An explicit validation-resource and timing comparison

An explicit validation-resource and timing comparison

This comparison uses the frozen fourteen-copy adaptive policy and the fourteen-copy repeated bisector. It does not optimize over all nonadaptive readers. Both protocols already attain the nominated one-percent deployment target. The extra error margin can nevertheless reduce the cost of verifying that target from trusted whole-experiment trials.

A checked fixed-reader error enclosure

At the nominal shared detector, a clicked bisector observation has correct-sign probability

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p = 1 + v/
√
2
2
.

Conditional on clicks, the fair-tie majority rule has error

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(︄
n
j
)︄
(︄
n
n/2
)︄
pj(1 −p)n−j + 12|n
fn(p) =
∑︂
pn/2(1 −p)n/2.
2
j<n/2

Thus its actual error at

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14
∑︂
(︄
14
n
)︄
ηn(1 −η)14−nfn(p).
qN =
n=0

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The success event with a fair terminal tie coin is increasing in all Bernoulli observations. Coupling
those observations with common uniform random variables proves that fn is nonincreasing in p. The
verifier brackets
√

2 by adjacent 100-bit dyadic rationals, computes the corresponding endpoint sums exactly, and certifies

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.008866 ≤qN ≤.00887,
qN = .00886634707138013 . . . .

All four labels have this same error for the fixed bisector. This check is independent of the point-based search. The supplied research11/sources/resource_comparison.py and research11/audit/check _resource_comparison.py reproduce the appendix’s stock, finite-phase and timing inequalities. Their declared research11/nominal/results.json input is the already certified frozen policy-risk output, preserved with its exact hash; the reader’s policy and shared-response risk are also independently evaluated by the trajectory checks in this paper.

A strict comparison within the terminal-risk interface

Set01 and false-certificate probability. The adaptive policy’s certified error is at most 00875 for every label after its admitted symmetry randomization. The sequential intersection–union gate gives an expected source-stock upper bound

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UA ≤9,423,519.353

under that good-risk promise. There is no finite worst-case stopping budget. The comparator may choose its trusted labels adaptively, stop adaptively, and use a different terminal validation test. Require it to be false-certificate safe over the same unrestricted Bernoulli error alternatives and to accept the actual nominal bisector with probability at least95. The validation interface exposes only the trusted label and terminal error bit, together with the validator’s own randomness. The experiment’s settings, nulls, controller seed and timing records remain in an append-only archive but are not readable for validation stratification or feedback. The lower bound does not cover a validator admitted to those additional records without a further complete-law change-of-measure argument. For a one-bad-label alternative whose error tends down to, the validation information bound gives, for each label

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ENNh ≥kl(.95, α)
kl(qN, ε) .

Since kl) decreases for, the certified lower bound008866 can replace in the denominator conservatively. Summing the four labels and charging fourteen source copies per terminal experiment yields

The adaptive upper bound is strictly below this fixed-reader validation lower bound, by at least 3988%. If the comparator must instead accept almost surely, its lower bound becomes 11 and the guaranteed fraction is at least 18 This is a comparison of expected validation resources at the specified good devices, under a strong black-box validity requirement. It does not prove a universal gain over structured Pauli calibration, other nonadaptive readers, or every physical implementation. Its two good devices have different errors; that difference is the mechanism of the certification saving. The positivity and stopping arguments, not a comparison of two loose cost upper bounds, establish the strict result.

A finite-precision variant

Suppose actual conditional measurement angles differ from the commanded angles by at most

radians, with the specified detector parameters and independent preparation contract otherwise unchanged. For any label and accessible history, the one-copy outcome total variation changes by at most2. Sequential coupling and the union bound therefore increase any fixed policy’s error by at most

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2
= .0000504.

This bound permits conditional angle errors and hence does not require them to be freshly independent. The certified nominal adaptive error plus this allowance remains below A nearest 19-bit command on the quarter circle has rounding error at most radians. The remaining error budget is a separately required actuator and reference promise; the bit count does not demonstrate that promise in hardware. The quarter-circle encoding also requires its coarse quadrant or equivalent outcome/setting relabelling bits. For the subsequent validation-cost comparison, the declared independent whole-experiment terminal-trial contract remains required; a within-experiment perturbation bound alone does not supply fresh independence across repeated experiments. Using0088 as the sequential gate’s good-risk parameter gives

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Uphase
A
≤10,243,551.605 < LN.

It retains a strict expected terminal-validation stock comparison of at least 47758%, against the ideal nominal fixed-bisector validation lower bound with95 power. A device/model-specific calibration could consume fewer resources than either black-box validation interface, so this is not asserted as a universal physical frontier.

Timing, setup and reuse

Let a full source trial take seconds, including common preparation, detection and reset costs. Charge the adaptive reader up to thirteen additional feedback waits ofseconds per validation experiment. Let be an upper bound on additional per-experiment adaptive phase-switching, controller or reference time; let be an unavoidable lower bound for the comparator. In particular, an upper bound on its switching time cannot be inserted as a compulsory tax. Let be an adaptive setup upper bound and an unavoidable comparator setup lower bound (or their actual fixed setup times). A sufficient condition for a strict expected validation-time improvement is

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DA + UA
14 (14τ + 13l+ sA) < DN + LN
14 (14τ + sN).

With and = 0, the certified bounds imply

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l/τ < .15242554

for the ideal-command adaptive gate, or

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l/τ < .05401187

for the finite-phase variant. These are sufficient model-dependent conditions, not measured latency claims. If feedback waits are fewer than thirteen on some paths, the bound is conservative. After validation, both displayed protocols consume fourteen source copies per deployed decision. Hence this comparison gives a finite validation saving, not a deployment copy saving that grows with the number of uses. If later deployments consume an extra adaptive overhead 0 seconds each, the validation-time margin can absorb at most; any required lifetime or drift guarantee over those uses remains a separate premise. Conversely, a one-time calibration overhead can be amortized only under such a reuse promise. Append-only archives, causal feedback memory, trusted label preparations, independent ready coins, command precision and reference access remain separately charged resources. No energy or hardware-size inference follows.