order was fixed. The choice of order has no deductive grounds and is a convention, so it must be fixed in advance (Remark 7.1).

(4)

A limitations section was placed in Chapter 9 (Section 9.4). Of the deductive chapters it alone had none, and its limitations were written in Chapter 17. The theoretical limitations were moved to that chapter and the empirical ones left in Chapter 17.

(5)

Open problems were reorganized from 12 items to 21. Nine items in the limitations sections and records of abandonment across the chapters were not reflected in the list. Six of them fell under no obstacle at all and had simply not been started, so a subsection was separated out for “items with no obstacle”. The table of priorities was likewise split into what can be started and what will not advance.

This changed the close of Chapter 19. It had read “no item remains that can be advanced by retrieving public data alone”; six items have no obstacle.

E.23 Placing the empirical design first

Chapter 13 recorded that the four-way classification should be placed ahead of the design of an investigation, because Part III treated selection bias in samples while having no procedure for judging “is verification possible at all?”.

This was carried out at stage 12 as Chapter 11. The four-way classification itself is not placed first, however. The classification was extended to six through the empirical work, and placing it first would put findings not yet obtained ahead of their derivation. What was brought forward is only the stages to be checked; the classification of causes for failing to pass them was left in Chapter 13 (Remark 11.1).

The distinction between comparison and testing was also stated as a discipline (Section 11.4). This work contains several consistency checks not resting on pre-registration, such as the family-3 comparison of Chapter 8, but no passage stated their standing; each chapter merely noted individually that “this is a retrospective comparison and does not establish causation”. The discipline is needed because a comparison requires no declaration of a population and so can be carried out while skipping stages 2 and 3 (Remark 11.2).

E.24 Implication Z tested

Up to stage 22, implication Z was left untested as “the sample is already retrieved; what is missing is only the strength of the refund scheme”. It was tested at stage 23 and rejected.

E.24.1 Implications $\mathrm{\Phi }$1–$\mathrm{\Phi }$4: three different failures

Registered at stage 16 and tested at stage 24. Of the four, only $\mathrm{\Phi }$2 is supported, and the forms of failure all differ.

$\mathrm{\Phi }$1: an error in the design of the estimate. At registration it was thought that keeping the identifier as the unit of observation and using standard errors robust to clustering by implementing address would handle the concentration of replicates. This is wrong. Robust standard errors correct for within-cluster correlation but not for bias in the point estimate when one cluster accounts for 95% of the exposure. Only on reporting an estimate excluding the dominant cluster did it emerge that the sign of the coefficient reverses and becomes insignificant (Remark 18.2).

$\mathrm{\Phi }$3: an error in the choice of control. “The distribution of $\mathrm{CCC}$ is skewed non-negative” was designed as a contrast with the corporate sector, but the corporate sector’s $\mathrm{CCC}$ is also non-negative throughout at the level of industry aggregates. Negative $\mathrm{CCC}$ appears in individual firms and cancels out in aggregation. The implication and its alternative say the same thing, and observation cannot tell them apart. This is ill-posedness of the same form as implication B in Section 15.3.5.0 (Remark 18.3).

$\mathrm{\Phi }$4: mistaking the implication. Proposition 10.6 states that a $\mathrm{\Phi }$ requiring identification of the beneficiary cannot be implemented. From this “only 7-4 survives among family 7” was derived, but the proposition speaks of implementability, not frequency of appearance. The quantitative part — that family 7 is rare — was right; the part about the surviving type was wrong (Remark 18.4).

What the three have in common is that each was noticeable at the time of registration. All are problems of design that could have been examined before measurement; the data were not lacking.

E.24.2 Dropping the symbol for the production function

Up to stage 26 the technology side was called a production function $f$. But no functional form was ever given, and the notation list recorded that “the domain is not stated”. There were five mentions in the body, none of which used $f$ itself; it appeared only through $C_{\mathrm{prod}}$.

At stage 27, $f$ was dropped and the technology side unified into $C_{\mathrm{prod}}$ (the correspondence from delivery to cost). The definition of production surplus (Definition 6.2) already referred to $c(\delta )=d{C}_{\mathrm{prod}}∕\mathit{𝑑𝛿}$ and holds without $f$.

A symbol with a name and no substance had been left in place.

E.24.3 Defining the relation between classes of agents and individuals

The primitives merely listed customers $C$, suppliers $S$ and so on as “the breakdown of agents”, without giving the relation between an individual customer and the ${\kappa }_C$ denoting customers as a whole. ${\kappa }_C$ occurs frequently in the body, and in Example 2.6 the writing wavered between classes and individuals.

It was added to the definitions that a class is a subset of $N$, that an individual agent is written as $i\in C$ and as an element of $N$, and that a class as a subscript denotes the sum ${\kappa }_C={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i\in C}{\kappa }_i$. The index $j$ is used for identifiers in Chapter 10 and is therefore not used for individual agents.

E.24.4 Revising the verdict on implication D

Implication D was registered at stage 13 as “the fewer the dependencies of a configuration, the smaller ${\lambda }_{\mathit{𝑑𝑒𝑝}}$ and the longer the period of unattended operation”. Implication $\mathrm{\Phi }$1, registered at stage 16, reads “the fewer the other procedures a procedure calls, the longer the period before it is repaired” — the same relation. The duplication went unnoticed at registration.

Up to stage 24 the table of verdicts recorded D as “partial support”. But no section of the body supported that verdict, and Chapter 17 recorded that “implication D has not been tested”. The table and the body contradicted each other.

The test of $\mathrm{\Phi }$1 at stage 24 is also a test of D. The verdict was changed to rejected.

Registering the same relation twice is itself a failure in operating the pre-registration. The cause was keeping the register by chapter and never reconciling it as a whole.

How strength was to be measured had not been laid down at registration. Deciding after reading the documents would be fitting after the fact, so it was fixed as equation (15.2) before they were read.

Extraction by regular expression was abandoned. Checking 8 documents produced errors in 3, and the errors were biased towards overstating strength. Not being random error, they do not vanish on averaging. All 907 were read by a person, and the 132 carrying no text layer were put through OCR. None was illegible.

The reading brought several facts about publication practice to light. Some documents paste the rate table as an image alone, so the body is legible while the content is not. Some offices have a refund-field link pointing to a fee schedule. Some display the scheme as “yes” while filing a rate of 0%. Filing a document and having its content legible are different things.

E.25 Giving deductive grounds for choosing families 2 and 5

Section 11.5 said only that “families 2 and 5 contain the theoretical issues most clearly” and did not show why those two. What could be derived from the framework was left underived.

It was rewritten in four steps. Since ${\varphi }_{\mathrm{prod}}$ in equation (6.1) does not depend on $\mathrm{\Phi }$, the shape of $\mathrm{\Phi }$ bears only on the remaining two terms. ${\varphi }_{\mathrm{cog}}$ requires degrees of freedom (5) and (6) and ${\varphi }_{\mathrm{barg}}$ requires (4), and the order of family assignment maps these onto families 2 and 5 respectively. The two families sit on the diagonal of Figure 2.1, spanned by the two dominant axes.

The reason for placing them side by side was also set out first, as a difference of institutional grounding. It does not anticipate the conclusion of Part IV (completeness and richness are independent) and is confined to the design-level reason that the two need separating.

E.26 Removing references to places that do not exist

Part IV contained “at the end of Part II it was expected that family 5 had the better conditions”, but no such statement exists in Part II or in Section 11.5. A sentence from an earlier version had been deleted, leaving only the sentence that answered it. The conclusion was kept and only the reference to the premise dropped.

Also, the conditions for divergence were changed from a proposition to a remark at stage 2, but two places still referred to it as a proposition. Both were corrected.

E.27 Resolving a contradiction in the description of ${\varphi }_{\mathit{𝑐𝑜𝑔}}$

Immediately before Section 14.6.1 it was stated that “the only corresponding measurement is the recovery figure for family 2”, while the section went on to say that “prior work contains direct instances of measurement”. Of the same quantity it was written both that no measurement exists and that measurements exist.

Correctly, it is measurable and measurements exist; what this work obtained on its own is only an upper bound. The family-2 recovery figure was folded in as one of the instances, and the passage recast to list four.

E.28 Extending the scope of inventories to 62 industries

Stage 14 measured only ten industries and eleven size strata, and recorded that “the elasticity improves on correcting the definition”. Extending to 62 industries and all sizes reverses the direction: the denominator effect rises slightly from 0.160 to 0.172.

The two are not error. Variation between size strata and variation between industries identify different things. The estimate of that chapter is identified by variation between industries and years, so it is the 62-industry figure that applies (Remark 16.8).

The range covered by the sample had been mistaken for the range reached by the conclusion. There was never any guarantee that a direction obtained on ten industries could be applied to a 45-industry estimate. All three verdicts remain “supported”, but each is weaker than at stage 14.

E.29 Stating the handling of per-firm data

The Jinzai Service General Site publishes the filings of individual offices rather than aggregates. These were included in the repository at stage 19, but using them as input to analysis and redistributing the collection are different things.

The policy was written into Appendix E.33 and the per-firm data removed from the repository. What remains is only the list of licence and filing-receipt numbers, containing none of the filed values. Re-retrieving with that as a seed reproduces the realized fee rates, turnover rates and placement counts. What had already been published was removed from the history.

E.30 Multiple definitions of a symbol

At stage 28 the whole text was swept and six places where the same letter carried different meanings were fixed.

$c$ had four meanings. Only the cohort $c$ of age–period–cohort was retained, following the notation of that literature, with a note to that effect in the body.

The three forms of degree of freedom (2) were also gathered into equation (2.10), $\pi =F+\mathit{𝑝𝛿}$: $p=0$ is flat, $F=0$ is usage-based, and both non-zero is a two-part tariff. What had stood as three forms became one family.

E.31 Correcting a count

The catalogue of Part II was described consistently from stage 1 as “27 types”, but the counts by family are 4, 5, 3, 3, 5, 4, 4, totalling 28. Eleven places in the body were corrected at stage 9. The composition of the families and the content of each type are unchanged.

E.31.1 Calling the types a “basis”

Up to stage 27 it was recorded of the catalogue of Chapter 7 that “the fact that a $\mathrm{\Phi }$ cannot be reduced to a single type means the types function as a basis”. But no addition was defined on the space of $\mathrm{\Phi }$, so the word had no meaning.

At stage 28 the additivity of $\kappa$ was laid down as Proposition 3.9 and, at the same time, the non-uniqueness of the decomposition shown (Remark 3.10). Without uniqueness the types are a generating set, not a basis. The word in the body was changed.

E.32 Considered but not adopted in the body

In writing Chapter 10 at stage 9, a structure presenting it as a novel contribution was initially considered. On checking the literature it emerged that the restriction of funding by a collateral constraint, the exclusion of agents holding no assets, and the divergence of identifier from agent all have existing counterparts, so it was recast to claim no novelty and a table of correspondences added to Appendix A.4.1 (Remark 10.13).

The chapter was retained because, in the position of showing how the theory of Part I functions under concrete conditions, an example varying the $\mathcal{𝒢}$ of Chapter 4 was missing. Its role is the same as that of Chapter 8 and Chapter 9.

E.32.1 Describing an allocation that is not an interior solution as one

Up to stage 27 it was recorded of equation (8.14) in Chapter 8 that “an interior solution exists”. The equation is linear in the allocation $\mathrm{Cap}\mathrm{cl}$ and, under $0<𝜃<\gamma$, strictly decreasing, so taken alone it is maximized at the corner $\mathrm{Cap}\mathrm{cl}=0$. The error arose from putting the cash that commissioned work brings in neither into the objective nor into the constraints.

At stage 28 the cash the fundamental inequality demands was made explicit as a constraint, and the optimal allocation re-derived as the edge of the constraint, $\mathrm{Cap}{\mathrm{cl}}^{\ast }=c_{\mathit{𝑙𝑖𝑣}}∕w$ (Proposition 8.10). The direction of the conclusion — that the criterion for selecting engagements moves from the fee to $𝜃$ — is unchanged.

E.33 Changes of structure

Apart from corrections of content, the structure was changed twice.

(1)

From the chronological order of the examination to the logical order of theory, catalogue, method, empirics and conclusion.

(2)

The deductions under constraints (Chapters 8 and 9) moved from a separate part on “applications” into Part II. Applications are justified only after a theory has passed through empirical work, and the empirical work here is not complete.

The second change was accompanied by fixing 64 inconsistent references (parts confused with chapters) and two logical inversions (a later chapter referred to in the past tense).