# Feedback and Robust Local Readout of a Fixed Four-State Quantum Source Jeremy Rodgers. 27 October 2026. Publication DOI: https://doi.org/10.5281/zenodo.23721114 This is the complete searchable PDF text transcription. Display geometry is preserved in the HTML vector formula panels. Linear PDF text does not encode fraction, matrix, or script topology. Consult the HTML formula rendering for exact notation. ## Source page 1 Feedback and Robust Local Readout of a Fixed Four-State Quantum Source Jeremy Rodgers Independent Researcher www.everythingequation.com 27 October 2026 Abstract For a fixed hidden label among four equatorial qubit preparations, we study its common-sign target with independent conditional copy stock and a fixed decision deadline. Adaptive and randomized nonadaptive equatorial readers have equal minimax risk through three copies for every shared efficiency and visibility. At four copies, exact rational policies and continuous nonadaptive certificates establish strict feedback advantages in both ideal and specified noisy parents. The nominal one-percent copy minima lie in [13, 14] for both classes; under the original asymmetric fixed- gain/phase uncertainty each lies in [16, 17]. A robust repeated-angle minimax frontier extends to arbitrary common-positive frozen-angle lotteries, and convex-source theorems identify when a single reader suffices globally. Compact policy graphs, exact dual certificates and retained-record witnesses are supplied. Copy savings are distinct from total apparatus savings, and no new Born-selection law is inferred. 1 Task, contributions and operational contract This paper develops local quantum-reader decision problems within Shadow Theory’s pre-existing source–readout programme [9]. The results below quantify target access under the declared preparation, measurement and retained-record conditions. Let h ∈{X+, X−, Y +, Y −} label ρX± = (I ± X)/2 and ρY ± = (I ± Y )/2. The target is the common sign. One label remains fixed for a complete experiment: the source provides ρ⊗B h , rather than redrawing the axis on each specimen. The preparation controller is unavailable. A trial destructively measures one fresh copy and all commanded settings, outcomes, nulls and specified timings remain retained. Every fixed deadline requires a sign; no postselection or abstention is allowed. Randomized nonadaptive protocols may choose a joint distribution of entire schedules before outcomes and retain their seed. Adaptive protocols choose settings from past accessible records and ready coins. Neither class has coherent quantum memory returning between specimens. The effects specified below and independent ready stock define the statistical comparison; they do not identify unread detector baths. Equatorial phase control is realized by the declared two-mode phase rotation, mixing and detection interface, with phase, clock and reference costs separate from specimen count. The results give constructive source distinctions inaccessible to separate-mode intensities, together with restrictions under which feedback, randomization and calibration actually change attainable readout risk. Adaptive multiple-copy discrimination and minimax quantum decision theory have established treatments[5, 7]. The known-binary comparison in the former does not supply a bound for the present grouped target or for all arbitrary preselected measurement schedules. Here the hidden axis stays fixed across the complete conditional-copy stock. Standard quantum discrimination, minimax symmetrization and representation methods have antecedents cited throughout; the contributions are certified portions of the declared-task frontiers, rational constructions and independently checkable 1 https://doi.org/10.5281/zenodo.23721114 ## Source page 2 continuous certificates. There is no worldwide novelty or empirical validation claim. Ordinary Born probabilities are assumptions. 1.1 Decision risks and the retained classical interface Write Cad and Cna for the two policy classes above. A policy Π has an independent, retained seed S, commands θt and a terminal sign DB. In Cad, θt may depend on (S, o1, . . . , ot−1); in Cna its entire schedule depends on S alone. The terminal decoder may use all recorded outcomes in either class. Its label risk and minimax value are rh(Π; η, v) = Pr h,Π;η,v{DB ̸= R(h)}, eC B(η, v) = inf Π∈C max h rh(Π; η, v). For a prior w, Bayes risk instead averages the label risks with weights wh; a Bayes-to-minimax equality is used only when a stated symmetry or least-favourable comparison proves it. In the robust task, the maximum additionally ranges over one fixed nuisance tuple for the complete experiment, and one policy is chosen before that tuple is revealed. The calibrated shared-noise comparisons use the conditional trial kernels Kθ,h(±) = η 2(1 ± v bh · nθ), Kθ,h(0) = 1 −η, where bh is the pure preparation’s equatorial Bloch vector and nθ = (cos θ, sin θ). Conditional on the fixed label and admitted nuisance, the fresh specimen and ready apparatus stock produce this kernel at each history. Equal one-copy marginals alone would not establish that sequential law. The retained decision record contains commanded settings, ternary outcomes, ready seeds and declared timings. Timings included in this statistical interface are determined by those records or have a source-independent conditional law; additional source-dependent arrival-time information would require an enlarged kernel and a new comparison. The theorem does not identify unread baths, time-resolved absorbing instruments or residual quantum outputs merely from their terminal effects. Organization and conventions. Sections 2 and 3 give the small-horizon continuous-menu optima and attainable four-copy policies. The nonadaptive certificates prove the cross-class separation. Sections 5 and 6 then treat the fixed asymmetric uncertainty; larger-horizon bounds and compact policies follow. The appendices explicitly restrict access to the X/Y menu. Angles are in radians, probabilities and Bloch gains are dimensionless, and physical pulse times and angular rates retain the units stated in their interaction models. All certificate paths below are relative to supplement/readers/; reproduction expands compressed inputs in a disposable tree. 2 Continuous equatorial access: exact small-horizon optima and an improved fixed reader Permit every equatorial setting θ, with σθ = cos θ X + sin θ Y , and keep the same shared calibrated efficiency and visibility: E±|θ = η 2(I ± vσθ), E0|θ = (1 −η)I, 0 ≤η, v ≤1. The source label h ∈{X+, X−, Y +, Y −} is fixed, its controller is unread, and independently prepared copies are supplied conditional on h. A policy selects one setting per fresh copy and always decides the sign target at the fixed deadline. All actual settings, signs, nulls and timing records remain retained. The expanded class includes randomized nonadaptive schedules and their terminal decoders, 2 ## Source page 3 as well as nonanticipating adaptive policies. No collective POVM, readable source register or returning environment is added. A phase rotation followed by a Hadamard beam splitter makes the bisector θ = π/4 a physically specified extension of the reader menu. For example, apply Uθ = exp(+iθZ/2) before the Hadamard and computational-path detection. Its sign effects are U† θ(I ± X)Uθ/2 = (I ± σθ)/2. A phase Hamiltonian Hz = ħΩzZ/2 realizes this command with signed pulse area ∫︁Ωz(t) dt = −θ and constant-drive duration |θ|/|Ωz|. This charges phase-reference availability, preparation, calibration and switching time; the following optima count copies rather than total apparatus cost. 2.1 A posterior formula and exact one/two-copy theorem The full continuous menu is closed under X/Y exchange and global sign reversal. These operations permute source labels transitively, transform settings as θ ↦→π/2 −θ or reverse outcomes, and transform the final sign consistently. The shared η, v noise law is invariant. Consequently averaging a uniform-prior Bayes-optimal policy over this group makes all four source risks equal. Uniform-prior Bayes lower bounds and randomized minimax optima therefore agree in both the adaptive and the randomized-nonadaptive continuous classes. This argument does not transfer unchanged to unequal axis responses. Lemma 2.1 (Exact one-step posterior risk). For unnormalized nonnegative hypothesis weights w = (wX+, wX−, wY +, wY −), put W = ∑︂ h wh, d = wX+ −wX−+ wY + −wY −, b = (wX+ + wX−, wY + + wY −). The optimal unnormalized Bayes loss with one remaining equatorial measurement is L1(w) = 1 2{W −(1 −η)|d| −η max(|d|, v∥b∥2)}. A reader parallel to b attains it; when b = 0 any reader suffices. Proof. The plus-minus target mass difference on a sign outcome r = ±1 is η[d + rv b · nθ]/2, while on a null it is (1 −η)d. For a two-target decision problem the Bayes error is half the total mass minus half the sum of absolute target-mass differences. The identity |d + c| + |d −c| = 2 max(|d|, |c|) reduces the optimization to maximizing |b · nθ|, whose maximum is ∥b∥2. This is an exact optimization over the entire circle, not a grid scan. Theorem 2.2 (Exact continuous optima at one and two copies). For the shared-noise continuous equatorial class, the randomized adaptive and randomized nonadaptive minimax optima coincide at horizons one and two: ead 1 = ena 1 = 1 2 −ηv 2 √ 2, ead 2 = ena 2 = 1 2 −η(2 −η)v 2 √ 2 . Repeated use of the fixed bisector θ = π/4, followed by majority of the nonnull signs and a fair tie decision, attains both optima. Proof. At the uniform prior, d = 0 and b = (1/2, 1/2), proving the one-copy statement through Lemma 2.1. For two copies, let the first angle be θ. After its sign r, the unnormalized posterior is wa,s = η(1 + rvsnθ,a)/8. Its total mass is η/2, its target difference is ηrv(cos θ + sin θ)/4, and its vector b = (η/4, η/4) is independent of that outcome and angle. Thus |d| ≤v∥b∥2 and the bisector is optimal 3 ## Source page 4 on either sign branch. The null branch retains the uniform prior scaled by 1 −η and also admits an optimal bisector. Adding all three optimal branch losses gives L2(θ) = 1 2 −ηv √ 2 4 −η(1 −η)v 4 | cos θ + sin θ|. Its minimum uses | cos θ + sin θ| = √ 2, giving the formula. At η = 1 the first angle is immaterial to the uniform Bayes value; a repeated bisector still attains the minimax optimum. When ηv = 0 all target information vanishes. Repeated bisector records have identical laws for X+ and Y +, and identical laws for X−and Y −, so its four source risks already agree. The symmetry argument upgrades the Bayes lower bound to minimax equality. Since this attaining protocol is nonadaptive, the two admitted classes have equal optima. The one-copy bisector is the familiar Breidbart direction for the unitarily equivalent BB84 subset/bit task[4]; it is not claimed as a new measurement discovery. The addition relative to the declared source–readout programme is the explicit noisy two-copy continuous-class theorem and its operational comparison. At η = v = 1, both errors are (1 −1/ √ 2)/2 ≃0.1464466, below the earlier X/Y value 1/4. At η = 4/5, v = 9/10, the exact errors are 1/2 −(9/50) √ 2 and 1/2 −(27/125) √ 2. 3 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. 3.1 The complete local class and exact three-copy theorem Use the four fixed source labels, their common sign target and the shared calibrated effects 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 qn(β) be fair-tie majority error for n independent signs of positive bias β. In particular q0 = 1 2, q1 = q2 = 1 2 −β 2 , q3 = 1 2 −3β 4 + β3 4 . Theorem 3.1 (Continuous equality through three copies). For the complete local equatorial class, all 0 ≤η, v ≤1, ead 3 = ena 3 = 3 ∑︂ n=0 (︄ 3 n )︄ ηn(1 −η)3−nqn(v/ √ 2). 4 ## Source page 5 Repeated measurement at θ = π/4, followed by majority of the nonnull signs and a fair tie, attains it. Equivalently, e3 = 1 2 −v(3η −3η2 + 3 2η3) 2 √ 2 + η3v3 8 √ 2 . At η = 4/5, v = 9/10 this is 1/2 −(4023/15625) √ 2 ≃0.135879605669. The theorem adds no assertion about four or more copies. 3.2 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 (c, s) and (p, q) be unit vectors, u ∈[0, 1], and put A = c + s, B = p + q, C = cp + sq, Q = c2p2 + s2q2. Then ∑︂ r=±1 max{|A + rB|, √︂ 2 + 2ruC + u2Q} ≤ √ 2(3 −u/2). Equality is attained when both directions are the positive bisector. Proof. There are four branches of the two maxima. If both choose absolute linear terms, their sum is 2 max(|A|, |B|) ≤2 √ 2. If both choose square roots, Cauchy–Schwarz and Q ≤1 give 2 √ 2 + u2. This is at most the claimed bound because 2(3 −u/2)2 −4(2 + u2) = 10 −6u −7u2/2 ≥1/2. For a mixed branch, changing (p, q) to its negative if necessary reduces the required estimate to M = |A + B| + √︂ 2 −2uC + u2Q ≤ √ 2(3 −u/2). Write (c, s) = (cos θ, sin θ), (p, q) = (cos φ, sin φ) and set α = θ−π/4, β = φ−π/4, x = | cos((α+β)/2)|, y = | cos((α −β)/2)|. Trigonometric identities give M √ 2 = G(x, y) = 2xy + R, R2 = H2 + u2x2(1 −x2), H = 1 + u/2 −uy2. Here x, y ∈[0, 1], R ≥H ≥1 −u/2 ≥1/2. If x ≥y, then ∂yG = 2x −2uyH/R ≥0, since xR ≥uyH. If y ≥x, then ∂xG = 2y + u2x(1 −2x2)/R ≥0. For x2 ≤1/2 this is immediate. Otherwise 2yR ≥y ≥x ≥u2x(2x2 −1). Thus increasing the smaller coordinate to the larger cannot decrease G. On the diagonal, with X = x2 = y2, G = 2X + √︂ (1 + u/2)2 −2uX. Its derivative is 2 −u/ √︁ (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. Proof of Theorem 3.1. First set η = 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 be (c, s). Its two equiprobable outcomes have posteriors related by global source-sign reversal, so their optimal continuation values agree. Let (p, q) be the second direction on the first-plus branch. After its outcome r, the unnormalized posterior weights have target mass difference v(A + rB)/8 and axis mass vector ((1 + rv2cp)/8, (1 + rv2sq)/8). 5 ## Source page 6 The preceding exact one-remaining-copy formula, obtained from |d + z| + |d −z| = 2 max(|d|, |z|), optimizes the third direction over the whole circle. Adding the two second outcomes and the equal first-sign branches gives three-copy Bayes loss 1 2 −v 8 ∑︂ r=±1 max{|A + rB|, √︂ 2 + 2rv2C + v4Q}. Lemma 3.2, with u = v2, bounds this below 1/2 −3v/(4 √ 2) + v3/(8 √ 2) = q3(v/ √ 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. 3.3 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 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 θ, and on a first nonnull outcome uses φ. 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 θ φ 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 6 ## Source page 7 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 (η, v) = (1, 1) the resulting minimax attainable upper error is strictly less than 0.030605 (actual exact-rational value approximately 0.0306041631615). For the nominal parent (4/5, 9/10) it is strictly less than 0.103471 (approximately 0.103470608280). These upper bounds are proved by summing each of the 34 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. 3.4 Verification and limits 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. 4 Certified continuous nonadaptive obstructions 4.1 Four ideal copies: an all-angle lower certificate Theorem 4.1 (Continuous nonadaptive four-copy lower bound). In the ideal parent η = v = 1, every randomized nonadaptive four-copy continuous equatorial reader, with arbitrary terminal decision and retained settings, has worst-label sign error at least 1/32. Proof. For a fixed schedule, an optimal uniform-prior decoder can be chosen odd on four sign bits. Its Walsh expansion has coefficients only on singletons and triples. There are 28 = 256 such odd decoders; exact enumeration divides them into seven orbits under independent outcome reversals and permutations of the four measurements, of respective sizes 8, 64, 32, 48, 64, 32, 8. For coefficients αi and βI, its target correlation is C = 1 2 ⎧ ⎨ ⎩ ∑︂ i αi(ci + si) + ∑︂ |I|=3 βI (︄∏︂ i∈I ci + ∏︂ i∈I si )︄⎫ ⎬ ⎭. Eliminate the fourth measurement analytically. With A = α4 + β124c1c2 + β134c1c3 + β234c2c3, B = α4 + β124s1s2 + β134s1s3 + β234s2s3, F0 = 3 ∑︂ i=1 αi(ci + si) + β123(c1c2c3 + s1s2s3), its largest possible correlation is (F0 + √ A2 + B2)/2. This maximization covers the full fourth circle, not a discretized setting. 7 ## Source page 8 For the first three directions use the rational circle (ci, si) = εi (︄ 1 −t2 i 1 + t2 i , 2ti 1 + t2 i )︄ , −1 ≤ti ≤1, εi ∈{−1, 1}. The hemisphere signs are explicitly enumerated: suppressing them after choosing a decoder represen- tative would not establish a full-circle bound. Together with seven decoder orbits this gives 56 cube calculations. The delivered certificate partitions each cube into a finite prefix-free set of dyadic octree boxes. On a box, exact endpoint formulas enclose each ci, si; ci additionally has maximum one when its interval contains zero. Rational interval products and sums give an upper bound on F0 and intervals for A, B. An integer-square-root enclosure bounds √ max A2 + max B2 upwards. Every resulting correlation bound is at most 15/16. The exact verifier checks all 143633 leaves, all orbit and sign variants, prefix-freeness and total cube volume one; the largest octree depth is eight. These finite exact inequalities prove the continuum bound. Floating calculations only constructed a candidate partition and are not accepted as proof of any leaf. Thus uniform-prior error is at least (1−15/16)/2 = 1/32 for every schedule. A randomized schedule is an average of such experiments with its retained seed and cannot lower this average Bayes bound. Worst-label error is no smaller than uniform-prior error, proving the claim. The delivered research10/nonadaptive/four_certificate.json.gz contains the finite proof object. The reproduction runner expands it without changing the underlying JSON content (the expanded hash is verified); research10/nonadaptive/verify_four.py recomputes every accepted inequality with standard-library rational arithmetic and integer square roots. The exploratory partition generator is retained in the historical evidence archive and is not required for verification. The bound 1/32 is deliberately sufficient for discrimination and is not claimed as the exact four-copy nonadaptive optimum. Floating exploratory optimization suggests a minimum near 0.03740607, but that number carries no global optimality claim. The certificate concerns the ideal parent only; it is not the principal nominal noisy four-copy optimum. 4.2 A noisy full-continuum separation at the principal benchmark The ideal separation is not the only surviving regime. A second exact certificate treats the principal shared detector parent, retaining all nulls and all measurement settings. Theorem 4.2 (Nominal continuous nonadaptive four-copy lower bound). At η = 4/5, v = 9/10, every randomized nonadaptive four-copy continuous equatorial reader has worst-label sign error at least 13/125 = 0.104. Proof. Condition on the entire source-independent erasure pattern and maximize the terminal decision separately on each pattern. Denote by Ck the optimal uniform-prior target correlation for the nominated k nonnull directions. For a single fixed direction i, put ai = ci + si; then C1(i) = v 2|ai|, C2(i, j) = v 2 max(|ai|, |aj|). For three specified directions, C3(1, 2, 3) = 1 8 ∑︂ r2,r3=±1 ⃓⃓⃓⃓⃓v 3 ∑︂ i=1 riai + v3r1r2r3(c1c2c3 + s1s2s3) ⃓⃓⃓⃓⃓, r1 = 1. The preceding exact three-copy theorem gives global upper correlations C1 = C2 = v/ √ 2 and C3 = v(6 −v2)/(4 √ 2). 8 ## Source page 9 Fix the first three directions, and grant separate fourth-direction optimization to the different erasure patterns. This relaxation need not be a realizable nonadaptive schedule; it only increases the upper correlation used for the lower error bound. With z = 1 −η, the resulting upper bound is U = ηz3{C1(1) + C1(2) + C1(3) + C1} + η2z2{C2(1, 2) + C2(1, 3) + C2(2, 3) + 3C1} + η3z{C3(1, 2, 3) + 3C3} + η4Cfree 4 (1, 2, 3). For Cfree 4 , enumerate the same odd four-bit decoders as in Theorem 4.1. Multiply every triple coefficient by v2 in F0, A, B, then its free-fourth correlation is v(F0 + √ A2 + B2)/2. The first-three conditional term can also be capped by C3. The original objective is invariant under simultaneous schedule/decoder outcome reversals and permutations. Apply these symmetries before choosing the relaxed fourth direction. The lower-click optimum terms are invariant under outcome reversals. Thus the seven decoder orbits and eight hemisphere sign choices again cover all original schedules. Relaxing first and then silently imposing a symmetry of the relaxed formula would not be justified. The delivered research10/nonadaptive/nominal_certificate.json.gz, expanded by the reproduction runner, bounds U ≤99/125 on all 56 cube variants. It contains 4230653 leaves, with maximum depth eleven. Every inequality is checked using outward dyadic intervals with scale 260. Interval sums, rational scalar products and general products use integer floor and ceiling. Integer square roots are rounded upwards. No floating number is used to accept a bound. Exact endpoint formulas for the rational circle give the starting intervals. For clarity, at the stated η, v let c1, c2, c3 denote upper interval numerators of the three braces above, and let f0, A, B be upper/intermediate numerators for the free-fourth decoder expression. All are expressed at the same dyadic scale. The total upper numerator is rounded from 20c1 + 80c2 + 320c3 + 576{f0 + √ max A2 + max B2} 3125 . The constants enclose C1 = 9 √ 2/20 and C3 = 4671 √ 2/8000 upwards. Prefix-freeness and exact cube-volume one establish full coverage, not merely a collection of point checks. The decoder orbit enumeration and all hemisphere signs are checked again. Hence every schedule has average error at least (1 −99/125)/2 = 13/125; retaining its random seed and mixing schedules cannot improve that average lower bound. Worst-label risk is no smaller. Combining this universal nonadaptive bound with the separately certified four-copy adaptive error < 0.103471 proves strict feedback benefit in the principal noisy parent over all continuous nonadaptive equatorial schedules. This is stronger than comparing selected angle allocations. The certificate is not an exact nonadaptive optimum; numerical exploration near 0.10476238 remains separately labelled. For the selected tolerance 519/5000 = 0.1038, the exact copy minima are four adaptive versus five randomized nonadaptive. The three-copy adaptive equality gives a lower risk ≃0.13587961, the four-copy adaptive construction passes, and the universal four-copy nonadaptive lower bound fails the target. Five repeated bisectors attain 1/2 −(11631897/39062500) √ 2 ≃0.07888035. This post hoc tolerance demonstrates an attained copy separation; it is not the original one-percent target. The strict separation also persists on a specified open parameter box. For a frozen policy, or uniformly for every nonadaptive competitor, changing shared parameters by (δη, δv) changes the one-trial record law by TV at most κ = |δη| + |δv|/2. Sequential coupling bounds any four-copy target risk change by 4κ. Thus the strict nominal gap of more than 0.000529 remains positive whenever 8κ < 0.000529. In particular |δη|, |δv| ≤10−5 leaves a certified gap of more than 0.000409. The adaptive 9 ## Source page 10 policy stays frozen; inaccessible parameters do not select a different policy. This is a shared-noise robustness statement, not a solution of the separate asymmetric fixed-unknown-gain problem. verify_nominal_integer.py carries the efficient exact replay; verify_nominal.py supplies a separately written Fraction interval reference. 961 selected leaf bounds are cross-checked with that reference. An independent audit checks every leaf with a root-free final squared allowance. The full certificate is large because it covers continuous settings rigorously, rather than claiming a finite grid solves the continuum. It is compressed in the supporting package. 4.3 The resulting ideal feedback separation and its limits The separately constructed adaptive four-copy policy has a certified ideal error strictly below 1/32. The preceding lower bound therefore covers every continuous nonadaptive competitor, rather than only the old X/Y menu or selected fixed-angle designs. This is a genuine continuous-access feedback separation under the stated preparation and record contract. At the explicitly selected tolerance 0.031, the exhibited adaptive policy passes with four copies. Every nonadaptive reader with at most four copies fails by the 1/32 bound and monotonicity. A five-copy repeated bisector has error 1/2 −(43/128) √ 2 ≃0.02491263, so five suffice nonadaptively. Thus the adaptive four-copy construction and exact nonadaptive five-copy minimum give a concrete task separation at that post hoc illustrative tolerance. The separate three-copy adaptive theorem gives the same ideal lower value 1/2 −(5/16) √ 2 > 0.031, so four is also the exact adaptive minimum. Thus the exact copy minima at this selected tolerance are four adaptive versus five randomized nonadaptive copies. The tolerance was chosen after the frontier calculation and is not the original one-percent target. Full large-horizon continuous optima, exact four-copy optimum values and general robust adaptation remain separate questions. 5 A robust frozen reader without X/Y symmetry Admit unequal contrast transfer, a common unknown phase offset and a broader nominal equatorial direction. One frozen setting and one parameter-independent decoder suffice for a sharp restricted minimax theorem. The restriction is consequential: it optimizes the common-positive, repeated-single- angle class, not all equatorial adaptive policies or arbitrary collective measurements. 5.1 Physical parent and uncertainty contract The source remains the four pure preparations ρX±, ρY ± and the nominated binary sign target. Conditional on its fixed inaccessible label, fresh copies and apparatus stock are independent. Before the phase reader apply a fixed, unknown Pauli channel with Bloch multipliers (gX, gY , gXgY ), where 0 ≤gX, gY ≤1. It has the explicit nonnegative Pauli probabilities (pI, pX, pY , pZ) = 1 4 (︁(1 + gX)(1 + gY ), (1 + gX)(1 −gY ), (1 −gX)(1 + gY ), (1 −gX)(1 −gY ) )︁. Their sum is one, so the channel is completely positive and trace preserving for every admitted gain pair. A coherent two-mode reader then measures the actual equatorial angle θ + δ, followed by independent sign-blind erasure. Thus E±|θ = η 2 [I ± (gX cos(θ + δ)X + gY sin(θ + δ)Y )] , E0|θ = (1 −η)I. The channel precedes the phase measurement; changing this order can change the model. All effects are positive because their Bloch-vector length is at most one. The phase rotation uses the specified 10 ## Source page 11 Hz = ħΩzZ/2 coupling, calibrated beam splitter and terminal detector. Phase and offset are in radians, Ωz is in radians per second and pulse durations are in seconds. The Pauli/environment ready stock and erasure port are physical resources, not free mathematical randomness. The unknown parameters are fixed throughout a deployment block and obey η ∈[η0, 1], gX ∈[a, 1], gY ∈[b, 1], |δ| ≤∆, 0 ≤∆< π/4, 0 < η0 ≤1, 0 < a, b ≤1. No prior distribution over these nuisance parameters is assumed. The controller fixes one nominal angle before learning their values. The admitted common-positive class is θ ∈[∆, π/2 −∆], so both axes have nonnegative target-aligned signal at every admitted offset. Every trial at the fixed deadline is counted; the decoder returns the majority of the nonnull signs, with an independent fair guess on a tie, including the all-null record. Nulls and all settings remain archived. No hidden label or calibration purifier is read. Under source label X± the nonnull sign has mean ±βX, where βX = gX cos(θ + δ); under Y ± it has mean ±βY , where βY = gY sin(θ + δ). The four labels need not have equal signal strengths. The proof below uses a hard binary pair for its lower bound rather than incorrectly reusing uniform-prior minimax equality from the symmetric experiment. 5.2 Exact minimax angle and full record risk Define the majority error for n successful signs of bias β by qn(β) = ∑︂ k