Nominal threshold: a checked obstruction and a continuous lower-backup method

Nominal threshold: a checked obstruction and a continuous lower-backup method

The nominal one-percent copy threshold remains [13 14] for each admitted reader class. This route does not establish thirteen-copy feasibility or impossibility. It supplies an exact thirteen-copy risk lower certificate and a convergent continuous-action lower-backup construction, while identifying why the tested cheaper relaxations do not settle the endpoint.

The thirteen-copy operator comparison

The local-effect grouped-target block representation and residual-corrected Gram dual bound also admit a thirteen-click certificate. Rational Gram factorization and exact residual charging give, at thirteen trials,

The exact rational in research12/threshold/verification_results.json is authoritative. It applies to both declared local classes and is below 001: it cannot exclude thirteen copies. The supplied baseline thirteen-copy policy has error 0; the additional child-pool policy research12 /nominal/policy_13_pool.json improves its certified error to 0, still above 01. The outward rational bounds are reproduced by research12/nominal/verify_nominal12.py. Fourteen copies attain 0. These are not matching lower and upper certificates. We also tested a stricter first-step relaxation: the first command is local and chosen before erasures, and each posterior continuation is relaxed to a general permutation/local-transpose-invariant effect. The posterior need not be sign symmetric, so all real Pauli coefficient degrees, including the identity, enter the continuation SDP. On nine sampled directions from zero to2, the smallest proposed bound was about 000949372508, at the bisector; the axis endpoints gave about 001001652347. These are floating explorations, not continuous certificates. Their failure to close the gap is recorded without treating a sample minimum or solver status as proof.

A rational lower dynamic programme for the full circle

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A finite angular menu normally restricts the reader and supplies upper, not lower, bounds on the
continuous optimum. The following outer-action construction reverses that inclusion in a controlled way.
Let w ∈R4
+ be an unnormalized label weight vector. Write Vt(w) for minimum expected terminal-error
mass with t trials remaining. Thus V0(w) = min(wX+ + wY +, wX−+ wY −). Every fixed continuation
has a four-component loss vector in [0, 1]4; Vt is their infimum. It is positively homogeneous, concave,
and on the unit simplex is 1/2-Lipschitz in l1 distance. For normalized w, the last statement follows
because any loss vector has range at most one and P
h(wh −w′
h) = 0.
For a real pair a = (c, s) define the nominal kernels

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K±,h(a) = η
2(1 ± vah),
K0,h(a) = 1 −η,
ah = (c, −c, s, −s)h.

These are physical for = 1. They are still nonnegative normalized classical label kernels throughout [, although points outside the unit disk are explicitly fictitious relaxations, not allowed phase commands. Define

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Qt(w, a) =
X
o
Vt−1(w ⊙Ko(a)).

It is concave in wherever these kernels remain nonnegative.

Take rational unit supporting normals

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1 −(j/N)2
!
1 + (j/N)2 ,
2j/N
1 + (j/N)2
(cj, sj) =
,
0 ≤j < N,

with their three quarter-turn rotations. Intersect successive supporting lines = 1 to obtain the 4 vertices of a circumscribed polygon. Cardinal normals ensure; all fictitious kernels are therefore valid classical kernels. Let be the simplex grid whose coordinates are integer multiples of 1

Theorem 3.1 (Certified outer-action/grid lower backups). Suppose at every For each grid point and polygon vertex, decompose each outcome weight as

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zo = w ⊙Ko(a) =
X
i
γo,iqo,i,
γo,i ≥0,
qo,i ∈GM,
X
i
γo,i = ∥zo∥1.

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Then
Lt(w) :=
min
a∈vert PN
X
o,i
γo,iLt−1(qo,i) ≤Vt(w).

Starting from exact on the grid gives valid full-circle adaptive Bayes-risk lower bounds at every horizon. They also lower-bound minimax risk and every randomized nonadaptive reader. They do not establish local attainment of any polygon vertex.

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Proof. Homogeneity and concavity of the true Vt−1 imply P
i γo,iLt−1(qo,i) ≤Vt−1(zo). Every phys-
ical circle point lies in the convex hull of the polygon vertices.
Concavity of the true Qt gives
mina∈vert PN Qt(w, a) ≤mina∈S1 Qt(w, a) = Vt(w). Combining the two inequalities proves the backup.
In particular, no concavity of a chosen piecewise-linear interpolant is asserted or needed. The argument
first compares the true Qt on the outer polygon, then bounds its vertex values individually. Random
schedules and feedback coins are covered because a Bayes minimum over randomized policies cannot
be smaller than the minimum over its deterministic components.

Proposition 3.2 (Explicit convergence allowance). Use the cumulative-coordinate simplex triangulation implemented in co nv ex _g ri d_ bo un d. py . At normalized grid beliefs,

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0 ≤Vt(w) −Lt(w) ≤t
 4
M + ηv

.
4N2

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Proof. In cumulative coordinates, each interpolating simplex vertex differs from the normalized child
by less than 1/M in each cumulative coordinate. The four original probability coordinates consequently
differ by at most 2/M each. The l1/2 Lipschitz bound charges at most 4/M for the interpolation;
child masses sum to one. For consecutive normals the half-gap has tangent at most 1/N, so any outer
vertex has radius at most
√

). Project it radially to the circle. For every label the outcome kernels change in total variation by at most). The same bound applies to any continuation’s terminal [0 1] loss and hence to its optimal value. Combining the one-step bounds with the preceding grid-value error and inducting proves the claim. This is an explicit conservative allowance, not a claim that it is sharp.

This is a finite-label specialization of established belief-grid bounding methods, with a rational outer-action interface for this continuous reader. Lovejoy’s grid-based upper/lower POMDP approximation[4] is a close antecedent (Operations Research 39(1), 1991, pp. 162–175, DOI 10.1287/opre.39.1.162); the theorem above is proved independently. No worldwide novelty claim is made.

Exact demonstrations and scalability ceiling

The standard-library implementation constructs rational intersections, checks every support inequality and kernel normalization, and verifies every child’s nonnegative exact barycentric reconstruction. At 16), the exact lower values at the uniform prior are

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L1 = 6/25,
L2 = 199/1250,
L3 = 18278/171875.

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They are deliberately coarse and do not improve the existing small-horizon closed forms. Replays
at (4, 8) and (8, 8) verify the same algorithm under coarser interpolation; exact results are supplied.
There are
M+3
3

 belief points, 4N actions and three outcomes; a straightforward backup has O(tNM3)
rational operations and O(NM3) cached interpolation storage. Each child needs at most four vertices.
Bit complexity is additional and depends on the exact rational directions and horizon.
To force the conservative allowance below 10−4 at thirteen trials, one sufficient choice is M >
1,040,000 together with a correspondingly large angular resolution. A full uniform grid is therefore
impractical as an immediate endpoint solver. Targeted refinements, stronger continuation bounds or a
different structure could improve it, but were not established here. We record this limit rather than
propose an enormous unperformed run. The branch’s current threshold statement remains [13, 14] for
each class.