Task-relative validation with a near-optimal expected stock bound

Task-relative validation with a near-optimal expected stock bound

Response-level calibration can certify more than one frozen reader needs. We validate its four terminal-error probabilities directly. Exact binomial gates give a finite-budget interface; likelihood-ratio gates give an anytime interface with a sharp expected-stock comparison. These are established testing methods, newly applied and fully proved for the declared source–reader contract, not a new quantum probability law.

The validation interface and indispensable stability promise

Fix a complete reader policy Π with horizon, root randomness, null handling and final decoder before opening validation data. A trusted calibration controller requests one of the four source labels , supplies fresh copies independent conditional on that fixed label, and records the terminal error Π decides the wrong sign. The label is available to the external auditor for scoring, but is not an input to the reader policy. Every ordered setting, sign, null and controller seed remains retained; no record is postselected. The physical stability contract is that the repeated complete experiments, conditional on the fixed device and trusted label, have one error probability and fresh independent stock. Predictably choosing the next label from past calibration data is allowed, provided the conditional error law remains . Deployment must have the same conditional policy-error probabilities. This promise is not inferred from reset, past calibration, or a detector’s average density equation. Drift, changed preparation, unread environment memory, policy changes and reference depletion require separate allowances. The target is max. The gate reads only the terminal error bits and trusted label identifiers for its decisions. Richer archives are appended but not queried. Thus it certifies one frozen policy’s loss, not a continuous response curve, arbitrary future interventions or a complete retained quantum instrument. This restricted information interface is also the class of the matching stock lower bound in Section 3.

Finite exact gates without a four-label Bonferroni penalty

For complete experiments on label, let be its error count and write PrBin

Theorem 2.1 (Intersection–union policy certificate). Choose fixed label budgets and count thresholds with. Accept the frozen policy only if for every label. Uniformly over all fixed risk vectors with, the probability of issuing this false certificate is at most. The per-label test size is, not

Proof. Pick any labelwhose true risk exceeds. The all-pass event is a subset of Binomial lower-tail probability is nonincreasing in its success parameter, by coupling Bernoulli variables with common uniforms. Therefore its probability is at most. No union over the labels is needed: the null is a union of bad-label hypotheses and acceptance is an intersection of their individual rejection events. Independence between label-gate events is unnecessary for this soundness argument.

For a nominated, the largest integer satisfying this inequality is the most permissive deterministic lower-count gate. It maximizes acceptance power within that count-threshold class for every The usual Bernoulli monotone likelihood ratio gives the broader Neyman–Pearson interpretation, with optional randomization at a boundary if that larger class is admitted. We do not claim that the selected finite budget below is the globally minimum finite budget over all sequential designs. Exact binomial inversion is established statistics[1, 5]; the stated inequality is independently implemented here. At, take

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nh = 365000,
ch = 3366
(h = X+, X−, Y +, Y −).

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Integer binomial sums certify
F365000,.01(3366) < 2−20 < F365000,.01(3367).

Thus the chosen count threshold is maximal at this budget. If every true100000, the same exact calculation gives80. A union bound on failure then gives all-label acceptance probability greater than 1920, without using an independence assumption between the four gate events. This consumes

365000 = 1460000 whole experiments 1460000 = 20440000 source copies

The planning risk00887 is a rational upper bound on the nominal fourteen-copy bisector error ; its exact quadratic-field comparison is replayed. This is a synthetic good-device planning case, not an observed validation run. The minimum sample count allowing deterministic acceptance of a zero-error record under the same per-label size is exactly 1380, because (. That is a minimum for accepting that record, not a useful high-power design at. Short calibration that can accept a very rare good-looking record is different from a procedure likely to certify a genuinely good reader.

Anytime validation with certified expected cost

Choose an alternative 0 and define for each label

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)︃Sh(n) (︃1 −p0
)︃n−Sh(n)
.
Eh(n) =
(︃p0
ε
1 −ε

Query labels predictably. Mark a label accepted irreversibly the first time; stop querying it. Issue the policy certificate only when all labels are marked. Testing one unaccepted label at a time is one simple implementable allocation. Nulls within each-copy experiment still enter its terminal error and remain archived.

Theorem 2.2 (Anytime intersection–union gate). The probability of ever issuing a false policy certificate is at most under every fixed risk vector with, including predictable label allocation and stopping. For the attaining allocation, process one unaccepted label at a time until it crosses, then continue through the remaining labels, without early rejection. If every, this serial exhaustive rule accepts all labels almost surely, and for each label’s crossing query count

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Eτh ≤A + b
D
,
A = log(1/α),
b = log 1 −p0
1 −ε ,
D = kl(p0∥ε) > 0.

Consequently that complete procedure has expected source consumption at most. Arbitrary predictable allocation, early rejection or stopping remains permitted for false-certificate control; it does not by itself carry this acceptance/attainment promise. No finite deterministic completion time is asserted.

Proof. For a bad label with, the expected multiplier on its next query is

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qh
p0
ε + (1 −qh)1 −p0
1 −ε = 1 + (qh −ε)(p0 −ε)
ε(1 −ε)
≤1.

Its nonnegative likelihood-ratio process, held constant when another label is queried, is therefore a supermartingale in the global calibration filtration. Stopping it at its threshold and applying the elementary nonnegative-supermartingale maximal inequality gives crossing probability at most. Any false all-label certificate requires a crossing by one fixed bad label, proving the intersection–union claim. The underlying anytime testing machinery is established[3]; it is proved here for this particular gate and filtration. For the expected-cost claim, use the serial exhaustive allocation and put log 0. On each good label query the log multiplier has expectation, since and. The log process stopped at is bounded above by: before crossing it is below, and an upward crossing adds only. Taking expectations of the finite stopped increment sum gives Monotone convergence proves the displayed finite expected bound. A positive probability of infinite would make that expectation infinite, so each label crosses almost surely. Summing four label expectations proves the source-stock bound.

At and = 14, outward rational logarithm enclosures give

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Eτh < 206802.173725,
E[source copies] < 11580921.729.

For the frozen adaptive fourteen-copy policy, its separate risk certificate permits the smaller planning bound00875. With the same target, confidence and horizon,

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Eτh < 168277.131290,
E[source copies] < 9423519.353.

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The matched expected upper budget decreases by more than 18.6%. A finite-command implementation
whose separate phase-error certificate gives risk at most .0088 instead has expected source stock below
10243551.605. That third entry is conditional on its independently checked physical-command allowance,
rather than silently treating real-valued commands as free. The smaller planning risk is not assumed
for an arbitrary uncalibrated device; it defines the good-device class on which the acceptance-cost
promise holds. False-certificate control remains valid outside that class. The second planning case is
used only after the frozen policy’s mathematical risk bound is independently checked.
Section 3 proves the matching terminal-score lower bound ENh ≥A/D for every validator with
false-certificate size α and almost-sure acceptance at the all-p0 good vector. Our per-label upper is
within the factor
1 + b
A < 1.0001

of that lower frontier in both planning cases. This is a restricted expected observation-cost statement, not a universal optimum over physically structured detector models, informative quantum archives, controllers or calibration preparations. A trusted physical relation between different label risks may permit a different frontier.