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.
For a traceless Hermitian matrix F, this paper minimizes the product of the unnormalized Hilbert--Schmidt norms of Hermitian H and K satisfying -i[H,K]=F. On the complete one-spike spectral cone with nonzero spectrum (P,-b_1,...,-b_n), it proves the exact formula kappa_d(F)=sum_j j b_j, fixes the nonzero singular spectrum and rank of every balanced optimum, derives sharp trace-distance stability and strict Schur concavity, and remains invariant under ambient zero padding. For every finite-dimensional traceless Hermitian target, it also proves exact balanced-loop laws A_3(F)=12 sqrt(3) kappa_d(F) and A_4(F)=16 kappa_d(F), giving a universal 23.02 percent fourth-kick reduction. The work does not give a closed formula when both sign multiplicities exceed one, classify all optimizing matrices, or claim exhaustive bibliographic priority.
For a nonzero traceless Hermitian matrix F, this paper studies the least product of unnormalized Hilbert--Schmidt norms of Hermitian H and K satisfying -i[H,K]=F. Writing rho=rank(F) and P(F)=||F||_1/2, it proves the sharp dimension-free bounds P(F)<=kappa_d(F)<=rho P(F)/2. Both equality loci are classified: the lower endpoint is attained exactly by centrally paired nonzero spectra, while the upper endpoint is attained exactly, up to positive scale and sign, by the one-spike spectrum (rho-1,-1,...,-1), with arbitrary zero padding. An exact sign-cut leakage identity identifies the full tax above the trace-norm floor. The upper bound uses an exact weighted-shift permutation average, and rho-1 explicit Horn--Littlewood--Richardson inequalities certify upper-endpoint sharpness. The work does not claim a closed formula for general interior spectra or an upper-endpoint stability modulus.
A sigmoid neuron induces two distinct spectral geometries: a population radial-tangential anisotropy controlled locally by the Schwarzian derivative, and an empirical extreme-saturation hierarchy controlled by samples nearest the transition hyperplane. For logistic powers h_p=sigma'^p, p>0, we prove exact saturation constants, sharp pointwise bilateral oracle-Gram complexity, and regularly varying spherical radius phases. We then make the empirical hierarchy quantitative. A deterministic exterior-power theorem resolves every leading eigenspace; an exact inverse-Gaussian angle law and a finite hierarchy perturbation lemma yield an explicit joint finite-radius, finite-sample guarantee for the bottom eigenspace and teacher direction. Finally, a two-dimensional tangent Gaussian process proves that uniform control over a full angular shell intrinsically incurs a square-root log-radius factor in the iterated high-probability limit. Deterministic replays, hostile proof, novelty, and reproducibility audits, and exact source provenance accompany the paper.
Given d >= 1, 1 <= s <= m, N = binom(d+m,d), and any N isolated complex hypersurface germs g_i of order at least s+1, we construct, for every D >= 1 + sum_i(tau(g_i)+2), a smooth integral degree-D hypersurface with a tangent hyperplane whose reduced Gauss fibre has exactly N points, is point-span order-s osculating-absorbing, and has the prescribed completed Tjurina algebras. The fibre length is sum_i tau(g_i), and the dual multiplicity is sum_i mu(g_i). For surfaces, the classical complete Tjurina spectrum of ordinary plane multiple points yields every integer fibre length between binom(m+2,2) floor((3s^2+4s-3)/4) and binom(m+2,2)s^2 at fixed dual multiplicity. We also give a direct strong-Lefschetz proof that x^(s+1)+y^(s+1)+z^(s+1)+(x+y+z)^(s+2) attains Wahl's classical three-variable value s(s+2)(2s-1)/3, producing sharp absorbed threefold fibres. All claims are over C; degree bounds are sufficient, not asserted minimal, and no exhaustive priority or human peer-review claim is made.
Let h in C[[x_1,...,x_d]] have order at least s+1 and finite Tjurina algebra. Euler cancellation replaces h, without changing its Tjurina ideal, by a generator of order at least s+2. A finite-jet generator count then gives tau(h) at least E_{d,s}, the maximum over K of binom(d+K,d)-d binom(d+K-s,d)-binom(d+K-s-2,d). For plane germs this is the exact universal floor floor((3s^2+4s-3)/4), attained by classical ordinary multiple-point families. Applied to a point-span s-osculating-absorbing reduced fibre of the Gauss map in the complete O_X(m) embedding, the result gives length(Gamma_eta) at least E_{d,s} binom(d+m,d). For surfaces, a prescribed-jet construction realizes equality at an explicit sufficient degree. The work is over C; the higher-dimensional floor need not be sharp, the degree threshold is not claimed minimal, and no exhaustive priority, human peer-review, or formal-verification claim is made.
Let X be a smooth non-linear complex hypersurface and let Gamma_eta be the scheme-theoretic fibre of its Gauss normalization over a tangent hyperplane. The completed local fibre algebra is the Tjurina algebra of the tangent-section germ, while the classical dual multiplicity is the sum of its Milnor numbers; hence their difference is the total Tjurina–Milnor defect. Under point-span order-s osculating absorption in the complete O_X(m) embedding, the fibre contains the order-s fat point at every reduced support and has length at least binom(d+s-1,d) binom(d+m,d). When d(s-1)>s+1, prescribed jets realize every integral defect from 0 through the extremal support size while keeping all Milnor numbers fixed. The degree threshold is sufficient, not minimal; no positive-characteristic extension, exhaustive priority claim, human peer review, or formal verification is claimed.
Let X be a smooth complex hypersurface and let Z be the complete reduced fibre of its Gauss map over a tangent hyperplane W. In the complete O_X(m) embedding, assume that the point span of Z contains the order-s osculating space at every support, with 1 <= s <= m. We prove the sharp floor mult_W(X^vee) >= s^d |Z| >= s^d binom(d+m,d). Absorption forces order-(s+1) contact; the classical multiplicity-Milnor formula and a local Milnor lower bound give the dual multiplicity estimate, while exact tangent absorption gives the branch floor. For every d,m,s we construct equality examples with proper point span, exact reduced Gauss fibre, ordinary section singularities, and the stated weighted tangent-cone cycle. The construction gives a sufficient nonminimal hypersurface degree. No equality classification, positive-characteristic extension, exhaustive priority claim, human peer review, or formal verification is claimed.
Let X be a smooth projective integral d-fold over an algebraically closed field, let H be very ample, and use the complete embedding defined by H^m. Fix 1 <= s <= m and suppose that the span of a nonempty finite reduced set Z contains the order-s affine osculating space at every support. We prove the exact characteristic-free floor dim(S_Z), |Z| >= binom(d+m,d), by reduction to exact tangent absorption. For every d, m, s and every characteristic, we construct a smooth integral hypersurface with a proper-span equality set whose normal coordinate vanishes to order s+1 at every support. If r_1(Z) is the degree-one evaluation rank and m >= 2s+1, we also prove the rank-sensitive term binom(d+s,d) r_1(Z); rational normal curves show the threshold is necessary for that uniform formula. The earlier mixed-jet certificate is retained for jet-ample polarizations but identified as numerically subordinate for tensor powers. No equality classification, minimal hypersurface degree, exhaustive priority claim, human peer review, or formal verification is claimed.
Let X be a smooth projective integral d-fold over an algebraically closed field of arbitrary characteristic, let L be very ample, let m be at least one, and let Z be a nonempty finite reduced set whose point span in the complete L^m-embedding contains every tangent space at its supports. We prove the exact codimension-free floor h_Z(m), |Z| >= binom(d+m,d). An adapted finite projection to P^d transfers absorption to equality between the degree-m pieces of the radical ideal of the projected points and its symbolic square. The standard perfect-field derivative/p-th-root gap forces that piece to vanish. Simplex-lattice interpolation and characteristic-zero Bertini yield proper-span equality examples over C in every dimension and degree. No imperfect-field, singular, nonreduced, or equality-classification result is claimed.
Let X be a smooth projective integral d-fold over an algebraically closed field and use the complete embedding defined by H^m, m at least 3. For a nonempty finite reduced point-span tangent-absorbing set Z, this paper proves in arbitrary characteristic the lower bound dim(S_Z), |Z| at least max{J(d,m), min{(d+1)(d+2), binom(d+m,d)}}. The positive-characteristic Gauss-branch argument replaces the characteristic-zero derivative contradiction by Frobenius and radicality. The paper also proves injectivity of the original Gauss map for suitable complete factorized polarizations and, over the complex numbers, constructs smooth hypersurfaces attaining the binomial estimate inside the exceptional Gauss branch for every d at least 2 and m at least 3. The construction is existential; no positive-characteristic Bertini realization, equality classification, human peer review, or exhaustive priority claim is made. A related ARR paper proves a sharper exact global binomial floor in characteristic zero.
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.
Let G be a compact connected Lie group with a bi-invariant metric. For the two-dimensional heat-kernel Yang--Mills model, this paper gives an explicit finite chain from a boundary-conditioned edge integral on an arbitrary regular annular cellulation to the gauge-invariant class-function transfer kernel. Conditional Haar coordinates make the boundary constraints and boundary-edge subdivision precise. Edge and face subdivision invariance, an explicit PL radial-cut construction, and a primal-tree/dual-cotree disk-elimination schedule yield a cellulation-independent orbital heat-kernel integral. Peter--Weyl theory then diagonalizes the transfer operator with multipliers exp(-t c_lambda). The exact mean-zero norm is exp(-t c_*(G)), the gap of I-T_t is 1-exp(-t c_*(G)), and the equality sector consists of all irreducible characters with minimum positive Casimir. In the stated SU(2) normalization the sharp exponent is 3/4, while the induced SO(3) quotient removes the fundamental channel and raises it to 2. The work assembles and audits classical two-dimensional Yang--Mills ingredients; it does not claim a new character solution, four-dimensional mass gap, continuum reconstruction, or clustering for arbitrary bulk observables.
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.
Let P be a Haar-distributed complex Grassmann projector and consider the matrix--Bingham normalizer Z_A(s)=E exp{s tr(AP)} on the trace-zero, Frobenius-unit external sphere. We prove exact complementary results for its canonical two-level branch geometry. On every half Grassmannian Gr_C(r,2r), the balanced rank-r field strictly maximizes the normalized two-block moments of degrees 4, 6, 8, and 10. Exact endpoint analysis gives balanced dominance at small and large field, while a finite certification theorem reduces coefficientwise all-field order at each fixed rank to a terminating block of rational Hankel signs plus an analytic tail. At Gr_C(3,6), a closed all-degree argument proves strict balanced order for every field. At Gr_C(4,8), exact rational arithmetic through degree 268 and the analytic tail give a second all-field theorem. On Gr_C(2,5), compact overlap densities, Sturm arithmetic over Q(sqrt(6)), and strict total positivity prove a unique simple exchange in the complete oriented two-level family. The results do not classify arbitrary multi-level external spectra and do not claim an unrestricted all-distortion rate--distortion function.
Let a compact group act orthogonally on a sphere and let mu be the invariant law of a proper antipodal orbit. We study the exponential orbital potential. At a point of the source orbit, every spherical-harmonic coefficient of the orbit average is a squared norm, giving an exact negative spherical-Laplacian identity at every nonzero field. Provided the source orbit is critical--automatically so when the normal isotropy representation has no fixed vector--irreducibility, or a transitive symmetry of its irreducible normal blocks, upgrades this trace identity to strict Morse--Bott maximality for every field. An explicit torus-orbit counterexample shows why criticality cannot be omitted. Applied to centered projector embeddings of the real, complex, and quaternionic half-Grassmannians, the theorem proves all-field local rigidity of the balanced matrix--Bingham branch and excludes a radial spinodal. In the complex case, exact Jacobi-generator identities yield constrained-Hessian operators for every two-block external spectrum and a sharp weak-field metastability threshold at k=2r/3. A Stein inequality closes every sufficiently high Taylor degree at each fixed multiplicity. The results constrain unresolved intermediate phases but do not claim global spectral optimality, a complete finite-field phase diagram, or an all-distortion rate--distortion function.
We give a finite-sample test for whether one joint Gaussian sketch covariance is compatible with a positive-semidefinite matrix-valued measure supported on a prescribed interval. The observed vector contains several exponent blocks, so its population covariance is one block-Hankel Gram matrix rather than a list of unrelated moment estimates. The classical truncated matricial Hausdorff theorem supplies the exact parity-dependent null. A single Gaussian singular-value event gives a simultaneous Loewner band for the whole covariance; intersecting that band with the structured moment cone yields a nonasymptotic level-alpha semidefinite test without sample splitting. The full joint Gram is strictly more informative than the integer-time localizer used in the preceding finite-sample method: an explicit two-atom family satisfies the old population condition but violates the half-time condition. We prove opposing finite-sample power guarantees above an explicit threshold on the same acquisition. At regular boundary points, the constrained likelihood ratio converges to squared Gaussian distance from an explicit spectrahedral tangent cone, giving pointwise local power and a matching root-n separation boundary. We also give auditable dual semantics and an exact rational certificate for the strict fixture. Classical moment, concentration, and constrained-likelihood ingredients are attributed explicitly; the contribution is their structured joint-sketch integration and strict same-data separation.
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.
Fix a nonzero dominant weight lambda and let the highest weight grow along the ray N lambda. We study the exact classical Shannon rate-distortion function of the invariant phase-lifted coherent orbit in V_{N lambda} under squared ambient Hilbert-space loss. The finite-N scalar formula is known; the new problem is the coupled large-weight and distortion limit. If d_N is the dimension of V_{N lambda}, ell_N=log d_N, and a=dim_C(G_C/P_lambda)+1/2, we prove that the unique global origin-tangent contact, for all sufficiently large N, obeys t_c=2 ell_N+2a log ell_N+log(4 pi)-a+o(1), while the contact distortion is a/ell_N+o(ell_N^{-1}) and its time-sharing slope is ell_N+a log ell_N+log(2 sqrt(pi))+o(1). All root-system and Weyl leading constants cancel in these dimension variables. For fixed 0<D<=1, the exact unrestricted RDF satisfies R_N(D)/ell_N -> 1-D. On the noncommuting boundary scale D=x/ell_N, the centered rate converges locally uniformly to an explicit two-branch profile, with a curved high-fidelity branch tangent to a linear coexistence face at x=a. We also identify the soft-activation window and an exact fixed-radius Legendre expansion. The result is classical rate-distortion for an embedded coherent-amplitude source, not quantum rate-distortion or a derivation of Born's rule.
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.
Let V_lambda be a finite-dimensional irreducible unitary representation of a compact connected semisimple group, and let X be uniform on the phase-lifted orbit of a highest-weight vector. We determine the classical Shannon rate–distortion function under squared ambient loss for arbitrary standard-Borel descriptions and decoders unrestricted to the orbit. Sugita's known integer-moment coherent-state extremum, combined with positive phase–Bessel resummation, yields an all-field fixed-radius Laplace order and exact equality classification. The vector problem reduces to an exact scalar radial envelope. Covariant tilted channels attain exposed points and revealed binary flags attain nonexposed chords. A representation-dimensional criterion forces discontinuous directional-information onset, while Weyl dimensions give a sharp root-system high-fidelity constant. For exterior powers, the source is a phased fermionic Slater determinant: the normalizer is hypergeometric, squared Pluecker overlap is a beta product, and every interior family with n at least 6 has discontinuous first activation. A distinct projective corollary treats intrinsic Pluecker distortion with reproduction restricted to the coherent orbit. The results concern classical phase-sensitive amplitude reconstruction, not quantum rate–distortion, click-only Born statistics, or particle dynamics.
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.
Let W = x wedge y be the signed unit Pluecker coordinate of a Haar oriented two-plane in R^n, and let reproduction range over the full posterior-mean body under squared ambient loss. An all-field extremum theorem reduces its unrestricted Shannon rate--distortion function to phi_lambda(b) = K_q(2 lambda b) - lambda b^2, where q = n-2 and K_q = log(q! sum_{j>=0} kappa^(2j)/(q+2j)!). We solve the remaining global phase problem in every dimension. For J_q = L_q L_q' - kappa L_q L_q'' + kappa(L_q')^2, its coefficient of kappa^(2r-1) is a positive factor times r - E[d^2], where d follows a central parity-truncated binomial law. Likelihood-ratio ordering and an exact tail estimate prove one coefficient sign change for every q >= 4. Consequently the stationary curve has exactly one fold, the zero and positive phases have exactly one coexistence contact, and no later exchange or radial reentrance is possible. The RDF is therefore an explicit linear face followed by one covariant parametric branch for every n >= 6; together with the continuous cases n = 3, 4, 5, this completes the all-dimensional phase diagram. The unique contact has an exact large-dimension limit and an explicit logarithmic finite-size displacement. We additionally prove that the same RDF is the minimum worst-source channel capacity, that arbitrary joint classical memory costs exactly N R_n(D) on blocks of length N, and that transitivity makes the tilted information constant, giving a zero-dispersion strong converse. The results concern signed Pluecker-coordinate compression, not quantum rate--distortion or Born-probability prediction.
At distinct boundary frequencies, let a square rational-inner multiport route one fixed input ray to prescribed output rays. We prove that the minimum McMillan degree equals the least degree of a base-point-free projective curve through the ordered target rays, independently of ambient port count. For orthogonal target bands we solve line-to-subspace incidence constraints by full-support Lagrange kernels and derive a closed generic codimension law. Exact memory is the maximum shifted band cost, whereas border memory is the minimum and may be arbitrarily smaller. An intrinsic incidence matrix gives a calibrated singular-value error floor, a determinantal zero-error closure, and an all-data base-point deletion law with an incremental exact certificate. Finally, every zero-error sequence below exact memory has divergent peak dimensionless Wigner-Smith delay; any finite delay cap restores compactness, a positive attained error, and a three-phase operational classification. Explicit planar strata and exact rational certificates audit the results.
Suzuki's localized Weil form is represented by a lower-bounded self-adjoint operator A_a on L2(-a,a); nonnegativity for every a > 0 is equivalent to the Riemann hypothesis, while positivity at one fixed support is an unconditional and strictly weaker problem. We give a source-level interval certificate proving A_0.72 >= 5.890 x 10^-17 I > 0. Nested-core monotonicity gives the same scalar lower bound for every 0 < a <= 0.72. The proof decomposes the exact prime-power translation graph through n = 4 into thirteen intervals and bounds its infinite complement by a mode-sensitive Schur estimate. It isolates degrees 12 through 23 before controlling [24,176) and [176,infinity); both parity Schur matrices have 78 certified positive directions and no unresolved direction at 512-bit Arb precision. We also prove an exact Gauss-Stieltjes hierarchy for the logarithmic boundary potential, complete for strict positivity at every fixed support, and a multiband Loewner majorant requiring only O_epsilon(log log M) bands through degree M. The revised manuscript prints a formula-level source-to-Gram specification. A separately written program importing neither project modules nor python-flint reconstructs the prime-power graph and parity maps and independently reassembles the exported Schur balls with positive Weyl margins in both sectors. This is a bounded-support theorem, not a proof of the Riemann hypothesis.
We solve a finite-dimensional external-spectrum optimization problem for the complex matrix–Bingham law and use it to determine an unrestricted Shannon rate–distortion function. If P is Haar on Gr_C(2,4) and A is traceless Hermitian with fixed Frobenius norm, then E exp[s tr(AP)] is, for every s>0, uniquely maximized, up to unitary conjugacy, by the balanced spectrum (R/2,R/2,-R/2,-R/2). The exceptional model Gr_C(2,4)=Gr_2^+(R^6) turns the orbital integral into a positive series whose coefficients share one simplex-vertex maximizer. For the Haar rank-two state source rho_P=P/2, conditional least squares places arbitrary reports in the full body {sigma: 0<=sigma<=I/2, tr sigma=1}. We obtain its exact classical rate–distortion function for every 0<=D<=1/4. Beyond a scalar dual representation, we prove the complete radial phase diagram: one fold, one positive coexistence contact, no reentrance, and an exact two-piece frontier consisting of one time-sharing segment and one matrix–Bingham branch. The proof reduces the fold derivative to a power series with exactly one negative coefficient followed by strictly positive coefficients. Covariant channels attain the frontier, and the same value is the source-universal worst-state capacity; arbitrary joint n-block memories cost exactly n times the one-letter frontier. The result concerns classical memory for calibrated Born-probability prediction, not quantum rate–distortion, click simulation, or a derivation of Born's rule.
Finite Euclidean correlation matrices are routinely converted into spectral gaps by generalized eigenvalue, Prony, or Lanczos procedures. We determine a finite-sample boundary for what such data can falsify and what they cannot certify by non-rejection. Block Hankel pencils give monotone lower bounds on the visible transfer edge, while exact rank stabilization recovers the complete visible spectrum. An atom of weight w at any x*=exp(-g) changes every moment by at most w while forcing gap at most g; arbitrarily small strictly positive gaps therefore remain hidden even when an extra zero mode is forbidden. For a fixed nonsingular Gaussian experiment we compute the exact hidden-atom likelihood and derive its LAN information, sharp local power envelope, and powerless/local/consistent phase at the w sqrt(n) scale. Under a disclosed visibility floor, a fourth-kind Chebyshev filter solves the weighted localizer minimax exactly; a two-atom measure attains the bound, making the uniform sign threshold necessary and sufficient in the declared polynomial class. This margin feeds an exact-level two-coordinate Wishart test and a closed depth–sample resource law. A distribution-free companion uses paired differences and the exact quadratic range to handle bounded iid readouts with unknown mean and same-sample selection over a finite filter bank. A dependence-robust extension treats one stationary bounded beta-mixing trajectory: sparse pairing, Berbee coupling, and an explicit lag-covariance correction give finite-sample level, power, and total chain-horizon bounds under an externally certified mixing envelope. An interacting ANNNI-chain experiment through 2^16 states demonstrates exact symmetry blindness and multichannel recovery. Rejection falsifies an overstated gap; non-rejection alone is not a positive gap certificate.
We give a single exact state-counting theory for finite passive spectral routing tables. Fix distinct boundary frequencies, one input k-plane, and a word whose symbol a requests a target k-plane Y_a exactly n_a times. If the used targets are in direct-sum position—they need not be orthogonal and may have arbitrarily small principal angles—then the minimum McMillan degree among square finite rational-inner interpolants is k(L - min_a n_a). Thus the rarest target fixes generic passive memory, independently of node spacing and word order. The lower bound uses block-dual detectors: direct-sum geometry supplies a constant compression that is invertible on one target and annihilates every other target. Its determinant has k forced zeros at every wrong node, while a Blaschke–Potapov minor has no more zeros than the network has states. The upper bound is a matrix spectral compiler. Target-weighted node polynomials form a full-rank polynomial column; matrix Fejér–Riesz factorization normalizes it to a rational-inner column of degree at most the lower bound, and a degree-preserving lossless completion closes the network. Beyond the direct-sum locus we prove a detector-rank hierarchy, a maximum weighted hyperplane-occupancy bound for lines, and the complete three-line phase diagram. We also give a span sandwich, open-dense genericity, fail-closed noisy certification, an exact collision discontinuity, and the optimal integrated Wigner–Smith delay. Producer and independent verifiers audit nonorthogonal scalar and block tables, collision openings, and singular-incidence fixtures.
We give a unified exact rate–distortion theory for retaining the complete rank-one Born-probability field of a Haar-random pure state in complex projective space. A constrained root count, a sharp centered power-sum theorem, and Newton's expansion prove an all-degree spectral result: at fixed traceless Frobenius norm the projective Laplace transform is maximized by a one-positive-spike spectrum. This reduces the unrestricted Shannon rate–distortion function to an exact scalar complex-Bingham envelope in every dimension, with covariant channels and independently flagged coexistence mixtures attaining every distortion. For dimension at least three, directional information turns on discontinuously; the qubit curve and the all-dimensional high-fidelity constant are explicit. We then solve the full fixed-normalized-distortion limit. If y_* > 2 solves y_* - 1 = 2 log y_*, the limiting free energy has a unique coexistence point, and the information cost per dimension is a closed two-piece function with a nontrivial linear face. The finite-dimensional onset multiplier equals alpha_* d minus [alpha_*/(2 log y_*)] log d up to an O(1) remainder. Finally, a pointwise Hilbert projection converts every measurable scalar reporter into a physical density-matrix reporter without increasing risk for any pure input. Hence the exact worst-state channel capacity equals the Haar rate–distortion function, arbitrary joint n-state memories cost exactly n times the one-state frontier, and transitivity forces zero rate dispersion with an exponential finite-blocklength strong converse. These results concern reusable calibrated probability fields, not click-only simulation, state update, or intrinsic randomness.
Let P be Haar on the complex Grassmannian Gr(r,d) and set rho_P=P/r. We solve two coupled finite-dimensional problems. First, on the sphere of traceless Hermitian external fields with fixed Frobenius norm, we determine incompatible weak- and strong-field optimizers of the matrix-Bingham free energy log E exp{s tr(AP)}. For 1<r<d/2, an exact cubic Haar moment and a uniform C2 perturbation theorem make the positive one-spike spectrum uniquely optimal at weak field. A parameter-uniform differentiated Grassmann Laplace expansion upgrades the strong-field Ky Fan limit to exact eventual uniqueness of the rank-r two-block spectrum. The globally optimized free energy must therefore fail to be real analytic at a finite field; at its first exit from the one-spike branch there is either nonconjugate coexistence or transverse-Hessian degeneracy. The balanced case r=d/2 is selected by a negative quartic coefficient instead. Second, we determine the unrestricted classical Shannon rate–distortion function for squared-Frobenius reconstruction of rho_P on a nonempty open interval adjacent to zero distortion. Arbitrary standard-Borel classical memories and arbitrary reports reduce by conditional least squares to the full posterior-density body, not merely to the source Grassmannian. Exact eventual two-block optimality and radial localization reduce the complete Shannon dual to one variable. The frontier is attained by a covariant matrix-Bingham channel whose posterior mean is a depolarized rank-r projector. A Jacobi–Selberg calculation gives the exact high-rate constant, and complementation transfers the result to every nontrivial rank. The complete intermediate-phase classification and the full all-distortion rank-r curve remain open.
At L boundary frequencies, a passive lossless network must route one fixed k-dimensional input channel to prescribed output k-planes. For targets with joint span dimension r, every square finite rational-inner interpolant satisfies the sharp universal sandwich r-k <= d_min <= k(L-1). Direct-sum targets therefore have exact minimum McMillan degree k(L-1); when N >= Lk this maximal-memory locus is open and dense. The paper gives an explicit positive Pick completion, a realization- and inertia-based lower bound, a family of targets that collide while retaining maximal exact degree for every nonzero opening, an exact quadratic certification margin, and a fail-closed noisy projector test. For three scalar target lines it also proves the complete phase diagram: degree zero for one common line, degree one exactly for three distinct coplanar lines, and degree two otherwise. Deterministic certificates and an independent implementation accompany the proofs.
Four balanced spin kicks along the vertices of a regular tetrahedron generatean order-dependent eight-pulse echo. We prove that all 24 echo orders have oneconjugacy class in SU(2). Haar randomization of only the global orientationtherefore gives an order-law-independent, self-adjoint random-unitary channelon every spin-j matrix algebra. Its exact eigenvalues are normalized SU(2)characters, lambda_l=U_{2l}(q)/(2l+1), with multiplicity 2l+1 and an explicitfinite-pulse architecture polynomial q=1-A/2. The maximal link-reflection-positive interval connected to zero is alpha<=pi/(4j+1), with an exact endpointequation and a finite OS Hamiltonian under strict inequality. In the doublescaling N=4j+1, y=xi/sqrt(N), the full tower converges uniformly tosinc(8xi^2u). This gives the sharp full-tower boundary xi_c=sqrt(pi/8), acritical boundary law N lambda_top -> 3pi/8-2s, an explicit scaled mass profile,and an ultraviolet heat-trace integral. The fixed-rank limit recovers the fuzzy-sphere Laplacian. Poissonization yields a genuine CPTP semigroup and an exactmany-body connected nuclear profile exp(3e^-c)-1. Symbolic and deterministicmatrix certificates accompany the manuscript.The Haar average also has explicit exact finite implementations. A classicalpositive Gauss--Legendre/trapezoidal cubature at bandlimit 4j needs(4j+1)(2j+1) order-conditioned frames, or (4j+1)^2(2j+1) frames when the sameframe list must be order oblivious.The common-class proof reduces the 24 orders to the even and odd A4 orbits.For finite implementations, a separate stability bound controls l1 weighterror, orientation displacement, and pulse perturbations; total variationagainst Haar is not used for discrete frames.
A passive lossless network may route the same k-dimensional input channel to different output subspaces at different frequencies. Every two-frequency restriction can be cheap while the joint task is not. For L distinct boundary frequencies and mutually orthogonal prescribed output k-planes, we prove that the minimum McMillan degree among square finite rational-inner transfer matrices regular at the interpolation nodes is exactly k(L-1). Each pair alone requires degree k, so the largest two-node minimum underestimates the global memory by the unbounded factor L-1. The result is independent of node spacing. Necessity follows both from a zero budget for an analytic minor and from a Pick-Stein displacement identity whose negative inertia cannot exceed realization rank. Sufficiency follows from an explicit positive Pick completion. Beyond exact orthogonality, we derive a general cross-frequency inertia certificate, a gauge-independent span bound, an open robust region preserving the integer memory count, and a fail-closed noisy-eigenvalue certificate. Deterministic code tests arbitrary nodes, synthesizes minimal conservative realizations, and independently verifies the stated identities.
Under an explicit noncontextual homogeneous-response ansatz for rank-one quantum effects, we study normalization on every unitary rotation of a fixed finite POVM. The orbit-normalization operator diagonalizes on the projective harmonics of complex projective space. We derive its weighted all-degree singular spectrum and show that the smallest non-Born spectral value is the sharp condition number controlling distance from the affine space of trace-one Hermitian quadratic responses. For dimensions at least three, the high-degree spectrum of a finite seed converges to the sum of squared aggregate ray weights; for an outcome-simple equal-trace seed with N outcomes, the limit is d squared divided by N. Combining this ceiling with the forced geometry of the two smallest tight frames solves the global equal-trace design problem in every dimension d at least five. An orthonormal-basis seed uniquely maximizes the all-degree Born-rigidity gap, with value d minus 6 divided by d plus 1, up to unitary equivalence and outcome relabeling. Every distinct-outcome overcomplete seed in the stated class is strictly worse. By contrast, every weighted complex projective 2-design has a degree-two nullspace and therefore zero rigidity gap. Thus measurements optimal for state tomography can be maximally non-rigid for this covariant normalization objective. The result concerns one-step response fields and does not derive state update, intrinsic randomness, or microscopic particle dynamics.
For a prescribed traceless Hermitian five-level target F, we determine exactly the least product of Hilbert-Schmidt norms of Hermitian A,B satisfying -i[A,B]=F. If g_i=lambda_i-lambda_{i+1} are the ordered spectral gaps, the answer is the maximum of six explicit rational linear forms and their reversals. Exact Littlewood-Richardson enumeration gives 142 Horn inequalities. Twelve sparse dual identities prove the lower facets, and twelve rational singular-spectrum maps attain them; exact cone certificates prove completeness without sampling. In contrast with the two qutrit facets and four four-level facets, dimension five has twelve exposed facets and open chambers that force rank four. For every nonzero target, the least rank of an optimal factor is max(n_+(F),n_-(F)). We also prove the sharp tax 1 <= kappa_5/(||F||_1/2) <= 5/2 with complete equality loci, and the exact balanced triangular perimeter S_2^2=12 sqrt(3) kappa_5. Two rational certificates and an independent deterministic LP campaign accompany the paper.
For a prescribed traceless Hermitian four-level target F, we determine exactly the least product of Hilbert-Schmidt norms of Hermitian A,B satisfying -i[A,B]=F. With ordered eigenvalues lambda_1 >= ... >= lambda_4, the answer is the maximum of four explicit linear spectral forms. The lower bounds are Horn-Littlewood-Richardson certificates and four closed constructions attain them. On every nonzero spectral stratum, the least optimal rank is exactly max(n_+(F),n_-(F)), closing all chamber walls and degenerations. We also prove a finite-dimensional Horn linear-program reduction, the sharp tax 1 <= kappa_4/(||F||_1/2) <= 2 with complete equality cases, and the exact balanced three-kick action S_2^2=12 sqrt(3) kappa_4. Exact symbolic and deterministic LP certificates accompany the manuscript.
A balanced sequence of short Hamiltonian kicks has no first-order drift, but its order-dependent commutator holonomy survives when the order record is discarded. We prove that a reversal-symmetric average of complete order/reverse echoes is a self-adjoint random-unitary transfer operator and obeys the nonperturbative bound cos(4|y|S) I <= T_y <= I for |y|S < pi/8. This yields site and link reflection positivity and a finite-dimensional Osterwalder-Schrader Hamiltonian. The physical transfer approaches the curvature-frame generator with an explicit sixth-order remainder, giving quantitative O(y^2) convergence of its mass gap. For the regular tetrahedral qubit loop, we derive the exact finite-pulse depolarizing eigenvalue, isolate its first positivity zero at y = 0.455698535295322..., and obtain the mass expansion. We then prove a complete finite-depth Hausdorff criterion for visible transfer edges, with uniformly sharp depth r-1, and give an exact rational four-kick certificate: the true frame gap 10/3 is certified while the false claim 4 has rational witness value -1/64. A Wishart-Loewner band supplies finite-sample type-I control for independent Gaussian correlator sketches, including adaptive witnesses. Exact symbolic and deterministic numerical certificates accompany the manuscript.
A passive lossless network pays two different costs when it rotates a signal subspace over a frequency arc: a continuous Wigner—Smith action and an integer McMillan memory. Known Grassmannian length bounds and rational-inner degree identities constrain these resources separately. For the stated finite rational-inner class, we determine their joint attainable region exactly. Let two rank-k subspaces have principal angles βj, put B = Σj βj and let r be the number of nonzero angles. For a square finite rational inner transfer matrix of degree at most d, define the arc action A = ∫I tr Q(θ)dθ, where Q = −iS*∂θS is positive semidefinite. If B > 0, the exact feasible set is d ≥ r and 2B ≤ A ≤ 2πd − 2B. Both faces are attained. Equivalently, dmin(A) = max{r, ⌈(A + 2B)/(2π)⌉}. The upper face is a return cost: the complementary arc must rotate the subspace back, while the full-circle trace action is exactly 2π times the degree. We prove sufficiency by an explicit compiler of normalized rank-one Blaschke—Potapov gates; it realizes every interior point and both faces, including rank-deficient and orthogonal cases. When the endpoint subspaces coincide, the region changes discontinuously to A ∈ [0, 2πd), with the upper endpoint open. We derive fail-closed noisy degree certificates and give a multi-frequency warning using classical boundary Nevanlinna—Pick theory: three pairwise degree-one routing tasks can require degree two jointly. For orthogonal-line data the obstruction is a frame-independent cycle phase, exhibited by an exact positive semidefinite Pick completion with spectrum (2,1,0). Deterministic code compiles random points of the diamond, constructs the degree-two colligation, and is replayed by an independent verifier.
A balanced sequence of short Hamiltonian kicks has zero first-order drift but retains an order-dependent commutator holonomy. For a traceless qutrit target with ordered eigenvalues, we solve the inverse Hermitian-commutator problem exactly: its minimum product Hilbert-Schmidt norm is the larger adjacent spectral gap, equivalently half the trace norm plus the absolute middle eigenvalue. This yields the exact minimum action of a balanced three-kick realization and identifies a middle-eigenvalue cost tax between 1 and 3/2. Among four equal-norm balanced qubit kicks, the forgotten-order curvature gap is at most `S^4/108`, with equality only for a regular tetrahedron. Its 24 orders generate exactly the six Pauli holonomies, so the finite twirl is depolarizing and an explicit inverse-cosine schedule realizes any depolarizing semigroup exactly at every finite step count. We then derive the diffusion limit from complete echoed, pinched physical words, obtaining an explicit `O(n^-1/2)` diamond-norm bound. Off-block pulses of size `y^(1+beta)` produce a sharp trichotomy: irrelevant for `beta>1`, additive at `beta=1`, and Zeno-projective for `0<beta<1`; a quantitative compression bound gives uniform convergence away from zero and an explicit initial-layer profile. The complete implementation has serial action proportional to `n^(3/4)`. Exact symbolic and deterministic numerical certificates accompany the manuscript.
A balanced ordered loop of retained Hamiltonians generates a second-order geometric Hamiltonian. We solve its higher-rank action problem and determine what remains when the order record is erased. For every matrix dimension and every number of kicks `m >= 3`, a gauge-invariant operator action obeys a sharp, dimension-free polygonal bound on the spectral diameter of the geometric Hamiltonian; regular two-level polygons attain the constant in every dimension. A target-sensitive Hilbert--Schmidt companion bound is controlled by the trace norm and has equality exactly for sign-paired nonzero spectra. These laws yield exact diamond-norm ceilings with a physical state witness. For a uniformly forgotten order, we derive an exact `1/12` commutator-frame covariance. The resulting reversal-symmetric random-unitary product converges in diamond norm at rate `O(n^-1)` to a GKLS frame Laplacian whose fixed algebra is the joint commutant of the pair commutators. A tetrahedral Pauli architecture gives exact primitive depolarization with gap `64/3`. Finally, fixed laboratory time `T` and bounded kick amplitude `Lambda` impose a sharp curvature ceiling proportional to `T^2 Lambda^2/n`; maintaining nonzero holonomy requires `Lambda=Omega(sqrt(n))`. Exact symbolic and deterministic numerical certificates accompany the paper.
Lower bounds do not determine which resource profiles are physically attainable. We close the continuous inverse problem for passive subspace transport. Let two rank-k subspaces in C^N, N >= 2k, have ordered canonical angles beta, and let q_j be the integrated leading eigenvalues of a positive generator Q = -iS* dS/dt. We prove the exact equivalence: q is attainable if and only if 2 beta is weakly majorized by q. Every feasible profile has a compiler using at most k+1 mutually commuting positive generators of rank at most k, with constant ordered top spectrum. Analytic rationality adds an integer-valued resource: for two boundary frequencies, the least McMillan degree of an inner matrix sending a fixed input subspace to two prescribed output subspaces equals the number of nonzero canonical angles. We also give a fail-closed finite-error certificate for both resources from noisy projectors.At multiple frequencies, Wigner-Smith delay is only the block diagonal of a boundary de Branges-Rovnyak Pick matrix. This positive matrix is the Gram matrix of frequency-excited internal states and has rank bounded by the McMillan degree. Its spectrum yields robust degree certificates and unavoidable model-reduction tails. An exact example exhibits identical local proper delays for degree-one and degree-two devices while their Pick ranks distinguish them. Deterministic code reconstructs the compilers, tests 2,240 randomized theorem interfaces, checks the strict separation, and independently replays the certificate.
The order of short operations is normally a microscopic detail. We show that a fixed degenerate measurement turns it into an operational curvature with an exact macroscopic readout. Let E retain a block matrix algebra, and let two words apply the same short Hamiltonian kicks in different orders, with the same nonselective pinching after every kick. Their leading difference is `-i tau^2 ad(F) E`, where F is the sum of `i[h_j,h_k]` over pairwise inversions. This gives a discrete non-Abelian Stokes law. Its diamond-norm coefficient is exactly the largest spectral diameter of a block of F, and therefore fixes the leading optimal channel-discrimination advantage. Under first-order balance, all permutations have the same dissipator while their geometric Hamiltonians differ exactly by F, so local curvature integrates into distinct diffusive quantum Markov semigroups. We characterize contextual invisibility, the rank-one classical boundary, and inverse-success amplification under postselection. Every retained Hamiltonian modulo the block center is realizable by a balanced three-kick loop. For a qubit block we prove the sharp action law `kappa <= S^2/[m tan(pi/m)]`, attained by regular planar kick polygons. An integer-Pauli four-kick architecture saturates the bound and supports primitive dissipation. Symbolic, semidefinite, and convergence certificates accompany the paper.
Repeated projective measurements are usually discussed either in the ballistic Zeno scaling or after reduction to classical outcome probabilities. We study a different regime in which each nonselective measurement retains a full matrix algebra inside every degenerate outcome block. Let E(X)=sum_a P_a X P_a, let E_tau be its rotation by exp(-i tau H), and close one cycle by Phi_tau=E E_tau E. We prove, uniformly on compact time intervals and in diamond norm, that Phi_{sqrt(t/n)}^n converges to exp(-tK)E, where K=E ad_H(1-E)ad_H E=C_H^* C_H. The limit is a genuinely quantum Markov semigroup on the direct sum of the block matrix algebras, with explicit jumps sqrt(2) P_b H P_a. Beyond upper bounds, we identify three exact local obstructions for the unprocessed physical product. The cubic map M_H produces a nonzero n^{-1/2} coefficient; after it vanishes, the quartic product obstruction J_H produces a nonzero n^{-1} coefficient; after both vanish, the quintic obstruction P_H produces a nonzero n^{-3/2} coefficient. If all three vanish, the error is O(n^{-2}). All three coefficient transforms are injective. An exact three-record algebraic example has M_H=J_H=0 but P_H nonzero, proving that the third branch is attained. The fixed algebra is A intersect {H_off}', and the least singular value of the commutator frame C_H defines a noncommutative connectivity gap. In the primitive case that gap is the exact exponential mixing exponent in diamond norm and is Lipschitz robust under Hamiltonian perturbations. Rank-one blocks reduce exactly to twice the squared-coupling graph Laplacian. An eight-dimensional rational architecture with four qubit blocks is certified irreducible; its gap is the unique root in (0.4545,0.4547) of an explicit quartic. Symbolic certificates, semidefinite diamond-norm replays, and convergence tests accompany the paper.
A subspace quantum speed limit usually records only the largest canonical angle. Passive scattering theory usually records either the largest proper delay or their total trace. Both compressions discard how a rank-k routing event is distributed across modes. We derive the missing vector law: the integrated spectral-spread vector weakly majorizes twice the canonical-angle vector. More strongly, for every symmetric gauge Φ and every pair of k-subspaces in N ≥ 2k dimensions, we solve the variational problem exactly: the minimum spectral-spread action is 2Φ(β). The same value holds for the leading-proper-delay action under positive generators, and one constant coupled-mode path minimizes all gauges simultaneously. The Ky Fan members recover the bandwidth limit, strengthen the trace-action law, and expose every intermediate modal budget. An exact nonnegative slack decomposition separates common-mode delay, inefficient spectral coupling, and nongeodesic subspace motion. Robust corollaries convert heterogeneous pass/stop leakage spectra and finite tomography errors into certified Lorenz curves for delay and, for rational inner networks, McMillan-degree lower bounds. A deterministic artifact tests 8,192 random generators, 1,536 time-dependent paths, sharp equality families, and 2,048 noisy tomography instances.
Exact interpolation can force the internal degree of a passive network, but laboratory calibrations are approximate. We prove a sharp frequency-domain speed limit requiring neither exact zeros nor analytic continuation away from the measured frequency axis. Let (S(e^{itheta})) be an absolutely continuous unitary scattering path with positive Wigner—Smith generator (Q(theta)=-iS(e^{itheta})^astpartial_theta S(e^{itheta})succeq0). If a fixed (k)-dimensional input subspace is routed approximately between complementary output sectors with amplitude leakages (varepsilon_p,varepsilon_s), define (alpha=[pi/2-arcsinvarepsilon_p-arcsinvarepsilon_s]_+). Every transition then requires Wigner—Smith trace action at least (2kalpha) and largest-proper-delay action at least (2alpha). Costs add over disjoint frequency arcs. For a rational inner network of McMillan degree (n), (M) alternating pass/stop pairs imply (nge 2Mkalpha/pi), recovering (nge Mk) at zero error. An explicit (2k)-port interferometric family attains the bounds for every admissible error pair. A tomography-error corollary converts finite scattering measurements directly into certified degree and delay lower bounds. Reproducible certificates audit equality cases, positive-block inequalities and random Blaschke—Potapov products.
We study whether passive reservoir filtering can suppress decoherence more effectively when its channel mixing is irreducible, even after fixing the passband responses and rational complexity. For every integer S ≥ 5, we construct an explicit causal inner six-port network with three signal and three vacuum-loss ports. Its signal block is a rational Schur transfer satisfying 3(S−1) delayed full-spark tangential calibrations and exhibiting exponentially small leakage on two stop arcs. In contrast, every transfer of the same bidegree possessing a constant nontrivial reducing channel and satisfying the same calibrations retains unit stopband norm.We strengthen this exact separation with a quantitative finite-error obstruction: for calibration defect δ and sampled reducing-line defect β, comparator leakage is bounded below by [1−C_S(δ+β)]_+, with C_S given explicitly by finite singular-value margins. A scalar Schur construction proves that every bound of this form must deteriorate at least as 2 exp(9S/20)(1+o(1)); hence uniform robustness is impossible for the chosen clustered calibrations.For uniformly nondegenerate bath spectra, the signal-level separation is squared at the Kossakowski-rate level. A closed Markov pure-dephasing model includes all auxiliary vacuum ports exactly, producing an architecture-independent measurable baseline and an explicit total Ramsey-rate advantage. Finally, we prove that strong passive suppression requires large dwell time: under a peak-delay budget D, the rate-improvement factor is asymptotically at most quadratic in D/S. The construction therefore moves the coherence-maintenance resource into passive memory, vacuum noise and conditioning rather than eliminating it. All certificates, figures and numerical audits are publicly reproducible.
Scalar Chebyshev filtering treats every vector in a block Krylov iterate with the same polynomial, while simultaneously diagonalizable matrix coefficients amount to independent scalar filters after a fixed channel rotation. We study the larger class of Hermitian matrix-polynomial filters under tangential pass constraints. Already at fixed channel dimension d=3, we construct rationally generated signatures with quantitative full-spark margin for which an irreducible symmetric filter has stopband leakage O(exp(-cN)). In contrast, every exactly calibrated symmetric coefficient family with any common nontrivial invariant channel subspace has leakage at least one; this comparator strictly contains the pairwise-commuting class and permits a fully noncommutative 2 by 2 block. A quantitative theorem covers approximately reducible filters by charging their sampled off-block coupling together with calibration error. The signature margin and the admissible combined error are both exp(-O(N)), not exp(-N^3). This exponential scale is unavoidable: for arbitrary pass nodes and signatures in the same separated bands, an explicit scalar binomial-tail polynomial has both pass error and stopband leakage at most exp(-2N/81). Thus constant-error separation is impossible, while the irreducible construction achieves the correct exponential scale class. An affine cosine substitution gives the same robust law for fixed-latency reciprocal linear-phase MIMO FIR filters. We also give an exact rational five-tap certificate with stopband norm at most 25/32 and prove that its advantage survives reducible calibration errors delta < 7/1920. A second exact-arithmetic certificate is calibrated from a public triaxial pump-vibration data set: it gives leakage below 0.96 versus one for every exactly calibrated reducible symmetric class, while its maximum directional error on six held-out records is 0.00385. In a frequency-domain-decomposition test on the frozen held-out cospectra, filtering rotates the leading modal direction by at most 0.222 degrees and changes its leading spectral ordinate by at most 2.4 × 10^-5 relatively. Numerical programs test the mechanism and expose an ordinary graph-denoising setting in which scalar Chebyshev filtering is instead preferable. Tangential matrix interpolation, MIMO filtering, scalar two-band approximation and convex FIR design are not claimed as new; the contribution is the fixed-order reducible/irreducible separation together with its two-sided calibration scale.
For the spatial Z_2 (Ising-slice) system inside the Dobrushin window 2 tanh|beta| + 2 tanh|gamma| <= alpha < 1, we machine-check in Lean 4 an end-to-end chain from the Gibbs measure to the spectrum of the reconstructed transfer operator. (i) The Osterwalder-Schrader (site-form) reconstruction of the transfer operator is unitarily conjugate, by the explicit sqrt(w) boundary dressing, to the symmetrised Dobrushin kernel. (ii) The unnormalised Gibbs sums themselves are exact matrix elements of that operator's powers: gibbsPathSum(w,beta,N,A,B) = lambda^N , with the partition function the same shape at the dressed constant. These are identities, not bounds, and they hold at every real beta and every positive weight. (iii) There is one mass m > 0 such that for every spatial extent L the projected operator norm is at most e^{-m} and every mixed connected correlator obeys | - | <= ||u|| ||v|| (e^{-m})^n, the zero-time case included. (iv) The connecte two-point function of the normalised Gibbs measure decays at that same rate with a constant independent of the time depth; dividing by the partition function is licensed by a denominator floor uniform in N, which the positive cone supplies and the spectrum does not, since the spectral route controls only the even powers. (v) The N -> infinity limit state exists, is the vacuum state of the reconstructed operator, and does not depend on the strictly positive observable terminating the chain. (vi) The reconstructed operator is a reversible Markov chain -- stochastic and in detailed balance for pi = Omega^2, both proved -- and in that stationary state the connected correlator of bounded observables obeys |E_pi[f P^N g] - E_pi[f] E_pi[g]| <= K_f K_g (e^{-m})^N, with quantifier order "there exists m, for all L": no factor depending on the spatial extent. Summing over time separations gives a susceptibility bound K_f K_g / (1 - e^{-m}), independent of the cut-off and of the extent. The window is non-empty at an interacting point (beta = gamma = 1/10, alpha = 1/2), machine-checked, so none of these conditionals is vacuous. The analytic input is inherited: the mass is the one the Dobrushin corollary already produced, and the window is not widened. What the reconstruction contributes is the identification, the exact identities, and the normalisation in which both the rate and the constant lose their dependence on the volume.
We determine when rapid exact maintenance of a rank-deficient quantum target produces logarithmically divergent free-energy restoration power in algebraic quantum field theory. On every sigma-finite properly infinite factor we construct a bounded quantum Markov semigroup, a nonfaithful normal target and a faithful invariant reference state for which Araki relative entropy reduces exactly to a binary divergence. We derive the semigroup first from localized thermal fermionic probe collisions and then from one autonomous finite-bandwidth Dirac KMS reservoir with fixed smooth coupling. Exact memory equations yield an explicit finite-coupling Davies bound on finite van Hove windows. For a displayed compactly supported massless Dirac form factor, Araki—Wyss regularity, threshold behaviour and Fermi-golden-rule positivity give a completely bounded Davies approximation uniformly for all times with error of order (O(|lambda|)); the sharper (O(lambda^2)) result is isolated under additional reduced-resonance hypotheses. We construct a background-covariant two-Dirac-field completion using Green operators, Møller maps and relative Cauchy scattering, proving naturality, causal factorization and exact spacelike triviality. A locality obstruction shows why a strictly local multiplier cannot coincide exactly with the solvable rank-one reservoir coupling, while a Feshbach reduction quantifies the correction. Finally, we prove that no fixed faithful vacuum or KMS restriction can exhibit the rank-boundary mechanism, but faithful families with a vanishing spectral floor recover its complete coefficient. The results separate algebra type, target faithfulness, microscopic realizability and regulator uniformity, and provide reproducible numerical audits of the finite-dimensional identities and explicit Dirac form factor.
We present a Lean 4 formalization of a thermodynamic-limit construction for the anisotropic nearest-neighbour Ising model on the two-dimensional integer lattice. A telescoping comparison argument and the classical Dobrushin resolvent produce a volume-uniform expectation bound. Exact restriction and reindexing maps between finite Gibbs measures yield convergence of the complete free-boundary sequence, independence of auxiliary envelopes and cofinal samplings, stability under receding boundary perturbations, and equality of free and periodic limits.Finite-support cylinder presentations are quotiented by equality of their represented functions on the full spin space. The resulting local algebra carries a genuine lattice-translation action, and the limiting functional is positive, normalized, real-linear, and invariant under every integer translation. We equip this algebra with its intrinsic uniform norm, construct its complex star-algebra representation, take the corresponding commutative C*-closure, and extend the limiting functional to a positive complex-linear functional of norm one. We also formalize normalized Gibbs conditional kernels for arbitrary finite conditioning sets and prove positivity, exterior locality, idempotence, and the exact finite-volume Gibbs tower identity.All results remain within the classical anisotropic Dobrushin region. The infinite-volume DLR fixed-point equation for the completed state is not claimed.
We study whether a flat lattice-gauge Hodge form supplemented by a penalty depending only on a line-integral block variable can control the full fine one-cochain norm uniformly in the block side. The constant-field sector selects the exponent (d-2)/2 only under exact scale neutrality, but this scaling test does not address the kernel of the block map. We construct transverse, divergence-free, block-periodic Fourier cochains that are annihilated exactly by the block map. Their first-mode Hodge Rayleigh quotient is 4 sin²(pi/L), forcing every admissible full-space Poincaré constant to grow at least quadratically with L. The obstruction applies to every scale-dependent scalar functional factored through the block variable and satisfying only that it vanishes at the coarse zero field; no linearity, continuity, positivity, or growth condition is required. For linear postconditioners, higher Fourier frequencies yield an orthogonal real subspace of dimension 2R and a min—max bound on the 2R-th eigenvalue, establishing a growing low-energy spectral cluster. We also derive an exact repair identity and a necessary witness-channel budget for modified block measurements and additional fine-space terms. The kernel construction, Hodge energy, factorized no-go theorem, repair identity, and necessary repair budget are formalized in Lean 4. The result concerns a finite periodic flat full-domain form and makes no claim about interacting coercivity, gauge quotients, continuum limits, or the Yang—Mills mass gap.
