From the continuous feedback gap to a physical resource certificate

From the continuous feedback gap to a physical resource certificate

The continuous four-copy construction and universal nonadaptive lower certificate support a resource statement: a feedback advantage in the same complete equatorial menu, including a nonzero operating region of asymmetric gain, erasure and phase errors. We also replace a crude setting-switch count by an exact actuator-path account and give a finite calibration interface covering the whole circle. All these are restricted operational claims, not a universal apparatus, energy or thermodynamic optimum.

A nonzero noise and control-error operating region

Let denote the delivered exact rational ideal risk of the frozen four-copy adaptive tree, including its two root symmetry bits. Its value is. All forty action nodes and eighty-one ternary leaves remain specified, including nominally impossible ideal null histories. Keep the four fixed-label preparations and inaccessible preparation controller. At a fixed unknown detector tuple use the asymmetric parent

with the fixed Pauli channel of Bloch multipliers ( ) before the phase reader. Every commanded angle has total phase error of absolute value at most 1 radian, including the common fixed offset and declared control inaccuracy. The phase-error bound must apply to every command used by the policy; it is not inferred from earlier calibration alone. Fresh apparatus stock and copies are independent conditional on the fixed hidden source and nuisance parameters. For the adaptive/nonadaptive comparison, and the common offset are fixed throughout the block. Any residual optical/control error or quantization is a response to the command and trial index fixed in advance, never a function of deployment outcomes or an uncounted feedback controller. Independent stateless fresh noise may be included if declared separately. The uniform TV upper bound tolerates broader conditional errors, but that fact does not license outcome-dependent hidden feedback in the nonadaptive comparator.

Theorem 6.1 (Frozen-tree operating certificate). The same implementable four-copy policy, fixed before the nuisance parameters are revealed, has worst sign error at most

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eA + 1 −(1 −κ)4,
κ = (1 −η0) + (1 −η0g0) + η0g0d
2
.

The coupling bound controls the complete retained classical deployment record, including com- manded/logical setting identifiers, nulls and controller coins, for the same frozen policy. Unknown actual analog phases are part of the response kernel, not additional accessible phase logs. It is not a norm bound on the residual quantum instrument or its bath.

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Proof. For a source sign along either axis the ideal signed mean is one nominated component nh of the
unit direction. In the perturbed apparatus it is μh = ηghn′
h. The Euclidean chord is at most the phase
error, so
|nh −μh| ≤1 −ηgh + ηghd ≤1 −η0g0 + η0g0d.

For ternary rows parametrized by efficiency and signed mean,

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p = (1
2(η + μ), 1
2(η −μ), 1 −η),

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the exact row distance is

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TV(p, p′) = 1
2{|η −η′| + max(|η −η′|, |μ −μ′|)}.

The displayed mean allowance dominates 1, giving row distance at most. Couple identical root coins and, until the first record mismatch, use the same history-dependent command in both experiments. The probability of no mismatch is at least (1 by the conditional row bound. Every terminal event, including a wrong decision, therefore differs in probability by at most 1. The bound is uniform in the four source labels and preserves one fixed nuisance tuple throughout the tree. The nominal final minimax risk is, which proves the assertion.

For the explicit operating box

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η0 = g0 = 99999/100000,
d = 1/10000,

we obtain the exact rational200000000000000 and

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eA + 1 −(1 −κ)4 < 0.030865 < 31/1000 < 1/32.

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The total phase allowance is about 0.00573 degrees. This is a narrow, precisely declared operating region
near the ideal apparatus, not an operating certificate for the principal noisy benchmark η = .8, v = .9
(whose separate finite-horizon comparison is handled in the continuous-reader paper), nor for the
specified larger asymmetric box a = .9, b = .8, η0 = .8.
The uncertainty set includes the ideal apparatus. Hence every robust randomized-nonadaptive
protocol still has worst error at least the universal ideal lower bound 1/32 at four copies. A stronger
pointwise statement is also available: for any fixed nuisance tuple, each nonnull unsharp equatorial
measurement is an ideal equatorial projector at its effective Bloch direction followed by an independent
sign flip, together with the sign-blind erasure. A nonadaptive schedule remains nonadaptive under
this simulation. Thus adding these detector imperfections cannot improve the full-menu nonadaptive
optimum. This is a statistical simulation, not an identification of inaccessible environments.
An explicit five-copy nonadaptive upper construction repeats the bisector and takes fair-tie majority.
Its ideal error is 1/2 −43
√

100000. Applying the same row comparison at five copies gives

Together with the independent ideal three-copy adaptive lower bound in the continuous-reader paper, these results give exact minimum budgets of four adaptive versus five nonadaptive at the selected target 311000 throughout this operating box. The target was selected transparently inside a certified gap, not presented as a preregistered experimental tolerance. The fixed unknown detector tuple is never learned by an inaccessible oracle. The error guarantee holds for the same published four-copy policy over the entire box.

Actual phase travel, rather than a generic switch charge

There are two distinct physical timing parents. In the serial pulse parent, each fresh copy receives 2 withΩ. A fresh-copy phase command costs its required rotation angle divided by Ω. In the held-actuator parent, a calibrated phase element obeys ̇Ω, and its current setting is retained between trials. Its time is the total actual phase travel divided by Ω. This

is an additional apparatus assumption, not a consequence of a single qubit’s Hamiltonian pulse. Both start at phase zero. No terminal reset is included; any required reset is a separately charged operation. Equatorial axes have period if the controller keeps track of an outcome-polarity relabelling. This does not permit an instantaneous physical jump. A continuous shortest axis path is lifted to the actual phase coordinate, and its known polarity bit recodes the recorded signs. Hardware without that permission instead pays for its actual angle range. Rates are in radians per second; all timing expressions below are in seconds.

Theorem 6.2 (Nonadaptive schedule ordering). For independent copies and time-homogeneous fixed nuisance parameters, a nonadaptive schedule’s decision risk is unchanged by permuting its settings and correspondingly permuting the labelled outcome data supplied to its decoder. Let be the set of required projective-axis settings and include the actuator’s initial point. On the angle circle of circumference , delete each consecutive gap in turn. If the remaining arc has length and has distances and from its endpoints, then the exact minimum open actuator travel over schedule orders is

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L(S; q0) = min
gaps{L + min(u, L −u)}.

Repeated settings require no additional travel when measured consecutively. In the common-positive quadrant with, increasing angle order attains = max

Proof. The conditional likelihood is a product of setting-labelled factors, so a permutation with the inverse data relabelling leaves every source-label error unchanged. The physical timestamps change and are retained; this is not equality of the original timed apparatus channels. An optimal open actuator route can be shorter than a full circle, since starting at and visiting all settings in one circular direction already has length strictly below for a nontrivial finite set. Its travelled support therefore lies in a proper closed arc containing all required points and. Shortening either unused endpoint to a required point shows that the omitted gap may be chosen consecutive. On an interval of length, a route starting at coordinate must visit both endpoints. It pays at least min), attained by visiting the nearer endpoint first and then traversing to the farther endpoint. Minimize over omitted gaps. Conversely every such candidate route is implementable by the bounded-speed actuator. In the common-positive quadrant, direct increasing order from zero attains the farthest point’s unavoidable distance.

This theorem gives the nonadaptive comparator its best setting order; it cannot be charged an arbitrary back-and-forth order to manufacture a feedback advantage. It does not apply to arbitrary drift or correlated future stock. An adaptive schedule generally cannot be reordered without changing its observation-dependent actions. For the actual forty-node adaptive tree, evaluate every ternary history and both root axis-swap controls. The sign-reversal root operation is implemented by logical sign and decision recoding and adds no phase motion. Rational outward arctangent intervals give worst total phase costs

(held actuator) (per-copy pulse)

The enclosed values are respectively 5 and 4 radians. All 162 ternary-history/root-swap cases are covered, so rare perturbed nulls have not been discarded. A finite command interface can use fifteen-bit indices for 32768 equally spaced levels on [0 Nearest rounding has error at most40000 radian, using7. Allocate at most 120000 to the fixed offset and at most 140000 to additional optical phase error; the total remains inside10000. The archived command identifiers and finite packets are identical in the reference and perturbed record comparison; quantization enters their response kernel. No TV comparison of disjoint exact-analog-angle logs is asserted. Quantization adds at most seven rounding

errors to a four-trial held path and four to the pulse sum. Thus 740000 and 440000 respectively still fit inside the conservative 556 and 47 radian caps. The assumed slew model concerns the commanded actuator trajectory; bounded optical phase error alone would not bound arbitrary unmodelled actuator wandering or settling time. The policy already uses a useful physical tie-break: on twenty-one final nodes no nonnull sign can change the nominated decision, and it keeps the previous setting. Twenty of those nodes have the same decision on all three outcomes. One has a different default decision on the ideal-impossible null, but its null probability is angle-independent in this parent. The retained terminal trial remains present. The repair preserves ideal risk and avoids unnecessary held-actuator movement; the operating certificate directly covers the resulting frozen tree rather than assuming all residual records are unchanged.

A full-menu copy advantage with a conditional time advantage

Let 0 be the serial per-copy duration for preparation, propagation, detection and matched terminal handling, excluding the separately modelled phase operation. Let0 upper-bound each of the three noninitial causal lookup/feedback delays. Sufficient adaptive time bounds are

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TA,s ≤4τ + 3l+ 139
25Ωs
,
TA,p ≤4τ + 3l+
47
10Ωp
.

Every nonadaptive protocol meeting the same target within the same continuous menu and operating box needs at least five fresh copies, so its time is at least 5, even before its optimized phase travel is charged. Consequently either

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3l+ 139
25Ωs
< τ,
or
3l+
47
10Ωp
< τ

suffices for a strict clock advantage in its corresponding serial parent. These are conservative sufficient conditions, not exact hardware optima. The two rates refer to different controls and cannot be substituted for one another by reusing a single apparatus certificate. For Ω = 20 and20, the held implementation saves at least 143 per experiment relative to the universal nonadaptive lower time. In the separately specified pulse parent with Ω = 20 the lower saving is 123. These are illustrative declared rates, not measured device data. If an actual extra net setup/calibration time is charged to the adaptive apparatus, repeated uses certify a benefit only when times that lower saving exceeds. For the illustrative, the sufficient counts are 18 and 17 uses, respectively. Reuse requires a calibration/reference hold contract over the whole block. The nonadaptive reader is not forced to calibrate settings it never uses. Failed sufficient inequalities do not prove that feedback is slower. The constructive adaptive controller may store a seven-bit pointer over the 121 internal/terminal prefixes; the remaining deadline is encoded in the pointer’s depth. Two root group bits remain separate. A direct finite-command ROM account uses at most 80 15 = 1200 angle bits for both root axis controls, plus 81 terminal decision bits; indexed tree arithmetic avoids a separate child-pointer table. Arithmetic workspace, clock precision, the held actuator’s physical state, phase tracking and the reference remain additional resources. A fully explicit ternary-prefix archive can be retained independently. These are upper resource accounts, not minimum-controller theorems. They cannot be turned into a thermodynamic work, energy or apparatus-volume conclusion.

Finite calibration covering all continuous angles

A finite calibration route is available under an explicit structural promise: shared sign-blind efficiency, unbiased outcomes, and a time-homogeneous sinusoidal effect law. For each trusted positive source

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h = X+, Y + write
μh(θ) = Ah cos θ + Bh sin θ,

with the negative source having mean). The asymmetric Pauli/phase parent with fixed gains and a common fixed offset has exactly this form about the known realized command angle. This calibration theorem is a separate structural promise: arbitrary command-dependent optical errors or nonlinear quantization residuals need not preserve the sinusoid. They require a separately validated uniform row-error allowance, additional command calibration, or a justified response model; four probe batches do not certify them automatically. Use independent trusted probes for each of the four pairs

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(X+, 0), (X+, π/2), (Y +, 0), (Y +, π/2).

The recorded variable estimates the four coefficients; the pooled click fraction over 4 probes estimates. Hoeffding and a union bound give the simultaneous predeployment event

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Pr

max
h {| bAh −Ah|, | bBh −Bh|} > u or |bη −η| > w

≤8e−Mu2/2 + 2e−8Mw2.

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For completeness, the bounded-variable inequality follows by writing K(t) = log Eet(Z−EZ). Under
exponential tilting, K′′(t) is a variance and is at most one quarter of the squared support width;
K(0) = K′(0) = 0 then bounds the moment generating function by et2(b−a)2/8. Chernoff optimization
for an independent sample gives the two-sided bound 2e−2Mu2/(b−a)2. Apply widths two and one to
the signed means and pooled clicks respectively, then union the five events. For α = 2−k the exact
conservative choice
M =

max
2(k + 4)
u2
, k + 2
8w2


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makes the failure probability at most α; total consumed stock is 4M. This bound covers every angle,
because the coefficient error has absolute value at most u(| cos θ| + | sin θ|) ≤
√

. It cannot validate an arbitrary detector between unmeasured settings; without the sinusoidal/common-efficiency promise additional coverage is required. Trusted source preparation and independence are paid premises. For any nominated valid reference kernel whose four coefficients and efficiency lie inside those same calibration intervals, the actual and reference kernels differ by at most

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κcal = min{1, w + max(w,
√
2u)}

in every source/angle row. Thus the same policy at horizon has retained classical commanded-setting/outcome record distance at most 1. It excludes an extra unread analog-phase log. This includes policies and angles selected from the calibration data: the event covers the whole continuous family before deployment. It does not identify the full residual quantum instrument or unread environment. If calibration acceptance is, a selected reference policy has verified reference error and transfer allowance, then error)

An accepted-run conditional error bound must retain the corresponding Pr) term, unless a separate acceptance-probability guarantee justifies a smaller expression. Bad calibration is charged once for a stable shared block; past samples do not prove future stability or a reference-reuse lifetime. This is a completed finite interface, not a practical calibration-cost claim. For example, 01 consumes 1,120,000 probes but does not by itself certify any advertised reader accuracy. A much tighter illustration,88000, consumes 92,928,000,000 probes. Its conservative coupling allowance certifies Prerror01 for the nominal fourteen-copy bisector under the stated reference-acceptance contract; this is not an unqualified conditional-on-acceptance bound.

The stock cost plainly overwhelms a few saved copies. Better device-specific likelihood calibration can be pursued; it is not hidden inside the present resource claim. Phase-reference preparation, conditional update, reset, holding and consumption remain separately declared resources. Established reference-degradation analyses may bound an average even when records are kept; that does not automatically supply a uniform conditional guarantee or the adaptive interaction model used here[2]. No reference lifetime is transferred without matching those contracts.

Reproducible checks and remaining limits

The delivered research10/resources/verify_resources.py and research10/resources/policy_ resources.py, with the independent research10/audit/check_resources.py, check 84 exhaustive schedule-tour instances against independent permutation enumeration, 873 individual quadrant orders, 81 exact ternary row distances, the two exact calibration allocations, and the rational operating and clock inequalities. The policy cost evaluator uses the delivered tree and outward arctangent intervals rather than a floating-point maximum. The independent checker can verify these certificates without trusting the exploratory angle search. The nominal9 continuous one-percent threshold, the larger asymmetric robust frontier and universal total-resource optimization remain unresolved. The source–readout contribution is a typed full-menu reader, calibration and cost interface, not a Born-selector, source-ontology or consciousness mechanism.