18 Evidence From Protocol Records: M(t) and the Rate of Modification

English日本語|PDF (EN)PDF (JA)

v0.28.9 — This text is under construction. The structure of the theory, the propositions, and the empirical conclusions may all change. Overview

Chapter 18
Evidence From Protocol Records: $M (t)$ and the Rate of Modification

In the “inside” partition of Chapter 10, $M (t)$ , $κ_{j}$ , $\bar{δ} - δ$ and $B$ are published per identifier (Section 10.6). These are quantities that other chapters cannot measure for want of per-firm data. This chapter observes them.

The population is every $Φ$ operating at the reference date, and the sample was not chosen after the fact.

18.1 The implications tested

Following the procedure of Chapter 11.5, four implications were fixed before acquisition. They are numbered with $Φ$ to distinguish them.

Implication $Φ$ 1: the fewer other procedures a procedure calls, the longer until it is modified. This is the $λ_{𝑑𝑒𝑝}$ side of Proposition 9.2.

Alternative: the frequency of modification is fixed by usage, not by the number of dependencies. Few dependencies but heavy use still leads to modification.

Implication $Φ$ 2: for a $Φ$ whose transactions cease, $M (t)$ declines monotonically before the cessation. This follows from $τ = \inf {t : M (t) < 0}$ of Section 4.2 being absorbing.

Alternative: cessation occurs independently of $M (t)$ , which does not change until immediately before.

Implication $Φ$ 3: the distribution of $CCC$ among $Φ$ operating in the “inside” partition is skewed non-negative, because by Corollary 10.12 realizing $CCC < 0$ requires $E > 0$ .

Alternative: the distribution of $CCC$ resembles that of the corporate sector in Chapter 16, with no skew specific to the “inside” partition.

Implication $Φ$ 4: assigning operating $Φ$ by the rule of Section 7.9, family 7 barely appears except for 7-4. This follows from Proposition 10.6.

Alternative: family 7 types appear with about the same weight as other families.

18.1.1 What was fixed in advance

Item	Content
Population	every $Φ$ operating in the protocol record at the reference date; not chosen after the fact
Floor	those with fewer than a set number of transactions in the reference period are excluded; the threshold is fixed before acquisition
Definition of cessation	no transaction over a set period. Because the protocol itself does not stop in the “inside” partition, cessation rather than stoppage is used
Steady state	$CCC$ presumes a steady state, so only periods with stable $r$ are used (Section 12.4)
Unit of observation	the identifier. No aggregation to parties (Remark 10.5)
Assignment of types	the order of Section 7.9, unchanged

Table 18.1: What was fixed in advance for the four implications.

Remark 18.1 ( $λ_{𝑑𝑒𝑝}$ and usage can be correlated). The alternative to implication $Φ$ 1 is hard to reject, because procedures with many dependencies tend to be feature-rich and also heavily used. The test must control for usage, and whether an effect of dependencies survives in the residual cannot be known in advance. The control procedure is fixed before acquisition.

18.2 Results

One implication was supported.

Implication	Sample	Result	Verdict
$Φ$ 1	1.93m modifiable procedures	the number of dependencies does not explain time to modification	rejected
$Φ$ 2	427 with both fees and balances	declines over the 30 days before cessation ( $p = 3.2 \times 1 0^{- 9}$ )	supported
$Φ$ 3	52 operating $Φ$	the alternative is not identified	untestable
$Φ$ 4	as above	family 7 appears once, but as 7-2; 7-4 does not appear	rejected

Table 18.2: Results for the four implications.

Remark 18.2 (Concentration of clones dominates the estimate). The 1.93 million identifiers point to only 76,609 distinct implementations, and 1.55 million of them — 80% of the population — point to a single one. These are not independent $Φ$ but copies of one $Φ$ .

Over the whole sample the coefficient on the number of dependencies is positive and significant ( $p = 0.004$ ), but removing that one cluster reverses the sign and destroys significance ( $p = 0.45$ ). Cluster-robust standard errors do not protect against this. They correct the correlation within a cluster but not the bias in a point estimate when one cluster accounts for 95% of the exposure.

The loss of additivity in Remark 10.5 concerned the divergence between parties and identifiers; the concentration of identifiers pointing to one implementation is a different form of divergence. When assembling a sample from a ledger, the concentration of clones must be counted first.

Remark 18.3 (The alternative is not identified). $CCC < 0$ means payment in advance, which by Corollary 10.12 requires $E > 0$ . But the $CCC$ of the comparison group, the corporate sector, is non-negative in all 58 industries of Chapter 16 (minimum 4.9 days). A negative $CCC$ appears in individual firms and cancels out on aggregation by industry.

The implication and the alternative therefore say the same thing, and no observation distinguishes them. The ill-posedness has the same form as that of the measurability hypothesis in Section 15.3.5.0. In addition, $τ$ in the “inside” partition is measured in seconds against days in the corporate sector — five orders of magnitude apart.

Remark 18.4 (Being implementable and appearing are different). Family 7 is 1 of 52, so the quantitative part — that it is rare — was right. But what appeared was 7-2 (cross-subsidy), and 7-4 appeared zero times.

The implication derived “only 7-4 remains” from Proposition 10.6, but the proposition speaks of whether implementation is possible, not of how often something appears. Deriving appearance from implementability was the error.

The single 7-2 selects its beneficiaries by signature, outside the protocol. That is consistent with Proposition 10.6, but since the wording of the implication missed, it is not treated as supporting evidence.

Remark 18.5 (A step is needed before the assignment order). Table 7.8 takes $Φ$ as given. Applied to ledger identifiers, 131 of a sample of 200 were not $Φ$ at all: 74 were token ledgers (a medium of settlement with no pair of $δ$ and $π$ ), 27 were infrastructure with no consideration, and 30 were identifiers imitating the name of a settlement medium.

The last satisfy Definition 10.1 but have no delivery. The unobservability of $ι$ in Section 10.3 appears here as misrepresentation of a name.

18.3 Quantities measurable from the same records

Three quantities that other chapters could not measure can be measured from the records used for these tests.

18.3.1 The rate of response to unforeseen states

Proposition 9.2 put the unmanned operating period at $1 ∕ λ_{𝑛𝑜𝑣𝑒𝑙}$ . Two rates can be measured in the “inside” partition.

Quantity	Content	$λ$ (per year)	$1 ∕ λ$
Modification rate $λ_{𝑓𝑖𝑥}$	rate at which the implementation was replaced; 1.93m records, excluding the clone concentration	0.048	20.6 years
Cessation rate $λ_{𝑠𝑡𝑜𝑝}$	rate at which operation ceased; all 1,718	0.030	33.8 years

Table 18.3: Response rates measured in the “inside” partition.

Remark 18.6 (What is measured is not $λ_{𝑛𝑜𝑣𝑒𝑙}$ ). Both entries in Table 18.3 count responses to arrivals, not arrivals of unforeseen states. A modification is a case where the operator responded; a cessation is a case where they did not, or could not.

Arrivals that required no response appear in neither. Hence

λ_{𝑛𝑜𝑣𝑒𝑙} \geq λ_{𝑓𝑖𝑥} + λ_{𝑠𝑡𝑜𝑝}

and what was measured is a lower bound. The two are not additive because their populations differ, but an upper bound follows: the $1 ∕ λ_{𝑛𝑜𝑣𝑒𝑙}$ of Proposition 9.2 is shorter than 20 years.

The distribution of time to modification is extremely skewed. The median of the 20,732 that were modified is 42 days; the median of the 1.9 million that ended the observation without modification is 4.6 years. What gets touched is touched at once; what does not goes untouched for years. A mean $1 ∕ λ$ conceals this bimodality.

18.3.2 Duration of the procedures

Chapter 17 abandoned direct measurement of the unmanned operating period for want of a population frame. As Section 10.6.1 notes, that reason disappears in the “inside” partition.

Of all 1,718 that operated, 143 (8.3%) ceased and 1,575 continue. Among those that ceased, the median duration was 1.62 years, the quartiles 0.64–2.82 years, and the maximum 6.0 years. Those continuing have a median of 2.75 years. Because the ceased were counted from the full population rather than collected after the fact, no conditioning on survival (Section 12.2) arises.

Cessation rates are ordered by kind: services $1 ∕ λ = 11.1$ years, high-leverage operation 11.5, issuance support 15.3, indices 18.1, bridging 20.1, insurance 22.3, lending 22.6, and yield 22.7. The direction — kinds with lighter dependence on other procedures last longer — can be read off, but since implication $Φ$ 1 was rejected, this ordering is not explained by the number of dependencies.

18.3.3 Locked state $B$

Table 10.1 recorded that this text has no quantity corresponding to $B$ . In the “inside” partition it is directly observable as assets placed on the protocol, and daily series for 1,718 records are available. This is what served as the proxy for $M (t)$ in implication $Φ$ 2.

18.3.4 Rates

Once $B$ is measurable, a rate can be defined as annual fees divided by $B$ . But the denominator differs by family. For family 3 (renting assets) $B$ is the right denominator, whereas for family 6 (intermediation) the denominator is transaction value and $B$ is merely inventory. Mixing them makes the ratio diverge.

Restricting to family 3 types with $B$ of at least one million dollars, 144 records give the following.

Kind	$n$	Median	Quartiles
Arbitrage	7	4.97%	2.00–9.31%
Real assets	20	3.85%	2.35–5.84%
Collateralized debt	15	3.53%	0.99–11.86%
Liquid staking	19	3.12%	2.58–4.21%
Lending	30	2.97%	1.77–4.68%
Risk management	21	2.83%	2.02–4.47%
Yield	15	2.72%	0.76–8.01%
Yield aggregation	14	1.69%	0.90–4.19%

Table 18.4: Rates for family 3 types (annual fees

\div

locked state).

The overall median is 3.07% with quartiles of 1.77–5.84%. The ordering has the same form as the ladder of rates in Chapter 17, but the two cannot be compared: the rates there are on transaction value, the rates here on balances. The dimensions differ.

Chapter 18 Evidence From Protocol Records: M(t) and the Rate of Modification