For degree-$n$ polynomials with all zeros on the unit circle, the least normalized residual $\|P'-\lambda P\|_{H^2}/\|P\|_{H^2}$ tends to $1/\sqrt8$, both for arbitrary complex $\lambda$ and for the restricted slope $\lambda=P'(0)$, $P(0)=1$. We prove the sharp lower limit through the classical Mathieu ground state and give an explicit circle-root family attaining it in every large degree. In contrast, exactly one zero strictly inside the disk, with all remaining zeros on the circle, permits restricted residual $O(n^{-1/6})$, giving the exact zero-count threshold for collapse. More generally, for each fixed $A>0$, a fixed number $M_A$ of interior zeros permits $O(n^{-A})$. Their distance and radial gap from 1 have order $n^{-1/3}$; explicit algebraic parameters in the one-root case, and an implicit-function construction in general, correct both degree rounding and the exact slope. For all disk-root polynomials and arbitrary slope, we also prove a uniform lower bound $\exp(-C\sqrt n)$ for every $C>\sqrt{(8/3)\log17}$, explicitly $\exp(-3\sqrt n)$ for $n\ge10^4$. This refines the known residual method for exterior power sums; the negative answer to Erdos Problem 973 was already known. The optimal disk and one-root decay rates, and small exterior power-sum constructions, remain unresolved.
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We determine exactly when the globally asymptotically stable cubic family introduced by Liu and Fitzsimmons admits polynomial Lyapunov certificates. In their normalization, with damping $a>0$ and angular forcing $b\in \mathbb{R}$, the criterion is $|b|^{2/3}<1+a^{2/3}$. The same strict inequality characterizes positive homogeneous polynomial certificates with weak nonpositive Lie derivative, strict polynomial certificates, and local real-analytic certificates. Equality admits none of them. Below the threshold we construct a certificate as a finite product of positive quadratic forms. The proof pairs positive harmonic functions with an explicit analytic square-root test and constructs a complementary limiting harmonic pair at the boundary. Poisson smoothing and equal-weight quantiles then produce a polynomial with a strict margin. At a fixed nonzero boundary point, an explicit sufficient degree is $O_b(\delta^{-3})$, where $\delta$ is the damping excess, while the minimum degree diverges; no matching degree rate is asserted. The threshold persists for every single even angular harmonic and nonzero angular speed. Classical binary-form factorization gives the corresponding sum-of-squares consequence without increasing the Lyapunov degree. The family, its global stability and the original counterexample to a polynomial converse theorem are prior results. Our classification concerns certificates, not a change in dynamical stability.
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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.
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We classify the numbers of interior zeros that permit vanishing and superalgebraic differentiation residuals for polynomials whose zeros lie in the closed unit disk. If a degree-$n$ polynomial has exactly $M$ strictly interior zeros, then, for every complex slope, its normalized $H^2$ residual is at least $n/[2\sqrt216^M(n+2M+4)^{2M+2}]$. Consequently a prescribed budget $b_n$ permits a superalgebraically small residual sequence if and only if $b_n\to\infty$. Sufficiency holds with real coefficients, normalization at the origin, the slope fixed by that normalization, and exactly $b_n$ interior zeros. In contrast, one interior zero already permits residuals $O(n^{-1/6})$, with family-specific asymptotic constant $\sqrt{15}/4^{4/3}$, whereas the circle-only infimum tends to $1/\sqrt8$. For one interior zero we also prove a universal $c n^{-13/6}$ lower bound; the optimal rate remains open. The finite-budget obstruction combines a reciprocal identity with classical Mahler, truncated-shift and Poisson-kernel estimates, and extends to a stable factor times a self-inversive cofactor. For unrestricted interior counts we prove $\exp(-C\sqrt n)$ lower bounds for every $C>\sqrt{(8/3)\log17}$. The upper construction corrects both the exact slope and degree rounding at every large degree. The resulting refinement of exterior power-sum bounds uses a known residual method; the negative answer to Erdos Problem 973 was already known. No matching fixed-budget rates or stretched-exponential upper bounds are claimed.
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We characterize polynomial Lyapunov existence over all degrees for planar homogeneous polynomial fields with nonvanishing angular velocity. A moment threshold separates strict polynomial certificates from the absence of local analytic weak certificates in the stable case. Classical confluent Nevanlinna-Pick interpolation gives a finite convex semialgebraic formulation, with exact boundary decision for algebraic coefficients. For each odd field degree, the certificate region is open and semialgebraic, and compact admissible families have a computable common certificate degree. For rational fields in dense binary encoding, we prove polynomial-time additive approximation of the threshold and polynomial-time decision under a supplied positive gap promise. Explicit inner and outer radii permit a classical weak convex optimization reduction; a uniform functional perturbation argument handles algebraic poles without constructing their common number field. This approximates a value and does not bound the output size of a Lyapunov polynomial polynomially. Jackson bounds and a constructive sufficient-degree estimate complement the finite formulation. An exact cubic example has an algebraic threshold and minimum Lyapunov degree eight at one rational parameter. Beyond the rotating class, a boundary-factor obstruction produces a full twelve-dimensional rational box of stable quintic fields with sign-changing angular velocity and no local analytic weak certificate. One explicit rational C1 Lyapunov function works for the entire box. The latter result is a sufficient obstruction, not a general classification with angular zeros. General certified solver implementations are not supplied.
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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.
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Let F be a finite connected simple graph with at least one edge such that deleting any edge leaves its independence number unchanged. For every integer s at least two, we prove that the clique blowup F[K_s] admits a strictly positive rational rank-two antiferromagnetic target on six states whose graph partition polynomial is not Lorentzian. The proof classifies minimum-induced-edge subsets using equality in Turan's theorem, identifies an exact four-profile block of inertia (2,2,0), and compresses all higher profiles by a scaled generalized Vandermonde matrix. This yields obstructions for connected biregular bipartite bases other than K2 and for connected bases with a unique maximum independent set and at least one edge. Exact computations check 994 base graphs, 720 stable blowups and four positive rational six-state Hessian witnesses. Six is an upper bound; optimality, uniform target parameters and the general connected-source characterization remain open.
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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.
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We give rank-two antiferromagnetic obstructions to the Lorentzian property of graph partition functions. A connected noncomplete source of maximum degree Delta has a strictly positive rational obstruction on Delta+1 spins whenever no two maximum-degree vertices have equal closed neighbourhoods. This covers graphs without true twins, unique maximum-degree vertices and connected noncomplete edge-regular graphs. Conditioned-set Hessian kernels also yield a neighbourhood-independence-gap criterion. One fixed nonnegative four-spin target characterizes cliques among connected diamond-free and complete multipartite sources; a fixed five-spin target does so for joins of a nonempty clique with disjoint nonempty cliques. Positive perturbations may depend on the source. Exact certificates cover all 989 connected noncliques of orders three through seven. An explicit ten-vertex graph shows that the small-parameter criterion is incomplete. The connected-source conjecture and completeness of the full hierarchy remain open.
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For a finite tuple of square complex matrices, let F_infinity be the supremum of the normalized determinant over all matrix amplifications and Hilbert-Schmidt unit coefficient tuples, and let C be the maximum operator capacity over coefficient covariances of trace one. The previously established comparison C/e <= F_infinity <= C has the optimal universal constant: we construct finite tuples with C/F_infinity arbitrarily close to e. The tuples may have exactly identity Hilbert-Schmidt Gram matrix; their capacities tend to one and their amplified determinant values tend to e^(-1). The construction uses independent complex Gaussian coefficients with a fixed number of coefficients before the matrix dimension tends to infinity. A balance property of determinant maximizers makes polynomial moment comparison uniform over every amplification. A separate cut criterion turns Gaussian expansion estimates into a dimension-independent capacity lower bound. For each fixed number of Gaussian coefficients, scalar and arbitrarily amplified optima both converge in probability to e^(-1).
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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.
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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.
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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.
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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.
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For an arbitrary finite tuple of complex square matrices, we maximize the geometric mean of the squared singular values of its linear pencil over auxiliary matrices with unit normalized Hilbert-Schmidt energy and unrestricted size. We prove that this optimum lies between C/e and C, where C is the maximum, over coefficient covariances, of the dimension-normalized operator capacity. No Hermitian or positive-span assumption is imposed on this comparison. An invertible finite evaluation supplies a perturbation inside the original matrix space, subharmonicity selects its phase, and quantitative resolvent comparison, algebraic spectral regularity and Han’s least-singular-value theorem control the entire logarithmic tail. This transfers the Mai–Speicher free determinant formula to feasible finite witnesses. Separately, for Hermitian coefficients whose real span contains the identity, we give an explicit polynomial auxiliary-size bound and a Gaussian sampling guarantee. We do not assert an optimal constant or a general polynomial-time witness algorithm.
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For a matrix tuple, we maximize the geometric mean of the squared singular values of its linear pencil over auxiliary matrices with unit normalized Hilbert-Schmidt energy, allowing arbitrary auxiliary size. We compare this regularized optimum with an optimization of Gurvits capacity over coefficient covariance matrices. The capacity optimization is an upper bound for every tuple. If the coefficients are Hermitian and their real span contains a positive definite matrix, it also bounds the optimum from below up to the universal factor e. The lower bound combines the established Mai-Speicher free determinant formula with finite-dimensional regularization. An imaginary shift is represented inside the original coefficient tuple, and its energy cost is computed exactly. Real covariance suffices for the capacity optimum, giving normal auxiliary witnesses without size doubling. A separate dilation transfers general bounded moment approximants to feasible witnesses. Only convergence of a bounded, regularized logarithm is needed; convergence of unregularized Gaussian log determinants is not asserted. The result gives finite witnesses approaching the bound, without a quantitative dimension estimate. We state separately the unresolved extension to arbitrary matrix tuples and the question of an effective construction.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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This paper establishes a discrete calculus framework acting on the internal structure of formal mathematical expressions. The fundamental premise is that mathematical expressions possess not only numerical semantics but also an internal syntax determined by their construction. Consequently, two expressions that are numerically equivalent are not necessarily identical as differential objects. To capture this distinction, we construct a free term algebra and define a structural derivative $\D$ upon it. Unlike classical calculus, $\D$ does not rely on limits; rather, it is recursively determined by local structural rules on expression trees. We establish four fundamental axioms of structural calculus. The cornerstone of this framework is the \textbf{power node rule}, \[ \boxed{\D(U^V) = U^{V-1} + (U-1)^V,} \] which highlights exponentiation as the true bedrock of the theory by decomposing a power structure into two parallel discrete evolution directions. Under a strict structural normalization principle, trivial multiplications by the identity element are forbidden, while atomic integers are fundamentally embedded as degenerate power structures (e.g., $8 = 8^1$). This leads to a striking departure from classical calculus: the structural derivative of an integer is not zero, but possesses its own non-trivial value (e.g., $\D(8) = 8$). By introducing two commuting discrete shift operators, we express the structural derivative as their sum and derive a binomial expansion for its higher-order iterations. As an application, we obtain an explicit formula for the $n$-th order structural derivative of $x^x$, yielding a novel combinatorial integer sequence. Furthermore, we extend the scope from individual expressions to the complete structural decompositions of integers. We define the \textit{structural spectrum} $\Sigma(n)$ and the \textit{total structural response} $R(n)$ for an integer $n$. This reveals a profound structural dichotomy between primes and composites. Finally, we propose two natural arithmetic conjectures concerning the integrality and critical self-consistency of $R(n)$, opening a new pathway from local structural rules to the global arithmetic properties of integers.
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For a traceless Hermitian $F\in M_d(\mathbb C)$ let $\kappa_d(F)=\tfrac12\min\{\|C\|_{HS}^2:CC^*-C^*C=2F\}$. On the inertia-$(m,n)$ stratum ($m$ positive and $n$ negative eigenvalues, $z$ zeros) we prove three results valid for all $m,n,z$: a dimension-free lower bound $\kappa_d\ge\operatorname{cost}(L)$ for every ordered full-capacity layering $L$, certified by Lidskii–Wielandt and Weyl inequalities alone; an exact additive formula on the aligned-spike cone, where an optimizer is a direct sum of weighted shifts; and the closed form $\kappa_d=\operatorname{cost}(L_k)$ on explicit full-dimensional regions $R_k$, $1\le k\le m-n+1$, obtained from one rank-one interlacing step. Open question 1 of the inertia-$(m,2)$ paper is refuted by an exact weak-duality certificate: at inertia $(5,4)$ a Horn tile with $n=4$ boxes is necessary, and one valid form cannot be certified using Pieri-shaped tiles alone. The $d=8$ rank-onset example is explained by a late negative. Conjectures for general $(m,n)$ are stated with their numerical evidence.
For a traceless Hermitian $F\in M_d(\mathbb C)$ let $\kappa_d(F)=\tfrac12\min\{\|C\|_{HS}^2: CC^*-C^*C=2F\}$. On the inertia-$(m,3)$ stratum ($m$ positive eigenvalues, three negative ones, $z$ zeros) we determine $\kappa_d$ as the maximum of an explicit finite set $S(m,z)$ of linear forms for $m=3,\dots,9$ and $z\le2$, by exact Horn "tiling" certificates below and $3\times3$-block weighted shifts above, the latter verified at every vertex of every chamber together with a convexity argument. The forms are costs of layerings; for the ordered family A the lower bound is proved for all $m$ and $z$ by Lidskii–Wielandt tail inequalities. Unlike inertia $(m,2)$, the value is not padding-invariant: for $m\equiv1,2\pmod3$ exactly one extra exposed form exists at $z=0$, so $\kappa_{m+3}>\kappa_{m+4}$ on an open set within the stated finite range. At an exact $m=4$ example, padding raises the minimum optimal rank from $4$ to $5$ while $\kappa_7=74/61>\kappa_8=73/61$. Six combinatorial families are conjectured to describe $S(m,z\ge1)$ for all $m$, with quadratic growth; they are tested numerically out of sample for $m=10,11$.
For an equal-trace rank-one POVM seed $F$ on $\mathbb{CP}^{d-1}$ the all-degree Born-rigidity gap is $\gamma(F)=\inf_{k\ge 2}g_k(F)$, where $g_k$ is the squared singular value of the orbit-normalisation operator on the projective harmonic sector of degree $k$, and $\Gamma_d$ is its supremum over outcome-simple seeds. A previous record proved $\Gamma_d=d-6/(d+1)$, attained only by an orthonormal basis, for $d\ge 5$, and left $d=3,4$ open. We prove an explicit tail bound $|Q_{n,d}(t)|\le(1-t)^{-(d-2)}/\binom{n+d-2}{n}$ for the normalised zonal function (a two-line corollary of Koornwinder's Laplace-type integral) and use it to settle $d=4$ exactly: $\Gamma_4=22018975/7340032=2.999847\ldots$, attained only by the regular five-outcome simplex, with the minimising degree $k=6$. In $d=3$ we prove $77/45<\Gamma_3\le 9/5$, certify $\Gamma_3\ge 1.776009937964733$ by an explicit five-outcome frame in exact arithmetic, and locate numerically a strict local maximum $1.7760099429996636\ldots$ with five tied degrees $4,8,9,12,16$.
Let $L_q(\kappa)=q!\sum_{j\ge0}\kappa^{2j}/(q+2j)!$, $q=n-2$, be the normalizer of Haar oriented two-planes in $\mathbb R^n$, $m_q=L_q'/L_q$, and $H_q=m_q-\kappa m_q'$ the fold numerator of the stationary multiplier $\lambda_q=\kappa/(2m_q)$. The record ARR-2026-7H9FAPTBZA897AMJ proved that $H_q$ has exactly one positive zero for every $q\ge4$ through a one-change law for the coefficients of $L_q^2H_q$. We give a short independent proof by a different method: in the coordinates $(X,u)=(\kappa m_q,1/L_q)$ the inhomogeneous second-order equation of $L_q$ makes $\kappa H_q$ a polynomial, the transversality of a crossing is a second polynomial $T$, and $Q=T-(q-\tfrac12)\kappa H_q$ obeys a first-order linear equation whose forcing, on the zero set of $H_q$, changes sign exactly where $L_q=q(q-1)/8$. An integrating factor then excludes a zero before that point and a second zero after the first. Probabilistically, $\kappa H_q=4(\mathbb EJ-\operatorname{Var}J)$ for the index law of the series. The same argument gives no fold for $q=2,3$. A now redundant interval-arithmetic certificate and numerics are included.
Let $d=p^k$ and let $W_{(a,b)}=X^aZ^b$ be the cyclic Weyl operators on $\mathbb C^d$. Write $w_p(r)$ for the base-$p$ digit weight $\prod_t(r_t+1)$ and $\Omega_k(m)=\max\{r<p^k:w_p(r)\le m\}$. We give a valuation proof of the classical Delvaux–Van Barel scalar law: (Z) a polynomial with at most $m$ terms and degree below $d$ annihilates at most $\Omega_k(m)$ of the $d$-th roots of unity, and this is sharp. Our operator extension is: (B) every nonzero combination $C$ of $\ell\le2p-1$ Weyl operators satisfies $\dim\ker C\le\Omega_k(\ell)$, i.e. $\operatorname{rank}C\ge p^k-(\ell-1)p^{k-1}$ for $\ell\le p$ (sharp) and $\operatorname{rank}C\ge p^{k-1}$ for $p\le\ell\le2p-1$. The commutative ingredient is the Massey–Costello–Justesen weight-retaining property; the operator ingredient is a Radon-type lemma showing that first-order obstructions have weight at least $2p$. For composite $d$ the maximal nullity is $d/p_{\min}$ at sparsity two and, conditionally on a published trinomial rank rule, $\max(2d/r_3,\,d/2\,[d\text{ even}])$ at sparsity three. The conjecture $\dim\ker C\le\Omega_k(s_W(C))$ for all sparsities is proved for $d\in\{4,8,9\}$ and for torsion cosets; its first open case is $d=16$, $\ell=6$.
For a traceless Hermitian $F$ with $N$ positive, $N$ negative and $d-2N$ zero eigenvalues, the inverse self-commutator cost $\kappa_d(F)$ satisfies $\delta_d\ge\gamma_{N,d}D_*$ with an optimal rational constant $\gamma_{N,d}$; the values $\gamma_2,\dots,\gamma_5$ were known and $\gamma_6,\dots,\gamma_{10}$ only numerically. We give exact, independently machine-checked certificates proving $\gamma_{N,d}=\gamma_N$ for $N=3,\dots,12$ and every $d\ge2N$: $17/36$, $4/7$, $113/152$, $9/10$, $781/740$, $317/264$, $23/17$, $3/2$, $773/468$, $289/160$. The minimising pairs switch at $N=7$ from $(A_N,B)$ to $(u_3,z_N)$. The certificates are exact rational hive primals at every positive-distance vertex of the finite reduction and matching duals at the listed minimisers, and dimension-independent Horn combinations (Weyl, two size-two and Lidskii–Wielandt inequalities) at the minimisers, written out in full. We also prove the $(A_N,B)$ cost law for $N\le14$ and the exact six-level formula $\kappa_6=\max$ of twenty-two explicit linear forms, all exposed. The supplied package reports a separate adversarial replay; fresh checks using separately written code are documented in Section 7, with provenance limits stated explicitly.
For $M={}_1F_1(a;c;\kappa)$, the moment generating function of $X\sim\mathrm{Beta}(a,c-a)$, the stationary multiplier $\lambda(\kappa)=\kappa/(2b(\kappa))$ of the Curie–Weiss-type radial problem $\max_{0\le b\le1}\{K(2\lambda b)-\lambda R^2b^2\}$, $K=\log M-(a/c)\kappa$, folds where $H=K'-\kappa K''$ vanishes. We show that the logarithmic derivative $m=M'/M$ satisfies the Riccati equation $\kappa m'=a-(c-\kappa)m-\kappa m^2$, so that $H=m(1-m)\,(\Psi(m)-\kappa)$ with $\Psi$ an explicit rational function, and a transversality argument in the plane $(m,\kappa)$ proves: $H$ has exactly one positive zero if $c>2a$ (right-skewed overlap law) and none if $c\le2a$. This settles the second open question of ARR-2026-68BH5JDJCQ8XW8GT, gives the complete scalar radial phase and two-piece envelope for every $(a,c)$ with explicit fold bounds for $a\ge1$ (false for $a<1$), covers the Bessel family ${}_0F_1(;c;\kappa^2/4)$ (no fold), and yields "one fold iff $d\ge3$" for rank-one sources over $\mathbb R$, $\mathbb C$, $\mathbb H$. For the oriented-two-plane normalizer, which is third order, we give an exact identity and numerics only.
Inside a Lean 4 formalization programme for four-dimensional SU(Nc) lattice Yang–Mills, this paper studies the operator-theoretic criterion between exponential decay of a connected correlator and a spectral gap. Let T be a bounded self-adjoint operator on a Hilbert space, let Ω be a fixed unit vector, and set S = T − |Ω⟩⟨Ω|. The dense-family criterion states that if the vectors whose S-orbits decay at one common exponential rate r form a dense set, then ‖S‖ ≤ r, even when each vector has its own arbitrary finite prefactor. The decay domain is a linear subspace, so a densely spanning observable family suffices. The corrected manuscript supplies an elementary dyadic proof alongside the historical Lean functional-calculus formalization. The underlying criterion is classical; the contribution is its mechanization and packaging for observable families. The paper constructs no gauge-theory Hilbert space, proves no reflection positivity of the Wilson measure, and identifies no physical correlator with an operator matrix element. It makes no continuum-limit or Clay-problem claim. The release preserves historical formal evidence and distinguishes it from the paper-level additions; no fresh complete Lean build is certified.
For the one-parameter family B(x) = x/(nu+c+sqrt((nu+c)^2+x^2)) of Amos-type expressions, whose member c = 1/2 is the classical Amos-type upper bound for the modified Bessel ratio I_{nu+1}/I_nu, we formalize in Lean 4, at every real order nu >= 0 over the Gamma-power series, the classification of the parameter: B is a uniform upper bound for the ratio exactly when c <= 1/2, and a uniform lower bound for every c >= 1, with explicit rational counterexample witnesses (the classification itself is known mathematics, due to Ruiz-Antolin and Segura; we claim only the machine-checking). The contribution is the regime between the ends: for every nu >= 0 and c strictly between 1/2 and 1 we prove that the fixed family member crosses the ratio exactly once on (0, infinity) -- a transversal crossing in an explicit finite window, strictly above an explicit threshold, with globally determined sign on both sides and a two-sided scale law; degenerate contact is excluded by an exact second-derivative identity. The chain carries the axiom oracle [propext, Classical.choice, Quot.sound] and no analytic hypothesis beyond nu >= 0, x > 0; a pre-registered certified interval-arithmetic companion verifies the crossing phenomenon independently of the crossing theorems at 30 parameter pairs spanning the hard regimes, all passing at 128 bits.
The Amos-type upper bound for the modified Bessel function ratio, rho_nu(x) = I_{nu+1}(x)/I_nu(x) < x/(nu + 1/2 + sqrt((nu+1/2)^2 + x^2)), is classical, and its derivation through the qualitative theory of the associated Riccati equation is an established technique. A companion paper formalized the bound at integer order over a factorial power series. This paper extends the formalization to every real order nu >= 0: the function I_nu is defined by its Gamma-power series (real exponents via rpow), and the complete chain — convergence, positivity, the three-term recurrence, termwise differentiation with a dominated-derivative argument that must treat the leading term separately (its exponent nu-1 is negative for nu < 1), the Riccati equation, a small-argument zone bound uniform in nu, and a first-crossing barrier — is machine-checked in Lean 4 with axiom oracle [propext, Classical.choice, Quot.sound] and no analytic hypothesis beyond nu >= 0, x > 0. Two structural locks tie the result to the integer development: an identification theorem proves that at nu = n the Gamma-series object coincides with the factorial-series object, so the integer-order theorem of the companion development is recovered as a corollary in three rewrites; and a genuinely non-integer instance at nu = 1/2 witnesses that the endpoint lives outside the natural-number embedding. The theorem is proved for the in-core Gamma-series definition; no identification with an external special-functions library object is claimed.
How many internal state coordinates in a specified rational realization are required to transmit prescribed quantum noise channels without loss while suppressing all signal directions elsewhere? This paper synthesizes classical degree and phase-volume mechanisms in a protected signal/loss formulation for finite rational inner networks. For a rational inner scattering matrix of McMillan degree n, exact lossless calibration subspaces at distinct regular boundary frequencies have nodewise total dimension at most n if the signal block is a strict contraction at another boundary frequency. Local nullity forces multiplicity in a nonzero complementary loss minor; exterior powers of minimal Blaschke-Potapov factors bound its zeros by n. The same factorization identifies degree with integrated Wigner-Smith trace delay in the unit-circle/sample-delay coordinate; an explicit rotated Cayley map supplies a continuous-frequency convention and chain-rule delay conversion; a physical oscillator count is conditional on a minimal annihilation-only passive realization. An explicit finite inner family attains the bound. The Rouche extension preserves complex zero counts in disjoint pole-free closed regions for rational inner perturbed transfers holomorphic there and satisfying a strict contour margin for the same loss minor. Sampled approximate calibration alone cannot imply a degree bound. A previously constructed irreducible six-port reservoir filter has 3S exact calibration zeros and degree 3S+6, within six states of the optimum among finite rational inner completions with the same exact calibration data and a strict stop. Its exact integrated trace delay is 3S+6 and its peak grows exponentially. Under a specified stationary weak-coupling Markov model, the rate interpretation counts nodewise preserved input dimensions at distinct frequencies. Numerical degree entries are analytic formula values, not computed minimal realizations; quadrature and root audits test the explicit family. Sources, corrections and dated executions remain distinguished.
For a traceless Hermitian $F\in M_d(\mathbb C)$ let $\kappa_d(F)=\tfrac12\min\{\|C\|_{HS}^2: CC^*-C^*C=2F\}$. We give the exact value of $\kappa_d$ on the whole inertia-$(m,2)$ stratum (positive eigenvalues $a_1\ge\dots\ge a_m>0$, two negative eigenvalues $-b_2\ge -b_1$, any number of zeros) for every $m\ge2$ and every $d$: $\kappa_d=\max(F_0,\dots,F_{m-1},G_1,G_3,\dots)$ for explicit linear forms. The lower bound sums Lidskii–Wielandt and Weyl inequalities with one Pieri-type Horn triple (Littlewood–Richardson coefficient $1$); the upper bound is an explicit $2\times2$-block weighted shift whose feasibility is a one-parameter interval recursion. Both bounds are dimension-free, so the value is invariant under zero padding, and the chamber structure is explicit. Every optimizer has rank $m$ or $m+1$, the minimal optimal rank equals $\max(n_+,n_-)=m$ in every dimension, and rank $m+1$ occurs exactly in one open chamber after padding, where the optimal spectrum is not unique. By $F\mapsto-F$ the same formula covers inertia $(2,n)$.
A passive lossless $N$-port network must send one fixed $k$-dimensional input channel to prescribed output $k$-planes $Y_1,\dots,Y_L$ at $L$ distinct frequencies on the unit circle. We prove that the minimum McMillan degree $d_{\min}$ of a rational inner $N\times N$ matrix performing this task equals $\delta_{\mathrm{Gr}}$, the least degree of a rational curve $\mathbb P^1\to\mathrm{Gr}(k,H)$ through the data, $H=\sum_iY_i$, for every $k$, $N$ and $L$. The lower bound projects the Plücker vector of the routed column onto $\Lambda^kH$. The upper bound shows that the lossless lift $MH_o^{-1}$ of a Forney minimal basis $M$ has exactly $\sum_j\deg m_j$ states, through the backward-shift identity $PK_M=K_{F_0}$, and completes it without new states. The published laws ($k=1$ projective degree, direct-sum occupancy $k(L-n_*)$, generic $k(L-1)$) are special cases. We also prove the additivity $\deg(F_0U)=\deg F_0+\deg U$, a base-point law for arbitrary witness pencils $P$ (the compiled router has $\operatorname{fdeg}(P)$ states plus one per base point in the open disc, so it is minimal iff $\operatorname{fdeg}(P)=\delta_{\mathrm{Gr}}$ and no base point lies in the disc), the generic bound $\delta_{\mathrm{Gr}}\ge k\lfloor L(r-k)/r\rfloor$ and, for every $L$, the generic value $k\lfloor L/2\rfloor$ in $\mathrm{Gr}(k,2k)$, in particular $2\lfloor L/2\rfloor$ in $\mathrm{Gr}(2,4)$.
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".
For rational-inner passive routers with exactly two internal states and three prescribed nodes, we give a complete three-parameter description of achievable trace-delay functions. A global delay cap is equivalent to nonnegativity of one quartic, with an explicit small Gram certificate. We characterize exactly when the winding lower bound two is attained, prove a positive gap elsewhere, and construct a sharper cubic-root upper bound for symmetric acute configurations. An explicit two-port completion realizes each admissible triple. The global acute optimum and general asymmetric minimum remain open. The results assume orthogonal output rays, freely chosen phases and dimensionless angular trace delay.
We determine the optimal forward stability coefficient 113/152 for inverse Hilbert-Schmidt self-commutator cost at balanced inertia (5,5), uniformly for every fixed number of ambient zero eigenvalues. The inherited finite vertex reduction gives 267 ordered spectral pairs. Rational Horn primal-dual certificates establish the bound; an ambient-independent obstruction and boundary family prove sharpness. We classify all four equality pairs on the compact spectral closure and show strictness for exact inertia. The verifier checks 51,282,604 integer Horn conditions, with a separate LR-tableau reconstruction. Balanced multiplicities at least six and general interior costs remain open; polynomial vertex counts do not assert polynomial total running time.
In cyclic Weyl dimension d=3^k, k at least two, we classify every reduced pure-probe state of rank strictly below 7d/9 permitting exact channel discrimination with lists of at most three labels. The low-rank stratum consists of 150 relative interiors indexed by affine lines over the field of three elements, with explicit ranks, spectra, unique positive decompositions and minimum factor counts. Exactly twelve states permit lists of two. We obtain exact-rank list minima at ranks 4d/9 and 5d/9, and show that the structural cutoff 7d/9 is sharp. The result concerns a fixed cyclic Weyl basis and exact zeros; higher-rank and approximate classifications remain open. Exact finite-geometry and matrix certificates accompany the written proofs.
We determine the complete minimax error versus peak angular delay law for rational-inner passive routing of degree at most one at three asymmetric boundary nodes, with one exceptional orthogonal target and two repeated targets. The result supplies both exact branches, the positivity transition, positive lower certificates, and an attaining two-port construction. We then invert the law: a prescribed chordal error determines the minimum delay through one uniquely isolated cubic root with an explicit positivity constraint, or an elementary expression in the active regime. Constant and degree-one branches, the jump at unit peak delay, the transition target and the unattainability of zero error at finite delay are included. Exact rational and symbolic certificates distinguish the continuum proof from supplementary numerical constructions. The result concerns a dimensionless one-state, three-node model and does not claim general circuit-component synthesis or higher-degree optimality.
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.
For every balanced sign multiplicity N and fixed ambient dimension, we reduce the optimal forward stability coefficient for inverse self-commutator cost to a rational list of O(N^5) spectral pairs. This counts Horn linear programs, not their size or total running time. At inertia (4,4), exact rational certificates for 89 ordered vertices establish the optimal coefficient 4/7 in every fixed ambient dimension at least eight. An ambient-independent Horn obstruction and a boundary perturbation prove sharpness. We classify the full equality set on the compact spectral closure and show strictness throughout the exact inertia stratum. Together with the previously sharp reverse coefficient 3/2, this completes both stability constants at multiplicity four. Constants for N at least five remain unevaluated. All finite certificates and replay programs are supplied; classical Horn sufficiency and written polyhedral arguments are explicit dependencies.
We determine the sharp worst inverse self-commutator cost for every prescribed inertia, including arbitrary ambient zeros, and quantify its extremal geometry. For unequal sign multiplicities we obtain both optimal constants relating the normalized cost deficit to distance from the one-spike, opposite-flat boundary. The reverse coefficient is also optimal for every balanced inertia. For three positive and three negative eigenvalues we close the other direction as well: the optimal forward coefficient is 17/36. Its proof reduces a piecewise-affine distance to 22 rational vertices and supplies exact Horn witnesses; a boundary spectrum gives ambient-independent sharpness. Thus every nearly extremal sequence is classified. A transportation refinement gives computable primal-dual upper certificates, while an exact example shows a 12.5% gap from the actual matrix cost. A sharp deficit threshold also reduces rank-ceiling stability to the known one-spike case. Classical Horn sufficiency is imported; all finite certificates are supplied for replay. The optimal balanced forward coefficient for multiplicity at least four and the general interior cost remain open here.
For F_(p,q)=diag(q repeated p times, -p repeated q times), this paper studies the least rank among factors minimizing one half of the squared Hilbert--Schmidt norm subject to CC^*-C^*C=2F. On q=2p+1, exact rational hive primal-dual certificates determine the complete finite cost-and-rank frontier for 4<=p<=32: the rank excess above q is 0 on p=4..7, 1 on p=8..14, 2 on p=15..26, 3 at p=27, and 4 on p=28..32. The resulting consecutive rank transitions at p=27 and p=28 are followed by a distinct cost-slope transition at p=29. Separately, exact certificates prove kappa(F_(53,107))=8847 with minimum attaining rank 115; its optimum is half-integral, while an earlier integer candidate of trace 8843 is refuted by an integral Farkas certificate. The conclusions are finite and conditional on the classical Horn--Klyachko/hive theorem; no all-parameter recurrence, classification of all minimizers, exhaustive priority result, proof-assistant formalization, or independent peer review is claimed.
For a traceless Hermitian matrix F in M_d(C), define kappa_d(F) = (1/2) min{||C||_HS^2 : CC^* - C^*C = 2F} and let r_*(F) be the least rank among its minimizers. Inertia gives the universal lower bound r_0(F) = max{n_+(F), n_-(F)}. We prove the exact low-dimensional classification r_*(F) = r_0(F) for d <= 7, including every singular spectrum. The bound fails sharply in dimension eight. For the entire explicit two-ray cone lambda(b,c) = (4c-3b,b,b,b,-c,-c,-c,-c), 0 <= b <= c, we determine the unrestricted and rank-four costs: their difference is min{b,c-b}, and every interior point has r_* = 5 > 4 = r_0. In particular, the integer spectrum (5,1,1,1,-2,-2,-2,-2) has unrestricted cost 13 and rank-four cost 14. Thus eight is the least dimension in which Hilbert--Schmidt norm optimality can force factor rank strictly above inertia. The proof is an exact computer-assisted Horn-polyhedral argument. Thirty-three sign--zero strata through dimension seven yield 813 exact face rays and 272 projected facets; every facet has a rational nonnegative Farkas certificate on the unrestricted cone. A separate exact replay verifies the dimension-eight phase and a two-valued dimension-nine witness. No floating point output is used as a proof object. Finally, for the target G_t with spectrum (5^(12t),-4^(15t)) in dimension 27t, we prove kappa_27t(G_t) = 87t and 17t+1 <= r_*(G_t) <= 18t. Since r_0(G_t) = 15t, the additive excess is at least 2t+1 and is therefore unbounded. The amplification is symbolic: a coarse-grid restriction of any fine hive transfers a frozen order-27 rank-17 lower certificate to every t by telescoping elementary rhombi.
