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Serviceability
Status: open / unregistered; realization material. The results below are paper-derived with exact rational fixtures, from a research round that is merged but registers no claim. They are not part of the canonical generic theory on Legitimacy; they are how a realization services what that theory exports.
The program treats this layer as closed for research sequencing — settled enough to build on unless a contradiction appears. That is a statement about where effort should go, not an evidence class: almost none of it is formally verified, and the repository's status ledger is what governs any claim about how strongly a particular result is established.
A reason that has been raised and not answered is owed something. Serviceability asks the boring, indispensable question underneath that: can the owing actually be discharged? Not whether the reason is correct, and not whether the answer is good — whether a bounded reasoner with finite room to manoeuvre can get to it at all, and get to it in time.
Two measures, and almost everything here is about keeping them apart.
The obligation measure records what is owed and when it was incurred. If a reason
r accumulates exposure c_t at each date, the obligation measure weights dates by
that exposure. This is Answerability's object; it is fixed by history and no
scheduler touches it.
The service measure records what was actually delivered. Service is the broad
Answerability-facing relation — an obligation receives the answering work it is owed
— and in the traderized realization the learner supplies it by allocating some
quantity of enforcement intensity a_t at each date. The service measure weights
dates by that allocation. This is the scheduler's object. Service is not identical
to enforcement intensity: intensity is how a market-embedded reasoner happens to pay
for service, and the force actually exerted is decided later, at the market's fixed
point.
A learner that drives its service-weighted error to zero has done something, but possibly not the thing that was asked. It may have been servicing the dates where nothing was owed. The gap between the two measures is where a great deal of bad behaviour can hide, and one of the round's more useful findings is that a natural fairness condition — service is spread evenly across the surfaces that need it — does not close it. A two-surface rotation can satisfy that condition, have service-weighted error exactly zero at every horizon, and have obligation-weighted error tending to one half.
The naive repair is to insist that service happen at the date the obligation was incurred. That is too strong, and obviously so: answering a challenge next week is not a failure of answerability. What is needed is Service Transport: a transport plan — a bookkeeping of which obligation mass, owed when, was discharged when — together with three conditions: the plan is feasible against the enforcement intensity actually available; it does not spend wildly more service than there was obligation, which is the service-to-obligation cap; and the reason does not change too much between being owed and being answered, which is semantic stability.
That last condition is the interesting one, and it is the one the mathematics cannot yet supply for itself. See What is still missing.
There is an exact characterization of when transport with a fixed deadline is possible at all — for every window, the obligation mass incurred in it must fit inside the enforcement intensity available over that window extended by the deadline — and serving the oldest outstanding obligation first turns out to be optimal.
Neither is new. The feasibility criterion is a specialization of a classical network-flow theorem, and the repository's proof cites that theorem rather than proving it; the serve-oldest-first argument is the classical exchange argument, independently written out here. The prior-art ledger records which is which, because the distinction between using a result and resembling one is exactly what such a ledger is for.
The round's most quotable result is a separation.
- Persistence — the reasoner keeps allocating enforcement intensity to this reason forever.
- Eventual full service — every obligation is eventually discharged, with no deadline.
- Uniform bounded delay — every obligation is discharged within some fixed time.
The first two turn out to be the same question. If a reasoner can afford to keep allocating, it can afford to eventually answer everything — by a diagonal argument that gives each obligation its own cheap moment rather than batching them together. The third is strictly stronger, and no amount of "cheap moments happen often enough" substitutes for the exact criterion.
Unlimited deferral makes "eventually answer every persistent obligation" no harder than maintaining divergent effort. The substantive constraint enters only when delay itself matters.
That is worth sitting with. It means that if a practice's answerability demands are satisfiable at all by a reasoner willing to keep working, then the whole content of timeliness is in the deadlines — and a theory of answerability that does not say when an answer is late has not yet said anything a scheduler can fail.
Once delay matters, budget and timeliness trade against each other, and the trade has two named pieces.
The critical delay H*(B) is the shortest deadline a lifetime budget B can
actually meet. It is a scalar, and it may be infinite even for a reason that is
perfectly affordable under unlimited deferral — being able to answer eventually says
nothing about being able to answer by a date.
The monotone relation B |-> H*(B) is the liability–timeliness tradeoff: paying
more buys a shorter delay, a shorter delay leaves the reason less time to change, and
semantic stability converts that into a smaller residual. An earlier draft treated
this relation as a primitive object in its own right; it is not one. The primitive is
the critical delay, and the tradeoff is what the scalar does as the budget moves.
Putting the pieces together — this is Sharp Timely Service — if the obligations can be transported onto enforcement dates that are both timely and affordable in a specific sharp sense, then the same liability budget that keeps the learner from being exploited also drives the settlement-friction residual to zero, leaving only the obligation-normalized semantic error: how much the reason changed while it waited to be answered.
sharp there names the sharp robust liability charge, not tightness or
exactness in any generic sense. It is precisely because that charge is computed from
the same deficit the residual averages that one budget controls both.
This is the strongest statement the fixed-era theory makes. It is also conditional in a way worth being precise about: it assumes a bound on that semantic change, and nothing in the program yet supplies one.
Two kinds of impossibility, and only one of them is certifiable.
If the obligations already on the books cannot be answered by their deadlines, that is a finite fact about a bounded stretch of dates, and there is an exact deadline insolvency certificate for it: the minimum cost of serving those obligations on time, against the reasoner's remaining budget. Every field of it is settled record. It is the Answerability-facing specialization of the broader finite-horizon infeasibility certificate — which may witness a capacity failure, a liability failure, or both. The relation is a typing, not a theorem the repository states, and the two names are worth keeping distinct.
If a reason can never be persistently afforded, that is an assertion about the infinite future, and no finite observation establishes it. Watching the cost stay high for a long time is not a certificate. The honest response to that situation is deferral and inquiry, not a record that the reason is unanswerable.
The asymmetry has a practical edge: a scheduler that can only bound its costs from above — which is the safe direction for acting — is in the useless direction for certifying impossibility.
There are two entirely different reasons a moment can be cheap to enforce at. The norm may be nearly satisfied already, or nobody may be pushing back against it.
The first is the good case: cheap because there is little left to correct. The second is not, and a reasoner that persists only on such moments has bought the appearance of sustained normative pressure without buying any conformance at all.
Cheap enforcement is not always conforming enforcement.
The endpoint theorem ends in a single term — the obligation-normalized semantic error, how much the reason changed while waiting — and nothing certifies it. Within a single era it is a technical gap. Across a self-revision it is the entire problem, because "how much of a reason survives a revision" is that constant. This is the program's highest-value open problem and it is where Diachronic Answerability picks up.
Also missing: whether a schedule can be timely and sharply affordable at the same time; whether any of this survives when the cost of enforcing responds to the enforcement; and whether the settlement-friction residual is ever actually zero for a norm a real practice produces.
Evidence. The round is
2026-08-31-normative-affordability,
with the endpoint theorem in
SHARP_TIMELY_SERVICE.md
and a list of everything the round withdrew in
CONSISTENCY_AUDIT.md.
The dry canonical statement is the
September checkpoint.