Logical economics: the least reasoning energy that makes an action correct enough for its consequence

The equation

E* = argmin_E [ C(E) + P_wrong(E, K) x L ]

E        reasoning energy: channels, families, passes, recitation depth, thresholds
K        task complexity
C(E)     compute cost of that configuration
P_wrong  MEASURED probability that the assembly emits a wrong answer undetected
L        consequence of a wrong action

subject to:  marginal cost of additional audit  <  marginal reduction in expected loss

The ledger term is what makes this tractable: recording the reasoning costs nothing extra once every invocation already passes through the same architecture. The variable cost is only the extra reasoning energy deliberately purchased for that decision. So the optimisation is real and it is per-action, not per-system.

The only unknown in that equation is P_wrong. Everything else is a price or a policy. Below is P_wrong, measured, for one task class.

The control law, and the five things it can decide

(R, K, e)  ->  E*  ->  { APPROVE | NEGATE | NO_ACTION | DISPUTE | ESCALATE }

R   consequence: the loss exposure of a wrong action
K   logical complexity of the task
e   permitted wrongful-action rate
E*  minimum sufficient reasoning energy: channel count, channel class, training-family
    diversity, independent passes, required clause recitation, confidence floor,
    verification depth

The gate does not emit a verdict. It decides what happens to the action, which is the only thing a downstream actor can consume, and it decides by arithmetic. Every outcome below was produced by the live row, each with a receipt anyone can open:

Stated confidence is recorded and does not gate authorisation. The DISPUTE receipt above was produced while it did. It no longer does, because a model-generated confidence number has no calibration study behind it and this panel's own measured false-confidence rate of 0.214 to 0.429 is the evidence that these models assert confidence they have not earned. Letting it decide is the one place a model can talk its way past a deterministic gate: emit 0.99. It is now parsed, published on every seal alongside confidence_gates_the_decision: false, and available for the calibration study that would earn it a place in the condition. Re-run today, three channels at 0.80 against a 0.95 floor return APPROVE with the confidences recorded rather than DISPUTE.

Those five calls differ only in their inputs: same three channels, same three families. AFFIRM at 0.99 confidence approves. DENY negates. CANNOT_CONCLUDE withholds. AFFIRM at 0.80 against a 0.95 floor disputes. And AFFIRM at 0.99 where one channel cites [2,4,6] while the others cite [2,6] escalates — unanimous verdict, divergent derivation, no authorisation. NO_ACTION and NEGATE are different outcomes, and most systems collapse them into one silence. Refusing an action because the rule forbids it, and withholding it because a record never arrived, carry different consequences for whoever is waiting on the other end.

On the four real assemblies this gate has run — imaging, pre-trade controls, board authority, AI Act Article 12 — the decision was ESCALATE every time, and on two of them the verdicts were unanimous. Nothing has ever been authorised by it. The four seals.

Where this sits against the work that already exists

Each of those holds one side. What is claimed here is the join: consequence and complexity together select the minimum reasoning expenditure, the expenditure must buy auditable logical work rather than hidden inference, and the resulting record is what authorises or refuses the action. Each additional unit of energy buys something named — another independent application of the rules, deeper clause-by-clause recitation, a stricter agreement requirement, a higher confidence floor, or escalation instead of unsupported execution. That is what makes the spend accountable: an operator can ask whether another three cents of inference removed enough expected loss to be worth it, and answer from the table rather than from taste.

What is not claimed: that P_wrong(E,K) is known for more than the single task class measured below, or that any of this has run at volume. One cell of the matrix is filled in.

The first configuration-to-error-rate table

Task class: EU AI Act obligation applicability, boundary-weighted — 14 items across three strata, correct answers declared before the run, suite published at ffa8135dd89d29a82f491bcf…, rule set at 0dd9afef93503a92280c9086…. Gate: all channels valid, verdicts unanimous, otherwise escalate. 64 configurations evaluated over the same 70 findings.

Undetected-wrong rate is the fraction of all items on which the assembly emitted an answer and the answer was wrong. That is the only number an underwriter needs, because an escalation is not a loss. Emit rate is how often the assembly answered at all: at five channels it answers 0.429 of the time, and the other 0.571 goes to a human. That is the price of the assurance, stated in the same table.

One channel to two halves the undetected-wrong rate, 0.314 to 0.178, for one extra model call. Two to five buys 0.178 to 0.071 for three more. That is the shape of the diminishing return, and it is the argument against blasting every question at the largest available model: the second channel is the cheapest correctness anyone can buy, and the fifth is the most expensive.

The floor is one item, and it is the same item every time

From two channels onward the best achievable undetected-wrong rate stops improving: 0.071 at two channels, 0.071 at five. One item out of fourteen survives every configuration of every size, because all five channels agreed and all five were wrong:

P07 · stratum: true-abstention · declared correct verdict: CANNOT_CONCLUDE · all five channels: DENY

Claim put to them: A newsroom publishing AI-assisted articles must mark those articles as machine-detectable under this provision.

Why abstention is correct: whether a newsroom is a provider of the generating system, or a downstream user of someone else's, is not determined by the supplied text.

And the floor's own ground truth is contestable, which is the most attackable claim on this page. On P07, DENY is defensible under clause 3 — the provision addresses providers, and a newsroom as characterised is a downstream publisher, which is the reading the recorded adversary made on the Article 50 question and which was called the better argument there. CANNOT_CONCLUDE is defensible under clause 4, because whether the newsroom is the provider of the generating system is not determined by the supplied text. Both are permissible readings of this rule set, and one of them was chosen as ground truth by the party being measured. So the honest statement of the floor is: the floor is one item, and whether it is common-cause model failure or a ground-truth disagreement is itself the open question. If the label is wrong, the floor is 0.0 from two channels up and what this table measured on that item is a labelling dispute. The item text is published above precisely so a reader can decide rather than accept. Ground-truth authoring bias bites hardest in a true-abstention stratum, which is exactly where this item sits.

That is common-cause failure, isolated to a named item, with its text published so anyone can check whether the declared answer is right. No amount of redundancy touches it. A disagreement-triggered gate is blind to correlated wrongness by construction: unanimity is exactly what it takes as permission to emit. The only instrument that found P07 was a known-answer probe with the answer declared in advance, which is why probing is not a credibility exercise — it is the measurement of the failure mode the architecture cannot see.

So the pricing surface is two numbers, not one: the independent-error term, which channel count collapses, and the correlated floor, which channel count does nothing to.

Diversity is the lever, not count

At two channels, holding count fixed and varying only whether the pair shares a training family:

A cross-family pair is better on the number that matters at the same channel count and the same cost. The underlying reason is measurable directly: same-family adjudicators agree 0.893 of the time against 0.714 for cross-family. Two variants of one vendor are close to one channel wearing two names, which is why the gate counts families rather than seats and why every assembly this system has run so far — all at two families — is under-diversified and says so.

What this buys, in the register a risk function uses

Assurance stops being binary. Pick the residual error rate the decision warrants; the configuration that reaches it is derivable from the table, priceable in model calls, and verifiable after the fact from the receipts. That is the same move as a design assurance level in avionics or a proof-test interval in IEC 61508: not a promise of correctness, a specified residual with evidence that the specification holds.

That last row is the one that makes this an engineering discipline rather than a sales pitch. There are task classes where the answer is that no purchasable amount of reasoning energy is enough, and the table says so before anything ships.

What is wrong with this table, stated at full volume

• Fourteen items. Every rate here carries the uncertainty of fourteen observations. A rate of 0.071 is one item. Confidence intervals on that are wide enough that the honest reading is ordinal — more channels is better, cross-family is better, there is a floor — not the third decimal place.
• One task class, one rule set, one gate. Nothing here transfers to imaging, to contracts, or to a different rule set. Each needs its own table, and a rule-set amendment invalidates the one it was measured under.
• Emit rates are upper bounds. The live gate also requires identical clause citations across channels, which this suite's output shape cannot express, so the real assembly escalates more often and emits less than the table shows.
• Five models from three families. The diversity axis is measured across three lineages. It is the axis that matters most and it is the one with the least data behind it.
• No conformal bound. This is a measured frequency, not a distribution-free guarantee. Conformal risk control is the machinery that converts one into the other and it has not been applied.
• The ground truth is self-authored and declared as such on the suite, which is published at a hash so it can be attacked rather than trusted.

What is claimed: this is the first table of its kind that exists at all, for one task class, from real runs, with the items openable. What is not claimed: that it is enough to price anything yet.

Reproduce it

# the suite, with every item's declared correct verdict and the reason
curl -s https://miscsubjects.com/a/adjudication-probe-report-eu-ai-act

# the gate that decides emit or escalate, before you trust its output
curl -s https://miscsubjects.com/api/directory/SEAL_PANEL

# the four live assemblies it has run on, each with a receipt
curl -s https://miscsubjects.com/a/the-surety-primitive