What a person at the handoff supplies, which half of it is already free, and the region where no chain length is enough
A working session here looks like three turns in a chain with a person reading every handoff. The question this paper makes answerable is whether six turns of nobody watching would rival it, and the answer turns out to be a quantity rather than an opinion.
Direction supplies two goods and they are separately purchasable. The first is context,
which a person carries across a handoff; WP2 §10c already
prices that as λ, and briefed buys λ = 1 structurally for no extra turns. The
second is correction, which nothing in the wiring supplies. This paper models only the
second.
Writing g for the share of live defects an unattended turn removes, a defect made k handoffs upstream is removed with probability g·λk−1, so its chance of surviving m downstream turns is a product. Two consequences follow and both are arithmetic. The defect introduction rate cancels, leaving three parameters where there seemed to be four. And the product converges whenever λ < 1, so an unattended chain has a floor it never beats, while a directed chain, whose context does not decay, has none.
The exchange rate between the two is therefore a small number, a large one, or infinite, and 30.5% of a plausible parameter grid holds the third. Six is the answer in a specific place: over almost the whole range of λ, but only for g between 0.35 and 0.65.
H9 measures it by seeding unannounced defects and scoring removals by gap. Simulation did not change its price so much as its reported outcome: near the feasibility boundary the exchange rate is not estimable at any affordable size, because it is genuinely steep there, so the design reports which of three bands it falls in.
Having a person in the loop is not one thing, and separating it is what makes the question arithmetic. A person at a handoff supplies context, carrying the thread so the next turn starts where the last one stopped, and correction, seeing a defect and saying so.
The first is already cheap. WP2 gives it a name, λ, the share
of an input surviving one summarisation hop, and briefed in the catalogue reaches
sink ken 1.000 for zero extra turns by copying every intermediate turn to the last one. Wiring
buys context. That is the whole of H5 and it is the cheapest manipulation the programme has.
Correction has no such trick. A turn removes a defect only if it can see one, and this paper is about what that costs.
The split is not new in substance. Dessein sets the loss from delegating a decision against the loss from communicating it imperfectly, and shows which dominates as the two parties' interests converge. Garicano organises production so that problems a worker cannot solve escalate to someone who can, which is a catch rate arranged by design. What follows keeps that decomposition and replaces the agents' incentives with two rates, because the agents here have no interests to diverge and the question is entirely about what reaches whom.
Let every turn introduce defects at some rate r. A defect introduced at turn i is removed by a later turn j with probability
The λ factor carries the argument. A turn removes only what it can still see, and what it can see decays with the number of handoffs since the defect was made. Write S(m) for the chance a defect survives m downstream turns:
After n turns the run holds r·∑m<n S(m) defects and has delivered n turns of work, so defect density is r times the running mean of S. Two things follow, and both are consequences rather than assumptions.
The introduction rate cancels. Comparing two densities divides r out, so how buggy a turn is drops out of the comparison entirely. Three parameters remain where there appeared to be four, and one of them already has an instrument in H8.
An unattended chain has a density it never beats, and a directed one does not. At λ = 1 the product S(m) = (1−q)m falls to zero, so enough turns drive density to nothing. At λ < 1 the product converges to a positive number: the later chances are discounted so heavily that they stop helping, and some share of defects becomes permanently invisible.
That is the difference between watched and unwatched work in this model, and it is worth being precise about what it is not. It is not that a person is sharper per turn; h and g could be equal and the asymmetry would remain. It is that a person's context does not decay, so their later chances stay worth something, while a chain's decay geometrically and the sum of them is finite.
That structure has an empirical ancestor. Bavelas and Leavitt wired small groups into fixed communication patterns and measured errors as well as speed, finding both to depend on the pattern rather than on the people in it. This model makes the same move with one difference: their patterns were undirected and could carry a message back, and a turn graph cannot.
Given the three rates, ask for the smallest n at which an unattended chain reaches the density of three directed turns. When the unattended floor sits above that target the answer is that no n does, and it is reported as never rather than as a large number, because those are different claims and only one of them is true.
| Quantity | Value | Where it comes from |
|---|---|---|
| λ, context surviving a handoff | 0.6 | assumed — H8 would measure it |
| g, unattended catch rate | 0.45 | assumed — H9's seeded arm would measure it |
| h, directed catch rate | 0.7 | assumed — H9 observes the directed arm's density instead |
| target: density of 3 directed turns | 0.4633 | computed |
| floor: density no chain length beats | 0.2618 | computed |
| exchange rate | 7 | computed |
| λ \ g | 0.20 | 0.35 | 0.50 | 0.65 | 0.80 |
|---|---|---|---|---|---|
| 0.20 | never | never | 26 | 5 | 3 |
| 0.40 | never | never | 8 | 4 | 3 |
| 0.60 | never | 14 | 6 | 4 | 3 |
| 0.80 | 22 | 8 | 5 | 4 | 3 |
| 0.95 | 11 | 6 | 5 | 4 | 3 |
Across a 19 × 19 sweep of the two rates, 30.5% of cells are never. That is the result this paper would most like to be wrong about, and it is not an artefact of extreme parameters: it is the whole lower-left region, where a weak gate meets a lossy handoff.
Six is the answer somewhere specific. Sweeping at a step of 0.05, the cells whose exchange rate falls between five and seven span λ from 0.05 to 0.95 but g only from 0.30 to 0.70, and the cells that give exactly six span g from 0.35 to 0.65. So the six-turn intuition is a claim about the catch rate rather than about attenuation: it says an unattended turn removes somewhere around half of what a directed one would, and at that rate six of them buy three.
The line between a finite exchange rate and none is a curve in the two rates, and it falls steeply. Attenuation and gate quality substitute for each other. With context surviving almost intact, even a weak gate eventually finds everything, because its later chances are still worth having. With context evaporating, only a gate nearly as good as the person keeps up.
The engineering consequence is immediate and it inverts the obvious priority. At g = 0.35 and λ = 0.4 the answer is never; at the same g with λ raised to 0.8 it is 8 turns. Briefing, which costs no turns, can move a design from impossible to affordable, while a better gate at the same λ may not. The cheapest way to make an unattended run more capable is to stop its context decaying, and only then to sharpen what it checks.
The three rates below are yours to set, because nobody has measured them. The page runs
graph/equivalence.mjs directly, so this widget and the gate cannot disagree.
This section needs JavaScript. Tables 1 and 2 carry the same arithmetic without it.
| What it is | What would measure it | What stands behind it now | |
|---|---|---|---|
| lambda attenuation | how much of a defect`s context survives one handoff | H8, the incidental-residue probe, priced at 36 turns | designed, not run |
| g automated catch rate | the share of live defects one unattended turn removes | a seeded-defect run: plant known faults, count removals by depth | nothing. The loop ledger records 17 gate failures over 89 turns, which counts what the gate CAUGHT and says nothing about what it missed |
| h directed catch rate | the same, for a turn a person is reading | the same seeded-defect run with a person at the handoff | nothing, and it is the expensive one to get |
| r defect introduction rate | defects made per turn | not needed | CANCELS. It scales both densities equally and leaves the ratio alone |
The loop ledger records 17 gate failures across 89 turns, and that number is not g. It counts what the gate caught and is silent about what went past it, which is the quantity the model needs. Measuring g requires knowing the denominator, and the only way to know it is to put the defects there yourself.
Seed k defects into the setup artefact. Run the chain. Score removals by the gap between the seeding turn and the turn that removed them; pooling over gaps gives a method-of-moments estimate needing no likelihood, where the denominator is exposure discounted by λ.
The seeds must not be announced. A turn told the artefact holds twelve planted faults hunts for twelve planted faults, and what comes back is a hunting rate. This is the seeded-defect twin of the failure that sank H8's first design, so it is a stated precondition of H9 rather than a caveat at the end.
Following R13, whose practice is Burton's and Morris's , the design was simulated before being proposed, and the simulation changed its outcome rather than its price. The exchange rate is a step function of g with a discontinuity at the feasibility boundary, so an estimator good enough for a figure can be useless for the decision.
| Seeds | Runs | Turns | Far: width on g | Far: band right | Near: width on g | Near: band right | Near: rate right |
|---|---|---|---|---|---|---|---|
| 6 | 1 | 6 | 0.648 | 84% | 0.608 | 43.8% | 0.8% |
| 12 | 1 | 6 | 0.498 | 92.7% | 0.431 | 55.1% | 9.7% |
| 12 | 2 | 12 | 0.374 | 97.9% | 0.293 | 66.3% | 14.4% |
| 12 | 3 | 18 | 0.297 | 99.3% | 0.241 | 74.5% | 15.4% |
| 20 | 3 | 18 | 0.232 | 100% | 0.187 | 77.5% | 19.8% |
| 20 | 6 | 36 | 0.165 | 100% | 0.137 | 87.3% | 25.7% |
| 40 | 6 | 36 | 0.111 | 100% | 0.096 | 96.3% | 37.4% |
Read the last two columns together. Forty seeded defects over six chains, 36 turns, call the band right 96.3% of the time near the boundary and 100% far from it. The same study gets the exchange rate itself right 37.4% of the time near the boundary, and no affordable study does better, because a swing of 0.05 in g moves the rate from 40 turns to 9. The instability is a property of the question rather than a defect of the instrument, and the design's answer is to report the side rather than the number.
A study sized on parameter precision alone would have bought a figure nobody could act on. That is the second time on this site that simulating before running changed a design rather than merely costing it, and both times the correction went in the same direction: the quantity worth reporting was coarser than the one first proposed.
H9 therefore costs 72 turns, half of them read by a person: forty seeded defects across six six-turn chains in each arm. The directed arm's density is observed rather than modelled, so h is never estimated and one unknown disappears from the comparison.
The obvious way to make unattended runs stronger is to give the turns more capability: let them compile, let them search, let them run the thing they wrote. In this model all of that lands on g and none of it on λ. A turn that can compile catches compile errors it could already see the code for; it recovers nothing about a decision three handoffs upstream that no longer reaches it.
That gives two levers with different prices and different effects, and Figure 2 says which to pull. Capability raises g, costs engineering, and moves you along the horizontal. Wiring raises λ, costs nothing per WP2, and moves you along the vertical, which is the steeper direction near the boundary. The prediction is testable and falsifiable: tool upgrades should move the measured g and leave the measured λ where it was. If a compiler moves λ, this decomposition is wrong.