Uniform angular approximation

Uniform angular approximation

A certified angular-net transfer, not a grid optimum

Theorem 5.1 (Uniform finite-horizon setting-rounding bound). Let a rational menu contain the circle points

Searchable transcription

M2 −k2
!
M2 + k2 ,
2Mk
M2 + k2
nk =
,
−M ≤k ≤M,

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their negatives and their axis swaps, where M ≥1. For the fixed shared η, v parent, write econt
B
and egrid
B
for the minimax error in either the adaptive class or the randomized nonadaptive class, consistently on
both sides. Then

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B
≤Bηv
econt
B
≤egrid
B
≤econt
B
+ εB,
εB = 1 −

1 −ηv
2M .
2M

Thus an independently certified grid interval gives the continuous interval [max for the same policy class.

Proof. Up to sign, every direction is)) for 1]. Round to , within 1). The chord identity

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∥n(t) −n(t′)∥2 =
2|t −t′|
p
(1 + t2)(1 + t′2) ≤1
M

implies source-specific one-trial outcome-law TV at most). For an adaptive policy the rounded controller maintains its virtual nominal history and applies the rounded setting at each branch. Coupling the outcome records sequentially gives probability of any divergence at most The terminal target error changes by at most that amount on every source label. This rounds every continuous policy to a legal grid policy, without converting feedback to a nonadaptive schedule. Root mixtures are rounded componentwise. The converse inclusion of classes gives the other inequality.

The coupling compares outcome histories and their logical policy commands. It does not claim TV closeness of literal physical setting archives: a different real angle can make those raw setting records disjoint. All actual settings remain retained, and this theorem certifies target-decision risk rather than full retained quantum-channel equivalence. It assumes the same calibrated noise model at the omitted and retained angles; unrelated setting-dependent losses do not satisfy it automatically. The code constructs the rational menu and an interval-transfer adapter, but it does not solve a large grid or advertise its illustrative interval as a measured continuous optimum. Required grid size, policy search cost and coefficient precision must all be charged by any consumer. This keeps the continuous-angle guarantee distinct from an uncertified finite scan.

Verification and exact status

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The all-posterior formula is also exposed as a reusable terminal-control certificate: rational weights
specify a direction numerator, squared norm and an exact rational-plus-square-root loss. It can eliminate
the last continuous setting search in a policy optimizer; it does not by itself solve the preceding controls
or establish a new minimax prior.
The standard-library implementation provides exact arithmetic in Q(
√

2), the fixed-reader risks, small-horizon theorem values, the rational circle menu and the transfer allowance. Independent formulas check 85 click-mixture sums, 125 two-copy ordered-record calculations and 720 exact rational chord-rounding cases, plus 114 arbitrary-posterior terminal checks with rational Pythagorean directions.

Analytical proofs, rather than those finite tests, cover every angle and the all-horizon transfer. The angular-net interface alone establishes no large-horizon continuous optimum, collective optimum or total energy/apparatus advantage. Exact continuous equalities through three copies and four-copy comparison certificates are supplied separately in the reader paper. The physical interaction remains the declared controlled reader under ordinary Born probabilities. No trine, Born-selection or consciousness theorem is added.