What richer records can and cannot certify

What richer records can and cannot certify

A full trajectory is a richer observation than its terminal error, but the number of recorded fields alone does not imply a cheaper certificate. This section supplies the missing interface-dependent lower-bound comparison. It also gives an exact trusted-seed repair. These are specializations of classical change-of-measure and safe-testing methods, not new general statistical principles; the programme contribution is a checked account of which source–readout promises permit a resource reduction.

An unrestricted full-record no-free-lunch result

Let be the entire classical record of one completed frozen-copy experiment, including any available commands, seed and outcomes, and let indicate a wrong terminal sign. At label its good law is, with). Suppose the admitted model of record laws is closed under changing the error mass while keeping both conditional record laws ) fixed. This is an explicit unrestricted observation model, not a claim about the Pauli apparatus.

Theorem 4.1 (Error-mass tilt preserves the terminal information floor). For define

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{︄
q/ph,
f(z) = 1,
(1 −q)/(1 −ph),
f(z) = 0.
dQh,q
dPh
(z) =

Then is a probability law with error and

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D(Ph∥Qh,q) = kl(ph∥q).

Consequently every sequential validator with access to the full, global false-acceptance probability at most whenever some label error is above, and good-world acceptance probability at least obeys

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EP Nh ≥kl(1 −β∥α)
kl(ph∥ε)
.

Here completed experiments are fresh and independent conditional on their selected labels; label allocation and stopping may be adaptive. The bound allows an infinite stopping time in a bad world.

Proof. The two density values normalize because the two event masses under are and 1 Integrating their logarithms gives the binary relative entropy exactly. Change only label to this alternative, with. At a finite global query cap the transcript KL chain rule gives kl allocation coins have the same conditional law in both worlds. Data processing through acceptance by that cap, followed by monotone convergence as the cap increases and then, proves the result. This is the same finite-prefix argument used in the earlier terminal-only analysis and the established sequential change-of-measure literature.

Thus merely reading an unconstrained archive does not defeat the previous information limit. A useful improvement must exploit a restriction on the alternative record laws, not their representation length. Conversely, closure under this tilt must not be silently assumed for a physically specified detector. The tilt can violate a trusted seed marginal, a known erasure law or a declared response family. Such restrictions are genuine additional information and can yield a different frontier.

A trusted-seed information projection and attaining gate

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Assume each experiment begins with an independently trusted seed S ∈{1, . . . , m}, with known fixed
weights ws > 0, ∑︁
s ws = 1. The validator observes S and Y after the complete experiment. It neither
selects a seed retrospectively nor discards an inconvenient seed. Deployment uses the same seed law.
The seed weights hold conditional on the completed validation past and selected label; completed
experiments use the declared fresh stock. Thus the following likelihood and drift statements are
conditional on the common proof filtration, without giving the reader access to the hidden preparation
label. For one label, nominate a good conditional-error vector ps ∈(0, 1) with  ̄p = ∑︁
s wsps < ε < 1.
The bad class consists of all conditional risks qs whose average exceeds ε, with no hidden response-model
constraint.

Theorem 4.2 (Seed-informed lower boundary and safe likelihood gate). There is a unique

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minimizing
Iw(p, ε) =
min
∑︁
∑︂
s
ws kl(ps∥qs).
s wsqs≥ε

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It satisfies ∑︁
s wsq∗
s = ε and, for one λ > 0,

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(1 −λ)2 + 4λps
q∗
s −ps = λq∗
s(1 −q∗
s),
q∗
s = λ −1 +
√︁
2λ
.

For the weights trusted above, define

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fs(1) = ps/q∗
s,
fs(0) = (1 −ps)/(1 −q∗
s).

The product of these observed multipliers is a nonnegative supermartingale under every bad conditional- risk vector. Stop and certify the label when it first reaches. Under the exact good vector, or under a coordinatewise smaller conditional-risk vector, it crosses almost surely with expected experiment count bounded by

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ENh ≤log(1/α) + bw
Iw(p, ε)
,
bw = max
s
log fs(0).

Every seed-informed validator with false-certificate probability at most and good acceptance at least instead satisfies

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EpNh ≥kl(1 −β∥α)
Iw(p, ε)
.

For multiple labels, an intersection–union gate may certify only when every label passes; no confidence split over labels is required.

Proof. The objective is strictly convex in for interior. Its unconstrained minimum is, which is infeasible, and moving any upward toward improves the objective and the constraint. The optimum is therefore at the average-risk boundary with. Differentiation gives the common positive multiplier equation. The displayed positive quadratic root increases continuously from as increases, so there is one multiplier giving the nominated boundary.

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The same equation implies

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fs(1) −fs(0) = −λ,
∑︂
s
wsfs(0) = 1 + λε.

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Hence at any alternative vector q,

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(︄
)︄
EqfS(Y ) = 1 + λ
ε −
∑︂
s
wsqs
≤1.

This proves composite-null validity, including predictable label allocation. Ville’s inequality gives the crossing-size bound. At, expected log increment is, and reducing any increases that drift since (0). Before crossing the log product is below log); an upward step adds at most Applying the finite stopped increment sum yields, and monotone convergence proves finite expectation and almost-sure crossing. For the lower bound, preserve the seed weights, choose errors just above, and preserve all other record shapes conditional on ( ). Per-experiment divergence decreases to as that alternative approaches the boundary. The finite-prefix acceptance change-of-measure proof above then applies. An all-label certificate with a bad label requires that particular bad label’s crossing; this proves the intersection–union claim.

This construction specializes established reverse-information-projection safe testing[2] and sequential change-of-measure bounds[4] to the declared reader-seed interface. The contribution is the explicit specialization and verification, rather than a new general information projection or likelihood-testing principle. Log-sum/data processing gives

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For interior, heterogeneous ps the inequality is strict. Equality would require both ps/q∗
s and (1 −
ps)/(1−q∗
s) to be constant across seeds. Two distinct ps then force both constants to one, contradicting
 ̄p < ε. Known seed marginal information can therefore improve the terminal-only information floor. It
does not ensure a large improvement for a particular reader.

Actual fourteen-copy check: seed heterogeneity is too small

The verified unsymmetrized fourteen-copy base risks, in label order, are approxi-mately

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(.008642797684269258, .009021802859698374, .008473934585934470, .008858815447399504).

For each fixed physical label the four trusted root symmetries permute this vector. Thus every label has the same mean. The code constructs a nearby exact rational good-design vector dominating the independently verified outward bounds, and a rational boundary vector with mean exactly01. It chooses one rational so the quadratic relation is exact. This avoids presenting numerical root finding as a proof. The resulting certified information ratio is

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Iw
kl( ̄p∥.01) = 1.00000546458 . . . .

It implies only about 0000546% improvement in the matched information floor. The upper expected source-stock account is below 9413292 million for four labels and fourteen copies per complete experiment. This is not the earlier00875 rounded planning case: comparisons must use the same

good-risk promise. Root-seed stratification alone is therefore not a meaningful practical advance for this frozen reader. An exact two-seed illustration uses weights (1019) and has mean good error00875 and an information ratio 1. This checks the strict-gain construction but is not claimed as a new quantum-reader protocol. Reading an even richer archive can help under a justified detector-response model, as in the separate structured protocol. It is that restricted model, not a universal claim that more fields always lower certification cost, that must support any substantive savings. All tests here certify terminal risk only; neither the unread bath nor the complete retained quantum instrument is identified. Expected stopping costs have no finite worst-case completion bound. A selected policy must be frozen before fresh validation, or selection multiplicity must be controlled separately.