Continuous equatorial readers: exact equality through three copies

Continuous equatorial readers: exact equality through three copies

We establish equality of adaptive and randomized nonadaptive continuous-angle readers through three copies for every shared efficiency and visibility, then construct genuinely angle-adaptive four-copy readers. The latter are attainable upper bounds; universal nonadaptive comparison requires the separate lower certificates. No finite angular scan is used as a surrogate for the continuous class.

The complete local class and exact three-copy theorem

Use the four fixed source labels, their common sign target and the shared calibrated effects

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E±|θ = η
2(I ± vσθ),
E0|θ = (1 −η)I,
0 ≤η, v ≤1.

All actual angles, signs and nulls remain retained. The label controller is inaccessible; copies are independent conditional on that one fixed label. Adaptive settings are nonanticipating functions of accessible records and private coins. Nonadaptive schedules may be jointly randomized in advance, with arbitrary terminal decoders and their seed retained. A fixed deadline requires a sign decision, with neither postselection nor abstention. The full equatorial menu and common detector law are closed under axis exchange and global source-sign reversal. Uniform-prior Bayes risk is a minimax lower bound; symmetrization promotes an attaining Bayes reader to minimax. The repeated bisector already has equal error on all four labels. Let) be fair-tie majority error for independent signs of positive bias. In particular

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4 + β3
q0 = 1
2,
q1 = q2 = 1
2 −β
2 ,
q3 = 1
2 −3β
4 .

Theorem 3.1 (Continuous equality through three copies). For the complete local equatorial class, all

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3
∑︂
(︄
3
n
)︄
ηn(1 −η)3−nqn(v/
√
ead
3 = ena
3 =
2).
n=0

Repeated measurement at, followed by majority of the nonnull signs and a fair tie, attains it. Equivalently,

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2 −v(3η −3η2 + 3
2η3)
2
√
2
+ η3v3
e3 = 1
8
√
2 .

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2 ≃0.135879605669. The theorem adds no assertion
about four or more copies.
At η = 4/5, v = 9/10 this is 1/2 −(4023/15625)
√

A continuous two-control inequality

The following lemma supplies the load-bearing continuous optimization. It covers negative as well as positive direction components.

Lemma 3.2. Let and be unit vectors,, and put

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A = c + s, B = p + q, C = cp + sq, Q = c2p2 + s2q2.

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Then
∑︂
2 + 2ruC + u2Q} ≤
√
r=±1
max{|A + rB|,
√︂
2(3 −u/2).

Equality is attained when both directions are the positive bisector.

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Proof. There are four branches of the two maxima. If both choose absolute linear terms, their sum is
2 max(|A|, |B|) ≤2
√

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2. If both choose square roots, Cauchy–Schwarz and Q ≤1 give 2
√

. This is at most the claimed bound because 2(3 For a mixed branch, changing () to its negative if necessary reduces the required estimate to

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2 −2uC + u2Q ≤
√
M = |A + B| +
√︂
2(3 −u/2).

Write (cos sincos sin) and set cos cos((. Trigonometric identities give

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M
√
2 = G(x, y) = 2xy + R,
R2 = H2 + u2x2(1 −x2),
H = 1 + u/2 −uy2.

Here, then0, since If, then For2 this is immediate. Otherwise 1). Thus increasing the smaller coordinate to the larger cannot decrease On the diagonal, with

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G = 2X +
√︂
(1 + u/2)2 −2uX.

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(1 + u/2)2 −2uX ≥0 because the square root is at least 1 −u/2 and u ≤1.
Hence its maximum is G(1, 1) = 3 −u/2. This proves both mixed branches and the lemma.
Its derivative is 2 −u/
√︁

Proof of Theorem 3.1. First set = 1. It suffices to lower-bound the uniform Bayes risk of every deterministic adaptive policy: independent random seeds only average such risks. Let the first direction ). Its two equiprobable outcomes have posteriors related by global source-sign reversal, so their optimal continuation values agree. Let () be the second direction on the first-plus branch. After its outcome, the unnormalized posterior weights have target mass difference8 and axis mass vector ((1 +

The preceding exact one-remaining-copy formula, obtained from max optimizes the third direction over the whole circle. Adding the two second outcomes and the equal first-sign branches gives three-copy Bayes loss

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1
2 −v
r=±1
max{|A + rB|,
√︂
∑︂
2 + 2rv2C + v4Q}.
8

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Lemma 3.2, with u = v2, bounds this below 1/2 −3v/(4
√
2) + v3/(8
√
2) = q3(v/
√

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2). The repeated
bisector attains this value, including arbitrary terminal decision rules in the comparison.
For 0 ≤η ≤1, condition on the entire erasure pattern of the three trials. The pattern is independent
of the source label and signs. A policy conditioned on a nominated pattern with n clicks induces an
admitted n-copy adaptive ideal-efficiency reader; giving the controller that pattern in advance can only
make the lower bound more permissive. The preceding n = 1, 2 optima and the just-proved n = 3
optimum therefore lower-bound each conditional risk by qn(v/
√

2). Averaging over the binomial click count yields the claimed lower bound. The fixed bisector attains each conditional value simultaneously. Its four errors are equal, so the Bayes lower bound is an attained minimax value for both classes.

The conditional-erasure argument uses independent sign-blind erasures. It does not assert that a fresh reset removes arbitrary correlations, or that an unknown future pattern is physically accessible to the deployed reader. Advance pattern disclosure is used only as a lower-bound relaxation.

Constructive four-copy angle adaptation

At four copies we provide explicit frozen policy tables, not numerical claims of globally optimal adaptive controls. Both the ideal and nominal policies use forty declared measurement nodes and eighty-one terminal ternary records, including impossible ideal-null branches with harmless defaults. Each direction is an exactly unit rational pair

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n(t) = ((1 −t2)/(1 + t2), 2t/(1 + t2)),
t ∈Q.

Numerical search selected candidate angles; rational conversion and exact full-word verification establish the delivered performance. Search failure or floating convergence is not an optimality certificate. The base policy first uses a fixed angle, and on a first nonnull outcome uses. Its third setting depends on whether the second sign agrees with the first, disagrees, or is null. A first-null branch uses the bisector candidate. At the last node a rational approximation to the posterior axis-mass direction is used. Where the nominated posterior gives no possible Bayes-risk improvement from the last measurement, its setting is instead held at the previous setting. The published leaf decisions remain specified on every record, including zero-probability ideal-null defaults. This is a flat nominal-risk choice, not a claim that every leaf decision is identical under a changed noise model. The final trial still occurs and is fully archived. This saves potential actuator motion; it does not silently convert to an optional-stopping or reduced-copy experiment. Approximate design angles, in radians, are:

Parent same sign opposite sign null Ideal .17468079 1.39611555 .01091177 .78539702 .79955276 Nominal .75070933 1.09092805 1.28465859 .49454527 .79086328

The JSON files, not these rounded displays, define the actual rational commands, all branches and terminal decisions. Root symmetrization uses two fair bits: exchange X and Y in every setting, and/or reverse every recorded sign and the final decision. This group acts transitively on the four labels and respects the complete admitted continuous class and common noise law. Averaging the base policy’s

four exact errors therefore gives the same error on each label. The seed is available to the decoder and counted as controller stock; it does not reveal the hidden source label. For the ideal parent ( 1) the resulting minimax attainable upper error is strictly less than 0030605 (actual exact-rational value approximately 00306041631615). For the nominal parent 10) it is strictly less than 0103471 (approximately 0103470608280). These upper bounds are proved by summing each of the 3 retained-record probabilities for each source label. They do not by themselves exclude better nonadaptive readers or establish the adaptive optimum. The separate continuous nonadaptive certificate must be used for any cross-class separation.

Verification and limits

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The independent standard-library checker verifies 630 rational instances of the mixed geometry inequality
using exact signs in Q(
√

2), then checks all 81 words for each policy, normalization on each source, all root symmetry permutations and the stated rational upper bounds. The analytical inequality proves the entire continuous three-copy theorem; finite tests alone do not do so. Frozen rational policy tables provide independently checkable attainability without rerunning the exploratory optimizer. Optional exploratory scripts use SciPy; core certification uses only the Python standard library. The equality theorem and policy constructions extend the specified source–readout decision problem. The fixed inaccessible source label, conditional-copy independence, common calibrated detector law, controller coins, phase reference, clock and complete retained records remain premises. No complete residual-instrument equivalence, Born-selection law, consciousness criterion, trine-resource theorem or universal physical cost advantage is claimed by these decision-risk results.