One calibration event and a finite adaptive record

One calibration event and a finite adaptive record

The calibration certificate can be used over a finite sequence without charging its failure probability at every use. This is an application of established martingale concentration, not a new probability law or an independence theorem. It makes the batch-calibration and robust-reader constructions compatible at the level of the complete declared classical record. Let contain all original calibration settings, counts, nulls, chosen reader settings and admitted calibration metadata. For the fixed true calibration parameters, let be the simultaneous confidence event with Pr. The drift, access and preparation conditions of the calibration section are physical assumptions: on, every future actual input conditional on its preceding declared history must remain in the certified set. Neither an unchanged wave, a reset flag nor a null outcome proves this condition for a reused source. For use, let indicate a conclusive flag and a wrong conclusive flag; hence 0 The source-label truth needed to score must be defined in the test model and recorded in a validation experiment; an unknown production label does not become observable from this theorem. For take to contain the completed-trial outcomes, scoring truths and original calibration record. The physical controller is adapted only to its permitted accessible-record subfiltration. The conditional contracts below are assumed on this augmented mathematical past; they do not follow merely by enlarging an observed-record filtration. Scoring access in the proof grants the reader no access to an unread preparation label. Let 1] and 1] be predictable, calibrated bounds for the selected reader. On suppose

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E[Wt | Ft−1] ≤εtE[Ct | Ft−1],
E[Ct | Ft−1] ≥qt.
(49)

The constructed symmetric robust reader supplies these inequalities label by label under the calibrated conditional-state contract. Consequently its certificate permits arbitrary, history-dependent label

priors when both label-conditioned source marginals satisfy that contract. This does not transfer an optimality claim from a different prior-weighted task. Arbitrary inert quantum references may be retained. Their additional measurement or feedback is outside the contract unless the augmented conditional input bounds have also been established.

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Theorem 7.1 (Finite-use certificate with a single calibration fee). For any predetermined horizon N
and 0 < β < 1, put bN =
p

2) log(2. With probability at least, simultaneously

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N
X
N
X
t=1
Wt ≤
t=1
εtCt + bN,
(50)
N
X
N
X
t=1
Ct ≥
t=1
qt −bN.
(51)

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No independence of the future trials is required. With fixed εt = ε and qt = q, if Nq > bN this implies
P
t Wt
P
t Ct
≤ε +
bN
Nq −bN
.

Every null and loss event remains in the record and in the horizon cost.

Proof. On each calibration outcome in, set. It has nonpositive conditional mean and its conditional range is contained in [], of length one. The conditional Hoeffding lemma therefore gives for 0. Induction and exponential Markov give

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X
!
≤e−2b2/N.
Pr
t
Zt ≥b | F0

Apply the same argument to, whose conditional mean is nonpositive and whose range also has length one. Taking bounds each failure by2. The union bound supplies on; the original calibration failure adds once. This proof permits adaptation to all admitted preceding classical outcomes, not retroactive selection of a better calibration confidence event.

The range-dependent lemma can be checked directly: among bounded variables with fixed mean, convexity maximizes the moment generating function on the two endpoints. The centered endpoint Bernoulli log-mgf has second derivative at most 14; integrating twice from zero bounds it by This proves the concentration constant used above. An anytime version is also available. For any fixed 0, with failure at most

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n
X
t=1
(Wt −εtCt) < θn
8 + log(1/β)
θ
for every n ≥1.

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Indeed exp[θ P
t≤n Zt −nθ2/8] is a nonnegative supermartingale starting at one; stopping it at its first
crossing and using the expectation bound proves the displayed probability inequality. This supports an
admitted stopping policy without falsely applying a fixed-horizon bound to a data-selected horizon.