Original robust task: fixed-nuisance certification and an unresolved sixteen-copy endpoint

Original robust task: fixed-nuisance certification and an unresolved sixteen-copy endpoint

Throughout the generic domain statements, 01; the benchmark uses5. The gain floors and phase bound retain the stated physical-parent constraints. The original robust one-percent intervals remain [16 17], separately for adaptive and randomized nonadaptive continuous local readers. A larger endpoint-policy search did not attain sixteen copies. We do not count its very small endpoint-risk improvement as a substantive threshold advance. The useful supporting result is an exact low-dimensional representation of any frozen finite policy’s entire nuisance-dependent risk, together with a sound, strict-feasibility-complete certificate procedure. It does not prove that sixteen copies are impossible.

A three-variable exact interface for a frozen reader

A finite policy is a directed acyclic graph, with one rational unit direction () and three outcome children at each internal node, a fixed deadline, and sign decisions at leaves. It may be compiled from a larger history tree. A fixed rational root mixture is allowed by taking the corresponding weighted sum of the deterministic-root polynomials below. Classical root sign symmetrization is treated in the same way. No axis-exchange symmetry is assumed for the asymmetric parent. For a fixed source label, let be its relevant gain ( or ), and introduce

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e = η,
u = ηg cos δ,
w = ηg sin δ.

These parameters remain fixed throughout the complete graph. They are a coordinate change for the specified physical parent, not a new dynamics or an adversary choosing fresh values at each node. For a positive X source, the three outcome probabilities at direction () are

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(e + cu −sw)/2,
(e −cu + sw)/2,
1 −e.

For a positive Y source replace by; the negative label reverses the signed terms.

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Theorem 4.1 (Exact frozen-policy nuisance polynomial). For every such B-trial reader, each label risk
is a rational polynomial qh(e, u, w) of total degree at most B. Its coefficients are computed bottom-up
by the same three-child recurrence as the risk itself, with polynomial rather than scalar arithmetic. The
number of possible monomials per label and node is at most
B+3
3

. For gain floor ah > 0, the exact
physical parameter domain is

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Kh = {(e, u, w) : e ∈[η0, 1], u ≥0, |w| ≤tu, a2
he2 ≤u2 + w2 ≤e2},
t = tan ∆< 1.

Consequently uniform certification of this frozen reader is equivalent to on each. The unused other-axis gain need not be introduced into a label’s risk calculation.

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Proof. At a leaf the error polynomial is zero or one. Each kernel is affine in (e, u, w). Multiplying the
three child risks by their kernels and adding raises the degree by at most one, proving the assertion by
induction. For the domain, a specified physical point plainly satisfies the stated inequalities. Conversely
the positive lower radial bound and |w| ≤tu force u > 0. Define g =
√

and arctan Then 1] and∆, and the original coordinates are recovered. Every point of is therefore admitted. Maximizing separately over labels is valid because the original robust objective is a supremum over label and one nuisance choice fixed for that experiment. This does not authorize changing nuisance values within a single trajectory.

This reduces certification from a superficial five-parameter search to four three-variable polynomial problems. It does not reduce the unknown policy’s optimization dimension, assume that the gain floors alone are worst cases, or certify arbitrary command-dependent errors outside the specified parent. Coefficient growth, graph compilation, precision and the cost of the resulting polynomial optimization still have to be charged. The bound is a monomial count, not a practical polynomial-time robust-design claim.

Exact Bernstein cover certificates

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For a rational box Q = Q
i[li, ri], map its coordinates affinely to the unit cube. If eq(t) = P
β aβtβ has
coordinate degrees di, its tensor Bernstein coefficients are

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ki
βi


bk =
X
β≤k
aβ
Y
di
βi

.
i

The Bernstein basis functions are nonnegative and sum to one, hence minmax on All entries are rational for the compiled policy and rational box. This standard polynomial-enclosure method is used here as an exact frozen-reader certificate, not claimed as a new approximation technique. Start with]. Discard boxes whose interval radial or angular inequalities exclude; otherwise accept a box only when every Bernstein coefficient is strictly below. Bisect unresolved boxes, cycling through their nontrivial coordinates. A finite accepted cover proves the uniform risk guarantee; a cap or interrupted search reports inconclusive.

Proposition 4.2 (Soundness and strict-feasibility completeness). Every finite cover accepted by this procedure is a valid original-parent risk certificate. For a fixed polynomial with max, the uncapped subdivision procedure terminates. No termination is asserted when the maximum equals the target or exceeds it.

Proof. Pruning only removes boxes disjoint from the physical set by necessary inequalities. Accepted boxes have a polynomial upper bound below the target, so their union proves the claim. For completeness, the compact set has a positive strict risk margin. Continuity supplies a neighborhood on which the

margin persists. Bernstein coefficient ranges converge to the point value as a box shrinks. Every point outside violates at least one defining closed inequality strictly; the elementary interval exclusions eventually recognize a sufficiently small box around it. Compactness of the initial box then yields a finite subdivision depth at which every surviving box is either excluded or risk-certified. Cycling nontrivial coordinates ensures shrinking box diameters. The argument is a completeness statement for strict certification, not a complexity bound or unrestricted quantum-reader theorem.

polynomial_certificate.py implements exact polynomial compilation, Bernstein bounds, physical-domain exclusions and a capped cover search. Independent ordered-word enumeration verifies 108 polynomial identities on a three-trial genuinely adaptive graph; 500 exact grid evaluations check the Bernstein enclosures. The proof, rather than those finite tests, covers arbitrary degrees. Constant safe and unsafe examples exercise the stopping and inconclusive statuses. This package does not claim that the cover code certified a new sixteen-copy operating region.

Why gain-endpoint checks alone are invalid

Even a legal fixed reader can have an interior maximum in one source label’s gain. Set use a direction with5 for sixteen trials, and decide minus exactly when fourteen plus outcomes occur. For X+ the error is

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16
14
!
p(g)14[1 −p(g)]2,
p(g) = (1 + 4g/5)/2.
q(g) =

On 1] it is maximized at16, where, larger than the endpoint values and. The maximizer follows from 14) = 0, and all three comparisons are exact rational inequalities in the verifier. This is a counterexample to unjustified per-label gain-endpoint certification, not a low-error reader or a counterexample to the correctly scoped frozen-majority theorem. Other labels of this deliberately poor reader can have still larger errors.

Searches and exact updated status

The specified transpose-invariant operator relaxation, evaluated at sixteen copies under the existing fixed hard prior, gives only approximately00838194185. That value is exploratory and too weak to exclude sixteen; we did not repeat the unchanged solve or call its optimizer a physical reader. A new point-based policy proposal used 24 rational directions, including rational approximations to the optimized frozen direction and nearby commands, with 150 sampled hard-endpoint trajectories and a design-belief bank. It retained approximately 12,214 candidate nodes at level sixteen. A separate linear programme mixed two roots and used a fresh sign-symmetry coin. The resulting pruned graph has 1,467 nodes and floating endpoint errors approximately01023427595453 on each hard axis. This is still above one percent. Its tiny change from0102368487113 does not settle an adjacent threshold and is not advertised as a robust advance. The full original nuisance continuum was not certified for this unsuccessful candidate. Four separately optimized two-setting nonadaptive schedules used splits (1 the exact endpoint offset arctan01), and an optimized terminal Bayes-decoder prior. Their best observed endpoint worst errors remained near the frozen benchmark. Such searches cannot exclude all schedules, all feedback policies, or the original continuous uncertainty set. They remain reproducible exploratory evidence. Both full continuous robust thresholds therefore remain in [16 17]. Seventeen is still the exact minimum only in the established random-frozen common-positive-angle class. Further repetition of these same policy families is not supported by the current search outcomes; a new structural lower bound or substantially different control construction would be needed.