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Liability and Affordability
Status: open / unregistered; realization material, except where noted. The
enforcement layer's three narrow results are registered and carry lean-proved; the
affordability theory built on top of them is how a realization produces the
bounded-liability hypothesis of the conditional Normative Inductor
theorem, not part of the canonical generic theory.
Suppose you have decided what a reasoner ought to do, and you have a mechanism that makes it actually do so. There is still a question left, and it is easy to miss: what does the making cost?
Not a metaphor. In the traderized construction, a normative constraint acquires force by adding a trading position to the market the reasoner's beliefs are priced in — a position that pushes back whenever the constraint is violated. That position can lose money. If it loses enough, the reasoner is no longer a good reasoner: the whole point of the underlying construction is that no cheap trader can systematically beat it, and an enforcement position bleeding without limit is exactly such a trader.
So normative authority is not free, and unbounded authority does not mean unbounded legitimate expenditure.
This is the interface distinction the program guards most carefully.
The enforcement mechanism gives a conformance guarantee: at any price the market maker will accept, the constraint is violated by no more than a declared tolerance. That part is kernel-checked.
It does not give a safety certificate. What it emits alongside the guarantee is a liability obligation — a quantity that some surrounding layer has to discharge. A reader who takes the conformance certificate as a promise that the reasoner survives has misread the interface, and the interface note says so explicitly.
What discharges the obligation is a bound on the enforcement position's cumulative value across the worlds settlement has not yet ruled out. Bound it, and the preservation theorem applies: no efficiently computable trader exploits the modified market.
Preservation is substrate preservation. It says the learner survives being made to obey. It does not say the norm was taken up, and it is not a normative result.
A natural simplification is to charge each date its worst-case loss and require every date to pay for itself. It is wrong, and the program got it wrong once before correcting it.
The account is signed and cumulative. A date that loses can be paid for by a date that gains, and there are norms every one of whose dates is robustly loss-making while the running account stays comfortably inside a fixed band. Worst- case-per-date underwriting is sufficient for safety and is strictly, unboundedly, more conservative than what safety actually requires.
There is a related trap. A reason the reasoner is currently satisfying takes a zero position, and therefore costs nothing at all — however deeply that norm excludes the worlds still in play. Cheapness of enforcement is a fact about the reasoner's current conduct, not about the severity of the norm.
If several reasons are being enforced at once, is it enough that the aggregate account stays healthy?
No, and the counterexample is a single line: two books whose increments are +1
and -1 have an identically zero aggregate account and unbounded values
individually. What makes reason-level safety available is a floor on each row
separately, with the floors summing to something finite; then every group of rows
inherits a uniform ceiling.
There is a second, subtler failure. A policy that respects its per-date spending limit at every single date can still have an account that falls without bound, if the limits themselves do not sum. Local enforcement capacity is not lifetime safety, and conflating them is the sort of mistake that looks fine in every finite check.
Affordability is the schematic question: can the mechanism meet the normative demand? Bounded cumulative liability is one concrete way to certify it — a realization of the concept, not its definition. The conceptual answerability theory makes this typing itself, and the program has adopted it unchanged.
The sharp result is a criterion. Enforcing a reason forever, on a finite lifetime budget, is possible exactly when the cost of a fixed unit of enforcement dips arbitrarily close to zero infinitely often. Not "often enough", not "on average" — just infinitely often, however rarely. The budget doesn't appear in the criterion; any positive budget buys unbounded persistence when the criterion holds, and none buys it when it fails.
Two consequences that are not obvious.
Persistence does not create competition between reasons. Every individually sustainable reason can be sustained simultaneously with all the others. There is no scarcity condition to check. Timeliness does: once deadlines are imposed, the minimum cost is a definite positive number, budgets add, and reasons genuinely compete for a finite resource.
A reasoner deciding in real time loses nothing on the persistence question. A simple rule that waits for cheap moments and spends geometrically shrinking amounts achieves persistence whenever a scheduler with perfect foresight could. It loses everything on the quantitative question: there is no positive competitive ratio for how much enforcement intensity gets accumulated.
Affordability is the feasibility side of Serviceability, and it is one of the three candidate pillars of Legitimacy — though the checkpoint argues it is better read as a side condition than a conjunct, since a cheaper norm is not thereby a more legitimate one.
It also has a downstream consumer on the deference side. The corrigibility theorem's
service residual — the adverse mass of discovered reasons an independent supplier
cannot route before commitment — has two resources: a capacity resource in release
windows, where the exact criterion is the suffix-cut condition (Hall on nested
neighbourhoods) and the least miss is the cut excess, and a budget resource, where
the criterion is this page's sliding-window sum D4 taken over the release window and
budgets add across reasons (M1). A supplier is affordable when both hold
(Corrigibility §5; the non-capture round's
AFFORDABILITY_REFINEMENT.md).
Whether bounded liability is necessary for preservation or merely sufficient. Until that is settled, every "unaffordable" verdict in this line means the known route to safety no longer applies, not no safe policy exists — a materially weaker statement than the prose usually suggests.
Whether any of it survives when the cost of enforcing responds to the enforcement. Every existence result assumes the prices are handed down from outside.
What a scheduler needs to know to operate safely on the signed account. The running slack is provably not enough, because the world attaining the minimum can itself be settled away.
Evidence. The force interface is
TRADERIZED_FORCE_INTERFACE.md;
the liability theory is
2026-08-30-liability-theory;
the affordability characterization is in
SHARP_PERSISTENCE.md.