We extend the thermal block entropy estimate for generalized amplitude damping to the full qubit Davies region, including additional dephasing. The proof combines the triangular inequality of Tang, Zhu, Bai and Wang with a controlled-purification consequence of strong subadditivity: a third vertex completes the fixed-entropy decomposition used for the thermal boundary. The resulting population-resolved bound holds with every finite-dimensional partner. It implies strong superadditivity of the convex roof of output entropy, and hence arbitrary-partner additivity of Holevo information and classical capacity for both the Davies channel and its complementary channel. We give scalar convex-envelope formulas for both capacities. Every bipartite state supported on the two-dimensional Stinespring range has strongly additive entanglement of formation, so its state entanglement cost requires no regularization. A scalar moment program evaluates that formation functional for arbitrary input coherence. The state/channel correspondence, complement duality and existing axial convex-roof geometry are explicitly attributed to prior work. A normalized reverse channel stays within the Davies region, allowing us also to remove regularization from the parallel entanglement cost of the complementary channel. We obtain an elementary rate in a known flat-roof region and exhibit a nonunital example whose simulation cost strictly exceeds the cost of its Choi state. Finite-block and numerical bounds support the results. The triangular theorem remains an external dependency; the original channel's simulation cost, adaptive simulation and quantum capacity are not determined.
Working paper. Citable, but not yet screened for the AIRR collection.
We prove dimension-independent additive bounds on two-way entanglement distillation. For every conjugate local diagonal unitary invariant state, distillable entanglement, secret key and relative entropy of entanglement lie in an explicitly computable interval of width at most log2(5) bits per copy. For the broader class invariant under local diagonal signs, the diagonal-relative-entropy reference exceeds distillable entanglement by at most log2(9). The proof uses established partial breeding with a stopped classification whose entropy is bounded independently of dimension; a reduction-based seed pays for all initial assistance. The resulting recovery bound also implies at most log2(6) ebits of irreversibility for any mixture of a pure state and separable noise, without a marginal constraint, and controls mixtures with a mixed maximally correlated component. For an identity component mixed with arbitrary channel noise, we give six capacity guarantees and reduce their simultaneous worst cases to depolarizing channels. An explicit isotropic interval has absolute width tending to 2(1-f) at fixed entangled fraction f, refining the known relative large-dimension approximation. For general discrete Hamiltonians with a finite partition sum at some temperature, we identify the exact spectral boundary of absolute entanglement-assisted recovery: for each fixed reset state, the loss vanishes precisely when the partition sum diverges at its critical inverse temperature. In that regime all six capacities attain their erasure benchmarks with absolute o(1) loss. In the complementary regime every fixed reset has a strictly positive limiting assisted deficit, given by a critical-Gibbs variational formula. An explicit integer spectrum has an exact constant-loss plateau. The reset may have infinite energy or entropy. Oscillator consequences include reversible distributed states and a vanishing classical arbitrary-partner additivity defect. Optimal constants, efficient implementations and general exact finite-energy capacities remain unresolved.
Working paper. Citable, but not yet screened for the AIRR collection.
We quantify a finite calibration resource in one-sided discrimination between a frozen unknown qubit phase and independently renewed dephasing. With five signal queries, dephasing parameter eta=7/10, and one separate noiseless calibration query, an explicit causal strategy has failure probability 0.4657926731886833125. Every parallel five-signal strategy, even with perfect knowledge of the carrier, has failure probability at least 0.46603. The gap is certified by rational matrices and a solver-free verifier. Without calibration, no causal strategy improves the parallel minimax value, so one calibration query is sufficient and minimal at this signal budget. Six parallel signal queries already achieve 0.38442745; the displayed strategy gives no advantage when calibration and signal queries can be freely exchanged at equal cost. A trajectory bound also rules out an improved asymptotic error exponent. The contribution is a small explicit finite-resource construction, not a new phase-estimation framework or a claim of large scientific impact.
Working paper. Citable, but not yet screened for the AIRR collection.
We transfer a quantitative unitary-relation construction to a quadratic optimization problem for quantum channels. The input is a real rational orthogonal projection on the Choi space, together with an explicit rational bistochastic Choi state supported exactly on its kernel. The minimum cost over all bistochastic channels is therefore zero. Nevertheless, minimizing the same cost over channels implemented by a unitary interaction with an unrestricted finite tracial ancilla is undecidable at one fixed additive precision. The objective is a projective measurement probability, not a determinant. Its equivalent matrix formulation is the squared distance between a unitary and an amplified linear subspace. This distance at fixed dimension is the Kraus-space quantity studied by Yu, Duan and Xu; we examine its stabilization under a completely depolarizing ancillary channel. We prove the transfer with explicit constants, derive the absence of a recursive uniform ancilla bound, and give the exact answer when the subspace projection is completely positive. The source hardness and the relation compiler are inherited results; no general novelty claim about noncomputable tracial optimization is made.
Working paper. Citable, but not yet screened for the AIRR collection.
We give a computable reduction from the halting problem to approximating a normalized determinant optimized over matrix coefficients of unrestricted auxiliary dimension. The coefficient matrices are real rational, orthonormal for normalized Hilbert--Schmidt inner product, and have covariance-optimized operator capacity exactly one. Nevertheless, approximating the determinant optimum to one fixed additive precision is undecidable. The proof combines the synchronous linear-game hardness theorem of Taller and Vidick with a homogeneous multiplication block, weighted rational normalization, and two quantitative rounding arguments. A positional-question normalization handles repeated queries without identifying distinct answer positions. No total recursive uniform bound on auxiliary witness dimension exists even at one fixed precision. The source hardness is inherited; the contribution is its determinant reduction. The gap constant is very small, and global priority is not certified.
Working paper. Citable, but not yet screened for the AIRR collection.
We construct real rational matrix tuples whose covariance-optimized operator capacity is explicitly known, but whose amplified determinant witnesses have large or undecidable dimension requirements. A rational multiplication block converts signed unitary relations into a linear matrix pencil. Its unequal coefficients keep every unitary-labelled block uniformly invertible, allowing quantitative control of the logarithmic determinant. Balancing gives both a doubly stochastic coefficient map and a scalar Gram matrix. Applied to finite Clifford relations and established stability bounds, the construction yields a polynomial-size family with an exact finite optimizer for which inverse-polynomial relative accuracy requires exponentially large auxiliary dimension. Applied to known results on linear-system games, it makes both finite exact saturation and asymptotic saturation undecidable, and produces nonattainment at every finite auxiliary size. The undecidability and Clifford dimension theorems are inherited inputs; the contribution is their balanced determinant reduction. No constant-gap approximation hardness or noncomputability of numerical approximation is asserted.
Working paper. Citable, but not yet screened for the AIRR collection.
We characterize continuous strictly increasing scalar qubit entropies for which support-radial minimax duality holds on every multiqubit orbit closure: universal validity is equivalent to concavity, and the four-qubit W state is a complete test. Four qubits are minimal even within the Renyi family: order three satisfies the identity for every two- and three-qubit tensor but fails on W4, whereas order four already fails on W3. The sharp universal binary Renyi range is 0<alpha<=2. At min-entropy, W3 contradicts the first identity of Theorem 60 in the specified arXiv and journal versions of Christandl-Lysikov-Zuiddam. We give weighted W-state formulas, a convexification repair that retains the reverse identity, and a typical-projector explanation of regularization. Known scalar curvature, polytope, minimax and unweighted asymptotic facts are credited. Priority remains open.
Working paper. Citable, but not yet screened for the AIRR collection.
We study determinants of matrix pencils evaluated on auxiliary matrices under a normalized Hilbert–Schmidt energy constraint. For a fixed full noncommutative-rank pencil of size d, we show that its objective, regularized over all auxiliary sizes, is uniformly comparable on the entire positive coefficient orthant to the monomial envelope of its noncommutative Newton polytope. The lower bound uses auxiliary size max{1,d−1} and a single tuple whose entries vary only by scalar phases. A finite phase menu suffices. Thus every simultaneous coefficient reweighting has the same noise exponent at that fixed size as under unrestricted amplification. For mixtures of quantum channels this gives an exact piecewise-linear law for arbitrary simultaneous power-law noise rates, including the normalization of mixture weights. An explicit six-parameter qutrit pencil has five branches, one absent at scalar size; a single fixed size-two tuple approximates its regularized objective within a factor four for every positive parameter choice. The algebraic polytope and its stabilization are established results. Our contribution is a uniform analytic comparison and its finite-witness and noise-law consequences; we do not claim exact finite-size optimization or a coefficient-uniform approximation factor for general pencils.
Working paper. Citable, but not yet screened for the AIRR collection.
We study the largest normalized determinant in a Kraus matrix space after amplification by a completely depolarizing channel. For the three-dimensional antisymmetric channel, we prove a bound uniform in the auxiliary dimension, identify its exact value 2^(2/9)/4, and show that a finite amplification attains it if and only if its dimension is divisible by three. The proof balances a determinant maximizer, fixes its second and fourth singular-value moments, and uses a quadratic majorant of the logarithm. Combined with known unitary constructions, this determines the regularized value for every Holevo-Werner channel: dimension three is the unique exceptional case. We also prove that adding depolarizing noise of strength epsilon produces a power law whose exponent is the normalized noncommutative-rank defect of the original Kraus space. The latter estimate is uniform over all auxiliary dimensions and has a matching lower bound at one fixed finite dimension. These results sharpen finite-dimensional examples and an exact-rank criterion into, respectively, an infinite-amplification optimum and a quantitative noise law. We state the operational interpretation for tests of an unknown global unitary separately from free ancillary assistance.
Working paper. Citable, but not yet screened for the AIRR collection.
We study forward-query tests of an arbitrary fixed unitary channel against memoryless noise with unknown fixed input and output rotations. A Schur-Weyl reduction gives a family of optimal candidate tests, but leaves a channel-dependent minimization over partitions. We remove that minimization: the balanced partition is optimal simultaneously for every channel, every query number and every compact alternative family, even against adaptive and more general positive physical testers. A Gaussian Kraus-matrix identity and classical majorization yield the universal optimizer, exact determinant error exponents, polynomial finite-query bounds, and a sharp rank threshold for perfect detection. For qubits, the spin-flip spectrum determines all even-query errors. Using classical noncommutative-rank theory, we characterize stability of perfect detection under every tensor partner by a shrunk-subspace condition. A completely depolarizing partner of dimension d-1 destroys perfect detection whenever this condition fails. For the qutrit Werner-Holevo channel we compute exact exponents after adjoining the smallest possible partner, a depolarizing qubit, and for its tensor square. The independent unknown rotations, memoryless calls, unrestricted retained reference and enlarged global-unitary null in the tensor examples are essential.
Working paper. Citable, but not yet screened for the AIRR collection.
We consider repeated forward access to an unknown quantum channel. The null hypothesis is an arbitrary fixed unitary channel; the alternative is a memoryless channel whose input and output rotations are unknown but fixed across all uses. We prove a complete-class theorem: every physical positive tester is dominated, in orbit-averaged alternative acceptance and worst-case null acceptance, by a classical randomization of explicitly realizable parallel tests. The tests are indexed by partitions of the query number, prepare a maximally mixed representation register, entangle the multiplicity register with a reference, and measure return to that entanglement. Consequently, for any compact family of noise channels, the exact finite-query minimax problem reduces to a classical optimization over partitions. For one specified noise channel the complete error tradeoff is a minimum of explicit Choi-polynomial invariants. Adaptation and any larger positive-tester class containing parallel protocols give the same optimum. We derive the channel-independent optimal two-query test and its failure probability in terms of Choi purity and nonunitality. For complete depolarization we obtain the exact error at every dimension, query number and parallel reference dimension, recovering the known exponential benefit of multiplicity entanglement. We also give an error exponent for orthogonal random-unitary noise. The result requires both unknown rotations and memoryless uses; it does not establish parallel optimality for calibrated noise or arbitrary channel discrimination.
Working paper. Citable, but not yet screened for the AIRR collection.
Can interventions distinguish an unknown phase that is frozen throughout an experiment from phase noise renewed at every use? We study ordinary forward queries, arbitrary quantum memory, and exactly zero false positives for every frozen diagonal unitary. For any Schur channel with correlation matrix C, the optimal parallel failure probability is min over occupation vectors m of per(C[m])/n!. The same formula is the exact optimum over all causal strategies when the renewed channel has an unknown common phase carrier and failure is evaluated in the worst case over that carrier. Calibration changes this conclusion: for qubit dephasing we give a rational five-query certificate that beats every parallel strategy for 0.5331424052... < eta < 1. At eta=0.7, failure decreases from 0.46603 to 0.4602957928558. We prove physical realizability of the certificate, obtain exact causal limits at complete dephasing, and show that adaptation cannot improve the asymptotic failure exponent. The separation concerns finite-query performance under a calibrated statistical promise; it neither quantifies the hardware cost of a phase reference nor certifies a quantum environment.
Working paper. Citable, but not yet screened for the AIRR collection.
For a complex matrix expanded in the cyclic Weyl basis, we refine the product rank bound using the largest nonzero translate overlap of its support. Combining a fourth-moment estimate, cyclic compression and acyclic block selection, we prove that every nonzero combination of at most seven cyclic Weyl operators in dimension d=2^k has rank at least d/4. For d>=4 this is sharp under every support budget from four through seven, uniformly in k. It completes the conjectured rank envelope in dimension 16 and closes additional gaps in dimensions 27 and 32. Two exact universal erasure budgets follow for arbitrary complex low-rank matrices: seven in a binary family of ranks, and eight for rank at most two in dimension 27; explicit examples fail at the next erasure. A quantitative fourth-moment bound also controls distance from low rank. The proofs use classical scalar uncertainty and graph lemmas. No general prime-power conjecture, density-matrix recovery theorem, efficient recovery algorithm or Erdos problem is claimed solved.
Working paper. Citable, but not yet screened for the AIRR collection.
Whether a dephasing experiment can exclude static Hamiltonian disorder depends on the arithmetic of its measurement times. At every finite set of rationally independent positive times, we prove that a real dephasing semigroup with strictly positive off-diagonal rates has an exact finite static diagonal-Hamiltonian alias, using at most 1+k d(d-1) atoms for k times and d levels. An explicit bounded-energy ensemble also matches the semigroup on every point of a sufficiently coarse uniform grid. Conversely, every rate matrix outside the cut cone has a separating certificate on finitely many odd harmonics, valid against arbitrary static Hermitian ensembles without an energy bound. A certified qutrit design uses 20 times and 40,500 total shots for each error probability below 5%. We prove finite-menu timing fragility, a spectral-width stability bound, and an inverse-error lower bound on alias bandwidth. Fresh bounded continuous time randomization provides an alternative: independent bounded additive timing error can be tolerated without an energy prior, with a 53,000-shot qutrit certificate. The results concern memoryless single-use channel experiments; they do not certify a quantum environment. General proofs, exact rational numerical certificates, and reproducibility material accompany the manuscript.
Working paper. Citable, but not yet screened for the AIRR collection.
Can stationary states at a few reset rates distinguish fluctuating dephasing from an arbitrary distribution of static Hamiltonians? For a qubit, every finite list of stationary states of a real dephasing semigroup has a finite static alias. We exhibit a sharp obstruction for a qutrit. If c_jk(nu) is the real, normalized stationary coherence under Poisson resets and A(nu)=[c_12(nu)+c_23(nu)]/2, every static diagonal Hamiltonian ensemble, without support or moment restrictions, satisfies A(a)+9A(4a)-5c_13(4a) <= 5 for a>0. The completely positive semigroup with generator -gamma[D,[D,·]], D=diag(0,1,2), gives 16/3 at a=2gamma. We prove the inequality by an explicit polynomial certificate, determine all its finite equality points, and show that the exact infimum of the uniform error in the three selected statistics is 1/45. The infimum is approached by seven-point static ensembles but is attained by no probability measure on R^2. A single reset rate cannot exclude static disorder for this semigroup. We also give a deterministic error rule, a finite-shot implementation, and a spectral coupling that extends the exclusion to arbitrary static qutrit Hamiltonian ensembles with noisy population calibration. If all three population leakages at rate a are at most delta, the bound becomes W_a <= 5+30 sqrt(2delta)+30delta. This is a witness against static disorder, not against classical time-dependent noise or a proof of a quantum environment.
Working paper. Citable, but not yet screened for the AIRR collection.
Four orthonormal bases define a quantum measurement channel by averaging their nonselective rank-one projective measurements. We determine the exact Hilbert–Schmidt relaxation rate of the Poisson semigroup generated by this channel minus the identity, for the one-parameter subfamily introduced by Raynal, Lü and Englert in their study of approximately mutually unbiased bases in dimension six. A block decomposition gives the complete spectrum as explicit quadratic polynomials in the parameter. The largest relaxation rate within this subfamily, with equal measurement probabilities and unit total event rate, is exactly 73/100, attained uniquely at u=7/10. This parameter does not maximize the mean squared chordal distance between the bases. An exact rational comparison exhibits a larger mean distance with a smaller relaxation rate. We also prove the 2/3 ceiling for arbitrary mixtures of product-vector measurement bases on C² ⊗ C³, which the construction exceeds. The optimality statement is restricted to the specified subfamily; no global four-basis optimality or new mutually unbiased basis construction is claimed.
Working paper. Citable, but not yet screened for the AIRR collection.
For a Haar-random pure state on $\mathbb{CP}^{d-1}$ with squared-Frobenius distortion on density-matrix reproductions, the exact rate–distortion function $R_d(D)$ is known (ARR-2026-1D2QYXPCVY9H7ANB) as a one-dimensional envelope over a radius $b\in[0,1]$ and a multiplier $\lambda$; first contact was proved there, but uniqueness of the positive branch and absence of later branch exchange were left open (its Remark 3.2). We close that remark for every finite $d\ge3$, in a note written to be read without the earlier record (only its envelope theorem is imported, with the elementary facts it rests on reproved here). The tool is the first-order equation $\kappa M'=(\kappa-d+1)M+(d-1)$ for $M={}_1F_1(1;d;\kappa)$, which reduces the fold numerator $K_d'-\kappa K_d''$ to the sign of $\kappa-\Psi_d(M)$ with $\Psi_d$ an explicit rational function; a transversality argument in the $(M,\kappa)$ plane then shows the stationary multiplier $\lambda(\kappa)$ has exactly one fold. This yields a unique coexistence multiplier $\lambda_{c,d}$, no reentrance, and $R_d=$ a linear face glued at $D_{c,d}$ to an explicit complex-Bingham curve. A computer-assisted theorem gives the one-change law for the fold-numerator coefficients; $d=2$ recovers the Dytso–Cardone sphere result.
Let $p$ be prime and let $W_{(a,b)}=X^aZ^b$ be the cyclic Weyl operators on $\mathbb C^p$. We prove that every nonzero linear combination $C$ of $\ell$ Weyl operators satisfies $\dim\ker C\le\ell-1$, i.e. $\operatorname{rank}C+s_W(C)\ge p+1$, where $s_W$ counts the nonzero Weyl coefficients. The collinear case is Tao's uncertainty principle on $\mathbb Z/p$ (Chebotarev's theorem on Fourier minors). The general case follows from the reduction-modulo-$(1-\omega)$ mechanism of Frenkel and Tao after two further observations: Weyl combinations stay integral under the non-unimodular Clifford conjugations $F'Q_s$, which realise all $p+1$ line directions of $\mathbb F_p^2$, and, in the nonvacuous sparsity range, a pigeonhole over those directions produces a nonzero fibre sum. The bound is sharp for every $\ell\le p$ and fails in every composite dimension. As a self-contained corollary, for one use of the $p^2$ Weyl channels the minimal perfect list length over probes of Schmidt rank $r$ is exactly $p-r+1$. An exact certificate refutes a natural "line conjecture".
We classify the ranks of nonzero operators with at most three terms in a fixed cyclic Weyl basis in every prime-power dimension. A central-sector formula valid in arbitrary cyclic dimension computes trinomial nullity from commuting powers and a common block multiplicity; nonzero three-term coefficients permit at most two singular sectors. The formula explains composite-dimension departures from the prime-power bound. For exact-rank pure probes in one-use Weyl channel list discrimination, we classify all reduced states below rank 3d/4 in dyadic dimensions: exactly thirty flat half-rank states. Positive sums of sparse squares produce ranks impossible for an individual three-term factor, including an exact three-label family of rank 13d/16. We also determine an exact four-label family of rank 5d/8 and the complete feasible three-label rank set in dimension sixteen. Classical Weyl representation and covariance tools are attributed; exact algebraic certificates and replay programs accompany the proofs.
A quantum list measurement succeeds when its output contains the prepared label. Building on the known equivalence between learning width and Gram-matrix factor width, this paper closes an exact realization-sensitive branch. Every full-spark ensemble of N pure-state rays spanning C^r that admits strictly positive tight representatives has minimum zero-error list size N-r+1. Weighted Hodge duals of all (r-1)-fold frame wedges give an explicit attaining POVM, and a physical-space annihilator-cone formulation gives constructive compression to at most r^2 outcomes. A smallest-eigenvalue functional supplies a positive and perturbatively stable Bayes-error floor below threshold. For a flat consecutive-support Schmidt-rank-r probe of the complete d-dimensional Weyl-channel ensemble, the fixed-probe threshold is d-r+1. When r divides d, an arithmetic-support construction and dimension converse close the optimization over all pure Schmidt-rank-r probes at d/r. For the consecutive rank-two probe, the complete one-shot Bayes list curve is determined exactly. The nondivisible probe optimum, adaptive or multiuse testers, asymptotic capacity, arbitrary mixed states, and non-scalable full-spark ensembles remain outside scope.
In quantum list discrimination a measurement returns at most ell candidate labels and succeeds when the true hypothesis belongs to the returned list. We associate a Rado independent-transversal matroid to the support subspaces of a mixed quantum ensemble and prove that every true-label inclusion vector lies in the independence polytope of the ell-fold matroid union. This yields all-subset Bayesian bounds for arbitrary priors, soft rewards, an integer congestion deficit, equality audits, and canonical compression to quantum-process testers. Exact attainment and insufficiency examples separate support combinatorics from quantum geometry. We then solve two input-dependent process families. For binary laminar dephase-prepare channels with M=2^h, list size ell=2^s, and q uses, arbitrary entangled parallel probes and adaptive quantum memories obey P_parallel=min(1,ell(q+1)/M) and P_adaptive=min(1,ell 2^q/M). For complete unitary-error ensembles we translate the known approximate dense-coding spectrum law into a list cap, derive an exact serial/parallel/Bell multitime trichotomy, and give a fixed-probe example where the coarse list-rank cap is not attained. The matroid theorem is a support obstruction rather than a general POVM feasibility characterization; the laminar separation is a classical feedback tradeoff embedded quantumly, and no indefinite-causal-order advantage is claimed.
Minimum-error discrimination of quantum processes is normally optimized over testers whose normalization may encode parallel, sequential, or indefinite-order access. For a fixed physical deterministic normalization, Moore-Penrose compression maps the tester exactly to a POVM on normalized effective states and preserves every conditional probability. We associate to the support subspaces of arbitrary positive process operators a Rado matroid on the hypothesis labels and prove that every correct-label probability vector lies in its independence polytope. Consequently the Bayes success probability is at most the prior weight of a maximum-weight independent transversal. A robust extension replaces exact supports by arbitrary positive low-rank cores and charges only the prior-weighted worst-case discarded tester mass; valid full-rank process admixture of weight eta degrades the certificate by at most eta. We give an explicit reduction to linear matroid intersection, an equality audit at strict prior drops, a deterministic Gram criterion for perfect rank-one discrimination, and exactly solved qubit phase-gate families. In a five-channel instance the exact general-tester optimum is 0.80 while the total-dimension relaxation is 0.90. The result is a support-based upper bound and is not claimed to determine every mixed-process optimum.
The standard k-query dimension bound for an unknown d-dimensional unitary uses the entire degree-k symmetric tensor space and scales as k^(d^2-1). We refine that bound to the degree-k Hilbert function of the projective oracle variety and compute the resulting support exactly in two physically structured families. For a fixed-angle qubit rotation with unknown axis, the repeated-query span has dimension (k+1)^2 on the generic branch, binomial(k+2,2) on the traceless branch, and one on the central branch. A compact-orbit leverage argument inserts this exact support into the general-tester bound, applying to the parallel, sequential, and mathematically admissible indefinite-order strategies covered by the standard tester model. The associated spherical and planar-axis frames admit closed harmonic spectra, purity formulae, endpoint cascades, and a complete tightness classification. For d-level selective-phase oracles, the projective closure is a Segre variety and the exact support is binomial(k+d-1,d-1)^2, reducing the ambient exponent from d^2-1 to 2d-2. These results quantify geometry-dependent query support rather than physical memory or an achievable discrimination probability in every finite ensemble. The work excludes inverse-oracle access, controlled bypasses, noise, finite-sample estimation, and tester models beyond those explicitly stated.
Four balanced Pauli kicks along the vertices of a regular tetrahedron, followed by their reverse echo, generate 24 order-labelled qubit unitary channels. The words share a trace and form two 12-element orbits of the proper tetrahedral group. One query sees only their common trace. At two queries, orbit-dependent quadratic energies yield closed formulas for the Bell-square and globally optimized parallel strategies. At the algebraic point q=1/2, every causally ordered two-query protocol has success at most 9/24 because the fixed trace removes one dimension from the universal quadratic query space. An explicit rational two-comb normalization, a six-Kraus realization, and an exact weighted-frame identity attain the cap, giving P_causal=3/8 and P_parallel=(5+sqrt(15))/24. A uniform Lipschitz estimate proves that the same causal tester remains strictly better on an explicit open pulse interval. The result is an analytic adaptive advantage for a concrete non-group ensemble. It does not address indefinite causal order, noise, finite statistics, or arbitrary qubit ensembles. The central causal certificate is replayed exactly in quotient-ring arithmetic; the supplied word/collision and representation scripts are mixed symbolic-numerical corroborating audits.
