A predictable response-tube guarantee

A predictable response-tube guarantee

The exact sinusoidal family is a physical promise, not something inferred from the observed command archive. A conservative enlargement permits a specified discrepancy without pretending to certify arbitrary response curves. Let each actual conditional outcome law, given the augmented proof past defined in the introduction and the fixed label, be within total variation of the declared kernel at that command. A single fixed visibility 1] applies across calibration and deployment. Ready symmetry bits remain fresh independent uniform conditional on the completed past and chosen label. Blockwise drift is averaged before their new draw; one-trial validity holds after the seed and command are known. This conditional tube is required even across experiment boundaries; neither reset nor past successful validation proves it. It can allow correlated conditional discrepancies, but not violations of the tube itself.

Theorem 6.1 (A safe robustified full-record likelihood). Write. For a recorded local outcome use. Then

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10/11 ≤f ≤190/189,
κ = 200ζ
2079.

Replace each multiplier by, retain all records, and check for crossing at complete-experiment boundaries. Under every and every allowed conditional tube discrepancy, the resulting product is

a nonnegative supermartingale. Its crossing probability is at most. If the model risk is uniformly bounded by for, every actual conditional deployment risk there is at most

Proof. For a trusted label, put. Nulls have multiplier one. Direct algebra gives

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EKvf = 1 + η(v −vc)(v0 −vc)x2
1 −v2cx2
≤1
(v ≤vc).

The two extreme nonnull ratios occur at1, giving the displayed range and its oscillation 200 For bounded, a TV perturbation of size changes its expectation by at most osc. Normalization proves the conditional supermartingale inequality; Ville’s inequality controls every crossing, including the selected boundary times. Sequential coupling of the actual outcomes with their model outcomes, using the same controller and ready coins, fails with probability at most. On the complementary event the terminal sign agrees. The same coupling applies conditional on the past, proving the deployment claim.

The action-dependent normalizer in Section 5 gives the sharper nominal tube bound below 255044. The present construction uses one global normalizer, independent of the action, and a conservative uniform-good drift bound. It is a different, simpler robustified test; its larger budgets below do not supersede that sharper nominal result. For expected costs let log log189), and log189). Let be the certified nominal mean log growth per complete experiment, and let be the certified path-minimum lower growth for every9 in the exact family, as computed in the structured certificate. For an exact nominal good device, the normalized experiment drift is at least

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d0 = D0 −14 log(1 + κ).

For any good device in the entire conditional tube with9, it is at least

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dtube = Dmin −196ζR −14 log(1 + κ).

Indeed the complete-experiment log likelihood has range at most 14 and the two conditional experiment laws differ by TV at most 14. The normalization is deterministic. If either drift lower bound is positive, truncating the boundary stopping time gives

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E[source copies] ≤14(A + b)/d.

There is no finite worst-case stopping duration. Positive drift also proves almost-sure acceptance in the stated good class. This argument permits a correlated tube because it lower-bounds each experiment’s drift conditional on its entire preceding history. At and, the exact directed-log replay gives

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qc + 14ζ < .009906709 < .01,
C0 < 255194,
Ctube < 627692.

These are expected source-stock upper bounds; trusted efficiency, preparation, controller, reference and timing contracts remain additional. The conditional TV tube is separately supplied, not established by the likelihood test. The same physical family with zero discrepancy is included in this tube, so the explicit bad visibility used for the terminal-only lower bound remains an admitted alternative. The structured same-family comparison therefore continues to demonstrate a record-access benefit at the exact nominal good device, even when false-certificate validity is required over this enlarged model. This does not extend the claim to arbitrary command-dependent errors.